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/*-
* Copyright (c) 2009-2012 Steven G. Kargl
* 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 unmodified, 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.
*
* THIS SOFTWARE IS PROVIDED BY THE AUTHOR ``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 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.
*
* Optimized by Bruce D. Evans.
*/
#include <sys/cdefs.h>
__FBSDID("$FreeBSD$");
/*-
* Compute the exponential of x for Intel 80-bit format. This is based on:
*
* PTP Tang, "Table-driven implementation of the exponential function
* in IEEE floating-point arithmetic," ACM Trans. Math. Soft., 15,
* 144-157 (1989).
*
* where the 32 table entries have been expanded to INTERVALS (see below).
*/
#include <float.h>
#ifdef __i386__
#include <ieeefp.h>
#endif
#include "fpmath.h"
#include "math.h"
#include "math_private.h"
#define INTERVALS 128
#define BIAS (LDBL_MAX_EXP - 1)
static const long double
huge = 0x1p10000L,
twom10000 = 0x1p-10000L;
/* XXX Prevent gcc from erroneously constant folding this: */
static volatile const long double tiny = 0x1p-10000L;
static const union IEEEl2bits
/* log(2**16384 - 0.5) rounded towards zero: */
o_threshold = LD80C(0xb17217f7d1cf79ab, 13, 11356.5234062941439488L),
/* log(2**(-16381-64-1)) rounded towards zero: */
u_threshold = LD80C(0xb21dfe7f09e2baa9, 13, -11399.4985314888605581L);
static const double
/*
* ln2/INTERVALS = L1+L2 (hi+lo decomposition for multiplication). L1 must
* have at least 22 (= log2(|LDBL_MIN_EXP-extras|) + log2(INTERVALS)) lowest
* bits zero so that multiplication of it by n is exact.
*/
INV_L = 1.8466496523378731e+2, /* 0x171547652b82fe.0p-45 */
L1 = 5.4152123484527692e-3, /* 0x162e42ff000000.0p-60 */
L2 = -3.2819649005320973e-13, /* -0x1718432a1b0e26.0p-94 */
/*
* Domain [-0.002708, 0.002708], range ~[-5.7136e-24, 5.7110e-24]:
* |exp(x) - p(x)| < 2**-77.2
* (0.002708 is ln2/(2*INTERVALS) rounded up a little).
*/
P2 = 0.5,
P3 = 1.6666666666666119e-1, /* 0x15555555555490.0p-55 */
P4 = 4.1666666666665887e-2, /* 0x155555555554e5.0p-57 */
P5 = 8.3333354987869413e-3, /* 0x1111115b789919.0p-59 */
P6 = 1.3888891738560272e-3; /* 0x16c16c651633ae.0p-62 */
/*
* 2^(i/INTERVALS) for i in [0,INTERVALS] is represented by two values where
* the first 53 bits of the significand are stored in hi and the next 53
* bits are in lo. Tang's paper states that the trailing 6 bits of hi must
* be zero for his algorithm in both single and double precision, because
* the table is re-used in the implementation of expm1() where a floating
* point addition involving hi must be exact. Here hi is double, so
* converting it to long double gives 11 trailing zero bits.
