src/math/log2l.c

   1 /* origin: OpenBSD /usr/src/lib/libm/src/ld80/e_log2l.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  *      Base 2 logarithm, long double precision
  19  *
  20  *
  21  * SYNOPSIS:
  22  *
  23  * long double x, y, log2l();
  24  *
  25  * y = log2l( x );
  26  *
  27  *
  28  * DESCRIPTION:
  29  *
  30  * Returns the base 2 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 (natural)
  34  * logarithm 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) = 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     30000      9.8e-20     2.7e-20
  48  *    IEEE     exp(+-10000)  70000      5.4e-20     2.3e-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 log2l(long double x)
  59 {
  60         return log2(x);
  61 }
  62 #elif LDBL_MANT_DIG == 64 && LDBL_MAX_EXP == 16384
  63 /* Coefficients for ln(1+x) = x - x**2/2 + x**3 P(x)/Q(x)
  64  * 1/sqrt(2) <= x < sqrt(2)
  65  * Theoretical peak relative error = 6.2e-22
  66  */
  67 static const long double P[] = {
  68  4.9962495940332550844739E-1L,
  69  1.0767376367209449010438E1L,
  70  7.7671073698359539859595E1L,
  71  2.5620629828144409632571E2L,
  72  4.2401812743503691187826E2L,
  73  3.4258224542413922935104E2L,
  74  1.0747524399916215149070E2L,
  75 };
  76 static const long double Q[] = {
  77 /* 1.0000000000000000000000E0,*/
  78  2.3479774160285863271658E1L,
  79  1.9444210022760132894510E2L,
  80  7.7952888181207260646090E2L,
  81  1.6911722418503949084863E3L,
  82  2.0307734695595183428202E3L,
  83  1.2695660352705325274404E3L,
  84  3.2242573199748645407652E2L,
  85 };
  86
  87 /* Coefficients for log(x) = z + z^3 P(z^2)/Q(z^2),
  88  * where z = 2(x-1)/(x+1)
  89  * 1/sqrt(2) <= x < sqrt(2)
  90  * Theoretical peak relative error = 6.16e-22
  91  */
  92 static const long double R[4] = {
  93  1.9757429581415468984296E-3L,
  94 -7.1990767473014147232598E-1L,
  95  1.0777257190312272158094E1L,
  96 -3.5717684488096787370998E1L,
  97 };
  98 static const long double S[4] = {
  99 /* 1.00000000000000000000E0L,*/
 100 -2.6201045551331104417768E1L,
 101  1.9361891836232102174846E2L,
 102 -4.2861221385716144629696E2L,
 103 };
 104 /* log2(e) - 1 */
 105 #define LOG2EA 4.4269504088896340735992e-1L
 106
 107 #define SQRTH 0.70710678118654752440L
 108
 109 long double log2l(long double x)
 110 {
 111         long double y, z;
 112         int e;
 113
 114         if (isnan(x))
 115                 return x;
 116         if (x == INFINITY)
 117                 return x;
 118         if (x <= 0.0) {
 119                 if (x == 0.0)
 120                         return -1/(x*x); /* -inf with divbyzero */
 121                 return 0/0.0f; /* nan with invalid */
 122         }
 123
 124         /* separate mantissa from exponent */
 125         /* Note, frexp is used so that denormal numbers
 126          * will be handled properly.
 127          */
 128         x = frexpl(x, &e);
 129
 130         /* logarithm using log(x) = z + z**3 P(z)/Q(z),
 131          * where z = 2(x-1)/x+1)
 132          */
 133         if (e > 2 || e < -2) {
 134                 if (x < SQRTH) {  /* 2(2x-1)/(2x+1) */
 135                         e -= 1;
 136                         z = x - 0.5;
 137                         y = 0.5 * z + 0.5;
 138                 } else {  /*  2 (x-1)/(x+1)   */
 139                         z = x - 0.5;
 140                         z -= 0.5;
 141                         y = 0.5 * x + 0.5;
 142                 }
 143                 x = z / y;
 144                 z = x*x;
 145                 y = x * (z * __polevll(z, R, 3) / __p1evll(z, S, 3));
 146                 goto done;
 147         }
 148
 149         /* logarithm using log(1+x) = x - .5x**2 + x**3 P(x)/Q(x) */
 150         if (x < SQRTH) {
 151                 e -= 1;
 152                 x = 2.0*x - 1.0;
 153         } else {
 154                 x = x - 1.0;
 155         }
 156         z = x*x;
 157         y = x * (z * __polevll(x, P, 6) / __p1evll(x, Q, 7));
 158         y = y - 0.5*z;
 159
 160 done:
 161         /* Multiply log of fraction by log2(e)
 162          * and base 2 exponent by 1
 163          *
 164          * ***CAUTION***
 165          *
 166          * This sequence of operations is critical and it may
 167          * be horribly defeated by some compiler optimizers.
 168          */
 169         z = y * LOG2EA;
 170         z += x * LOG2EA;
 171         z += y;
 172         z += x;
 173         z += e;
 174         return z;
 175 }
 176 #elif LDBL_MANT_DIG == 113 && LDBL_MAX_EXP == 16384
 177 // TODO: broken implementation to make things compile
 178 long double log2l(long double x)
 179 {
 180         return log2(x);
 181 }
 182 #endif