sysdeps/ieee754/ldbl-128ibm/e_expl.c

   1 /* Quad-precision floating point e^x.
   2    Copyright (C) 1999-2014 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| < -LDBL_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 #include <float.h>
  63 #include <ieee754.h>
  64 #include <math.h>
  65 #include <fenv.h>
  66 #include <inttypes.h>
  67 #include <math_private.h>
  68 #include <sysdeps/ieee754/ldbl-128/t_expl.h>
  69
  70 static const long double C[] = {
  71 /* Smallest integer x for which e^x overflows.  */
  72 #define himark C[0]
  73  709.78271289338399678773454114191496482L,
  74
  75 /* Largest integer x for which e^x underflows.  */
  76 #define lomark C[1]
  77 -744.44007192138126231410729844608163411L,
  78
  79 /* 3x2^96 */
  80 #define THREEp96 C[2]
  81  59421121885698253195157962752.0L,
  82
  83 /* 3x2^103 */
  84 #define THREEp103 C[3]
  85  30423614405477505635920876929024.0L,
  86
  87 /* 3x2^111 */
  88 #define THREEp111 C[4]
  89  7788445287802241442795744493830144.0L,
  90
  91 /* 1/ln(2) */
  92 #define M_1_LN2 C[5]
  93  1.44269504088896340735992468100189204L,
  94
  95 /* first 93 bits of ln(2) */
  96 #define M_LN2_0 C[6]
  97  0.693147180559945309417232121457981864L,
  98
  99 /* ln2_0 - ln(2) */
 100 #define M_LN2_1 C[7]
 101 -1.94704509238074995158795957333327386E-31L,
 102
 103 /* very small number */
 104 #define TINY C[8]
 105  1.0e-308L,
 106
 107 /* 2^16383 */
 108 #define TWO1023 C[9]
 109  8.988465674311579538646525953945123668E+307L,
 110
 111 /* 256 */
 112 #define TWO8 C[10]
 113  256.0L,
 114
 115 /* 32768 */
 116 #define TWO15 C[11]
 117  32768.0L,
 118
 119 /* Chebyshev polynom coefficients for (exp(x)-1)/x */
 120 #define P1 C[12]
 121 #define P2 C[13]
 122 #define P3 C[14]
 123 #define P4 C[15]
 124 #define P5 C[16]
 125 #define P6 C[17]
 126  0.5L,
 127  1.66666666666666666666666666666666683E-01L,
 128  4.16666666666666666666654902320001674E-02L,
 129  8.33333333333333333333314659767198461E-03L,
 130  1.38888888889899438565058018857254025E-03L,
 131  1.98412698413981650382436541785404286E-04L,
 132 };
 133
 134 long double
 135 __ieee754_expl (long double x)
 136 {
 137   long double result, x22;
 138   union ibm_extended_long_double ex2_u, scale_u;
 139   int unsafe;
 140
 141   /* Check for usual case.  */
 142   if (isless (x, himark) && isgreater (x, lomark))
 143     {
 144       int tval1, tval2, n_i, exponent2;
 145       long double n, xl;
 146
 147       SET_RESTORE_ROUND (FE_TONEAREST);
 148
 149       n = __roundl (x*M_1_LN2);
 150       x = x-n*M_LN2_0;
 151       xl = n*M_LN2_1;
 152
 153       tval1 = __roundl (x*TWO8);
 154       x -= __expl_table[T_EXPL_ARG1+2*tval1];
 155       xl -= __expl_table[T_EXPL_ARG1+2*tval1+1];
 156
 157       tval2 = __roundl (x*TWO15);
 158       x -= __expl_table[T_EXPL_ARG2+2*tval2];
 159       xl -= __expl_table[T_EXPL_ARG2+2*tval2+1];
 160
 161       x = x + xl;
 162
 163       /* Compute ex2 = 2^n_0 e^(argtable[tval1]) e^(argtable[tval2]).  */
 164       ex2_u.ld = (__expl_table[T_EXPL_RES1 + tval1]
 165                   * __expl_table[T_EXPL_RES2 + tval2]);
 166       n_i = (int)n;
 167       /* 'unsafe' is 1 iff n_1 != 0.  */
 168       unsafe = fabsl(n_i) >= -LDBL_MIN_EXP - 1;
 169       ex2_u.d[0].ieee.exponent += n_i >> unsafe;
 170       /* Fortunately, there are no subnormal lowpart doubles in
 171          __expl_table, only normal values and zeros.
