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bcc9 ................\n-00001d90: 2e8e 8e93 8d8d 9102 bcc9 2e8e 8e93 8d8d ................\n-00001da0: 9102 bcc9 2e8e 8e93 8d8d 9102 bcc9 2e8e ................\n-00001db0: 8e93 8d8d 9102 bcc9 2e8e 8e93 8d8d 9102 ................\n-00001dc0: bcc9 2e8e 8e93 8d8d 9102 bcc9 2e8e 8e93 ................\n-00001dd0: 8d8d 9102 bcc9 2e8e 8e93 8d8d 9102 bcc9 ................\n-00001de0: 2e8e 8e93 8d8d 9102 bcc9 2e8e 8e93 8d8d ................\n-00001df0: 9102 bcc9 2e8e 8e93 8d8d 9102 bcc9 2e8e ................\n-00001e00: 8e93 8d8d 9102 bcc9 2e8e 8e93 8d8d 9102 ................\n-00001e10: bcc9 2e8e 8e93 8d8d 9102 bcc9 2e8e 8e93 ................\n-00001e20: 8d8d 9102 bcc9 2e8e 8e93 8d8d 9102 bcc9 ................\n-00001e30: 2e8e 8e93 8d8d 9102 bcc9 2e8e 8e93 8d8d ................\n-00001e40: 9102 bcc9 2e8e 8e93 8d8d 9102 bcc9 2e8e ................\n-00001e50: 8e93 8d8d 9102 bcc9 2e8e 8e93 8d8d 9102 ................\n-00001e60: bcc9 2e8e 8e93 8d8d 9102 bcc9 2e8e 8e93 ................\n-00001e70: 8d8d 9102 bcc9 2e8e 8e93 8d8d 9102 bcc9 ................\n-00001e80: 2e8e 8e93 8d8d 9102 bcc9 2e8e 8e93 8d8d ................\n-00001e90: 9102 bcc9 2e8e 8e93 8d8d 9102 bcc9 2e8e ................\n-00001ea0: 8e93 8d8d 9102 bcc9 2e8e 8e93 8d8d 9102 ................\n-00001eb0: bcc9 2e8e 8e93 8d8d 9102 bcc9 2e8e 8e93 ................\n-00001ec0: 8d8d 9102 bcc9 2e8e 8e93 8d8d 9102 bcc9 ................\n-00001ed0: 2e8e 8e93 8d8d 9102 bcc9 2e8e 8e93 8d8d ................\n-00001ee0: 9102 bcc9 2e8e 8e93 8d8d 9102 bcc9 2e8e ................\n-00001ef0: 8e93 8d8d 9102 bcc9 2e8e 8e93 8d8d 9102 ................\n-00001f00: bcc9 2e8e 8e8d 910f f1af 3137 8e8e a18d ..........17....\n-00001f10: 8d91 106c cf35 2e32 8e91 299c 314c 6f67 ...l.5.2..).1Log\n-00001f20: 6172 6974 686d 9102 018e 8d8d 9102 bcc9 arithm..........\n-00001f30: 2e8e 8e96 0884 3d8d 8d91 02bc c92e 8e8e ......=.........\n-00001f40: 938d 8d91 02bc c92e 8e8e 938d 8d91 02bc ................\n-00001f50: c92e 8e8e 938d 8d91 02bc c92e 8e8e 938d ................\n-00001f60: 8d91 02bc c92e 8e8e 938d 8d91 02bc c92e ................\n-00001f70: 8e8e 938d 8d91 02bc c92e 8e8e 938d 8d91 ................\n-00001f80: 02bc c92e 8e8e 938d 8d91 02bc c92e 8e8e ................\n-00001f90: 938d 8d91 02bc c92e 8e8e 938d 8d91 02bc ................\n-00001fa0: c92e 8e8e 938d 8d91 02bc c92e 8e8e 938d ................\n-00001fb0: 8d91 02bc c92e 8e8e 938d 8d91 02bc c92e ................\n-00001fc0: 8e8e 938d 8d91 02bc c92e 8e8e 938d 8d91 ................\n-00001fd0: 02bc c92e 8e8e 938d 8d91 02bc c92e 8e8e ................\n-00001fe0: 938d 8d91 02bc c92e 8e8e 938d 8d91 02bc ................\n-00001ff0: c92e 8e8e 938d 8d91 02bc c92e 8e8e 938d ................\n-00002000: 8d91 02bc c92e 8e8e 938d 8d91 02bc c92e ................\n-00002010: 8e8e 938d 8d91 02bc c92e 8e8e 938d 8d91 ................\n-00002020: 02bc c92e 8e8e 938d 8d91 02bc c92e 8e8e ................\n-00002030: 938d 8d91 02bc c92e 8e8e 938d 8d91 02bc ................\n-00002040: c92e 8e8e 938d 8d91 02bc c92e 8e8e 938d ................\n-00002050: 8d91 02bc c92e 8e8e 938d 8d91 02bc c92e ................\n-00002060: 8e8e 938d 8d91 02bc c92e 8e8e 938d 8d91 ................\n-00002070: 02bc c92e 8e8e 938d 8d91 02bc c92e 8e8e ................\n-00002080: 938d 8d91 02bc c92e 8e8e 938d 8d91 02bc ................\n-00002090: c92e 8e8e 938d 8d91 02bc c92e 8e8e 938d ................\n-000020a0: 8d91 02bc c92e 8e8e 938d 8d91 02bc c92e ................\n-000020b0: 8e8e 938d 8d91 02bc c92e 8e8e 938d 8d91 ................\n-000020c0: 02bc c92e 8e8e 938d 8d91 02bc c92e 8e8e ................\n-000020d0: 8d91 0ff1 af31 378e 8ea1 8d8d 9110 6ccf .....17.......l.\n-000020e0: 352e 338e 9129 9c31 5491 ff16 6572 6967 5.3..).1T...erig\n-000020f0: 6f6e 6f6d 6574 7269 6373 9106 0550 8d8d onometrics...P..\n-00002100: 9102 bcc9 2e8e 8e96 0884 3d8d 8d91 02bc ..........=.....\n-00002110: c92e 8e8e 938d 8d91 02bc c92e 8e8e 938d ................\n-00002120: 8d91 02bc c92e 8e8e 938d 8d91 02bc c92e ................\n-00002130: 8e8e 938d 8d91 02bc c92e 8e8e 938d 8d91 ................\n-00002140: 02bc c92e 8e8e 938d 8d91 02bc c92e 8e8e ................\n-00002150: 938d 8d91 02bc c92e 8e8e 938d 8d91 02bc ................\n-00002160: c92e 8e8e 938d 8d91 02bc c92e 8e8e 938d ................\n-00002170: 8d91 02bc c92e 8e8e 938d 8d91 02bc c92e ................\n-00002180: 8e8e 938d 8d91 02bc c92e 8e8e 938d 8d91 ................\n-00002190: 02bc c92e 8e8e 938d 8d91 02bc c92e 8e8e ................\n-000021a0: 938d 8d91 02bc c92e 8e8e 938d 8d91 02bc ................\n-000021b0: c92e 8e8e 938d 8d91 02bc c92e 8e8e 938d ................\n-000021c0: 8d91 02bc c92e 8e8e 938d 8d91 02bc c92e ................\n-000021d0: 8e8e 938d 8d91 02bc c92e 8e8e 938d 8d91 ................\n-000021e0: 02bc c92e 8e8e 938d 8d91 02bc c92e 8e8e ................\n-000021f0: 938d 8d91 02bc c92e 8e8e 938d 8d91 02bc ................\n-00002200: c92e 8e8e 938d 8d91 02bc c92e 8e8e 938d ................\n-00002210: 8d91 02bc c92e 8e8e 938d 8d91 02bc c92e ................\n-00002220: 8e8e 938d 8d91 02bc c92e 8e8e 938d 8d91 ................\n-00002230: 02bc c92e 8e8e 938d 8d91 02bc c92e 8e8e ................\n-00002240: 938d 8d91 02bc c92e 8e8e 938d 8d91 02bc ................\n-00002250: c92e 8e8e 938d 8d91 02bc c92e 8e8e 938d ................\n-00002260: 8d91 02bc c92e 8e8e 938d 8d91 02bc c92e ................\n-00002270: 8e8e 938d 8d91 02bc c92e 8e8e 938d 8d91 ................\n-00002280: 02bc c92e 8e8e 8d91 0ff1 af31 378e 8ea1 ...........17...\n-00002290: 8d8d 9110 6ccf 352e 348e 9129 9c31 496e ....l.5.4..).1In\n-000022a0: 95b2 2176 9365 7273 6591 03a6 6654 91ff ..!v.erse...fT..\n-000022b0: 1665 7269 676f 6e6f 6d65 7472 6963 7391 .erigonometrics.\n-000022c0: 02e3 2c8d 8d91 02bc c92e 8e8e 9608 843d ..,............=\n-000022d0: 8d8d 9102 bcc9 2e8e 8e93 8d8d 9102 bcc9 ................\n-000022e0: 2e8e 8e93 8d8d 9102 bcc9 2e8e 8e93 8d8d ................\n-000022f0: 9102 bcc9 2e8e 8e93 8d8d 9102 bcc9 2e8e ................\n-00002300: 8e93 8d8d 9102 bcc9 2e8e 8e93 8d8d 9102 ................\n-00002310: bcc9 2e8e 8e93 8d8d 9102 bcc9 2e8e 8e93 ................\n-00002320: 8d8d 9102 bcc9 2e8e 8e93 8d8d 9102 bcc9 ................\n-00002330: 2e8e 8e93 8d8d 9102 bcc9 2e8e 8e93 8d8d ................\n-00002340: 9102 bcc9 2e8e 8e93 8d8d 9102 bcc9 2e8e ................\n-00002350: 8e93 8d8d 9102 bcc9 2e8e 8e93 8d8d 9102 ................\n-00002360: bcc9 2e8e 8e93 8d8d 9102 bcc9 2e8e 8e93 ................\n-00002370: 8d8d 9102 bcc9 2e8e 8e93 8d8d 9102 bcc9 ................\n-00002380: 2e8e 8e93 8d8d 9102 bcc9 2e8e 8e93 8d8d ................\n-00002390: 9102 bcc9 2e8e 8e93 8d8d 9102 bcc9 2e8e ................\n-000023a0: 8e93 8d8d 9102 bcc9 2e8e 8e93 8d8d 9102 ................\n-000023b0: bcc9 2e8e 8e93 8d8d 9102 bcc9 2e8e 8e93 ................\n-000023c0: 8d8d 9102 bcc9 2e8e 8e93 8d8d 9102 bcc9 ................\n-000023d0: 2e8e 8e93 8d8d 9102 bcc9 2e8e 8e93 8d8d ................\n-000023e0: 9102 bcc9 2e8e 8e93 8d8d 9102 bcc9 2e8e ................\n-000023f0: 8e93 8d8d 9102 bcc9 2e8e 8e93 8d8d 9102 ................\n-00002400: bcc9 2e8e 8e93 8d8d 9102 bcc9 2e8e 8e93 ................\n-00002410: 8d8d 9102 bcc9 2e8e 8e93 8d8d 9102 bcc9 ................\n-00002420: 2e8e 8e8d 910f f1af 3138 8e8e a18d 8d91 ........18......\n-00002430: 106c cf35 2e35 8e91 299c 3148 7970 954d .l.5.5..).1Hyp.M\n-00002440: de65 7262 936f 6c69 6391 03a6 6646 91ff .erb.olic...fF..\n-00002450: 1665 756e 6374 696f 6e73 9100 ba55 8d8d .eunctions...U..\n-00002460: 9102 bcc9 2e8e 8e96 0884 3d8d 8d91 02bc ..........=.....\n-00002470: c92e 8e8e 938d 8d91 02bc c92e 8e8e 938d ................\n-00002480: 8d91 02bc c92e 8e8e 938d 8d91 02bc c92e ................\n-00002490: 8e8e 938d 8d91 02bc c92e 8e8e 938d 8d91 ................\n-000024a0: 02bc c92e 8e8e 938d 8d91 02bc c92e 8e8e ................\n-000024b0: 938d 8d91 02bc c92e 8e8e 938d 8d91 02bc ................\n-000024c0: c92e 8e8e 938d 8d91 02bc c92e 8e8e 938d ................\n-000024d0: 8d91 02bc c92e 8e8e 938d 8d91 02bc c92e ................\n-000024e0: 8e8e 938d 8d91 02bc c92e 8e8e 938d 8d91 ................\n-000024f0: 02bc c92e 8e8e 938d 8d91 02bc c92e 8e8e ................\n-00002500: 938d 8d91 02bc c92e 8e8e 938d 8d91 02bc ................\n-00002510: c92e 8e8e 938d 8d91 02bc c92e 8e8e 938d ................\n-00002520: 8d91 02bc c92e 8e8e 938d 8d91 02bc c92e ................\n-00002530: 8e8e 938d 8d91 02bc c92e 8e8e 938d 8d91 ................\n-00002540: 02bc c92e 8e8e 938d 8d91 02bc c92e 8e8e ................\n-00002550: 938d 8d91 02bc c92e 8e8e 938d 8d91 02bc ................\n-00002560: c92e 8e8e 938d 8d91 02bc c92e 8e8e 938d ................\n-00002570: 8d91 02bc c92e 8e8e 938d 8d91 02bc c92e ................\n-00002580: 8e8e 938d 8d91 02bc c92e 8e8e 938d 8d91 ................\n-00002590: 02bc c92e 8e8e 938d 8d91 02bc c92e 8e8e ................\n-000025a0: 938d 8d91 02bc c92e 8e8e 938d 8d91 02bc ................\n-000025b0: c92e 8e8e 938d 8d91 02bc c92e 8e8e 938d ................\n-000025c0: 8d91 02bc c92e 8e8e 8d91 0ff1 af31 388e .............18.\n-000025d0: 8ea6 8d8d dd36 8e91 106c cf4d 6973 6365 .....6...l.Misce\n-000025e0: 6c6c 616e 656f 7573 9104 328d 526f 7574 llaneous..2.Rout\n-000025f0: 696e 6573 8d92 0137 be8c 3138 8e8e a18d ines...7..18....\n-00002600: 8d91 106c cfc6 362e 318e 9129 9c31 496e ...l..6.1..).1In\n-00002610: 7075 7496 03a6 662f 934f 7574 7075 7491 put...f/.Output.\n-00002620: 0319 c88d 8d91 02bc c92e 8e8e 9608 843d ...............=\n-00002630: 8d8d 9102 bcc9 2e8e 8e93 8d8d 9102 bcc9 ................\n-00002640: 2e8e 8e93 8d8d 9102 bcc9 2e8e 8e93 8d8d ................\n-00002650: 9102 bcc9 2e8e 8e93 8d8d 9102 bcc9 2e8e ................\n-00002660: 8e93 8d8d 9102 bcc9 2e8e 8e93 8d8d 9102 ................\n-00002670: bcc9 2e8e 8e93 8d8d 9102 bcc9 2e8e 8e93 ................\n-00002680: 8d8d 9102 bcc9 2e8e 8e93 8d8d 9102 bcc9 ................\n-00002690: 2e8e 8e93 8d8d 9102 bcc9 2e8e 8e93 8d8d ................\n-000026a0: 9102 bcc9 2e8e 8e93 8d8d 9102 bcc9 2e8e ................\n-000026b0: 8e93 8d8d 9102 bcc9 2e8e 8e93 8d8d 9102 ................\n-000026c0: bcc9 2e8e 8e93 8d8d 9102 bcc9 2e8e 8e93 ................\n-000026d0: 8d8d 9102 bcc9 2e8e 8e93 8d8d 9102 bcc9 ................\n-000026e0: 2e8e 8e93 8d8d 9102 bcc9 2e8e 8e93 8d8d ................\n-000026f0: 9102 bcc9 2e8e 8e93 8d8d 9102 bcc9 2e8e ................\n-00002700: 8e93 8d8d 9102 bcc9 2e8e 8e93 8d8d 9102 ................\n-00002710: bcc9 2e8e 8e93 8d8d 9102 bcc9 2e8e 8e93 ................\n-00002720: 8d8d 9102 bcc9 2e8e 8e93 8d8d 9102 bcc9 ................\n-00002730: 2e8e 8e93 8d8d 9102 bcc9 2e8e 8e93 8d8d ................\n-00002740: 9102 bcc9 2e8e 8e93 8d8d 9102 bcc9 2e8e ................\n-00002750: 8e93 8d8d 9102 bcc9 2e8e 8e93 8d8d 9102 ................\n-00002760: bcc9 2e8e 8e93 8d8d 9102 bcc9 2e8e 8e93 ................\n-00002770: 8d8d 9102 bcc9 2e8e 8e93 8d8d 9102 bcc9 ................\n-00002780: 2e8e 8e93 8d8d 9102 bcc9 2e8e 8e93 8d8d ................\n-00002790: 9102 bcc9 2e8e 8e93 8d8d 9102 bcc9 2e8e ................\n-000027a0: 8e93 8d8d 9102 bcc9 2e8e 8e8d 910f f1af ................\n-000027b0: 3138 8e8e a18d 8d91 106c cf36 2e32 8e91 18.......l.6.2..\n-000027c0: 299c 3143 6f6d 7061 7269 736f 6e73 9107 ).1Comparisons..\n-000027d0: 82e3 8d8d 9102 bcc9 2e8e 8e96 0884 3d8d ..............=.\n-000027e0: 8d91 02bc c92e 8e8e 938d 8d91 02bc c92e ................\n-000027f0: 8e8e 938d 8d91 02bc c92e 8e8e 938d 8d91 ................\n-00002800: 02bc c92e 8e8e 938d 8d91 02bc c92e 8e8e ................\n-00002810: 938d 8d91 02bc c92e 8e8e 938d 8d91 02bc ................\n-00002820: c92e 8e8e 938d 8d91 02bc c92e 8e8e 938d ................\n-00002830: 8d91 02bc c92e 8e8e 938d 8d91 02bc c92e ................\n-00002840: 8e8e 938d 8d91 02bc c92e 8e8e 938d 8d91 ................\n-00002850: 02bc c92e 8e8e 938d 8d91 02bc c92e 8e8e ................\n-00002860: 938d 8d91 02bc c92e 8e8e 938d 8d91 02bc ................\n-00002870: c92e 8e8e 938d 8d91 02bc c92e 8e8e 938d ................\n-00002880: 8d91 02bc c92e 8e8e 938d 8d91 02bc c92e ................\n-00002890: 8e8e 938d 8d91 02bc c92e 8e8e 938d 8d91 ................\n-000028a0: 02bc c92e 8e8e 938d 8d91 02bc c92e 8e8e ................\n-000028b0: 938d 8d91 02bc c92e 8e8e 938d 8d91 02bc ................\n-000028c0: c92e 8e8e 938d 8d91 02bc c92e 8e8e 938d ................\n-000028d0: 8d91 02bc c92e 8e8e 938d 8d91 02bc c92e ................\n-000028e0: 8e8e 938d 8d91 02bc c92e 8e8e 938d 8d91 ................\n-000028f0: 02bc c92e 8e8e 938d 8d91 02bc c92e 8e8e ................\n-00002900: 938d 8d91 02bc c92e 8e8e 938d 8d91 02bc ................\n-00002910: c92e 8e8e 938d 8d91 02bc c92e 8e8e 938d ................\n-00002920: 8d91 02bc c92e 8e8e 938d 8d91 02bc c92e ................\n-00002930: 8e8e 938d 8d91 02bc c92e 8e8e 938d 8d91 ................\n-00002940: 02bc c92e 8e8e 938d 8d91 02bc c92e 8e8e ................\n-00002950: 938d 8d91 02bc c92e 8e8e 938d 8d91 02bc ................\n-00002960: c92e 8e8e 8d91 0ff1 af31 388e 8ea1 8d8d .........18.....\n-00002970: 9110 6ccf 362e 338e 9129 9c31 5261 6e64 ..l.6.3..).1Rand\n-00002980: 6f6d 9603 a666 4e75 6d90 b221 6290 4dde om...fNum..!b.M.\n-00002990: 6572 9347 656e 6572 6174 6f72 9102 c7d0 er.Generator....\n-000029a0: 8d8d 9102 bcc9 2e8e 8e96 0884 3d8d 8d91 ............=...\n-000029b0: 02bc c92e 8e8e 938d 8d91 02bc c92e 8e8e ................\n-000029c0: 938d 8d91 02bc c92e 8e8e 938d 8d91 02bc ................\n-000029d0: c92e 8e8e 938d 8d91 02bc c92e 8e8e 938d ................\n-000029e0: 8d91 02bc c92e 8e8e 938d 8d91 02bc c92e ................\n-000029f0: 8e8e 938d 8d91 02bc c92e 8e8e 938d 8d91 ................\n-00002a00: 02bc c92e 8e8e 938d 8d91 02bc c92e 8e8e ................\n-00002a10: 938d 8d91 02bc c92e 8e8e 938d 8d91 02bc ................\n-00002a20: c92e 8e8e 938d 8d91 02bc c92e 8e8e 938d ................\n-00002a30: 8d91 02bc c92e 8e8e 938d 8d91 02bc c92e ................\n-00002a40: 8e8e 938d 8d91 02bc c92e 8e8e 938d 8d91 ................\n-00002a50: 02bc c92e 8e8e 938d 8d91 02bc c92e 8e8e ................\n-00002a60: 938d 8d91 02bc c92e 8e8e 938d 8d91 02bc ................\n-00002a70: c92e 8e8e 938d 8d91 02bc c92e 8e8e 938d ................\n-00002a80: 8d91 02bc c92e 8e8e 938d 8d91 02bc c92e ................\n-00002a90: 8e8e 938d 8d91 02bc c92e 8e8e 938d 8d91 ................\n-00002aa0: 02bc c92e 8e8e 938d 8d91 02bc c92e 8e8e ................\n-00002ab0: 938d 8d91 02bc c92e 8e8e 938d 8d91 02bc ................\n-00002ac0: c92e 8e8e 938d 8d91 02bc c92e 8e8e 938d ................\n-00002ad0: 8d91 02bc c92e 8e8e 938d 8d91 02bc c92e ................\n-00002ae0: 8e8e 8d91 0ff1 af31 388e 8ea6 8d8d dd37 .......18......7\n-00002af0: 8e91 106c cf43 2b2b 9104 328d 496d 706c ...l.C++..2.Impl\n-00002b00: 656d 656e 90a6 7474 6174 696f 6e8d 9201 emen..ttation...\n-00002b10: 40b4 e231 388e 8ea6 8d8d 388e 9110 6ccf @..18.....8...l.\n-00002b20: 5090 a674 6572 666f 726d 616e 6365 8d92 P..terformance..\n-00002b30: 0173 368c 3230 8e8e a68d 8d39 8e91 106c .s6.20.....9...l\n-00002b40: cf46 96fe f35c 7574 7572 6591 0432 8d57 .F...\\uture..2.W\n-00002b50: 936f 726b 8d92 0171 b903 3231 8e8e a68d .ork...q..21....\n-00002b60: 8d31 308e 9110 6ccf 4163 95a6 746b 6e6f .10...l.Ac..tkno\n-00002b70: 9377 6c65 6467 656d 656e 9374 738d 9201 .wledgemen.ts...\n-00002b80: 502d ee32 318e 8ea6 8d8d 418e 9110 6ccf P-.21.....A...l.\n-00002b90: 5072 6f90 598b 6f66 9604 328d 6f66 9351 Pro.Y.of..2.of.Q\n-00002ba0: 7561 642d 446f 7562 6c65 9341 6464 6974 uad-Double.Addit\n-00002bb0: 696f 6e93 4572 726f 7293 426f 756e 6493 ion.Error.Bound.\n-00002bc0: 284c 656d 6d61 9331 3229 8d91 77bc c832 (Lemma.12)..w..2\n-00002bd0: 328e 8e8e 9f1e 0000 8d92 00e8 23d7 c632 2...........#..2\n-00002be0: 8e8e 8c8b 0000 0003 0000 0000 0000 0000 ................\n+00000840: 0000 0000 0000 0000 0000 0000 0000 002a ...............*\n+00000850: a002 9ad7 5b8d a0fd 62b4 7ba0 027f 4b85 ....[...b.{...K.\n+00000860: 8da0 fda4 b47b 8df3 31c2 d64e a000 0e66 .....{..1..N...f\n+00000870: 6600 0c00 0000 0663 6d62 7831 32dc 436f f......cmbx12.Co\n+00000880: 6e95 8ccc 7465 6e93 7473 8ea4 188c cf8d n...ten.ts......\n+00000890: 8df3 321a f222 5600 0af3 3300 0a00 0000 ..2..\"V...3.....\n+000008a0: 0663 6d62 7831 30dd 318e 9110 6ccf 496e .cmbx10.1...l.In\n+000008b0: 90a6 7474 726f 9059 8b64 7563 7469 6f6e ..ttro.Y.duction\n+000008c0: 8d92 0179 9bb1 338e 8ea1 8d8d 328e 9110 ...y..3.....2...\n+000008d0: 6ccf 5072 656c 696d 696e 6172 6965 738d l.Preliminaries.\n+000008e0: 9201 75f1 0133 8e8e a18d 8d33 8e91 106c ..u..3.....3...l\n+000008f0: cf42 6173 6963 9104 328d 4f70 9059 8b65 .Basic..2.Op.Y.e\n+00000900: 7261 7469 6f6e 738d 9201 6128 6c35 8e8e rations...a(l5..\n+00000910: a40d 999a 8d8d 9110 6ccf c633 2e31 8e91 ........l..3.1..\n+00000920: 299c 3152 656e 6f72 6d61 6c69 7a61 7469 ).1Renormalizati\n+00000930: 6f6e 9107 7b09 8d8d 9102 bcc9 2e8e 8e96 on..{...........\n+00000940: 0884 3d8d 8d91 02bc c92e 8e8e 938d 8d91 ..=.............\n+00000950: 02bc c92e 8e8e 938d 8d91 02bc c92e 8e8e ................\n+00000960: 938d 8d91 02bc c92e 8e8e 938d 8d91 02bc ................\n+00000970: c92e 8e8e 938d 8d91 02bc c92e 8e8e 938d ................\n+00000980: 8d91 02bc c92e 8e8e 938d 8d91 02bc c92e ................\n+00000990: 8e8e 938d 8d91 02bc c92e 8e8e 938d 8d91 ................\n+000009a0: 02bc c92e 8e8e 938d 8d91 02bc c92e 8e8e ................\n+000009b0: 938d 8d91 02bc c92e 8e8e 938d 8d91 02bc ................\n+000009c0: c92e 8e8e 938d 8d91 02bc c92e 8e8e 938d ................\n+000009d0: 8d91 02bc c92e 8e8e 938d 8d91 02bc c92e ................\n+000009e0: 8e8e 938d 8d91 02bc c92e 8e8e 938d 8d91 ................\n+000009f0: 02bc c92e 8e8e 938d 8d91 02bc c92e 8e8e ................\n+00000a00: 938d 8d91 02bc c92e 8e8e 938d 8d91 02bc ................\n+00000a10: c92e 8e8e 938d 8d91 02bc c92e 8e8e 938d ................\n+00000a20: 8d91 02bc c92e 8e8e 938d 8d91 02bc c92e ................\n+00000a30: 8e8e 938d 8d91 02bc c92e 8e8e 938d 8d91 ................\n+00000a40: 02bc c92e 8e8e 938d 8d91 02bc c92e 8e8e ................\n+00000a50: 938d 8d91 02bc c92e 8e8e 938d 8d91 02bc ................\n+00000a60: c92e 8e8e 938d 8d91 02bc c92e 8e8e 938d ................\n+00000a70: 8d91 02bc c92e 8e8e 938d 8d91 02bc c92e ................\n+00000a80: 8e8e 938d 8d91 02bc c92e 8e8e 938d 8d91 ................\n+00000a90: 02bc c92e 8e8e 938d 8d91 02bc c92e 8e8e ................\n+00000aa0: 938d 8d91 02bc c92e 8e8e 938d 8d91 02bc ................\n+00000ab0: c92e 8e8e 8d91 156b 4935 8e8e a18d 8d91 .......kI5......\n+00000ac0: 106c cf33 2e32 8e91 299c 3141 6464 6974 .l.3.2..).1Addit\n+00000ad0: 696f 6e91 0146 b58d 8d91 02bc c92e 8e8e ion..F..........\n+00000ae0: 9608 843d 8d8d 9102 bcc9 2e8e 8e93 8d8d ...=............\n+00000af0: 9102 bcc9 2e8e 8e93 8d8d 9102 bcc9 2e8e ................\n+00000b00: 8e93 8d8d 9102 bcc9 2e8e 8e93 8d8d 9102 ................\n+00000b10: bcc9 2e8e 8e93 8d8d 9102 bcc9 2e8e 8e93 ................\n+00000b20: 8d8d 9102 bcc9 2e8e 8e93 8d8d 9102 bcc9 ................\n+00000b30: 2e8e 8e93 8d8d 9102 bcc9 2e8e 8e93 8d8d ................\n+00000b40: 9102 bcc9 2e8e 8e93 8d8d 9102 bcc9 2e8e ................\n+00000b50: 8e93 8d8d 9102 bcc9 2e8e 8e93 8d8d 9102 ................\n+00000b60: bcc9 2e8e 8e93 8d8d 9102 bcc9 2e8e 8e93 ................\n+00000b70: 8d8d 9102 bcc9 2e8e 8e93 8d8d 9102 bcc9 ................\n+00000b80: 2e8e 8e93 8d8d 9102 bcc9 2e8e 8e93 8d8d ................\n+00000b90: 9102 bcc9 2e8e 8e93 8d8d 9102 bcc9 2e8e ................\n+00000ba0: 8e93 8d8d 9102 bcc9 2e8e 8e93 8d8d 9102 ................\n+00000bb0: bcc9 2e8e 8e93 8d8d 9102 bcc9 2e8e 8e93 ................\n+00000bc0: 8d8d 9102 bcc9 2e8e 8e93 8d8d 9102 bcc9 ................\n+00000bd0: 2e8e 8e93 8d8d 9102 bcc9 2e8e 8e93 8d8d ................\n+00000be0: 9102 bcc9 2e8e 8e93 8d8d 9102 bcc9 2e8e ................\n+00000bf0: 8e93 8d8d 9102 bcc9 2e8e 8e93 8d8d 9102 ................\n+00000c00: bcc9 2e8e 8e93 8d8d 9102 bcc9 2e8e 8e93 ................\n+00000c10: 8d8d 9102 bcc9 2e8e 8e93 8d8d 9102 bcc9 ................\n+00000c20: 2e8e 8e93 8d8d 9102 bcc9 2e8e 8e93 8d8d ................\n+00000c30: 9102 bcc9 2e8e 8e93 8d8d 9102 bcc9 2e8e ................\n+00000c40: 8e93 8d8d 9102 bcc9 2e8e 8e93 8d8d 9102 ................\n+00000c50: bcc9 2e8e 8e93 8d8d 9102 bcc9 2e8e 8e93 ................\n+00000c60: 8d8d 9102 bcc9 2e8e 8e93 8d8d 9102 bcc9 ................\n+00000c70: 2e8e 8e93 8d8d 9102 bcc9 2e8e 8e93 8d8d ................\n+00000c80: 9102 bcc9 2e8e 8e8d 9115 6b49 378e 8ea1 ..........kI7...\n+00000c90: 8d8d 9129 9c31 332e 322e 318e 914c a672 ...).13.2.1..L.r\n+00000ca0: 5175 6164 2d44 6f75 626c 6596 03a6 662b Quad-Double...f+\n+00000cb0: 9344 6f75 626c 6591 05c6 bb8d 8d91 02bc .Double.........\n+00000cc0: c92e 8e8e 9608 843d 8d8d 9102 bcc9 2e8e .......=........\n+00000cd0: 8e93 8d8d 9102 bcc9 2e8e 8e93 8d8d 9102 ................\n+00000ce0: bcc9 2e8e 8e93 8d8d 9102 bcc9 2e8e 8e93 ................\n+00000cf0: 8d8d 9102 bcc9 2e8e 8e93 8d8d 9102 bcc9 ................\n+00000d00: 2e8e 8e93 8d8d 9102 bcc9 2e8e 8e93 8d8d ................\n+00000d10: 9102 bcc9 2e8e 8e93 8d8d 9102 bcc9 2e8e ................\n+00000d20: 8e93 8d8d 9102 bcc9 2e8e 8e93 8d8d 9102 ................\n+00000d30: bcc9 2e8e 8e93 8d8d 9102 bcc9 2e8e 8e93 ................\n+00000d40: 8d8d 9102 bcc9 2e8e 8e93 8d8d 9102 bcc9 ................\n+00000d50: 2e8e 8e93 8d8d 9102 bcc9 2e8e 8e93 8d8d ................\n+00000d60: 9102 bcc9 2e8e 8e93 8d8d 9102 bcc9 2e8e ................\n+00000d70: 8e93 8d8d 9102 bcc9 2e8e 8e93 8d8d 9102 ................\n+00000d80: bcc9 2e8e 8e93 8d8d 9102 bcc9 2e8e 8e93 ................\n+00000d90: 8d8d 9102 bcc9 2e8e 8e93 8d8d 9102 bcc9 ................\n+00000da0: 2e8e 8e93 8d8d 9102 bcc9 2e8e 8e93 8d8d ................\n+00000db0: 9102 bcc9 2e8e 8e93 8d8d 9102 bcc9 2e8e ................\n+00000dc0: 8e93 8d8d 9102 bcc9 2e8e 8e93 8d8d 9102 ................\n+00000dd0: bcc9 2e8e 8e93 8d8d 9102 bcc9 2e8e 8e93 ................