compiler/stdc/math/ld80/e_logl.c

   1 /*      $OpenBSD: e_logl.c,v 1.3 2013/11/12 20:35:19 martynas Exp $     */
   2
   3 /*
   4  * Copyright (c) 2008 Stephen L. Moshier <steve@moshier.net>
   5  *
   6  * Permission to use, copy, modify, and distribute this software for any
   7  * purpose with or without fee is hereby granted, provided that the above
   8  * copyright notice and this permission notice appear in all copies.
   9  *
  10  * THE SOFTWARE IS PROVIDED "AS IS" AND THE AUTHOR DISCLAIMS ALL WARRANTIES
  11  * WITH REGARD TO THIS SOFTWARE INCLUDING ALL IMPLIED WARRANTIES OF
  12  * MERCHANTABILITY AND FITNESS. IN NO EVENT SHALL THE AUTHOR BE LIABLE FOR
  13  * ANY SPECIAL, DIRECT, INDIRECT, OR CONSEQUENTIAL DAMAGES OR ANY DAMAGES
  14  * WHATSOEVER RESULTING FROM LOSS OF USE, DATA OR PROFITS, WHETHER IN AN
  15  * ACTION OF CONTRACT, NEGLIGENCE OR OTHER TORTIOUS ACTION, ARISING OUT OF
  16  * OR IN CONNECTION WITH THE USE OR PERFORMANCE OF THIS SOFTWARE.
  17  */
  18
  19 /*                                                      logl.c
  20  *
  21  *      Natural logarithm, long double precision
  22  *
  23  *
  24  *
  25  * SYNOPSIS:
  26  *
  27  * long double x, y, logl();
  28  *
  29  * y = logl( x );
  30  *
  31  *
  32  *
  33  * DESCRIPTION:
  34  *
  35  * Returns the base e (2.718...) logarithm of x.
  36  *
  37  * The argument is separated into its exponent and fractional
  38  * parts.  If the exponent is between -1 and +1, the logarithm
  39  * of the fraction is approximated by
  40  *
  41  *     log(1+x) = x - 0.5 x**2 + x**3 P(x)/Q(x).
  42  *
  43  * Otherwise, setting  z = 2(x-1)/x+1),
  44  *
  45  *     log(x) = z + z**3 P(z)/Q(z).
  46  *
  47  *
  48  *
  49  * ACCURACY:
  50  *
  51  *                      Relative error:
  52  * arithmetic   domain     # trials      peak         rms
  53  *    IEEE      0.5, 2.0    150000      8.71e-20    2.75e-20
  54  *    IEEE     exp(+-10000) 100000      5.39e-20    2.34e-20
  55  *
  56  * In the tests over the interval exp(+-10000), the logarithms
  57  * of the random arguments were uniformly distributed over
  58  * [-10000, +10000].
  59  *
  60  * ERROR MESSAGES:
  61  *
  62  * log singularity:  x = 0; returns -INFINITY
  63  * log domain:       x < 0; returns NAN
  64  */
  65
  66 #include "math.h"
  67
  68 #include "math_private.h"
  69
  70 /* Coefficients for log(1+x) = x - x**2/2 + x**3 P(x)/Q(x)
  71  * 1/sqrt(2) <= x < sqrt(2)
  72  * Theoretical peak relative error = 2.32e-20
  73  */
  74 static const long double P[] = {
  75  4.5270000862445199635215E-5L,
  76  4.9854102823193375972212E-1L,
  77  6.5787325942061044846969E0L,
  78  2.9911919328553073277375E1L,
  79  6.0949667980987787057556E1L,
  80  5.7112963590585538103336E1L,
  81  2.0039553499201281259648E1L,
  82 };
  83 static const long double Q[] = {
  84 /* 1.0000000000000000000000E0,*/
  85  1.5062909083469192043167E1L,
  86  8.3047565967967209469434E1L,
  87  2.2176239823732856465394E2L,
  88  3.0909872225312059774938E2L,
  89  2.1642788614495947685003E2L,
  90  6.0118660497603843919306E1L,
  91 };
  92
  93 /* Coefficients for log(x) = z + z^3 P(z^2)/Q(z^2),
  94  * where z = 2(x-1)/(x+1)
  95  * 1/sqrt(2) <= x < sqrt(2)
  96  * Theoretical peak relative error = 6.16e-22
  97  */
  98
  99 static const long double R[4] = {
 100  1.9757429581415468984296E-3L,
 101 -7.1990767473014147232598E-1L,
 102  1.0777257190312272158094E1L,
 103 -3.5717684488096787370998E1L,
 104 };
 105 static const long double S[4] = {
 106 /* 1.00000000000000000000E0L,*/
 107 -2.6201045551331104417768E1L,
 108  1.9361891836232102174846E2L,
 109 -4.2861221385716144629696E2L,
 110 };
 111 static const long double C1 = 6.9314575195312500000000E-1L;
 112 static const long double C2 = 1.4286068203094172321215E-6L;
 113
 114 #define SQRTH 0.70710678118654752440L
 115
 116 long double
 117 logl(long double x)
 118 {
 119 long double y, z;
 120 int e;
 121
 122 if( isnan(x) )
 123         return(x);
 124 if( x == INFINITY )
 125         return(x);
 126 /* Test for domain */
 127 if( x <= 0.0L )
 128         {
 129         if( x == 0.0L )
 130                 return( -INFINITY );
 131         else
 132                 return( NAN );
 133         }
 134
 135 /* separate mantissa from exponent */
 136
 137 /* Note, frexp is used so that denormal numbers
 138  * will be handled properly.
 139  */
 140 x = frexpl( x, &e );
 141
 142 /* logarithm using log(x) = z + z**3 P(z)/Q(z),
 143  * where z = 2(x-1)/x+1)
 144  */
 145 if( (e > 2) || (e < -2) )
 146 {
 147 if( x < SQRTH )
 148         { /* 2( 2x-1 )/( 2x+1 ) */
 149         e -= 1;
 150         z = x - 0.5L;
 151         y = 0.5L * z + 0.5L;
 152         }
 153 else
 154         { /*  2 (x-1)/(x+1)   */
 155         z = x - 0.5L;
 156         z -= 0.5L;
 157         y = 0.5L * x  + 0.5L;
 158         }
 159 x = z / y;
 160 z = x*x;
 161 z = x * ( z * __polevll( z, (void *)R, 3 ) / __p1evll( z, (void *)S, 3 ) );
 162 z = z + e * C2;
 163 z = z + x;
 164 z = z + e * C1;
 165 return( z );
 166 }
 167
 168
 169 /* logarithm using log(1+x) = x - .5x**2 + x**3 P(x)/Q(x) */
 170
 171 if( x < SQRTH )
 172         {
 173         e -= 1;
 174         x = ldexpl( x, 1 ) - 1.0L; /*  2x - 1  */
 175         }
 176 else
 177         {
 178         x = x - 1.0L;
 179         }
 180 z = x*x;
 181 y = x * ( z * __polevll( x, (void *)P, 6 ) / __p1evll( x, (void *)Q, 6 ) );
 182 y = y + e * C2;
 183 z = y - ldexpl( z, -1 );   /*  y - 0.5 * z  */
 184 /* Note, the sum of above terms does not exceed x/4,
 185  * so it contributes at most about 1/4 lsb to the error.
 186  */
 187 z = z + x;
 188 z = z + e * C1; /* This sum has an error of 1/2 lsb. */
 189 return( z );
 190 }