src/math/logl.c

   1 /* origin: OpenBSD /usr/src/lib/libm/src/ld80/e_logl.c */
   2 /*
   3  * Copyright (c) 2008 Stephen L. Moshier <steve@moshier.net>
   4  *
   5  * Permission to use, copy, modify, and distribute this software for any
   6  * purpose with or without fee is hereby granted, provided that the above
   7  * copyright notice and this permission notice appear in all copies.
   8  *
   9  * THE SOFTWARE IS PROVIDED "AS IS" AND THE AUTHOR DISCLAIMS ALL WARRANTIES
  10  * WITH REGARD TO THIS SOFTWARE INCLUDING ALL IMPLIED WARRANTIES OF
  11  * MERCHANTABILITY AND FITNESS. IN NO EVENT SHALL THE AUTHOR BE LIABLE FOR
  12  * ANY SPECIAL, DIRECT, INDIRECT, OR CONSEQUENTIAL DAMAGES OR ANY DAMAGES
  13  * WHATSOEVER RESULTING FROM LOSS OF USE, DATA OR PROFITS, WHETHER IN AN
  14  * ACTION OF CONTRACT, NEGLIGENCE OR OTHER TORTIOUS ACTION, ARISING OUT OF
  15  * OR IN CONNECTION WITH THE USE OR PERFORMANCE OF THIS SOFTWARE.
  16  */
  17 /*
  18  *      Natural logarithm, long double precision
  19  *
  20  *
  21  * SYNOPSIS:
  22  *
  23  * long double x, y, logl();
  24  *
  25  * y = logl( x );
  26  *
  27  *
  28  * DESCRIPTION:
  29  *
  30  * Returns the base e (2.718...) logarithm of x.
  31  *
  32  * The argument is separated into its exponent and fractional
  33  * parts.  If the exponent is between -1 and +1, the logarithm
  34  * of the fraction is approximated by
  35  *
  36  *     log(1+x) = x - 0.5 x**2 + x**3 P(x)/Q(x).
  37  *
  38  * Otherwise, setting  z = 2(x-1)/(x+1),
  39  *
  40  *     log(x) = log(1+z/2) - log(1-z/2) = z + z**3 P(z)/Q(z).
  41  *
  42  *
  43  * ACCURACY:
  44  *
  45  *                      Relative error:
  46  * arithmetic   domain     # trials      peak         rms
  47  *    IEEE      0.5, 2.0    150000      8.71e-20    2.75e-20
  48  *    IEEE     exp(+-10000) 100000      5.39e-20    2.34e-20
  49  *
  50  * In the tests over the interval exp(+-10000), the logarithms
  51  * of the random arguments were uniformly distributed over
  52  * [-10000, +10000].
  53  */
  54
  55 #include "libm.h"
  56
  57 #if LDBL_MANT_DIG == 53 && LDBL_MAX_EXP == 1024
  58 long double logl(long double x)
  59 {
  60         return log(x);
  61 }
  62 #elif LDBL_MANT_DIG == 64 && LDBL_MAX_EXP == 16384
  63 /* Coefficients for log(1+x) = x - x**2/2 + x**3 P(x)/Q(x)
  64  * 1/sqrt(2) <= x < sqrt(2)
  65  * Theoretical peak relative error = 2.32e-20
  66  */
  67 static const long double P[] = {
  68  4.5270000862445199635215E-5L,
  69  4.9854102823193375972212E-1L,
  70  6.5787325942061044846969E0L,
  71  2.9911919328553073277375E1L,
  72  6.0949667980987787057556E1L,
  73  5.7112963590585538103336E1L,
  74  2.0039553499201281259648E1L,
  75 };
  76 static const long double Q[] = {
  77 /* 1.0000000000000000000000E0,*/
  78  1.5062909083469192043167E1L,
  79  8.3047565967967209469434E1L,
  80  2.2176239823732856465394E2L,
  81  3.0909872225312059774938E2L,
  82  2.1642788614495947685003E2L,
  83  6.0118660497603843919306E1L,
  84 };
  85
  86 /* Coefficients for log(x) = z + z^3 P(z^2)/Q(z^2),
  87  * where z = 2(x-1)/(x+1)
  88  * 1/sqrt(2) <= x < sqrt(2)
  89  * Theoretical peak relative error = 6.16e-22
  90  */
  91 static const long double R[4] = {
  92  1.9757429581415468984296E-3L,
  93 -7.1990767473014147232598E-1L,
  94  1.0777257190312272158094E1L,
  95 -3.5717684488096787370998E1L,
  96 };
  97 static const long double S[4] = {
  98 /* 1.00000000000000000000E0L,*/
  99 -2.6201045551331104417768E1L,
 100  1.9361891836232102174846E2L,
 101 -4.2861221385716144629696E2L,
 102 };
 103 static const long double C1 = 6.9314575195312500000000E-1L;
 104 static const long double C2 = 1.4286068203094172321215E-6L;
 105
 106 #define SQRTH 0.70710678118654752440L
 107
 108 long double logl(long double x)
 109 {
 110         long double y, z;
 111         int e;
 112
 113         if (isnan(x))
 114                 return x;
 115         if (x == INFINITY)
 116                 return x;
 117         if (x <= 0.0) {
 118                 if (x == 0.0)
 119                         return -1/(x*x); /* -inf with divbyzero */
 120                 return 0/0.0f; /* nan with invalid */
 121         }
 122
 123         /* separate mantissa from exponent */
 124         /* Note, frexp is used so that denormal numbers
 125          * will be handled properly.
 126          */
 127         x = frexpl(x, &e);
 128
 129         /* logarithm using log(x) = z + z**3 P(z)/Q(z),
 130          * where z = 2(x-1)/(x+1)
 131          */
 132         if (e > 2 || e < -2) {
 133                 if (x < SQRTH) {  /* 2(2x-1)/(2x+1) */
 134                         e -= 1;
 135                         z = x - 0.5;
 136                         y = 0.5 * z + 0.5;
 137                 } else {  /*  2 (x-1)/(x+1)   */
 138                         z = x - 0.5;
 139                         z -= 0.5;
 140                         y = 0.5 * x  + 0.5;
 141                 }
 142                 x = z / y;
 143                 z = x*x;
 144                 z = x * (z * __polevll(z, R, 3) / __p1evll(z, S, 3));
 145                 z = z + e * C2;
 146                 z = z + x;
 147                 z = z + e * C1;
 148                 return z;
 149         }
 150
 151         /* logarithm using log(1+x) = x - .5x**2 + x**3 P(x)/Q(x) */
 152         if (x < SQRTH) {
 153                 e -= 1;
 154                 x = 2.0*x - 1.0;
 155         } else {
 156                 x = x - 1.0;
 157         }
 158         z = x*x;
 159         y = x * (z * __polevll(x, P, 6) / __p1evll(x, Q, 6));
 160         y = y + e * C2;
 161         z = y - 0.5*z;
 162         /* Note, the sum of above terms does not exceed x/4,
 163          * so it contributes at most about 1/4 lsb to the error.
 164          */
 165         z = z + x;
 166         z = z + e * C1; /* This sum has an error of 1/2 lsb. */
 167         return z;
 168 }
 169 #elif LDBL_MANT_DIG == 113 && LDBL_MAX_EXP == 16384
 170 // TODO: broken implementation to make things compile
 171 long double logl(long double x)
 172 {
 173         return log(x);
 174 }
 175 #endif