/**************************************************************************** tanh.s by Stephen Canon Copyright © 2007, Apple Inc. All Rights Reserved. Implementation of the C99 tanh function for the MacOS X i386 and x86_64 ABIs. ****************************************************************************/ #include <machine/asm.h> #include "abi.h" /**************************************************************************** Tables and Constants ****************************************************************************/ .const .align 4 abs_mask: .quad 0x7fffffffffffffff, 0x0000000000000000 four_exp: .quad 0x0020000000000000, 0x0000000000000000 one: .quad 0x3ff0000000000000, 0x0000000000000000 // Coefficients for minimax approximation in 2 exp(-2x) on [2,22) .quad 0xbf8dc165dad1eab3 .quad 0x3f9fef26c0bbebc1 .quad 0xbfafffbdaa5e7368 .quad 0x3fbfffff6ee83673 .quad 0xbfcfffffff536d53 .quad 0x3fdfffffffff9b22 .quad 0x3ce6000000000000 // Coefficients for the 4th order minimax polynomial // approximation of exp2( ) on the interval [-0x1p-27, 0x1p-7] .quad 0x3f55e52272e0eaec // c4 .quad 0x3f83b2a8abda3d8f // c3 .quad 0x3fac6b08d78310b8 // c2 .quad 0x3fcebfbdff82bda7 // c1 .quad 0x3fe62e42fefa39ef // c0 // table of values of x for which exp2(x) is very precise in // double precision constructed by J. Kidder. // x exp2(x) exp2table: .quad 0xbfe000000005f3f1, 0x3fe6a09e667c50df // x = -0x1.000000005f3f1p-1, exp2(x) = 0x1.6a09e667c50dfp-1 to 76 bits .quad 0xbfdf800000504e67, 0x3fe6c01274f81141 // x = -0x1.f800000504e67p-2, exp2(x) = 0x1.6c01274f81141p-1 to 75 bits .quad 0xbfdf000000413114, 0x3fe6dfb23c54f3bf // x = -0x1.f000000413114p-2, exp2(x) = 0x1.6dfb23c54f3bfp-1 to 75 bits .quad 0xbfde800000026750, 0x3fe6ff7df950fb41 // x = -0x1.e800000026750p-2, exp2(x) = 0x1.6ff7df950fb41p-1 to 75 bits .quad 0xbfde00000001f188, 0x3fe71f75e8ebe2db // x = -0x1.e00000001f188p-2, exp2(x) = 0x1.71f75e8ebe2dbp-1 to 75 bits .quad 0xbfdd800000060771, 0x3fe73f9a48a3fcd4 // x = -0x1.d800000060771p-2, exp2(x) = 0x1.73f9a48a3fcd4p-1 to 75 bits .quad 0xbfdd000000027151, 0x3fe75feb5641c97b // x = -0x1.d000000027151p-2, exp2(x) = 0x1.75feb5641c97bp-1 to 75 bits .quad 0xbfdc800000133b3c, 0x3fe780694fd97820 // x = -0x1.c800000133b3cp-2, exp2(x) = 0x1.780694fd97820p-1 to 77 bits .quad 0xbfdc000000006742, 0x3fe7a11473eae71a // x = -0x1.c000000006742p-2, exp2(x) = 0x1.7a11473eae71ap-1 to 76 bits .quad 0xbfdb8000000026c4, 0x3fe7c1ed0130b739 // x = -0x1.b8000000026c4p-2, exp2(x) = 0x1.7c1ed0130b739p-1 to 75 bits .quad 0xbfdb00000010b0f8, 0x3fe7e2f336cafcf6 // x = -0x1.b00000010b0f8p-2, exp2(x) = 0x1.7e2f336cafcf6p-1 to 75 bits .quad 0xbfda800000006fca, 0x3fe80427543dfcfe // x = -0x1.a800000006fcap-2, exp2(x) = 0x1.80427543dfcfep-1 to 75 bits .quad 0xbfda0000000a60b7, 0x3fe82589994a174b // x = -0x1.a0000000a60b7p-2, exp2(x) = 0x1.82589994a174bp-1 to 75 bits .quad 0xbfd98000000f3180, 0x3fe8471a461fc8fa // x = -0x1.98000000f3180p-2, exp2(x) = 0x1.8471a461fc8fap-1 to 82 bits .quad 0xbfd9000000065897, 0x3fe868d99b42e570 // x = -0x1.9000000065897p-2, exp2(x) = 0x1.868d99b42e570p-1 to 76 bits .quad 0xbfd8800000057289, 0x3fe88ac7d988f3ee // x = -0x1.8800000057289p-2, exp2(x) = 0x1.88ac7d988f3eep-1 to 76 bits .quad 0xbfd800000005c28b, 0x3fe8ace5422916cb // x = -0x1.800000005c28bp-2, exp2(x) = 0x1.8ace5422916cbp-1 to 75 bits .quad 0xbfd780000060838c, 0x3fe8cf32169b55bb // x = -0x1.780000060838cp-2, exp2(x) = 0x1.8cf32169b55bbp-1 to 75 bits .quad 0xbfd7000000160f6a, 0x3fe8f1ae990f8189 // x = -0x1.7000000160f6ap-2, exp2(x) = 0x1.8f1ae990f8189p-1 to 75 bits .quad 0xbfd6800000015814, 0x3fe9145b0b91a250 // x = -0x1.6800000015814p-2, exp2(x) = 0x1.9145b0b91a250p-1 to 79 bits .quad 0xbfd600000005b1dc, 0x3fe93737b0cc37c2 // x = -0x1.600000005b1dcp-2, exp2(x) = 0x1.93737b0cc37c2p-1 to 75 bits .quad 0xbfd58000000ced09, 0x3fe95a44cbc4c577 // x = -0x1.58000000ced09p-2, exp2(x) = 0x1.95a44cbc4c577p-1 to 75 bits .quad 0xbfd500000004055e, 0x3fe97d829fdd3222 // x = -0x1.500000004055ep-2, exp2(x) = 0x1.97d829fdd3222p-1 to 76 bits .quad 0xbfd4800000044514, 0x3fe9a0f170c8d852 // x = -0x1.4800000044514p-2, exp2(x) = 0x1.9a0f170c8d852p-1 to 75 bits .quad 0xbfd400000062eb27, 0x3fe9c49182885576 // x = -0x1.400000062eb27p-2, exp2(x) = 0x1.9c49182885576p-1 to 76 bits .quad 0xbfd380000025f5a3, 0x3fe9e86319d87c72 // x = -0x1.380000025f5a3p-2, exp2(x) = 0x1.9e86319d87c72p-1 to 75 bits .quad 0xbfd300000026d9fb, 0x3fea0c667b52ef7a // x = -0x1.300000026d9fbp-2, exp2(x) = 0x1.a0c667b52ef7ap-1 to 75 bits .quad 0xbfd28000000ef1d6, 0x3fea309bec45f003 // x = -0x1.28000000ef1d6p-2, exp2(x) = 0x1.a309bec45f003p-1 to 75 bits .quad 0xbfd200000028a35e, 0x3fea5503b2328e73 // x = -0x1.200000028a35ep-2, exp2(x) = 0x1.a5503b2328e73p-1 to 75 bits .quad 0xbfd18000003892f0, 0x3fea799e13207a90 // x = -0x1.18000003892f0p-2, exp2(x) = 0x1.a799e13207a90p-1 to 77 bits .quad 0xbfd100000036ebd4, 0x3fea9e6b556a2865 // x = -0x1.100000036ebd4p-2, exp2(x) = 0x1.a9e6b556a2865p-1 to 75 bits .quad 0xbfd08000001df4d6, 0x3feac36bbfcb4499 // x = -0x1.08000001df4d6p-2, exp2(x) = 0x1.ac36bbfcb4499p-1 to 75 bits .quad 0xbfd00000007109bc, 0x3feae89f9939e248 // x = -0x1.00000007109bcp-2, exp2(x) = 0x1.ae89f9939e248p-1 to 76 bits .quad 0xbfcf0000000d0ce0, 0x3feb0e07298bccef // x = -0x1.f0000000d0ce0p-3, exp2(x) = 0x1.b0e07298bccefp-1 to 76 bits .quad 0xbfce000000038571, 0x3feb33a2b84e9132 // x = -0x1.e000000038571p-3, exp2(x) = 0x1.b33a2b84e9132p-1 to 75 bits .quad 0xbfcd0000003291c8, 0x3feb59728ddddbeb // x = -0x1.d0000003291c8p-3, exp2(x) = 0x1.b59728ddddbebp-1 to 77 bits .quad 0xbfcc00000009a474, 0x3feb7f76f2f9eeb5 // x = -0x1.c00000009a474p-3, exp2(x) = 0x1.b7f76f2f9eeb5p-1 to 77 bits .quad 0xbfcb000000041435, 0x3feba5b030a069f4 // x = -0x1.b000000041435p-3, exp2(x) = 0x1.ba5b030a069f4p-1 to 77 bits .quad 0xbfca0000001c5bde, 0x3febcc1e90477d01 // x = -0x1.a0000001c5bdep-3, exp2(x) = 0x1.bcc1e90477d01p-1 to 76 bits .quad 0xbfc90000000fd3f7, 0x3febf2c25bd4b8c9 // x = -0x1.90000000fd3f7p-3, exp2(x) = 0x1.bf2c25bd4b8c9p-1 to 76 bits .quad 0xbfc800000042ee2b, 0x3fec199bdd7b2358 // x = -0x1.800000042ee2bp-3, exp2(x) = 0x1.c199bdd7b2358p-1 to 76 bits .quad 0xbfc700000012ef60, 0x3fec40ab5ffceadc // x = -0x1.700000012ef60p-3, exp2(x) = 0x1.c40ab5ffceadcp-1 to 78 bits .quad 0xbfc600000017f9f5, 0x3fec67f12e542120 // x = -0x1.600000017f9f5p-3, exp2(x) = 0x1.c67f12e542120p-1 to 75 bits .quad 0xbfc50000001c6b03, 0x3fec8f6d9402828e // x = -0x1.50000001c6b03p-3, exp2(x) = 0x1.c8f6d9402828ep-1 to 75 bits .quad 0xbfc40000004dc028, 0x3fecb720dce37952 // x = -0x1.40000004dc028p-3, exp2(x) = 0x1.cb720dce37952p-1 to 75 bits .quad 0xbfc300000002c957, 0x3fecdf0b555d5473 // x = -0x1.300000002c957p-3, exp2(x) = 0x1.cdf0b555d5473p-1 to 80 bits .quad 0xbfc20000000307a2, 0x3fed072d4a070f8f // x = -0x1.20000000307a2p-3, exp2(x) = 0x1.d072d4a070f8fp-1 to 77 bits .quad 0xbfc100000008225f, 0x3fed2f87080c40d5 // x = -0x1.100000008225fp-3, exp2(x) = 0x1.d2f87080c40d5p-1 to 78 bits .quad 0xbfc00000000d35cc, 0x3fed5818dcf98b25 // x = -0x1.00000000d35ccp-3, exp2(x) = 0x1.d5818dcf98b25p-1 to 75 bits .quad 0xbfbe00000000a23c, 0x3fed80e316c976a2 // x = -0x1.e00000000a23cp-4, exp2(x) = 0x1.d80e316c976a2p-1 to 75 bits .quad 0xbfbc0000001a43df, 0x3feda9e603d9167a // x = -0x1.c0000001a43dfp-4, exp2(x) = 0x1.da9e603d9167ap-1 to 75 bits .quad 0xbfba0000000a59b6, 0x3fedd321f300de67 // x = -0x1.a0000000a59b6p-4, exp2(x) = 0x1.dd321f300de67p-1 to 78 bits .quad 0xbfb800000004084f, 0x3fedfc97337b478e // x = -0x1.800000004084fp-4, exp2(x) = 0x1.dfc97337b478ep-1 to 75 bits .quad 0xbfb60000000cc2e4, 0x3fee264614f49679 // x = -0x1.60000000cc2e4p-4, exp2(x) = 0x1.e264614f49679p-1 to 75 bits .quad 0xbfb40000002a7000, 0x3fee502ee787c449 // x = -0x1.40000002a7000p-4, exp2(x) = 0x1.e502ee787c449p-1 to 75 bits .quad 0xbfb20000001b557e, 0x3fee7a51fbc50b11 // x = -0x1.20000001b557ep-4, exp2(x) = 0x1.e7a51fbc50b11p-1 to 79 bits .quad 0xbfb0000000067db9, 0x3feea4afa2a406fa // x = -0x1.0000000067db9p-4, exp2(x) = 0x1.ea4afa2a406fap-1 to 75 bits .quad 0xbfac0000004cac7c, 0x3feecf482d8b353b // x = -0x1.c0000004cac7cp-5, exp2(x) = 0x1.ecf482d8b353bp-1 to 79 bits .quad 0xbfa80000002255b2, 0x3feefa1bee5fe98b // x = -0x1.80000002255b2p-5, exp2(x) = 0x1.efa1bee5fe98bp-1 to 75 bits .quad 0xbfa40000000fbb0b, 0x3fef252b376b10cb // x = -0x1.40000000fbb0bp-5, exp2(x) = 0x1.f252b376b10cbp-1 to 75 bits .quad 0xbfa000000013426b, 0x3fef50765b6d743c // x = -0x1.000000013426bp-5, exp2(x) = 0x1.f50765b6d743cp-1 to 75 bits .quad 0xbf980000001b1b4b, 0x3fef7bfdad9c2a30 // x = -0x1.80000001b1b4bp-6, exp2(x) = 0x1.f7bfdad9c2a30p-1 to 75 bits .quad 0xbf90000000108597, 0x3fefa7c1819e3637 // x = -0x1.0000000108597p-6, exp2(x) = 0x1.fa7c1819e3637p-1 to 75 bits .quad 0xbf800000001446bb, 0x3fefd3c22b8f3a07 // x = -0x1.00000001446bbp-7, exp2(x) = 0x1.fd3c22b8f3a07p-1 to 78 bits .quad 0x0000000000000000, 0x3ff0000000000000 // x = 0x0.0000000000000p+0, exp2(x) = 