*/
static const struct {
double hi;
double lo;
/* XXX should rename 's'. */
} s[INTERVALS] = {
0x1p+0, 0x0p+0,
0x1.0163da9fb3335p+0, 0x1.b61299ab8cdb7p-54,
0x1.02c9a3e778060p+0, 0x1.dcdef95949ef4p-53,
0x1.04315e86e7f84p+0, 0x1.7ae71f3441b49p-53,
0x1.059b0d3158574p+0, 0x1.d73e2a475b465p-55,
0x1.0706b29ddf6ddp+0, 0x1.8db880753b0f6p-53,
0x1.0874518759bc8p+0, 0x1.186be4bb284ffp-57,
0x1.09e3ecac6f383p+0, 0x1.1487818316136p-54,
0x1.0b5586cf9890fp+0, 0x1.8a62e4adc610bp-54,
0x1.0cc922b7247f7p+0, 0x1.01edc16e24f71p-54,
0x1.0e3ec32d3d1a2p+0, 0x1.03a1727c57b53p-59,
0x1.0fb66affed31ap+0, 0x1.e464123bb1428p-53,
0x1.11301d0125b50p+0, 0x1.49d77e35db263p-53,
0x1.12abdc06c31cbp+0, 0x1.f72575a649ad2p-53,
0x1.1429aaea92ddfp+0, 0x1.66820328764b1p-53,
0x1.15a98c8a58e51p+0, 0x1.2406ab9eeab0ap-55,
0x1.172b83c7d517ap+0, 0x1.b9bef918a1d63p-53,
0x1.18af9388c8de9p+0, 0x1.777ee1734784ap-53,
0x1.1a35beb6fcb75p+0, 0x1.e5b4c7b4968e4p-55,
0x1.1bbe084045cd3p+0, 0x1.3563ce56884fcp-53,
0x1.1d4873168b9aap+0, 0x1.e016e00a2643cp-54,
0x1.1ed5022fcd91cp+0, 0x1.71033fec2243ap-53,
0x1.2063b88628cd6p+0, 0x1.dc775814a8495p-55,
0x1.21f49917ddc96p+0, 0x1.2a97e9494a5eep-55,
0x1.2387a6e756238p+0, 0x1.9b07eb6c70573p-54,
0x1.251ce4fb2a63fp+0, 0x1.ac155bef4f4a4p-55,
0x1.26b4565e27cddp+0, 0x1.2bd339940e9d9p-55,
0x1.284dfe1f56380p+0, 0x1.2d9e2b9e07941p-53,
0x1.29e9df51fdee1p+0, 0x1.612e8afad1255p-55,
0x1.2b87fd0dad98fp+0, 0x1.fbbd48ca71f95p-53,
0x1.2d285a6e4030bp+0, 0x1.0024754db41d5p-54,
0x1.2ecafa93e2f56p+0, 0x1.1ca0f45d52383p-56,
0x1.306fe0a31b715p+0, 0x1.6f46ad23182e4p-55,
0x1.32170fc4cd831p+0, 0x1.a9ce78e18047cp-55,
0x1.33c08b26416ffp+0, 0x1.32721843659a6p-54,
0x1.356c55f929ff0p+0, 0x1.928c468ec6e76p-53,
0x1.371a7373aa9cap+0, 0x1.4e28aa05e8a8fp-53,
0x1.38cae6d05d865p+0, 0x1.0b53961b37da2p-53,
0x1.3a7db34e59ff6p+0, 0x1.d43792533c144p-53,
0x1.3c32dc313a8e4p+0, 0x1.08003e4516b1ep-53,
0x1.3dea64c123422p+0, 0x1.ada0911f09ebcp-55,
0x1.3fa4504ac801bp+0, 0x1.417ee03548306p-53,
0x1.4160a21f72e29p+0, 0x1.f0864b71e7b6cp-53,
0x1.431f5d950a896p+0, 0x1.b8e088728219ap-53,
0x1.44e086061892dp+0, 0x1.89b7a04ef80d0p-59,
0x1.46a41ed1d0057p+0, 0x1.c944bd1648a76p-54,
0x1.486a2b5c13cd0p+0, 0x1.3c1a3b69062f0p-56,
0x1.4a32af0d7d3dep+0, 0x1.9cb62f3d1be56p-54,
0x1.4bfdad5362a27p+0, 0x1.d4397afec42e2p-56,
0x1.4dcb299fddd0dp+0, 0x1.8ecdbbc6a7833p-54,
0x1.4f9b2769d2ca6p+0, 0x1.5a67b16d3540ep-53,
0x1.516daa2cf6641p+0, 0x1.8225ea5909b04p-53,
0x1.5342b569d4f81p+0, 0x1.be1507893b0d5p-53,
0x1.551a4ca5d920ep+0, 0x1.8a5d8c4048699p-53,
0x1.56f4736b527dap+0, 0x1.9bb2c011d93adp-54,
