sysdeps/ieee754/ldbl-128/e_expl.c

   1 /* Quad-precision floating point e^x.
   2    Copyright (C) 1999 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, write to the Free
  20    Software Foundation, Inc., 59 Temple Place, Suite 330, Boston, MA
  21    02111-1307 USA.  */
  22
  23 /* The basic design here is from
  24    Abraham Ziv, "Fast Evaluation of Elementary Mathematical Functions with
  25    Correctly Rounded Last Bit", ACM Trans. Math. Soft., 17 (3), September 1991,
  26    pp. 410-423.
  27
  28    We work with number pairs where the first number is the high part and
  29    the second one is the low part. Arithmetic with the high part numbers must
  30    be exact, without any roundoff errors.
  31
  32    The input value, X, is written as
  33    X = n * ln(2)_0 + arg1[t1]_0 + arg2[t2]_0 + x
  34        - n * ln(2)_1 + arg1[t1]_1 + arg2[t2]_1 + xl
  35
  36    where:
  37    - n is an integer, 16384 >= n >= -16495;
  38    - ln(2)_0 is the first 93 bits of ln(2), and |ln(2)_0-ln(2)-ln(2)_1| < 2^-205
  39    - t1 is an integer, 89 >= t1 >= -89
  40    - t2 is an integer, 65 >= t2 >= -65
  41    - |arg1[t1]-t1/256.0| < 2^-53
  42    - |arg2[t2]-t2/32768.0| < 2^-53
  43    - x + xl is whatever is left, |x + xl| < 2^-16 + 2^-53
  44
  45    Then e^x is approximated as
  46
  47    e^x = 2^n_1 ( 2^n_0 e^(arg1[t1]_0 + arg1[t1]_1) e^(arg2[t2]_0 + arg2[t2]_1)
  48                + 2^n_0 e^(arg1[t1]_0 + arg1[t1]_1) e^(arg2[t2]_0 + arg2[t2]_1)
  49                  * p (x + xl + n * ln(2)_1))
  50    where:
  51    - p(x) is a polynomial approximating e(x)-1
  52    - e^(arg1[t1]_0 + arg1[t1]_1) is obtained from a table
  53    - e^(arg2[t2]_0 + arg2[t2]_1) likewise
  54    - n_1 + n_0 = n, so that |n_0| < -LDBL_MIN_EXP-1.
  55
  56    If it happens that n_1 == 0 (this is the usual case), that multiplication
  57    is omitted.
  58    */
  59
  60 #ifndef _GNU_SOURCE
  61 #define _GNU_SOURCE
  62 #endif
  63 #include <float.h>
  64 #include <ieee754.h>
  65 #include <math.h>
  66 #include <fenv.h>
  67 #include <inttypes.h>
  68 #include <math_private.h>
  69 #include <stdlib.h>
  70 #include "t_expl.h"
  71
  72 static const long double C[] = {
  73 /* Smallest integer x for which e^x overflows.  */
  74 #define himark C[0]
  75  11356.523406294143949491931077970765L,
  76
  77 /* Largest integer x for which e^x underflows.  */
  78 #define lomark C[1]
  79 -11433.4627433362978788372438434526231L,
  80
  81 /* 3x2^96 */
  82 #define THREEp96 C[2]
  83  59421121885698253195157962752.0L,
  84
  85 /* 3x2^103 */
  86 #define THREEp103 C[3]
  87  30423614405477505635920876929024.0L,
  88
  89 /* 3x2^111 */
  90 #define THREEp111 C[4]
  91  7788445287802241442795744493830144.0L,
  92
  93 /* 1/ln(2) */
  94 #define M_1_LN2 C[5]
  95  1.44269504088896340735992468100189204L,
  96
  97 /* first 93 bits of ln(2) */
  98 #define M_LN2_0 C[6]
  99  0.693147180559945309417232121457981864L,
 100
 101 /* ln2_0 - ln(2) */
 102 #define M_LN2_1 C[7]
 103 -1.94704509238074995158795957333327386E-31L,
 104
 105 /* very small number */
 106 #define TINY C[8]
 107  1.0e-4900L,
 108
 109 /* 2^16383 */
 110 #define TWO16383 C[9]
 111  5.94865747678615882542879663314003565E+4931L,
 112
 113 /* 256 */
 114 #define TWO8 C[10]
 115  256.0L,
 116
 117 /* 32768 */
 118 #define TWO15 C[11]
 119  32768.0L,
 120
 121 /* Chebyshev polynom coeficients for (exp(x)-1)/x */
 122 #define P1 C[12]
 123 #define P2 C[13]
 124 #define P3 C[14]
 125 #define P4 C[15]
 126 #define P5 C[16]
 127 #define P6 C[17]
 128  0.5L,
 129  1.66666666666666666666666666666666683E-01L,
 130  4.16666666666666666666654902320001674E-02L,
 131  8.33333333333333333333314659767198461E-03L,
 132  1.38888888889899438565058018857254025E-03L,
 133  1.98412698413981650382436541785404286E-04L,
 134 };
 135
 136 long double
