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Cider Isn't Darwin Emulation, Really
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#include <machine/asm.h>#include "abi.h"
.const
.align 4 .quad 0x3e607933f5c9e7de, 0xbfb348d7f4d2c2c1 // c0, d .quad 0xbec93b87a2f58777, 0x0000000000000000 // c1, 0 .quad 0x3ffb9a832f4eb8b5, 0x3ffec80a0deff17f // a0, b0 .quad 0x3ff7460b0c70ba83, 0xbff9ffff4aa91ba4 // a1, b1
// Table used to compute log(2x) for the "large" case. Copied from logf.sacoshf_hi_table: .quad 0x0000000000000000, 0x3ff0000000000000 // log(1.00000), 1/1.00000 .quad 0x3f6ff00aa2b10bc0, 0x3fefe01fe01fe020 // log(1.00391), 1/1.00391 .quad 0x3f7fe02a6b106789, 0x3fefc07f01fc07f0 // log(1.00781), 1/1.00781 .quad 0x3f87dc475f810a77, 0x3fefa11caa01fa12 // log(1.01172), 1/1.01172 .quad 0x3f8fc0a8b0fc03e4, 0x3fef81f81f81f820 // log(1.01562), 1/1.01562 .quad 0x3f93cea44346a575, 0x3fef6310aca0dbb5 // log(1.01953), 1/1.01953 .quad 0x3f97b91b07d5b11b, 0x3fef44659e4a4271 // log(1.02344), 1/1.02344 .quad 0x3f9b9fc027af9198, 0x3fef25f644230ab5 // log(1.02734), 1/1.02734 .quad 0x3f9f829b0e783300, 0x3fef07c1f07c1f08 // log(1.03125), 1/1.03125 .quad 0x3fa1b0d98923d980, 0x3feee9c7f8458e02 // log(1.03516), 1/1.03516 .quad 0x3fa39e87b9febd60, 0x3feecc07b301ecc0 // log(1.03906), 1/1.03906 .quad 0x3fa58a5bafc8e4d5, 0x3feeae807aba01eb // log(1.04297), 1/1.04297 .quad 0x3fa77458f632dcfc, 0x3fee9131abf0b767 // log(1.04688), 1/1.04688 .quad 0x3fa95c830ec8e3eb, 0x3fee741aa59750e4 // log(1.05078), 1/1.05078 .quad 0x3fab42dd711971bf, 0x3fee573ac901e574 // log(1.05469), 1/1.05469 .quad 0x3fad276b8adb0b52, 0x3fee3a9179dc1a73 // log(1.05859), 1/1.05859 .quad 0x3faf0a30c01162a6, 0x3fee1e1e1e1e1e1e // log(1.06250), 1/1.06250 .quad 0x3fb075983598e471, 0x3fee01e01e01e01e // log(1.06641), 1/1.06641 .quad 0x3fb16536eea37ae1, 0x3fede5d6e3f8868a // log(1.07031), 1/1.07031 .quad 0x3fb253f62f0a1417, 0x3fedca01dca01dca // log(1.07422), 1/1.07422 .quad 0x3fb341d7961bd1d1, 0x3fedae6076b981db // log(1.07812), 1/1.07812 .quad 0x3fb42edcbea646f0, 0x3fed92f2231e7f8a // log(1.08203), 1/1.08203 .quad 0x3fb51b073f06183f, 0x3fed77b654b82c34 // log(1.08594), 1/1.08594 .quad 0x3fb60658a93750c4, 0x3fed5cac807572b2 // log(1.08984), 1/1.08984 .quad 0x3fb6f0d28ae56b4c, 0x3fed41d41d41d41d // log(1.09375), 1/1.09375 .quad 0x3fb7da766d7b12cd, 0x3fed272ca3fc5b1a // log(1.09766), 1/1.09766 .quad 0x3fb8c345d6319b21, 0x3fed0cb58f6ec074 // log(1.10156), 1/1.10156 .quad 0x3fb9ab42462033ad, 0x3fecf26e5c44bfc6 // log(1.10547), 1/1.10547 .quad 0x3fba926d3a4ad563, 0x3fecd85689039b0b // log(1.10938), 1/1.10938 .quad 0x3fbb78c82bb0eda1, 0x3fecbe6d9601cbe7 // log(1.11328), 1/1.11328 .quad 0x3fbc5e548f5bc743, 0x3feca4b3055ee191 // log(1.11719), 1/1.11719 .quad 0x3fbd4313d66cb35d, 0x3fec8b265afb8a42 // log(1.12109), 1/1.12109 .quad 0x3fbe27076e2af2e6, 0x3fec71c71c71c71c // log(1.12500), 1/1.12500 .quad 0x3fbf0a30c01162a6, 0x3fec5894d10d4986 // log(1.12891), 1/1.12891 .quad 0x3fbfec9131dbeabb, 0x3fec3f8f01c3f8f0 // log(1.13281), 1/1.13281 .quad 0x3fc0671512ca596e, 0x3fec26b5392ea01c // log(1.13672), 1/1.13672 .quad 0x3fc0d77e7cd08e59, 