0x1.58d12d497c7fdp+0, 0x1.295e15b9a1de8p-55,
0x1.5ab07dd485429p+0, 0x1.6324c054647adp-54,
0x1.5c9268a5946b7p+0, 0x1.c4b1b816986a2p-60,
0x1.5e76f15ad2148p+0, 0x1.ba6f93080e65ep-54,
0x1.605e1b976dc08p+0, 0x1.60edeb25490dcp-53,
0x1.6247eb03a5584p+0, 0x1.63e1f40dfa5b5p-53,
0x1.6434634ccc31fp+0, 0x1.8edf0e2989db3p-53,
0x1.6623882552224p+0, 0x1.224fb3c5371e6p-53,
0x1.68155d44ca973p+0, 0x1.038ae44f73e65p-57,
0x1.6a09e667f3bccp+0, 0x1.21165f626cdd5p-53,
0x1.6c012750bdabep+0, 0x1.daed533001e9ep-53,
0x1.6dfb23c651a2ep+0, 0x1.e441c597c3775p-53,
0x1.6ff7df9519483p+0, 0x1.9f0fc369e7c42p-53,
0x1.71f75e8ec5f73p+0, 0x1.ba46e1e5de15ap-53,
0x1.73f9a48a58173p+0, 0x1.7ab9349cd1562p-53,
0x1.75feb564267c8p+0, 0x1.7edd354674916p-53,
0x1.780694fde5d3fp+0, 0x1.866b80a02162dp-54,
0x1.7a11473eb0186p+0, 0x1.afaa2047ed9b4p-53,
0x1.7c1ed0130c132p+0, 0x1.f124cd1164dd6p-54,
0x1.7e2f336cf4e62p+0, 0x1.05d02ba15797ep-56,
0x1.80427543e1a11p+0, 0x1.6c1bccec9346bp-53,
0x1.82589994cce12p+0, 0x1.159f115f56694p-53,
0x1.8471a4623c7acp+0, 0x1.9ca5ed72f8c81p-53,
0x1.868d99b4492ecp+0, 0x1.01c83b21584a3p-53,
0x1.88ac7d98a6699p+0, 0x1.994c2f37cb53ap-54,
0x1.8ace5422aa0dbp+0, 0x1.6e9f156864b27p-54,
0x1.8cf3216b5448bp+0, 0x1.de55439a2c38bp-53,
0x1.8f1ae99157736p+0, 0x1.5cc13a2e3976cp-55,
0x1.9145b0b91ffc5p+0, 0x1.114c368d3ed6ep-53,
0x1.93737b0cdc5e4p+0, 0x1.e8a0387e4a814p-53,
0x1.95a44cbc8520ep+0, 0x1.d36906d2b41f9p-53,
0x1.97d829fde4e4fp+0, 0x1.173d241f23d18p-53,
0x1.9a0f170ca07b9p+0, 0x1.7462137188ce7p-53,
0x1.9c49182a3f090p+0, 0x1.c7c46b071f2bep-56,
0x1.9e86319e32323p+0, 0x1.824ca78e64c6ep-56,
0x1.a0c667b5de564p+0, 0x1.6535b51719567p-53,
0x1.a309bec4a2d33p+0, 0x1.6305c7ddc36abp-54,
0x1.a5503b23e255cp+0, 0x1.1684892395f0fp-53,
0x1.a799e1330b358p+0, 0x1.bcb7ecac563c7p-54,
0x1.a9e6b5579fdbfp+0, 0x1.0fac90ef7fd31p-54,
0x1.ac36bbfd3f379p+0, 0x1.81b72cd4624ccp-53,
0x1.ae89f995ad3adp+0, 0x1.7a1cd345dcc81p-54,
0x1.b0e07298db665p+0, 0x1.2108559bf8deep-53,
0x1.b33a2b84f15fap+0, 0x1.ed7fa1cf7b290p-53,
0x1.b59728de55939p+0, 0x1.1c7102222c90ep-53,
0x1.b7f76f2fb5e46p+0, 0x1.d54f610356a79p-53,
0x1.ba5b030a10649p+0, 0x1.0819678d5eb69p-53,
0x1.bcc1e904bc1d2p+0, 0x1.23dd07a2d9e84p-55,
0x1.bf2c25bd71e08p+0, 0x1.0811ae04a31c7p-53,
0x1.c199bdd85529cp+0, 0x1.11065895048ddp-55,
0x1.c40ab5fffd07ap+0, 0x1.b4537e083c60ap-54,
0x1.c67f12e57d14bp+0, 0x1.2884dff483cadp-54,
0x1.c8f6d9406e7b5p+0, 0x1.1acbc48805c44p-56,
0x1.cb720dcef9069p+0, 0x1.503cbd1e949dbp-56,
0x1.cdf0b555dc3f9p+0, 0x1.889f12b1f58a3p-53,
0x1.d072d4a07897bp+0, 0x1.1a1e45e4342b2p-53,
0x1.d2f87080d89f1p+0, 0x1.15bc247313d44p-53,
