From 388b059643f6ccc72d82dda7ac2d768584c876c0 Mon Sep 17 00:00:00 2001 From: Martin Fouilleul Date: Sun, 24 Sep 2023 18:44:32 +0200 Subject: [PATCH] adding exp/expf --- src/libc-shim/include/math.h | 3 + src/libc-shim/src/exp.c | 136 ++++++++++++++++++++++++++ src/libc-shim/src/exp_data.c | 182 +++++++++++++++++++++++++++++++++++ src/libc-shim/src/expf.c | 80 +++++++++++++++ 4 files changed, 401 insertions(+) create mode 100644 src/libc-shim/src/exp.c create mode 100644 src/libc-shim/src/exp_data.c create mode 100644 src/libc-shim/src/expf.c diff --git a/src/libc-shim/include/math.h b/src/libc-shim/include/math.h index 7e186ee..737b629 100644 --- a/src/libc-shim/include/math.h +++ b/src/libc-shim/include/math.h @@ -84,6 +84,9 @@ float log2f(float); double pow(double, double); float powf(float, float); +double exp(double); +float expf(float); + double scalbn(double, int); double sin(double); diff --git a/src/libc-shim/src/exp.c b/src/libc-shim/src/exp.c new file mode 100644 index 0000000..2d28a94 --- /dev/null +++ b/src/libc-shim/src/exp.c @@ -0,0 +1,136 @@ +/* + * Double-precision e^x function. + * + * Copyright (c) 2018, Arm Limited. + * SPDX-License-Identifier: MIT + */ + +#include +#include +#include "libm.h" +#include "exp_data.h" + +#define N (1 << EXP_TABLE_BITS) +#define InvLn2N __exp_data.invln2N +#define NegLn2hiN __exp_data.negln2hiN +#define NegLn2loN __exp_data.negln2loN +#define Shift __exp_data.shift +#define T __exp_data.tab +#define C2 __exp_data.poly[5 - EXP_POLY_ORDER] +#define C3 __exp_data.poly[6 - EXP_POLY_ORDER] +#define C4 __exp_data.poly[7 - EXP_POLY_ORDER] +#define C5 __exp_data.poly[8 - EXP_POLY_ORDER] + +/* Handle cases that may overflow or underflow when computing the result that + is scale*(1+TMP) without intermediate rounding. The bit representation of + scale is in SBITS, however it has a computed exponent that may have + overflown into the sign bit so that needs to be adjusted before using it as + a double. (int32_t)KI is the k used in the argument reduction and exponent + adjustment of scale, positive k here means the result may overflow and + negative k means the result may underflow. */ +static inline double specialcase(double_t tmp, uint64_t sbits, uint64_t ki) +{ + double_t scale, y; + + if ((ki & 0x80000000) == 0) { + /* k > 0, the exponent of scale might have overflowed by <= 460. */ + sbits -= 1009ull << 52; + scale = asdouble(sbits); + y = 0x1p1009 * (scale + scale * tmp); + return eval_as_double(y); + } + /* k < 0, need special care in the subnormal range. */ + sbits += 1022ull << 52; + scale = asdouble(sbits); + y = scale + scale * tmp; + if (y < 1.0) { + /* Round y to the right precision before scaling it into the subnormal + range to avoid double rounding that can cause 0.5+E/2 ulp error where + E is the worst-case ulp error outside the subnormal range. So this + is only useful if the goal is better