\n+00000de0: 8d8d 9102 bcc9 2e8e 8e8d 9115 6b49 378e ............kI7.\n+00000df0: 8ea1 8d8d 9129 9c31 332e 322e 328e 914c .....).13.2.2..L\n+00000e00: a672 5175 6164 2d44 6f75 626c 6596 03a6 .rQuad-Double...\n+00000e10: 662b 9351 7561 642d 446f 7562 6c65 9101 f+.Quad-Double..\n+00000e20: 8480 8d8d 9102 bcc9 2e8e 8e96 0884 3d8d ..............=.\n+00000e30: 8d91 02bc c92e 8e8e 938d 8d91 02bc c92e ................\n+00000e40: 8e8e 938d 8d91 02bc c92e 8e8e 938d 8d91 ................\n+00000e50: 02bc c92e 8e8e 938d 8d91 02bc c92e 8e8e ................\n+00000e60: 938d 8d91 02bc c92e 8e8e 938d 8d91 02bc ................\n+00000e70: c92e 8e8e 938d 8d91 02bc c92e 8e8e 938d ................\n+00000e80: 8d91 02bc c92e 8e8e 938d 8d91 02bc c92e ................\n+00000e90: 8e8e 938d 8d91 02bc c92e 8e8e 938d 8d91 ................\n+00000ea0: 02bc c92e 8e8e 938d 8d91 02bc c92e 8e8e ................\n+00000eb0: 938d 8d91 02bc c92e 8e8e 938d 8d91 02bc ................\n+00000ec0: c92e 8e8e 938d 8d91 02bc c92e 8e8e 938d ................\n+00000ed0: 8d91 02bc c92e 8e8e 938d 8d91 02bc c92e ................\n+00000ee0: 8e8e 938d 8d91 02bc c92e 8e8e 938d 8d91 ................\n+00000ef0: 02bc c92e 8e8e 938d 8d91 02bc c92e 8e8e ................\n+00000f00: 938d 8d91 02bc c92e 8e8e 938d 8d91 02bc ................\n+00000f10: c92e 8e8e 938d 8d91 02bc c92e 8e8e 938d ................\n+00000f20: 8d91 02bc c92e 8e8e 938d 8d91 02bc c92e ................\n+00000f30: 8e8e 8d91 156b 4937 8e8e a18d 8d91 106c .....kI7.......l\n+00000f40: cf33 2e33 8e91 299c 3153 7562 7472 6163 .3.3..).1Subtrac\n+00000f50: 7469 6f6e 9104 9764 8d8d 9102 bcc9 2e8e tion...d........\n+00000f60: 8e96 0884 3d8d 8d91 02bc c92e 8e8e 938d ....=...........\n+00000f70: 8d91 02bc c92e 8e8e 938d 8d91 02bc c92e ................\n+00000f80: 8e8e 938d 8d91 02bc c92e 8e8e 938d 8d91 ................\n+00000f90: 02bc c92e 8e8e 938d 8d91 02bc c92e 8e8e ................\n+00000fa0: 938d 8d91 02bc c92e 8e8e 938d 8d91 02bc ................\n+00000fb0: c92e 8e8e 938d 8d91 02bc c92e 8e8e 938d ................\n+00000fc0: 8d91 02bc c92e 8e8e 938d 8d91 02bc c92e ................\n+00000fd0: 8e8e 938d 8d91 02bc c92e 8e8e 938d 8d91 ................\n+00000fe0: 02bc c92e 8e8e 938d 8d91 02bc c92e 8e8e ................\n+00000ff0: 938d 8d91 02bc c92e 8e8e 938d 8d91 02bc ................\n+00001000: c92e 8e8e 938d 8d91 02bc c92e 8e8e 938d ................\n+00001010: 8d91 02bc c92e 8e8e 938d 8d91 02bc c92e ................\n+00001020: 8e8e 938d 8d91 02bc c92e 8e8e 938d 8d91 ................\n+00001030: 02bc c92e 8e8e 938d 8d91 02bc c92e 8e8e ................\n+00001040: 938d 8d91 02bc c92e 8e8e 938d 8d91 02bc ................\n+00001050: c92e 8e8e 938d 8d91 02bc c92e 8e8e 938d ................\n+00001060: 8d91 02bc c92e 8e8e 938d 8d91 02bc c92e ................\n+00001070: 8e8e 938d 8d91 02bc c92e 8e8e 938d 8d91 ................\n+00001080: 02bc c92e 8e8e 938d 8d91 02bc c92e 8e8e ................\n+00001090: 938d 8d91 02bc c92e 8e8e 938d 8d91 02bc ................\n+000010a0: c92e 8e8e 938d 8d91 02bc c92e 8e8e 938d ................\n+000010b0: 8d91 02bc c92e 8e8e 938d 8d91 02bc c92e ................\n+000010c0: 8e8e 938d 8d91 02bc c92e 8e8e 938d 8d91 ................\n+000010d0: 02bc c92e 8e8e 938d 8d91 02bc c92e 8e8e ................\n+000010e0: 938d 8d91 02bc c92e 8e8e 938d 8d91 02bc ................\n+000010f0: c92e 8e8e 8d91 0ff1 af31 318e 8ea1 8d8d .........11.....\n+00001100: 9110 6ccf 332e 348e 9129 9c31 4d75 6c74 ..l.3.4..).1Mult\n+00001110: 6970 6c69 6361 7469 6f6e 9101 469c 8d8d iplication..F...\n+00001120: 9102 bcc9 2e8e 8e96 0884 3d8d 8d91 02bc ..........=.....\n+00001130: c92e 8e8e 938d 8d91 02bc c92e 8e8e 938d ................\n+00001140: 8d91 02bc c92e 8e8e 938d 8d91 02bc c92e ................\n+00001150: 8e8e 938d 8d91 02bc c92e 8e8e 938d 8d91 ................\n+00001160: 02bc c92e 8e8e 938d 8d91 02bc c92e 8e8e ................\n+00001170: 938d 8d91 02bc c92e 8e8e 938d 8d91 02bc ................\n+00001180: c92e 8e8e 938d 8d91 02bc c92e 8e8e 938d ................\n+00001190: 8d91 02bc c92e 8e8e 938d 8d91 02bc c92e ................\n+000011a0: 8e8e 938d 8d91 02bc c92e 8e8e 938d 8d91 ................\n+000011b0: 02bc c92e 8e8e 938d 8d91 02bc c92e 8e8e ................\n+000011c0: 938d 8d91 02bc c92e 8e8e 938d 8d91 02bc ................\n+000011d0: c92e 8e8e 938d 8d91 02bc c92e 8e8e 938d ................\n+000011e0: 8d91 02bc c92e 8e8e 938d 8d91 02bc c92e ................\n+000011f0: 8e8e 938d 8d91 02bc c92e 8e8e 938d 8d91 ................\n+00001200: 02bc c92e 8e8e 938d 8d91 02bc c92e 8e8e ................\n+00001210: 938d 8d91 02bc c92e 8e8e 938d 8d91 02bc ................\n+00001220: c92e 8e8e 938d 8d91 02bc c92e 8e8e 938d ................\n+00001230: 8d91 02bc c92e 8e8e 938d 8d91 02bc c92e ................\n+00001240: 8e8e 938d 8d91 02bc c92e 8e8e 938d 8d91 ................\n+00001250: 02bc c92e 8e8e 938d 8d91 02bc c92e 8e8e ................\n+00001260: 938d 8d91 02bc c92e 8e8e 938d 8d91 02bc ................\n+00001270: c92e 8e8e 938d 8d91 02bc c92e 8e8e 938d ................\n+00001280: 8d91 02bc c92e 8e8e 938d 8d91 02bc c92e ................\n+00001290: 8e8e 938d 8d91 02bc c92e 8e8e 938d 8d91 ................\n+000012a0: 02bc c92e 8e8e 938d 8d91 02bc c92e 8e8e ................\n+000012b0: 8d91 0ff1 af31 318e 8ea1 8d8d 9129 9c31 .....11......).1\n+000012c0: 332e 342e 318e 914c a672 5175 6164 2d44 3.4.1..L.rQuad-D\n+000012d0: 6f75 626c 6596 03a6 66f3 3421 222c 9a00 ouble...f.4!\",..\n+000012e0: 0af3 3300 0a00 0000 0663 6d73 7931 30df ..3......cmsy10.\n+000012f0: 0293 c644 6f75 626c 6591 05c6 bb8d 8d91 ...Double.......\n+00001300: 02bc c92e 8e8e 9608 843d 8d8d 9102 bcc9 .........=......\n+00001310: 2e8e 8e93 8d8d 9102 bcc9 2e8e 8e93 8d8d ................\n+00001320: 9102 bcc9 2e8e 8e93 8d8d 9102 bcc9 2e8e ................\n+00001330: 8e93 8d8d 9102 bcc9 2e8e 8e93 8d8d 9102 ................\n+00001340: bcc9 2e8e 8e93 8d8d 9102 bcc9 2e8e 8e93 ................\n+00001350: 8d8d 9102 bcc9 2e8e 8e93 8d8d 9102 bcc9 ................\n+00001360: 2e8e 8e93 8d8d 9102 bcc9 2e8e 8e93 8d8d ................\n+00001370: 9102 bcc9 2e8e 8e93 8d8d 9102 bcc9 2e8e ................\n+00001380: 8e93 8d8d 9102 bcc9 2e8e 8e93 8d8d 9102 ................\n+00001390: bcc9 2e8e 8e93 8d8d 9102 bcc9 2e8e 8e93 ................\n+000013a0: 8d8d 9102 bcc9 2e8e 8e93 8d8d 9102 bcc9 ................\n+000013b0: 2e8e 8e93 8d8d 9102 bcc9 2e8e 8e93 8d8d ................\n+000013c0: 9102 bcc9 2e8e 8e93 8d8d 9102 bcc9 2e8e ................\n+000013d0: 8e93 8d8d 9102 bcc9 2e8e 8e93 8d8d 9102 ................\n+000013e0: bcc9 2e8e 8e93 8d8d 9102 bcc9 2e8e 8e93 ................\n+000013f0: 8d8d 9102 bcc9 2e8e 8e93 8d8d 9102 bcc9 ................\n+00001400: 2e8e 8e93 8d8d 9102 bcc9 2e8e 8e93 8d8d ................\n+00001410: 9102 bcc9 2e8e 8e93 8d8d 9102 bcc9 2e8e ................\n+00001420: 8e93 8d8d 9102 bcc9 2e8e 8e8d 910f f1af ................\n+00001430: 3131 8e8e a18d 8d91 299c 3133 2e34 2e32 11......).13.4.2\n+00001440: 8e91 4ca6 7251 7561 642d 446f 7562 6c65 ..L.rQuad-Double\n+00001450: 9603 a666 df02 93c6 5175 6164 2d44 6f75 ...f....Quad-Dou\n+00001460: 626c 6591 0184 808d 8d91 02bc c92e 8e8e ble.............\n+00001470: 9608 843d 8d8d 9102 bcc9 2e8e 8e93 8d8d ...=............\n+00001480: 9102 bcc9 2e8e 8e93 8d8d 9102 bcc9 2e8e ................\n+00001490: 8e93 8d8d 9102 bcc9 2e8e 8e93 8d8d 9102 ................\n+000014a0: bcc9 2e8e 8e93 8d8d 9102 bcc9 2e8e 8e93 ................\n+000014b0: 8d8d 9102 bcc9 2e8e 8e93 8d8d 9102 bcc9 ................\n+000014c0: 2e8e 8e93 8d8d 9102 bcc9 2e8e 8e93 8d8d ................\n+000014d0: 9102 bcc9 2e8e 8e93 8d8d 9102 bcc9 2e8e ................\n+000014e0: 8e93 8d8d 9102 bcc9 2e8e 8e93 8d8d 9102 ................\n+000014f0: bcc9 2e8e 8e93 8d8d 9102 bcc9 2e8e 8e93 ................\n+00001500: 8d8d 9102 bcc9 2e8e 8e93 8d8d 9102 bcc9 ................\n+00001510: 2e8e 8e93 8d8d 9102 bcc9 2e8e 8e93 8d8d ................\n+00001520: 9102 bcc9 2e8e 8e93 8d8d 9102 bcc9 2e8e ................\n+00001530: 8e93 8d8d 9102 bcc9 2e8e 8e93 8d8d 9102 ................\n+00001540: bcc9 2e8e 8e93 8d8d 9102 bcc9 2e8e 8e93 ................\n+00001550: 8d8d 9102 bcc9 2e8e 8e93 8d8d 9102 bcc9 ................\n+00001560: 2e8e 8e93 8d8d 9102 bcc9 2e8e 8e93 8d8d ................\n+00001570: 9102 bcc9 2e8e 8e8d 910f f1af 3131 8e8e ............11..\n+00001580: a18d 8d91 106c cf33 2e35 8e91 299c 3144 .....l.3.5..).1D\n+00001590: 6976 6973 696f 6e91 0468 bc8d 8d91 02bc ivision..h......\n+000015a0: c92e 8e8e 9608 843d 8d8d 9102 bcc9 2e8e .......=........\n+000015b0: 8e93 8d8d 9102 bcc9 2e8e 8e93 8d8d 9102 ................\n+000015c0: bcc9 2e8e 8e93 8d8d 9102 bcc9 2e8e 8e93 ................\n+000015d0: 8d8d 9102 bcc9 2e8e 8e93 8d8d 9102 bcc9 ................\n+000015e0: 2e8e 8e93 8d8d 9102 bcc9 2e8e 8e93 8d8d ................\n+000015f0: 9102 bcc9 2e8e 8e93 8d8d 9102 bcc9 2e8e ................\n+00001600: 8e93 8d8d 9102 bcc9 2e8e 8e93 8d8d 9102 ................\n+00001610: bcc9 2e8e 8e93 8d8d 9102 bcc9 2e8e 8e93 ................\n+00001620: 8d8d 9102 bcc9 2e8e 8e93 8d8d 9102 bcc9 ................\n+00001630: 2e8e 8e93 8d8d 9102 bcc9 2e8e 8e93 8d8d ................\n+00001640: 9102 bcc9 2e8e 8e93 8d8d 9102 bcc9 2e8e ................\n+00001650: 8e93 8d8d 9102 bcc9 2e8e 8e93 8d8d 9102 ................\n+00001660: bcc9 2e8e 8e93 8d8d 9102 bcc9 2e8e 8e93 ................\n+00001670: 8d8d 9102 bcc9 2e8e 8e93 8d8d 9102 bcc9 ................\n+00001680: 2e8e 8e93 8d8d 9102 bcc9 2e8e 8e93 8d8d ................\n+00001690: 9102 bcc9 2e8e 8e93 8d8d 9102 bcc9 2e8e ................\n+000016a0: 8e93 8d8d 9102 bcc9 2e8e 8e93 8d8d 9102 ................\n+000016b0: bcc9 2e8e 8e93 8d8d 9102 bcc9 2e8e 8e93 ................\n+000016c0: 8d8d 9102 bcc9 2e8e 8e93 8d8d 9102 bcc9 ................\n+000016d0: 2e8e 8e93 8d8d 9102 bcc9 2e8e 8e93 8d8d ................\n+000016e0: 9102 bcc9 2e8e 8e93 8d8d 9102 bcc9 2e8e ................\n+000016f0: 8e93 8d8d 9102 bcc9 2e8e 8e93 8d8d 9102 ................\n+00001700: bcc9 2e8e 8e93 8d8d 9102 bcc9 2e8e 8e93 ................\n+00001710: 8d8d 9102 bcc9 2e8e 8e93 8d8d 9102 bcc9 ................\n+00001720: 2e8e 8e93 8d8d 9102 bcc9 2e8e 8e93 8d8d ................\n+00001730: 9102 bcc9 2e8e 8e93 8d8d 9102 bcc9 2e8e ................\n+00001740: 8e93 8d8d 9102 bcc9 2e8e 8e8d 910f f1af ................\n+00001750: 3136 8e8e a918 8ccf 8d8d dd34 8e91 106c 16.........4...l\n+00001760: cf41 6c67 6562 7261 6963 9104 328d 4f70 .Algebraic..2.Op\n+00001770: 9059 8b65 7261 7469 6f6e 738d 9201 4384 .Y.erations...C.\n+00001780: c131 368e 8ea1 8d8d 9110 6ccf c634 2e31 .16.......l..4.1\n+00001790: 8e91 299c 31f3 330b a062 3e00 0af3 3300 ..).1.3..b>...3.\n+000017a0: 0a00 0000 0663 6d6d 6931 30de 4e91 0131 .....cmmi10.N..1\n+000017b0: 9fc6 2d74 6891 03a6 6650 95b2 216f 9377 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8e8e 938d 8d91 02bc c92e 8e8e 938d 8d91 ................\n+00001ff0: 02bc c92e 8e8e 938d 8d91 02bc c92e 8e8e ................\n+00002000: 938d 8d91 02bc c92e 8e8e 938d 8d91 02bc ................\n+00002010: c92e 8e8e 938d 8d91 02bc c92e 8e8e 938d ................\n+00002020: 8d91 02bc c92e 8e8e 938d 8d91 02bc c92e ................\n+00002030: 8e8e 938d 8d91 02bc c92e 8e8e 938d 8d91 ................\n+00002040: 02bc c92e 8e8e 938d 8d91 02bc c92e 8e8e ................\n+00002050: 938d 8d91 02bc c92e 8e8e 938d 8d91 02bc ................\n+00002060: c92e 8e8e 938d 8d91 02bc c92e 8e8e 938d ................\n+00002070: 8d91 02bc c92e 8e8e 938d 8d91 02bc c92e ................\n+00002080: 8e8e 938d 8d91 02bc c92e 8e8e 938d 8d91 ................\n+00002090: 02bc c92e 8e8e 938d 8d91 02bc c92e 8e8e ................\n+000020a0: 938d 8d91 02bc c92e 8e8e 938d 8d91 02bc ................\n+000020b0: c92e 8e8e 938d 8d91 02bc c92e 8e8e 938d ................\n+000020c0: 8d91 02bc c92e 8e8e 938d 8d91 02bc c92e ................\n+000020d0: 8e8e 8d91 0ff1 af31 378e 8ea1 8d8d 9110 .......17.......\n+000020e0: 6ccf 352e 338e 9129 9c31 5491 ff16 6572 l.5.3..).1T...er\n+000020f0: 6967 6f6e 6f6d 6574 7269 6373 9106 0550 igonometrics...P\n+00002100: 8d8d 9102 bcc9 2e8e 8e96 0884 3d8d 8d91 ............=...\n+00002110: 02bc c92e 8e8e 938d 8d91 02bc c92e 8e8e ................\n+00002120: 938d 8d91 02bc c92e 8e8e 938d 8d91 02bc ................\n+00002130: c92e 8e8e 938d 8d91 02bc c92e 8e8e 938d ................\n+00002140: 8d91 02bc c92e 8e8e 938d 8d91 02bc c92e ................\n+00002150: 8e8e 938d 8d91 02bc c92e 8e8e 938d 8d91 ................\n+00002160: 02bc c92e 8e8e 938d 8d91 02bc c92e 8e8e ................\n+00002170: 938d 8d91 02bc c92e 8e8e 938d 8d91 02bc ................\n+00002180: c92e 8e8e 938d 8d91 02bc c92e 8e8e 938d ................\n+00002190: 8d91 02bc c92e 8e8e 938d 8d91 02bc c92e ................\n+000021a0: 8e8e 938d 8d91 02bc c92e 8e8e 938d 8d91 ................\n+000021b0: 02bc c92e 8e8e 938d 8d91 02bc c92e 8e8e ................\n+000021c0: 938d 8d91 02bc c92e 8e8e 938d 8d91 02bc ................\n+000021d0: c92e 8e8e 938d 8d91 02bc c92e 8e8e 938d ................\n+000021e0: 8d91 02bc c92e 8e8e 938d 8d91 02bc c92e ................\n+000021f0: 8e8e 938d 8d91 02bc c92e 8e8e 938d 8d91 ................\n+00002200: 02bc c92e 8e8e 938d 8d91 02bc c92e 8e8e ................\n+00002210: 938d 8d91 02bc c92e 8e8e 938d 8d91 02bc ................\n+00002220: c92e 8e8e 938d 8d91 02bc c92e 8e8e 938d ................\n+00002230: 8d91 02bc c92e 8e8e 938d 8d91 02bc c92e ................\n+00002240: 8e8e 938d 8d91 02bc c92e 8e8e 938d 8d91 ................\n+00002250: 02bc c92e 8e8e 938d 8d91 02bc c92e 8e8e ................\n+00002260: 938d 8d91 02bc c92e 8e8e 938d 8d91 02bc ................\n+00002270: c92e 8e8e 938d 8d91 02bc c92e 8e8e 938d ................\n+00002280: 8d91 02bc c92e 8e8e 8d91 0ff1 af31 378e .............17.\n+00002290: 8ea1 8d8d 9110 6ccf 352e 348e 9129 9c31 ......l.5.4..).1\n+000022a0: 496e 95b2 2176 9365 7273 6591 03a6 6654 In..!v.erse...fT\n+000022b0: 91ff 1665 7269 676f 6e6f 6d65 7472 6963 ...erigonometric\n+000022c0: 7391 02e3 2c8d 8d91 02bc c92e 8e8e 9608 s...,...........\n+000022d0: 843d 8d8d 9102 bcc9 2e8e 8e93 8d8d 9102 .=..............\n+000022e0: bcc9 2e8e 8e93 8d8d 9102 bcc9 2e8e 8e93 ................\n+000022f0: 8d8d 9102 bcc9 2e8e 8e93 8d8d 9102 bcc9 ................\n+00002300: 2e8e 8e93 8d8d 9102 bcc9 2e8e 8e93 8d8d ................\n+00002310: 9102 bcc9 2e8e 8e93 8d8d 9102 bcc9 2e8e ................\n+00002320: 8e93 8d8d 9102 bcc9 2e8e 8e93 8d8d 9102 ................\n+00002330: bcc9 2e8e 8e93 8d8d 9102 bcc9 2e8e 8e93 ................\n+00002340: 8d8d 9102 bcc9 2e8e 8e93 8d8d 9102 bcc9 ................\n+00002350: 2e8e 8e93 8d8d 9102 bcc9 2e8e 8e93 8d8d ................\n+00002360: 9102 bcc9 2e8e 8e93 8d8d 9102 bcc9 2e8e ................\n+00002370: 8e93 8d8d 9102 bcc9 2e8e 8e93 8d8d 9102 ................\n+00002380: bcc9 2e8e 8e93 8d8d 9102 bcc9 2e8e 8e93 ................\n+00002390: 8d8d 9102 bcc9 2e8e 8e93 8d8d 9102 bcc9 ................\n+000023a0: 2e8e 8e93 8d8d 9102 bcc9 2e8e 8e93 8d8d ................\n+000023b0: 9102 bcc9 2e8e 8e93 8d8d 9102 bcc9 2e8e ................\n+000023c0: 8e93 8d8d 9102 bcc9 2e8e 8e93 8d8d 9102 ................\n+000023d0: bcc9 2e8e 8e93 8d8d 9102 bcc9 2e8e 8e93 ................\n+000023e0: 8d8d 9102 bcc9 2e8e 8e93 8d8d 9102 bcc9 ................\n+000023f0: 2e8e 8e93 8d8d 9102 bcc9 2e8e 8e93 8d8d ................\n+00002400: 9102 bcc9 2e8e 8e93 8d8d 9102 bcc9 2e8e ................\n+00002410: 8e93 8d8d 9102 bcc9 2e8e 8e93 8d8d 9102 ................\n+00002420: bcc9 2e8e 8e8d 910f f1af 3138 8e8e a18d ..........18....\n+00002430: 8d91 106c cf35 2e35 8e91 299c 3148 7970 ...l.5.5..).1Hyp\n+00002440: 954d de65 7262 936f 6c69 6391 03a6 6646 .M.erb.olic...fF\n+00002450: 91ff 1665 756e 6374 696f 6e73 9100 ba55 ...eunctions...U\n+00002460: 8d8d 9102 bcc9 2e8e 8e96 0884 3d8d 8d91 ............=...\n+00002470: 02bc c92e 8e8e 938d 8d91 02bc c92e 8e8e ................\n+00002480: 938d 8d91 02bc c92e 8e8e 938d 8d91 02bc ................\n+00002490: c92e 8e8e 938d 8d91 02bc c92e 8e8e 938d ................\n+000024a0: 8d91 02bc c92e 8e8e 938d 8d91 02bc c92e ................\n+000024b0: 8e8e 938d 8d91 02bc c92e 8e8e 938d 8d91 ................\n+000024c0: 02bc c92e 8e8e 938d 8d91 02bc c92e 8e8e ................\n+000024d0: 938d 8d91 02bc c92e 8e8e 938d 8d91 02bc ................\n+000024e0: c92e 8e8e 938d 8d91 02bc c92e 8e8e 938d ................\n+000024f0: 8d91 02bc c92e 8e8e 938d 8d91 02bc c92e ................\n+00002500: 8e8e 938d 8d91 02bc c92e 8e8e 938d 8d91 ................\n+00002510: 02bc c92e 8e8e 938d 8d91 02bc c92e 8e8e ................\n+00002520: 938d 8d91 02bc c92e 8e8e 938d 8d91 02bc ................\n+00002530: c92e 8e8e 938d 8d91 02bc c92e 8e8e 938d ................\n+00002540: 8d91 02bc c92e 8e8e 938d 8d91 02bc c92e ................\n+00002550: 8e8e 938d 8d91 02bc c92e 8e8e 938d 8d91 ................\n+00002560: 02bc c92e 8e8e 938d 8d91 02bc c92e 8e8e ................\n+00002570: 938d 8d91 02bc c92e 8e8e 938d 8d91 02bc ................\n+00002580: c92e 8e8e 938d 8d91 02bc c92e 8e8e 938d ................\n+00002590: 8d91 02bc c92e 8e8e 938d 8d91 02bc c92e ................\n+000025a0: 8e8e 938d 8d91 02bc c92e 8e8e 938d 8d91 ................\n+000025b0: 02bc c92e 8e8e 938d 8d91 02bc c92e 8e8e ................\n+000025c0: 938d 8d91 02bc c92e 8e8e 8d91 0ff1 af31 ...............1\n+000025d0: 388e 8ea6 8d8d dd36 8e91 106c cf4d 6973 8......6...l.Mis\n+000025e0: 6365 6c6c 616e 656f 7573 9104 328d 526f cellaneous..2.Ro\n+000025f0: 7574 696e 6573 8d92 0137 be8c 3138 8e8e utines...7..18..\n+00002600: a18d 8d91 106c cfc6 362e 318e 9129 9c31 .....l..6.1..).1\n+00002610: 496e 7075 7496 03a6 662f 934f 7574 7075 Input...f/.Outpu\n+00002620: 7491 0319 c88d 8d91 02bc c92e 8e8e 9608 t...............\n+00002630: 843d 8d8d 9102 bcc9 2e8e 8e93 8d8d 9102 .=..............\n+00002640: bcc9 2e8e 8e93 8d8d 9102 bcc9 2e8e 8e93 ................\n+00002650: 8d8d 9102 bcc9 2e8e 8e93 8d8d 9102 bcc9 ................\n+00002660: 2e8e 8e93 8d8d 9102 bcc9 2e8e 8e93 8d8d ................\n+00002670: 9102 bcc9 2e8e 8e93 8d8d 9102 bcc9 2e8e ................\n+00002680: 8e93 8d8d 9102 bcc9 2e8e 8e93 8d8d 9102 ................\n+00002690: bcc9 2e8e 8e93 8d8d 9102 bcc9 2e8e 8e93 ................\n+000026a0: 8d8d 9102 bcc9 2e8e 8e93 8d8d 9102 bcc9 ................\n+000026b0: 2e8e 8e93 8d8d 9102 bcc9 2e8e 8e93 8d8d ................\n+000026c0: 9102 bcc9 2e8e 8e93 8d8d 9102 bcc9 2e8e ................\n+000026d0: 8e93 8d8d 9102 bcc9 2e8e 8e93 8d8d 9102 ................\n+000026e0: bcc9 2e8e 8e93 8d8d 9102 bcc9 2e8e 8e93 ................\n+000026f0: 8d8d 9102 bcc9 2e8e 8e93 8d8d 9102 bcc9 ................\n+00002700: 2e8e 8e93 8d8d 9102 bcc9 2e8e 8e93 8d8d ................\n+00002710: 9102 bcc9 2e8e 8e93 8d8d 9102 bcc9 2e8e ................\n+00002720: 8e93 8d8d 9102 bcc9 2e8e 8e93 8d8d 9102 ................\n+00002730: bcc9 2e8e 8e93 8d8d 9102 bcc9 2e8e 8e93 ................\n+00002740: 8d8d 9102 bcc9 2e8e 8e93 8d8d 9102 bcc9 ................\n+00002750: 2e8e 8e93 8d8d 9102 bcc9 2e8e 8e93 8d8d ................\n+00002760: 9102 bcc9 2e8e 8e93 8d8d 9102 bcc9 2e8e ................\n+00002770: 8e93 8d8d 9102 bcc9 2e8e 8e93 8d8d 9102 ................\n+00002780: bcc9 2e8e 8e93 8d8d 9102 bcc9 2e8e 8e93 ................\n+00002790: 8d8d 9102 bcc9 2e8e 8e93 8d8d 9102 bcc9 ................\n+000027a0: 2e8e 8e93 8d8d 9102 bcc9 2e8e 8e8d 910f ................\n+000027b0: f1af 3138 8e8e a18d 8d91 106c cf36 2e32 ..18.......l.6.2\n+000027c0: 8e91 299c 3143 6f6d 7061 7269 736f 6e73 ..).1Comparisons\n+000027d0: 9107 82e3 8d8d 9102 bcc9 2e8e 8e96 0884 ................\n+000027e0: 3d8d 8d91 02bc c92e 8e8e 938d 8d91 02bc =...............\n+000027f0: c92e 8e8e 938d 8d91 02bc c92e 8e8e 938d ................\n+00002800: 8d91 02bc c92e 8e8e 938d 8d91 02bc c92e ................\n+00002810: 8e8e 938d 8d91 02bc c92e 8e8e 938d 8d91 ................\n+00002820: 02bc c92e 8e8e 938d 8d91 02bc c92e 8e8e ................\n+00002830: 938d 8d91 02bc c92e 8e8e 938d 8d91 02bc ................\n+00002840: c92e 8e8e 938d 8d91 02bc c92e 8e8e 938d ................\n+00002850: 8d91 02bc c92e 8e8e 938d 8d91 02bc c92e ................\n+00002860: 8e8e 938d 8d91 02bc c92e 8e8e 938d 8d91 ................\n+00002870: 02bc c92e 8e8e 938d 8d91 02bc c92e 8e8e ................\n+00002880: 938d 8d91 02bc c92e 8e8e 938d 8d91 02bc ................\n+00002890: c92e 8e8e 938d 8d91 02bc c92e 8e8e 938d ................\n+000028a0: 8d91 02bc c92e 8e8e 938d 8d91 02bc c92e ................\n+000028b0: 8e8e 938d 8d91 02bc c92e 8e8e 938d 8d91 ................\n+000028c0: 02bc c92e 8e8e 938d 8d91 02bc c92e 8e8e ................\n+000028d0: 938d 8d91 02bc c92e 8e8e 938d 8d91 02bc ................\n+000028e0: c92e 8e8e 938d 8d91 02bc c92e 8e8e 938d ................\n+000028f0: 8d91 02bc c92e 8e8e 938d 8d91 02bc c92e ................\n+00002900: 8e8e 938d 8d91 02bc c92e 8e8e 938d 8d91 ................\n+00002910: 02bc c92e 8e8e 938d 8d91 02bc c92e 8e8e ................\n+00002920: 938d 8d91 02bc c92e 8e8e 938d 8d91 02bc ................\n+00002930: c92e 8e8e 938d 8d91 02bc c92e 8e8e 938d ................\n+00002940: 8d91 02bc c92e 8e8e 938d 8d91 02bc c92e ................\n+00002950: 8e8e 938d 8d91 02bc c92e 8e8e 938d 8d91 ................\n+00002960: 02bc c92e 8e8e 8d91 0ff1 af31 388e 8ea1 ...........18...\n+00002970: 8d8d 9110 6ccf 362e 338e 9129 9c31 5261 ....l.6.3..).1Ra\n+00002980: 6e64 6f6d 9603 a666 4e75 6d90 b221 6290 ndom...fNum..!b.\n+00002990: 4dde 6572 9347 656e 6572 6174 6f72 9102 M.er.Generator..\n+000029a0: c7d0 8d8d 9102 bcc9 2e8e 8e96 0884 3d8d ..............=.\n+000029b0: 8d91 02bc c92e 8e8e 938d 8d91 02bc c92e ................\n+000029c0: 8e8e 938d 8d91 02bc c92e 8e8e 938d 8d91 ................\n+000029d0: 02bc c92e 8e8e 938d 8d91 02bc c92e 8e8e ................\n+000029e0: 938d 8d91 02bc c92e 8e8e 938d 8d91 02bc ................\n+000029f0: c92e 8e8e 938d 8d91 02bc c92e 8e8e 938d ................