0x3fec0e070381c0e0 // log(1.14062), 1/1.14062 .quad 0x3fc14785846742ac, 0x3febf583ee868d8b // log(1.14453), 1/1.14453 .quad 0x3fc1b72ad52f67a0, 0x3febdd2b899406f7 // log(1.14844), 1/1.14844 .quad 0x3fc2266f190a5acb, 0x3febc4fd65883e7b // log(1.15234), 1/1.15234 .quad 0x3fc29552f81ff523, 0x3febacf914c1bad0 // log(1.15625), 1/1.15625 .quad 0x3fc303d718e47fd3, 0x3feb951e2b18ff23 // log(1.16016), 1/1.16016 .quad 0x3fc371fc201e8f74, 0x3feb7d6c3dda338b // log(1.16406), 1/1.16406 .quad 0x3fc3dfc2b0ecc62a, 0x3feb65e2e3beee05 // log(1.16797), 1/1.16797 .quad 0x3fc44d2b6ccb7d1e, 0x3feb4e81b4e81b4f // log(1.17188), 1/1.17188 .quad 0x3fc4ba36f39a55e5, 0x3feb37484ad806ce // log(1.17578), 1/1.17578 .quad 0x3fc526e5e3a1b438, 0x3feb2036406c80d9 // log(1.17969), 1/1.17969 .quad 0x3fc59338d9982086, 0x3feb094b31d922a4 // log(1.18359), 1/1.18359 .quad 0x3fc5ff3070a793d4, 0x3feaf286bca1af28 // log(1.18750), 1/1.18750 .quad 0x3fc66acd4272ad51, 0x3feadbe87f94905e // log(1.19141), 1/1.19141 .quad 0x3fc6d60fe719d21d, 0x3feac5701ac5701b // log(1.19531), 1/1.19531 .quad 0x3fc740f8f54037a5, 0x3feaaf1d2f87ebfd // log(1.19922), 1/1.19922 .quad 0x3fc7ab890210d909, 0x3fea98ef606a63be // log(1.20312), 1/1.20312 .quad 0x3fc815c0a14357eb, 0x3fea82e65130e159 // log(1.20703), 1/1.20703 .quad 0x3fc87fa06520c911, 0x3fea6d01a6d01a6d // log(1.21094), 1/1.21094 .quad 0x3fc8e928de886d41, 0x3fea574107688a4a // log(1.21484), 1/1.21484 .quad 0x3fc9525a9cf456b4, 0x3fea41a41a41a41a // log(1.21875), 1/1.21875 .quad 0x3fc9bb362e7dfb83, 0x3fea2c2a87c51ca0 // log(1.22266), 1/1.22266 .quad 0x3fca23bc1fe2b563, 0x3fea16d3f97a4b02 // log(1.22656), 1/1.22656 .quad 0x3fca8becfc882f19, 0x3fea01a01a01a01a // log(1.23047), 1/1.23047 .quad 0x3fcaf3c94e80bff3, 0x3fe9ec8e951033d9 // log(1.23438), 1/1.23438 .quad 0x3fcb5b519e8fb5a4, 0x3fe9d79f176b682d // log(1.23828), 1/1.23828 .quad 0x3fcbc286742d8cd6, 0x3fe9c2d14ee4a102 // log(1.24219), 1/1.24219 .quad 0x3fcc2968558c18c1, 0x3fe9ae24ea5510da // log(1.24609), 1/1.24609 .quad 0x3fcc8ff7c79a9a22, 0x3fe999999999999a // log(1.25000), 1/1.25000 .quad 0x3fccf6354e09c5dc, 0x3fe9852f0d8ec0ff // log(1.25391), 1/1.25391 .quad 0x3fcd5c216b4fbb91, 0x3fe970e4f80cb872 // log(1.25781), 1/1.25781 .quad 0x3fcdc1bca0abec7d, 0x3fe95cbb0be377ae // log(1.26172), 1/1.26172 .quad 0x3fce27076e2af2e6, 0x3fe948b0fcd6e9e0 // log(1.26562), 1/1.26562 .quad 0x3fce8c0252aa5a60, 0x3fe934c67f9b2ce6 // log(1.26953), 1/1.26953 .quad 0x3fcef0adcbdc5936, 0x3fe920fb49d0e229 // log(1.27344), 1/1.27344 .quad 0x3fcf550a564b7b37, 0x3fe90d4f120190d5 // log(1.27734), 1/1.27734 .quad 0x3fcfb9186d5e3e2b, 0x3fe8f9c18f9c18fa // log(1.28125), 1/1.28125 .quad 0x3fd00e6c45ad501d, 0x3fe8e6527af1373f // log(1.28516), 1/1.28516 .quad 0x3fd0402594b4d041, 0x3fe8d3018d3018d3 // log(1.28906), 1/1.28906 .quad 0x3fd071b85fcd590d, 0x3fe8bfce8062ff3a // log(1.29297), 1/1.29297 .quad 0x3fd0a324e27390e3, 0x3fe8acb90f6bf3aa // log(1.29688), 1/1.29688 .quad 0x3fd0d46b579ab74b, 0x3fe899c0f601899c // log(1.30078), 1/1.30078 .quad 0x3fd1058bf9ae4ad5, 0x3fe886e5f0abb04a // log(1.30469), 1/1.30469 .quad 0x3fd136870293a8b0, 0x3fe87427bcc092b9 // log(1.30859), 1/1.30859 .quad 0x3fd1675cababa60e, 0x3fe8618618618618 // log(1.31250), 1/1.31250 .quad 0x3fd1980d2dd4236f, 0x3fe84f00c2780614 // log(1.31641), 1/1.31641 .quad 0x3fd1c898c16999fb, 0x3fe83c977ab2bedd // log(1.32031), 1/1.32031 .quad 0x3fd1f8ff9e48a2f3, 0x3fe82a4a0182a4a0 // log(1.32422), 1/1.32422 .quad 0x3fd22941fbcf7966, 0x3fe8181818181818 // log(1.32812), 1/1.32812 .quad 0x3fd2596010df763a, 0x3fe8060180601806 // log(1.33203), 1/1.33203 .quad 0x3fd2895a13de86a3, 0x3fe7f405fd017f40 // log(1.33594), 1/1.33594 .quad 0x3fd2b9303ab89d25, 0x3fe7e225515a4f1d // log(1.33984), 1/1.33984 .quad 0x3fd2e8e2bae11d31, 0x3fe7d05f417d05f4 // log(1.34375), 1/1.34375 .quad 0x3fd31871c9544185, 0x3fe7beb3922e017c // log(1.34766), 1/1.34766 .quad 0x3fd347dd9a987d55, 0x3fe7ad2208e0ecc3 // log(1.35156), 1/1.35156 .quad 0x3fd3772662bfd85b, 0x3fe79baa6bb6398b // log(1.35547), 1/1.35547 .quad 0x3fd3a64c556945ea, 0x3fe78a4c8178a4c8 // log(1.35938), 1/1.35938 .quad 0x3fd3d54fa5c1f710, 0x3fe77908119ac60d // log(1.36328), 1/1.36328 .quad 0x3fd404308686a7e4, 0x3fe767dce434a9b1 // log(1.36719), 1/1.36719 .quad 0x3fd432ef2a04e814, 0x3fe756cac201756d // log(1.37109), 1/1.37109 .quad 0x3fd4618bc21c5ec2, 0x3fe745d1745d1746 // log(1.37500), 1/1.37500 .quad 0x3fd49006804009d1, 0x3fe734f0c541fe8d // log(1.37891), 1/1.37891 .quad 0x3fd4be5f957778a1, 0x3fe724287f46debc // log(1.38281), 1/1.38281 .quad 0x3fd4ec973260026a, 0x3fe713786d9c7c09 // log(1.38672), 1/1.38672 .quad 0x3fd51aad872df82d, 0x3fe702e05c0b8170 // log(1.39062), 1/1.39062 .quad 0x3fd548a2c3add263, 0x3fe6f26016f26017 // log(1.39453), 1/1.39453 .quad 0x3fd5767717455a6c, 0x3fe6e1f76b4337c7 // log(1.39844), 1/1.39844 .quad 0x3fd5a42ab0f4cfe2, 0x3fe6d1a62681c861 // log(1.40234), 1/1.40234 .quad 0x3fd5d1bdbf5809ca, 0x3fe6c16c16c16c17 // log(1.40625), 1/1.40625 .quad 0x3fd5ff3070a793d4, 0x3fe6b1490aa31a3d // log(1.41016), 1/1.41016 .quad 0x3fd62c82f2b9c795, 0x3fe6a13cd1537290 // log(1.41406), 1/1.41406 .quad 0x3fd659b57303e1f3, 0x3fe691473a88d0c0 // log(1.41797), 1/1.41797 .quad 0x3fd686c81e9b14af, 0x3fe6816816816817 // log(1.42188), 1/1.42188 .quad 0x3fd6b3bb2235943e, 0x3fe6719f3601671a // log(1.42578), 1/1.42578 .quad 0x3fd6e08eaa2ba1e4, 0x3fe661ec6a5122f9 // log(1.42969), 1/1.42969 .quad 0x3fd70d42e2789236, 0x3fe6524f853b4aa3 // log(1.43359), 1/1.43359 .quad 0x3fd739d7f6bbd007, 0x3fe642c8590b2164 // log(1.43750), 1/1.43750 .quad 0x3fd7664e1239dbcf, 0x3fe63356b88ac0de // log(1.44141), 1/1.44141 .quad 0x3fd792a55fdd47a2, 0x3fe623fa77016240 // log(1.44531), 1/1.44531 .quad 0x3fd7bede0a37afc0, 0x3fe614b36831ae94 // log(1.44922), 1/1.44922 .quad 0x3fd7eaf83b82afc3, 0x3fe6058160581606 // log(1.45312), 1/1.45312 .quad 0x3fd816f41da0d496, 0x3fe5f66434292dfc // log(1.45703), 1/1.45703 .quad 0x3fd842d1da1e8b17, 0x3fe5e75bb8d015e7 // log(1.46094), 1/1.46094 .quad 0x3fd86e919a330ba0, 0x3fe5d867c3ece2a5 // log(1.46484), 1/1.46484 .quad 0x3fd89a3386c1425b, 0x3fe5c9882b931057 // log(1.46875), 1/1.46875 .quad 0x3fd8c5b7c858b48b, 0x3fe5babcc647fa91 // log(1.47266), 1/1.47266 .quad 0x3fd8f11e873662c7, 0x3fe5ac056b015ac0 // log(1.47656), 1/1.47656 .quad 