than 1 ulp worst-case error. */ + double_t hi, lo; + lo = scale - y + scale * tmp; + hi = 1.0 + y; + lo = 1.0 - hi + y + lo; + y = eval_as_double(hi + lo) - 1.0; + /* Avoid -0.0 with downward rounding. */ + if (WANT_ROUNDING && y == 0.0) + y = 0.0; + /* The underflow exception needs to be signaled explicitly. */ + //WARN(orca): we don't have fp_barrier in wasm + //fp_force_eval(fp_barrier(0x1p-1022) * 0x1p-1022); + fp_force_eval((0x1p-1022) * 0x1p-1022); + } + y = 0x1p-1022 * y; + return eval_as_double(y); +} + +/* Top 12 bits of a double (sign and exponent bits). */ +static inline uint32_t top12(double x) +{ + return asuint64(x) >> 52; +} + +double exp(double x) +{ + uint32_t abstop; + uint64_t ki, idx, top, sbits; + double_t kd, z, r, r2, scale, tail, tmp; + + abstop = top12(x) & 0x7ff; + if (predict_false(abstop - top12(0x1p-54) >= top12(512.0) - top12(0x1p-54))) { + if (abstop - top12(0x1p-54) >= 0x80000000) + /* Avoid spurious underflow for tiny x. */ + /* Note: 0 is common input. */ + return WANT_ROUNDING ? 1.0 + x : 1.0; + if (abstop >= top12(1024.0)) { + if (asuint64(x) == asuint64(-INFINITY)) + return 0.0; + if (abstop >= top12(INFINITY)) + return 1.0 + x; + if (asuint64(x) >> 63) + return __math_uflow(0); + else + return __math_oflow(0); + } + /* Large x is special cased below. */ + abstop = 0; + } + + /* exp(x) = 2^(k/N) * exp(r), with exp(r) in [2^(-1/2N),2^(1/2N)]. */ + /* x = ln2/N*k + r, with int k and r in [-ln2/2N, ln2/2N]. */ + z = InvLn2N * x; +#if TOINT_INTRINSICS + kd = roundtoint(z); + ki = converttoint(z); +#elif EXP_USE_TOINT_NARROW + /* z - kd is in [-0.5-2^-16, 0.5] in all rounding modes. */ + kd = eval_as_double(z + Shift); + ki = asuint64(kd) >> 16; + kd = (double_t)(int32_t)ki; +#else + /* z - kd is in [-1, 1] in non-nearest rounding modes. */ + kd = eval_as_double(z + Shift); + ki = asuint64(kd); + kd -= Shift; +#endif + r = x + kd * NegLn2hiN + kd * NegLn2loN; + /* 2^(k/N) ~= scale * (1 + tail). */ + idx = 2 * (ki % N); + top = ki << (52 - EXP_TABLE_BITS); + tail = asdouble(T[idx]); + /* This is only a valid scale when -1023*N < k < 1024*N. */ + sbits = T[idx + 1] + top; + /* exp(x) = 2^(k/N) * exp(r) ~= scale + scale * (tail + exp(r) - 1). */ + /* Evaluation is optimized assuming superscalar pipelined execution. */ + r2 = r * r; + /* Without fma the worst case error is 0.25/N ulp larger. */ + /* Worst case error is less than 0.5+1.11/N+(abs poly error * 2^53) ulp. */ + tmp = tail + r + r2 * (C2 + r * C3) + r2 * r2 * (C4 + r * C5); + if (predict_false(abstop == 0)) + return specialcase(tmp, sbits, ki); + scale = asdouble(sbits); + /* Note: tmp == 0 or |tmp| > 2^-200 and scale > 2^-739, so there + is no spurious underflow here even without fma. */ + return eval_as_double(scale + scale * tmp); +} diff --git a/src/libc-shim/src/exp_data.c b/src/libc-shim/src/exp_data.c new file mode 100644 index 0000000..21be014 --- /dev/null +++ b/src/libc-shim/src/exp_data.c @@ -0,0 +1,182 @@ +/* + * Shared data between exp, exp2 and pow. + * + * Copyright (c) 2018, Arm Limited. + * SPDX-License-Identifier: MIT + */ + +#include "exp_data.h" + +#define N (1 << EXP_TABLE_BITS) + +const struct exp_data __exp_data = { +// N/ln2 +.invln2N = 0x1.71547652b82fep0 * N, +// -ln2/N +.negln2hiN = -0x1.62e42fefa0000p-8, +.negln2loN = -0x1.cf79abc9e3b3ap-47, +// Used for rounding when !TOINT_INTRINSICS +#if EXP_USE_TOINT_NARROW +.shift = 0x1800000000.8p0, +#else +.shift = 0x1.8p52, +#endif +// exp polynomial coefficients. +.poly = { +// abs error: 1.555*2^-66 +// ulp error: 0.509 (0.511 without fma) +// if |x| < ln2/256+eps +// abs error if |x| < ln2/256+0x1p-15: 1.09*2^-65 +// abs error if |x| < ln2/128: 1.7145*2^-56 +0x1.ffffffffffdbdp-2, +0x1.555555555543cp-3, +0x1.55555cf172b91p-5, +0x1.1111167a4d017p-7, +}, +.exp2_shift = 0x1.8p52 / N, +// exp2 polynomial coefficients. +.exp2_poly = { +// abs error: 1.2195*2^-65 +// ulp error: 0.507 (0.511 without fma) +// if |x| < 1/256 +// abs error if |x| < 1/128: 1.9941*2^-56 +0x1.62e42fefa39efp-1, +0x1.ebfbdff82c424p-3, +0x1.c6b08d70cf4b5p-5, +0x1.3b2abd24650ccp-7, +0x1.5d7e09b4e3a84p-10, +}, +// 2^(k/N) ~= H[k]*(1 + T[k]) for int k in [0,N) +// tab[2*k] = asuint64(T[k]) +// tab[2*k+1] = asuint64(H[k]) - (k << 52)/N +.tab = { +0x0, 0x3ff0000000000000, +0x3c9b3b4f1a88bf6e, 0x3feff63da9fb3335, +0xbc7160139cd8dc5d, 0x3fefec9a3e778061, +0xbc905e7a108766d1, 0x3fefe315e86e7f85, +0x3c8cd2523567f613, 0x3fefd9b0d3158574, +0xbc8bce8023f98efa, 0x3fefd06b29ddf6de, +0x3c60f74e61e6c861, 0x3fefc74518759bc8, +0x3c90a3e45b33d399, 0x3fefbe3ecac6f383, +0x3c979aa65d837b6d, 0x3fefb5586cf9890f, +0x3c8eb51a92fdeffc, 0x3fefac922b7247f7, +0x3c3ebe3d702f9cd1, 0x3fefa3ec32d3d1a2, +0xbc6a033489906e0b, 0x3fef9b66affed31b, +0xbc9556522a2fbd0e, 0x3fef9301d0125b51, +0xbc5080ef8c4eea55, 0x3fef8abdc06c31cc, +0xbc91c923b9d5f416, 0x3fef829aaea92de0, +0x3c80d3e3e95c55af, 0x3fef7a98c8a58e51, +0xbc801b15eaa59348, 0x3fef72b83c7d517b, +0xbc8f1ff055de323d, 0x3fef6af9388c8dea, +0x3c8b898c3f1353bf, 0x3fef635beb6fcb75, +0xbc96d99c7611eb26, 0x3fef5be084045cd4, +0x3c9aecf73e3a2f60, 0x3fef54873168b9aa, +0xbc8fe782cb86389d, 0x3fef4d5022fcd91d, +0x3c8a6f4144a6c38d, 0x3fef463b88628cd6, +0x3c807a05b0e4047d, 0x3fef3f49917ddc96, +0x3c968efde3a8a894, 0x3fef387a6e756238, +0x3c875e18f274487d, 0x3fef31ce4fb2a63f, +0x3c80472b981fe7f2, 0x3fef2b4565e27cdd, +0xbc96b87b3f71085e, 0x3fef24dfe1f56381, +0x3c82f7e16d09ab31, 0x3fef1e9df51fdee1, +0xbc3d219b1a6fbffa, 0x3fef187fd0dad990, +0x3c8b3782720c0ab4, 0x3fef1285a6e4030b, +0x3c6e149289cecb8f, 0x3fef0cafa93e2f56, +0x3c834d754db0abb6, 0x3fef06fe0a31b715, +0x3c864201e2ac744c, 0x3fef0170fc4cd831, +0x3c8fdd395dd3f84a, 0x3feefc08b26416ff, +0xbc86a3803b8e5b04, 0x3feef6c55f929ff1, +0xbc924aedcc4b5068, 0x3feef1a7373aa9cb, +0xbc9907f81b512d8e, 0x3feeecae6d05d866, +0xbc71d1e83e9436d2, 