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249 lines
6.8 KiB
C
249 lines
6.8 KiB
C
/* Quad-precision floating point e^x.
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Copyright (C) 1999-2018 Free Software Foundation, Inc.
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This file is part of the GNU C Library.
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Contributed by Jakub Jelinek <jj@ultra.linux.cz>
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Partly based on double-precision code
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by Geoffrey Keating <geoffk@ozemail.com.au>
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The GNU C Library is free software; you can redistribute it and/or
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modify it under the terms of the GNU Lesser General Public
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License as published by the Free Software Foundation; either
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version 2.1 of the License, or (at your option) any later version.
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The GNU C Library is distributed in the hope that it will be useful,
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but WITHOUT ANY WARRANTY; without even the implied warranty of
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MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the GNU
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Lesser General Public License for more details.
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You should have received a copy of the GNU Lesser General Public
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License along with the GNU C Library; if not, see
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<http://www.gnu.org/licenses/>. */
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/* The basic design here is from
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Abraham Ziv, "Fast Evaluation of Elementary Mathematical Functions with
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Correctly Rounded Last Bit", ACM Trans. Math. Soft., 17 (3), September 1991,
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pp. 410-423.
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We work with number pairs where the first number is the high part and
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the second one is the low part. Arithmetic with the high part numbers must
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be exact, without any roundoff errors.
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The input value, X, is written as
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X = n * ln(2)_0 + arg1[t1]_0 + arg2[t2]_0 + x
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- n * ln(2)_1 + arg1[t1]_1 + arg2[t2]_1 + xl
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where:
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- n is an integer, 16384 >= n >= -16495;
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- ln(2)_0 is the first 93 bits of ln(2), and |ln(2)_0-ln(2)-ln(2)_1| < 2^-205
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- t1 is an integer, 89 >= t1 >= -89
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- t2 is an integer, 65 >= t2 >= -65
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- |arg1[t1]-t1/256.0| < 2^-53
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- |arg2[t2]-t2/32768.0| < 2^-53
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- x + xl is whatever is left, |x + xl| < 2^-16 + 2^-53
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Then e^x is approximated as
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e^x = 2^n_1 ( 2^n_0 e^(arg1[t1]_0 + arg1[t1]_1) e^(arg2[t2]_0 + arg2[t2]_1)
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+ 2^n_0 e^(arg1[t1]_0 + arg1[t1]_1) e^(arg2[t2]_0 + arg2[t2]_1)
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* p (x + xl + n * ln(2)_1))
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where:
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- p(x) is a polynomial approximating e(x)-1
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- e^(arg1[t1]_0 + arg1[t1]_1) is obtained from a table
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- e^(arg2[t2]_0 + arg2[t2]_1) likewise
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- n_1 + n_0 = n, so that |n_0| < -FLT128_MIN_EXP-1.
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If it happens that n_1 == 0 (this is the usual case), that multiplication
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is omitted.
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*/
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#ifndef _GNU_SOURCE
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#define _GNU_SOURCE
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#endif
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#include "quadmath-imp.h"
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#include "expq_table.h"
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static const __float128 C[] = {
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/* Smallest integer x for which e^x overflows. */
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#define himark C[0]
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11356.523406294143949491931077970765Q,
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/* Largest integer x for which e^x underflows. */
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#define lomark C[1]
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-11433.4627433362978788372438434526231Q,
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/* 3x2^96 */
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#define THREEp96 C[2]
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59421121885698253195157962752.0Q,
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/* 3x2^103 */
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#define THREEp103 C[3]
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30423614405477505635920876929024.0Q,
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/* 3x2^111 */
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#define THREEp111 C[4]
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7788445287802241442795744493830144.0Q,
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/* 1/ln(2) */
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#define M_1_LN2 C[5]
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1.44269504088896340735992468100189204Q,
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/* first 93 bits of ln(2) */
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#define M_LN2_0 C[6]
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0.693147180559945309417232121457981864Q,
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/* ln2_0 - ln(2) */
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#define M_LN2_1 C[7]
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-1.94704509238074995158795957333327386E-31Q,
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/* very small number */
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#define TINY C[8]
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1.0e-4900Q,
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/* 2^16383 */
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#define TWO16383 C[9]
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5.94865747678615882542879663314003565E+4931Q,
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/* 256 */
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#define TWO8 C[10]
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256,
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/* 32768 */
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#define TWO15 C[11]
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32768,
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/* Chebyshev polynom coefficients for (exp(x)-1)/x */
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#define P1 C[12]
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#define P2 C[13]
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#define P3 C[14]
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#define P4 C[15]
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#define P5 C[16]
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#define P6 C[17]
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0.5Q,
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1.66666666666666666666666666666666683E-01Q,