0x3fd91c67eb45a83e, 0x3fe59d61f123ccaa // log(1.48047), 1/1.48047 .quad 0x3fd947941c2116fb, 0x3fe58ed2308158ed // log(1.48438), 1/1.48438 .quad 0x3fd972a341135158, 0x3fe5805601580560 // log(1.48828), 1/1.48828 .quad 0x3fd99d958117e08b, 0x3fe571ed3c506b3a // log(1.49219), 1/1.49219 .quad 0x3fd9c86b02dc0863, 0x3fe56397ba7c52e2 // log(1.49609), 1/1.49609 .quad 0x3fd9f323ecbf984c, 0x3fe5555555555555 // log(1.50000), 1/1.50000 .quad 0x3fda1dc064d5b995, 0x3fe54725e6bb82fe // log(1.50391), 1/1.50391 .quad 0x3fda484090e5bb0a, 0x3fe5390948f40feb // log(1.50781), 1/1.50781 .quad 0x3fda72a4966bd9ea, 0x3fe52aff56a8054b // log(1.51172), 1/1.51172 .quad 0x3fda9cec9a9a084a, 0x3fe51d07eae2f815 // log(1.51562), 1/1.51562 .quad 0x3fdac718c258b0e4, 0x3fe50f22e111c4c5 // log(1.51953), 1/1.51953 .quad 0x3fdaf1293247786b, 0x3fe5015015015015 // log(1.52344), 1/1.52344 .quad 0x3fdb1b1e0ebdfc5b, 0x3fe4f38f62dd4c9b // log(1.52734), 1/1.52734 .quad 0x3fdb44f77bcc8f63, 0x3fe4e5e0a72f0539 // log(1.53125), 1/1.53125 .quad 0x3fdb6eb59d3cf35e, 0x3fe4d843bedc2c4c // log(1.53516), 1/1.53516 .quad 0x3fdb9858969310fb, 0x3fe4cab88725af6e // log(1.53906), 1/1.53906 .quad 0x3fdbc1e08b0dad0a, 0x3fe4bd3edda68fe1 // log(1.54297), 1/1.54297 .quad 0x3fdbeb4d9da71b7c, 0x3fe4afd6a052bf5b // log(1.54688), 1/1.54688 .quad 0x3fdc149ff115f027, 0x3fe4a27fad76014a // log(1.55078), 1/1.55078 .quad 0x3fdc3dd7a7cdad4d, 0x3fe49539e3b2d067 // log(1.55469), 1/1.55469 .quad 0x3fdc66f4e3ff6ff8, 0x3fe4880522014880 // log(1.55859), 1/1.55859 .quad 0x3fdc8ff7c79a9a22, 0x3fe47ae147ae147b // log(1.56250), 1/1.56250 .quad 0x3fdcb8e0744d7aca, 0x3fe46dce34596066 // log(1.56641), 1/1.56641 .quad 0x3fdce1af0b85f3eb, 0x3fe460cbc7f5cf9a // log(1.57031), 1/1.57031 .quad 0x3fdd0a63ae721e64, 0x3fe453d9e2c776ca // log(1.57422), 1/1.57422 .quad 0x3fdd32fe7e00ebd5, 0x3fe446f86562d9fb // log(1.57812), 1/1.57812 .quad 0x3fdd5b7f9ae2c684, 0x3fe43a2730abee4d // log(1.58203), 1/1.58203 .quad 0x3fdd83e7258a2f3e, 0x3fe42d6625d51f87 // log(1.58594), 1/1.58594 .quad 0x3fddac353e2c5954, 0x3fe420b5265e5951 // log(1.58984), 1/1.58984 .quad 0x3fddd46a04c1c4a1, 0x3fe4141414141414 // log(1.59375), 1/1.59375 .quad 0x3fddfc859906d5b5, 0x3fe40782d10e6566 // log(1.59766), 1/1.59766 .quad 0x3fde24881a7c6c26, 0x3fe3fb013fb013fb // log(1.60156), 1/1.60156 .quad 0x3fde4c71a8687704, 0x3fe3ee8f42a5af07 // log(1.60547), 1/1.60547 .quad 0x3fde744261d68788, 0x3fe3e22cbce4a902 // log(1.60938), 1/1.60938 .quad 0x3fde9bfa659861f5, 0x3fe3d5d991aa75c6 // log(1.61328), 1/1.61328 .quad 0x3fdec399d2468cc0, 0x3fe3c995a47babe7 // log(1.61719), 1/1.61719 .quad 0x3fdeeb20c640ddf4, 0x3fe3bd60d9232955 // log(1.62109), 1/1.62109 .quad 0x3fdf128f5faf06ed, 0x3fe3b13b13b13b14 // log(1.62500), 1/1.62500 .quad 0x3fdf39e5bc811e5c, 0x3fe3a524387ac822 // log(1.62891), 1/1.62891 .quad 0x3fdf6123fa7028ac, 0x3fe3991c2c187f63 // log(1.63281), 1/1.63281 .quad 0x3fdf884a36fe9ec2, 0x3fe38d22d366088e // log(1.63672), 1/1.63672 .quad 0x3fdfaf588f78f31f, 0x3fe3813813813814 // log(1.64062), 1/1.64062 .quad 0x3fdfd64f20f61572, 0x3fe3755bd1c945ee // log(1.64453), 1/1.64453 .quad 0x3fdffd2e0857f498, 0x3fe3698df3de0748 // log(1.64844), 1/1.64844 .quad 0x3fe011fab125ff8a, 0x3fe35dce5f9f2af8 // log(1.65234), 1/1.65234 .quad 0x3fe02552a5a5d0ff, 0x3fe3521cfb2b78c1 // log(1.65625), 1/1.65625 .quad 0x3fe0389eefce633b, 0x3fe34679ace01346 // log(1.66016), 1/1.66016 .quad 0x3fe04bdf9da926d2, 0x3fe33ae45b57bcb2 // log(1.66406), 1/1.66406 .quad 0x3fe05f14bd26459c, 0x3fe32f5ced6a1dfa // log(1.66797), 1/1.66797 .quad 0x3fe0723e5c1cdf40, 0x3fe323e34a2b10bf // log(1.67188), 1/1.67188 .quad 0x3fe0855c884b450e, 0x3fe3187758e9ebb6 // log(1.67578), 1/1.67578 .quad 0x3fe0986f4f573521, 0x3fe30d190130d190 // log(1.67969), 1/1.67969 .quad 0x3fe0ab76bece14d2, 0x3fe301c82ac40260 // log(1.68359), 1/1.68359 .quad 0x3fe0be72e4252a83, 0x3fe2f684bda12f68 // log(1.68750), 1/1.68750 .quad 0x3fe0d163ccb9d6b8, 0x3fe2eb4ea1fed14b // log(1.69141), 1/1.69141 .quad 0x3fe0e44985d1cc8c, 0x3fe2e025c04b8097 // log(1.69531), 1/1.69531 .quad 0x3fe0f7241c9b497d, 0x3fe2d50a012d50a0 // log(1.69922), 1/1.69922 .quad 0x3fe109f39e2d4c97, 0x3fe2c9fb4d812ca0 // log(1.70312), 1/1.70312 .quad 0x3fe11cb81787ccf8, 0x3fe2bef98e5a3711 // log(1.70703), 1/1.70703 .quad 0x3fe12f719593efbc, 0x3fe2b404ad012b40 // log(1.71094), 1/1.71094 .quad 0x3fe1422025243d45, 0x3fe2a91c92f3c105 // log(1.71484), 1/1.71484 .quad 0x3fe154c3d2f4d5ea, 0x3fe29e4129e4129e // log(1.71875), 1/1.71875 .quad 0x3fe1675cababa60e, 0x3fe293725bb804a5 // log(1.72266), 1/1.72266 .quad 0x3fe179eabbd899a1, 0x3fe288b01288b013 // log(1.72656), 1/1.72656 .quad 0x3fe18c6e0ff5cf06, 0x3fe27dfa38a1ce4d // log(1.73047), 1/1.73047 .quad 0x3fe19ee6b467c96f, 0x3fe27350b8812735 // log(1.73438), 1/1.73438 .quad 0x3fe1b154b57da29f, 0x3fe268b37cd60127 // log(1.73828), 1/1.73828 .quad 0x3fe1c3b81f713c25, 0x3fe25e22708092f1 // log(1.74219), 1/1.74219 .quad 0x3fe1d610fe677003, 0x3fe2539d7e9177b2 // log(1.74609), 1/1.74609 .quad 0x3fe1e85f5e7040d0, 0x3fe2492492492492 // log(1.75000), 1/1.75000 .quad 0x3fe1faa34b87094c, 0x3fe23eb79717605b // log(1.75391), 1/1.75391 .quad 0x3fe20cdcd192ab6e, 0x3fe23456789abcdf // log(1.75781), 1/1.75781 .quad 0x3fe21f0bfc65beec, 0x3fe22a0122a0122a // log(1.76172), 1/1.76172 .quad 0x3fe23130d7bebf43, 0x3fe21fb78121fb78 // log(1.76562), 1/1.76562 .quad 0x3fe2434b6f483934, 0x3fe21579804855e6 // log(1.76953), 1/1.76953 .quad 0x3fe2555bce98f7cb, 0x3fe20b470c67c0d9 // log(1.77344), 1/1.77344 .quad 0x3fe26762013430e0, 0x3fe2012012012012 // log(1.77734), 1/1.77734 .quad 0x3fe2795e1289b11b, 0x3fe1f7047dc11f70 // log(1.78125), 1/1.78125 .quad 0x3fe28b500df60783, 0x3fe1ecf43c7fb84c // log(1.78516), 1/1.78516 .quad 0x3fe29d37fec2b08b, 0x3fe1e2ef3b3fb874 // log(1.78906), 1/1.78906 .quad 0x3fe2af15f02640ad, 0x3fe1d8f5672e4abd // log(1.79297), 1/1.79297 .quad 0x3fe2c0e9ed448e8c, 0x3fe1cf06ada2811d // log(1.79688), 1/1.79688 .quad 0x3fe2d2b4012edc9e, 0x3fe1c522fc1ce059 // log(1.80078), 1/1.80078 .quad 0x3fe2e47436e40268, 0x3fe1bb4a4046ed29 // log(1.80469), 1/1.80469 .quad 0x3fe2f62a99509546, 0x3fe1b17c67f2bae3 // log(1.80859), 1/1.80859 .quad 0x3fe307d7334f10be, 0x3fe1a7b9611a7b96 // log(1.81250), 1/1.81250 .quad 0x3fe3197a0fa7fe6a, 0x3fe19e0119e0119e // log(1.81641), 