0x3feee7db34e59ff7, +0xbc991919b3ce1b15, 0x3feee32dc313a8e5, +0x3c859f48a72a4c6d, 0x3feedea64c123422, +0xbc9312607a28698a, 0x3feeda4504ac801c, +0xbc58a78f4817895b, 0x3feed60a21f72e2a, +0xbc7c2c9b67499a1b, 0x3feed1f5d950a897, +0x3c4363ed60c2ac11, 0x3feece086061892d, +0x3c9666093b0664ef, 0x3feeca41ed1d0057, +0x3c6ecce1daa10379, 0x3feec6a2b5c13cd0, +0x3c93ff8e3f0f1230, 0x3feec32af0d7d3de, +0x3c7690cebb7aafb0, 0x3feebfdad5362a27, +0x3c931dbdeb54e077, 0x3feebcb299fddd0d, +0xbc8f94340071a38e, 0x3feeb9b2769d2ca7, +0xbc87deccdc93a349, 0x3feeb6daa2cf6642, +0xbc78dec6bd0f385f, 0x3feeb42b569d4f82, +0xbc861246ec7b5cf6, 0x3feeb1a4ca5d920f, +0x3c93350518fdd78e, 0x3feeaf4736b527da, +0x3c7b98b72f8a9b05, 0x3feead12d497c7fd, +0x3c9063e1e21c5409, 0x3feeab07dd485429, +0x3c34c7855019c6ea, 0x3feea9268a5946b7, +0x3c9432e62b64c035, 0x3feea76f15ad2148, +0xbc8ce44a6199769f, 0x3feea5e1b976dc09, +0xbc8c33c53bef4da8, 0x3feea47eb03a5585, +0xbc845378892be9ae, 0x3feea34634ccc320, +0xbc93cedd78565858, 0x3feea23882552225, +0x3c5710aa807e1964, 0x3feea155d44ca973, +0xbc93b3efbf5e2228, 0x3feea09e667f3bcd, +0xbc6a12ad8734b982, 0x3feea012750bdabf, +0xbc6367efb86da9ee, 0x3fee9fb23c651a2f, +0xbc80dc3d54e08851, 0x3fee9f7df9519484, +0xbc781f647e5a3ecf, 0x3fee9f75e8ec5f74, +0xbc86ee4ac08b7db0, 0x3fee9f9a48a58174, +0xbc8619321e55e68a, 0x3fee9feb564267c9, +0x3c909ccb5e09d4d3, 0x3feea0694fde5d3f, +0xbc7b32dcb94da51d, 0x3feea11473eb0187, +0x3c94ecfd5467c06b, 0x3feea1ed0130c132, +0x3c65ebe1abd66c55, 0x3feea2f336cf4e62, +0xbc88a1c52fb3cf42, 0x3feea427543e1a12, +0xbc9369b6f13b3734, 0x3feea589994cce13, +0xbc805e843a19ff1e, 0x3feea71a4623c7ad, +0xbc94d450d872576e, 0x3feea8d99b4492ed, +0x3c90ad675b0e8a00, 0x3feeaac7d98a6699, +0x3c8db72fc1f0eab4, 0x3feeace5422aa0db, +0xbc65b6609cc5e7ff, 0x3feeaf3216b5448c, +0x3c7bf68359f35f44, 0x3feeb1ae99157736, +0xbc93091fa71e3d83, 0x3feeb45b0b91ffc6, +0xbc5da9b88b6c1e29, 0x3feeb737b0cdc5e5, +0xbc6c23f97c90b959, 0x3feeba44cbc8520f, +0xbc92434322f4f9aa, 0x3feebd829fde4e50, +0xbc85ca6cd7668e4b, 0x3feec0f170ca07ba, +0x3c71affc2b91ce27, 0x3feec49182a3f090, +0x3c6dd235e10a73bb, 0x3feec86319e32323, +0xbc87c50422622263, 0x3feecc667b5de565, +0x3c8b1c86e3e231d5, 0x3feed09bec4a2d33, +0xbc91bbd1d3bcbb15, 0x3feed503b23e255d, +0x3c90cc319cee31d2, 0x3feed99e1330b358, +0x3c8469846e735ab3, 0x3feede6b5579fdbf, +0xbc82dfcd978e9db4, 0x3feee36bbfd3f37a, +0x3c8c1a7792cb3387, 0x3feee89f995ad3ad, +0xbc907b8f4ad1d9fa, 0x3feeee07298db666, +0xbc55c3d956dcaeba, 0x3feef3a2b84f15fb, +0xbc90a40e3da6f640, 0x3feef9728de5593a, +0xbc68d6f438ad9334, 0x3feeff76f2fb5e47, +0xbc91eee26b588a35, 0x3fef05b030a1064a, +0x3c74ffd70a5fddcd, 0x3fef0c1e904bc1d2, +0xbc91bdfbfa9298ac, 0x3fef12c25bd71e09, +0x3c736eae30af0cb3, 0x3fef199bdd85529c, +0x3c8ee3325c9ffd94, 0x3fef20ab5fffd07a, +0x3c84e08fd10959ac, 0x3fef27f12e57d14b, +0x3c63cdaf384e1a67, 0x3fef2f6d9406e7b5, +0x3c676b2c6c921968, 