0x1.0000000000000p+0 exactly .quad 0x3f80000000046787, 0x3ff0163da9fb3959 // x = 0x1.0000000046787p-7, exp2(x) = 0x1.0163da9fb3959p+0 to 75 bits .quad 0x3f90000000413bdf, 0x3ff02c9a3e783737 // x = 0x1.0000000413bdfp-6, exp2(x) = 0x1.02c9a3e783737p+0 to 75 bits .quad 0x3f9800000009e3ad, 0x3ff04315e86e9b63 // x = 0x1.800000009e3adp-6, exp2(x) = 0x1.04315e86e9b63p+0 to 75 bits .quad 0x3fa00000000c4c0e, 0x3ff059b0d315cb23 // x = 0x1.00000000c4c0ep-5, exp2(x) = 0x1.059b0d315cb23p+0 to 77 bits .quad 0x3fa40000001dce0d, 0x3ff0706b29dea0ad // x = 0x1.40000001dce0dp-5, exp2(x) = 0x1.0706b29dea0adp+0 to 76 bits .quad 0x3fa80000002def51, 0x3ff087451876a2e9 // x = 0x1.80000002def51p-5, exp2(x) = 0x1.087451876a2e9p+0 to 79 bits .quad 0x3fac000000110f6a, 0x3ff09e3ecac755c5 // x = 0x1.c000000110f6ap-5, exp2(x) = 0x1.09e3ecac755c5p+0 to 75 bits .quad 0x3fb0000000042de0, 0x3ff0b5586cf9b976 // x = 0x1.0000000042de0p-4, exp2(x) = 0x1.0b5586cf9b976p+0 to 75 bits .quad 0x3fb20000000af8a9, 0x3ff0cc922b72c7b8 // x = 0x1.20000000af8a9p-4, exp2(x) = 0x1.0cc922b72c7b8p+0 to 75 bits .quad 0x3fb40000000120ca, 0x3ff0e3ec32d3ded7 // x = 0x1.40000000120cap-4, exp2(x) = 0x1.0e3ec32d3ded7p+0 to 77 bits .quad 0x3fb600000008f69e, 0x3ff0fb66afff3c9d // x = 0x1.600000008f69ep-4, exp2(x) = 0x1.0fb66afff3c9dp+0 to 75 bits .quad 0x3fb8000000305124, 0x3ff11301d0149725 // x = 0x1.8000000305124p-4, exp2(x) = 0x1.11301d0149725p+0 to 76 bits .quad 0x3fba0000000021c9, 0x3ff12abdc06c335e // x = 0x1.a0000000021c9p-4, exp2(x) = 0x1.12abdc06c335ep+0 to 75 bits .quad 0x3fbc0000000b765e, 0x3ff1429aaea9b702 // x = 0x1.c0000000b765ep-4, exp2(x) = 0x1.1429aaea9b702p+0 to 75 bits .quad 0x3fbe0000000561d9, 0x3ff15a98c8a5cf0f // x = 0x1.e0000000561d9p-4, exp2(x) = 0x1.15a98c8a5cf0fp+0 to 75 bits .quad 0x3fc00000000d9993, 0x3ff172b83c7e9a70 // x = 0x1.00000000d9993p-3, exp2(x) = 0x1.172b83c7e9a70p+0 to 76 bits .quad 0x3fc10000001aa7db, 0x3ff18af9388f162b // x = 0x1.10000001aa7dbp-3, exp2(x) = 0x1.18af9388f162bp+0 to 76 bits .quad 0x3fc2000000023143, 0x3ff1a35beb700111 // x = 0x1.2000000023143p-3, exp2(x) = 0x1.1a35beb700111p+0 to 76 bits .quad 0x3fc300000035902b, 0x3ff1bbe0840981a5 // x = 0x1.300000035902bp-3, exp2(x) = 0x1.1bbe0840981a5p+0 to 78 bits .quad 0x3fc40000000c261e, 0x3ff1d4873169e5f6 // x = 0x1.40000000c261ep-3, exp2(x) = 0x1.1d4873169e5f6p+0 to 76 bits .quad 0x3fc50000000f8bff, 0x3ff1ed5022fe5b7c // x = 0x1.50000000f8bffp-3, exp2(x) = 0x1.1ed5022fe5b7cp+0 to 75 bits .quad 0x3fc600000019e18b, 0x3ff2063b88651387 // x = 0x1.600000019e18bp-3, exp2(x) = 0x1.2063b88651387p+0 to 78 bits .quad 0x3fc700000010f9d2, 0x3ff21f49917f8711 // x = 0x1.700000010f9d2p-3, exp2(x) = 0x1.21f49917f8711p+0 to 78 bits .quad 0x3fc800000021f3d9, 0x3ff2387a6e78bbd5 // x = 0x1.800000021f3d9p-3, exp2(x) = 0x1.2387a6e78bbd5p+0 to 76 bits .quad 0x3fc900000007aca6, 0x3ff251ce4fb36926 // x = 0x1.900000007aca6p-3, exp2(x) = 0x1.251ce4fb36926p+0 to 77 bits .quad 0x3fca0000000834d3, 0x3ff26b4565e34e68 // x = 0x1.a0000000834d3p-3, exp2(x) = 0x1.26b4565e34e68p+0 to 76 bits .quad 0x3fcb0000000f383c, 0x3ff284dfe1f6ea3c // x = 0x1.b0000000f383cp-3, exp2(x) = 0x1.284dfe1f6ea3cp+0 to 75 bits .quad 0x3fcc000000295470, 0x3ff29e9df52409b2 // x = 0x1.c000000295470p-3, exp2(x) = 0x1.29e9df52409b2p+0 to 75 bits .quad 0x3fcd000000aa2014, 0x3ff2b87fd0ec18b9 // x = 0x1.d000000aa2014p-3, exp2(x) = 0x1.2b87fd0ec18b9p+0 to 75 bits .quad 0x3fce00000089ee06, 0x3ff2d285a6f21217 // x = 0x1.e00000089ee06p-3, exp2(x) = 0x1.2d285a6f21217p+0 to 77 bits .quad 0x3fcf00000011e56f, 0x3ff2ecafa94004d8 // x = 0x1.f00000011e56fp-3, exp2(x) = 0x1.2ecafa94004d8p+0 to 76 bits .quad 0x3fd00000002450b9, 0x3ff306fe0a3932e5 // x = 0x1.00000002450b9p-2, exp2(x) = 0x1.306fe0a3932e5p+0 to 78 bits .quad 0x3fd0800000018109, 0x3ff32170fc4d27f8 // x = 0x1.0800000018109p-2, exp2(x) = 0x1.32170fc4d27f8p+0 to 75 bits .quad 0x3fd10000000f3232, 0x3ff33c08b2674165 // x = 0x1.10000000f3232p-2, exp2(x) = 0x1.33c08b2674165p+0 to 75 bits .quad 0x3fd18000000237f0, 0x3ff356c55f9316e5 // x = 0x1.18000000237f0p-2, exp2(x) = 0x1.356c55f9316e5p+0 to 76 bits .quad 0x3fd20000000199c8, 0x3ff371a7373b0016 // x = 0x1.20000000199c8p-2, exp2(x) = 0x1.371a7373b0016p+0 to 75 bits .quad 0x3fd28000000b48f9, 0x3ff38cae6d083c14 // x = 0x1.28000000b48f9p-2, exp2(x) = 0x1.38cae6d083c14p+0 to 75 bits .quad 0x3fd30000004c6957, 0x3ff3a7db34f5e42c // x = 0x1.30000004c6957p-2, exp2(x) = 0x1.3a7db34f5e42cp+0 to 77 bits .quad 