1/* Quad-precision floating point e^x. 2 Copyright (C) 1999-2018 Free Software Foundation, Inc. 3 This file is part of the GNU C Library. 4 Contributed by Jakub Jelinek <jj@ultra.linux.cz> 5 Partly based on double-precision code 6 by Geoffrey Keating <geoffk@ozemail.com.au> 7 8 The GNU C Library is free software; you can redistribute it and/or 9 modify it under the terms of the GNU Lesser General Public 10 License as published by the Free Software Foundation; either 11 version 2.1 of the License, or (at your option) any later version. 12 13 The GNU C Library is distributed in the hope that it will be useful, 14 but WITHOUT ANY WARRANTY; without even the implied warranty of 15 MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the GNU 16 Lesser General Public License for more details. 17 18 You should have received a copy of the GNU Lesser General Public 19 License along with the GNU C Library; if not, see 20 <http://www.gnu.org/licenses/>. */ 21 22/* The basic design here is from 23 Abraham Ziv, "Fast Evaluation of Elementary Mathematical Functions with 24 Correctly Rounded Last Bit", ACM Trans. Math. Soft., 17 (3), September 1991, 25 pp. 410-423. 26 27 We work with number pairs where the first number is the high part and 28 the second one is the low part. Arithmetic with the high part numbers must 29 be exact, without any roundoff errors. 30 31 The input value, X, is written as 32 X = n * ln(2)_0 + arg1[t1]_0 + arg2[t2]_0 + x 33 - n * ln(2)_1 + arg1[t1]_1 + arg2[t2]_1 + xl 34 35 where: 36 - n is an integer, 16384 >= n >= -16495; 37 - ln(2)_0 is the first 93 bits of ln(2), and |ln(2)_0-ln(2)-ln(2)_1| < 2^-205 38 - t1 is an integer, 89 >= t1 >= -89 39 - t2 is an integer, 65 >= t2 >= -65 40 - |arg1[t1]-t1/256.0| < 2^-53 41 - |arg2[t2]-t2/32768.0| < 2^-53 42 - x + xl is whatever is left, |x + xl| < 2^-16 + 2^-53 43 44 Then e^x is approximated as 45 46 e^x = 2^n_1 ( 2^n_0 e^(arg1[t1]_0 + arg1[t1]_1) e^(arg2[t2]_0 + arg2[t2]_1) 47 + 2^n_0 e^(arg1[t1]_0 + arg1[t1]_1) e^(arg2[t2]_0 + arg2[t2]_1) 48 * p (x + xl + n * ln(2)_1)) 49 where: 50 - p(x) is a polynomial approximating e(x)-1 51 - e^(arg1[t1]_0 + arg1[t1]_1) is obtained from a table 52 - e^(arg2[t2]_0 + arg2[t2]_1) likewise 53 - n_1 + n_0 = n, so that |n_0| < -FLT128_MIN_EXP-1. 54 55 If it happens that n_1 == 0 (this is the usual case), that multiplication 56 is omitted. 57 */ 58 59#ifndef _GNU_SOURCE 60#define _GNU_SOURCE 61#endif 62 63#include "quadmath-imp.h" 64#include "expq_table.h" 65 66static const __float128 C[] = { 67/* Smallest integer x for which e^x overflows. */ 68#define himark C[0] 69 11356.523406294143949491931077970765Q, 70 71/* Largest integer x for which e^x underflows. */ 72#define lomark C[1] 73-11433.4627433362978788372438434526231Q, 74 75/* 3x2^96 */ 76#define THREEp96 C[2] 77 59421121885698253195157962752.0Q, 78 79/* 3x2^103 */ 80#define THREEp103 C[3] 81 30423614405477505635920876929024.0Q, 82 83/* 3x2^111 */ 84#define THREEp111 C[4] 85 7788445287802241442795744493830144.0Q, 86 87/* 1/ln(2) */ 88#define M_1_LN2 C[5] 89 1.44269504088896340735992468100189204Q, 90 91/* first 93 bits of ln(2) */ 92#define M_LN2_0 C[6] 93 0.693147180559945309417232121457981864Q, 94 95/* ln2_0 - ln(2) */ 96#define M_LN2_1 C[7] 97-1.94704509238074995158795957333327386E-31Q, 98 99/* very small number */ 100#define TINY C[8] 101 1.0e-4900Q, 102 103/* 2^16383 */ 104#define TWO16383 C[9] 105 5.94865747678615882542879663314003565E+4931Q, 106 107/* 256 */ 108#define TWO8 C[10] 109 256, 110 111/* 32768 */ 112#define TWO15 C[11] 113 32768, 114 115/* Chebyshev polynom