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

   1 /* Quad-precision floating point e^x.
   2    Copyright (C) 1999,2004,2006, 2008 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 <sysdeps/ieee754/ldbl-128/t_expl.h>
  70
  71 static const long double C[] = {
  72 /* Smallest integer x for which e^x overflows.  */
  73 #define himark C[0]
  74  709.08956571282405153382846025171462914L,
  75
  76 /* Largest integer x for which e^x underflows.  */
  77 #define lomark C[1]
  78 -709.08956571282405153382846025171462914L,
  79
  80 /* 3x2^96 */
  81 #define THREEp96 C[2]
  82  59421121885698253195157962752.0L,
  83
  84 /* 3x2^103 */
  85 #define THREEp103 C[3]
  86  30423614405477505635920876929024.0L,
  87
  88 /* 3x2^111 */
  89 #define THREEp111 C[4]
  90  7788445287802241442795744493830144.0L,
  91
  92 /* 1/ln(2) */
  93 #define M_1_LN2 C[5]
  94  1.44269504088896340735992468100189204L,
  95
  96 /* first 93 bits of ln(2) */
  97 #define M_LN2_0 C[6]
  98  0.693147180559945309417232121457981864L,
  99
 100 /* ln2_0 - ln(2) */
 101 #define M_LN2_1 C[7]
 102 -1.94704509238074995158795957333327386E-31L,
 103
 104 /* very small number */
 105 #define TINY C[8]
 106  1.0e-308L,
 107
 108 /* 2^16383 */
 109 #define TWO1023 C[9]
 110  8.988465674311579538646525953945123668E+307L,
 111
 112 /* 256 */
 113 #define TWO8 C[10]
 114  256.0L,
 115
 116 /* 32768 */
 117 #define TWO15 C[11]
 118  32768.0L,
 119
 120 /* Chebyshev polynom coeficients for (exp(x)-1)/x */
 121 #define P1 C[12]
 122 #define P2 C[13]
 123 #define P3 C[14]
 124 #define P4 C[15]
 125 #define P5 C[16]
 126 #define P6 C[17]
 127  0.5L,
 128  1.66666666666666666666666666666666683E-01L,
 129  4.16666666666666666666654902320001674E-02L,
 130  8.33333333333333333333314659767198461E-03L,
 131  1.38888888889899438565058018857254025E-03L,
 132  1.98412698413981650382436541785404286E-04L,
 133 };
 134
 135 long double
 136 __ieee754_expl (long double x)
 137 {
 138   /* Check for usual case.  */
 139   if (isless (x, himark) && isgreater (x, lomark))
 140     {
 141       int tval1, tval2, unsafe, n_i, exponent2;
 142       long double x22, n, result, xl;
 143       union ibm_extended_long_double ex2_u, scale_u;
 144       fenv_t oldenv;
 145
 146       feholdexcept (&oldenv);
 147 #ifdef FE_TONEAREST
 148       fesetround (FE_TONEAREST);
 149 #endif
 150
 151       n = __roundl (x*M_1_LN2);
 152       x = x-n*M_LN2_0;
 153       xl = n*M_LN2_1;
 154
 155       tval1 = __roundl (x*TWO8);
 156       x -= __expl_table[T_EXPL_ARG1+2*tval1];
 157       xl -= __expl_table[T_EXPL_ARG1+2*tval1+1];
 158
 159       tval2 = __roundl (x*TWO15);
 160       x -= __expl_table[T_EXPL_ARG2+2*tval2];
 161       xl -= __expl_table[T_EXPL_ARG2+2*tval2+1];
 162
 163       x = x + xl;
 164
 165       /* Compute ex2 = 2^n_0 e^(argtable[tval1]) e^(argtable[tval2]).  */
 166       ex2_u.d = __expl_table[T_EXPL_RES1 + tval1]
 167                 * __expl_table[T_EXPL_RES2 + tval2];
 168       n_i = (int)n;
 169       /* 'unsafe' is 1 iff n_1 != 0.  */
 170       unsafe = fabsl(n_i) >= -LDBL_MIN_EXP - 1;
 171       ex2_u.ieee.exponent += n_i >> unsafe;
 172       /* Fortunately, there are no subnormal lowpart doubles in
 173          __expl_table, only normal values and zeros.