\n+00002a00: 8d91 02bc c92e 8e8e 938d 8d91 02bc c92e ................\n+00002a10: 8e8e 938d 8d91 02bc c92e 8e8e 938d 8d91 ................\n+00002a20: 02bc c92e 8e8e 938d 8d91 02bc c92e 8e8e ................\n+00002a30: 938d 8d91 02bc c92e 8e8e 938d 8d91 02bc ................\n+00002a40: c92e 8e8e 938d 8d91 02bc c92e 8e8e 938d ................\n+00002a50: 8d91 02bc c92e 8e8e 938d 8d91 02bc c92e ................\n+00002a60: 8e8e 938d 8d91 02bc c92e 8e8e 938d 8d91 ................\n+00002a70: 02bc c92e 8e8e 938d 8d91 02bc c92e 8e8e ................\n+00002a80: 938d 8d91 02bc c92e 8e8e 938d 8d91 02bc ................\n+00002a90: c92e 8e8e 938d 8d91 02bc c92e 8e8e 938d ................\n+00002aa0: 8d91 02bc c92e 8e8e 938d 8d91 02bc c92e ................\n+00002ab0: 8e8e 938d 8d91 02bc c92e 8e8e 938d 8d91 ................\n+00002ac0: 02bc c92e 8e8e 938d 8d91 02bc c92e 8e8e ................\n+00002ad0: 938d 8d91 02bc c92e 8e8e 938d 8d91 02bc ................\n+00002ae0: c92e 8e8e 8d91 0ff1 af31 388e 8ea6 8d8d .........18.....\n+00002af0: dd37 8e91 106c cf43 2b2b 9104 328d 496d .7...l.C++..2.Im\n+00002b00: 706c 656d 656e 90a6 7474 6174 696f 6e8d plemen..ttation.\n+00002b10: 9201 40b4 e231 388e 8ea6 8d8d 388e 9110 ..@..18.....8...\n+00002b20: 6ccf 5090 a674 6572 666f 726d 616e 6365 l.P..terformance\n+00002b30: 8d92 0173 368c 3230 8e8e a68d 8d39 8e91 ...s6.20.....9..\n+00002b40: 106c cf46 96fe f35c 7574 7572 6591 0432 .l.F...\\uture..2\n+00002b50: 8d57 936f 726b 8d92 0171 b903 3231 8e8e .W.ork...q..21..\n+00002b60: a68d 8d31 308e 9110 6ccf 4163 95a6 746b ...10...l.Ac..tk\n+00002b70: 6e6f 9377 6c65 6467 656d 656e 9374 738d no.wledgemen.ts.\n+00002b80: 9201 502d ee32 318e 8ea6 8d8d 418e 9110 ..P-.21.....A...\n+00002b90: 6ccf 5072 6f90 598b 6f66 9604 328d 6f66 l.Pro.Y.of..2.of\n+00002ba0: 9351 7561 642d 446f 7562 6c65 9341 6464 .Quad-Double.Add\n+00002bb0: 6974 696f 6e93 4572 726f 7293 426f 756e ition.Error.Boun\n+00002bc0: 6493 284c 656d 6d61 9331 3229 8d91 77bc d.(Lemma.12)..w.\n+00002bd0: c832 328e 8e8e 9f1e 0000 8d92 00e8 23d7 .22...........#.\n+00002be0: c632 8e8e 8c8b 0000 0003 0000 0000 0000 .2..............\n 00002bf0: 0000 0000 0000 0000 0000 0000 0000 0000 ................\n-00002c00: 0000 0000 0000 0000 0000 0000 0000 0821 ...............!\n-00002c10: a002 9ad7 5b8d a0fd 62b4 7ba0 027f 4b85 ....[...b.{...K.\n-00002c20: 8da0 fda4 b47b 8d8d dc31 8e91 184c cb49 .....{...1...L.I\n-00002c30: 6e90 8ccc 7472 6f90 7333 6475 6374 696f n...tro.s3ductio\n-00002c40: 6e8e a918 718b 8dc6 4d75 6c74 6970 7265 n...q...Multipre\n-00002c50: 6369 7369 6f6e 9603 9160 636f 6d70 7574 cision...`comput\n-00002c60: 6174 696f 6e93 6861 7391 0391 6161 9376 ation.has...aa.v\n-00002c70: 91ff 6444 6172 6965 749a b221 7993 6f66 ..dDariet..!y.of\n-00002c80: 9361 7070 6c69 6361 7469 6f6e 9361 7265 .application.are\n-00002c90: 6173 2c91 0395 9573 7563 9868 9361 7393 as,....suc.h.as.\n-00002ca0: 7075 7265 936d 6174 6865 6d61 7469 6373 pure.mathematics\n-00002cb0: 2c91 0395 9573 7475 6479 936f 668e a40d ,....study.of...\n-00002cc0: 999a 8d6d 6174 6865 6d61 7469 6361 6c9b ...mathematical.\n-00002cd0: 03fa 0e63 6f6e 7374 616e 95b2 2174 732c ...constan..!ts,\n-00002ce0: 9104 0ef9 6372 7970 746f 6772 6170 6893 ....cryptograph.\n-00002cf0: 7996 ff16 652c 9104 0ef8 616e 6491 03fa y...e,....and...\n-00002d00: 0f63 6f6d 7075 7461 7469 6f6e 616c 9867 .computational.g\n-00002d10: 656f 6d65 7472 7993 2e91 05d8 d642 6563 eometry......Bec\n-00002d20: 6175 7365 9103 fa0f 6f66 9874 6869 732c ause....of.this,\n-00002d30: 9104 0ef9 6d61 6e90 b221 7998 6172 6269 ....man..!y.arbi\n-00002d40: 2d8e a18d 7472 6172 7996 0421 c770 7265 -...trary..!.pre\n-00002d50: 6369 7369 6f6e 9361 6c67 6f72 6974 686d cision.algorithm\n-00002d60: 7393 616e 6493 6c69 6272 6172 6965 7393 s.and.libraries.\n-00002d70: 6861 95b2 2176 9365 9104 21c7 629a 4dde ha..!v.e..!.b.M.\n-00002d80: 6565 6e91 0421 c664 6576 9365 6c6f 7098 een..!.dev.elop.\n-00002d90: 6564 9604 21c7 7573 696e 6793 6f6e 6c79 ed..!.using.only\n-00002da0: 9374 6865 930c 7865 6493 7072 6563 6973 .the..xed.precis\n-00002db0: 696f 6e93 6172 6974 682d 8ea1 8d6d 6574 ion.arith-...met\n-00002dc0: 6963 2e91 0564 8354 6865 7996 03d3 4863 ic...d.They...Hc\n-00002dd0: 616e 9103 d347 6290 4dde 6593 6469 7669 an...Gb.M.e.divi\n-00002de0: 6465 6493 696e 9ab2 2174 6f93 7498 7798 ded.in..!to.t.w.\n-00002df0: 6f93 6772 6f75 7073 9362 6173 6564 936f o.groups.based.o\n-00002e00: 6e93 7468 6591 03d3 4777 9861 9879 9370 n.the...Gw.a.y.p\n-00002e10: 7265 6369 7369 6f6e 936e 9875 6d98 6290 recision.n.um.b.\n-00002e20: 4dde 6572 7393 6172 6593 7265 7072 6573 M.ers.are.repres\n-00002e30: 656e 9874 6564 2e8e a18d 536f 6d65 9602 en.ted....Some..\n-00002e40: d5ef 6c69 6272 6172 6965 7393 7374 6f72 ..libraries.stor\n-00002e50: 659b 02d5 f06e 95b2 2175 6d93 6290 4dde 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7265 6369 7369 6f6e 9370 6163 90b2 .precision.pac..\n+00003800: 216b 91ff 6444 6167 6598 4d50 4655 4e2e !k..dDage.MPFUN.\n+00003810: 934f 7572 8ea1 8d71 7561 642d 7072 6563 .Our...quad-prec\n+00003820: 6973 696f 6e96 0516 136c 6962 7261 7279 ision....library\n+00003830: 9b05 1614 6973 9361 90b2 2176 91ff 6444 ....is.a..!v..dD\n+00003840: 6169 6c61 626c 6598 6174 93e1 6874 7470 ailable.at..http\n+00003850: 3a2f 2f77 7777 2e6e 6572 7363 2e67 6f76 ://www.nersc.gov\n+00003860: 2f7e 6468 6261 696c 6579 2f6d 7064 6973 /~dhbailey/mpdis\n+00003870: 742f 6d70 6469 7374 2e68 746d 6cc6 2e8e t/mpdist.html...\n+00003880: a18d 4f75 7296 03ff 4971 7561 642d 646f ..Our...Iquad-do\n+00003890: 7562 6c65 936c 6962 7261 7279 9368 6173 uble.library.has\n+000038a0: 9362 904d de65 656e 9373 7563 6365 7373 .b.M.een.success\n+000038b0: 6675 6c6c 7991 03ff 4a69 6e9a b221 7465 fully...Jin..!te\n+000038c0: 6772 6174 6564 9369 6e98 746f 9361 9370 grated.in.to.a.p\n+000038d0: 6172 616c 6c65 6c93 7698 6f72 7465 7893 arallel.v.ortex.\n+000038e0: 726f 6c6c 2d75 7093 7369 6d98 756c 6174 roll-up.sim.ulat\n+000038f0: 696f 6e8e a18d 636f 9a4d de64 653b 9603 ion...co.M.de;..\n+00003900: a666 7468 6973 9369 7393 6272 6965 0d79 .fthis.is.brie.y\n+00003910: 9364 6573 6372 6962 9865 6493 696e 935b .describ.ed.in.[\n+00003920: 8d38 8e91 0579 9a5d 2e8e a18d 9111 0000 .8...y.]........\n+00003930: 5468 6596 0507 9672 6573 7493 6f66 9105 The....rest.of..\n+00003940: 0795 7468 6593 7061 709a 4dde 6572 9369 ..the.pap.M.er.i\n+00003950: 7393 6f72 6761 6e69 7a65 6493 6173 9366 s.organized.as.f\n+00003960: 6f6c 6c6f 90b2 2177 732e 9109 016c 5365 ollo..!ws....lSe\n+00003970: 6374 696f 6e93 3293 6465 7363 7269 6298 ction.2.describ.\n+00003980: 6573 9373 6f6d 6591 0507 9562 6173 6963 es.some....basic\n+00003990: 9370 726f 7098 6572 7469 6573 936f 668e .prop.erties.of.\n+000039a0: a18d 4945 4545 9104 1e31 0d6f 6174 696e ..IEEE...1.oatin\n+000039b0: 6791 041e 5170 9a4d de6f 696e 90b2 2174 g...Qp.M.oin..!t\n+000039c0: 9604 1e50 6172 6974 686d 6574 6963 9361 ...Parithmetic.a\n+000039d0: 6e64 9362 7569 6c64 696e 6793 626c 6f98 nd.building.blo.\n+000039e0: 6390 b221 6b73 9104 1e51 666f 7293 6f75 c..!ks...Qfor.ou\n+000039f0: 7293 7175 6164 2d64 6f75 626c 6593 616c r.quad-double.al\n+00003a00: 676f 7269 7468 6d73 2e91 0645 9c49 6e93 gorithms...E.In.\n+00003a10: 5365 6374 696f 6e93 338e a18d 7795 b221 Section.3...w..!\n+00003a20: 659b 0536 d870 7265 7365 6e93 7498 7468 e..6.presen.t.th\n+00003a30: 6598 7175 6164 2d64 6f75 626c 6598 616c e.quad-double.al\n+00003a40: 676f 7269 7468 6d73 9105 36d9 666f 7298 gorithms..6.for.\n+00003a50: 6261 7369 6398 6f70 904d de65 7261 7469 basic.op.M.erati\n+00003a60: 6f6e 732c 9605 9af4 696e 636c 7564 696e ons,....includin\n+00003a70: 6798 7265 6e6f 726d 697a 6174 696f 6e2c g.renormization,\n+00003a80: 9361 6464 6974 696f 6e2c 8ea1 8d6d 9ab2 .addition,...m..\n+00003a90: 2175 6c74 6970 6c69 6361 7469 6f6e 9603 !ultiplication..\n+00003aa0: ee8c 616e 6493 6469 7669 7369 6f6e 2e91 ..and.division..\n+00003ab0: 05b6 4e53 6563 7469 6f6e 9334 9361 6e64 ..NSection.4.and\n+00003ac0: 9335 9370 7265 7365 6e98 7493 7468 6593 .5.presen.t.the.\n+00003ad0: 616c 676f 7269 7468 6d73 9103 ee8b 666f algorithms....fo\n+00003ae0: 7293 736f 6d65 9361 6c67 6562 7261 6963 r.some.algebraic\n+00003af0: 936f 7090 4dde 6572 6174 696f 6e73 8ea1 .op.M.erations..\n+00003b00: 8d61 6e64 9605 788d 7472 616e 7363 656e .and..x.transcen\n+00003b10: 6465 6e90 b221 7461 6c93 6675 6e63 7469 den..!tal.functi\n+00003b20: 6f6e 732e 9b0a 5453 5365 6374 696f 6e93 ons...TSSection.\n+00003b30: 3693 6465 7363 7269 6290 4dde 6573 9105 6.describ.M.es..\n+00003b40: 788e 736f 6d65 9361 7578 696c 6961 7279 x.some.auxiliary\n+00003b50: 9366 756e 6374 696f 6e73 2e98 5365 6374 .functions..Sect\n+00003b60: 696f 6e93 3793 6272 6965 0d79 8ea1 8d64 ion.7.brie.y...d\n+00003b70: 6573 6372 6962 904d de65 7396 0546 6c6f escrib.M.es..Flo\n+00003b80: 7572 9b05 466b 432b 2b93 6c69 6272 6172 ur..FkC++.librar\n+00003b90: 7998 696d 706c 656d 656e 90b2 2174 696e y.implemen..!tin\n+00003ba0: 6793 7468 6598 6162 904d de6f 95b2 2176 g.the.ab.M.o..!v\n+00003bb0: 9365 9605 466c 616c 676f 7269 7468 6d73 .e..Flalgorithms\n+00003bc0: 2e91 09bd ed53 6563 7469 6f6e 9338 9870 .....Section.8.p\n+00003bd0: 7265 7365 6e90 b221 7473 9374 6865 9874 resen..!ts.the.t\n+00003be0: 696d 696e 678e a18d 7265 7375 6c74 7396 iming...results.\n+00003bf0: 03a6 666f 6693 7468 6593 6b9a b221 6572 ..fof.the.k..!er\n+00003c00: 6e65 6c93 6f70 904d de65 7261 7469 6f6e nel.op.M.eration\n+00003c10: 7393 6f6e 9364 690b 6572 656e 9874 9361 s.on.di.eren.t.a\n+00003c20: 7263 9868 6974 6563 7475 7265 732e 9104 rc.hitectures...\n+00003c30: dddd 5365 6374 696f 6e93 3993 6469 7363 ..Section.9.disc\n+00003c40: 7573 7365 7393 6675 7475 7265 9377 986f usses.future.w.o\n+00003c50: 726b 2e8e 9f22 8041 8d8d dc32 8e91 184c rk...\".A...2...L\n+00003c60: cb50 7265 6c69 6d69 6e61 7269 6573 8ea6 .Preliminaries..\n+00003c70: 8dc6 496e 9603 9577 7468 6973 9373 6563 ..In...wthis.sec\n+00003c80: 7469 6f6e 2c91 0398 d977 9ab2 2165 9370 tion,....w..!e.p\n+00003c90: 7265 7365 6e98 7493 736f 6d65 9362 6173 resen.t.some.bas\n+00003ca0: 6963 9370 726f 709a 4dde 6572 7469 6573 ic.prop.M.erties\n+00003cb0: 9103 9576 616e 6493 616c 676f 7269 7468 ...vand.algorith\n+00003cc0: 6d73 936f 6693 4945 4545 9103 9572 0d6f ms.of.IEEE...r.o\n+00003cd0: 6174 696e 6793 7098 6f69 6e90 b221 7493 ating.p.oin..!t.\n+00003ce0: 6172 6974 686d 6574 6963 8ea1 8d75 7365 arithmetic...use\n+00003cf0: 6496 04f8 c269 6e93 7175 6164 2d64 6f75 d....in.quad-dou\n+00003d00: 626c 6593 6172 6974 686d 6574 6963 2e91 ble.arithmetic..\n+00003d10: 08d4 f154 6865 7365 9372 6573 756c 7473 ...These.results\n+00003d20: 9104 f8c3 6172 6593 6261 7365 6493 6f6e ....are.based.on\n+00003d30: 9344 656b 6b9a b221 6572 935b 8d36 8e91 .Dekk..!er.[.6..\n+00003d40: 0579 9a5d 2c91 054d 584b 6e98 7574 6893 .y.],..MXKn.uth.\n+00003d50: 5b8d 398e 9105 799a 5d2c 9105 4d59 5072 [.9...y.],..MYPr\n+00003d60: 6965 7374 935b 8d31 308e 910a f334 5d2c iest.[.10....4],\n+00003d70: 8ea1 8d53 6865 7795 b221 6393 6893 756b ...Shew..!c.h.uk\n+00003d80: 9603 c1fb 5b8d 3131 8e91 0af3 345d 2c9b ....[.11....4],.\n+00003d90: 03c8 df61 6e64 936f 7468 6572 732e 9105 ...and.others...\n+00003da0: 309a 496e 9366 6163 742c 986d 616e 90b2 0.In.fact,.man..\n+00003db0: 2179 936f 669b 03c1 fa74 6865 9361 6c67 !y.of....the.alg\n+00003dc0: 6f72 6974 686d 7398 616e 6493 6469 6167 orithms.and.diag\n+00003dd0: 7261 6d73 9861 7265 9364 6972 6563 746c rams.are.directl\n+00003de0: 7998 7461 6b90 b221 656e 9366 726f 6d2c y.tak..!en.from,\n+00003df0: 8ea1 8d6f 7296 03a6 6662 6173 6564 936f ...or...fbased.o\n+00003e00: 6e2c 9353 6865 7795 b221 6393 6893 756b n,.Shew..!c.h.uk\n+00003e10: 2773 9103 a666 7061 7090 4dde 6572 2e8e 's...fpap.M.er..\n+00003e20: a18d 9111 0000 416c 6c96 0397 d962 6173 ......All....bas\n+00003e30: 6963 9b03 97da 6172 6974 686d 6574 6963 ic....arithmetic\n+00003e40: 7393 6172 6593 6173 7375 6d65 6493 746f s.are.assumed.to\n+00003e50: 9862 9a4d de65 9370 9865 7266 6f72 6d65 .b.M.e.p.erforme\n+00003e60: 6493 696e 9103 97da 4945 4545 9103 97d5 d.in....IEEE....\n+00003e70: 646f 7562 6c65 9366 6f72 6d61 742c 9103 double.format,..\n+00003e80: 9ac3 7769 7468 9372 6f75 6e64 2d74 6f2d ..with.round-to-\n+00003e90: 6576 90b2 2165 6e8e a18d 726f 756e 6469 ev..!en...roundi\n+00003ea0: 6e67 9604 1bba 6f6e 9374 6965 732e 9106 ng....on.ties...\n+00003eb0: 3dd9 4691 ff16 656f 7293 616e 9ab2 2179 =.F...eor.an..!y\n+00003ec0: 9362 696e 6172 7993 6f70 904d de65 7261 .binary.op.M.era\n+00003ed0: 746f 7293 df01 9103 ce33 3291 03ce 3266 tor......32...2f\n+00003ee0: c62b de3b 9601 d331 df00 de3b 93df 02de .+.;...1...;....\n+00003ef0: 3b93 3ddf 67c6 2c91 0439 0f77 9865 9604 ;.=.g.,..9.w.e..\n+00003f00: 1bba 7573 658d 930d 8e91 0a31 1028 de61 ..use......1.(.a\n+00003f10: 9602 bd23 df01 9b02 bd24 de62 c629 9103 ...#.....$.b.)..\n+00003f20: ce32 3d91 03ce 33de 6193 df0c 98de 6296 .2=...3.a.....b.\n+00003f30: 041b bac6 746f 9364 656e 6f74 6593 7468 ....to.denote.th\n+00003f40: 658e a18d 0d6f 6174 696e 6796 0457 2970 e....oating..W)p\n+00003f50: 904d de6f 696e 90b2 2174 9b04 5728 7265 .M.oin..!t..W(re\n+00003f60: 7375 6c74 936f 6698 de61 9102 e4c3 df01 sult.of..a......\n+00003f70: 9102 e4c2 de62 c62c 9104 8359 616e 6493 .....b.,...Yand.\n+00003f80: 6465 0c6e 658d 9865 7272 8e91 11c8 de28 de.ne..err.....(\n+00003f90: de61 9102 e4c3 df01 9102 e4c2 de62 c629 .a...........b.)\n+00003fa0: 9361 7398 de61 9b02 e4c3 df01 9602 e4c2 .as..a..........\n+00003fb0: de62 9104 313f c63d 8d91 0431 400d 8e91 .b..1?.=...1@...\n+00003fc0: 0a46 9628 de61 93df 0198 de62 c629 932b .F.(.a.....b.).+\n+00003fd0: 8d98 6572 728e 9110 5679 28de 6193 df01 ..err...Vy(.a...\n+00003fe0: 98de 62c6 292e 9106 f024 5468 726f 7567 ..b.)....$Throug\n+00003ff0: 686f 7574 9104 5729 7468 6973 8e8e 9f1e hout..W)this....\n+00004000: 0000 8d92 00e8 23d7 338e 8e8c 8b00 0000 ......#.3.......\n+00004010: 0400 0000 0000 0000 0000 0000 0000 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1593 6ade 62df 6ae0 2e91 05ca b054 .U..j.b.j......T\n-000041f0: 6865 6e8e a18d df6a 8dc6 6572 728e 910d hen....j..err...\n-00004200: 71b6 28de 6196 026e ecc6 2b93 de62 c629 q.(.a..n..+..b.)\n-00004210: df6a 9603 0aa7 1493 6ade 62df 6a93 1493 .j......j.b.j...\n-00004220: 6ade 61df 6ae0 2e8e a68d dd4c 656d 6d61 j.a.j......Lemma\n-00004230: 9105 f955 322e 9105 d8b2 c65b 8d31 318e ...U2......[.11.\n-00004240: 910a f334 2c91 0594 bb70 2e91 0980 4333 ...4,....p....C3\n-00004250: 3131 5d9b 0556 bde0 4c91 ff70 b965 7496 11]..V..L..p.et.\n-00004260: 0556 bede 6193 e061 6e64 98de 6293 e062 .V..a..and..b..b\n-00004270: 91ff 70b9 6593 7477 6f98 de70 e02d 6269 ..p.e.two..p.-bi\n-00004280: 7493 0d6f 9bff 70b9 6174 696e 6793 7098 t..o..p.ating.p.\n-00004290: 6f69 6e74 9105 56bd 6e75 6d62 9865 7273 oint..V.numb.ers\n-000042a0: 2e91 094c f154 6865 6e8d 93c6 6572 728e ...L.Then...err.\n-000042b0: 9112 c874 28de 6191 0376 90c6 2b91 0376 ...t(.a..v..+..v\n-000042c0: 8fde 62c6 2991 059d c13d 8ea1 8d28 de61 ..b.)....=...(.a\n-000042d0: 9602 6eec c62b 93de 62c6 2993 df00 8d93 ..n..+..b.).....\n-000042e0: c60d 8e91 0884 4228 de61 93c6 2b93 de62 ......B(.a..+..b\n-000042f0: c629 9603 eaea e069 7393 7296 ff70 b965 .).....is.r..p.e\n-00004300: 7072 9365 7365 6e74 6162 6c65 9603 eaea pr.esentable....\n-00004310: 6173 9361 93de 70e0 2d62 6974 930d 6f9b as.a..p.-bit..o.\n-00004320: ff70 b961 7469 6e67 9370 986f 696e 7493 .p.ating.p.oint.\n-00004330: 6e75 6d62 9865 722e 8ea6 8ddd 416c 676f numb.er.....Algo\n-00004340: 7269 7468 6d91 0515 b033 2e91 056c 3ac6 rithm....3...l:.\n-00004350: 5b8d 3131 8e91 0af3 342c 9104 9d4a 702e [.11....4,...Jp.\n-00004360: 9107 2e67 3331 325d 9604 6be9 5468 659b ...g312]..k.The.\n-00004370: 046b ea66 6f6c 6c6f 90b2 2177 696e 6793 .k.follo..!wing.\n-00004380: 616c 676f 7269 7468 6d93 636f 6d70 7574 algorithm.comput\n-00004390: 6573 98de 7391 0453 d6c6 3d8d 9104 53d5 es..s..S..=...S.\n-000043a0: 0d8e 910a 692b 28de 6191 02f2 99c6 2b91 ....i+(.a.....+.\n-000043b0: 02f2 98de 62c6 2993 616e 6498 de65 9104 ....b.).and..e..\n-000043c0: 53d6 c63d 8d91 0453 d565 7272 8e91 11c5 S..=...S.err....\n-000043d0: 8b28 de61 9102 f299 c62b 9102 f298 de62 .(.a.....+.....b\n-000043e0: c629 2c8e a18d 6173 7375 6d69 6e67 9103 .),...assuming..\n-000043f0: a666 df6a de61 df6a 9603 0aa7 1593 6ade .f.j.a.j......j.\n-00004400: 62df 6ac6 2e8e a914 d3d3 8d8d 8d92 00b4 b.j.............\n-00004410: accd f337 8c2d f814 000a f333 000a 0000 ...7.-.....3....\n-00004420: 0007 636d 6373 6331 30e2 5175 6963 6b2d ..cmcsc10.Quick-\n-00004430: 5477 90c1 b46f 2d53 756d c628 de61 3b91 Tw...o-Sum.(.a;.\n-00004440: 01d3 3162 c629 8e9f 1adf fe8d 8d8d 9ff2 ..1b.)..........\n-00004450: 61ea 8d8d 9200 baac ce31 2e8e 8d92 00cf a........1......\n-00004460: 3114 de73 9603 0aa7 df20 93de 6196 026e 1..s..... ..a..n\n-00004470: ecdf 0893 de62 8e8e a40d 9999 8d8d 9200 .....b..........\n-00004480: baac cec6 322e 8e8d 9200 cf31 14de 6596 ....2......1..e.\n-00004490: 030a a7df 2093 de62 9602 6eec df09 93c6 .... ..b..n.....\n-000044a0: 28de 7393 df09 93de 61c6 298e 8ea1 8d8d (.s.....a.).....\n-000044b0: 9200 baac ce33 2e8e 8d92 00cf 3114 dd72 .....3......1..r\n-000044c0: 6574 7572 6e91 03a6 66c6 28de 733b 9101 eturn...f.(.s;..\n-000044d0: d331 65c6 298e 8e8e 8e8e 8e8e 9f3e bffd .1e.)........>..\n-000044e0: 8ddd 416c 676f 7269 7468 6d91 03a2 6534 ..Algorithm...e4\n-000044f0: 2e91 0498 f9c6 5b8d 3131 8e91 0af3 342c ......[.11....4,\n-00004500: 9103 421e 702e 9104 b414 3331 345d 9603 ..B.p.....314]..\n-00004510: 290c 5468 6593 666f 6c6c 6f90 b221 7769 ).The.follo..!wi\n-00004520: 6e67 9361 6c67 6f72 6974 686d 9363 6f6d ng.algorithm.com\n-00004530: 7075 7465 7393 de73 9603 0aa7 c63d 8d93 putes..s.....=..\n-00004540: 0d8e 9109 1ffd 28de 6191 0174 38c6 2b9b ......(.a..t8.+.\n-00004550: 0174 39de 62c6 2991 0329 0c61 6e64 9103 .t9.b.)..).and..\n-00004560: 290b de65 93c6 3d8d 9365 7272 8e91 107c )..e..=..err...|\n-00004570: 5d28 de61 98c6 2b98 de62 c629 2e91 04b4 ](.a..+..b.)....\n-00004580: 1454 6869 738e a18d 616c 676f 7269 7468 .This...algorith\n-00004590: 6d96 03a6 6675 7365 7393 7468 7265 6593 m...fuses.three.\n-000045a0: 6d6f 7265 930d 6f61 7469 6e67 9370 9a4d more..oating.p.M\n-000045b0: de6f 696e 90b2 2174 936f 7098 6572 6174 .oin..!t.op.erat\n-000045c0: 696f 6e73 9369 6e73 7465 6164 936f 6693 ions.instead.of.\n-000045d0: 6193 6272 616e 6390 b221 682e 8ea6 8d8d a.branc..!h.....\n-000045e0: 8d92 00b4 accd e254 7790 c1b4 6f2d 5375 .......Tw...o-Su\n-000045f0: 6dc6 28de 613b 9101 d331 62c6 298e 9f21 m.(.a;...1b.)..!\n-00004600: acca 8d8d 8d9f eb95 1e8d 8d92 00ba acce ................\n-00004610: 312e 8e8d 9200 cf31 14de 7396 030a a7df 1......1..s.....\n-00004620: 2093 de61 9602 6eec df08 93de 628e 8ea4 ..a..n.....b...\n-00004630: 0d99 998d 8d92 00ba acce c632 2e8e 8d92 ...........2....\n-00004640: 00cf 3114 de76 9103 6f3a df20 9103 0aa7 ..1..v..o:. ....\n-00004650: de73 9602 6eec df09 93de 618e 8ea1 8d8d .s..n.....a.....\n-00004660: 9200 baac cec6 332e 8e8d 9200 cf31 14de ......3......1..\n-00004670: 6596 030a a7df 2093 c628 de61 9602 6eec e..... ..(.a..n.\n-00004680: df09 93c6 28de 7393 df09 93de 769a 6493 ....(.s.....v.d.\n-00004690: c629 2993 df08 93c6 28de 6293 df09 93de .)).....(.b.....\n-000046a0: 7698 c629 8e8e a18d 8d92 00ba acce 342e v..)..........4.\n-000046b0: 8e8d 9200 cf31 14dd 7265 7475 726e 9103 .....1..return..\n-000046c0: a666 c628 de73 3b91 01d3 3165 c629 8e8e .f.(.s;...1e.)..\n-000046d0: 8e8e 8e8e 8e9f 4c59 968d dd41 6c67 6f72 ......LY...Algor\n-000046e0: 6974 686d 9104 4ac4 352e 9105 0b89 c65b ithm..J.5......[\n-000046f0: 8d31 318e 910a f334 2c91 03c0 b870 2e91 .11....4,....p..\n-00004700: 051d 0733 3235 5d96 03bb 7454 6865 9366 ...325]...tThe.f\n-00004710: 6f6c 6c6f 90b2 2177 696e 6793 616c 676f ollo..!wing.algo\n-00004720: 7269 7468 6d93 7370 6c69 7473 9361 9335 rithm.splits.a.5\n-00004730: 332d 6269 7493 4945 4545 9103 bb6f 646f 3-bit.IEEE...odo\n-00004740: 7562 6c65 9370 7265 6369 7369 6f6e 930d uble.precision..\n-00004750: 6f61 7469 6e67 8ea1 8d70 9a4d de6f 696e oating...p.M.oin\n-00004760: 95b2 2174 9104 51b8 6e93 756d 9362 9865 ..!t..Q.n.um.b.e\n-00004770: 729b 0451 b769 6e93 746f 9604 51b8 de61 r..Q.in.to..Q..a\n-00004780: 8d9f 01c8 aece 6869 8e91 0be7 14c6 616e ......hi......an\n-00004790: 6493 de61 8d9f 01c8 aece 6c6f 8e91 071c d..a......lo....\n-000047a0: 78c6 2c91 047c 8b65 6163 90b2 2168 9377 x.,..|.eac..!h.w\n-000047b0: 6974 6893 3236 9362 6974 7398 6f66 9373 ith.26.bits.of.s\n-000047c0: 6967 6e69 0c63 616e 642c 9104 7c8c 7375 igni.cand,..|.su\n-000047d0: 6390 b221 6893 7468 6174 98de 6196 0428 c..!h.that..a..(\n-000047e0: 2ec6 3d93 de61 8d9f 01c8 aece 6869 8e91 ..=..a......hi..\n-000047f0: 0a76 7ec6 2b91 02e1 22de 618d 9f01 c8ae .v~.+...\".a.....\n-00004800: ce6c 6f8e 9107 1c78 c62e 9106 dfd2 de61 .lo....x.......a\n-00004810: 8d9f 01c8 aece 6869 8e91 0be7 14c6 7769 ......hi......wi\n-00004820: 6c6c 8ea1 8d63 6f6e 9ab2 2174 6169 6e96 ll...con..!tain.\n-00004830: 03a6 6674 6865 930c 7273 7493 3236 9362 ..fthe..rst.26.b\n-00004840: 6974 732c 9377 6869 6c65 93de 618d 9f01 its,.while..a...\n-00004850: c8ae ce6c 6f8e 910a c2de c677 696c 6c93 ...lo......will.\n-00004860: 636f 6e98 7461 696e 9374 6865 936c 6f98 con.tain.the.lo.\n-00004870: 7798 6572 9332 3693 6269 7473 2e8e a68d w.er.26.bits....\n-00004880: 8d8d 9200 b4ac cde2 5370 6c69 74c6 28de ........Split.(.\n-00004890: 61c6 298e 9f21 acca 8d8d 8d9f eb95 1e8d a.)..!..........\n-000048a0: 8d92 00ba acce 312e 8e8d 9200 cf31 14de ......1......1..\n-000048b0: 7496 030a a7df 2093 c628 328d 9ffc 06be t..... ..(2.....\n-000048c0: ce32 378e 910b 6ef4 c62b 9602 6eec 3129 .27...n..+..n.1)\n-000048d0: 93df 0a93 de61 8e8e a40d 9999 8d8d 9200 .....a..........\n-000048e0: baac cec6 322e 8e8d 9200 cf31 14de 618d ....2......1..a.\n-000048f0: 9f01 c8ae ce68 698e 910a a003 df20 9103 .....hi...... ..\n-00004900: 0aa7 de74 9602 6eec df09 93c6 28de 7493 ...t..n.....(.t.\n-00004910: df09 93de 61c6 298e 8ea1 8d8d 9200 baac ....a.).........\n-00004920: ce33 2e8e 8d92 00cf 3114 de61 8d9f 01c8 .3......1..a....\n-00004930: aece 6c6f 8e91 0a27 1fdf 2091 030a a7de ..lo...'.. .....\n-00004940: 6196 026e ecdf 0993 de61 8d9f 01c8 aece a..n.....a......