1/1.81641 .quad 0x3fe32b1339121d71, 0x3fe19453808ca29c // log(1.82031), 1/1.82031 .quad 0x3fe33ca2ba328995, 0x3fe18ab083902bdb // log(1.82422), 1/1.82422 .quad 0x3fe34e289d9ce1d3, 0x3fe1811811811812 // log(1.82812), 1/1.82812 .quad 0x3fe35fa4edd36ea0, 0x3fe1778a191bd684 // log(1.83203), 1/1.83203 .quad 0x3fe37117b54747b6, 0x3fe16e0689427379 // log(1.83594), 1/1.83594 .quad 0x3fe38280fe58797f, 0x3fe1648d50fc3201 // log(1.83984), 1/1.83984 .quad 0x3fe393e0d3562a1a, 0x3fe15b1e5f75270d // log(1.84375), 1/1.84375 .quad 0x3fe3a5373e7ebdfa, 0x3fe151b9a3fdd5c9 // log(1.84766), 1/1.84766 .quad 0x3fe3b68449fffc23, 0x3fe1485f0e0acd3b // log(1.85156), 1/1.85156 .quad 0x3fe3c7c7fff73206, 0x3fe13f0e8d344724 // log(1.85547), 1/1.85547 .quad 0x3fe3d9026a7156fb, 0x3fe135c81135c811 // log(1.85938), 1/1.85938 .quad 0x3fe3ea33936b2f5c, 0x3fe12c8b89edc0ac // log(1.86328), 1/1.86328 .quad 0x3fe3fb5b84d16f42, 0x3fe12358e75d3033 // log(1.86719), 1/1.86719 .quad 0x3fe40c7a4880dce9, 0x3fe11a3019a74826 // log(1.87109), 1/1.87109 .quad 0x3fe41d8fe84672ae, 0x3fe1111111111111 // log(1.87500), 1/1.87500 .quad 0x3fe42e9c6ddf80bf, 0x3fe107fbbe011080 // log(1.87891), 1/1.87891 .quad 0x3fe43f9fe2f9ce67, 0x3fe0fef010fef011 // log(1.88281), 1/1.88281 .quad 0x3fe4509a5133bb0a, 0x3fe0f5edfab325a2 // log(1.88672), 1/1.88672 .quad 0x3fe4618bc21c5ec2, 0x3fe0ecf56be69c90 // log(1.89062), 1/1.89062 .quad 0x3fe472743f33aaad, 0x3fe0e40655826011 // log(1.89453), 1/1.89453 .quad 0x3fe48353d1ea88df, 0x3fe0db20a88f4696 // log(1.89844), 1/1.89844 .quad 0x3fe4942a83a2fc07, 0x3fe0d24456359e3a // log(1.90234), 1/1.90234 .quad 0x3fe4a4f85db03ebb, 0x3fe0c9714fbcda3b // log(1.90625), 1/1.90625 .quad 0x3fe4b5bd6956e274, 0x3fe0c0a7868b4171 // log(1.91016), 1/1.91016 .quad 0x3fe4c679afccee3a, 0x3fe0b7e6ec259dc8 // log(1.91406), 1/1.91406 .quad 0x3fe4d72d3a39fd00, 0x3fe0af2f722eecb5 // log(1.91797), 1/1.91797 .quad 0x3fe4e7d811b75bb1, 0x3fe0a6810a6810a7 // log(1.92188), 1/1.92188 .quad 0x3fe4f87a3f5026e9, 0x3fe09ddba6af8360 // log(1.92578), 1/1.92578 .quad 0x3fe50913cc01686b, 0x3fe0953f39010954 // log(1.92969), 1/1.92969 .quad 0x3fe519a4c0ba3446, 0x3fe08cabb37565e2 // log(1.93359), 1/1.93359 .quad 0x3fe52a2d265bc5ab, 0x3fe0842108421084 // log(1.93750), 1/1.93750 .quad 0x3fe53aad05b99b7d, 0x3fe07b9f29b8eae2 // log(1.94141), 1/1.94141 .quad 0x3fe54b2467999498, 0x3fe073260a47f7c6 // log(1.94531), 1/1.94531 .quad 0x3fe55b9354b40bcd, 0x3fe06ab59c7912fb // log(1.94922), 1/1.94922 .quad 0x3fe56bf9d5b3f399, 0x3fe0624dd2f1a9fc // log(1.95312), 1/1.95312 .quad 0x3fe57c57f336f191, 0x3fe059eea0727586 // log(1.95703), 1/1.95703 .quad 0x3fe58cadb5cd7989, 0x3fe05197f7d73404 // log(1.96094), 1/1.96094 .quad 0x3fe59cfb25fae87e, 0x3fe04949cc1664c5 // log(1.96484), 1/1.96484 .quad 0x3fe5ad404c359f2d, 0x3fe0410410410410 // log(1.96875), 1/1.96875 .quad 0x3fe5bd7d30e71c73, 0x3fe038c6b78247fc // log(1.97266), 1/1.97266 .quad 0x3fe5cdb1dc6c1765, 0x3fe03091b51f5e1a // log(1.97656), 1/1.97656 .quad 0x3fe5ddde57149923, 0x3fe02864fc7729e9 // log(1.98047), 1/1.98047 .quad 0x3fe5ee02a9241675, 0x3fe0204081020408 // log(1.98438), 1/1.98438 .quad 0x3fe5fe1edad18919, 0x3fe0182436517a37 // log(1.98828), 1/1.98828 .quad 0x3fe60e32f44788d9, 0x3fe0101010101010 // log(1.99219), 1/1.99219 .quad 0x3fe61e3efda46467, 0x3fe0080402010080 // log(1.99609), 1/1.99609