 174          But after scaling it can be subnormal.  */
 175       exponent2 = ex2_u.ieee.exponent2 + (n_i >> unsafe);
 176       if (ex2_u.ieee.exponent2 == 0)
 177         /* assert ((ex2_u.ieee.mantissa2|ex2_u.ieee.mantissa3) == 0) */;
 178       else if (exponent2 > 0)
 179         ex2_u.ieee.exponent2 = exponent2;
 180       else if (exponent2 <= -54)
 181         {
 182           ex2_u.ieee.exponent2 = 0;
 183           ex2_u.ieee.mantissa2 = 0;
 184           ex2_u.ieee.mantissa3 = 0;
 185         }
 186       else
 187         {
 188           static const double
 189             two54 = 1.80143985094819840000e+16, /* 4350000000000000 */
 190             twom54 = 5.55111512312578270212e-17; /* 3C90000000000000 */
 191           ex2_u.dd[1] *= two54;
 192           ex2_u.ieee.exponent2 += n_i >> unsafe;
 193           ex2_u.dd[1] *= twom54;
 194         }
 195
 196       /* Compute scale = 2^n_1.  */
 197       scale_u.d = 1.0L;
 198       scale_u.ieee.exponent += n_i - (n_i >> unsafe);
 199
 200       /* Approximate e^x2 - 1, using a seventh-degree polynomial,
 201          with maximum error in [-2^-16-2^-53,2^-16+2^-53]
 202          less than 4.8e-39.  */
 203       x22 = x + x*x*(P1+x*(P2+x*(P3+x*(P4+x*(P5+x*P6)))));
 204
 205       /* Return result.  */
 206       fesetenv (&oldenv);
 207
 208       result = x22 * ex2_u.d + ex2_u.d;
 209
 210       /* Now we can test whether the result is ultimate or if we are unsure.
 211          In the later case we should probably call a mpn based routine to give
 212          the ultimate result.
 213          Empirically, this routine is already ultimate in about 99.9986% of
 214          cases, the test below for the round to nearest case will be false
 215          in ~ 99.9963% of cases.
 216          Without proc2 routine maximum error which has been seen is
 217          0.5000262 ulp.
 218
 219           union ieee854_long_double ex3_u;
 220
 221           #ifdef FE_TONEAREST
 222             fesetround (FE_TONEAREST);
 223           #endif
 224           ex3_u.d = (result - ex2_u.d) - x22 * ex2_u.d;
 225           ex2_u.d = result;
 226           ex3_u.ieee.exponent += LDBL_MANT_DIG + 15 + IEEE854_LONG_DOUBLE_BIAS
 227                                  - ex2_u.ieee.exponent;
 228           n_i = abs (ex3_u.d);
 229           n_i = (n_i + 1) / 2;
 230           fesetenv (&oldenv);
 231           #ifdef FE_TONEAREST
 232           if (fegetround () == FE_TONEAREST)
 233             n_i -= 0x4000;
 234           #endif
 235           if (!n_i) {
 236             return __ieee754_expl_proc2 (origx);
 237           }
 238        */
 239       if (!unsafe)
 240         return result;
 241       else
 242         return result * scale_u.d;
 243     }
 244   /* Exceptional cases:  */
 245   else if (isless (x, himark))
 246     {
 247       if (__isinfl (x))
 248         /* e^-inf == 0, with no error.  */
 249         return 0;
 250       else
 251         /* Underflow */
 252         return TINY * TINY;
 253     }
 254   else
 255     /* Return x, if x is a NaN or Inf; or overflow, otherwise.  */
 256     return TWO1023*x;
 257 }