\n-00004950: 6869 8e8e 8ea1 8d8d 9200 baac cec6 342e hi............4.\n-00004960: 8e8d 9200 cf31 14dd 7265 7475 726e 9103 .....1..return..\n-00004970: a666 c628 de61 8d9f 01c8 aece 6869 8e91 .f.(.a......hi..\n-00004980: 0795 5cde 3b91 01d3 3161 8d9f 01c8 aece ..\\.;...1a......\n-00004990: 6c6f 8e91 071c 78c6 298e 8e8e 8e8e 8e8e lo....x.).......\n-000049a0: 9f4c 5996 8ddd 416c 676f 7269 7468 6d91 .LY...Algorithm.\n-000049b0: 0432 8d36 2e91 0500 00c6 5b8d 3131 8e91 .2.6......[.11..\n-000049c0: 0af3 342c 9603 a666 702e 9104 dddd 3332 ..4,...fp.....32\n-000049d0: 365d 9354 6865 9366 6f6c 6c6f 90b2 2177 6].The.follo..!w\n-000049e0: 696e 6793 616c 676f 7269 7468 6d93 636f ing.algorithm.co\n-000049f0: 6d70 7574 6573 93de 7096 030a a7c6 3d8d mputes..p.....=.\n-00004a00: 930d 8e91 091f fd28 de61 9602 6eec df02 .......(.a..n...\n-00004a10: 93de 62c6 2996 03a6 6661 6e64 93de 6596 ..b.)...fand..e.\n-00004a20: 030a a7c6 3d8d 9365 7272 8e91 107c 5d28 ....=..err...|](\n-00004a30: de61 9602 6eec df02 93de 62c6 292e 8ea6 .a..n.....b.)...\n-00004a40: 8d8d 8d91 6c67 aee2 5477 95c1 b46f 2d50 ....lg..Tw...o-P\n-00004a50: 7293 6f64 c628 de61 3b91 01d3 3162 c629 r.od.(.a;...1b.)\n-00004a60: 8e9f 2879 978d 8d8d 9fe4 c851 8d8d 9172 ..(y.......Q...r\n-00004a70: 67af 312e 8e8d 9200 86eb f5de 7096 030a g.1.........p...\n-00004a80: a7df 2093 de61 9602 6eec df0a 93de 628e .. ..a..n.....b.\n-00004a90: 8ea4 0d99 998d 8d91 7267 afc6 322e 8e8d ........rg..2...\n-00004aa0: 9200 86eb f528 de61 8d9f 01c8 aece 6869 .....(.a......hi\n-00004ab0: 8e91 0795 5cde 3b91 01d3 3161 8d9f 01c8 ....\\.;...1a....\n-00004ac0: aece 6c6f 8e91 071c 78c6 2991 030a a7df ..lo....x.).....\n-00004ad0: 2091 03a6 66e2 5370 6c69 74c6 28de 61c6 ...f.Split.(.a.\n-00004ae0: 298e 8ea1 8d8d 9172 67af 332e 8e8d 9200 )......rg.3.....\n-00004af0: 86eb f528 de62 8d9f 01c8 aece 6869 8e91 ...(.b......hi..\n-00004b00: 0795 5cde 3b91 01d3 3162 8d9f 01c8 aece ..\\.;...1b......\n-00004b10: 6c6f 8e91 071c 78c6 2991 030a a7df 2091 lo....x.)..... .\n-00004b20: 03a6 66e2 5370 6c69 74c6 28de 62c6 298e ..f.Split.(.b.).\n-00004b30: 8ea1 8d8d 9172 67af 342e 8e8d 9200 86eb .....rg.4.......\n-00004b40: f5de 6596 030a a7df 2093 c628 28de 618d ..e..... ..((.a.\n-00004b50: 9f01 c8ae ce68 698e 960a 0448 df0a 9b02 .....hi....H....\n-00004b60: 6eec de62 8d9f 01c8 aece 6869 8e93 df09 n..b......hi....\n-00004b70: 98de 70c6 2998 df08 98de 618d 9f01 c8ae ..p.).....a.....\n-00004b80: ce68 698e 93df 0a98 de62 8d9f 01c8 aece .hi......b......\n-00004b90: 6c6f 8e96 098b 64df 0898 de61 8d9f 01c8 lo....d....a....\n-00004ba0: aece 6c6f 8e93 df0a 98de 628d 9f01 c8ae ..lo......b.....\n-00004bb0: ce68 698e 9107 955c c629 98df 0898 de61 .hi....\\.).....a\n-00004bc0: 8d9f 01c8 aece 6c6f 8e93 df0a 98de 628d ......lo......b.\n-00004bd0: 9f01 c8ae ce6c 6f8e 8e8e a18d 8d91 7267 .....lo.......rg\n-00004be0: afc6 352e 8e8d 9200 86eb f5dd 7265 7475 ..5.........retu\n-00004bf0: 726e 9103 a666 c628 de70 3b91 01d3 3165 rn...f.(.p;...1e\n-00004c00: c629 8e8e 8e8e 8e8e 8e9f 59f3 2f8d 9111 .)........Y./...\n-00004c10: 0000 536f 6d65 9b03 3c0a 6d61 6395 b221 ..Some..<.mac..!\n-00004c20: 6869 6e65 7391 033c 0b68 6193 7693 6598 hines..<.ha.v.e.\n-00004c30: 6191 033c 0b66 7573 6564 986d 9375 6c74 a..<.fused.m.ult\n-00004c40: 6970 6c79 2d61 6464 9603 3c0b 696e 7374 iply-add..<.inst\n-00004c50: 7275 6374 696f 6e98 2846 4d41 2991 033b ruction.(FMA)..;\n-00004c60: f074 6861 7493 6361 6e98 6576 91ff 6444 .that.can.ev..dD\n-00004c70: 616c 7561 7465 9365 7870 7265 7373 696f aluate.expressio\n-00004c80: 6e98 7375 6390 b221 688e a18d 6173 9b04 n.suc..!h...as..\n-00004c90: 3518 de61 9102 ce0e df02 9602 ce0d de62 5..a...........b\n-00004ca0: 93df 0693 de63 9604 3519 c677 6974 6898 .....c..5..with.\n-00004cb0: 6193 7369 6e67 6c65 9872 6f75 6e64 696e a.single.roundin\n-00004cc0: 6798 6572 726f 722e 9106 89f5 5791 ff16 g.error.....W...\n-00004cd0: 6565 9863 616e 9874 616b 9ab2 2165 9361 ee.can.tak..!e.a\n-00004ce0: 6476 91ff 6444 616e 9874 6167 659b 0435 dv..dDan.tage..5\n-00004cf0: 186f 6693 7468 6973 9869 6e73 7472 7563 .of.this.instruc\n-00004d00: 7469 6f6e 9374 6f98 636f 6d70 7574 658e tion.to.compute.\n-00004d10: a18d 6578 6163 7496 02e9 c670 726f 9a4d ..exact....pro.M\n-00004d20: de64 7563 7493 6f66 9102 e9c7 7495 b221 .duct.of....t..!\n-00004d30: 7793 6f96 02e9 c60d 6f61 7469 6e67 9370 w.o.....oating.p\n-00004d40: 986f 696e 9ab2 2174 936e 9875 6d98 6290 .oin..!t.n.um.b.\n-00004d50: 4dde 6572 7393 6d98 7563 9868 9102 e9c7 M.ers.m.uc.h....\n-00004d60: 6661 7374 6572 2e91 049e fd54 6865 7365 faster.....These\n-00004d70: 936d 6163 9868 696e 6573 9369 6e63 6c75 .mac.hines.inclu\n-00004d80: 6465 9349 424d 9102 e997 5098 6f98 7798 de.IBM....P.o.w.\n-00004d90: 6572 9373 6572 6965 738e a18d 2869 6e63 er.series...(inc\n-00004da0: 6c75 6469 6e67 9603 a666 7468 6593 5095 luding...fthe.P.\n-00004db0: b221 6f93 7793 6572 5043 292c 9603 a666 .!o.w.erPC),...f\n-00004dc0: 6f6e 9377 6869 6390 b221 6893 7468 6973 on.whic..!h.this\n-00004dd0: 9373 696d 706c 690c 6361 7469 6f6e 9369 .simpli.cation.i\n-00004de0: 7393 7465 7374 6564 2e8e 8e9f 1e00 008d s.tested........\n-00004df0: 9200 e823 d734 8e8e 8c8b 0000 0005 0000 ...#.4..........\n+00004030: 0000 0000 0000 002b e5a0 029a d75b 8da0 .......+.....[..\n+00004040: fd62 b47b a002 7f4b 858d a0fd a4b4 7b8d .b.{...K......{.\n+00004050: c670 6170 904d de65 722c 9103 ba8f de22 .pap.M.er,.....\"\n+00004060: 9603 2588 c63d 9332 8d9f fc06 bef3 27be ..%..=.2......'.\n+00004070: 4bc8 0b00 0800 0000 0800 0000 0563 6d73 K............cms\n+00004080: 7938 d200 ce35 338e 9113 5307 c669 7396 y8...53...S..is.\n+00004090: 03b6 8774 6865 936d 6163 90b2 2168 696e ...the.mac..!hin\n+000040a0: 659b 03b6 8665 7073 696c 6f6e 9366 6f72 e....epsilon.for\n+000040b0: 9349 4545 4591 03b6 8364 6f75 626c 6593 .IEEE....double.\n+000040c0: 7072 6563 6973 696f 6e98 6e95 b221 756d precision.n..!um\n+000040d0: 9362 904d de65 7273 2c91 03ba 8f61 6e64 .b.M.ers,....and\n+000040e0: 9603 b687 de22 8d9f 01c8 aece 7164 8e91 .....\"......qd..\n+000040f0: 0cda e6c6 3d91 0325 8832 8d9f fc06 bed2 ....=..%.2......\n+00004100: 00ce 3231 318e 9117 930b c669 7393 7468 ..211......is.th\n+00004110: 658e a40d 999a 8d70 7265 6369 7369 6f6e e......precision\n+00004120: 9603 a666 6f6e 6593 6578 709a 4dde 6563 ...fone.exp.M.ec\n+00004130: 7473 9366 6f72 9371 7561 642d 646f 7562 ts.for.quad-doub\n+00004140: 6c65 936e 95b2 2175 6d93 6298 6572 732e le.n..!um.b.ers.\n+00004150: 8ea9 1699 9a8d dd4c 656d 6d61 9104 8310 .......Lemma....\n+00004160: 312e 9105 265c c65b 8d31 318e 910a f334 1...&\\.[.11....4\n+00004170: 2c91 03fd e970 2e91 05af e533 3130 5d9b ,....p.....310].\n+00004180: 042b 52e0 4c91 ff70 b965 7496 042b 53de .+R.L..p.et..+S.\n+00004190: 6193 e061 6e64 98de 6293 e062 91ff 70b9 a..and..b..b..p.\n+000041a0: 6598 7477 6f93 de70 e02d 6269 7493 0d6f e.two..p.-bit..o\n+000041b0: 96ff 70b9 6174 696e 6798 7093 6f69 6e74 ..p.ating.p.oint\n+000041c0: 9104 2b53 6e75 6d62 9365 7273 9873 7563 ..+Snumb.ers.suc\n+000041d0: 6896 042b 5374 6861 7493 df6a de61 df6a h..+Sthat..j.a.j\n+000041e0: 9603 7f55 1593 6ade 62df 6ae0 2e91 05ca ...U..j.b.j.....\n+000041f0: b054 6865 6e8e a18d df6a 8dc6 6572 728e .Then....j..err.\n+00004200: 910d 71b6 28de 6196 026e ecc6 2b93 de62 ..q.(.a..n..+..b\n+00004210: c629 df6a 9603 0aa7 1493 6ade 62df 6a93 .).j......j.b.j.\n+00004220: 1493 6ade 61df 6ae0 2e8e a68d dd4c 656d ..j.a.j......Lem\n+00004230: 6d61 9105 f955 322e 9105 d8b2 c65b 8d31 ma...U2......[.1\n+00004240: 318e 910a f334 2c91 0594 bb70 2e91 0980 1....4,....p....\n+00004250: 4333 3131 5d9b 0556 bde0 4c91 ff70 b965 C311]..V..L..p.e\n+00004260: 7496 0556 bede 6193 e061 6e64 98de 6293 t..V..a..and..b.\n+00004270: e062 91ff 70b9 6593 7477 6f98 de70 e02d .b..p.e.two..p.-\n+00004280: 6269 7493 0d6f 9bff 70b9 6174 696e 6793 bit..o..p.ating.\n+00004290: 7098 6f69 6e74 9105 56bd 6e75 6d62 9865 p.oint..V.numb.e\n+000042a0: 7273 2e91 094c f154 6865 6e8d 93c6 6572 rs...L.Then...er\n+000042b0: 728e 9112 c874 28de 6191 0376 90c6 2b91 r....t(.a..v..+.\n+000042c0: 0376 8fde 62c6 2991 059d c13d 8ea1 8d28 .v..b.)....=...(\n+000042d0: de61 9602 6eec c62b 93de 62c6 2993 df00 .a..n..+..b.)...\n+000042e0: 8d93 c60d 8e91 0884 4228 de61 93c6 2b93 ........B(.a..+.\n+000042f0: de62 c629 9603 eaea e069 7393 7296 ff70 .b.).....is.r..p\n+00004300: b965 7072 9365 7365 6e74 6162 6c65 9603 .epr.esentable..\n+00004310: eaea 6173 9361 93de 70e0 2d62 6974 930d ..as.a..p.-bit..\n+00004320: 6f9b ff70 b961 7469 6e67 9370 986f 696e o..p.ating.p.oin\n+00004330: 7493 6e75 6d62 9865 722e 8ea6 8ddd 416c t.numb.er.....Al\n+00004340: 676f 7269 7468 6d91 0515 b033 2e91 056c gorithm....3...l\n+00004350: 3ac6 5b8d 3131 8e91 0af3 342c 9104 9d4a :.[.11....4,...J\n+00004360: 702e 9107 2e67 3331 325d 9604 6be9 5468 p....g312]..k.Th\n+00004370: 659b 046b ea66 6f6c 6c6f 90b2 2177 696e e..k.follo..!win\n+00004380: 6793 616c 676f 7269 7468 6d93 636f 6d70 g.algorithm.comp\n+00004390: 7574 6573 98de 7391 0453 d6c6 3d8d 9104 utes..s..S..=...\n+000043a0: 53d5 0d8e 910a 692b 28de 6191 02f2 99c6 S.....i+(.a.....\n+000043b0: 2b91 02f2 98de 62c6 2993 616e 6498 de65 +.....b.).and..e\n+000043c0: 9104 53d6 c63d 8d91 0453 d565 7272 8e91 ..S..=...S.err..\n+000043d0: 11c5 8b28 de61 9102 f299 c62b 9102 f298 ...(.a.....+....\n+000043e0: de62 c629 2c8e a18d 6173 7375 6d69 6e67 .b.),...assuming\n+000043f0: 9103 a666 df6a de61 df6a 9603 0aa7 1593 ...f.j.a.j......\n+00004400: 6ade 62df 6ac6 2e8e a914 d3d3 8d8d 8d92 j.b.j...........\n+00004410: 00b4 accd f337 8c2d f814 000a f333 000a .....7.-.....3..\n+00004420: 0000 0007 636d 6373 6331 30e2 5175 6963 ....cmcsc10.Quic\n+00004430: 6b2d 5477 90c1 b46f 2d53 756d c628 de61 k-Tw...o-Sum.(.a\n+00004440: 3b91 01d3 3162 c629 8e9f 1adf fe8d 8d8d ;...1b.)........\n+00004450: 9ff2 61ea 8d8d 9200 baac ce31 2e8e 8d92 ..a........1....\n+00004460: 00cf 3114 de73 9603 0aa7 df20 93de 6196 ..1..s..... ..a.\n+00004470: 026e ecdf 0893 de62 8e8e a40d 9999 8d8d .n.....b........\n+00004480: 9200 baac cec6 322e 8e8d 9200 cf31 14de ......2......1..\n+00004490: 6596 030a a7df 2093 de62 9602 6eec df09 e..... ..b..n...\n+000044a0: 93c6 28de 7393 df09 93de 61c6 298e 8ea1 ..(.s.....a.)...\n+000044b0: 8d8d 9200 baac ce33 2e8e 8d92 00cf 3114 .......3......1.\n+000044c0: dd72 6574 7572 6e91 03a6 66c6 28de 733b .return...f.(.s;\n+000044d0: 9101 d331 65c6 298e 8e8e 8e8e 8e8e 9f3e ...1e.)........>\n+000044e0: bffd 8ddd 416c 676f 7269 7468 6d91 03a2 ....Algorithm...\n+000044f0: 6534 2e91 0498 f9c6 5b8d 3131 8e91 0af3 e4......[.11....\n+00004500: 342c 9103 421e 702e 9104 b414 3331 345d 4,..B.p.....314]\n+00004510: 9603 290c 5468 6593 666f 6c6c 6f90 b221 ..).The.follo..!\n+00004520: 7769 6e67 9361 6c67 6f72 6974 686d 9363 wing.algorithm.c\n+00004530: 6f6d 7075 7465 7393 de73 9603 0aa7 c63d omputes..s.....=\n+00004540: 8d93 0d8e 9109 1ffd 28de 6191 0174 38c6 ........(.a..t8.\n+00004550: 2b9b 0174 39de 62c6 2991 0329 0c61 6e64 +..t9.b.)..).and\n+00004560: 9103 290b de65 93c6 3d8d 9365 7272 8e91 ..)..e..=..err..\n+00004570: 107c 5d28 de61 98c6 2b98 de62 c629 2e91 .|](.a..+..b.)..\n+00004580: 04b4 1454 6869 738e a18d 616c 676f 7269 ...This...algori\n+00004590: 7468 6d96 03a6 6675 7365 7393 7468 7265 thm...fuses.thre\n+000045a0: 6593 6d6f 7265 930d 6f61 7469 6e67 9370 e.more..oating.p\n+000045b0: 9a4d de6f 696e 90b2 2174 936f 7098 6572 .M.oin..!t.op.er\n+000045c0: 6174 696f 6e73 9369 6e73 7465 6164 936f ations.instead.o\n+000045d0: 6693 6193 6272 616e 6390 b221 682e 8ea6 f.a.branc..!h...\n+000045e0: 8d8d 8d92 00b4 accd e254 7790 c1b4 6f2d .........Tw...o-\n+000045f0: 5375 6dc6 28de 613b 9101 d331 62c6 298e Sum.(.a;...1b.).\n+00004600: 9f21 acca 8d8d 8d9f eb95 1e8d 8d92 00ba .!..............\n+00004610: acce 312e 8e8d 9200 cf31 14de 7396 030a ..1......1..s...\n+00004620: a7df 2093 de61 9602 6eec df08 93de 628e .. ..a..n.....b.\n+00004630: 8ea4 0d99 998d 8d92 00ba acce c632 2e8e .............2..\n+00004640: 8d92 00cf 3114 de76 9103 6f3a df20 9103 ....1..v..o:. ..\n+00004650: 0aa7 de73 9602 6eec df09 93de 618e 8ea1 ...s..n.....a...\n+00004660: 8d8d 9200 baac cec6 332e 8e8d 9200 cf31 ........3......1\n+00004670: 14de 6596 030a a7df 2093 c628 de61 9602 ..e..... ..(.a..\n+00004680: 6eec df09 93c6 28de 7393 df09 93de 769a n.....(.s.....v.\n+00004690: 6493 c629 2993 df08 93c6 28de 6293 df09 d..)).....(.b...\n+000046a0: 93de 7698 c629 8e8e a18d 8d92 00ba acce ..v..)..........\n+000046b0: 342e 8e8d 9200 cf31 14dd 7265 7475 726e 4......1..return\n+000046c0: 9103 a666 c628 de73 3b91 01d3 3165 c629 ...f.(.s;...1e.)\n+000046d0: 8e8e 8e8e 8e8e 8e9f 4c59 968d dd41 6c67 ........LY...Alg\n+000046e0: 6f72 6974 686d 9104 4ac4 352e 9105 0b89 orithm..J.5.....\n+000046f0: c65b 8d31 318e 910a f334 2c91 03c0 b870 .[.11....4,....p\n+00004700: 2e91 051d 0733 3235 5d96 03bb 7454 6865 .....325]...tThe\n+00004710: 9366 6f6c 6c6f 90b2 2177 696e 6793 616c .follo..!wing.al\n+00004720: 676f 7269 7468 6d93 7370 6c69 7473 9361 gorithm.splits.a\n+00004730: 9335 332d 6269 7493 4945 4545 9103 bb6f .53-bit.IEEE...o\n+00004740: 646f 7562 6c65 9370 7265 6369 7369 6f6e double.precision\n+00004750: 930d 6f61 7469 6e67 8ea1 8d70 9a4d de6f ..oating...p.M.o\n+00004760: 696e 95b2 2174 9104 51b8 6e93 756d 9362 in..!t..Q.n.um.b\n+00004770: 9865 729b 0451 b769 6e93 746f 9604 51b8 .er..Q.in.to..Q.\n+00004780: de61 8d9f 01c8 aece 6869 8e91 0be7 14c6 .a......hi......\n+00004790: 616e 6493 de61 8d9f 01c8 aece 6c6f 8e91 and..a......lo..\n+000047a0: 071c 78c6 2c91 047c 8b65 6163 90b2 2168 ..x.,..|.eac..!h\n+000047b0: 9377 6974 6893 3236 9362 6974 7398 6f66 .with.26.bits.of\n+000047c0: 9373 6967 6e69 0c63 616e 642c 9104 7c8c .signi.cand,..|.\n+000047d0: 7375 6390 b221 6893 7468 6174 98de 6196 suc..!h.that..a.\n+000047e0: 0428 2ec6 3d93 de61 8d9f 01c8 aece 6869 .(..=..a......hi\n+000047f0: 8e91 0a76 7ec6 2b91 02e1 22de 618d 9f01 ...v~.+...\".a...\n+00004800: c8ae ce6c 6f8e 9107 1c78 c62e 9106 dfd2 ...lo....x......\n+00004810: de61 8d9f 01c8 aece 6869 8e91 0be7 14c6 .a......hi......\n+00004820: 7769 6c6c 8ea1 8d63 6f6e 9ab2 2174 6169 will...con..!tai\n+00004830: 6e96 03a6 6674 6865 930c 7273 7493 3236 n...fthe..rst.26\n+00004840: 9362 6974 732c 9377 6869 6c65 93de 618d .bits,.while..a.\n+00004850: 9f01 c8ae ce6c 6f8e 910a c2de c677 696c .....lo......wil\n+00004860: 6c93 636f 6e98 7461 696e 9374 6865 936c l.con.tain.the.l\n+00004870: 6f98 7798 6572 9332 3693 6269 7473 2e8e o.w.er.26.bits..\n+00004880: a68d 8d8d 9200 b4ac cde2 5370 6c69 74c6 ..........Split.\n+00004890: 28de 61c6 298e 9f21 acca 8d8d 8d9f eb95 (.a.)..!........\n+000048a0: 1e8d 8d92 00ba acce 312e 8e8d 9200 cf31 ........1......1\n+000048b0: 14de 7496 030a a7df 2093 c628 328d 9ffc ..t..... ..(2...\n+000048c0: 06be ce32 378e 910b 6ef4 c62b 9602 6eec ...27...n..+..n.\n+000048d0: 3129 93df 0a93 de61 8e8e a40d 9999 8d8d 1).....a........\n+000048e0: 9200 baac cec6 322e 8e8d 9200 cf31 14de ......2......1..\n+000048f0: 618d 9f01 c8ae ce68 698e 910a a003 df20 a......hi...... \n+00004900: 9103 0aa7 de74 9602 6eec df09 93c6 28de .....t..n.....(.\n+00004910: 7493 df09 93de 61c6 298e 8ea1 8d8d 9200 t.....a.).......\n+00004920: baac ce33 2e8e 8d92 00cf 3114 de61 8d9f ...3......1..a..\n+00004930: 01c8 aece 6c6f 8e91 0a27 1fdf 2091 030a ....lo...'.. ...\n+00004940: a7de 6196 026e ecdf 0993 de61 8d9f 01c8 ..a..n.....a....\n+00004950: aece 6869 8e8e 8ea1 8d8d 9200 baac cec6 ..hi............\n+00004960: 342e 8e8d 9200 cf31 14dd 7265 7475 726e 4......1..return\n+00004970: 9103 a666 c628 de61 8d9f 01c8 aece 6869 ...f.(.a......hi\n+00004980: 8e91 0795 5cde 3b91 01d3 3161 8d9f 01c8 ....\\.;...1a....\n+00004990: aece 6c6f 8e91 071c 78c6 298e 8e8e 8e8e ..lo....x.).....\n+000049a0: 8e8e 9f4c 5996 8ddd 416c 676f 7269 7468 ...LY...Algorith\n+000049b0: 6d91 0432 8d36 2e91 0500 00c6 5b8d 3131 m..2.6......[.11\n+000049c0: 8e91 0af3 342c 9603 a666 702e 9104 dddd ....4,...fp.....\n+000049d0: 3332 365d 9354 6865 9366 6f6c 6c6f 90b2 326].The.follo..\n+000049e0: 2177 696e 6793 616c 676f 7269 7468 6d93 !wing.algorithm.\n+000049f0: 636f 6d70 7574 6573 93de 7096 030a a7c6 computes..p.....\n+00004a00: 3d8d 930d 8e91 091f fd28 de61 9602 6eec =........(.a..n.\n+00004a10: df02 93de 62c6 2996 03a6 6661 6e64 93de ....b.)...fand..\n+00004a20: 6596 030a a7c6 3d8d 9365 7272 8e91 107c e.....=..err...|\n+00004a30: 5d28 de61 9602 6eec df02 93de 62c6 292e ](.a..n.....b.).\n+00004a40: 8ea6 8d8d 8d91 6c67 aee2 5477 95c1 b46f ......lg..Tw...o\n+00004a50: 2d50 7293 6f64 c628 de61 3b91 01d3 3162 -Pr.od.(.a;...1b\n+00004a60: c629 8e9f 2879 978d 8d8d 9fe4 c851 8d8d .)..(y.......Q..\n+00004a70: 9172 67af 312e 8e8d 9200 86eb f5de 7096 .rg.1.........p.\n+00004a80: 030a a7df 2093 de61 9602 6eec df0a 93de .... ..a..n.....\n+00004a90: 628e 8ea4 0d99 998d 8d91 7267 afc6 322e b.........rg..2.\n+00004aa0: 8e8d 9200 86eb f528 de61 8d9f 01c8 aece .......(.a......\n+00004ab0: 6869 8e91 0795 5cde 3b91 01d3 3161 8d9f hi....\\.;...1a..\n+00004ac0: 01c8 aece 6c6f 8e91 071c 78c6 2991 030a ....lo....x.)...\n+00004ad0: a7df 2091 03a6 66e2 5370 6c69 74c6 28de .. ...f.Split.(.\n+00004ae0: 61c6 298e 8ea1 8d8d 9172 67af 332e 8e8d a.)......rg.3...\n+00004af0: 9200 86eb f528 de62 8d9f 01c8 aece 6869 .....(.b......hi\n+00004b00: 8e91 0795 5cde 3b91 01d3 3162 8d9f 01c8 ....\\.;...1b....\n+00004b10: aece 6c6f 8e91 071c 78c6 2991 030a a7df ..lo....x.).....\n+00004b20: 2091 03a6 66e2 5370 6c69 74c6 28de 62c6 ...f.Split.(.b.\n+00004b30: 298e 8ea1 8d8d 9172 67af 342e 8e8d 9200 )......rg.4.....\n+00004b40: 86eb f5de 6596 030a a7df 2093 c628 28de ....e..... ..((.\n+00004b50: 618d 9f01 c8ae ce68 698e 960a 0448 df0a a......hi....H..\n+00004b60: 9b02 6eec de62 8d9f 01c8 aece 6869 8e93 ..n..b......hi..\n+00004b70: df09 98de 70c6 2998 df08 98de 618d 9f01 ....p.).....a...\n+00004b80: c8ae ce68 698e 93df 0a98 de62 8d9f 01c8 ...hi......b....\n+00004b90: aece 6c6f 8e96 098b 64df 0898 de61 8d9f ..lo....d....a..\n+00004ba0: 01c8 aece 6c6f 8e93 df0a 98de 628d 9f01 ....lo......b...\n+00004bb0: c8ae ce68 698e 9107 955c c629 98df 0898 ...hi....\\.)....\n+00004bc0: de61 8d9f 01c8 aece 6c6f 8e93 df0a 98de .a......lo......\n+00004bd0: 628d 9f01 c8ae ce6c 6f8e 8e8e a18d 8d91 b......lo.......\n+00004be0: 7267 afc6 352e 8e8d 9200 86eb f5dd 7265 rg..5.........re\n+00004bf0: 7475 726e 9103 a666 c628 de70 3b91 01d3 turn...f.(.p;...\n+00004c00: 3165 c629 8e8e 8e8e 8e8e 8e9f 59f3 2f8d 1e.)........Y./.\n+00004c10: 9111 0000 536f 6d65 9b03 3c0a 6d61 6395 ....Some..<.mac.\n+00004c20: b221 6869 6e65 7391 033c 0b68 6193 7693 .!hines..<.ha.v.\n+00004c30: 6598 6191 033c 0b66 7573 6564 986d 9375 e.a..<.fused.m.u\n+00004c40: 6c74 6970 6c79 2d61 6464 9603 3c0b 696e ltiply-add..<.in\n+00004c50: 7374 7275 6374 696f 6e98 2846 4d41 2991 struction.(FMA).\n+00004c60: 033b f074 6861 7493 6361 6e98 6576 91ff .;.that.can.ev..\n+00004c70: 6444 616c 7561 7465 9365 7870 7265 7373 dDaluate.express\n+00004c80: 696f 6e98 7375 6390 b221 688e a18d 6173 ion.suc..!h...as\n+00004c90: 9b04 3518 de61 9102 ce0e df02 9602 ce0d ..5..a..........\n+00004ca0: de62 93df 0693 de63 9604 3519 c677 6974 .b.....c..5..wit\n+00004cb0: 6898 6193 7369 6e67 6c65 9872 6f75 6e64 h.a.single.round\n+00004cc0: 696e 6798 6572 726f 722e 9106 89f5 5791 ing.error.....W.\n+00004cd0: ff16 6565 9863 616e 9874 616b 9ab2 2165 ..ee.can.tak..!e\n+00004ce0: 9361 6476 91ff 6444 616e 9874 6167 659b .adv..dDan.tage.\n+00004cf0: 0435 186f 6693 7468 6973 9869 6e73 7472 .5.of.this.instr\n+00004d00: 7563 7469 6f6e 9374 6f98 636f 6d70 7574 uction.to.comput\n+00004d10: 658e a18d 6578 6163 7496 02e9 c670 726f e...exact....pro\n+00004d20: 9a4d de64 7563 7493 6f66 9102 e9c7 7495 .M.duct.of....t.\n+00004d30: b221 7793 6f96 02e9 c60d 6f61 7469 6e67 .!w.o.....oating\n+00004d40: 9370 986f 696e 9ab2 2174 936e 9875 6d98 .p.oin..!t.n.um.\n+00004d50: 6290 4dde 6572 7393 6d98 7563 9868 9102 b.M.ers.m.uc.h..\n+00004d60: e9c7 6661 7374 6572 2e91 049e fd54 6865 ..faster.....The\n+00004d70: 7365 936d 6163 9868 696e 6573 9369 6e63 se.mac.hines.inc\n+00004d80: 6c75 6465 9349 424d 9102 e997 5098 6f98 lude.IBM....P.o.\n+00004d90: 7798 6572 9373 6572 6965 738e a18d 2869 w.er.series...(i\n+00004da0: 6e63 6c75 6469 6e67 9603 a666 7468 6593 ncluding...fthe.\n+00004db0: 5095 b221 6f93 7793 6572 5043 292c 9603 P..!o.w.erPC),..\n+00004dc0: a666 6f6e 9377 6869 6390 b221 6893 7468 .fon.whic..!h.th\n+00004dd0: 6973 9373 696d 706c 690c 6361 7469 6f6e is.simpli.cation\n+00004de0: 9369 7393 7465 7374 6564 2e8e 8e9f 1e00 .is.tested......\n+00004df0: 008d 9200 e823 d734 8e8e 8c8b 0000 0005 .....#.4........\n 00004e00: 0000 0000 0000 0000 0000 0000 0000 0000 ................\n 00004e10: 0000 0000 0000 0000 0000 0000 0000 0000 ................\n-00004e20: 0000 0000 400a a002 9ad7 5b8d a0fd 62b4 ....@.....[...b.\n-00004e30: 7ba0 027f 4b85 8da0 fe37 edde 8d9f e645 {...K....7.....E\n-00004e40: 558d 8d8d 8d91 23dc 7def 4050 5366 696c U.....#.}.@PSfil\n-00004e50: 653d 2271 7569 636b 2d74 776f 2d73 756d e=\"quick-two-sum\n-00004e60: 2e65 7073 2220 6c6c 783d 3020 6c6c 793d .eps\" llx=0 lly=\n-00004e70: 3020 7572 783d 3133 3020 7572 793d 3133 0 urx=130 ury=13\n-00004e80: 3020 7277 693d 3133 3030 208e 8e9f 1799 0 rwi=1300 .....\n-00004e90: 9a8d 8d91 219e afc6 4669 6775 7265 9103 ....!...Figure..\n-00004ea0: a666 313a 9104 dddd e251 7569 636b 2d54 .f1:.....Quick-T\n-00004eb0: 7790 c1b4 6f2d 5375 6d8e 8e8e 8d8d 8d92 w...o-Sum.......\n-00004ec0: 00b8 0d20 ef3a 5053 6669 6c65 3d22 7477 ... .:PSfile=\"tw\n-00004ed0: 6f2d 7375 6d2e 6570 7322 206c 6c78 3d30 o-sum.eps\" llx=0\n-00004ee0: 206c 6c79 3d30 2075 7278 3d31 3330 2075 lly=0 urx=130 u\n-00004ef0: 7279 3d31 3330 2072 7769 3d31 3330 3020 ry=130 rwi=1300 \n-00004f00: 8e8e 9f17 999a 8d8d 9200 c83a e4c6 4669 ...........:..Fi\n-00004f10: 6775 7265 9103 a666 323a 9104 dddd e254 gure...f2:.....T\n-00004f20: 7790 c1b4 6f2d 5375 6d8e 8e8e 8d8d 8d92 w...o-Sum.......\n-00004f30: 014c be3d ef3b 5053 6669 6c65 3d22 7477 .L.=.;PSfile=\"tw\n-00004f40: 6f2d 7072 6f64 2e65 7073 2220 6c6c 783d o-prod.eps\" llx=\n-00004f50: 3020 6c6c 793d 3020 7572 783d 3132 3920 0 lly=0 urx=129 \n-00004f60: 7572 793d 3133 3220 7277 693d 3132 3930 ury=132 rwi=1290\n-00004f70: 208e 8e9f 1799 9a8d 8d92 0159 13c8 c646 ..........Y...F\n-00004f80: 6967 7572 6591 03a6 6633 3a91 04dd dde2 igure...f3:.....