acoshf_polys: .quad 0x40582c7cd80ad8f9, 0x4045c4c4d92795e2 // p01 q01 .quad 0x408b448d184fb3a1, 0x4070c2cc9992a6be // p00 q00 .quad 0x40122cdb2289b638, 0x401034288b4b598e // p11 q11 .quad 0x40119b927d52a518, 0x400d0166c1b45b07 // p10 q10 .quad 0xc00aaaaabc2aaaaf, 0x4013ffffed400007 // s1 s0 .quad 0x3fc316216fa0e2f6 // c
.literal8.align 3one: .quad 0x3ff0000000000000onehalf: .quad 0x3fe0000000000000onethird: .quad 0x3fd5555555555555log2: .quad 0x3fe62e42fefa39ef // ln(2)
.literal4onef: .long 0x3f800000
.text.align 4#if defined( __x86_64__ ) #define RELATIVE_ADDR( _a ) (_a)( %rip )#else #define RELATIVE_ADDR( _a ) (_a)-acoshf_body( %ecx )acoshf_pic: movl (%esp), %ecx // Copy address of this instruction to %ecx ret#endif
ENTRY(acoshf)#if defined(__i386__) movl FRAME_SIZE(STACKP), %eax movss FRAME_SIZE(STACKP), %xmm0 calll acoshf_picacoshf_body:#else movd %xmm0, %eax#endif subl $0x3f800001, %eax // eax <-- x - 1+ulp as integers cmpl $0x010fffff, %eax // if ( x > 4 or x < 1+ulp or isnan(x) ) ja 2f // goto 2 // 1 + ulp < |x| < 4 // approximation of the form c*sqrt(x-1)*p(x)/q(x) cvtss2sd %xmm0, %xmm3 // xmm3 <-- (double)x subss RELATIVE_ADDR(onef), %xmm0 // xmm0 <-- x - 1 cvtss2sd %xmm0, %xmm1 // xmm1 <-- x - 1 lea RELATIVE_ADDR(acoshf_polys), AX_P unpcklpd %xmm3, %xmm3
rsqrtss %xmm0, %xmm0 // xmm0 <-- e = approximate reciprical sqrt of (x-1) cvtss2sd %xmm0, %xmm0 movsd 64(AX_P), %xmm2 mulsd %xmm0, %xmm1 // xmm0 <-- e(x-1) mulsd %xmm1, %xmm0 // xmm1 <-- e^2(x-1) mulsd 80(AX_P), %xmm1 // xmm1 <-- c(e(x-1)) addsd %xmm0, %xmm2 // xmm2 <-- e^2(x-1) + s1 mulsd %xmm2, %xmm0 // xmm0 <-- (e^2(x-1))^2 + s1(e^2(x-1) addsd 72(AX_P), %xmm0 // xmm0 <-- (e^2(x-1))^2 + s1(e^2(x-1) + s0 mulsd %xmm1, %xmm0 // xmm0 <-- c*sqrt(x-1) movapd (AX_P), %xmm4 // xmm4 <-- [p01, q01] movapd 32(AX_P), %xmm5 // xmm5 <-- [p11, q11] addpd %xmm3, %xmm4 // xmm4 <-- [x + p01, x + q01] addpd %xmm3, %xmm5 // xmm5 <-- [x + p11, x + q11] mulpd %xmm3, %xmm4 // xmm4 <-- [x^2 + p01x, x^2 + q01x] mulpd %xmm3, %xmm5 // xmm5 <-- [x^2 + p11x, x^2 + q11x] addpd 16(AX_P), %xmm4 // xmm4 <-- [p0(x), q0(x)] addpd 48(AX_P), %xmm5 // xmm5 <-- [p1(x), q1(x)] mulpd %xmm5, %xmm4 // xmm4 <-- [ p(x), q(x) ] mulsd %xmm4, %xmm0 // xmm0 <-- c*sqrt(x-1)*p(x) movhlps %xmm4, %xmm1 // xmm1 <-- q(x) divsd %xmm1, %xmm0 // xmm0 <-- c*sqrt(x-1)*p(x) / q(x) cvtsd2ss %xmm0, %xmm0#if defined(__i386__) movss %xmm0, FRAME_SIZE( STACKP ) flds FRAME_SIZE( STACKP )#endif ret
2: cmpl $0x3fffffff, %eax // if ( x < 1 + ulp or x = ±∞ or isnan(x) ) jae 3f // goto 3
// Go off and compute the reciprocal of |x|. This will take a little while, but we have other stuff to do// while this is up in the air. cvtss2sd %xmm0, %xmm2 xorpd %xmm1, %xmm1 movsd RELATIVE_ADDR(one), %xmm1 divsd %xmm2, %xmm1 // xmm1 <-- 1/x // We need to take a logarithm for this case. This code is shamelessly taken from our log2f.s implementation.// Consult that file for more details of operation.
