libquadmath/math/expq.c

   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
  66 static 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
 131 expq (__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 }