\n-00004f90: 5477 95c1 b46f 2d50 7293 6f64 8e8e 8e8e Tw...o-Pr.od....\n-00004fa0: 8ea0 00a3 2725 8d9f da45 558d 8d92 0093 ....'%...EU.....\n-00004fb0: 0ca4 ef42 5053 6669 6c65 3d22 6e6f 726d ...BPSfile=\"norm\n-00004fc0: 616c 5f73 756d 5f70 726f 642e 6570 7322 al_sum_prod.eps\"\n-00004fd0: 206c 6c78 3d30 206c 6c79 3d30 2075 7278 llx=0 lly=0 urx\n-00004fe0: 3d31 3735 2075 7279 3d31 3133 2072 7769 =175 ury=113 rwi\n-00004ff0: 3d31 3735 3020 8e8e 9f17 999a 8d8d 915e =1750 .........^\n-00005000: d6cb c646 6967 7572 6596 03a6 6634 3a91 ...Figure...f4:.\n-00005010: 04dd dd4e 6f72 6d61 6c93 4945 4545 9364 ...Normal.IEEE.d\n-00005020: 6f75 626c 6593 7072 6563 6973 696f 6e93 ouble.precision.\n-00005030: 7375 6d93 616e 6493 7072 6f90 4dde 6475 sum.and.pro.M.du\n-00005040: 6374 8e8e 8e9f 1f00 008d dd41 6c67 6f72 ct.........Algor\n-00005050: 6974 686d 9104 cbcb 372e 9105 4904 c654 ithm....7...I..T\n-00005060: 6865 9604 2ba7 666f 6c6c 6f90 b221 7769 he..+.follo..!wi\n-00005070: 6e67 9104 2ba8 616c 676f 7269 7468 6d93 ng..+.algorithm.\n-00005080: 636f 6d70 7574 6573 93de 709b 03e8 bec6 computes..p.....\n-00005090: 3d8d 9103 e8bd 0d8e 9109 fe13 28de 6196 =...........(.a.\n-000050a0: 02c7 c2df 0293 de62 c629 9604 2ba7 616e .......b.)..+.an\n-000050b0: 6493 de65 98c6 3d8d 9865 7272 8e91 115a d..e..=..err...Z\n-000050c0: 7428 de61 9102 c7c1 df02 9102 c7c2 de62 t(.a...........b\n-000050d0: c629 936f 6e91 042b a861 936d 6163 90b2 .).on..+.a.mac..\n-000050e0: 2168 696e 658e a40d 999a 8d77 6974 689b !hine......with.\n-000050f0: 041a f661 9604 1af7 464d 4191 041a d869 ...a....FMA....i\n-00005100: 6e73 7472 7563 7469 6f6e 2e91 063b 8e4e nstruction...;.N\n-00005110: 6f74 6593 7468 6174 9873 6f6d 6598 636f ote.that.some.co\n-00005120: 6d70 696c 6572 7393 656d 6974 9846 4d41 mpilers.emit.FMA\n-00005130: 9104 1ad9 696e 7374 7275 6374 696f 6e73 ....instructions\n-00005140: 9866 6f72 98de 6191 02bc a2df 0296 02bc .for..a.........\n-00005150: a1de 6293 c62b 93de 6391 041a f7c6 6275 ..b..+..c.....bu\n-00005160: 7498 6e6f 748e a18d 666f 729b 03a6 66de t.not...for...f.\n-00005170: 6196 026e ecdf 0293 de62 93df 0093 de63 a..n.....b.....c\n-00005180: c63b 9869 6e98 7468 6973 9863 6173 652c .;.in.this.case,\n-00005190: 9873 6f6d 6598 7369 676e 9861 646a 7573 .some.sign.adjus\n-000051a0: 746d 656e 95b2 2174 7398 6d93 7573 7498 tmen..!ts.m.ust.\n-000051b0: 6290 4dde 6598 6d61 6465 2e8e 9f14 d3d3 b.M.e.made......\n-000051c0: 8d8d 8d92 00b4 accd e254 7795 c1b4 6f2d .........Tw...o-\n-000051d0: 5072 936f 642d 464d 41c6 28de 613b 9101 Pr.od-FMA.(.a;..\n-000051e0: d331 62c6 298e 9f1a dffe 8d8d 8d9f f261 .1b.)..........a\n-000051f0: ea8d 8d92 00ba acce 312e 8e8d 9200 cf31 ........1......1\n-00005200: 14de 7096 030a a7df 2093 de61 9602 6eec ..p..... ..a..n.\n-00005210: df0a 93de 628e 8ea4 0d99 998d 8d92 00ba ....b...........\n-00005220: acce c632 2e8e 8d92 00cf 3114 de65 9603 ...2......1..e..\n-00005230: 0aa7 df20 8d93 c60d 8e91 091f fd28 de61 ... .........(.a\n-00005240: 9602 6eec df02 93de 6293 df00 93de 70c6 ..n.....b.....p.\n-00005250: 298e 8ea1 8d8d 9200 baac ce33 2e8e 8d92 )..........3....\n-00005260: 00cf 3114 dd72 6574 7572 6e91 03a6 66c6 ..1..return...f.\n-00005270: 28de 703b 9101 d331 65c6 298e 8e8e 8e8e (.p;...1e.).....\n-00005280: 8e8e 9f3e 2441 8d91 1100 0054 6865 9605 ...>$A.....The..\n-00005290: 44e4 616c 676f 7269 7468 6d73 9b05 44e5 D.algorithms..D.\n-000052a0: 7072 6573 656e 90b2 2174 6564 9361 7265 presen..!ted.are\n-000052b0: 9374 6865 9862 6173 6963 9362 7569 6c64 .the.basic.build\n-000052c0: 696e 6793 626c 6f90 4dde 6390 b221 6b73 ing.blo.M.c..!ks\n-000052d0: 986f 6693 7175 6164 2d64 6f75 626c 6593 .of.quad-double.\n-000052e0: 6172 6974 686d 6574 6963 2c91 05ac 8461 arithmetic,....a\n-000052f0: 6e64 9361 7265 8ea1 8d72 6570 7265 7365 nd.are...represe\n-00005300: 6e9a b221 7465 6496 0429 4b69 6e93 4669 n..!ted..)Kin.Fi\n-00005310: 6775 7265 7393 312c 9104 4a04 322c 9104 gures.1,..J.2,..\n-00005320: 4a03 616e 6493 332e 9106 668c 5379 6d98 J.and.3...f.Sym.\n-00005330: 629a 4dde 6f6c 7393 666f 7293 6e6f 726d b.M.ols.for.norm\n-00005340: 616c 9364 6f75 626c 6593 7072 6563 6973 al.double.precis\n-00005350: 696f 6e91 0429 4a73 756d 9361 6e64 9370 ion..)Jsum.and.p\n-00005360: 726f 9864 7563 7493 6172 6593 696e 8ea1 ro.duct.are.in..\n-00005370: 8d46 6967 7572 6591 03a6 6634 2e8e 9f22 .Figure...f4...\"\n-00005380: 8041 8d8d dc33 8e91 184c cb42 6173 6963 .A...3...L.Basic\n-00005390: 9105 6666 4f70 9073 3365 7261 7469 6f6e ..ffOp.s3eration\n-000053a0: 738e 9f18 d7f1 8d8d f338 c2d6 4ea0 000c s........8..N...\n-000053b0: 0000 000c 0000 0006 636d 6278 3132 e333 ........cmbx12.3\n-000053c0: 2e31 8e91 1ec0 0052 656e 6f72 6d61 6c69 .1.....Renormali\n-000053d0: 7a61 7469 6f6e 8e9f 14ab ff8d c641 9104 zation.......A..\n-000053e0: a489 7175 6164 2d64 6f75 626c 659b 04a4 ..quad-double...\n-000053f0: cb6e 95b2 2175 6d93 6290 4dde 6572 9604 .n..!um.b.M.er..\n-00005400: a4ca 6973 9361 6e98 756e 6576 91ff 6444 ..is.an.unev..dD\n-00005410: 616c 7561 7465 6493 7375 6d98 6f66 9366 aluated.sum.of.f\n-00005420: 6f75 7298 4945 4545 9104 a489 646f 7562 our.IEEE....doub\n-00005430: 6c65 936e 95b2 2175 6d93 6290 4dde 6572 le.n..!um.b.M.er\n-00005440: 732e 9107 d90b 5468 6591 04a4 ca71 7561 s.....The....qua\n-00005450: 642d 646f 7562 6c65 8ea1 8d6e 95b2 2175 d-double...n..!u\n-00005460: 6d93 6290 4dde 6572 9104 66a8 28de 618d m.b.M.er..f.(.a.\n-00005470: 9f01 a47a ce30 8e96 04c0 04de 3b9b 01d3 ...z.0......;...\n-00005480: 3161 8d9f 01a4 7ace 318e 93de 3b98 618d 1a....z.1...;.a.\n-00005490: 9f01 a47a ce32 8e93 de3b 9861 8d9f 01a4 ...z.2...;.a....\n-000054a0: 7ace 338e 93c6 2996 0466 a872 6570 7265 z.3...)..f.repre\n-000054b0: 7365 6e90 b221 7473 9374 6865 9104 66a9 sen..!ts.the..f.\n-000054c0: 6578 6163 7493 7375 6d93 de61 9604 4b14 exact.sum..a..K.\n-000054d0: c63d 93de 618d 9f01 a47a ce30 8e96 07af .=..a....z.0....\n-000054e0: 1bc6 2b9b 02ef 18de 618d 9f01 a47a ce31 ..+.....a....z.1\n-000054f0: 8e93 c62b 98de 618d 9f01 a47a ce32 8e93 ...+..a....z.2..\n-00005500: c62b 98de 618d 9f01 a47a ce33 8e91 04c0 .+..a....z.3....\n-00005510: 04c6 2e91 071e a34e 6f74 6591 0466 a974 .......Note..f.t\n-00005520: 6861 7496 0466 a866 6f72 9361 6e9a b221 hat..f.for.an..!\n-00005530: 7993 6769 7698 656e 8ea1 8d72 6570 7265 y.giv.en...repre\n-00005540: 7365 6e95 b221 7461 626c 659b 033a c26e sen..!table..:.n\n-00005550: 9375 6d93 6290 4dde 6572 9603 3ac3 de78 .um.b.M.er..:..x\n-00005560: c62c 9103 5049 7468 6572 6593 6361 6e98 .,..PIthere.can.\n-00005570: 6290 4dde 6593 6d61 6e95 b221 7998 7265 b.M.e.man..!y.re\n-00005580: 7072 6573 656e 9374 6174 696f 6e73 9861 presen.tations.a\n-00005590: 7396 033a c361 6e98 756e 6576 91ff 6444 s..:.an.unev..dD\n-000055a0: 616c 7561 7465 6493 7375 6d98 6f66 9366 aluated.sum.of.f\n-000055b0: 6f75 7298 646f 7562 6c65 732e 8e8e 9f1e our.doubles.....\n-000055c0: 0000 8d92 00e8 23d7 358e 8e8c 8b00 0000 ......#.5.......\n-000055d0: 0600 0000 0000 0000 0000 0000 0000 0000 ................\n+00004e20: 0000 0000 0000 400c a002 9ad7 5b8d a0fd ......@.....[...\n+00004e30: 62b4 7ba0 027f 4b85 8da0 fe37 edde 8d9f b.{...K....7....\n+00004e40: e645 558d 8d8d 8d91 23dc 7def 4050 5366 .EU.....#.}.@PSf\n+00004e50: 696c 653d 2271 7569 636b 2d74 776f 2d73 ile=\"quick-two-s\n+00004e60: 756d 2e65 7073 2220 6c6c 783d 3020 6c6c um.eps\" llx=0 ll\n+00004e70: 793d 3020 7572 783d 3133 3020 7572 793d y=0 urx=130 ury=\n+00004e80: 3133 3020 7277 693d 3133 3030 208e 8e9f 130 rwi=1300 ...\n+00004e90: 1799 9a8d 8d91 219e afc6 4669 6775 7265 ......!...Figure\n+00004ea0: 9103 a666 313a 9104 dddd e251 7569 636b ...f1:.....Quick\n+00004eb0: 2d54 7790 c1b4 6f2d 5375 6d8e 8e8e 8d8d -Tw...o-Sum.....\n+00004ec0: 8d92 00b8 0d20 ef3a 5053 6669 6c65 3d22 ..... .:PSfile=\"\n+00004ed0: 7477 6f2d 7375 6d2e 6570 7322 206c 6c78 two-sum.eps\" llx\n+00004ee0: 3d30 206c 6c79 3d30 2075 7278 3d31 3330 =0 lly=0 urx=130\n+00004ef0: 2075 7279 3d31 3330 2072 7769 3d31 3330 ury=130 rwi=130\n+00004f00: 3020 8e8e 9f17 999a 8d8d 9200 c83a e4c6 0 ...........:..\n+00004f10: 4669 6775 7265 9103 a666 323a 9104 dddd Figure...f2:....\n+00004f20: e254 7790 c1b4 6f2d 5375 6d8e 8e8e 8d8d .Tw...o-Sum.....\n+00004f30: 8d92 014c be3d ef3b 5053 6669 6c65 3d22 ...L.=.;PSfile=\"\n+00004f40: 7477 6f2d 7072 6f64 2e65 7073 2220 6c6c two-prod.eps\" ll\n+00004f50: 783d 3020 6c6c 793d 3020 7572 783d 3132 x=0 lly=0 urx=12\n+00004f60: 3920 7572 793d 3133 3220 7277 693d 3132 9 ury=132 rwi=12\n+00004f70: 3930 208e 8e9f 1799 9a8d 8d92 0159 13c8 90 ..........Y..\n+00004f80: c646 6967 7572 6591 03a6 6633 3a91 04dd .Figure...f3:...\n+00004f90: dde2 5477 95c1 b46f 2d50 7293 6f64 8e8e ..Tw...o-Pr.od..\n+00004fa0: 8e8e 8ea0 00a3 2725 8d9f da45 558d 8d92 ......'%...EU...\n+00004fb0: 0093 0ca4 ef42 5053 6669 6c65 3d22 6e6f .....BPSfile=\"no\n+00004fc0: 726d 616c 5f73 756d 5f70 726f 642e 6570 rmal_sum_prod.ep\n+00004fd0: 7322 206c 6c78 3d30 206c 6c79 3d30 2075 s\" llx=0 lly=0 u\n+00004fe0: 7278 3d31 3735 2075 7279 3d31 3133 2072 rx=175 ury=113 r\n+00004ff0: 7769 3d31 3735 3020 8e8e 9f17 999a 8d8d wi=1750 ........\n+00005000: 915e d6cb c646 6967 7572 6596 03a6 6634 .^...Figure...f4\n+00005010: 3a91 04dd dd4e 6f72 6d61 6c93 4945 4545 :....Normal.IEEE\n+00005020: 9364 6f75 626c 6593 7072 6563 6973 696f .double.precisio\n+00005030: 6e93 7375 6d93 616e 6493 7072 6f90 4dde n.sum.and.pro.M.\n+00005040: 6475 6374 8e8e 8e9f 1f00 008d dd41 6c67 duct.........Alg\n+00005050: 6f72 6974 686d 9104 cbcb 372e 9105 4904 orithm....7...I.\n+00005060: c654 6865 9604 2ba7 666f 6c6c 6f90 b221 .The..+.follo..!\n+00005070: 7769 6e67 9104 2ba8 616c 676f 7269 7468 wing..+.algorith\n+00005080: 6d93 636f 6d70 7574 6573 93de 709b 03e8 m.computes..p...\n+00005090: bec6 3d8d 9103 e8bd 0d8e 9109 fe13 28de ..=...........(.\n+000050a0: 6196 02c7 c2df 0293 de62 c629 9604 2ba7 a........b.)..+.\n+000050b0: 616e 6493 de65 98c6 3d8d 9865 7272 8e91 and..e..=..err..\n+000050c0: 115a 7428 de61 9102 c7c1 df02 9102 c7c2 .Zt(.a..........\n+000050d0: de62 c629 936f 6e91 042b a861 936d 6163 .b.).on..+.a.mac\n+000050e0: 90b2 2168 696e 658e a40d 999a 8d77 6974 ..!hine......wit\n+000050f0: 689b 041a f661 9604 1af7 464d 4191 041a h....a....FMA...\n+00005100: d869 6e73 7472 7563 7469 6f6e 2e91 063b .instruction...;\n+00005110: 8e4e 6f74 6593 7468 6174 9873 6f6d 6598 .Note.that.some.\n+00005120: 636f 6d70 696c 6572 7393 656d 6974 9846 compilers.emit.F\n+00005130: 4d41 9104 1ad9 696e 7374 7275 6374 696f MA....instructio\n+00005140: 6e73 9866 6f72 98de 6191 02bc a2df 0296 ns.for..a.......\n+00005150: 02bc a1de 6293 c62b 93de 6391 041a f7c6 ....b..+..c.....\n+00005160: 6275 7498 6e6f 748e a18d 666f 729b 03a6 but.not...for...\n+00005170: 66de 6196 026e ecdf 0293 de62 93df 0093 f.a..n.....b....\n+00005180: de63 c63b 9869 6e98 7468 6973 9863 6173 .c.;.in.this.cas\n+00005190: 652c 9873 6f6d 6598 7369 676e 9861 646a e,.some.sign.adj\n+000051a0: 7573 746d 656e 95b2 2174 7398 6d93 7573 ustmen..!ts.m.us\n+000051b0: 7498 6290 4dde 6598 6d61 6465 2e8e 9f14 t.b.M.e.made....\n+000051c0: d3d3 8d8d 8d92 00b4 accd e254 7795 c1b4 ...........Tw...\n+000051d0: 6f2d 5072 936f 642d 464d 41c6 28de 613b o-Pr.od-FMA.(.a;\n+000051e0: 9101 d331 62c6 298e 9f1a dffe 8d8d 8d9f ...1b.).........\n+000051f0: f261 ea8d 8d92 00ba acce 312e 8e8d 9200 .a........1.....\n+00005200: cf31 14de 7096 030a a7df 2093 de61 9602 .1..p..... ..a..\n+00005210: 6eec df0a 93de 628e 8ea4 0d99 998d 8d92 n.....b.........\n+00005220: 00ba acce c632 2e8e 8d92 00cf 3114 de65 .....2......1..e\n+00005230: 9603 0aa7 df20 8d93 c60d 8e91 091f fd28 ..... .........(\n+00005240: de61 9602 6eec df02 93de 6293 df00 93de .a..n.....b.....\n+00005250: 70c6 298e 8ea1 8d8d 9200 baac ce33 2e8e p.)..........3..\n+00005260: 8d92 00cf 3114 dd72 6574 7572 6e91 03a6 ....1..return...\n+00005270: 66c6 28de 703b 9101 d331 65c6 298e 8e8e f.(.p;...1e.)...\n+00005280: 8e8e 8e8e 9f3e 2441 8d91 1100 0054 6865 .....>$A.....The\n+00005290: 9605 44e4 616c 676f 7269 7468 6d73 9b05 ..D.algorithms..\n+000052a0: 44e5 7072 6573 656e 90b2 2174 6564 9361 D.presen..!ted.a\n+000052b0: 7265 9374 6865 9862 6173 6963 9362 7569 re.the.basic.bui\n+000052c0: 6c64 696e 6793 626c 6f90 4dde 6390 b221 lding.blo.M.c..!\n+000052d0: 6b73 986f 6693 7175 6164 2d64 6f75 626c ks.of.quad-doubl\n+000052e0: 6593 6172 6974 686d 6574 6963 2c91 05ac e.arithmetic,...\n+000052f0: 8461 6e64 9361 7265 8ea1 8d72 6570 7265 .and.are...repre\n+00005300: 7365 6e9a b221 7465 6496 0429 4b69 6e93 sen..!ted..)Kin.\n+00005310: 4669 6775 7265 7393 312c 9104 4a04 322c Figures.1,..J.2,\n+00005320: 9104 4a03 616e 6493 332e 9106 668c 5379 ..J.and.3...f.Sy\n+00005330: 6d98 629a 4dde 6f6c 7393 666f 7293 6e6f m.b.M.ols.for.no\n+00005340: 726d 616c 9364 6f75 626c 6593 7072 6563 rmal.double.prec\n+00005350: 6973 696f 6e91 0429 4a73 756d 9361 6e64 ision..)Jsum.and\n+00005360: 9370 726f 9864 7563 7493 6172 6593 696e .pro.duct.are.in\n+00005370: 8ea1 8d46 6967 7572 6591 03a6 6634 2e8e ...Figure...f4..\n+00005380: 9f22 8041 8d8d dc33 8e91 184c cb42 6173 .\".A...3...L.Bas\n+00005390: 6963 9105 6666 4f70 9073 3365 7261 7469 ic..ffOp.s3erati\n+000053a0: 6f6e 738e 9f18 d7f1 8d8d f338 c2d6 4ea0 ons........8..N.\n+000053b0: 000c 0000 000c 0000 0006 636d 6278 3132 ..........cmbx12\n+000053c0: e333 2e31 8e91 1ec0 0052 656e 6f72 6d61 .3.1.....Renorma\n+000053d0: 6c69 7a61 7469 6f6e 8e9f 14ab ff8d c641 lization.......A\n+000053e0: 9104 a489 7175 6164 2d64 6f75 626c 659b ....quad-double.\n+000053f0: 04a4 cb6e 95b2 2175 6d93 6290 4dde 6572 ...n..!um.b.M.er\n+00005400: 9604 a4ca 6973 9361 6e98 756e 6576 91ff ....is.an.unev..\n+00005410: 6444 616c 7561 7465 6493 7375 6d98 6f66 dDaluated.sum.of\n+00005420: 9366 6f75 7298 4945 4545 9104 a489 646f .four.IEEE....do\n+00005430: 7562 6c65 936e 95b2 2175 6d93 6290 4dde uble.n..!um.b.M.\n+00005440: 6572 732e 9107 d90b 5468 6591 04a4 ca71 ers.....The....q\n+00005450: 7561 642d 646f 7562 6c65 8ea1 8d6e 95b2 uad-double...n..\n+00005460: 2175 6d93 6290 4dde 6572 9104 66a8 28de !um.b.M.er..f.(.\n+00005470: 618d 9f01 a47a ce30 8e96 04c0 04de 3b9b a....z.0......;.\n+00005480: 01d3 3161 8d9f 01a4 7ace 318e 93de 3b98 ..1a....z.1...;.\n+00005490: 618d 9f01 a47a ce32 8e93 de3b 9861 8d9f a....z.2...;.a..\n+000054a0: 01a4 7ace 338e 93c6 2996 0466 a872 6570 ..z.3...)..f.rep\n+000054b0: 7265 7365 6e90 b221 7473 9374 6865 9104 resen..!ts.the..\n+000054c0: 66a9 6578 6163 7493 7375 6d93 de61 9604 f.exact.sum..a..\n+000054d0: 4b14 c63d 93de 618d 9f01 a47a ce30 8e96 K..=..a....z.0..\n+000054e0: 07af 1bc6 2b9b 02ef 18de 618d 9f01 a47a ....+.....a....z\n+000054f0: ce31 8e93 c62b 98de 618d 9f01 a47a ce32 .1...+..a....z.2\n+00005500: 8e93 c62b 98de 618d 9f01 a47a ce33 8e91 ...+..a....z.3..\n+00005510: 04c0 04c6 2e91 071e a34e 6f74 6591 0466 .........Note..f\n+00005520: a974 6861 7496 0466 a866 6f72 9361 6e9a .that..f.for.an.\n+00005530: b221 7993 6769 7698 656e 8ea1 8d72 6570 .!y.giv.en...rep\n+00005540: 7265 7365 6e95 b221 7461 626c 659b 033a resen..!table..:\n+00005550: c26e 9375 6d93 6290 4dde 6572 9603 3ac3 .n.um.b.M.er..:.\n+00005560: de78 c62c 9103 5049 7468 6572 6593 6361 .x.,..PIthere.ca\n+00005570: 6e98 6290 4dde 6593 6d61 6e95 b221 7998 n.b.M.e.man..!y.\n+00005580: 7265 7072 6573 656e 9374 6174 696f 6e73 represen.tations\n+00005590: 9861 7396 033a c361 6e98 756e 6576 91ff .as..:.an.unev..\n+000055a0: 6444 616c 7561 7465 6493 7375 6d98 6f66 dDaluated.sum.of\n+000055b0: 9366 6f75 7298 646f 7562 6c65 732e 8e8e .four.doubles...\n+000055c0: 9f1e 0000 8d92 00e8 23d7 358e 8e8c 8b00 ........#.5.....\n+000055d0: 0000 0600 0000 0000 0000 0000 0000 0000 ................\n 000055e0: 0000 0000 0000 0000 0000 0000 0000 0000 ................\n-000055f0: 0000 0000 0000 004d f9a0 029a d75b 8da0 .......M.....[..\n-00005600: fd62 b47b a002 7f4b 858d a0fd a4b4 7b8d .b.{...K......{.\n-00005610: c648 656e 6365 9603 a666 7790 b221 6593 .Hence...fw..!e.\n-00005620: 7265 7175 6972 6593 7468 6174 9374 6865 require.that.the\n-00005630: 9371 7561 6472 7570 6c65 9328 de61 8d9f .quadruple.(.a..\n-00005640: 01a4 7ace 308e 9604 c004 de3b 9b01 d331 ..z.0......;...1\n-00005650: 618d 9f01 a47a ce31 8e93 de3b 9861 8d9f a....z.1...;.a..\n-00005660: 01a4 7ace 328e 93de 3b98 618d 9f01 a47a ..z.2...;.a....z\n-00005670: ce33 8e93 c629 9603 a666 746f 9373 6174 .3...)...fto.sat\n-00005680: 6973 6679 8e9f 1d33 af8d 9200 c21f 8adf isfy...3........\n-00005690: 6ade 618d 9f01 a47a f325 d701 1732 0008 j.a....z.%...2..\n-000056a0: 0000 0008 0000 0005 636d 6d69 38d0 69ce ........cmmi8.i.\n-000056b0: 2b31 8e91 0e41 56df 6a91 030a a714 8d8d +1...AV.j.......\n-000056c0: 8d9f f897 9e8d 9104 3dda c631 8e91 043d ........=..1...=\n-000056d0: da9f 04de c889 0000 6665 0005 799a 9f0a ........fe..y...\n-000056e0: 0c74 8d32 8e8e 8e8e 8d91 0aea a775 6c70 .t.2.........ulp\n-000056f0: 8e91 1a1f fe28 de61 8d9f 01a4 7ad0 698e .....(.a....z.i.\n-00005700: 9103 64da c629 8e9f 1bb9 418d 666f 729b ..d..)....A.for.\n-00005710: 036a c5de 6996 030a a7c6 3d93 30de 3b96 .j..i.....=.0.;.\n-00005720: 01d3 31c6 31de 3b93 c632 2c96 0376 b277 ..1.1.;..2,..v.w\n-00005730: 6974 6898 6571 7561 6c69 7490 b221 7998 ith.equalit..!y.\n-00005740: 6f90 4dde 6363 7572 7269 6e67 9103 6ac4 o.M.ccurring..j.\n-00005750: 6f6e 6c79 9869 6698 de61 8d9f 01a4 7ad0 only.if..a....z.\n-00005760: 698e 9106 6f81 c63d 9103 0aa7 3098 6f72 i...o..=....0.or\n-00005770: 9874 6865 986c 6173 7498 6269 7498 6f66 .the.last.bit.of\n-00005780: 98de 618d 9f01 a47a d069 8e91 06cf 9ec6 ..a....z.i......\n-00005790: 6973 9830 9828 7468 6174 9869 732c 9372 is.0.(that.is,.r\n-000057a0: 6f75 6e64 2d74 6f2d 6576 90b2 2165 6e8e ound-to-ev..!en.\n-000057b0: a40d 999a 8d69 7396 0452 be75 7365 649b .....is..R.used.\n-000057c0: 0452 bd69 6e93 6361 7365 936f 6698 7469 .R.in.case.of.ti\n-000057d0: 6573 292e 9106 e2e4 4e6f 7465 9374 6861 es).....Note.tha\n-000057e0: 7498 7468 6593 0c72 7374 9364 6f75 626c t.the..rst.doubl\n-000057f0: 6598 de61 8d9f 01a4 7ace 308e 9109 12c2 e..a....z.0.....\n-00005800: c669 7398 6193 646f 7562 6c65 2d70 7265 .is.a.double-pre\n-00005810: 6369 7369 6f6e 9361 7070 726f 90b2 2178 cision.appro..!x\n-00005820: 696d 6174 696f 6e98 746f 9374 6865 8ea1 imation.to.the..\n-00005830: 8d71 7561 642d 646f 7562 6c65 9b03 a666 .quad-double...f\n-00005840: 6e95 b221 756d 9362 904d de65 7298 de61 n..!um.b.M.er..a\n-00005850: c62c 9861 6363 7572 6174 6598 746f 9861 .,.accurate.to.a\n-00005860: 6c6d 6f73 7498 6861 6c66 9861 6e98 756c lmost.half.an.ul\n-00005870: 702e 8ea9 1699 9a8d dd4c 656d 6d61 9106 p........Lemma..\n-00005880: 64bc 382e 9106 0bde e046 91ff 2917 6f72 d.8......F..).or\n-00005890: 9605 acaa 616e 7993 7175 6164 2d64 6f75 ....any.quad-dou\n-000058a0: 626c 6591 05ac a96e 756d 6291 ff70 b965 ble....numb..p.e\n-000058b0: 7293 de61 9106 3968 c63d 9106 3967 28de r..a..9h.=..9g(.\n-000058c0: 618d 9f01 a47a ce30 8e96 04c0 04de 3b9b a....z.0......;.\n-000058d0: 01d3 3161 8d9f 01a4 7ace 318e 93de 3b98 ..1a....z.1...;.\n-000058e0: 618d 9f01 a47a ce32 8e93 de3b 9861 8d9f a....z.2...;.a..\n-000058f0: 01a4 7ace 338e 93c6 29e0 2c91 061d 1974 ..z.3...).,....t\n-00005900: 6865 9b05 acaa 6e6f 726d 616c 697a 6596 he....normalize.\n-00005910: ff70 b964 9105 aca9 7293 6570 7293 6573 .p.d....r.epr.es\n-00005920: 656e 7461 7469 6f6e 9869 738e a18d 756e entation.is...un\n-00005930: 6971 7565 2e8e a68d 9111 0000 c64d 6f73 ique.........Mos\n-00005940: 7496 048a d96f 669b 048a da74 6865 9361 t....of....the.a\n-00005950: 6c67 6f72 6974 686d 7393 6465 7363 7269 lgorithms.descri\n-00005960: 6290 4dde 6564 9868 6572 6593 7072 6f90 b.M.ed.here.pro.\n-00005970: 4dde 6475 6365 9361 6e98 6578 7061 6e73 M.duce.an.expans\n-00005980: 696f 6e93 7468 6174 9369 7398 6e6f 7493 ion.that.is.not.\n-00005990: 6f66 9363 616e 6f6e 6963 616c 9866 6f72 of.canonical.for\n-000059a0: 6d93 7b8e a18d 6f66 7465 6e96 0318 a968 m.{...often....h\n-000059b0: 619a b221 7669 6e67 936f 9876 9865 726c a..!ving.o.v.erl\n-000059c0: 6170 7069 6e67 9362 6974 732e 9104 ae9e apping.bits.....\n-000059d0: 5468 6572 6566 6f72 652c 9103 3502 6193 Therefore,..5.a.\n-000059e0: 0c76 9865 2d74 6572 6d93 6578 7061 6e73 .v.e-term.expans\n-000059f0: 696f 6e91 0318 a869 7393 7072 6f90 4dde ion....is.pro.M.\n-00005a00: 6475 6365 642c 9103 3502 616e 6493 7468 duced,..5.and.th\n-00005a10: 656e 9372 656e 6f72 6d61 6c69 7a65 648e en.renormalized.\n-00005a20: a18d 746f 9603 a666 666f 7572 9363 6f6d ..to...ffour.com\n-00005a30: 7090 4dde 6f6e 656e 90b2 2174 732e 8ea6 p.M.onen..!ts...\n-00005a40: 8ddd 416c 676f 7269 7468 6d91 03be b439 ..Algorithm....9\n-00005a50: 2e91 04ad 34c6 5468 6973 9603 41a9 7265 ....4.This..A.re\n-00005a60: 6e6f 726d 616c 697a 6174 696f 6e93 7072 normalization.pr\n-00005a70: 6f90 4dde 6365 6475 7265 9369 739b 0341 o.M.cedure.is..A\n-00005a80: aa61 9376 91ff 6444 6172 6961 6e90 b221 .a.v..dDarian..!\n-00005a90: 7493 6f66 9350 7269 6573 7427 7398 7265 t.of.Priest's.re\n-00005aa0: 6e6f 726d 616c 697a 6174 696f 6e93 6d65 normalization.me\n-00005ab0: 7468 6f90 4dde 6493 5b8d 3130 8e91 0af3 tho.M.d.[.10....\n-00005ac0: 342c 8ea1 8d70 2e91 0565 7031 3136 5d2e 4,...p...ep116].\n-00005ad0: 9105 6571 5468 6596 03d3 9769 6e70 7574 ..eqThe....input\n-00005ae0: 9369 7393 6193 0c76 9ab2 2165 2d74 6572 .is.a..v..!e-ter\n-00005af0: 6d91 03d3 9865 7870 616e 7369 6f6e 9377 m....expansion.w\n-00005b00: 6974 6893 6c69 6d69 7465 6493 6f98 7698 ith.limited.o.v.\n-00005b10: 6572 6c61 7070 696e 6793 6269 7473 2c91 erlapping.bits,.\n-00005b20: 03de e377 6974 6891 03d3 98de 618d 9f01 ...with.....a...\n-00005b30: a47a ce30 8e91 0893 9bc6 6290 4dde 6569 .z.0......b.M.ei\n-00005b40: 6e67 9374 6865 936d 6f73 748e a18d 7369 ng.the.most...si\n-00005b50: 676e 690c 6361 6e95 b221 7491 03a6 6663 gni.can..!t...fc\n-00005b60: 6f6d 7090 4dde 6f6e 656e 9374 2e8e 9f14 omp.M.onen.t....\n-00005b70: d3d3 8d8d 8d91 6c67 aee2 5265 6e6f 726d ......lg..Renorm\n-00005b80: 616c 697a 65c6 28de 618d 9f01 a47a ce30 alize.(.a....z.0\n-00005b90: 8e96 04c0 04de 3b9b 01d3 3161 8d9f 01a4 ......;...1a....\n-00005ba0: 7ace 318e 93de 3b98 618d 9f01 a47a ce32 z.1...;.a....z.2\n-00005bb0: 8e93 de3b 9861 8d9f 01a4 7ace 338e 93de ...;.a....z.3...\n-00005bc0: 3b98 618d 9f01 a47a ce34 8e93 c629 8e9f ;.a....z.4...)..