diff options
Diffstat (limited to 'src')
| -rw-r--r-- | src/internal/libm.h | 16 | ||||
| -rw-r--r-- | src/math/exp2f.c | 165 | ||||
| -rw-r--r-- | src/math/exp2f_data.c | 35 | ||||
| -rw-r--r-- | src/math/exp2f_data.h | 23 | ||||
| -rw-r--r-- | src/math/expf.c | 133 |
5 files changed, 193 insertions, 179 deletions
diff --git a/src/internal/libm.h b/src/internal/libm.h index 5212bab1..98bf5c68 100644 --- a/src/internal/libm.h +++ b/src/internal/libm.h @@ -64,6 +64,22 @@ union ldshape { /* Support signaling NaNs. */ #define WANT_SNAN 0 +#ifndef TOINT_INTRINSICS +#define TOINT_INTRINSICS 0 +#endif + +#if TOINT_INTRINSICS +/* Round x to nearest int in all rounding modes, ties have to be rounded + consistently with converttoint so the results match. If the result + would be outside of [-2^31, 2^31-1] then the semantics is unspecified. */ +static double_t roundtoint(double_t); + +/* Convert x to nearest int in all rounding modes, ties have to be rounded + consistently with roundtoint. If the result is not representible in an + int32_t then the semantics is unspecified. */ +static int32_t converttoint(double_t); +#endif + /* Helps static branch prediction so hot path can be better optimized. */ #ifdef __GNUC__ #define predict_true(x) __builtin_expect(!!(x), 1) diff --git a/src/math/exp2f.c b/src/math/exp2f.c index 296b6343..0360482c 100644 --- a/src/math/exp2f.c +++ b/src/math/exp2f.c @@ -1,126 +1,69 @@ -/* origin: FreeBSD /usr/src/lib/msun/src/s_exp2f.c */ -/*- - * Copyright (c) 2005 David Schultz <das@FreeBSD.ORG> - * All rights reserved. - * - * Redistribution and use in source and binary forms, with or without - * modification, are permitted provided that the following conditions - * are met: - * 1. Redistributions of source code must retain the above copyright - * notice, this list of conditions and the following disclaimer. - * 2. Redistributions in binary form must reproduce the above copyright - * notice, this list of conditions and the following disclaimer in the - * documentation and/or other materials provided with the distribution. +/* + * Single-precision 2^x function. * - * THIS SOFTWARE IS PROVIDED BY THE AUTHOR AND CONTRIBUTORS ``AS IS'' AND - * ANY EXPRESS OR IMPLIED WARRANTIES, INCLUDING, BUT NOT LIMITED TO, THE - * IMPLIED WARRANTIES OF MERCHANTABILITY AND FITNESS FOR A PARTICULAR PURPOSE - * ARE DISCLAIMED. IN NO EVENT SHALL THE AUTHOR OR CONTRIBUTORS BE LIABLE - * FOR ANY DIRECT, INDIRECT, INCIDENTAL, SPECIAL, EXEMPLARY, OR CONSEQUENTIAL - * DAMAGES (INCLUDING, BUT NOT LIMITED TO, PROCUREMENT OF SUBSTITUTE GOODS - * OR SERVICES; LOSS OF USE, DATA, OR PROFITS; OR BUSINESS INTERRUPTION) - * HOWEVER CAUSED AND ON ANY THEORY OF LIABILITY, WHETHER IN CONTRACT, STRICT - * LIABILITY, OR TORT (INCLUDING NEGLIGENCE OR OTHERWISE) ARISING IN ANY WAY - * OUT OF THE USE OF THIS SOFTWARE, EVEN IF ADVISED OF THE POSSIBILITY OF - * SUCH DAMAGE. + * Copyright (c) 