0x3fef3720dcef9069, +0xbc808a1883ccb5d2, 0x3fef3f0b555dc3fa, +0xbc8fad5d3ffffa6f, 0x3fef472d4a07897c, +0xbc900dae3875a949, 0x3fef4f87080d89f2, +0x3c74a385a63d07a7, 0x3fef5818dcfba487, +0xbc82919e2040220f, 0x3fef60e316c98398, +0x3c8e5a50d5c192ac, 0x3fef69e603db3285, +0x3c843a59ac016b4b, 0x3fef7321f301b460, +0xbc82d52107b43e1f, 0x3fef7c97337b9b5f, +0xbc892ab93b470dc9, 0x3fef864614f5a129, +0x3c74b604603a88d3, 0x3fef902ee78b3ff6, +0x3c83c5ec519d7271, 0x3fef9a51fbc74c83, +0xbc8ff7128fd391f0, 0x3fefa4afa2a490da, +0xbc8dae98e223747d, 0x3fefaf482d8e67f1, +0x3c8ec3bc41aa2008, 0x3fefba1bee615a27, +0x3c842b94c3a9eb32, 0x3fefc52b376bba97, +0x3c8a64a931d185ee, 0x3fefd0765b6e4540, +0xbc8e37bae43be3ed, 0x3fefdbfdad9cbe14, +0x3c77893b4d91cd9d, 0x3fefe7c1819e90d8, +0x3c5305c14160cc89, 0x3feff3c22b8f71f1, +}, +}; diff --git a/src/libc-shim/src/expf.c b/src/libc-shim/src/expf.c new file mode 100644 index 0000000..f9fbf8e --- /dev/null +++ b/src/libc-shim/src/expf.c @@ -0,0 +1,80 @@ +/* + * Single-precision e^x function. + * + * Copyright (c) 2017-2018, Arm Limited. + * SPDX-License-Identifier: MIT + */ + +#include +#include +#include "libm.h" +#include "exp2f_data.h" + +/* +EXP2F_TABLE_BITS = 5 +EXP2F_POLY_ORDER = 3 + +ULP error: 0.502 (nearest rounding.) +Relative error: 1.69 * 2^-34 in [-ln2/64, ln2/64] (before rounding.) +Wrong count: 170635 (all nearest rounding wrong results with fma.) +Non-nearest ULP error: 1 (rounded ULP error) +*/ + +#define N (1 << EXP2F_TABLE_BITS) +#define InvLn2N __exp2f_data.invln2_scaled +#define T __exp2f_data.tab +#define C __exp2f_data.poly_scaled + +static inline uint32_t top12(float x) +{ + return asuint(x) >> 20; +} + +float expf(float x) +{ + uint32_t abstop; + uint64_t ki, t; + double_t kd, xd, z, r, r2, y, s; + + xd = (double_t)x; + abstop = top12(x) & 0x7ff; + if (predict_false(abstop >= top12(88.0f))) { + /* |x| >= 88 or x is nan. */ + if (asuint(x) == asuint(-INFINITY)) + return 0.0f; + if (abstop >= top12(INFINITY)) + return x + x; + if (x > 0x1.62e42ep6f) /* x > log(0x1p128) ~= 88.72 */ + return __math_oflowf(0); + if (x < -0x1.9fe368p6f) /* x < log(0x1p-150) ~= -103.97 */ + return __math_uflowf(0); + } + + /* x*N/Ln2 = k + r with r in [-1/2, 1/2] and int k. */ + z = InvLn2N * xd; + + /* Round and convert z to int, the result is in [-150*N, 128*N] and + ideally ties-to-even rule is used, otherwise the magnitude of r + can be bigger which gives larger approximation error. */ +#if TOINT_INTRINSICS + kd = roundtoint(z); + ki = converttoint(z); +#else +# define SHIFT __exp2f_data.shift + kd = eval_as_double(z + SHIFT); + ki = asuint64(kd); + kd -= SHIFT; +#endif + r = z - kd; + + /* exp(x) = 2^(k/N) * 2^(r/N) ~= s * (C0*r^3 + C1*r^2 + C2*r + 1) */ + t = T[ki % N]; + t += ki << (52 - EXP2F_TABLE_BITS); + s = asdouble(t); + z = C[0] * r + C[1]; + r2 = r * r; + y = C[2] * r + 1; + y = z * r2 + y; + y = y * s; + return eval_as_float(y); +}