s_expl.c revision 271779
1238722Skargl/*- 2251315Skargl * Copyright (c) 2009-2013 Steven G. Kargl 3238722Skargl * All rights reserved. 4238722Skargl * 5238722Skargl * Redistribution and use in source and binary forms, with or without 6238722Skargl * modification, are permitted provided that the following conditions 7238722Skargl * are met: 8238722Skargl * 1. Redistributions of source code must retain the above copyright 9238722Skargl * notice unmodified, this list of conditions, and the following 10238722Skargl * disclaimer. 11238722Skargl * 2. Redistributions in binary form must reproduce the above copyright 12238722Skargl * notice, this list of conditions and the following disclaimer in the 13238722Skargl * documentation and/or other materials provided with the distribution. 14238722Skargl * 15238722Skargl * THIS SOFTWARE IS PROVIDED BY THE AUTHOR ``AS IS'' AND ANY EXPRESS OR 16238722Skargl * IMPLIED WARRANTIES, INCLUDING, BUT NOT LIMITED TO, THE IMPLIED WARRANTIES 17238722Skargl * OF MERCHANTABILITY AND FITNESS FOR A PARTICULAR PURPOSE ARE DISCLAIMED. 18238722Skargl * IN NO EVENT SHALL THE AUTHOR BE LIABLE FOR ANY DIRECT, INDIRECT, 19238722Skargl * INCIDENTAL, SPECIAL, EXEMPLARY, OR CONSEQUENTIAL DAMAGES (INCLUDING, BUT 20238722Skargl * NOT LIMITED TO, PROCUREMENT OF SUBSTITUTE GOODS OR SERVICES; LOSS OF USE, 21238722Skargl * DATA, OR PROFITS; OR BUSINESS INTERRUPTION) HOWEVER CAUSED AND ON ANY 22238722Skargl * THEORY OF LIABILITY, WHETHER IN CONTRACT, STRICT LIABILITY, OR TORT 23238722Skargl * (INCLUDING NEGLIGENCE OR OTHERWISE) ARISING IN ANY WAY OUT OF THE USE OF 24238722Skargl * THIS SOFTWARE, EVEN IF ADVISED OF THE POSSIBILITY OF SUCH DAMAGE. 25238722Skargl * 26238722Skargl * Optimized by Bruce D. Evans. 27238722Skargl */ 28238722Skargl 29238722Skargl#include <sys/cdefs.h> 30238722Skargl__FBSDID("$FreeBSD: stable/10/lib/msun/ld80/s_expl.c 271779 2014-09-18 15:10:22Z tijl $"); 31238722Skargl 32251316Skargl/** 33238722Skargl * Compute the exponential of x for Intel 80-bit format. This is based on: 34238722Skargl * 35238722Skargl * PTP Tang, "Table-driven implementation of the exponential function 36238722Skargl * in IEEE floating-point arithmetic," ACM Trans. Math. Soft., 15, 37238722Skargl * 144-157 (1989). 38238722Skargl * 39238784Skargl * where the 32 table entries have been expanded to INTERVALS (see below). 40238722Skargl */ 41238722Skargl 42238722Skargl#include <float.h> 43238722Skargl 44238722Skargl#ifdef __i386__ 45238722Skargl#include <ieeefp.h> 46238722Skargl#endif 47238722Skargl 48238783Skargl#include "fpmath.h" 49238722Skargl#include "math.h" 50238722Skargl#include "math_private.h" 51271779Stijl#include "k_expl.h" 52238722Skargl 53271779Stijl/* XXX Prevent compilers from erroneously constant folding these: */ 54271779Stijlstatic const volatile long double 55271779Stijlhuge = 0x1p10000L, 56271779Stijltiny = 0x1p-10000L; 57238722Skargl 58238722Skarglstatic const long double 59238722Skargltwom10000 = 0x1p-10000L; 60238722Skargl 61238722Skarglstatic const union IEEEl2bits 62238722Skargl/* log(2**16384 - 0.5) rounded towards zero: */ 