2017-2018, Arm Limited. + * SPDX-License-Identifier: MIT */ +#include <math.h> +#include <stdint.h> #include "libm.h" +#include "exp2f_data.h" -#define TBLSIZE 16 +/* +EXP2F_TABLE_BITS = 5 +EXP2F_POLY_ORDER = 3 -static const float -redux = 0x1.8p23f / TBLSIZE, -P1 = 0x1.62e430p-1f, -P2 = 0x1.ebfbe0p-3f, -P3 = 0x1.c6b348p-5f, -P4 = 0x1.3b2c9cp-7f; +ULP error: 0.502 (nearest rounding.) +Relative error: 1.69 * 2^-34 in [-1/64, 1/64] (before rounding.) +Wrong count: 168353 (all nearest rounding wrong results with fma.) +Non-nearest ULP error: 1 (rounded ULP error) +*/ -static const double exp2ft[TBLSIZE] = { - 0x1.6a09e667f3bcdp-1, - 0x1.7a11473eb0187p-1, - 0x1.8ace5422aa0dbp-1, - 0x1.9c49182a3f090p-1, - 0x1.ae89f995ad3adp-1, - 0x1.c199bdd85529cp-1, - 0x1.d5818dcfba487p-1, - 0x1.ea4afa2a490dap-1, - 0x1.0000000000000p+0, - 0x1.0b5586cf9890fp+0, - 0x1.172b83c7d517bp+0, - 0x1.2387a6e756238p+0, - 0x1.306fe0a31b715p+0, - 0x1.3dea64c123422p+0, - 0x1.4bfdad5362a27p+0, - 0x1.5ab07dd485429p+0, -}; +#define N (1 << EXP2F_TABLE_BITS) +#define T __exp2f_data.tab +#define C __exp2f_data.poly +#define SHIFT __exp2f_data.shift_scaled + +static inline uint32_t top12(float x) +{ + return asuint(x) >> 20; +} -/* - * exp2f(x): compute the base 2 exponential of x - * - * Accuracy: Peak error < 0.501 ulp; location of peak: -0.030110927. - * - * Method: (equally-spaced tables) - * - * Reduce x: - * x = k + y, for integer k and |y| <= 1/2. - * Thus we have exp2f(x) = 2**k * exp2(y). - * - * Reduce y: - * y = i/TBLSIZE + z for integer i near y * TBLSIZE. - * Thus we have exp2(y) = exp2(i/TBLSIZE) * exp2(z), - * with |z| <= 2**-(TBLSIZE+1). - * - * We compute exp2(i/TBLSIZE) via table lookup and exp2(z) via a - * degree-4 minimax polynomial with maximum error under 1.4 * 2**-33. - * Using double precision for everything except the reduction makes - * roundoff error insignificant and simplifies the scaling step. - * - * This method is due to Tang, but I do not use his suggested parameters: - * - * Tang, P. Table-driven Implementation of the Exponential Function - * in IEEE Floating-Point Arithmetic. TOMS 15(2), 144-157 (1989). - */ float exp2f(float x) { - double_t t, r, z; - union {float f; uint32_t i;} u = {x}; - union {double f; uint64_t i;} uk; - uint32_t ix, i0, k; + uint32_t abstop; + uint64_t ki, t; + double_t kd, xd, z, r, r2, y, s; - /* Filter out exceptional cases. */ - ix = u.i & 0x7fffffff; - if (ix > 0x42fc0000) { /* |x| > 126 */ - if (ix > 0x7f800000) /* NaN */ - return x; - if (u.i >= 0x43000000 && u.i < 0x80000000) { /* x >= 128 */ - x *= 0x1p127f; - return x; - } - if (u.i >= 0x80000000) { /* x < -126 */ - if (u.i >= 0xc3160000 || (u.i & 0x0000ffff)) - FORCE_EVAL(-0x1p-149f/x); - if (u.i >= 0xc3160000) /* x <= -150 */ - return 0; - } - } else if (ix <= 0x33000000) { /* |x| <= 0x1p-25 */ - return 1.0f + x; + xd = (double_t)x; + abstop = top12(x) & 0x7ff; + if (predict_false(abstop >= top12(128.0f))) { + /* |x| >= 128 or x is nan. */ + if (asuint(x) == asuint(-INFINITY)) + return 0.0f; + if (abstop >= top12(INFINITY)) + return x + x; + if (x > 0.0f) + return __math_oflowf(0); + if (x <= -150.0f) + return __math_uflowf(0); } - /* Reduce x, computing z, i0, and k. */ - u.f = x + redux; - i0 = u.i; - i0 += TBLSIZE / 2; - k = i0 / TBLSIZE; - uk.i = (uint64_t)(0x3ff + k)<<52; - i0 &= TBLSIZE - 1; - u.f -= redux; - z = x - u.f; - /* Compute r = exp2(y) = exp2ft[i0] * p(z). */ - r = exp2ft[i0]; - t = r * z; - r = r + t * (P1 + z * P2) + t * (z * z) * (P3 + z * P4); + /* x = k/N + r with r in [-1/(2N), 1/(2N)] and int k. */ + kd = eval_as_double(xd + SHIFT); + ki = asuint64(kd); + kd -= SHIFT; /* k/N for int k. */ + r = xd - kd; - /* Scale by 2**k */ - return r * uk.f; + /* exp2(x) = 2^(k/N) * 2^r ~= 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); } diff --git a/src/math/exp2f_data.c b/src/math/exp2f_data.c new file mode 100644 index 00000000..be324727 --- /dev/null +++ b/src/math/exp2f_data.c @@ -0,0 +1,35 @@ +/* + * Shared data between expf, exp2f and powf. + * + * Copyright (c) 2017-2018, Arm Limited. + * SPDX-License-Identifier: MIT + */ + +#include "exp2f_data.h" + +#define N (1 << EXP2F_TABLE_BITS) + +const struct exp2f_data __exp2f_data = { + /* tab[i] = uint(2^(i/N)) - (i << 52-BITS) + used for computing 2^(k/N) for an int |k| < 150 N as + double(tab[k%N] + (k << 52-BITS)) */ + .tab = { +0x3ff0000000000000, 0x3fefd9b0d3158574, 0x3fefb5586cf9890f, 0x3fef9301d0125b51, +0x3fef72b83c7d517b, 0x3fef54873168b9aa, 0x3fef387a6e756238, 0x3fef1e9df51fdee1, +0x3fef06fe0a31b715, 0x3feef1a7373aa9cb, 0x3feedea64c123422, 0x3feece086061892d, +0x3feebfdad5362a27, 0x3feeb42b569d4f82, 0x3feeab07dd485429, 0x3feea47eb03a5585, +0x3feea09e667f3bcd, 0x3fee9f75e8ec5f74, 0x3feea11473eb0187, 0x3feea589994cce13, +0x3feeace5422aa0db, 0x3feeb737b0cdc5e5, 0x3feec49182a3f090, 0x3feed503b23e255d, +0x3feee89f995ad3ad, 0x3feeff76f2fb5e47, 0x3fef199bdd85529c, 0x3fef3720dcef9069, +0x3fef5818dcfba487, 0x3fef7c97337b9b5f, 0x3fefa4afa2a490da, 0x3fefd0765b6e4540, + }, + .shift_scaled = 0x1.8p+52 / N, + .poly = { + 0x1.c6af84b912394p-5, 0x1.ebfce50fac4f3p-3, 0x1.62e42ff0c52d6p-1, + }, + .shift = 0x1.8p+52, + .invln2_scaled = 0x1.71547652b82fep+0 * N, + .poly_scaled = { + 0x1.c6af84b912394p-5/N/N/N, 0x1.ebfce50fac4f3p-3/N/N, 0x1.62e42ff0c52d6p-1/N, + }, +}; diff --git a/src/math/exp2f_data.h b/src/math/exp2f_data.h new file mode 100644 index 00000000..fe744f15 --- /dev/null +++ b/src/math/exp2f_data.h @@ -0,0 +1,23 @@ +/* + * Copyright (c) 2017-2018, Arm Limited. + * SPDX-License-Identifier: MIT + */ +#ifndef _EXP2F_DATA_H +#define _EXP2F_DATA_H + +#include <features.h> +#include <stdint.h> + +/* Shared between expf, exp2f and powf. */ +#define EXP2F_TABLE_BITS 5 +#define EXP2F_POLY_ORDER 3 +extern hidden const struct exp2f_data { + uint64_t tab[1 << EXP2F_TABLE_BITS]; + double