\n-00005bd0: 7346 608d 8d8d 9f99 fb88 8d8d 9177 e149 sF`..........w.I\n-00005be0: 312e 8e8d 9200 8c65 8f28 de73 3b9b 01d3 1......e.(.s;...\n-00005bf0: 3174 8d9f 01a4 7ace 348e 9604 c004 c629 1t....z.4......)\n-00005c00: 9103 0aa7 df20 9103 a666 e251 7569 636b ..... ...f.Quick\n-00005c10: 2d54 7790 c1b4 6f2d 5375 6dc6 28de 618d -Tw...o-Sum.(.a.\n-00005c20: 9f01 a47a ce33 8e93 de3b 9861 8d9f 01a4 ...z.3...;.a....\n-00005c30: 7ace 348e 93c6 298e 8ea4 0d99 998d 8d91 z.4...).........\n-00005c40: 77e1 4932 2e8e 8d92 008c 658f 28de 733b w.I2......e.(.s;\n-00005c50: 9b01 d331 748d 9f01 a47a ce33 8e96 04c0 ...1t....z.3....\n-00005c60: 04c6 2991 030a a7df 2091 03a6 66e2 5175 ..)..... ...f.Qu\n-00005c70: 6963 6b2d 5477 90c1 b46f 2d53 756d c628 ick-Tw...o-Sum.(\n-00005c80: de61 8d9f 01a4 7ace 328e 93de 3b98 73c6 .a....z.2...;.s.\n-00005c90: 298e 8ea1 8d8d 9177 e149 332e 8e8d 9200 )......w.I3.....\n-00005ca0: 8c65 8f28 de73 3b9b 01d3 3174 8d9f 01a4 .e.(.s;...1t....\n-00005cb0: 7ace 328e 9604 c004 c629 9103 0aa7 df20 z.2......)..... \n-00005cc0: 9103 a666 e251 7569 636b 2d54 7790 c1b4 ...f.Quick-Tw...\n-00005cd0: 6f2d 5375 6dc6 28de 618d 9f01 a47a ce31 o-Sum.(.a....z.1\n-00005ce0: 8e93 de3b 9873 c629 8e8e a18d 8d91 77e1 ...;.s.)......w.\n-00005cf0: 4934 2e8e 8d92 008c 658f 28de 748d 9f01 I4......e.(.t...\n-00005d00: a47a ce30 8e96 04c0 04de 3b9b 01d3 3174 .z.0......;...1t\n-00005d10: 8d9f 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... ...f.Quick-T\n-00005e00: 7790 c1b4 6f2d 5375 6dc6 28de 733b 9374 w...o-Sum.(.s;.t\n-00005e10: 8d9f 01a4 7ad0 698e 9103 64da c629 8e8e ....z.i...d..)..\n-00005e20: a18d 8d91 77e1 4939 2e8e 8d92 0097 58c4 ....w.I9......X.\n-00005e30: dd69 6691 03a6 66de 6596 030a a7df 36c6 .if...f.e.....6.\n-00005e40: 3d93 308e 8ea1 8d8d 9172 67af 3130 2e8e =.0......rg.10..\n-00005e50: 8d92 00a2 4bf9 de62 8d9f 01c8 aed0 6b8e ....K..b......k.\n-00005e60: 9108 2e39 df20 9103 0aa7 de73 8e8e a18d ...9. .....s....\n-00005e70: 8d91 7267 afc6 3131 2e8e 8d92 00a2 4bf9 ..rg..11......K.\n-00005e80: de73 9603 0aa7 df20 93de 658e 8ea1 8d8d .s..... ..e.....\n-00005e90: 9172 67af c631 322e 8e8d 9200 a24b f9de .rg..12......K..\n-00005ea0: 6b91 0362 e6df 2091 030a a7de 6b91 02c7 k..b.. .....k...\n-00005eb0: 2bc6 2b91 026e ec31 8e8e a18d 8d91 7267 +.+..n.1......rg\n-00005ec0: af31 332e 8e8d 9200 9758 c4dd 656e 6491 .13......X..end.\n-00005ed0: 0432 8d69 668e 8ea1 8d8d 9172 67af c631 .2.if......rg..1\n-00005ee0: 342e 8e8d 9200 8c65 8fdd 656e 6491 0432 4......e..end..2\n-00005ef0: 8d66 6f72 8e8e a18d 8d91 7267 afc6 3135 .for......rg..15\n-00005f00: 2e8e 8d92 008c 658f dd72 6574 7572 6e91 ......e..return.\n-00005f10: 03a6 66c6 28de 628d 9f01 a47a ce30 8e96 ..f.(.b....z.0..\n-00005f20: 04c0 04de 3b9b 01d3 3162 8d9f 01a4 7ace ....;...1b....z.\n-00005f30: 318e 93de 3b98 628d 9f01 a47a ce32 8e93 1...;.b....z.2..\n-00005f40: de3b 9862 8d9f 01a4 7ace 338e 93c6 298e .;.b....z.3...).\n-00005f50: 8e8e 8e8e 8e8e a000 eef1 068d 9111 0000 ................\n-00005f60: 4e65 6365 7373 6172 7996 0488 1563 6f6e Necessary....con\n-00005f70: 6469 7469 6f6e 7393 666f 729b 0488 1474 ditions.for....t\n-00005f80: 6869 7393 7265 6e6f 726d 616c 697a 6174 his.renormalizat\n-00005f90: 696f 6e93 616c 676f 7269 7468 6d93 746f ion.algorithm.to\n-00005fa0: 9877 90b2 216f 726b 9363 6f72 7265 6374 .w..!ork.correct\n-00005fb0: 6c79 9361 7265 2c91 04c0 8075 6e66 6f72 ly.are,....unfor\n-00005fc0: 7475 6e61 7465 6c79 91ff 1665 2c8e a18d tunately...e,...\n-00005fd0: 6e6f 7496 0529 906b 6e6f 9ab2 2177 6e2e not..).kno..!wn.\n-00005fe0: 9109 675b 5072 6965 7374 9370 726f 9876 ..g[Priest.pro.v\n-00005ff0: 9865 7393 7468 6174 9369 6693 7468 6593 .es.that.if.the.\n-00006000: 696e 7075 7493 6578 7061 6e73 696f 6e93 input.expansion.\n-00006010: 646f 904d de65 7393 6e6f 7493 6f98 7698 do.M.es.not.o.v.\n-00006020: 6572 6c61 7093 6298 7993 6d6f 7265 9374 erlap.b.y.more.t\n-00006030: 6861 6e93 3531 9362 6974 732c 8ea1 8d74 han.51.bits,...t\n-00006040: 6865 6e96 047f 0674 6865 9361 6c67 6f72 hen....the.algor\n-00006050: 6974 686d 9377 9ab2 216f 726b 7393 636f ithm.w..!orks.co\n-00006060: 7272 6563 746c 7991 ff16 652e 9107 67be rrectly...e...g.\n-00006070: 486f 9877 9865 7698 6572 2c91 04b5 2d74 Ho.w.ev.er,...-t\n-00006080: 6869 7391 047f 0763 6f6e 6469 7469 6f6e his....condition\n-00006090: 9369 7393 6298 7993 6e6f 936d 6561 6e73 .is.b.y.no.means\n-000060a0: 936e 6563 6573 7361 7279 3b91 04eb 5674 .necessary;...Vt\n-000060b0: 6861 7493 7468 658e a18d 7265 6e6f 726d hat.the...renorm\n-000060c0: 616c 697a 6174 696f 6e96 03e6 a061 6c67 alization....alg\n-000060d0: 6f72 6974 686d 9b03 e6a1 2841 6c67 6f72 orithm....(Algor\n-000060e0: 6974 686d 9339 2993 7790 b221 6f72 6b73 ithm.9).w..!orks\n-000060f0: 986f 6e93 616c 6c93 7468 6598 6578 7061 .on.all.the.expa\n-00006100: 6e73 696f 6e73 9370 726f 904d de64 7563 nsions.pro.M.duc\n-00006110: 6564 9362 90b2 2179 9874 6865 9361 6c67 ed.b..!y.the.alg\n-00006120: 6f72 6974 686d 738e a18d 629a 4dde 656c orithms...b.M.el\n-00006130: 6f90 b221 7796 03a6 6672 656d 6169 6e73 o..!w...fremains\n-00006140: 9374 6f93 6298 6593 7368 6f90 b221 776e .to.b.e.sho..!wn\n-00006150: 2e8e 9f07 6666 8900 0066 6600 bbe6 2d9f ....ff...ff...-.\n-00006160: 0a4c cd8d 8d8d 8d91 0c7b ba9f fc2d 3df3 .L.......{...-=.\n-00006170: 24b9 4161 a800 0600 0000 0600 0000 0463 $.Aa...........c\n-00006180: 6d72 36cf 318e 8e8e 9110 a661 d549 6e96 mr6.1......a.In.\n-00006190: 0315 5474 6865 9369 6d70 6c65 6d65 6e9a ..Tthe.implemen.\n-000061a0: be39 7461 7469 6f6e 2c93 7468 6973 936c .9tation,.this.l\n-000061b0: 6f90 41c7 6f70 9369 7393 756e 726f 6c6c o.A.op.is.unroll\n-000061c0: 6564 9374 6f93 7365 7698 6572 616c 93d9 ed.to.sev.eral..\n-000061d0: 6966 93d5 7374 6174 656d 656e 9874 732e if..statemen.ts.\n-000061e0: 8e8e 9f1e 0000 8d92 00e8 23d7 c636 8e8e ..........#..6..\n-000061f0: 8c8b 0000 0007 0000 0000 0000 0000 0000 ................\n+000055f0: 0000 0000 0000 0000 004d fba0 029a d75b .........M.....[\n+00005600: 8da0 fd62 b47b a002 7f4b 858d a0fd a4b4 ...b.{...K......\n+00005610: 7b8d c648 656e 6365 9603 a666 7790 b221 {..Hence...fw..!\n+00005620: 6593 7265 7175 6972 6593 7468 6174 9374 e.require.that.t\n+00005630: 6865 9371 7561 6472 7570 6c65 9328 de61 he.quadruple.(.a\n+00005640: 8d9f 01a4 7ace 308e 9604 c004 de3b 9b01 ....z.0......;..\n+00005650: d331 618d 9f01 a47a ce31 8e93 de3b 9861 .1a....z.1...;.a\n+00005660: 8d9f 01a4 7ace 328e 93de 3b98 618d 9f01 ....z.2...;.a...\n+00005670: a47a ce33 8e93 c629 9603 a666 746f 9373 .z.3...)...fto.s\n+00005680: 6174 6973 6679 8e9f 1d33 af8d 9200 c21f atisfy...3......\n+00005690: 8adf 6ade 618d 9f01 a47a f325 d701 1732 ..j.a....z.%...2\n+000056a0: 0008 0000 0008 0000 0005 636d 6d69 38d0 ..........cmmi8.\n+000056b0: 69ce 2b31 8e91 0e41 56df 6a91 030a a714 i.+1...AV.j.....\n+000056c0: 8d8d 8d9f f897 9e8d 9104 3dda c631 8e91 ..........=..1..\n+000056d0: 043d da9f 04de c889 0000 6665 0005 799a .=........fe..y.\n+000056e0: 9f0a 0c74 8d32 8e8e 8e8e 8d91 0aea a775 ...t.2.........u\n+000056f0: 6c70 8e91 1a1f fe28 de61 8d9f 01a4 7ad0 lp.....(.a....z.\n+00005700: 698e 9103 64da c629 8e9f 1bb9 418d 666f i...d..)....A.fo\n+00005710: 729b 036a c5de 6996 030a a7c6 3d93 30de r..j..i.....=.0.\n+00005720: 3b96 01d3 31c6 31de 3b93 c632 2c96 0376 ;...1.1.;..2,..v\n+00005730: b277 6974 6898 6571 7561 6c69 7490 b221 .with.equalit..!\n+00005740: 7998 6f90 4dde 6363 7572 7269 6e67 9103 y.o.M.ccurring..\n+00005750: 6ac4 6f6e 6c79 9869 6698 de61 8d9f 01a4 j.only.if..a....\n+00005760: 7ad0 698e 9106 6f81 c63d 9103 0aa7 3098 z.i...o..=....0.\n+00005770: 6f72 9874 6865 986c 6173 7498 6269 7498 or.the.last.bit.\n+00005780: 6f66 98de 618d 9f01 a47a d069 8e91 06cf of..a....z.i....\n+00005790: 9ec6 6973 9830 9828 7468 6174 9869 732c ..is.0.(that.is,\n+000057a0: 9372 6f75 6e64 2d74 6f2d 6576 90b2 2165 .round-to-ev..!e\n+000057b0: 6e8e a40d 999a 8d69 7396 0452 be75 7365 n......is..R.use\n+000057c0: 649b 0452 bd69 6e93 6361 7365 936f 6698 d..R.in.case.of.\n+000057d0: 7469 6573 292e 9106 e2e4 4e6f 7465 9374 ties).....Note.t\n+000057e0: 6861 7498 7468 6593 0c72 7374 9364 6f75 hat.the..rst.dou\n+000057f0: 626c 6598 de61 8d9f 01a4 7ace 308e 9109 ble..a....z.0...\n+00005800: 12c2 c669 7398 6193 646f 7562 6c65 2d70 ...is.a.double-p\n+00005810: 7265 6369 7369 6f6e 9361 7070 726f 90b2 recision.appro..\n+00005820: 2178 696d 6174 696f 6e98 746f 9374 6865 !ximation.to.the\n+00005830: 8ea1 8d71 7561 642d 646f 7562 6c65 9b03 ...quad-double..\n+00005840: a666 6e95 b221 756d 9362 904d de65 7298 .fn..!um.b.M.er.\n+00005850: de61 c62c 9861 6363 7572 6174 6598 746f .a.,.accurate.to\n+00005860: 9861 6c6d 6f73 7498 6861 6c66 9861 6e98 .almost.half.an.\n+00005870: 756c 702e 8ea9 1699 9a8d dd4c 656d 6d61 ulp........Lemma\n+00005880: 9106 64bc 382e 9106 0bde e046 91ff 2917 ..d.8......F..).\n+00005890: 6f72 9605 acaa 616e 7993 7175 6164 2d64 or....any.quad-d\n+000058a0: 6f75 626c 6591 05ac a96e 756d 6291 ff70 ouble....numb..p\n+000058b0: b965 7293 de61 9106 3968 c63d 9106 3967 .er..a..9h.=..9g\n+000058c0: 28de 618d 9f01 a47a ce30 8e96 04c0 04de (.a....z.0......\n+000058d0: 3b9b 01d3 3161 8d9f 01a4 7ace 318e 93de ;...1a....z.1...\n+000058e0: 3b98 618d 9f01 a47a ce32 8e93 de3b 9861 ;.a....z.2...;.a\n+000058f0: 8d9f 01a4 7ace 338e 93c6 29e0 2c91 061d ....z.3...).,...\n+00005900: 1974 6865 9b05 acaa 6e6f 726d 616c 697a .the....normaliz\n+00005910: 6596 ff70 b964 9105 aca9 7293 6570 7293 e..p.d....r.epr.\n+00005920: 6573 656e 7461 7469 6f6e 9869 738e a18d esentation.is...\n+00005930: 756e 6971 7565 2e8e a68d 9111 0000 c64d unique.........M\n+00005940: 6f73 7496 048a d96f 669b 048a da74 6865 ost....of....the\n+00005950: 9361 6c67 6f72 6974 686d 7393 6465 7363 .algorithms.desc\n+00005960: 7269 6290 4dde 6564 9868 6572 6593 7072 rib.M.ed.here.pr\n+00005970: 6f90 4dde 6475 6365 9361 6e98 6578 7061 o.M.duce.an.expa\n+00005980: 6e73 696f 6e93 7468 6174 9369 7398 6e6f nsion.that.is.no\n+00005990: 7493 6f66 9363 616e 6f6e 6963 616c 9866 t.of.canonical.f\n+000059a0: 6f72 6d93 7b8e a18d 6f66 7465 6e96 0318 orm.{...often...\n+000059b0: a968 619a b221 7669 6e67 936f 9876 9865 .ha..!ving.o.v.e\n+000059c0: 726c 6170 7069 6e67 9362 6974 732e 9104 rlapping.bits...\n+000059d0: ae9e 5468 6572 6566 6f72 652c 9103 3502 ..Therefore,..5.\n+000059e0: 6193 0c76 9865 2d74 6572 6d93 6578 7061 a..v.e-term.expa\n+000059f0: 6e73 696f 6e91 0318 a869 7393 7072 6f90 nsion....is.pro.\n+00005a00: 4dde 6475 6365 642c 9103 3502 616e 6493 M.duced,..5.and.\n+00005a10: 7468 656e 9372 656e 6f72 6d61 6c69 7a65 then.renormalize\n+00005a20: 648e a18d 746f 9603 a666 666f 7572 9363 d...to...ffour.c\n+00005a30: 6f6d 7090 4dde 6f6e 656e 90b2 2174 732e omp.M.onen..!ts.\n+00005a40: 8ea6 8ddd 416c 676f 7269 7468 6d91 03be ....Algorithm...\n+00005a50: b439 2e91 04ad 34c6 5468 6973 9603 41a9 .9....4.This..A.\n+00005a60: 7265 6e6f 726d 616c 697a 6174 696f 6e93 renormalization.\n+00005a70: 7072 6f90 4dde 6365 6475 7265 9369 739b pro.M.cedure.is.\n+00005a80: 0341 aa61 9376 91ff 6444 6172 6961 6e90 .A.a.v..dDarian.\n+00005a90: b221 7493 6f66 9350 7269 6573 7427 7398 .!t.of.Priest's.\n+00005aa0: 7265 6e6f 726d 616c 697a 6174 696f 6e93 renormalization.\n+00005ab0: 6d65 7468 6f90 4dde 6493 5b8d 3130 8e91 metho.M.d.[.10..\n+00005ac0: 0af3 342c 8ea1 8d70 2e91 0565 7031 3136 ..4,...p...ep116\n+00005ad0: 5d2e 9105 6571 5468 6596 03d3 9769 6e70 ]...eqThe....inp\n+00005ae0: 7574 9369 7393 6193 0c76 9ab2 2165 2d74 ut.is.a..v..!e-t\n+00005af0: 6572 6d91 03d3 9865 7870 616e 7369 6f6e erm....expansion\n+00005b00: 9377 6974 6893 6c69 6d69 7465 6493 6f98 .with.limited.o.\n+00005b10: 7698 6572 6c61 7070 696e 6793 6269 7473 v.erlapping.bits\n+00005b20: 2c91 03de e377 6974 6891 03d3 98de 618d ,....with.....a.\n+00005b30: 9f01 a47a ce30 8e91 0893 9bc6 6290 4dde ...z.0......b.M.\n+00005b40: 6569 6e67 9374 6865 936d 6f73 748e a18d eing.the.most...\n+00005b50: 7369 676e 690c 6361 6e95 b221 7491 03a6 signi.can..!t...\n+00005b60: 6663 6f6d 7090 4dde 6f6e 656e 9374 2e8e fcomp.M.onen.t..\n+00005b70: 9f14 d3d3 8d8d 8d91 6c67 aee2 5265 6e6f ........lg..Reno\n+00005b80: 726d 616c 697a 65c6 28de 618d 9f01 a47a rmalize.(.a....z\n+00005b90: ce30 8e96 04c0 04de 3b9b 01d3 3161 8d9f .0......;...1a..\n+00005ba0: 01a4 7ace 318e 93de 3b98 618d 9f01 a47a ..z.1...;.a....z\n+00005bb0: ce32 8e93 de3b 9861 8d9f 01a4 7ace 338e .2...;.a....z.3.\n+00005bc0: 93de 3b98 618d 9f01 a47a ce34 8e93 c629 ..;.a....z.4...)\n+00005bd0: 8e9f 7346 608d 8d8d 9f99 fb88 8d8d 9177 ..sF`..........w\n+00005be0: e149 312e 8e8d 9200 8c65 8f28 de73 3b9b .I1......e.(.s;.\n+00005bf0: 01d3 3174 8d9f 01a4 7ace 348e 9604 c004 ..1t....z.4.....\n+00005c00: c629 9103 0aa7 df20 9103 a666 e251 7569 .)..... ...f.Qui\n+00005c10: 636b 2d54 7790 c1b4 6f2d 5375 6dc6 28de ck-Tw...o-Sum.(.\n+00005c20: 618d 9f01 a47a ce33 8e93 de3b 9861 8d9f a....z.3...;.a..\n+00005c30: 01a4 7ace 348e 93c6 298e 8ea4 0d99 998d ..z.4...).......\n+00005c40: 8d91 77e1 4932 2e8e 8d92 008c 658f 28de ..w.I2......e.(.\n+00005c50: 733b 9b01 d331 748d 9f01 a47a ce33 8e96 s;...1t....z.3..\n+00005c60: 04c0 04c6 2991 030a a7df 2091 03a6 66e2 ....)..... ...f.\n+00005c70: 5175 6963 6b2d 5477 90c1 b46f 2d53 756d Quick-Tw...o-Sum\n+00005c80: c628 de61 8d9f 01a4 7ace 328e 93de 3b98 .(.a....z.2...;.\n+00005c90: 73c6 298e 8ea1 8d8d 9177 e149 332e 8e8d s.)......w.I3...\n+00005ca0: 9200 8c65 8f28 de73 3b9b 01d3 3174 8d9f ...e.(.s;...1t..\n+00005cb0: 01a4 7ace 328e 9604 c004 c629 9103 0aa7 ..z.2......)....\n+00005cc0: df20 9103 a666 e251 7569 636b 2d54 7790 . ...f.Quick-Tw.\n+00005cd0: c1b4 6f2d 5375 6dc6 28de 618d 9f01 a47a ..o-Sum.(.a....z\n+00005ce0: ce31 8e93 de3b 9873 c629 8e8e a18d 8d91 .1...;.s.)......\n+00005cf0: 77e1 4934 2e8e 8d92 008c 658f 28de 748d w.I4......e.(.t.\n+00005d00: 9f01 a47a ce30 8e96 04c0 04de 3b9b 01d3 ...z.0......;...\n+00005d10: 3174 8d9f 01a4 7ace 318e 93c6 2991 030a 1t....z.1...)...\n+00005d20: a7df 2091 03a6 66e2 5175 6963 6b2d 5477 .. ...f.Quick-Tw\n+00005d30: 90c1 b46f 2d53 756d c628 de61 8d9f 01a4 ...o-Sum.(.a....\n+00005d40: 7ace 308e 93de 3b98 73c6 298e 8ea1 a18d z.0...;.s.).....\n+00005d50: 8d91 77e1 4935 2e8e 8d92 008c 658f de73 ..w.I5......e..s\n+00005d60: 9603 0aa7 df20 93de 748d 9f01 a47a ce30 ..... ..t....z.0\n+00005d70: 8e8e 8ea1 8d8d 9177 e149 c636 2e8e 8d92 .......w.I.6....\n+00005d80: 008c 658f de6b 9103 62e6 df20 9103 0aa7 ..e..k..b.. ....\n+00005d90: c630 8e8e a18d 8d91 77e1 4937 2e8e 8d92 .0......w.I7....\n+00005da0: 008c 658f dd66 6f72 8d8d 9ffc 06be ce31 ..e..for.......1\n+00005db0: 8e8e 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1v.d.;.w.Kn;.r..\n-00006a60: 01a4 7ace 308e 9b04 c004 de3b 9372 8d9f ..z.0......;.r..\n-00006a70: 01a4 7ace 318e 98de 3b9b 03e8 c1e0 616e ..z.1...;.....an\n-00006a80: 6496 03e8 c0de 728d 9f01 a47a ce32 8e91 d.....r....z.2..\n-00006a90: 08a8 c5e0 6291 ff70 b965 9361 7398 696e ....b..p.e.as.in\n-00006aa0: 6469 6396 ff70 b961 7465 9364 9103 e8c0 dic..p.ate.d....\n-00006ab0: 696e 8ea1 8d46 6967 7572 96ff 70b9 659b in...Figur..p.e.\n-00006ac0: 03ea ea37 2e91 0509 764c 9365 7498 de4d ...7....vL.et..M\n-00006ad0: 9104 3c46 c63d 8d91 030a a76d 6178 8e91 ..PSfile=\"qd_ad\n+00006250: 645f 7164 5f64 2e65 7073 2220 6c6c 783d d_qd_d.eps\" llx=\n+00006260: 3020 6c6c 793d 3020 7572 783d 3330 3320 0 lly=0 urx=303 \n+00006270: 7572 793d 3138 3820 7277 693d 3330 3330 ury=188 rwi=3030\n+00006280: 208e 8e9f 1799 9a8d 8d92 009a 2e97 c646 ..............F\n+00006290: 6967 7572 6596 03a6 6635 3a91 04dd dd51 igure...f5:....Q\n+000062a0: 7561 642d 446f 7562 6c65 932b 9344 6f75 uad-Double.+.Dou\n+000062b0: 626c 658e 8e8e 9f1d d23a 8d8d e333 2e32 ble......:...3.2\n+000062c0: 8e91 1ec0 0041 6464 6974 696f 6e8e a414 .....Addition...\n+000062d0: abff 8d8d dd33 2e32 2e31 8e91 267a 1151 .....3.2.1..&z.Q\n+000062e0: 7561 642d 446f 7562 6c65 9604 328d 2b93 uad-Double..2.+.\n+000062f0: 446f 7562 6c65 8ea1 8dc6 5468 6596 053d Double....The..=\n+00006300: 8661 6464 6974 696f 6e93 6f66 9361 9364 .addition.of.a.d\n+00006310: 6f75 626c 6593 7072 6563 6973 696f 6e93 ouble.precision.\n+00006320: 6e95 b221 756d 9362 9a4d de65 7296 053d n..!um.b.M.er..=\n+00006330: 8674 6f93 6193 7175 6164 2d64 6f75 626c .to.a.quad-doubl\n+00006340: 6593 6e95 b221 756d 9362 9865 7296 053d e.n..!um.b.er..=\n+00006350: 8669 7393 7369 6d69 6c61 7293 746f 9353 .is.similar.to.S\n+00006360: 6865 7795 b221 6393 6893 756b 2773 8ea4 hew..!c.h.uk's..\n+00006370: 0d99 9a8d e247 7295 c1b4 6f93 772d 4578 .....Gr...o.w-Ex\n+00006380: 7091 ff45 1e61 6e73 696f 6e9b 04f1 50c6 p..E.ansion...P.\n+00006390: 5b8d 3131 8e91 0af3 342c 9605 440b 702e [.11....4,..D.p.\n+000063a0: 9108 be9b 3331 365d 2c93 6275 7498 7468 ....316],.but.th\n+000063b0: 6598 646f 7562 6c65 9870 7265 6369 7369 e.double.precisi\n+000063c0: 6f6e 9104 f151 6e95 b221 756d 9362 904d on...Qn..!um.b.M\n+000063d0: de65 7298 de62 98c6 6973 9861 6464 6564 .er..b..is.added\n+000063e0: 9104 f151 746f 9861 9871 7561 642d 646f ...Qto.a.quad-do\n+000063f0: 7562 6c65 8ea1 8d6e 95b2 2175 6d93 629a uble...n..!um.b.\n+00006400: 4dde 6572 9603 3cc5 de61 9103 3cc4 c666 M.er..<..a..<..f\n+00006410: 726f 6d93 6d6f 7374 9373 6967 6e69 0c63 rom.most.signi.c\n+00006420: 616e 90b2 2174 9363 6f6d 7098 6f6e 656e an..!t.comp.onen\n+00006430: 90b2 2174 9b03 3cc4 0c72 7374 9328 7261 ..!t..<..rst.(ra\n+00006440: 7468 6572 9374 6861 6e98 6672 6f6d 936c ther.than.from.l\n+00006450: 6561 7374 9373 6967 6e69 0c63 616e 90b2 east.signi.can..\n+00006460: 2174 292e 9104 baa7 5468 6973 9370 726f !t).....This.pro\n+00006470: 904d de64 7563 6573 8ea1 8d61 9603 a666 .M.duces...a...f\n+00006480: 0c76 9ab2 2165 2d74 6572 6d93 6578 7061 .v..!e-term.expa\n+00006490: 6e73 696f 6e93 7768 6963 9868 9369 7393 nsion.whic.h.is.\n+000064a0: 7468 6593 6578 6163 7493 7265 7375 6c74 the.exact.result\n+000064b0: 2c93 7768 6963 9868 9369 7393 7468 656e ,.whic.h.is.then\n+000064c0: 9372 656e 6f72 6d61 6c69 7a65 642e 9104 .renormalized...\n+000064d0: dddd 5365 6593 4669 6775 7265 9335 2e8e ..See.Figure.5..\n+000064e0: a18d 9111 0000 5369 6e63 6596 028e 7874 ......Since...xt\n+000064f0: 6865 9b02 8e77 6578 6163 7493 7265 7375 he...wexact.resu\n+00006500: 6c74 9869 7393 636f 6d70 7574 6564 2c91 lt.is.computed,.\n+00006510: 02c6 7474 6865 6e98 6e6f 726d 616c 697a ..tthen.normaliz\n+00006520: 6564 9374 6f93 666f 7572 9863 6f6d 7090 ed.to.four.comp.\n+00006530: 4dde 6f6e 656e 90b2 2174 732c 9102 c674 M.onen..!ts,...t\n+00006540: 7468 6973 9361 6464 6974 696f 6e98 6973 this.addition.is\n+00006550: 9361 6363 7572 6174 658e a18d 746f 9603 .accurate...to..\n+00006560: a666 6174 936c 6561 7374 9374 6865 930c .fat.least.the..\n+00006570: 7273 7493 3231 3293 6269 7473 936f 6693 rst.212.bits.of.\n+00006580: 7468 6593 7265 7375 6c74 2e8e 9f1c a4fc the.result......\n+00006590: 8d8d dd33 2e32 2e32 8e91 267a 1151 7561 ...3.2.2..&z.Qua\n+000065a0: 642d 446f 7562 6c65 9604 328d 2b93 5175 d-Double..2.+.Qu\n+000065b0: 6164 2d44 6f75 626c 658e 9f14 abff 8dc6 ad-Double.......\n+000065c0: 5791 ff16 6565 9b04 622d 6861 95b2 2176 W...ee..b-ha..!v\n+000065d0: 9365 9869 6d70 6c65 6d65 6e93 7465 6498 .e.implemen.ted.\n+000065e0: 7493 7793 6f96 0462 2c61 6c67 6f72 6974 t.w.o..b,algorit\n+000065f0: 686d 7398 666f 7298 6164 6469 7469 6f6e hms.for.addition\n+00006600: 2e91 0711 3254 6865 980c 7273 7493 6f6e ....2The..rst.on\n+00006610: 6598 6973 9866 6173 7465 722c 9104 911e e.is.faster,....\n+00006620: 6275 7498 6f6e 6c79 9873 6174 6973 0c65 but.only.satis.e\n+00006630: 7398 7468 658e a18d 7795 b221 6561 6b93 s.the...w..!eak.\n+00006640: 6572 9b03 a38c 2843 7261 9379 2d73 7493 er....(Cra.y-st.\n+00006650: 796c 6529 9603 a38d 6572 726f 7298 6290 yle)....error.b.\n+00006660: 4dde 6f75 6e64 93de 6191 0269 38df 089b M.ound..a..i8...\n+00006670: 0269 39de 6296 030a a7c6 3d93 2831 982b .i9.b.....=.(1.+\n+00006680: 98de 0e8d 9f01 a47a ce31 8e96 04c0 04c6 .......z.1......\n+00006690: 29de 6198 c62b 9102 6938 2831 982b 98de ).a..+..i8(1.+..\n+000066a0: 0e8d 9f01 a47a ce32 8e93 c629 de62 9603 .....z.2...).b..\n+000066b0: a38c c677 6865 7265 9b03 a38d 7468 6593 ...where....the.\n+000066c0: 6d61 676e 6974 7564 6593 6f66 98de 0e8d magnitude.of....\n+000066d0: 9f01 a47a ce31 8e96 0863 90c6 616e 6498 ...z.1...c..and.\n+000066e0: de0e 8d9f 01a4 7ace 328e 93c6 6973 8ea1 ......z.2...is..\n+000066f0: 8d62 904d de6f 756e 6465 6496 03a6 6662 .b.M.ounded...fb\n+00006700: 90b2 2179 93de 228d 9f01 c8ae ce71 648e ..!y..\"......qd.\n+00006710: 910c c005 c63d 9103 0aa7 328d 9ffc 06be .....=....2.....\n+00006720: d200 ce32 3131 8e91 13dc 84c6 2e8e a18d ...211..........\n+00006730: 9111 0000 4669 6775 7265 9602 964d 3691 ....Figure...M6.\n+00006740: 0296 4e62 9a4d de65 7374 9364 6573 6372 ..Nb.M.est.descr\n+00006750: 6962 9865 7393 7468 659b 0296 4e0c 7273 ib.es.the...N.rs\n+00006760: 7493 6164 6469 7469 6f6e 9361 6c67 6f72 t.addition.algor\n+00006770: 6974 686d 986f 6693 7495 b221 7793 6f98 ithm.of.t..!w.o.\n+00006780: 7175 6164 2d64 6f75 626c 6591 0296 4d6e quad-double...Mn\n+00006790: 9375 6d93 6290 4dde 6572 732e 9104 832a .um.b.M.ers....*\n+000067a0: 496e 9874 6865 9102 964d 6469 6167 7261 In.the...Mdiagra\n+000067b0: 6d2c 8ea1 8d74 6865 7265 9603 e9bf 6172 m,...there....ar\n+000067c0: 659b 03e9 c074 6872 6565 936c 6172 6765 e....three.large\n+000067d0: 9862 904d de6f 90b2 2178 6573 9377 6974 .b.M.o..!xes.wit\n+000067e0: 6893 7468 7265 6598 696e 7075 7473 9374 h.three.inputs.t\n+000067f0: 6f98 7468 656d 2e91 05a7 e954 6865 7365 o.them.....These\n+00006800: 9361 7265 9876 91ff 6444 6172 696f 7573 .are.v..dDarious\n+00006810: 93e2 5468 7265 652d 5375 6d93 c662 904d ..Three-Sum..b.M\n+00006820: de6f 90b2 2178 6573 2c91 03fa 9661 6e64 .o..!xes,....and\n+00006830: 8ea1 8d74 6865 6972 9603 a666 696e 9ab2 ...their...fin..\n+00006840: 2174 6572 6e61 6c73 9361 7265 9373 686f !ternals.are.sho\n+00006850: 9877 6e93 696e 9346 6967 7572 6593 372e .wn.in.Figure.7.\n+00006860: 8ea1 8d91 1100 004e 6f90 b221 7796 03a6 .......No..!w...\n+00006870: 6666 6f72 9361 9366 6577 936d 6f72 6593 ffor.a.few.more.\n+00006880: 6c65 6d6d 6173 2e8e 9f15 2063 8ddd 4c65 lemmas.... c..Le\n+00006890: 6d6d 6191 04f4 a431 302e 9105 5c7b e04c mma....10...