63251328Skargl/* log(2**16384 - 0.5 + 1) rounded towards zero for expm1l() is the same: */ 64251328Skarglo_thresholdu = LD80C(0xb17217f7d1cf79ab, 13, 11356.5234062941439488L), 65251328Skargl#define o_threshold (o_thresholdu.e) 66238722Skargl/* log(2**(-16381-64-1)) rounded towards zero: */ 67251328Skarglu_thresholdu = LD80C(0xb21dfe7f09e2baa9, 13, -11399.4985314888605581L); 68251328Skargl#define u_threshold (u_thresholdu.e) 69238722Skargl 70238722Skargllong double 71238722Skarglexpl(long double x) 72238722Skargl{ 73271779Stijl union IEEEl2bits u; 74271779Stijl long double hi, lo, t, twopk; 75271779Stijl int k; 76238722Skargl uint16_t hx, ix; 77238722Skargl 78271779Stijl DOPRINT_START(&x); 79271779Stijl 80238722Skargl /* Filter out exceptional cases. */ 81238722Skargl u.e = x; 82238722Skargl hx = u.xbits.expsign; 83238722Skargl ix = hx & 0x7fff; 84238722Skargl if (ix >= BIAS + 13) { /* |x| >= 8192 or x is NaN */ 85238722Skargl if (ix == BIAS + LDBL_MAX_EXP) { 86251335Skargl if (hx & 0x8000) /* x is -Inf, -NaN or unsupported */ 87271779Stijl RETURNP(-1 / x); 88271779Stijl RETURNP(x + x); /* x is +Inf, +NaN or unsupported */ 89238722Skargl } 90251328Skargl if (x > o_threshold) 91271779Stijl RETURNP(huge * huge); 92251328Skargl if (x < u_threshold) 93271779Stijl RETURNP(tiny * tiny); 94271779Stijl } else if (ix < BIAS - 75) { /* |x| < 0x1p-75 (includes pseudos) */ 95271779Stijl RETURN2P(1, x); /* 1 with inexact iff x != 0 */ 96238722Skargl } 97238722Skargl 98238722Skargl ENTERI(); 99238722Skargl 100271779Stijl twopk = 1; 101271779Stijl __k_expl(x, &hi, &lo, &k); 102271779Stijl t = SUM2P(hi, lo); 103238722Skargl 104238722Skargl /* Scale by 2**k. */ 105238722Skargl if (k >= LDBL_MIN_EXP) { 106238722Skargl if (k == LDBL_MAX_EXP) 107251339Skargl RETURNI(t * 2 * 0x1p16383L); 108271779Stijl SET_LDBL_EXPSIGN(twopk, BIAS + k); 109238722Skargl RETURNI(t * twopk); 110238722Skargl } else { 111271779Stijl SET_LDBL_EXPSIGN(twopk, BIAS + k + 10000); 112271779Stijl RETURNI(t * twopk * twom10000); 113238722Skargl } 114238722Skargl} 115251343Skargl 116251343Skargl/** 117251343Skargl * Compute expm1l(x) for Intel 80-bit format. This is based on: 118251343Skargl * 119251343Skargl * PTP Tang, "Table-driven implementation of the Expm1 function 120251343Skargl * in IEEE floating-point arithmetic," ACM Trans. Math. Soft., 18, 121251343Skargl * 211-222 (1992). 122251343Skargl */ 123251343Skargl 124251343Skargl/* 125251343Skargl * Our T1 and T2 are chosen to be approximately the points where method 126251343Skargl * A and method B have the same accuracy. Tang's T1 and T2 are the 127251343Skargl * points where method A's accuracy changes by a full bit. For Tang, 128251343Skargl * this drop in accuracy makes method A immediately less accurate than 129251343Skargl * method B, but our larger INTERVALS makes method A 2 bits more 130251343Skargl * accurate so it remains the most accurate method significantly 131251343Skargl * closer to the origin despite losing the full bit in our extended 132251343Skargl * range for it. 133251343Skargl */ 134251343Skarglstatic const double 135251343SkarglT1 = -0.1659, /* ~-30.625/128 * log(2) */ 136251343SkarglT2 = 0.1659; /* ~30.625/128 * log(2) */ 137251343Skargl 138251343Skargl/* 139271779Stijl * Domain [-0.1659, 0.1659], range ~[-2.6155e-22, 2.5507e-23]: 140271779Stijl * |(exp(x)-1-x-x**2/2)/x - p(x)| < 2**-71.6 141271779Stijl * 142271779Stijl * XXX the coeffs aren't very carefully rounded, and I get 2.8 more bits, 143271779Stijl * but unlike for ld128 we can't drop any terms. 