0x1.d5818dcfba487p+0, 0x1.2ed02d75b3707p-55,
0x1.d80e316c98397p+0, 0x1.7709f3a09100cp-53,
0x1.da9e603db3285p+0, 0x1.c2300696db532p-54,
0x1.dd321f301b460p+0, 0x1.2da5778f018c3p-54,
0x1.dfc97337b9b5ep+0, 0x1.72d195873da52p-53,
0x1.e264614f5a128p+0, 0x1.424ec3f42f5b5p-53,
0x1.e502ee78b3ff6p+0, 0x1.39e8980a9cc8fp-55,
0x1.e7a51fbc74c83p+0, 0x1.2d522ca0c8de2p-54,
0x1.ea4afa2a490d9p+0, 0x1.0b1ee7431ebb6p-53,
0x1.ecf482d8e67f0p+0, 0x1.1b60625f7293ap-53,
0x1.efa1bee615a27p+0, 0x1.dc7f486a4b6b0p-54,
0x1.f252b376bba97p+0, 0x1.3a1a5bf0d8e43p-54,
0x1.f50765b6e4540p+0, 0x1.9d3e12dd8a18bp-54,
0x1.f7bfdad9cbe13p+0, 0x1.1227697fce57bp-53,
0x1.fa7c1819e90d8p+0, 0x1.74853f3a5931ep-55,
0x1.fd3c22b8f71f1p+0, 0x1.2eb74966579e7p-57
};
long double
expl(long double x)
{
union IEEEl2bits u, v;
long double fn, q, r, r1, r2, t, t23, t45, twopk, twopkp10000, z;
int k, n, n2;
uint16_t hx, ix;
/* Filter out exceptional cases. */
u.e = x;
hx = u.xbits.expsign;
ix = hx & 0x7fff;
if (ix >= BIAS + 13) { /* |x| >= 8192 or x is NaN */
if (ix == BIAS + LDBL_MAX_EXP) {
if (hx & 0x8000 && u.xbits.man == 1ULL << 63)
return (0.0L); /* x is -Inf */
return (x + x); /* x is +Inf, NaN or unsupported */
}
if (x > o_threshold.e)
return (huge * huge);
if (x < u_threshold.e)
return (tiny * tiny);
} else if (ix < BIAS - 66) { /* |x| < 0x1p-66 */
/* includes pseudo-denormals */
if (huge + x > 1.0L) /* trigger inexact iff x != 0 */
return (1.0L + x);
}
ENTERI();
/* Reduce x to (k*ln2 + midpoint[n2] + r1 + r2). */
/* Use a specialized rint() to get fn. Assume round-to-nearest. */
fn = x * INV_L + 0x1.8p63 - 0x1.8p63;
r = x - fn * L1 - fn * L2; /* r = r1 + r2 done independently. */
#if defined(HAVE_EFFICIENT_IRINTL)
n = irintl(fn);
#elif defined(HAVE_EFFICIENT_IRINT)
n = irint(fn);
#else
n = (int)fn;
#endif
n2 = (unsigned)n % INTERVALS;
k = (n - n2) / INTERVALS;
r1 = x - fn * L1;
r2 = -fn * L2;
/* Prepare scale factors. */
v.xbits.man = 1ULL << 63;
if (k >= LDBL_MIN_EXP) {
v.xbits.expsign = BIAS + k;
twopk = v.e;
} else {
v.xbits.expsign = BIAS + k + 10000;
twopkp10000 = v.e;
}
/* Evaluate expl(midpoint[n2] + r1 + r2) = s[n2] * expl(r1 + r2). */
/* Here q = q(r), not q(r1), since r1 is lopped like L1. */
t45 = r * P5 + P4;
z = r * r;
t23 = r * P3 + P2;
q = r2 + z * t23 + z * z * t45 + z * z * z * P6;
t = (long double)s[n2].lo + s[n2].hi;
t = s[n2].lo + t * (q + r1) + s[n2].hi;
/* Scale by 2**k. */
if (k >= LDBL_MIN_EXP) {
if (k == LDBL_MAX_EXP)
RETURNI(t * 2.0L * 0x1p16383L);
RETURNI(t * twopk);
} else {
RETURNI(t * twopkp10000 * twom10000);
}
}
|