\\{.L\n+000068a0: 9bff 70b9 6574 9604 8630 de61 9104 862f ..p.et...0.a.../\n+000068b0: e061 6e64 93de 6293 e062 9865 9374 776f .and..b..b.e.two\n+000068c0: 9104 862f 646f 7562 6c65 9370 7298 6598 .../double.pr.e.\n+000068d0: 6369 7369 6f6e 930d 6f98 6174 696e 6793 cision..o.ating.\n+000068e0: 7098 6f69 6e74 9104 862f 6e75 6d62 9865 p.oint.../numb.e\n+000068f0: 7273 2e91 06db 484c 9865 7493 de4d 9105 rs....HL.et..M..\n+00006900: 558f c63d 8d91 0423 f16d 6178 8e91 1885 U..=...#.max....\n+00006910: 0528 df6a de61 df6a de3b 9101 d331 df6a .(.j.a.j.;...1.j\n+00006920: de62 df6a c629 e02e 8ea1 8d54 6865 6e9b .b.j.).....Then.\n+00006930: 03ea eadf 6a8d c60d 8e91 0615 5628 de61 ....j.......V(.a\n+00006940: 9602 6eec c62b 93de 62c6 29df 6a96 030a ..n..+..b.).j...\n+00006950: a714 93c6 32de 4d91 0131 9fe0 2c98 616e ....2.M..1..,.an\n+00006960: 6498 6396 ff70 b96f 6e73 6593 7175 656e d.c..p.onse.quen\n+00006970: 746c 792c 98df 6a8d c665 7272 8e91 0d71 tly,..j..err...q\n+00006980: b628 de61 9602 6eec c62b 93de 62c6 29df .(.a..n..+..b.).\n+00006990: 6a96 030a a714 8d8d 8d9f fbb0 4b8d 9104 j...........K...\n+000069a0: 3dda ce31 8e91 043d da9f 01c6 1b89 0000 =..1...=........\n+000069b0: 6665 0004 4004 9f06 5041 8d32 8e8e 8e8e fe..@...PA.2....\n+000069c0: 8d91 09b1 11c6 756c 708e 9118 e668 2832 ......ulp....h(2\n+000069d0: de4d 9b01 319f c629 93df 1493 c632 de22 .M..1..).....2.\"\n+000069e0: 4d98 e02e 8ea9 1520 628d dd4c 656d 6d61 M...... b..Lemma\n+000069f0: 9104 2fd9 3131 2e91 04fe 11e0 4c9b ff70 ../.11......L..p\n+00006a00: b965 7496 03e8 c1de 783b 9101 d331 7990 .et.....x;...1y.\n+00006a10: 6493 3b91 03e8 c0e0 616e 6493 de7a 9104 d.;.....and..z..\n+00006a20: 6409 e062 9865 9369 6e70 7574 7391 03e8 d..b.e.inputs...\n+00006a30: c074 6f93 e254 6872 6565 2d53 756d e02e .to..Three-Sum..\n+00006a40: 9105 08bd 4c98 6574 9103 e8c0 de75 3b96 ....L.et.....u;.\n+00006a50: 01d3 3176 9064 933b 9377 904b 6e3b 9372 ..1v.d.;.w.Kn;.r\n+00006a60: 8d9f 01a4 7ace 308e 9b04 c004 de3b 9372 ....z.0......;.r\n+00006a70: 8d9f 01a4 7ace 318e 98de 3b9b 03e8 c1e0 ....z.1...;.....\n+00006a80: 616e 6496 03e8 c0de 728d 9f01 a47a ce32 and.....r....z.2\n+00006a90: 8e91 08a8 c5e0 6291 ff70 b965 9361 7398 ......b..p.e.as.\n+00006aa0: 696e 6469 6396 ff70 b961 7465 9364 9103 indic..p.ate.d..\n+00006ab0: e8c0 696e 8ea1 8d46 6967 7572 96ff 70b9 ..in...Figur..p.\n+00006ac0: 659b 03ea ea37 2e91 0509 764c 9365 7498 e....7....vL.et.\n+00006ad0: de4d 9104 3c46 c63d 8d91 030a a76d 6178 .M..PSfi\n-00006ec0: 6c65 3d22 7468 7265 652d 7375 6d2d 322e le=\"three-sum-2.\n-00006ed0: 6570 7322 206c 6c78 3d30 206c 6c79 3d30 eps\" llx=0 lly=0\n-00006ee0: 2075 7278 3d31 3434 2075 7279 3d31 3530 urx=144 ury=150\n-00006ef0: 2072 7769 3d31 3434 3020 8e8d 9201 61f5 rwi=1440 ....a.\n-00006f00: 70ef 3e50 5366 696c 653d 2274 6872 6565 p.>PSfile=\"three\n-00006f10: 2d73 756d 2d33 2e65 7073 2220 6c6c 783d -sum-3.eps\" llx=\n-00006f20: 3020 6c6c 793d 3020 7572 783d 3130 3120 0 lly=0 urx=101 \n-00006f30: 7572 793d 3137 3020 7277 693d 3130 3130 ury=170 rwi=1010\n-00006f40: 208e 8e9f 1799 9a8d 8d92 00b2 d2a3 c646 ..............F\n-00006f50: 6967 7572 6591 03a6 6637 3a91 04dd dde2 igure...f7:.....\n-00006f60: 5468 7265 652d 5375 6dc6 738e 8e8e 9f1d Three-Sum.s.....\n-00006f70: 039c 8d91 1100 004e 6f74 6596 0392 a374 .......Note....t\n-00006f80: 6861 749b 0392 a274 6865 9374 95b2 2177 hat....the.t..!w\n-00006f90: 936f 986f 7468 6572 9603 92a3 e254 6872 .o.other.....Thr\n-00006fa0: 6565 2d53 756d c673 9373 686f 90b2 2177 ee-Sum.s.sho..!w\n-00006fb0: 6e98 6172 6593 7369 6d70 6c69 0c63 6174 n.are.simpli.cat\n-00006fc0: 696f 6e93 6f66 9874 6865 930c 7273 7498 ion.of.the..rst.\n-00006fd0: e254 6872 6565 2d53 756d c62c 9103 9697 .Three-Sum.,....\n-00006fe0: 7768 6572 658e a40d 999a 8d69 7496 03a6 where......it...\n-00006ff0: 666f 6e6c 7993 636f 6d70 7574 6573 936f fonly.computes.o\n-00007000: 6e65 936f 7293 7495 b221 7793 6f9b 03a6 ne.or.t..!w.o...\n-00007010: 6663 6f6d 7090 4dde 6f6e 656e 9374 732c fcomp.M.onen.ts,\n-00007020: 9869 6e73 7465 6164 986f 6698 7468 7265 .instead.of.thre\n-00007030: 653b 9874 6893 7573 9874 6865 9873 616d e;.th.us.the.sam\n-00007040: 6598 6290 4dde 6f75 6e64 7398 6170 706c e.b.M.ounds.appl\n-00007050: 7991 ff16 652e 8ea1 8d91 1100 0054 6865 y...e........The\n-00007060: 9b02 eed1 6162 904d de6f 95b2 2176 9365 ....ab.M.o..!v.e\n-00007070: 9862 904d de6f 756e 6498 6973 986e 6f74 .b.M.ound.is.not\n-00007080: 9102 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9f01 a47a ce32 8e93 de3b 9862 8d9f 01a4 ...z.2...;.b....\n-00008190: 7ace 338e 93c6 292e 9104 760b 4261 7369 z.3...)...v.Basi\n-000081a0: 6361 6c6c 7996 026e ef69 7493 6d65 7267 cally..n.it.merg\n-000081b0: 652d 736f 7274 7393 7468 6593 6569 6768 e-sorts.the.eigh\n-000081c0: 90b2 2174 9364 6f75 626c 6573 2c91 02ad ..!t.doubles,...\n-000081d0: 3a61 6e64 9370 904d de65 7266 6f72 6d73 :and.p.M.erforms\n-000081e0: 93e2 446f 7562 6c65 2d41 90ae 3d63 6375 ..Double-A..=ccu\n-000081f0: 6d75 6c61 91ff 451e 7465 8ea1 8dc6 756e mula..E.te....un\n-00008200: 9ab2 2174 696c 9603 a666 666f 7572 9363 ..!til...ffour.c\n-00008210: 6f6d 7090 4dde 6f6e 656e 9874 7393 6172 omp.M.onen.ts.ar\n-00008220: 6593 6f62 7461 696e 6564 2e8e 9f14 d3d3 e.obtained......\n-00008230: 8d8d 8d91 6c67 aee2 5144 2d41 6464 2d41 ....lg..QD-Add-A\n-00008240: 90ae 3d63 6375 7261 91ff 451e 7465 c628 ..=ccura..E.te.(\n-00008250: de61 3b91 01d3 3162 c629 8e9f 7346 608d .a;...1b.)..sF`.\n-00008260: 8d8d 9f99 fb88 8d8d 9177 e149 312e 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788d E.te.(.u;.v.;.x.\n-00008440: 9f01 a47a d069 8e91 0364 dac6 298e 8ea1 ...z.i...d..)...\n-00008450: 8d8d 9177 e149 382e 8e8d 9200 9758 c4dd ...w.I8......X..\n-00008460: 6966 9103 a666 de73 9603 0aa7 df36 c63d if...f.s.....6.=\n-00008470: 9330 8e8e a18d 8d91 77e1 4939 2e8e 8d92 .0......w.I9....\n-00008480: 00a2 4bf9 de63 8d9f 01c8 aed0 6b8e 9108 ..K..c......k...\n-00008490: 2e39 df20 9103 0aa7 de73 8e8e a18d 8d91 .9. .....s......\n-000084a0: 7267 afc6 3130 2e8e 8d92 00a2 4bf9 de6b rg..10......K..k\n-000084b0: 9103 62e6 df20 9103 0aa7 de6b 9102 c72b ..b.. .....k...+\n-000084c0: c62b 9102 6eec 318e 8ea1 8d8d 9172 67af .+..n.1......rg.\n-000084d0: 3131 2e8e 8d92 0097 58c4 dd65 6e64 9104 11......X..end..\n-000084e0: 328d 6966 8e8e a18d 8d91 7267 afc6 3132 2.if......rg..12\n-000084f0: 2e8e 8d92 0097 58c4 de69 9603 0aa7 df20 ......X..i..... \n-00008500: 93de 6996 026e ecc6 2b93 318e 8ea1 8d8d ..i..n..+.1.....\n-00008510: 9172 67af 3133 2e8e 8d92 008c 658f dd65 .rg.13......e..e\n-00008520: 6e64 9104 328d 7768 696c 658e 8ea1 8d8d nd..2.while.....\n-00008530: 9172 67af c631 342e 8e8d 9200 8c65 8fdd .rg..14......e..\n-00008540: 6966 9603 a666 de6b 9103 62e6 3c9b 030a if...f.k..b.<...\n-00008550: a7c6 3293 dd74 6865 6e93 de63 8d9f 01c8 ..2..then..c....\n-00008560: aed0 6b90 3685 ce2b 318e 9113 0ab5 df20 ..k.6..+1...... \n-00008570: 98de 768e 8ea1 8d8d 9172 67af c631 352e ..v......rg..15.\n-00008580: 8e8d 9200 8c65 8fdd 6966 9603 a666 de6b .....e..if...f.k\n-00008590: 9103 62e6 3c9b 030a a7c6 3393 dd74 6865 ..b.<.....3..the\n-000085a0: 6e93 de63 8d9f 01c8 aed0 6b8e 9108 2e39 n..c......k....9\n-000085b0: df20 98de 758e 8ea1 8d8d 9172 67af c631 . ..u......rg..1\n-000085c0: 362e 8e8d 9200 8c65 8fdd 7265 7475 726e 6......e..return\n-000085d0: 9103 a666 e252 656e 6f72 6d61 6c69 7a65 ...f.Renormalize\n-000085e0: c628 de63 8d9f 01a4 7ace 308e 9604 c004 .(.c....z.0.....\n-000085f0: de3b 9b01 d331 638d 9f01 a47a ce31 8e93 .;...1c....z.1..\n-00008600: de3b 9863 8d9f 01a4 7ace 328e 93de 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a40d 999a 8d91 1100 0054 6865 9604 8339 .........The...9\n+00007b10: 616c 676f 7269 7468 6d9b 0483 3a69 7393 algorithm...:is.\n+00007b20: 7369 6d69 6c61 7293 746f 9853 6865 7795 similar.to.Shew.\n+00007b30: b221 6393 6893 756b 2773 9604 8339 e246 .!c.h.uk's...9.F\n+00007b40: 91fe b8f6 6173 742d 4578 7091 ff45 1e61 ....ast-Exp..E.a\n+00007b50: 6e73 696f 6e2d 5375 6d98 c65b 8d31 318e nsion-Sum..[.11.\n+00007b60: 910a f334 2c91 04ba 6d70 2e91 0774 5733 ...4,...mp...tW3\n+00007b70: 3230 5d2c 9104 ba6e 7768 6572 6598 6974 20],...nwhere.it\n+00007b80: 936d 6572 6765 2d8e a18d 736f 7274 739b .merge-...sorts.\n+00007b90: 03c5 1574 6865 9103 c514 7495 b221 7793 ...the....t..!w.\n+00007ba0: 6f98 6578 7061 6e73 696f 6e73 2e91 0539 o.expansions...9\n+00007bb0: e954 91ff 1665 6f98 7072 6576 9365 6e93 .T...eo.prev.en.\n+00007bc0: 7498 636f 6d70 904d de6f 6e65 6e93 7473 t.comp.M.onen.ts\n+00007bd0: 9603 c514 7769 7468 986f 6e6c 7998 6193 ....with.only.a.\n+00007be0: 6665 7798 7369 676e 690c 6361 6e90 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9ad7 5b8d a0fd 62b4 .....%....[...b.\n+00009930: 7ba0 027f 4b85 8d9f b180 378d 9fda 4555 {...K.....7...EU\n+00009940: 8d8d 914f 4bd7 ef3f 5053 6669 6c65 3d22 ...OK..?PSfile=\"\n+00009950: 7164 5f6d 756c 5f61 6363 756d 2e65 7073 qd_mul_accum.eps\n+00009960: 2220 6c6c 783d 3020 6c6c 793d 3020 7572 \" llx=0 lly=0 ur\n+00009970: 783d 3331 3020 7572 793d 3431 3820 7277 x=310 ury=418 rw\n+00009980: 693d 3331 3030 208e 8e9f 1799 9a8d 8d91 i=3100 .........\n+00009990: 5a6d b8c6 4669 6775 7265 9603 a666 393a Zm..Figure...f9:\n+000099a0: 9104 dddd 5175 6164 2d44 6f75 626c 6593 ....Quad-Double.\n+000099b0: df02 93c6 5175 6164 2d44 6f75 626c 6593 ....Quad-Double.\n+000099c0: 6163 6375 6d90 b221 756c 6174 696f 6e93 accum..!ulation.\n+000099d0: 7068 6173 658e 8e8e 8e9f 1e00 008d 9200 phase...........\n+000099e0: e567 0a31 338e 8e8c 8b00 0000 0e00 0000 .g.13...........\n 000099f0: 0000 0000 0000 0000 0000 0000 0000 0000 ................\n 00009a00: 0000 0000 0000 0000 0000 0000 0000 0000 ................\n-00009a10: 0098 f7a0 029a d75b 8da0 fd62 b47b a002 .......[...b.{..\n-00009a20: 7f4b 858d a0ff 36d6 308d 9fda 4555 8d8d .K....6.0...EU..\n-00009a30: 9157 d400 ef40 5053 6669 6c65 3d22 7369 .W...@PSfile=\"si\n-00009a40: 782d 7468 7265 652d 7375 6d2e 6570 7322 x-three-sum.eps\"\n-00009a50: 206c 6c78 3d30 206c 6c79 3d30 2075 7278 llx=0 lly=0 urx\n-00009a60: 3d32 3933 2075 7279 3d33 3734 2072 7769 =293 ury=374 rwi\n-00009a70: 3d32 3933 3020 8e8e 9f17 999a 8d8d 9200 =2930 ..........\n-00009a80: a7d1 72c6 4669 6775 7265 9103 a666 3130 ..r.Figure...f10\n-00009a90: 3a91 04dd dde2 5369 782d 5468 7265 652d :.....Six-Three-\n-00009aa0: 5375 6d8e 8e8e 9f1f 0000 8d91 1100 0053 Sum............S\n-00009ab0: 6978 2d54 6872 6565 2d53 756d 9604 3a52 ix-Three-Sum..:R\n-00009ac0: c663 6f6d 7075 7465 739b 043a 5174 6865 .computes..:Qthe\n-00009ad0: 9373 756d 936f 6693 7369 7898 646f 7562 .sum.of.six.doub\n-00009ae0: 6c65 7393 746f 9374 6872 6565 9364 6f75 les.to.three.dou\n-00009af0: 626c 6598 7790 b221 6f72 7468 936f 6693 ble.w..!orth.of.\n-00009b00: 6163 6375 7261 6379 9328 692e 652e 2c91 accuracy.(i.e.,.\n-00009b10: 045f 4c74 6f8e a40d 999a 8d72 656c 6174 ._Lto......relat\n-00009b20: 6976 90b2 2165 9602 b4c3 6572 726f 729b iv..!e....error.\n-00009b30: 02b4 c26f 6693 de4f 904d dec6 28de 228d ...of..O.M..(.\".\n-00009b40: 9ffc 06be ce33 8e91 04c0 04c6 2929 2e91 .....3......))..\n-00009b50: 048d 5154 6869 7393 6973 9864 6f6e 6593 ..QThis.is.done.\n-00009b60: 6290 b221 7998 6469 7669 6469 6e67 9374 b..!y.dividing.t\n-00009b70: 6865 9869 6e70 7574 7393 696e 95b2 2174 he.inputs.in..!t\n-00009b80: 6f98 7493 7793 6f96 02b4 c367 726f 7570 o.t.w.o....group\n-00009b90: 7398 6f66 9374 6872 6565 2c91 02e5 1661 s.of.three,....a\n-00009ba0: 6e64 9370 904d de65 7266 6f72 6d69 6e67 nd.p.M.erforming\n-00009bb0: 8ea1 8de2 5468 7265 652d 5375 6d96 03b1 ....Three-Sum...\n-00009bc0: 27c6 6f6e 9365 6163 9ab2 2168 9103 b126 '.on.eac..!h...&\n-00009bd0: 6772 6f75 702e 9104 fe20 5468 656e 9374 group.... Then.t\n-00009be0: 6865 9374 9877 986f 9b03 b126 7375 6d73 he.t.w.o...&sums\n-00009bf0: 9361 7265 9361 6464 6564 9374 6f67 6574 .are.added.toget\n-00009c00: 6865 722c 9103 b3d7 696e 9361 936d 616e her,....in.a.man\n-00009c10: 6e65 7298 7369 6d69 6c61 7293 746f 9371 ner.similar.to.q\n-00009c20: 7561 642d 8ea1 8d64 6f75 626c 6596 03a6 uad-...double...\n-00009c30: 6661 6464 6974 696f 6e2e 9104 dddd 5365 faddition.....Se\n-00009c40: 6593 4669 6775 7265 9331 302e 8ea1 8d91 e.Figure.10.....\n-00009c50: 1100 00e2 4e69 6e65 2d54 7790 c1b4 6f2d ....Nine-Tw...o-\n-00009c60: 5375 6d96 0363 cbc6 636f 6d70 7574 6573 Sum..c..computes\n-00009c70: 9374 6865 9373 756d 9103 63cc 6f66 936e .the.sum..c.of.n\n-00009c80: 696e 6593 646f 7562 6c65 7393 746f 9364 ine.doubles.to.d\n-00009c90: 6f75 626c 652d 646f 7562 6c65 9361 6363 ouble-double.acc\n-00009ca0: 7572 6163 7991 ff16 652e 9104 c7aa 5468 uracy...e.....Th\n-00009cb0: 6973 9369 7393 646f 6e65 9362 90b2 2179 is.is.done.b..!y\n-00009cc0: 8ea1 8d70 6169 7269 6e67 9603 40d3 7468 ...pairing..@.th\n-00009cd0: 6593 696e 7075 7473 9b03 40d4 746f 9363 e.inputs..@.to.c\n-00009ce0: 7265 6174 6593 666f 7572 9364 6f75 626c reate.four.doubl\n-00009cf0: 652d 646f 7562 6c65 936e 95b2 2175 6d93 e-double.n..!um.\n-00009d00: 6290 4dde 6572 7396 0340 d361 6e64 9861 b.M.ers..@.and.a\n-00009d10: 9373 696e 676c 6593 646f 7562 6c65 9370 .single.double.p\n-00009d20: 7265 6369 7369 6f6e 936e 95b2 2175 6d93 recision.n..!um.\n-00009d30: 6290 4dde 6572 2c91 0355 2461 6e64 8ea1 b.M.er,..U$and..\n-00009d40: 8d70 9a4d de65 7266 6f72 6d69 6e67 9603 .p.M.erforming..\n-00009d50: 6ca3 6164 6469 7469 6f6e 936f 6691 036c l.addition.of..l\n-00009d60: a474 95b2 2177 936f 9603 6ca3 646f 7562 .t..!w.o..l.doub\n-00009d70: 6c65 2d64 6f75 626c 6593 6e95 b221 756d le-double.n..!um\n-00009d80: 9362 9865 7273 9b03 6ca3 7265 6375 7273 .b.ers..l.recurs\n-00009d90: 6976 9365 6c79 9103 6ca4 756e 9374 696c iv.ely..l.un.til\n-00009da0: 986f 6e65 9861 7272 6976 9365 7398 6174 .one.arriv.es.at\n-00009db0: 9103 6ca4 6198 646f 7562 6c65 2d64 6f75 ..l.a.double-dou\n-00009dc0: 626c 658e a18d 6f75 7470 7574 2e91 06ae ble...output....\n-00009dd0: 4b54 6865 9b04 4136 646f 7562 6c65 2d64 KThe..A6double-d\n-00009de0: 6f75 626c 6596 0441 3561 6464 6974 696f ouble..A5additio\n-00009df0: 6e93 2874 6865 986c 6172 6765 9373 7175 n.(the.large.squ\n-00009e00: 6172 6598 6290 4dde 6f90 b221 7893 696e are.b.M.o..!x.in\n-00009e10: 9874 6865 9364 6961 6772 616d 2993 6973 .the.diagram).is\n-00009e20: 9874 6865 9373 616d 6598 6173 9344 6190 .the.same.as.Da.\n-00009e30: b221 7669 648e a18d 4261 696c 6579 2773 .!vid...Bailey's\n-00009e40: 9603 a666 616c 676f 7269 7468 6d93 5b8d ...falgorithm.[.\n-00009e50: 318e 9105 799a 5d2e 9104 dddd 5365 6593 1...y.].....See.\n-00009e60: 4669 6775 7265 9331 312e 8ea1 8d91 1100 Figure.11.......\n-00009e70: 0049 6696 054c 0b6f 6e65 9b05 4c0c 7769 .If..L.one..L.wi\n-00009e80: 7368 6573 9374 6f98 7472 6164 6593 6665 shes.to.trade.fe\n-00009e90: 7798 6269 7473 936f 6698 6163 6375 7261 w.bits.of.accura\n-00009ea0: 6379 9366 6f72 9873 7090 4dde 6565 642c cy.for.sp.M.eed,\n-00009eb0: 9105 b574 7790 b221 6598 646f 6e27 7493 ...tw..!e.don't.\n-00009ec0: 6576 90b2 2165 6e98 6e65 6564 9374 6f98 ev..!en.need.to.\n-00009ed0: 636f 6d70 7574 6593 7468 658e a18d de4f compute.the....O\n-00009ee0: 904d dec6 28de 228d 9ffc 06be ce34 8e91 .M..(.\"......4..\n-00009ef0: 04c0 04c6 2996 0439 0474 6572 6d73 3b91 ....)..9.terms;.\n-00009f00: 0482 5474 6865 7993 6361 6e9b 0439 0561 ..Tthey.can..9.a\n-00009f10: 0b65 6374 9374 6865 980c 7273 7493 3231 .ect.the..rst.21\n-00009f20: 3298 6269 7473 936f 6e6c 7993 6290 b221 2.bits.only.b..!\n-00009f30: 7998 6361 7272 6965 7393 6475 7269 6e67 y.carries.during\n-00009f40: 9861 6363 756d 90b2 2175 6c61 7469 6f6e .accum..!ulation\n-00009f50: 2e91 0695 b849 6e98 7468 6973 9363 6173 .....In.this.cas\n-00009f60: 652c 8ea1 8d77 9ab2 2165 9604 63d5 6361 e,...w..!e..c.ca\n-00009f70: 6e93 636f 6d70 7574 6593 7468 6593 de4f n.compute.the..O\n-00009f80: 904d dec6 28de 228d 9ffc 06be ce33 8e91 .M..(.\"......3..\n-00009f90: 04c0 04c6 2993 7465 726d 7393 7573 696e ....).terms.usin\n-00009fa0: 6793 6e6f 726d 616c 9364 6f75 626c 6593 g.normal.double.\n-00009fb0: 7072 6563 6973 696f 6e93 6172 6974 686d precision.arithm\n-00009fc0: 6574 6963 2c91 0493 3074 6865 7265 6298 etic,...0thereb.\n-00009fd0: 7993 7370 904d de65 6564 696e 6793 7570 y.sp.M.eeding.up\n-00009fe0: 8ea1 8d6d 90b2 2175 6c74 6970 6c69 6361 ...m..!ultiplica\n-00009ff0: 7469 6f6e 9103 a666 636f 6e73 6964 6572 tion...fconsider\n-0000a000: 6162 6c79 91ff 1665 2e8e 8e9f 1e00 008d ably...e........\n-0000a010: 9200 e567 0a31 348e 8e8c 8b00 0000 0f00 ...g.14.........\n-0000a020: 0000 0000 0000 0000 0000 0000 0000 0000 ................\n+00009a10: 0000 0098 f9a0 029a d75b 8da0 fd62 b47b .........[...b.{\n+00009a20: a002 7f4b 858d a0ff 36d6 308d 9fda 4555 ...K....6.0...EU\n+00009a30: 8d8d 9157 d400 ef40 5053 6669 6c65 3d22 ...W...@PSfile=\"\n+00009a40: 7369 782d 7468 7265 652d 7375 6d2e 6570 six-three-sum.ep\n+00009a50: 7322 206c 6c78 3d30 206c 6c79 3d30 2075 s\" llx=0 lly=0 u\n+00009a60: 7278 3d32 3933 2075 7279 3d33 3734 2072 rx=293 ury=374 r\n+00009a70: 7769 3d32 3933 3020 8e8e 9f17 999a 8d8d wi=2930 ........\n+00009a80: 9200 a7d1 72c6 4669 6775 7265 9103 a666 ....r.Figure...f\n+00009a90: 3130 3a91 04dd dde2 5369 782d 5468 7265 10:.....Six-Thre\n+00009aa0: 652d 5375 6d8e 8e8e 9f1f 0000 8d91 1100 e-Sum...........\n+00009ab0: 0053 6978 2d54 6872 6565 2d53 756d 9604 .Six-Three-Sum..\n+00009ac0: 3a52 c663 6f6d 7075 7465 739b 043a 5174 :R.computes..:Qt\n+00009ad0: 6865 9373 756d 936f 6693 7369 7898 646f he.sum.of.six.do\n+00009ae0: 7562 6c65 7393 746f 9374 6872 6565 9364 ubles.to.three.d\n+00009af0: 6f75 626c 6598 7790 b221 6f72 7468 936f ouble.w..!orth.o\n+00009b00: 6693 6163 6375 7261 6379 9328 692e 652e f.accuracy.(i.e.\n+00009b10: 2c91 045f 4c74 6f8e a40d 999a 8d72 656c ,.._Lto......rel\n+00009b20: 6174 6976 90b2 2165 9602 b4c3 6572 726f ativ..!e....erro\n+00009b30: 729b 02b4 c26f 6693 de4f 904d dec6 28de r....of..O.M..(.\n+00009b40: 228d 9ffc 06be ce33 8e91 04c0 04c6 2929 \"......3......))\n+00009b50: 2e91 048d 5154 6869 7393 6973 9864 6f6e ....QThis.is.don\n+00009b60: 6593 6290 b221 7998 6469 7669 6469 6e67 e.b..!y.dividing\n+00009b70: 9374 6865 9869 6e70 7574 7393 696e 95b2 .the.inputs.in..\n+00009b80: 2174 6f98 7493 7793 6f96 02b4 c367 726f !to.t.w.o....gro\n+00009b90: 7570 7398 6f66 9374 6872 6565 2c91 02e5 ups.of.three,...\n+00009ba0: 1661 6e64 9370 904d de65 7266 6f72 6d69 .and.p.M.erformi\n+00009bb0: 6e67 8ea1 8de2 5468 7265 652d 5375 6d96 ng....Three-Sum.\n+00009bc0: 03b1 27c6 6f6e 9365 6163 9ab2 2168 9103 ..'.on.eac..!h..\n+00009bd0: b126 6772 6f75 702e 9104 fe20 5468 656e .&group.... Then\n+00009be0: 9374 6865 9374 9877 986f 9b03 b126 7375 .the.t.w.o...&su\n+00009bf0: 6d73 9361 7265 9361 6464 6564 9374 6f67 ms.are.added.tog\n+00009c00: 6574 6865 722c 9103 b3d7 696e 9361 936d ether,....in.a.m\n+00009c10: 616e 6e65 7298 7369 6d69 6c61 7293 746f anner.similar.to\n+00009c20: 9371 7561 642d 8ea1 8d64 6f75 626c 6596 .quad-...double.\n+00009c30: 03a6 6661 6464 6974 696f 6e2e 9104 dddd ..faddition.....\n+00009c40: 5365 6593 4669 6775 7265 9331 302e 8ea1 See.Figure.10...\n+00009c50: 8d91 1100 00e2 4e69 6e65 2d54 7790 c1b4 ......Nine-Tw...\n+00009c60: 6f2d 5375 6d96 0363 cbc6 636f 6d70 7574 o-Sum..c..comput\n+00009c70: 6573 9374 6865 9373 756d 9103 63cc 6f66 es.the.sum..c.of\n+00009c80: 936e 696e 6593 646f 7562 6c65 7393 746f .nine.doubles.to\n+00009c90: 9364 6f75 626c 652d 646f 7562 6c65 9361 .double-double.a\n+00009ca0: 6363 7572 6163 7991 ff16 652e 9104 c7aa ccuracy...e.....\n+00009cb0: 5468 6973 9369 7393 646f 6e65 9362 90b2 This.is.done.b..\n+00009cc0: 2179 8ea1 8d70 6169 7269 6e67 9603 40d3 !y...pairing..@.\n+00009cd0: 7468 6593 696e 7075 7473 9b03 40d4 746f the.inputs..@.to\n+00009ce0: 9363 7265 6174 6593 666f 7572 9364 6f75 .create.four.dou\n+00009cf0: 626c 652d 646f 7562 6c65 936e 95b2 2175 ble-double.n..!u\n+00009d00: 6d93 6290 4dde 6572 7396 0340 d361 6e64 m.b.M.ers..@.and\n+00009d10: 9861 9373 696e 676c 6593 646f 7562 6c65 .a.single.double\n+00009d20: 9370 7265 6369 7369 6f6e 936e 95b2 2175 .precision.n..!u\n+00009d30: 6d93 6290 4dde 6572 2c91 0355 2461 6e64 m.b.M.er,..U$and\n+00009d40: 8ea1 8d70 9a4d de65 7266 6f72 6d69 6e67 ...p.M.erforming\n+00009d50: 9603 6ca3 6164 6469 7469 6f6e 936f 6691 ..l.addition.of.\n+00009d60: 036c a474 95b2 2177 936f 9603 6ca3 646f .l.t..!w.o..l.do\n+00009d70: 7562 6c65 2d64 6f75 626c 6593 6e95 b221 uble-double.n..!\n+00009d80: 756d 9362 9865 7273 9b03 6ca3 7265 6375 um.b.ers..l.recu\n+00009d90: 7273 6976 9365 6c79 9103 6ca4 756e 9374 rsiv.ely..l.un.t\n+00009da0: 696c 986f 6e65 9861 7272 6976 9365 7398 il.one.arriv.es.\n+00009db0: 6174 9103 6ca4 6198 646f 7562 6c65 2d64 at..l.a.double-d\n+00009dc0: 6f75 626c 658e a18d 6f75 7470 7574 2e91 ouble...output..\n+00009dd0: 06ae 4b54 6865 9b04 4136 646f 7562 6c65 ..KThe..A6double\n+00009de0: 2d64 6f75 626c 6596 0441 3561 6464 6974 -double..A5addit\n+00009df0: 696f 6e93 2874 6865 986c 6172 6765 9373 ion.(the.large.s\n+00009e00: 7175 6172 6598 6290 4dde 6f90 b221 7893 quare.b.M.o..!x.\n+00009e10: 696e 9874 6865 9364 6961 6772 616d 2993 in.the.diagram).\n+00009e20: 6973 9874 6865 9373 616d 6598 6173 9344 is.the.same.as.D\n+00009e30: 6190 b221 7669 648e a18d 4261 696c 6579 a..!vid...Bailey\n+00009e40: 2773 9603 a666 616c 676f 7269 7468 6d93 's...falgorithm.\n+00009e50: 5b8d 318e 9105 799a 5d2e 9104 dddd 5365 [.1...y.].....Se\n+00009e60: 6593 4669 6775 7265 9331 312e 8ea1 8d91 e.Figure.11.....\n+00009e70: 1100 0049 6696 054c 0b6f 6e65 9b05 4c0c ...If..L.one..L.\n+00009e80: 7769 7368 6573 9374 6f98 7472 6164 6593 wishes.to.trade.\n+00009e90: 6665 7798 6269 7473 936f 6698 6163 6375 few.bits.of.accu\n+00009ea0: 7261 6379 9366 6f72 9873 7090 4dde 6565 racy.for.sp.M.ee\n+00009eb0: 642c 9105 b574 7790 b221 6598 646f 6e27 d,...tw..!e.don'\n+00009ec0: 7493 6576 90b2 2165 6e98 6e65 6564 9374 t.ev..!en.need.t\n+00009ed0: 6f98 636f 6d70 7574 6593 7468 658e a18d o.compute.the...\n+00009ee0: de4f 904d dec6 28de 228d 9ffc 06be ce34 .O.M..(.\"......4\n+00009ef0: 8e91 04c0 04c6 2996 0439 0474 6572 6d73 ......)..9.terms\n+00009f00: 3b91 0482 5474 6865 7993 6361 6e9b 0439 ;...Tthey.can..9\n+00009f10: 0561 0b65 6374 9374 6865 980c 7273 7493 .a.ect.the..rst.\n+00009f20: 3231 3298 6269 7473 936f 6e6c 7993 6290 212.bits.only.b.\n+00009f30: b221 7998 6361 7272 6965 7393 6475 7269 .!y.carries.duri\n+00009f40: 6e67 9861 6363 756d 90b2 2175 6c61 7469 ng.accum..!ulati\n+00009f50: 6f6e 2e91 0695 b849 6e98 7468 6973 9363 on.....In.this.c\n+00009f60: 6173 652c 8ea1 8d77 9ab2 2165 9604 63d5 ase,...w..!e..c.