 137 __ieee754_expl (long double x)
 138 {
 139   /* Check for usual case.  */
 140   if (isless (x, himark) && isgreater (x, lomark))
 141     {
 142       int tval1, tval2, unsafe, n_i;
 143       long double x22, n, t, result, xl;
 144       union ieee854_long_double ex2_u, scale_u;
 145       fenv_t oldenv;
 146
 147       feholdexcept (&oldenv);
 148 #ifdef FE_TONEAREST
 149       fesetround (FE_TONEAREST);
 150 #endif
 151
 152       /* Calculate n.  */
 153       n = x * M_1_LN2 + THREEp111;
 154       n -= THREEp111;
 155       x = x - n * M_LN2_0;
 156       xl = n * M_LN2_1;
 157
 158       /* Calculate t/256.  */
 159       t = x + THREEp103;
 160       t -= THREEp103;
 161
 162       /* Compute tval1 = t.  */
 163       tval1 = (int) (t * TWO8);
 164
 165       x -= __expl_table[T_EXPL_ARG1+2*tval1];
 166       xl -= __expl_table[T_EXPL_ARG1+2*tval1+1];
 167
 168       /* Calculate t/32768.  */
 169       t = x + THREEp96;
 170       t -= THREEp96;
 171
 172       /* Compute tval2 = t.  */
 173       tval2 = (int) (t * TWO15);
 174
 175       x -= __expl_table[T_EXPL_ARG2+2*tval2];
 176       xl -= __expl_table[T_EXPL_ARG2+2*tval2+1];
 177
 178       x = x + xl;
 179
 180       /* Compute ex2 = 2^n_0 e^(argtable[tval1]) e^(argtable[tval2]).  */
 181       ex2_u.d = __expl_table[T_EXPL_RES1 + tval1]
 182                 * __expl_table[T_EXPL_RES2 + tval2];
 183       n_i = (int)n;
 184       /* 'unsafe' is 1 iff n_1 != 0.  */
 185       unsafe = abs(n_i) >= -LDBL_MIN_EXP - 1;
 186       ex2_u.ieee.exponent += n_i >> unsafe;
 187
 188       /* Compute scale = 2^n_1.  */
 189       scale_u.d = 1.0L;
 190       scale_u.ieee.exponent += n_i - (n_i >> unsafe);
 191
 192       /* Approximate e^x2 - 1, using a seventh-degree polynomial,
 193          with maximum error in [-2^-16-2^-53,2^-16+2^-53]
 194          less than 4.8e-39.  */
 195       x22 = x + x*x*(P1+x*(P2+x*(P3+x*(P4+x*(P5+x*P6)))));
 196
 197       /* Return result.  */
 198       fesetenv (&oldenv);
 199
 200       result = x22 * ex2_u.d + ex2_u.d;
 201
 202       /* Now we can test whether the result is ultimate or if we are unsure.
 203          In the later case we should probably call a mpn based routine to give
 204          the ultimate result.
 205          Empirically, this routine is already ultimate in about 99.9986% of
 206          cases, the test below for the round to nearest case will be false
 207          in ~ 99.9963% of cases.
 208          Without proc2 routine maximum error which has been seen is
 209          0.5000262 ulp.
 210
 211           union ieee854_long_double ex3_u;
 212
 213           #ifdef FE_TONEAREST
 214             fesetround (FE_TONEAREST);
 215           #endif
 216           ex3_u.d = (result - ex2_u.d) - x22 * ex2_u.d;
 217           ex2_u.d = result;
 218           ex3_u.ieee.exponent += LDBL_MANT_DIG + 15 + IEEE854_LONG_DOUBLE_BIAS
 219                                  - ex2_u.ieee.exponent;
 220           n_i = abs (ex3_u.d);
 221           n_i = (n_i + 1) / 2;
 222           fesetenv (&oldenv);
 223           #ifdef FE_TONEAREST
 224           if (fegetround () == FE_TONEAREST)
 225             n_i -= 0x4000;
 226           #endif
 227           if (!n_i) {
 228             return __ieee754_expl_proc2 (origx);
 229           }
 230        */
 231       if (!unsafe)
 232         return result;
 233       else
 234         return result * scale_u.d;
 235     }
 236   /* Exceptional cases:  */
 237   else if (isless (x, himark))
 238     {
 239       if (__isinfl (x))
 240         /* e^-inf == 0, with no error.  */
 241         return 0;
 242       else
 243         /* Underflow */
 244         return TINY * TINY;
 245     }
 246   else
 247     /* Return x, if x is a NaN or Inf; or overflow, otherwise.  */
 248     return TWO16383*x;
 249 }