// begin by factoring x as 2^n*(mantissa)
movd %xmm0, %eax movl %eax, %edx shrl $23, %edx // right-align exponent bits subl $126, %edx // subtract exponent bias, but add one to get log(2x) instead of log(x) // i.e. we subtract 126 instead of 127 (the actual bias). cvtsi2sd %edx, %xmm2 // xmm2 <-- trunc(lg |x|) = n mulsd RELATIVE_ADDR(log2), %xmm2 // xmm2 <-- log(2^n) // now, let mantissa = 1 + a + b = (1 + a)(1 + r), if r = b/(1+a), and//// log(mantissa) = log(1+a) + log(1+r).//// we look up log(1+a) and 1/(1+a) in a table, keyed from a, and we compute log(1+r) via taylor series. movl %eax, %edx andl $0x00007fff, %eax // eax <-- b andl $0x007f8000, %edx // edx <-- a << 15 orl $0x3f800000, %eax // eax <-- 1 + b shrl $(15-4), %edx // edx <-- a << 4, lookup key for log(1+a) movd %eax, %xmm3 // xmm3 <-- 1 + b cvtss2sd %xmm3, %xmm3 subsd RELATIVE_ADDR(one), %xmm3 // xmm3 <-- b lea RELATIVE_ADDR(acoshf_hi_table), AX_P mulsd 8(AX_P, DX_P, 1), %xmm3 // xmm3 <-- r = b * 1/(1+a) movapd %xmm3, %xmm4 mulsd RELATIVE_ADDR(onethird),%xmm3 // xmm3 <-- 1/3 r movapd %xmm4, %xmm5 // xmm5 <-- r mulsd %xmm4, %xmm4 // xmm4 <-- r^2 subsd RELATIVE_ADDR(onehalf), %xmm3 // xmm3 <-- 1/3 r - 1/2 mulsd %xmm4, %xmm3 // xmm3 <-- 1/3 r^3 - 1/2 r^2 addsd %xmm5, %xmm3 // xmm3 <-- 1/3 r^3 - 1/2 r^2 + r ~ log(1+r) addsd (AX_P, DX_P, 1), %xmm3 // xmm3 <-- log(a) + log(1+r) addsd %xmm2, %xmm3 // xmm3 <-- log(2x) with error < .5002 ulps // Now we compute a correction from a polynomial in u = 1/|x|, which conveniently has landed in %xmm1 about now. movapd -16(AX_P), %xmm6#if defined( __SSE3__ ) movddup %xmm1, %xmm0#else movapd %xmm1, %xmm0 unpcklpd %xmm0, %xmm0#endif addpd %xmm0, %xmm6 // xmm6 <-- [ u + a1 , u + b1 ] addsd -48(AX_P), %xmm1 // xmm1 <-- [ u + c1 , 0 ] mulpd %xmm0, %xmm6 // xmm6 <-- [ u^2 + a1u , u^2 + b1u ] mulpd %xmm1, %xmm0 // xmm0 <-- [ u^2 + c1u , 0 ] addpd -32(AX_P), %xmm6 // xmm6 <-- [ u^2 + a1u + a0 , u^2 + b1u + b0 ] addpd -64(AX_P), %xmm0 // xmm0 <-- [ u^2 + c1u + c0 , d ] mulpd %xmm6, %xmm0 movhlps %xmm0, %xmm1 mulsd %xmm1, %xmm0 addsd %xmm0, %xmm3 cvtsd2ss %xmm3, %xmm0#if defined(__i386__) movss %xmm0, FRAME_SIZE( STACKP ) flds FRAME_SIZE( STACKP )#endif ret
3: addl $1, %eax // eax <-- x - 1 cmpl $0x40000000, %eax jae 4f // Handle x = 1 case here, zero out xmm0 and return. xorps %xmm0, %xmm0 #if defined(__i386__) movss %xmm0, FRAME_SIZE( STACKP ) flds FRAME_SIZE( STACKP )#endif ret
4: je 6f // if x == Inf, goto 6 ucomiss %xmm0, %xmm0 // check for NaN jp 5f pcmpeqb %xmm0, %xmm0 pslld $23, %xmm05: movaps %xmm0, %xmm1 subss %xmm1, %xmm06:#if defined(__i386__) movss %xmm0, FRAME_SIZE( STACKP ) flds FRAME_SIZE( STACKP )#endif ret