\n+00009f70: 6361 6e93 636f 6d70 7574 6593 7468 6593 can.compute.the.\n+00009f80: de4f 904d dec6 28de 228d 9ffc 06be ce33 .O.M..(.\"......3\n+00009f90: 8e91 04c0 04c6 2993 7465 726d 7393 7573 ......).terms.us\n+00009fa0: 696e 6793 6e6f 726d 616c 9364 6f75 626c ing.normal.doubl\n+00009fb0: 6593 7072 6563 6973 696f 6e93 6172 6974 e.precision.arit\n+00009fc0: 686d 6574 6963 2c91 0493 3074 6865 7265 hmetic,...0there\n+00009fd0: 6298 7993 7370 904d de65 6564 696e 6793 b.y.sp.M.eeding.\n+00009fe0: 7570 8ea1 8d6d 90b2 2175 6c74 6970 6c69 up...m..!ultipli\n+00009ff0: 6361 7469 6f6e 9103 a666 636f 6e73 6964 cation...fconsid\n+0000a000: 6572 6162 6c79 91ff 1665 2e8e 8e9f 1e00 erably...e......\n+0000a010: 008d 9200 e567 0a31 348e 8e8c 8b00 0000 .....g.14.......\n+0000a020: 0f00 0000 0000 0000 0000 0000 0000 0000 ................\n 0000a030: 0000 0000 0000 0000 0000 0000 0000 0000 ................\n-0000a040: 0000 0000 0099 e6a0 029a d75b 8da0 fd62 ...........[...b\n-0000a050: b47b a002 7f4b 858d 9f80 512d 8d9f da45 .{...K....Q-...E\n-0000a060: 558d 8d91 2f2d 1fef 3f50 5366 696c 653d U.../-..?PSfile=\n-0000a070: 226e 696e 652d 7477 6f2d 7375 6d2e 6570 \"nine-two-sum.ep\n-0000a080: 7322 206c 6c78 3d30 206c 6c79 3d30 2075 s\" llx=0 lly=0 u\n-0000a090: 7278 3d33 3734 2075 7279 3d33 3230 2072 rx=374 ury=320 r\n-0000a0a0: 7769 3d33 3734 3020 8e8e 9f17 999a 8d8d wi=3740 ........\n-0000a0b0: 9200 a87b f9c6 4669 6775 7265 9103 a666 ...{..Figure...f\n-0000a0c0: 3131 3a91 04dd dde2 4e69 6e65 2d54 7790 11:.....Nine-Tw.\n-0000a0d0: c1b4 6f2d 5375 6d8e 8e8e 8e9f 1e00 008d ..o-Sum.........\n-0000a0e0: 9200 e567 0ac6 3135 8e8e 8c8b 0000 0010 ...g..15........\n-0000a0f0: 0000 0000 0000 0000 0000 0000 0000 0000 ................\n+0000a040: 0000 0000 0000 0099 e8a0 029a d75b 8da0 .............[..\n+0000a050: fd62 b47b a002 7f4b 858d 9f80 512d 8d9f .b.{...K....Q-..\n+0000a060: da45 558d 8d91 2f2d 1fef 3f50 5366 696c .EU.../-..?PSfil\n+0000a070: 653d 226e 696e 652d 7477 6f2d 7375 6d2e e=\"nine-two-sum.\n+0000a080: 6570 7322 206c 6c78 3d30 206c 6c79 3d30 eps\" llx=0 lly=0\n+0000a090: 2075 7278 3d33 3734 2075 7279 3d33 3230 urx=374 ury=320\n+0000a0a0: 2072 7769 3d33 3734 3020 8e8e 9f17 999a rwi=3740 ......\n+0000a0b0: 8d8d 9200 a87b f9c6 4669 6775 7265 9103 .....{..Figure..\n+0000a0c0: a666 3131 3a91 04dd dde2 4e69 6e65 2d54 .f11:.....Nine-T\n+0000a0d0: 7790 c1b4 6f2d 5375 6d8e 8e8e 8e9f 1e00 w...o-Sum.......\n+0000a0e0: 008d 9200 e567 0ac6 3135 8e8e 8c8b 0000 .....g..15......\n+0000a0f0: 0010 0000 0000 0000 0000 0000 0000 0000 ................\n 0000a100: 0000 0000 0000 0000 0000 0000 0000 0000 ................\n-0000a110: 0000 0000 0000 a01a a002 9ad7 5b8d a0fd ............[...\n-0000a120: 62b4 7ba0 027f 4b85 8da0 fda4 b47b 8d91 b.{...K......{..\n-0000a130: 1100 00c6 5371 7561 7269 6e67 9603 4371 ....Squaring..Cq\n-0000a140: 6191 0343 7271 7561 642d 646f 7562 6c65 a..Crquad-double\n-0000a150: 936e 95b2 2175 6d93 629a 4dde 6572 9603 .n..!um.b.M.er..\n-0000a160: 4371 6361 6e93 6298 659b 0343 7264 6f6e Cqcan.b.e..Crdon\n-0000a170: 6593 7369 676e 690c 6361 6e90 b221 746c e.signi.can..!tl\n-0000a180: 7993 6661 7374 6572 9873 696e 6365 9374 y.faster.since.t\n-0000a190: 6865 936e 95b2 2175 6d93 6290 4dde 6572 he.n..!um.b.M.er\n-0000a1a0: 9603 4371 6f66 9874 6572 6d73 9374 6861 ..Cqof.terms.tha\n-0000a1b0: 748e a40d 999a 8d6e 6565 6473 9603 a666 t......needs...f\n-0000a1c0: 746f 9362 9a4d de65 9361 6363 756d 90b2 to.b.M.e.accum..\n-0000a1d0: 2175 6c61 7465 6493 6361 6e93 6298 6593 !ulated.can.b.e.\n-0000a1e0: 7265 6475 6365 6493 6475 6593 746f 9373 reduced.due.to.s\n-0000a1f0: 796d 6d65 7472 7991 ff16 652e 8e9f 1d43 ymmetry...e....C\n-0000a200: 498d 8de3 332e 358e 911e c000 4469 7669 I...3.5.....Divi\n-0000a210: 7369 6f6e 8ea9 14ab ff8d c644 6976 6973 sion.......Divis\n-0000a220: 696f 6e96 0273 1569 739b 0273 1464 6f6e ion..s.is..s.don\n-0000a230: 6593 6290 b221 7993 7468 6593 6661 6d69 e.b..!y.the.fami\n-0000a240: 6c69 6172 986c 6f6e 6793 6469 7669 7369 liar.long.divisi\n-0000a250: 6f6e 9361 6c67 6f72 6974 686d 2e91 0477 on.algorithm...w\n-0000a260: 6d4c 6574 93de 6196 030a a7c6 3d93 28de mLet..a.....=.(.\n-0000a270: 618d 9f01 a47a ce30 8e96 04c0 04de 3b9b a....z.0......;.\n-0000a280: 01d3 3161 8d9f 01a4 7ace 318e 93de 3b98 ..1a....z.1...;.\n-0000a290: 618d 9f01 a47a ce32 8e93 de3b 9861 8d9f a....z.2...;.a..\n-0000a2a0: 01a4 7ace 338e 93c6 2991 0273 1461 6e64 ..z.3...)..s.and\n-0000a2b0: 9102 7315 de62 9603 0aa7 c63d 9328 de62 ..s..b.....=.(.b\n-0000a2c0: 8d9f 01a4 7ace 308e 9604 c004 de3b 9862 ....z.0......;.b\n-0000a2d0: 8d9f 01a4 7ace 318e 93de 3b98 628d 9f01 ....z.1...;.b...\n-0000a2e0: a47a ce32 8e93 de3b 9862 8d9f 01a4 7ace .z.2...;.b....z.\n-0000a2f0: 338e 93c6 298e a18d 629a 4dde 6591 049d 3...)...b.M.e...\n-0000a300: 4a71 7561 642d 646f 7562 6c65 9104 9d49 Jquad-double...I\n-0000a310: 6e95 b221 756d 9362 9865 7273 2e91 07c2 n..!um.b.ers....\n-0000a320: 8857 91ff 1665 6596 049d 4963 616e 9b04 .W...ee...Ican..\n-0000a330: 9d4a 0c72 7374 9363 6f6d 7075 7465 9861 .J.rst.compute.a\n-0000a340: 6e93 6170 7072 6f95 b221 7869 6d61 7465 n.appro..!ximate\n-0000a350: 9871 756f 7469 656e 9374 9104 9d49 de71 .quotien.t...I.q\n-0000a360: 8d9f 01a4 7ace 308e 9109 6625 c63d 9104 ....z.0...f%.=..\n-0000a370: a620 de61 8d9f 01a4 7ace 308e 9604 c004 . .a....z.0.....\n-0000a380: de3d 628d 9f01 a47a ce30 8e93 c62e 9107 .=b....z.0......\n-0000a390: c287 5791 ff16 6565 9874 6865 6e8e a18d ..W...ee.then...\n-0000a3a0: 636f 6d70 7574 6596 04dc 4d74 6865 9372 compute...Mthe.r\n-0000a3b0: 656d 6169 6e64 6572 93de 7291 055d 04c6 emainder..r..]..\n-0000a3c0: 3d91 050f 25de 6196 033d 85df 0091 033d =...%.a..=.....=\n-0000a3d0: 86de 718d 9f01 a47a ce30 8e91 07fd 89df ..q....z.0......\n-0000a3e0: 0293 de62 c62c 9105 29c6 616e 6496 04dc ...b.,..).and...\n-0000a3f0: 4d63 6f6d 7075 7465 9374 6865 9363 6f72 Mcompute.the.cor\n-0000a400: 7265 6374 696f 6e93 7465 726d 93de 718d rection.term..q.\n-0000a410: 9f01 a47a ce31 8e91 09cf 29c6 3d91 050f ...z.1....).=...\n-0000a420: 26de 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2993 616e 648d 9363 6f73 8e91 1301 24).and..cos....\n+0000bc10: 0328 de6d 1998 3dc6 3130 3234 299b 0457 .(.m..=.1024)..W\n+0000bc20: d561 7265 9370 7265 636f 6d70 7574 6564 .are.precomputed\n+0000bc30: 9362 90b2 2179 9375 7369 6e67 9861 7262 .b..!y.using.arb\n+0000bc40: 6974 7261 7279 9370 7265 6369 7369 6f6e itrary.precision\n+0000bc50: 8ea1 8d70 6163 95b2 216b 91ff 6444 6167 ...pac..!k..dDag\n+0000bc60: 659b 03a6 6673 7563 9368 9861 7398 4d50 e...fsuc.h.as.MP\n+0000bc70: 4655 4e98 5b8d 328e 9105 799a 5d98 7573 FUN.[.2...y.].us\n+0000bc80: 696e 6798 7468 6598 666f 726d 9375 6c61 ing.the.form.ula\n+0000bc90: 8e9f 1c16 cd8d 8d92 00b0 1999 7369 6e8e ............sin.\n+0000bca0: 8d8d 9200 bf5e 809f f1a9 92b6 128e 8d8d .....^..........\n+0000bcb0: 8d9f f897 9e8d 9200 c7f2 0cde 128e 9200 ................\n+0000bcc0: c7ee 279f 04de c889 0000 6665 0005 799a ..'.......fe..y.\n+0000bcd0: 9f0a 0c74 8dc6 328e 8e8e 8e8d 9200 ce9a ...t..2.........\n+0000bce0: f49f f1a9 92b6 138e 8e92 00d9 020f c63d 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......x.........\n-0000c090: 6eec de65 8d9f fc06 bed2 00d0 788e 8e91 n..e........x...\n-0000c0a0: 043d da9f 04de c889 0000 6665 0028 c2e0 .=........fe.(..\n-0000c0b0: 9f0a 0c74 8d91 11a4 a3c6 328e 8e8e 8e8d ...t......2.....\n-0000c0c0: 9145 ed88 636f 7368 8e91 5c7f 3cde 7893 .E..cosh..\\.<.x.\n-0000c0d0: c63d 8d8d 8d9f f897 9e8d 9104 3dda de65 .=..........=..e\n-0000c0e0: 8d9f fc06 bed0 788e 9107 b7d3 c62b 9102 ......x......+..\n-0000c0f0: 6eec de65 8d9f fc06 bed2 00d0 788e 8e91 n..e........x...\n-0000c100: 043d da9f 04de c889 0000 6665 0028 c2e0 .=........fe.(..\n-0000c110: 9f0a 0c74 8d91 11a4 a3c6 328e 8e8e 8e8e ...t......2.....\n-0000c120: 9f1b b941 8d48 6f95 b221 7793 6576 9365 ...A.Ho..!w.ev.e\n-0000c130: 722c 9104 1a6e 7768 656e 9604 0339 de78 r,...nwhen...9.x\n-0000c140: 93c6 6973 9373 6d61 6c6c 9328 7361 9ab2 ..is.small.(sa..\n-0000c150: 2179 93df 6ade 78df 6a96 03a5 5c14 93c6 !y..j.x.j...\\...\n-0000c160: 30de 3ac6 3031 292c 9104 1a6d 7468 6596 0.:.01),...mthe.\n-0000c170: 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7561 642d sion....of.quad-\n-0000c260: 646f 7562 6c65 936e 95b2 2175 6d93 6290 double.n..!um.b.\n-0000c270: 4dde 6572 9603 dc1f de78 98c6 6973 9364 M.er.....x..is.d\n-0000c280: 6f6e 6593 6290 b221 7998 6465 7465 726d one.b..!y.determ\n-0000c290: 696e 696e 6793 7468 6593 696e 95b2 2174 ining.the.in..!t\n-0000c2a0: 6567 6572 98de 6b91 0434 5ec6 7375 6393 eger..k..4^.suc.\n-0000c2b0: 688e a18d 7468 6174 9b03 be92 3196 0332 h...that....1..2\n-0000c2c0: f1df 1491 0332 f06a de78 c631 308d 9ffc .....2.j.x.10...\n-0000c2d0: 06be d200 d06b 8e91 0bc0 0adf 6a93 de3c .....k......j..<\n-0000c2e0: 93c6 3130 2c91 03c4 9d61 6e64 9872 6570 ..10,....and.rep\n-0000c2f0: 904d de65 6174 6564 6c79 9865 7874 7261 .M.eatedly.extra\n-0000c300: 6374 696e 6791 03be 9364 6967 6974 7398 cting....digits.\n-0000c310: 616e 6498 6d95 b221 756c 7469 706c 7969 and.m..!ultiplyi\n-0000c320: 6e67 9862 9379 9603 be93 3130 2e91 0526 ng.b.y....10...&\n-0000c330: 6154 91ff 1665 6f93 6d69 6e69 6d69 7a65 aT...eo.minimize\n-0000c340: 9865 7272 6f72 8ea1 8d61 6363 756d 9ab2 .error...accum..\n-0000c350: 2175 6c61 7469 6f6e 2c91 0444 ca61 9604 !ulation,..D.a..\n-0000c360: 251d 7461 626c 6593 6f66 9361 6363 7572 %.table.of.accur\n-0000c370: 6174 656c 7993 7072 6563 6f6d 7075 7465 ately.precompute\n-0000c380: 6493 7090 4dde 6f98 7798 6572 7393 6f66 d.p.M.o.w.ers.of\n-0000c390: 9331 3093 6973 9375 7365 642e 9106 5a01 .10.is.used...Z.\n-0000c3a0: 5468 6973 9374 6162 6c65 9369 7393 616c This.table.is.al\n-0000c3b0: 736f 9375 7365 6493 696e 8ea1 8d64 6563 so.used.in...dec\n-0000c3c0: 696d 616c 9603 a666 746f 9362 696e 6172 imal...fto.binar\n-0000c3d0: 7993 636f 6e95 b221 7693 6572 7369 6f6e y.con..!v.ersion\n-0000c3e0: 2e8e 9f1d 5285 8d8d e336 2e32 8e91 1ec0 ....R....6.2....\n-0000c3f0: 0043 6f6d 7061 7269 736f 6e73 8ea6 8dc6 .Comparisons....\n-0000c400: 5369 6e63 659b 02ca 7d71 7561 642d 646f Since...}quad-do\n-0000c410: 7562 6c65 9102 ca7c 6e95 b221 756d 9362 uble...|n..!um.b\n-0000c420: 904d de65 7273 9861 7265 9866 756c 6c79 .M.ers.are.fully\n-0000c430: 9102 ca7c 7265 6e6f 726d 616c 697a 6564 ...|renormalized\n-0000c440: 9861 6674 6572 9865 6163 9368 9602 ca7c .after.eac.h...|\n-0000c450: 6f70 904d de65 7261 7469 6f6e 2c91 02f6 op.M.eration,...\n-0000c460: 7963 6f6d 7061 7269 6e67 9374 95b2 2177 ycomparing.t..!w\n-0000c470: 936f 9871 7561 642d 646f 7562 6c65 8ea1 .o.quad-double..\n-0000c480: 8d6e 95b2 2175 6d93 629a 4dde 6572 9604 .n..!um.b.M.er..\n-0000c490: e4ac 666f 7293 6571 7561 6c69 7490 b221 ..for.equalit..!\n-0000c4a0: 7993 6361 6e91 04e4 ad62 9865 9364 6f6e y.can....b.e.don\n-0000c4b0: 6593 636f 6d70 986f 6e65 6e90 b221 742d e.comp.onen..!t-\n-0000c4c0: 7769 7365 2e91 0898 af43 6f6d 7061 7269 wise.....Compari\n-0000c4d0: 6e67 9374 6865 9373 697a 6591 04e4 ad63 ng.the.size....c\n-0000c4e0: 616e 9362 9865 9364 6f6e 6593 6672 6f6d an.b.e.done.from\n-0000c4f0: 936d 6f73 748e a18d 7369 676e 690c 6361 .most...signi.ca\n-0000c500: 6e95 b221 749b 045f a163 6f6d 7090 4dde n..!t.._.comp.M.\n-0000c510: 6f6e 656e 9374 9604 5fa2 0c72 7374 2c91 onen.t.._..rst,.\n-0000c520: 048d f073 696d 696c 6172 9874 6f93 6469 ...similar.to.di\n-0000c530: 6374 696f 6e61 7279 986f 7264 6572 696e ctionary.orderin\n-0000c540: 6793 6f66 9845 6e67 6c69 7368 9377 90b2 g.of.English.w..\n-0000c550: 216f 7264 732e 9107 098f 436f 6d70 6172 !ords.....Compar\n-0000c560: 6973 6f6e 9374 6f98 7a65 726f 8ea1 8d63 ison.to.zero...c\n-0000c570: 616e 9603 a666 6290 4dde 6593 646f 6e65 an...fb.M.e.done\n-0000c580: 936a 7573 7493 629a b221 7993 6398 6865 .just.b..!y.c.he\n-0000c590: 6398 6b69 6e67 9374 6865 936d 6f73 7493 c.king.the.most.\n-0000c5a0: 7369 676e 690c 6361 6e98 7493 7798 6f72 signi.can.t.w.or\n-0000c5b0: 642e 8e9f 1d52 858d 8de3 362e 338e 911e d....R....6.3...\n-0000c5c0: c000 5261 6e64 6f6d 9604 8000 4e75 6d90 ..Random....Num.\n-0000c5d0: a000 6290 6000 6572 9347 656e 6572 6174 ..b.`.er.Generat\n-0000c5e0: 6f72 8ea6 8dc6 5468 6596 0493 3c71 7561 or....The........\n-0000cc80: 7f54 2369 6e63 6c75 6465 9105 bfaa 3c71 .T#include........\n-0000cca0: 911c 7f54 7573 696e 6791 05bf aa73 7464 ...Tusing....std\n-0000ccb0: 3a3a 636f 7574 3b8e a18d 911c 7f54 7573 ::cout;......Tus\n-0000ccc0: 696e 6791 05bf aa73 7464 3a3a 656e 646c ing....std::endl\n-0000ccd0: 3b8e a1a1 8d91 1c7f 5469 6e74 9605 bfaa ;.......Tint....\n-0000cce0: 6d61 696e 2829 937b 8ea1 8d91 27fe a875 main().{....'..u\n-0000ccf0: 6e73 6967 6e65 6496 05bf aa69 6e74 936f nsigned....int.o\n-0000cd00: 6c64 6377 3b8e a18d 9127 fea8 6670 755f ldcw;....'..fpu_\n-0000cd10: 6669 785f 7374 6172 7428 266f 6c64 6377 fix_start(&oldcw\n-0000cd20: 293b 911c be52 2f2f 9605 bfaa 7365 6593 );...R//....see.\n-0000cd30: 6e6f 7465 7393 6f6e 9378 3836 936d 6163 notes.on.x86.mac\n-0000cd40: 6869 6e65 7393 6265 6c6f 772e 8ea1 a18d hines.below.....\n-0000cd50: 9127 fea8 7164 5f72 6561 6c96 05bf aa61 .'..qd_real....a\n-0000cd60: 933d 9322 332e 3134 3135 3932 3635 3335 .=.\"3.1415926535\n-0000cd70: 3839 3739 3332 3338 3436 3236 3433 3338 8979323846264338\n-0000cd80: 3332 3739 3530 3238 3834 3139 3731 3639 3279502884197169\n-0000cd90: 3339 3933 3735 3130 3538 3230 223b 8ea1 399375105820\";..\n-0000cda0: 8d91 27fe a864 645f 7265 616c 9605 bfaa ..'..dd_real....\n-0000cdb0: 6293 3d93 2232 2e32 3439 3737 3537 3234 b.=.\"2.249775724\n-0000cdc0: 3730 3933 3639 3939 3539 3537 223b 8ea1 709369995957\";..\n-0000cdd0: 8d91 27fe a871 645f 7265 616c 9105 bfaa ..'..qd_real....\n-0000cde0: 723b 8ea1 a18d 9127 fea8 7296 05bf aa3d r;.....'..r....=\n-0000cdf0: 9373 7172 7428 6193 2a93 6293 2b93 312e .sqrt(a.*.b.+.1.\n-0000ce00: 3029 3b8e a18d 9127 fea8 636f 7574 9605 0);....'..cout..\n-0000ce10: bfaa 3c3c 9322 9373 7172 7428 6193 2a93 ..<<.\".sqrt(a.*.\n-0000ce20: 6293 2b93 3129 933d 9322 933c 3c93 7293 b.+.1).=.\".<<.r.\n-0000ce30: 3c3c 9365 6e64 6c3b 8ea1 a18d 9127 fea8 <<.endl;.....'..\n-0000ce40: 6670 755f 6669 785f 656e 6428 266f 6c64 fpu_fix_end(&old\n-0000ce50: 6377 293b 9128 3da6 2f2f 9605 bfaa 7365 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7869 6d61 7469 6f6e 9369 7393 6173 .!ximation.is.as\n-0000cf50: 7369 676e 6564 2e91 04ac ac46 91ff 1665 signed.....F...e\n-0000cf60: 6f72 9365 7861 6d70 6c65 2c91 0330 57e1 or.example,..0W.\n-0000cf70: 6196 05bf aa3d 9330 2e31 9603 12d3 c664 a....=.0.1.....d\n-0000cf80: 6f90 4dde 6573 986e 6f74 9361 7373 6967 o.M.es.not.assig\n-0000cf90: 6e93 7175 6164 2d64 6f75 626c 658e a18d n.quad-double...\n-0000cfa0: 7072 6563 6973 696f 6e96 03a6 6630 2e31 precision...f0.1\n-0000cfb0: 2c93 6275 7493 7261 7468 6572 9361 9364 ,.but.rather.a.d\n-0000cfc0: 6f75 626c 6593 7072 6563 6973 696f 6e93 ouble.precision.\n-0000cfd0: 6e95 b221 756d 9362 904d de65 7296 03a6 n..!um.b.M.er...\n-0000cfe0: 6630 2e31 2e91 04dd dd49 6e73 7465 6164 f0.1.....Instead\n-0000cff0: 2c93 7573 6593 e161 9605 bfaa 3d93 2230 ,.use..a....=.\"0\n-0000d000: 2e31 22c6 2e8e a18d 9111 0000 436f 6d6d .1\".........Comm\n-0000d010: 6f6e 9603 60d9 636f 6e73 7461 6e9a b221 on..`.constan..!\n-0000d020: 7473 9373 7563 9868 9103 60da 6173 93de ts.suc.h..`.as..\n-0000d030: 1995 6493 c62c 9b03 6ec2 de19 933d c632 ..d..,..n....=.2\n-0000d040: 2c98 de19 933d c634 2c91 036e c3de 65c6 ,....=.4,..n..e.\n-0000d050: 2c8d 986c 6f67 8e91 1366 c032 9603 60d9 ,..log...f.2..`.\n-0000d060: 6172 6593 7072 6f90 b221 7669 6465 6491 are.pro..!vided.\n-0000d070: 0360 da61 7393 e171 648d 9100 b097 8900 .`.as..qd.......\n-0000d080: 0066 6600 0373 028e 9604 2399 7265 616c .ff..s....#.real\n-0000d090: 3a3a 8d91 00b0 9789 0000 6666 0003 7302 ::........ff..s.\n-0000d0a0: 8e93 7069 c62c 98e1 7164 8d91 00b0 9789 ..pi.,..qd......\n-0000d0b0: 0000 6666 0003 7302 8e93 7265 616c 3a3a ..ff..s...real::\n-0000d0c0: 8d91 00b0 9789 0000 6666 0003 7302 8e93 ........ff..s...\n-0000d0d0: 7069 32c6 2c8e a18d e171 648d 9100 b097 pi2.,....qd.....\n-0000d0e0: 8900 0066 6600 0373 028e 9604 2399 7265 ...ff..s....#.re\n-0000d0f0: 616c 3a3a 8d91 00b0 9789 0000 6666 0003 al::........ff..\n-0000d100: 7302 8e93 7069 34c6 2c9b 03cd cae1 7164 s...pi4.,.....qd\n-0000d110: 8d91 00b0 9789 0000 6666 0003 7302 8e93 ........ff..s...\n-0000d120: 7265 616c 3a3a 8d91 00b0 9789 0000 6666 real::........ff\n-0000d130: 0003 7302 8e93 65c6 2c98 616e 649b 03c5 ..s...e.,.and...\n-0000d140: e9e1 7164 8d91 00b0 9789 0000 6666 0003 ..qd........ff..\n-0000d150: 7302 8e93 7265 616c 3a3a 8d91 00b0 9789 s...real::......\n-0000d160: 0000 6666 0003 7302 8e93 6c6f 6732 c62e ..ff..s...log2..\n-0000d170: 9105 3c67 5468 6573 6598 7790 b221 6572 .....\n+0000cc80: 911c 7f54 2369 6e63 6c75 6465 9105 bfaa ...T#include....\n+0000cc90: 3c71 642f 7164 5f72 6561 6c2e 683e 8ea1 ..\n+0000cca0: a18d 911c 7f54 7573 696e 6791 05bf aa73 .....Tusing....s\n+0000ccb0: 7464 3a3a 636f 7574 3b8e a18d 911c 7f54 td::cout;......T\n+0000ccc0: 7573 696e 6791 05bf aa73 7464 3a3a 656e using....std::en\n+0000ccd0: 646c 3b8e a1a1 8d91 1c7f 5469 6e74 9605 dl;.......Tint..\n+0000cce0: bfaa 6d61 696e 2829 937b 8ea1 8d91 27fe ..main().{....'.\n+0000ccf0: a875 6e73 6967 6e65 6496 05bf aa69 6e74 .unsigned....int\n+0000cd00: 936f 6c64 6377 3b8e a18d 9127 fea8 6670 .oldcw;....'..fp\n+0000cd10: 755f 6669 785f 7374 6172 7428 266f 6c64 u_fix_start(&old\n+0000cd20: 6377 293b 911c be52 2f2f 9605 bfaa 7365 cw);...R//....se\n+0000cd30: 6593 6e6f 7465 7393 6f6e 9378 3836 936d e.notes.on.x86.m\n+0000cd40: 6163 6869 6e65 7393 6265 6c6f 772e 8ea1 achines.below...\n+0000cd50: a18d 9127 fea8 7164 5f72 6561 6c96 05bf ...'..qd_real...\n+0000cd60: aa61 933d 9322 332e 3134 3135 3932 3635 .a.=.\"3.14159265\n+0000cd70: 3335 3839 3739 3332 3338 3436 3236 3433 3589793238462643\n+0000cd80: 3338 3332 3739 3530 3238 3834 3139 3731 3832795028841971\n+0000cd90: 3639 3339 3933 3735 3130 3538 3230 223b 69399375105820\";\n+0000cda0: 8ea1 8d91 27fe a864 645f 7265 616c 9605 ....'..dd_real..\n+0000cdb0: bfaa 6293 3d93 2232 2e32 3439 3737 3537 ..b.=.\"2.2497757\n+0000cdc0: 3234 3730 3933 3639 3939 3539 3537 223b 24709369995957\";\n+0000cdd0: 8ea1 8d91 27fe a871 645f 7265 616c 9105 ....'..qd_real..\n+0000cde0: bfaa 723b 8ea1 a18d 9127 fea8 7296 05bf ..r;.....'..r...\n+0000cdf0: aa3d 9373 7172 7428 6193 2a93 6293 2b93 .=.sqrt(a.*.b.+.\n+0000ce00: 312e 3029 3b8e a18d 9127 fea8 636f 7574 1.0);....'..cout\n+0000ce10: 9605 bfaa 3c3c 9322 9373 7172 7428 6193 ....<<.\".sqrt(a.\n+0000ce20: 2a93 6293 2b93 3129 933d 9322 933c 3c93 *.b.+.1).=.\".<<.\n+0000ce30: 7293 3c3c 9365 6e64 6c3b 8ea1 a18d 9127 r.<<.endl;.....'\n+0000ce40: fea8 6670 755f 6669 785f 656e 6428 266f ..fpu_fix_end(&o\n+0000ce50: 6c64 6377 293b 9128 3da6 2f2f 9605 bfaa ldcw);.(=.//....\n+0000ce60: 7365 6593 6e6f 7465 7393 6f6e 9378 3836 see.notes.on.x86\n+0000ce70: 936d 6163 6869 6e65 7393 6265 6c6f 772e .machines.below.\n+0000ce80: 8ea1 8d91 27fe a872 6574 7572 6e91 05bf ....'..return...\n+0000ce90: aa30 3b8e a18d 911c 7f54 7d8e 8e8e a000 .0;......T}.....\n+0000cea0: 977a 258d 9111 0000 c64e 6f74 6596 0286 .z%......Note...\n+0000ceb0: b874 6861 7493 7374 7269 6e67 739b 0286 .that.strings...\n+0000cec0: b76d 90b2 2175 7374 9362 904d de65 9375 .m..!ust.b.M.e.u\n+0000ced0: 7365 6493 746f 9861 7373 6967 6e93 746f sed.to.assign.to\n+0000cee0: 9361 9371 7561 642d 646f 7562 6c65 9828 .a.quad-double.(\n+0000cef0: 6f72 9364 6f75 626c 652d 646f 7562 6c65 or.double-double\n+0000cf00: 2993 6e95 b221 756d 9362 904d de65 7273 ).n..!um.b.M.ers\n+0000cf10: 3b91 02e6 9d6f 7468 6572 7769 7365 8ea1 ;....otherwise..\n+0000cf20: 8d74 6865 9603 12d3 646f 7562 6c65 9370 .the....double.p\n+0000cf30: 7265 6369 7369 6f6e 9b03 12d4 6170 7072 recision....appr\n+0000cf40: 6f90 b221 7869 6d61 7469 6f6e 9369 7393 o..!ximation.is.\n+0000cf50: 6173 7369 676e 6564 2e91 04ac ac46 91ff assigned.....F..\n+0000cf60: 1665 6f72 9365 7861 6d70 6c65 2c91 0330 .eor.example,..0\n+0000cf70: 57e1 6196 05bf aa3d 9330 2e31 9603 12d3 W.a....=.0.1....\n+0000cf80: c664 6f90 4dde 6573 986e 6f74 9361 7373 .do.M.es.not.ass\n+0000cf90: 6967 6e93 7175 6164 2d64 6f75 626c 658e ign.quad-double.\n+0000cfa0: a18d 7072 6563 6973 696f 6e96 03a6 6630 ..precision...f0\n+0000cfb0: 2e31 2c93 6275 7493 7261 7468 6572 9361 .1,.but.rather.a\n+0000cfc0: 9364 6f75 626c 6593 7072 6563 6973 696f .double.precisio\n+0000cfd0: 6e93 6e95 b221 756d 9362 904d de65 7296 n.n..!um.b.M.er.\n+0000cfe0: 03a6 6630 2e31 2e91 04dd dd49 6e73 7465 ..f0.1.....Inste\n+0000cff0: 6164 2c93 7573 6593 e161 9605 bfaa 3d93 ad,.use..a....=.\n+0000d000: 2230 2e31 22c6 2e8e a18d 9111 0000 436f \"0.1\".........Co\n+0000d010: 6d6d 6f6e 9603 60d9 636f 6e73 7461 6e9a mmon..`.constan.\n+0000d020: b221 7473 9373 7563 9868 9103 60da 6173 .!ts.suc.h..`.as\n+0000d030: 93de 1995 6493 c62c 9b03 6ec2 de19 933d ....d..,..n....=\n+0000d040: c632 2c98 de19 933d c634 2c91 036e c3de .2,....=.4,..n..\n+0000d050: 65c6 2c8d 986c 6f67 8e91 1366 c032 9603 e.,..log...f.2..\n+0000d060: 60d9 6172 6593 7072 6f90 b221 7669 6465 `.are.pro..!vide\n+0000d070: 6491 0360 da61 7393 e171 648d 9100 b097 d..`.as..qd.....\n+0000d080: 8900 0066 6600 0373 028e 9604 2399 7265 ...ff..s....#.re\n+0000d090: 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"unified_diff": null, "details": [{"source1": "pdftotext {} -", "source2": "pdftotext {} -", "unified_diff": "@@ -1,15 +1,15 @@\n Library for Double-Double and Quad-Double Arithmetic\u2217\n Yozo Hida\u2020\n \n Xiaoye S. Li\u2021\n \n David H. Bailey\u2021\n \n-January 6, 2024\n+February 9, 2025\n \n Abstract\n A double-double number is an unevaluated sum of two IEEE double precision numbers,\n capable of representing at least 106 bits of significand. Similarly, a quad-double number is an\n unevaluated sum of four IEEE double precision numbers, capable of representing at least 212 bits\n of significand. Algorithms for various arithmetic operations (including the four basic operations\n and various algebraic and transcendental operations) are presented. A C++ implementation of\n"}]}]}]}]}]}]}