shift_scaled; + double poly[EXP2F_POLY_ORDER]; + double shift; + double invln2_scaled; + double poly_scaled[EXP2F_POLY_ORDER]; +} __exp2f_data; + +#endif diff --git a/src/math/expf.c b/src/math/expf.c index feee2b0e..f9fbf8e7 100644 --- a/src/math/expf.c +++ b/src/math/expf.c @@ -1,83 +1,80 @@ -/* origin: FreeBSD /usr/src/lib/msun/src/e_expf.c */ /* - * Conversion to float by Ian Lance Taylor, Cygnus Support, ian@cygnus.com. - */ -/* - * ==================================================== - * Copyright (C) 1993 by Sun Microsystems, Inc. All rights reserved. + * Single-precision e^x function. * - * Developed at SunPro, a Sun Microsystems, Inc. business. - * Permission to use, copy, modify, and distribute this - * software is freely granted, provided that this notice - * is preserved. - * ==================================================== + * Copyright (c) 2017-2018, Arm Limited. + * SPDX-License-Identifier: MIT */ +#include <math.h> +#include <stdint.h> #include "libm.h" +#include "exp2f_data.h" -static const float -half[2] = {0.5,-0.5}, -ln2hi = 6.9314575195e-1f, /* 0x3f317200 */ -ln2lo = 1.4286067653e-6f, /* 0x35bfbe8e */ -invln2 = 1.4426950216e+0f, /* 0x3fb8aa3b */ /* - * Domain [-0.34568, 0.34568], range ~[-4.278e-9, 4.447e-9]: - * |x*(exp(x)+1)/(exp(x)-1) - p(x)| < 2**-27.74 - */ -P1 = 1.6666625440e-1f, /* 0xaaaa8f.0p-26 */ -P2 = -2.7667332906e-3f; /* -0xb55215.0p-32 */ +EXP2F_TABLE_BITS = 5 +EXP2F_POLY_ORDER = 3 -float expf(float x) +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) { - float_t hi, lo, c, xx, y; - int k, sign; - uint32_t hx; + return asuint(x) >> 20; +} - GET_FLOAT_WORD(hx, x); - sign = hx >> 31; /* sign bit of x */ - hx &= 0x7fffffff; /* high word of |x| */ +float expf(float x) +{ + uint32_t abstop; + uint64_t ki, t; + double_t kd, xd, z, r, r2, y, s; - /* special cases */ - if (hx >= 0x42aeac50) { /* if |x| >= -87.33655f or NaN */ - if (hx > 0x7f800000) /* NaN */ - return x; - if (hx >= 0x42b17218 && !sign) { /* x >= 88.722839f */ - /* overflow */ - x *= 0x1p127f; - return x; - } - if (sign) { - /* underflow */ - FORCE_EVAL(-0x1p-149f/x); - if (hx >= 0x42cff1b5) /* x <= -103.972084f */ - return 0; - } + 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); } - /* argument reduction */ - if (hx > 0x3eb17218) { /* if |x| > 0.5 ln2 */ - if (hx > 0x3f851592) /* if |x| > 1.5 ln2 */ - k = invln2*x + half[sign]; - else - k = 1 - sign - sign; - hi = x - k*ln2hi; /* k*ln2hi is exact here */ - lo = k*ln2lo; - x = hi - lo; - } else if (hx > 0x39000000) { /* |x| > 2**-14 */ - k = 0; - hi = x; - lo = 0; - } else { - /* raise inexact */ - FORCE_EVAL(0x1p127f + x); - return 1 + x; - } + /* 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; - /* x is now in primary range */ - xx = x*x; - c = x - xx*(P1+xx*P2); - y = 1 + (x*c/(2-c) - lo + hi); - if (k == 0) - return y; - return scalbnf(y, k); + /* 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); } |