0x3fd38000004e208e, 0x3ff3c32dc32461b8 // x = 0x1.38000004e208ep-2, exp2(x) = 0x1.3c32dc32461b8p+0 to 75 bits .quad 0x3fd4000000013df2, 0x3ff3dea64c12788e // x = 0x1.4000000013df2p-2, exp2(x) = 0x1.3dea64c12788ep+0 to 75 bits .quad 0x3fd480000015af05, 0x3ff3fa4504b13129 // x = 0x1.480000015af05p-2, exp2(x) = 0x1.3fa4504b13129p+0 to 78 bits .quad 0x3fd500000009a838, 0x3ff4160a21f947f9 // x = 0x1.500000009a838p-2, exp2(x) = 0x1.4160a21f947f9p+0 to 75 bits .quad 0x3fd5800000234f8c, 0x3ff431f5d95861bd // x = 0x1.5800000234f8cp-2, exp2(x) = 0x1.431f5d95861bdp+0 to 77 bits .quad 0x3fd600000003a7aa, 0x3ff44e08606256f0 // x = 0x1.600000003a7aap-2, exp2(x) = 0x1.44e08606256f0p+0 to 78 bits .quad 0x3fd68000000e661d, 0x3ff46a41ed202f5b // x = 0x1.68000000e661dp-2, exp2(x) = 0x1.46a41ed202f5bp+0 to 75 bits .quad 0x3fd700000047e5a0, 0x3ff486a2b5d13877 // x = 0x1.700000047e5a0p-2, exp2(x) = 0x1.486a2b5d13877p+0 to 75 bits .quad 0x3fd780000013fc02, 0x3ff4a32af0dc4b5b // x = 0x1.780000013fc02p-2, exp2(x) = 0x1.4a32af0dc4b5bp+0 to 75 bits .quad 0x3fd800000018514b, 0x3ff4bfdad53ba122 // x = 0x1.800000018514bp-2, exp2(x) = 0x1.4bfdad53ba122p+0 to 75 bits .quad 0x3fd88000004b65dc, 0x3ff4dcb29a0ee638 // x = 0x1.88000004b65dcp-2, exp2(x) = 0x1.4dcb29a0ee638p+0 to 76 bits .quad 0x3fd90000002a7909, 0x3ff4f9b276a6d2b4 // x = 0x1.90000002a7909p-2, exp2(x) = 0x1.4f9b276a6d2b4p+0 to 76 bits .quad 0x3fd98000000e9ddf, 0x3ff516daa2d2bcec // x = 0x1.98000000e9ddfp-2, exp2(x) = 0x1.516daa2d2bcecp+0 to 77 bits .quad 0x3fda0000001165b7, 0x3ff5342b56a14e49 // x = 0x1.a0000001165b7p-2, exp2(x) = 0x1.5342b56a14e49p+0 to 76 bits .quad 0x3fda800000347522, 0x3ff551a4ca69aec0 // x = 0x1.a800000347522p-2, exp2(x) = 0x1.551a4ca69aec0p+0 to 76 bits .quad 0x3fdb000000163445, 0x3ff56f4736ba4f70 // x = 0x1.b000000163445p-2, exp2(x) = 0x1.56f4736ba4f70p+0 to 75 bits .quad 0x3fdb800000173fe4, 0x3ff58d12d49d3534 // x = 0x1.b800000173fe4p-2, exp2(x) = 0x1.58d12d49d3534p+0 to 75 bits .quad 0x3fdc0000000bbaca, 0x3ff5ab07dd4b14d7 // x = 0x1.c0000000bbacap-2, exp2(x) = 0x1.5ab07dd4b14d7p+0 to 76 bits .quad 0x3fdc80000016a2c1, 0x3ff5c9268a5e9dfb // x = 0x1.c80000016a2c1p-2, exp2(x) = 0x1.5c9268a5e9dfbp+0 to 75 bits .quad 0x3fdd0000004564c2, 0x3ff5e76f15bd979e // x = 0x1.d0000004564c2p-2, exp2(x) = 0x1.5e76f15bd979ep+0 to 75 bits .quad 0x3fdd8000001d2bb6, 0x3ff605e1b97dd138 // x = 0x1.d8000001d2bb6p-2, exp2(x) = 0x1.605e1b97dd138p+0 to 75 bits .quad 0x3fde000000122703, 0x3ff6247eb03eafef // x = 0x1.e000000122703p-2, exp2(x) = 0x1.6247eb03eafefp+0 to 76 bits .quad 0x3fde8000001fd876, 0x3ff6434634d470cf // x = 0x1.e8000001fd876p-2, exp2(x) = 0x1.6434634d470cfp+0 to 76 bits .quad 0x3fdf0000004a6dab, 0x3ff6623882672d39 // x = 0x1.f0000004a6dabp-2, exp2(x) = 0x1.6623882672d39p+0 to 76 bits .quad 0x3fdf80000033c91f, 0x3ff68155d45948c1 // x = 0x1.f80000033c91fp-2, exp2(x) = 0x1.68155d45948c1p+0 to 75 bits .quad 0x3fe000000006d6f2, 0x3ff6a09e668295fd // x = 0x1.000000006d6f2p-1, exp2(x) = 0x1.6a09e668295fdp+0 to 76 bits piecewise: /*-0.2864515168839901090209747108579654875605616e-3*(x+.4933464573244525103789200064459993631425989)* (x-0.1182415726241881770954331861172954748535105e-1)*(x-.1019569932000777092899138256855970885981370)* (x-1.707457006002457111723639323617400555673950)*(x-17.55700120861493418461246339928875474977667)* (x^2+2.770717309872218703235489009413553490784394*x+2.118123962269468702798445003445138300343937)* (x^2+1.838419944636830507881531448964308951713899*x+2.273811442636048173850054792255851453680793)* (x^2-1.510861832472913549784965349644347645304161*x+2.694330951091002235734548632049431595168001)* (x^2-2.891286467193815429903104107548024238154762*x+2.855309648308176377344446946752634506917244)*/ .quad 0x3feef20000000000, 0x3f6e000000000000 // 3961/4096 x + 15/4096 .quad 0xbfbd20c2f0233edb, 0xc03343b38896f0b4 // a_1, a_2 .quad 0x40062a6dd642e5f2, 0x3ffd6a2b08287816 // a_3, a_4 .quad 0xbff82c7d74f3eec6, 0xc007215acc9fb7f0 // a_5, a_6 .quad 0x3f53c077627b845c, 0x403dfa52b8813743 // b_1, b_2 .quad 0x4000f1eaf9d69209, 0x400230c40dbb23a6 // b_3, b_4 .quad 0x40058dfd62bc4a9c, 0x4006d7ac95bb7ebb // b_5, b_6 .quad 0xbf32c5dbe59ba738, 0xbf2285e830f09106 // a_0, b_0 /* -0.2375878883133903234332653094979448890904333e-2*(x+1.141393572200027712958325552880364704841667)* (x+0.9432391007626355646599517236151910406352282e-1)*(x-0.8037992537205077203224291088009425075443164e-1)* (x^2+2.922334131587764592096900045673310598662462*x+2.343651060006581810841903559429949385193724)* (x^2+2.164423024969638405314605029797899150324956*x+2.538857443423985682428496892587673324322619)* (x^2-.8978880630510944949367543166122444149618204*x+2.647052449400489220824414241183706204517465)* (x^2-2.338357501383843298875790789553535972448012*x+2.678582971901324051327760583919669959590076)* (x^2-3.175478809229660118488076975757657039109498*x+2.696893643567215337517965055644178016901177)*/ .quad 0x3febfc0000000000, 0x3f9cc00000000000 // 3582/4096 x + 115/4096 .quad 0x3f8c8ea9f23e804c, 0x400760f0b7993b69 // a_1, a_2 .quad 0x400150bd04d7a1c5, 0xbfecbb7fbf48be5b // a_3, a_4 .quad 0xc002b4f4c7166628, 0xc00967616f164262 // a_5, a_6 .quad 0xbf7f0e0a35e743ee, 0x4002bfcc207fb647 // b_1, b_2 .quad 0x40044f947dc5b290, 0x40052d29d5a7c63d // b_3, b_4 .quad 0x40056dbce8bf8267, 0x4005933cf97f4ab1 // b_5, b_6 .quad 0xbf63769443462e8e, 0xbf66371574fff23b // a_0, b_0 /* 0.6821180293172612295640961212495022937011365e-3*(x+0.9386570864065908327923776423721617419905786e-1)* (x-0.8589816406847459765176398027517267526480442e-1)*(x-2.166095361145909667282101208582047402400105)* (x^2+2.983194540901171801414314552018551787362570*x+2.327924268962062274660991496770128609481354)* (x^2+2.500812465242608806630929644245570226918524*x+2.604020578541220636907356420919440871778391)* (x^2-.4811225869073539084506952758557871069271204*x+3.113496944535685107372520625657988666598883)* (x^2-2.258568925224693781821970135740820585965534*x+3.443166841217303155561660113644021763452671)* (x^2-3.663633157183776833100177799273882127652364*x+3.986165335805472349372636980074778113068575)*/ .quad 0x3fe6400000000000, 0x3fbdf00000000000 // 2848/4096 x + 479/4096 .quad 0x3f805149baef85ef, 0x4007dd95197635cc // a_1, a_2 .quad 0x400401a9f73d2b4c, 0xbfdecab664089381 // a_3, a_4 .quad 0xc002118c95acd0c0, 0xc00d4f1ee6952a16 // a_5, a_6 .quad 0xbf8083470d069524, 0x40029f96c2560ddb // b_1, b_2 .quad 0x4004d508bdb79078, 0x4008e8711607d045 // b_3, b_4 .quad 0x400b8b9b0e8d997b, 0x400fe3aaa6cddded // b_5, b_6 .quad 0x3f465a05505fff51, 0xbf583538a7c2274b // a_0, b_0 /* 0.3675050415084445460952193551199634192284019e-3*(x+2.538253007068358766085970620345190291400062)* (x+1.891265936573533097652685331666438995456676)*(x+0.9156287399777636946014542863238438354098965e-1)* (x-0.8403859216027488106552551230547659805950613e-1)*(x-1.802954954543863782501756058330497972336359)* (x^2+3.119393795298718299316411466993714733886600*x+3.334575315228986457403114264913740785470140)* (x^2-0.2660697000023952164642638899375535969090176e-1*x+3.205893970965164102059321144392191513892765)* (x^2-1.805070720874556914625952264641807822799429*x+3.222143486665259634799789338307105717457736)* (x^2-3.121702245703248826686097843186573720076780*x+3.242552497083970905169346770196317588771562)*/ .quad 0x3fe0400000000000, 0x3fd0700000000000 // 2080/4096 x + 1052/4096 .quad 0x3ffcea67384b11e9, 0xbffb61dca98942f1 // a_1, a_2 .quad 0x4008f484bbf139fd, 0xbf9b3edb87fc9a2b // a_3, a_4 .quad 0xbffce191d611f94f, 0xc008f93f06e92964 // a_5, a_6 .quad 0xbfc4581fb52271d2, 0xbfc52176c275a3ab // b_1, b_2 .quad 0x400aad35d2a7a8bb, 0x4009a5abbcfde960 // b_3, b_4 .quad 0x4009c6f32a11f93f, 0x4009f0bf5d144f71 // b_5, b_6 .quad 0x3f3815b62267a35a, 0x3f4e91115d517686 // a_0, b_0 /* -0.1454916941539841281056690225872902433188312e-3*(x+1.895467253199031252686203675164860153418254)* (x+0.9130291828799857823935486078744448547578965e-1)*(x-0.8499583911816027391302764050755799469607102e-1)* (x^2+3.398145257103028458687443509002980550445610*x+3.636501613989623235844429443005415702389483)* (x^2+.5498410413574713586659146118099117563075050*x+3.861165354740107211770934343766878654423520)* (x^2-1.359498118602839703247623757217688185600174*x+4.035047519552677285194086315801939227901986)* (x^2-3.036050352195771266788909949443243206515610*x+4.189494379230451407340774930662591118810411)* (x^2-4.012340397489436305623606583590817514912250*x+4.276055559103643625266538933042186424497733)*/ .quad 0x3fd6400000000000, 0x3fdaa00000000000 // 1424/4096 x + 1704/4096 .quad 0x3f79d573ac48a801, 0x400b2f66c7d28529 // a_1, a_2 .quad 0x3fe1984c3d542023, 0xbff5c0811965f6af // a_3, a_4 .quad 0xc00849d4c45d855d, 0xc0100ca2f60e8c0d // a_5, a_6 .quad 0xbf7fc955f6cf414f, 0x400d178e287f7e41 // b_1, b_2 .quad 0x400ee3aaa95874bc, 0x401023e37f392380 // b_3, b_4 .quad 0x4010c20ad086484e, 0x40111aae4ef8eaa5 // b_5, b_6 .quad 0xbf2311e422ef2c53, 0xbf3212bb7f586ef1 // a_0, b_0 /* -0.5412652598572911167212102331230339557075992e-4*(x+2.629392008383213598304883848137312998666697)* (x+0.8789494961083907158478561801157544587360792e-1)*(x-0.8208279725873500595208981438214151574275247e-1)* (x^2+4.195478155052112358115820942477950941061380*x+4.966757346123734917905738242970175837409293)* (x^2+.8823263428901269743834670128774202678301578*x+4.118062244853954226344188393026476912078044)* (x^2-1.131877687016688662064358097058320698192151*x+4.076018964523053136439220066275491038611692)* (x^2-2.893472150157534957506432929430867039460720*x+4.093336646134720875078946869223410975744725)* (x^2-3.919024916687541429883627778453309015591906*x+4.107472917357761636545594014132211005272860)*/ .quad 0x3fcd280000000000, 0x3fe2160000000000 // 933/4096 x + 2315/4096 .quad 0x3f77ce7bc458b46c, 0x4010c82b6cec1ffd // a_1, a_2 .quad 0x3fec3c0474639899, 0xbff21c2bc70cf0da // a_3, a_4 .quad 0xc00725d4ba0681df, 0xc00f5a29bc4b0e92 // a_5, a_6 .quad 0xbf7d8d1f70cb1f6f, 0x4013ddf5a34313a8 // b_1, b_2 .quad 0x401078e54f22290a, 0x40104dd7ea5a025a // b_3, b_4 .quad 0x40105f93a44aaaf5, 0x40106e0d61650779 // b_5, b_6 .quad 0xbf0c60bd458a1834, 0xbf22a7763e951b37 // a_0, b_0 /* 0.4924053668989264983145283706733415056216404e-5*(x+0.8796471725749816598263629969503552203165658e-1)* (x-0.8243298434040044427057532032166616955244352e-1)*(x-5.247080327413196729391477686672343881798544)* (x^2+4.375087198300077517171213474173238111040758*x+5.113333656883580100278523194277394673560613)* (x^2+1.332348590031229077157574373887077497579276*x+5.115849567237085180807802824713706539664123)* (x^2-.9350409521747968106518680203179119052124734*x+5.390531195312283061461451786153366587344480)* (x^2-3.215852224589302541442409521049125766355344*x+5.835283059223468050039391207582881254538750)* (x^2-4.805142513405023792476061013566488171009830*x+6.284366447110418558084552080608121886245705)*/ .quad 0x3fc2900000000000, 0x3fe60a0000000000 // 594/4096 x + 2821/4096 .quad 0xc015516bf006503c, 0x40118016dbc7636e // a_1, a_2 .quad 0x3ff5514cc150e4b9, 0xbfedebdb00c05dd1 // a_3, a_4 .quad 0xc009ba10bb2b0a0a, 0xc0133877476ec678 // a_5, a_6 .quad 0x3fdbae9cc13f99d1, 0x4014740dbcf7677d // b_1, b_2 .quad 0x401476a144da27f9, 0x40158fe768dfbc76 // b_3, b_4 .quad 0x401757547139120d, 0x40192330f539ad04 // b_5, b_6 .quad 0x3ed4a72990f33fe0, 0x3e9d115702afdd8d // a_0, b_0 /* 0.6862565065994843133907783301321216634622772e-5*(x+0.9630374666835468858692155213887501416937462e-1)* (x-0.9263910186969991762077298562812547568924601e-1)*(x-2.323730050905880210881006882732927135598709)* (x^2+4.915343084278116341741257562832236772519938*x+6.366157435903189155603019667371637988996918)* (x^2+1.824272160555820074533801014349118587449441*x+5.502457749802857122655878777027393347399835)* (x^2-.3300561409680903303325074516366401983263062*x+5.332584863498335175639855633785690269440549)* (x^2-2.470967591354500287186505101999528533350514*x+5.359889068546726215460509020568700628842379)* (x^2-4.062368446227079298604593445606996624167424*x+5.389376533522531429130148833360296979673207)*/ .quad 0x3fb7300000000000, 0x3fe9120000000000 // 371/4096 x + 3209/4096 .quad 0xc00354b959b1dff0, 0x4013a94fb28e608b // a_1, a_2 .quad 0x3ffd3038014972b9, 0xbfd51fa3cad3528e // a_3, a_4 .quad 0xc003c48aa812bf6e, 0xc0103fdd839363fc // a_5, a_6 .quad 0x3fcb8de916d04f6d, 0x401976f1f99190cf // b_1, b_2 .quad 0x4016028448cc199f, 0x40155491205f7ada // b_3, b_4 .quad 0x40157086c28e6266, 0x40158eb8b8d53c8a // b_5, b_6 .quad 0x3edcc89f858814a5, 0x3ea62d021453335f // a_0, b_0 small_poly: /* -0.1912438313598201685992817228652558781857235e-3* (x^2+3.756675270173468426449695171076913404673252*x+6.166462828789874819735603927346200075670479)* (x^2+.7625866507820391697007654102468723536049672*x+6.364574963873789771039813999418889409285683)* (x^2-2.655819614791829218878773973800635875503330*x+6.590413750655284949335220033866164267136093)* (x^2-4.892571285931185215076239999657804121172708*x+6.738653176394183551008839845454158365598159)*/ .quad 0x400e0dabc398b233, 0x3fe8671c1eaf3374 // a_1, a_2 .quad 0xc0053f1e5aacd98c, 0xc01391fe3509b143 // a_3, a_4 .quad 0x4018aa753b569bd0, 0x4019755323ab1cfd // b_1, b_2 .quad 0x401a5c956c18b14e, 0x401af4617f8ecc54 // b_3, b_4 .quad 0xbf29111400a41fbf, 0x0000000000000000 // c .literal8 .align 3 m2_lge_p7: .quad 0xc0771547652b82fe m2_lge: .quad 0xc0071547652b82fe m2_lge_hi: .quad 0xc007154760000000 m2_lge_lo: .quad 0xbe64ae0bf85ddf44 two: .quad 0x4000000000000000 tiny_m_two: .quad 0xfe40000000000000 eighth_m_two: .quad 0xffc0000000000000 large_m_two: .quad 0x003c000000000000 split_mask: .quad 0xfffffffff8000000 one_half: .quad 0x3fe0000000000000 min_normal: .quad 0x0010000000000000 infinity: .quad 0x7ff0000000000000 one_p55: .quad 0x4360000000000000 one_n55: .quad 0x3c80000000000000 /**************************************************************************** PIC code ****************************************************************************/ #if defined( __x86_64__ ) #define REL_ADDR(_a) (_a)(%rip) #else #define REL_ADDR(_a) (_a)-0b(%ecx) #endif /**************************************************************************** Entry point for tanh( ) ****************************************************************************/ .text ENTRY( tanh ) #if defined( __x86_64__ ) movapd %xmm0, %xmm1 andpd REL_ADDR(abs_mask), %xmm0 // |x| movd %xmm0, %rax movapd %xmm1, %xmm7 sub REL_ADDR(two), %rax xorpd %xmm0, %xmm7 // signbit(x) cmp REL_ADDR(large_m_two), %rax ja 1f #elif defined( __i386__ ) movl 4+FRAME_SIZE(STACKP), %eax movsd FRAME_SIZE(STACKP), %xmm0 call 0f 0: pop %ecx and $0x7fffffff, %eax movapd %xmm0, %xmm1 andpd REL_ADDR(abs_mask), %xmm0 // |x| movapd %xmm1, %xmm7 sub $0x40000000, %eax xorpd %xmm0, %xmm7 // signbit(x) cmp $0x003c0000, %eax ja 1f #else #error UNKNOWN ARCH #endif // Code path for 2 < |x| < 22: // Code copied from exp( ) ----------------------- movapd %xmm0, %xmm1 mulsd REL_ADDR(m2_lge_p7), %xmm0 cvttsd2si %xmm0, AX_P add $63, AX_P mov AX_P, DX_P sar $7, AX_P and $0x7f, DX_P sal $4, DX_P movapd %xmm1, %xmm2 mulsd REL_ADDR(m2_lge), %xmm1 cvtsi2sd AX_P, %xmm0 movapd %xmm1, %xmm6 subsd %xmm0, %xmm1 movapd %xmm2, %xmm4 movsd REL_ADDR(split_mask), %xmm3 mulsd REL_ADDR(m2_lge_lo), %xmm4 movsd REL_ADDR(m2_lge_hi), %xmm5 lea REL_ADDR(exp2table), CX_P subsd (DX_P, CX_P, 1), %xmm1 andpd %xmm2, %xmm3 subsd %xmm3, %xmm2 mulsd %xmm5, %xmm3 mulsd %xmm5, %xmm2 subsd %xmm6, %xmm3 addsd %xmm3, %xmm2 addsd %xmm4, %xmm2 movsd -40(CX_P), %xmm6 mulsd %xmm1, %xmm6 addsd -32(CX_P), %xmm6 mulsd %xmm1, %xmm6 addsd -24(CX_P), %xmm6 addsd %xmm1, %xmm2 movapd %xmm2, %xmm3 mulsd %xmm2, %xmm6 mulsd %xmm3, %xmm3 addsd -16(CX_P), %xmm6 mulsd -8(CX_P), %xmm2 mulsd %xmm6, %xmm3 addsd %xmm3, %xmm2 movsd 8(DX_P, CX_P, 1), %xmm1 // End exp( ) code block ------------------------- // At this point, the following registers have useful values: // // AX_P n integer // xmm1 a double-precision number in the interval [exp(-1/2), exp(1/2)) // xmm2 b double-precision number < 2^-7 ish // xmm7 signbit(x) bit pattern // // where 2^(n+1)*(a + a*b) = 2 exp(-2x). inc AX_P movd AX_P, %xmm4 psllq $52, %xmm4 paddd %xmm4, %xmm1 // 2^(n+1) a mulsd %xmm1, %xmm2 // 2^(n+1) a b movapd %xmm2, %xmm4 addsd %xmm1, %xmm2 // z = 2 exp(-2x) = 2^(n+1)*(a + ab) movsd -96(CX_P), %xmm3 // c7 mulsd %xmm2, %xmm3 // c7z addsd -88(CX_P), %xmm3 // c6 + c7z mulsd %xmm2, %xmm3 // c6z + c7z^2 addsd -80(CX_P), %xmm3 // c5 + c6z + c7z^2 mulsd %xmm2, %xmm3 // c5z + c6z^2 + c7z^3 addsd -72(CX_P), %xmm3 // c4 + c5z + c6z^2 + c7z^3 mulsd %xmm2, %xmm3 // c4z + c5z^2 + c6z^3 + c7z^4 addsd -64(CX_P), %xmm3 // c3z + c4z + c5z^2 + c6z^3 + c7z^4 mulsd %xmm2, %xmm3 // c3z + c4z^2 + c5z^3 + c6z^4 + c7z^5 addsd -56(CX_P), %xmm3 // c2 + c3z + c4z^2 + c5z^3 + c6z^4 + c7z^5 mulsd %xmm2, %xmm3 // c2z + c3z^2 + c4z^3 + c5z^4 + c6z^5 + c7z^6 addsd -48(CX_P), %xmm3 // (c1 + 1.0) + c2z + c3z^2 + c4z^3 + c5z^4 + c6z^5 + c7z^6 mulsd %xmm2, %xmm3 // (c1 + 1.0)z + c2z^2 + c3z^3 + c4z^4 + c5z^5 + c6z^6 + c7z^7 movsd -112(CX_P), %xmm0 // 1.0 subsd %xmm1, %xmm0 // result_hi = 1.0 - 2^(n+1) a movapd %xmm0, %xmm5 subsd -112(CX_P), %xmm5 addsd %xmm1, %xmm5 // result_hi_err = (result_hi - 1.0) + 2^(n+1) a addsd %xmm5, %xmm4 // 2^(n+1) a b + result_hi_err subsd %xmm4, %xmm3 // result_lo = z p(z) - (2^(n+1) a b + result_hi_err) xorpd %xmm7, %xmm0 // or in signbit xorpd %xmm7, %xmm3 addsd %xmm3, %xmm0 // result_hi + result_lo // = 1.0 - 2^(n+1)a - 2^(n+1)a b + z p(z) #if defined(__i386__) movsd %xmm0, FRAME_SIZE(STACKP) fldl FRAME_SIZE(STACKP) #endif ret 1: jg 6f #if defined( __x86_64__ ) cmp REL_ADDR(eighth_m_two), %rax #else cmp $0xffc00000, %eax #endif jbe 2f // Code path for 0.125 < |x| < 2.0 movapd REL_ADDR(four_exp), %xmm2 paddd %xmm0, %xmm2 // 4|x| (adding 2 to the exponent bits effects // a multiplication by 4 because x has // exponent between -3 and 1). cvttsd2si %xmm2, AX_P // floor(4|x|) cvtsi2sd AX_P, %xmm2 addsd REL_ADDR(one_half), %xmm2 // floor(4|x|) + 0.5 psubd REL_ADDR(four_exp), %xmm2 // 0.25*floor(4|x|) + 0.125 // (again, we can use exponent subtraction // instead of multiplication because we // are guaranteed that floor(4|x|) + 0.5 // has an exponent between -1 and 3) subsd %xmm2, %xmm0 // u = |x| - 0.25*floor(4|x|) - 0.125 // u is in the interval [-0.125, 0.125] shl $7, AX_P // 16 * sizeof(double) * floor(4|x|) movlhps %xmm0, %xmm0 // [ u , u ] por REL_ADDR(one), %xmm7 movq REL_ADDR(split_mask), %xmm3 andpd %xmm1, %xmm3 // xhi -- top 26 bits of x subsd %xmm3, %xmm1 // xlo -- x - xhi lea REL_ADDR(piecewise), DX_P movsd 8(AX_P, DX_P, 1), %xmm6 mulsd (AX_P, DX_P, 1), %xmm3 // L1 xhi mulsd (AX_P, DX_P, 1), %xmm1 // L1 xlo mulsd %xmm7, %xmm6 addsd %xmm6, %xmm3 // L1 xhi + L0 /* Code for factored polynomial evaluation */ movapd %xmm0, %xmm4 movapd %xmm4, %xmm5 movapd %xmm5, %xmm6 addpd 16(AX_P, DX_P, 1), %xmm0 // [ u + a1 , u + a2 ] addpd 32(AX_P, DX_P, 1), %xmm4 // [ u + a3 , u + a4 ] addpd 48(AX_P, DX_P, 1), %xmm5 // [ u + a5 , u + a6 ] mulpd %xmm6, %xmm0 // [ (u + a1)u , (u + a2)u ] mulpd %xmm6, %xmm4 // [ (u + a3)u , (u + a4)u ] mulpd %xmm6, %xmm5 // [ (u + a5)u , (u + a6)u ] mulsd 112(AX_P, DX_P, 1), %xmm6 // a0 u addpd 64(AX_P, DX_P, 1), %xmm0 // [ (u + a1)u + b1 , (u + a2)u + b2 ] addpd 80(AX_P, DX_P, 1), %xmm4 // [ (u + a3)u + b3 , (u + a4)u + b4 ] addpd 96(AX_P, DX_P, 1), %xmm5 // [ (u + a5)u + b5 , (u + a6)u + b6 ] addsd 120(AX_P, DX_P, 1), %xmm6 // a0 u + b0 movlhps %xmm7, %xmm6 // [ lin , signbit ] mulpd %xmm6, %xmm0 // [ lin * quad1, signbit * quad2 ] mulpd %xmm5, %xmm4 // [ quad3 * quad5, quad4 * quad6 ] mulpd %xmm4, %xmm0 // [ lin * quad1 * quad3 * quad5, signbit * quad2 * quad4 * quad6 ] movhlps %xmm0, %xmm4 // signbit * quad2 * quad4 * quad6 mulsd %xmm4, %xmm0 // signbit * lin * quad1 * quad2 * quad3 * quad4 * quad5 * quad6 addsd %xmm1, %xmm0 // L1 xlo + p(u) addsd %xmm3, %xmm0 // (L1 xhi + L0) + (L1 xlo + p(u)) #if defined(__i386__) movsd %xmm0, FRAME_SIZE(STACKP) fldl FRAME_SIZE(STACKP) #endif ret 2: #if defined( __x86_64__ ) cmp REL_ADDR(tiny_m_two), %rax #else cmp $0xfe400000, %eax #endif jbe 3f // Polynomial for 2^-54 < |x| < 0.125 lea REL_ADDR(small_poly), DX_P mulsd %xmm0, %xmm0 // u = x^2 movapd %xmm1, %xmm2 // x mulsd 64(DX_P), %xmm1 // cx movlhps %xmm0, %xmm0 // [ u , u ] movapd %xmm0, %xmm3 // [ u , u ] movapd %xmm3, %xmm4 addpd (DX_P), %xmm0 // [ u + a1 , u + a2 ] addpd 16(DX_P), %xmm3 // [ u + a3 , u + a4 ] mulpd %xmm4, %xmm0 // [ (u + a1)u , (u + a2)u ] mulpd %xmm4, %xmm3 // [ (u + a3)u , (u + a4)u ] mulsd %xmm4, %xmm1 // cx^3 addpd 32(DX_P), %xmm0 // [ (u + a1)u + b1 , (u + a2)u + b2 ] addpd 48(DX_P), %xmm3 // [ (u + a3)u + b3 , (u + a4)u + b4 ] mulpd %xmm3, %xmm0 // [ quad1 * quad3 , quad2 * quad4 ] mulsd %xmm0, %xmm1 // cx^3 * quad1 * quad3 movhlps %xmm0, %xmm0 mulsd %xmm1, %xmm0 // cx^3 * quad1 * quad2 * quad3 * quad4 addsd %xmm2, %xmm0 // x + c x^3 p(x^2) #if defined(__i386__) movsd %xmm0, FRAME_SIZE(STACKP) fldl FRAME_SIZE(STACKP) #endif ret 3: // Handle |x| < 2^-27 #if defined(__x86_64__) movd %xmm0, %rax cmp REL_ADDR(min_normal), %rax #else psrlq $32, %xmm0 movd %xmm0, %eax cmp $0x00100000, %eax #endif jb 4f movsd REL_ADDR(one_p55), %xmm0 mulsd %xmm1, %xmm0 subsd %xmm1, %xmm0 mulsd REL_ADDR(one_n55), %xmm0 // Raises underflow in round-to-zero for x = ±smallest normal #if defined(__i386__) movsd %xmm0, FRAME_SIZE(STACKP) fldl FRAME_SIZE(STACKP) #endif ret 4: // Handle denormal inputs xorpd %xmm3, %xmm3 comisd %xmm1, %xmm3 jz 5f movsd REL_ADDR(one), %xmm0 orpd %xmm0, %xmm1 orpd %xmm7, %xmm0 // copysign(1.0, x) subsd %xmm0, %xmm1 // 2^(1022)*x movsd REL_ADDR(min_normal), %xmm2 orpd %xmm7, %xmm2 // copysign(tiny, x) subsd %xmm2, %xmm1 // 2^(1022)*x - copysign(tiny,x) movsd REL_ADDR(min_normal), %xmm0 mulsd %xmm0, %xmm1 // x - 2^(-1022)*copysign(tiny,x) 5: movapd %xmm1, %xmm0 #if defined(__i386__) movsd %xmm0, FRAME_SIZE(STACKP) fldl FRAME_SIZE(STACKP) #endif ret 6: // Handle x = NaN, |x| > 22 movapd %xmm0, %xmm2 movsd REL_ADDR(one), %xmm0 orpd %xmm7, %xmm0 // copysign(1.0, x) movsd REL_ADDR(infinity), %xmm1 ucomisd %xmm2, %xmm1 jp 8f jz 7f movsd REL_ADDR(one_n55), %xmm1 orpd %xmm7, %xmm1 // copysign(1.0p-55, x) subsd %xmm1, %xmm0 7: #if defined(__i386__) movsd %xmm0, FRAME_SIZE(STACKP) fldl FRAME_SIZE(STACKP) #endif ret 8: addsd %xmm2, %xmm2 // quiet NaN movapd %xmm2, %xmm0 #if defined(__i386__) movsd %xmm0, FRAME_SIZE(STACKP) fldl FRAME_SIZE(STACKP) #endif ret