144251343Skargl */ 145251343Skarglstatic const union IEEEl2bits 146251343SkarglB3 = LD80C(0xaaaaaaaaaaaaaaab, -3, 1.66666666666666666671e-1L), 147251343SkarglB4 = LD80C(0xaaaaaaaaaaaaaaac, -5, 4.16666666666666666712e-2L); 148251343Skargl 149251343Skarglstatic const double 150251343SkarglB5 = 8.3333333333333245e-3, /* 0x1.111111111110cp-7 */ 151251343SkarglB6 = 1.3888888888888861e-3, /* 0x1.6c16c16c16c0ap-10 */ 152251343SkarglB7 = 1.9841269841532042e-4, /* 0x1.a01a01a0319f9p-13 */ 153251343SkarglB8 = 2.4801587302069236e-5, /* 0x1.a01a01a03cbbcp-16 */ 154251343SkarglB9 = 2.7557316558468562e-6, /* 0x1.71de37fd33d67p-19 */ 155251343SkarglB10 = 2.7557315829785151e-7, /* 0x1.27e4f91418144p-22 */ 156251343SkarglB11 = 2.5063168199779829e-8, /* 0x1.ae94fabdc6b27p-26 */ 157251343SkarglB12 = 2.0887164654459567e-9; /* 0x1.1f122d6413fe1p-29 */ 158251343Skargl 159251343Skargllong double 160251343Skarglexpm1l(long double x) 161251343Skargl{ 162251343Skargl union IEEEl2bits u, v; 163251343Skargl long double fn, hx2_hi, hx2_lo, q, r, r1, r2, t, twomk, twopk, x_hi; 164251343Skargl long double x_lo, x2, z; 165251343Skargl long double x4; 166251343Skargl int k, n, n2; 167251343Skargl uint16_t hx, ix; 168251343Skargl 169271779Stijl DOPRINT_START(&x); 170271779Stijl 171251343Skargl /* Filter out exceptional cases. */ 172251343Skargl u.e = x; 173251343Skargl hx = u.xbits.expsign; 174251343Skargl ix = hx & 0x7fff; 175251343Skargl if (ix >= BIAS + 6) { /* |x| >= 64 or x is NaN */ 176251343Skargl if (ix == BIAS + LDBL_MAX_EXP) { 177251343Skargl if (hx & 0x8000) /* x is -Inf, -NaN or unsupported */ 178271779Stijl RETURNP(-1 / x - 1); 179271779Stijl RETURNP(x + x); /* x is +Inf, +NaN or unsupported */ 180251343Skargl } 181251343Skargl if (x > o_threshold) 182271779Stijl RETURNP(huge * huge); 183251343Skargl /* 184251343Skargl * expm1l() never underflows, but it must avoid 185251343Skargl * unrepresentable large negative exponents. We used a 186251343Skargl * much smaller threshold for large |x| above than in 187251343Skargl * expl() so as to handle not so large negative exponents 188251343Skargl * in the same way as large ones here. 189251343Skargl */ 190251343Skargl if (hx & 0x8000) /* x <= -64 */ 191271779Stijl RETURN2P(tiny, -1); /* good for x < -65ln2 - eps */ 192251343Skargl } 193251343Skargl 194251343Skargl ENTERI(); 195251343Skargl 196251343Skargl if (T1 < x && x < T2) { 197271779Stijl if (ix < BIAS - 74) { /* |x| < 0x1p-74 (includes pseudos) */ 198251343Skargl /* x (rounded) with inexact if x != 0: */ 199271779Stijl RETURNPI(x == 0 ? x : 200251343Skargl (0x1p100 * x + fabsl(x)) * 0x1p-100); 201251343Skargl } 202251343Skargl 203251343Skargl x2 = x * x; 204251343Skargl