coefficients for (exp(x)-1)/x */ 116#define P1 C[12] 117#define P2 C[13] 118#define P3 C[14] 119#define P4 C[15] 120#define P5 C[16] 121#define P6 C[17] 122 0.5Q, 123 1.66666666666666666666666666666666683E-01Q, 124 4.16666666666666666666654902320001674E-02Q, 125 8.33333333333333333333314659767198461E-03Q, 126 1.38888888889899438565058018857254025E-03Q, 127 1.98412698413981650382436541785404286E-04Q, 128}; 129 130__float128 131expq (__float128 x) 132{ 133 /* Check for usual case. */ 134 if (__builtin_isless (x, himark) && __builtin_isgreater (x, lomark)) 135 { 136 int tval1, tval2, unsafe, n_i; 137 __float128 x22, n, t, result, xl; 138 ieee854_float128 ex2_u, scale_u; 139 fenv_t oldenv; 140 141 feholdexcept (&oldenv); 142#ifdef FE_TONEAREST 143 fesetround (FE_TONEAREST); 144#endif 145 146 /* Calculate n. */ 147 n = x * M_1_LN2 + THREEp111; 148 n -= THREEp111; 149 x = x - n * M_LN2_0; 150 xl = n * M_LN2_1; 151 152 /* Calculate t/256. */ 153 t = x + THREEp103; 154 t -= THREEp103; 155 156 /* Compute tval1 = t. */ 157 tval1 = (int) (t * TWO8); 158 159 x -= __expq_table[T_EXPL_ARG1+2*tval1]; 160 xl -= __expq_table[T_EXPL_ARG1+2*tval1+1]; 161 162 /* Calculate t/32768. */ 163 t = x + THREEp96; 164 t -= THREEp96; 165 166 /* Compute tval2 = t. */ 167 tval2 = (int) (t * TWO15); 168 169 x -= __expq_table[T_EXPL_ARG2+2*tval2]; 170 xl -= __expq_table[T_EXPL_ARG2+2*tval2+1]; 171 172 x = x + xl; 173 174 /* Compute ex2 = 2^n_0 e^(argtable[tval1]) e^(argtable[tval2]). */ 175 ex2_u.value = __expq_table[T_EXPL_RES1 + tval1] 176 * __expq_table[T_EXPL_RES2 + tval2]; 177 n_i = (int)n; 178 /* 'unsafe' is 1 iff n_1 != 0. */ 179 unsafe = abs(n_i) >= 15000; 180 ex2_u.ieee.exponent += n_i >> unsafe; 181 182 /* Compute scale = 2^n_1. */ 183 scale_u.value = 1; 184 scale_u.ieee.exponent += n_i - (n_i >> unsafe); 185 186 /* Approximate e^x2 - 1, using a seventh-degree polynomial, 187 with maximum error in [-2^-16-2^-53,2^-16+2^-53] 188 less than 4.8e-39. */ 189 x22 = x + x*x*(P1+x*(P2+x*(P3+x*(P4+x*(P5+x*P6))))); 190 math_force_eval (x22); 191 192 /* Return result. */ 193 fesetenv (&oldenv); 194 195 result = x22 * ex2_u.value + ex2_u.value; 196 197 /* Now we can test whether the result is ultimate or if we are unsure. 198 In the later case we should probably call a mpn based routine to give 199 the ultimate result. 200 Empirically, this routine is already ultimate in about 99.9986% of 201 cases, the test below for the round to nearest case will be false 202 in ~ 99.9963% of cases. 203 Without proc2 routine maximum error which has been seen is 204 0.5000262 ulp. 205 206 ieee854_float128 ex3_u; 207 208 #ifdef FE_TONEAREST 209 fesetround (FE_TONEAREST); 210 #endif 211 ex3_u.value = (result - ex2_u.value) - x22 * ex2_u.value; 212 ex2_u.value = result; 213 ex3_u.ieee.exponent += FLT128_MANT_DIG + 15 + IEEE854_FLOAT128_BIAS 214 - ex2_u.ieee.exponent; 215 n_i = abs (ex3_u.value); 216 n_i = (n_i + 1) / 2; 217 fesetenv (&oldenv); 218 #ifdef FE_TONEAREST 219 if (fegetround () == FE_TONEAREST) 220 n_i -= 0x4000; 221 #endif 222 if (!n_i) { 223 return __ieee754_expl_proc2 (origx); 224 } 225 */ 226 if (!unsafe) 227 return result; 228 else 229 { 230 result *= scale_u.value; 231 math_check_force_underflow_nonneg (result); 232 return result; 233 } 234 } 235 /* Exceptional cases: */ 236 else if (__builtin_isless (x, himark)) 237 { 238 if (isinfq (x)) 239 /* e^-inf == 0, with no error. */ 240 return 0; 241 else 242 /* Underflow */ 243 return TINY * TINY; 244 } 245 else 246 /* Return x, if x is a NaN or Inf; or overflow, otherwise. */ 247 return TWO16383*x; 248} 249