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4.16666666666666666666654902320001674E-02Q,
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8.33333333333333333333314659767198461E-03Q,
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1.38888888889899438565058018857254025E-03Q,
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1.98412698413981650382436541785404286E-04Q,
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};
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__float128
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expq (__float128 x)
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{
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/* Check for usual case. */
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if (__builtin_isless (x, himark) && __builtin_isgreater (x, lomark))
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{
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int tval1, tval2, unsafe, n_i;
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__float128 x22, n, t, result, xl;
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ieee854_float128 ex2_u, scale_u;
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fenv_t oldenv;
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feholdexcept (&oldenv);
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#ifdef FE_TONEAREST
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fesetround (FE_TONEAREST);
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#endif
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/* Calculate n. */
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n = x * M_1_LN2 + THREEp111;
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n -= THREEp111;
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x = x - n * M_LN2_0;
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xl = n * M_LN2_1;
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/* Calculate t/256. */
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t = x + THREEp103;
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t -= THREEp103;
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/* Compute tval1 = t. */
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tval1 = (int) (t * TWO8);
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x -= __expq_table[T_EXPL_ARG1+2*tval1];
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xl -= __expq_table[T_EXPL_ARG1+2*tval1+1];
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/* Calculate t/32768. */
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t = x + THREEp96;
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t -= THREEp96;
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/* Compute tval2 = t. */
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tval2 = (int) (t * TWO15);
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x -= __expq_table[T_EXPL_ARG2+2*tval2];
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xl -= __expq_table[T_EXPL_ARG2+2*tval2+1];
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x = x + xl;
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/* Compute ex2 = 2^n_0 e^(argtable[tval1]) e^(argtable[tval2]). */
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ex2_u.value = __expq_table[T_EXPL_RES1 + tval1]
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* __expq_table[T_EXPL_RES2 + tval2];
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n_i = (int)n;
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/* 'unsafe' is 1 iff n_1 != 0. */
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unsafe = abs(n_i) >= 15000;
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ex2_u.ieee.exponent += n_i >> unsafe;
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/* Compute scale = 2^n_1. */
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scale_u.value = 1;
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scale_u.ieee.exponent += n_i - (n_i >> unsafe);
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/* Approximate e^x2 - 1, using a seventh-degree polynomial,
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with maximum error in [-2^-16-2^-53,2^-16+2^-53]
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less than 4.8e-39. */
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x22 = x + x*x*(P1+x*(P2+x*(P3+x*(P4+x*(P5+x*P6)))));
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math_force_eval (x22);
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/* Return result. */
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fesetenv (&oldenv);
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result = x22 * ex2_u.value + ex2_u.value;
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/* Now we can test whether the result is ultimate or if we are unsure.
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In the later case we should probably call a mpn based routine to give
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the ultimate result.
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Empirically, this routine is already ultimate in about 99.9986% of
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cases, the test below for the round to nearest case will be false
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in ~ 99.9963% of cases.
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Without proc2 routine maximum error which has been seen is
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0.5000262 ulp.
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ieee854_float128 ex3_u;
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#ifdef FE_TONEAREST
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fesetround (FE_TONEAREST);
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#endif
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ex3_u.value = (result - ex2_u.value) - x22 * ex2_u.value;
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ex2_u.value = result;
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ex3_u.ieee.exponent += FLT128_MANT_DIG + 15 + IEEE854_FLOAT128_BIAS
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- ex2_u.ieee.exponent;
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n_i = abs (ex3_u.value);
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n_i = (n_i + 1) / 2;
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fesetenv (&oldenv);
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#ifdef FE_TONEAREST
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if (fegetround () == FE_TONEAREST)
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n_i -= 0x4000;
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#endif
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if (!n_i) {
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return __ieee754_expl_proc2 (origx);
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}
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*/
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if (!unsafe)
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return result;
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else
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{
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result *= scale_u.value;
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math_check_force_underflow_nonneg (result);
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return result;
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}
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}
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/* Exceptional cases: */
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else if (__builtin_isless (x, himark))
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{
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if (isinfq (x))
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/* e^-inf == 0, with no error. */
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return 0;
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else
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/* Underflow */
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return TINY * TINY;
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}
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else
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/* Return x, if x is a NaN or Inf; or overflow, otherwise. */
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return TWO16383*x;
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}
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