compiler/stdc/math/ld80/e_log2l.c

   1 /*      $OpenBSD: e_log2l.c,v 1.2 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 /*                                                      log2l.c
  20  *
  21  *      Base 2 logarithm, long double precision
  22  *
  23  *
  24  *
  25  * SYNOPSIS:
  26  *
  27  * long double x, y, log2l();
  28  *
  29  * y = log2l( x );
  30  *
  31  *
  32  *
  33  * DESCRIPTION:
  34  *
  35  * Returns the base 2 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 (natural)
  39  * logarithm 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     30000      9.8e-20     2.7e-20
  54  *    IEEE     exp(+-10000)  70000      5.4e-20     2.3e-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 ln(1+x) = x - x**2/2 + x**3 P(x)/Q(x)
  71  * 1/sqrt(2) <= x < sqrt(2)
  72  * Theoretical peak relative error = 6.2e-22
  73  */
  74 static const long double P[] = {
  75  4.9962495940332550844739E-1L,
  76  1.0767376367209449010438E1L,
  77  7.7671073698359539859595E1L,
  78  2.5620629828144409632571E2L,
  79  4.2401812743503691187826E2L,
  80  3.4258224542413922935104E2L,
  81  1.0747524399916215149070E2L,
  82 };
  83 static const long double Q[] = {
  84 /* 1.0000000000000000000000E0,*/
  85  2.3479774160285863271658E1L,
  86  1.9444210022760132894510E2L,
  87  7.7952888181207260646090E2L,
  88  1.6911722418503949084863E3L,
  89  2.0307734695595183428202E3L,
  90  1.2695660352705325274404E3L,
  91  3.2242573199748645407652E2L,
  92 };
  93
  94 /* Coefficients for log(x) = z + z^3 P(z^2)/Q(z^2),
  95  * where z = 2(x-1)/(x+1)
  96  * 1/sqrt(2) <= x < sqrt(2)
  97  * Theoretical peak relative error = 6.16e-22
  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 /* log2(e) - 1 */
 112 #define LOG2EA 4.4269504088896340735992e-1L
 113
 114 #define SQRTH 0.70710678118654752440L
 115
 116 long double
 117 log2l(long double x)
 118 {
 119 volatile long double z;
 120 long double y;
 121 int e;
 122
 123 if( isnan(x) )
 124         return(x);
 125 if( x == INFINITY )
 126         return(x);
 127 /* Test for domain */
 128 if( x <= 0.0L )
 129         {
 130         if( x == 0.0L )
 131                 return( -INFINITY );
 132         else
 133                 return( NAN );
 134         }
 135
 136 /* separate mantissa from exponent */
 137
 138 /* Note, frexp is used so that denormal numbers
 139  * will be handled properly.
 140  */
 141 x = frexpl( x, &e );
 142
 143
 144 /* logarithm using log(x) = z + z**3 P(z)/Q(z),
 145  * where z = 2(x-1)/x+1)
 146  */
 147 if( (e > 2) || (e < -2) )
 148 {
 149 if( x < SQRTH )
 150         { /* 2( 2x-1 )/( 2x+1 ) */
 151         e -= 1;
 152         z = x - 0.5L;
 153         y = 0.5L * z + 0.5L;
 154         }
 155 else
 156         { /*  2 (x-1)/(x+1)   */
 157         z = x - 0.5L;
 158         z -= 0.5L;
 159         y = 0.5L * x  + 0.5L;
 160         }
 161 x = z / y;
 162 z = x*x;
 163 y = x * ( z * __polevll( z, (void *)R, 3 ) / __p1evll( z, (void *)S, 3 ) );
 164 goto done;
 165 }
 166
 167
 168 /* logarithm using log(1+x) = x - .5x**2 + x**3 P(x)/Q(x) */
 169
 170 if( x < SQRTH )
 171         {
 172         e -= 1;
 173         x = ldexpl( x, 1 ) - 1.0L; /*  2x - 1  */
 174         }
 175 else
 176         {
 177         x = x - 1.0L;
 178         }
 179 z = x*x;
 180 y = x * ( z * __polevll( x, (void *)P, 6 ) / __p1evll( x, (void *)Q, 7 ) );
 181 y = y - ldexpl( z, -1 );   /* -0.5x^2 + ... */
 182
 183 done:
 184
 185 /* Multiply log of fraction by log2(e)
 186  * and base 2 exponent by 1
 187  *
 188  * ***CAUTION***
 189  *
 190  * This sequence of operations is critical and it may
 191  * be horribly defeated by some compiler optimizers.
 192  */
 193 z = y * LOG2EA;
 194 z += x * LOG2EA;
 195 z += y;
 196 z += x;
 197 z += e;
 198 return( z );
 199 }