 172          But after scaling it can be subnormal.  */
 173       exponent2 = ex2_u.d[1].ieee.exponent + (n_i >> unsafe);
 174       if (ex2_u.d[1].ieee.exponent == 0)
 175         /* assert ((ex2_u.d[1].ieee.mantissa0|ex2_u.d[1].ieee.mantissa1) == 0) */;
 176       else if (exponent2 > 0)
 177         ex2_u.d[1].ieee.exponent = exponent2;
 178       else if (exponent2 <= -54)
 179         {
 180           ex2_u.d[1].ieee.exponent = 0;
 181           ex2_u.d[1].ieee.mantissa0 = 0;
 182           ex2_u.d[1].ieee.mantissa1 = 0;
 183         }
 184       else
 185         {
 186           static const double
 187             two54 = 1.80143985094819840000e+16, /* 4350000000000000 */
 188             twom54 = 5.55111512312578270212e-17; /* 3C90000000000000 */
 189           ex2_u.d[1].d *= two54;
 190           ex2_u.d[1].ieee.exponent += n_i >> unsafe;
 191           ex2_u.d[1].d *= twom54;
 192         }
 193
 194       /* Compute scale = 2^n_1.  */
 195       scale_u.ld = 1.0L;
 196       scale_u.d[0].ieee.exponent += n_i - (n_i >> unsafe);
 197
 198       /* Approximate e^x2 - 1, using a seventh-degree polynomial,
 199          with maximum error in [-2^-16-2^-53,2^-16+2^-53]
 200          less than 4.8e-39.  */
 201       x22 = x + x*x*(P1+x*(P2+x*(P3+x*(P4+x*(P5+x*P6)))));
 202
 203       /* Now we can test whether the result is ultimate or if we are unsure.
 204          In the later case we should probably call a mpn based routine to give
 205          the ultimate result.
 206          Empirically, this routine is already ultimate in about 99.9986% of
 207          cases, the test below for the round to nearest case will be false
 208          in ~ 99.9963% of cases.
 209          Without proc2 routine maximum error which has been seen is
 210          0.5000262 ulp.
 211
 212           union ieee854_long_double ex3_u;
 213
 214           #ifdef FE_TONEAREST
 215             fesetround (FE_TONEAREST);
 216           #endif
 217           ex3_u.d = (result - ex2_u.d) - x22 * ex2_u.d;
 218           ex2_u.d = result;
 219           ex3_u.ieee.exponent += LDBL_MANT_DIG + 15 + IEEE854_LONG_DOUBLE_BIAS
 220                                  - ex2_u.ieee.exponent;
 221           n_i = abs (ex3_u.d);
 222           n_i = (n_i + 1) / 2;
 223           fesetenv (&oldenv);
 224           #ifdef FE_TONEAREST
 225           if (fegetround () == FE_TONEAREST)
 226             n_i -= 0x4000;
 227           #endif
 228           if (!n_i) {
 229             return __ieee754_expl_proc2 (origx);
 230           }
 231        */
 232     }
 233   /* Exceptional cases:  */
 234   else if (isless (x, himark))
 235     {
 236       if (__isinfl (x))
 237         /* e^-inf == 0, with no error.  */
 238         return 0;
 239       else
 240         /* Underflow */
 241         return TINY * TINY;
 242     }
 243   else
 244     /* Return x, if x is a NaN or Inf; or overflow, otherwise.  */
 245     return TWO1023*x;
 246
 247   result = x22 * ex2_u.ld + ex2_u.ld;
 248   if (!unsafe)
 249     return result;
 250   return result * scale_u.ld;
 251 }
 252 strong_alias (__ieee754_expl, __expl_finite)