x4 = x2 * x2; 205251343Skargl q = x4 * (x2 * (x4 * 206251343Skargl /* 207251343Skargl * XXX the number of terms is no longer good for 208251343Skargl * pairwise grouping of all except B3, and the 209251343Skargl * grouping is no longer from highest down. 210251343Skargl */ 211251343Skargl (x2 * B12 + (x * B11 + B10)) + 212251343Skargl (x2 * (x * B9 + B8) + (x * B7 + B6))) + 213251343Skargl (x * B5 + B4.e)) + x2 * x * B3.e; 214251343Skargl 215251343Skargl x_hi = (float)x; 216251343Skargl x_lo = x - x_hi; 217251343Skargl hx2_hi = x_hi * x_hi / 2; 218251343Skargl hx2_lo = x_lo * (x + x_hi) / 2; 219251343Skargl if (ix >= BIAS - 7) 220271779Stijl RETURN2PI(hx2_hi + x_hi, hx2_lo + x_lo + q); 221251343Skargl else 222271779Stijl RETURN2PI(x, hx2_lo + q + hx2_hi); 223251343Skargl } 224251343Skargl 225251343Skargl /* Reduce x to (k*ln2 + endpoint[n2] + r1 + r2). */ 226251343Skargl /* Use a specialized rint() to get fn. Assume round-to-nearest. */ 227251343Skargl fn = x * INV_L + 0x1.8p63 - 0x1.8p63; 228251343Skargl#if defined(HAVE_EFFICIENT_IRINTL) 229251343Skargl n = irintl(fn); 230251343Skargl#elif defined(HAVE_EFFICIENT_IRINT) 231251343Skargl n = irint(fn); 232251343Skargl#else 233251343Skargl n = (int)fn; 234251343Skargl#endif 235251343Skargl n2 = (unsigned)n % INTERVALS; 236251343Skargl k = n >> LOG2_INTERVALS; 237251343Skargl r1 = x - fn * L1; 238251343Skargl r2 = fn * -L2; 239251343Skargl r = r1 + r2; 240251343Skargl 241251343Skargl /* Prepare scale factor. */ 242251343Skargl v.e = 1; 243251343Skargl v.xbits.expsign = BIAS + k; 244251343Skargl twopk = v.e; 245251343Skargl 246251343Skargl /* 247251343Skargl * Evaluate lower terms of 248251343Skargl * expl(endpoint[n2] + r1 + r2) = tbl[n2] * expl(r1 + r2). 249251343Skargl */ 250251343Skargl z = r * r; 251251343Skargl q = r2 + z * (A2 + r * A3) + z * z * (A4 + r * A5) + z * z * z * A6; 252251343Skargl 253251343Skargl t = (long double)tbl[n2].lo + tbl[n2].hi; 254251343Skargl 255251343Skargl if (k == 0) { 256271779Stijl t = SUM2P(tbl[n2].hi - 1, tbl[n2].lo * (r1 + 1) + t * q + 257271779Stijl tbl[n2].hi * r1); 258251343Skargl RETURNI(t); 259251343Skargl } 260251343Skargl if (k == -1) { 261271779Stijl t = SUM2P(tbl[n2].hi - 2, tbl[n2].lo * (r1 + 1) + t * q + 262271779Stijl tbl[n2].hi * r1); 263251343Skargl RETURNI(t / 2); 264251343Skargl } 265251343Skargl if (k < -7) { 266271779Stijl t = SUM2P(tbl[n2].hi, tbl[n2].lo + t * (q + r1)); 267251343Skargl RETURNI(t * twopk - 1); 268251343Skargl } 269251343Skargl if (k > 2 * LDBL_MANT_DIG - 1) { 270271779Stijl t = SUM2P(tbl[n2].hi, tbl[n2].lo + t * (q + r1)); 271251343Skargl if (k == LDBL_MAX_EXP) 272251343Skargl RETURNI(t * 2 * 0x1p16383L - 1); 273251343Skargl RETURNI(t * twopk - 1); 274251343Skargl } 275251343Skargl 276251343Skargl v.xbits.expsign = BIAS - k; 277251343Skargl twomk = v.e; 278251343Skargl 279251343Skargl if (k > LDBL_MANT_DIG - 1) 280271779Stijl t = SUM2P(tbl[n2].hi, tbl[n2].lo - twomk + t * (q + r1)); 281251343Skargl else 282271779Stijl t = SUM2P(tbl[n2].hi - twomk, tbl[n2].lo + t * (q + r1)); 283251343Skargl RETURNI(t * twopk); 284251343Skargl} 285