compiler/stdc/math/ld128/s_expm1l.c

   1 /*      $OpenBSD: s_expm1l.c,v 1.1 2011/07/06 00:02:42 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 /*                                                      expm1l.c
  20  *
  21  *      Exponential function, minus 1
  22  *      128-bit long double precision
  23  *
  24  *
  25  *
  26  * SYNOPSIS:
  27  *
  28  * long double x, y, expm1l();
  29  *
  30  * y = expm1l( x );
  31  *
  32  *
  33  *
  34  * DESCRIPTION:
  35  *
  36  * Returns e (2.71828...) raised to the x power, minus one.
  37  *
  38  * Range reduction is accomplished by separating the argument
  39  * into an integer k and fraction f such that
  40  *
  41  *     x    k  f
  42  *    e  = 2  e.
  43  *
  44  * An expansion x + .5 x^2 + x^3 R(x) approximates exp(f) - 1
  45  * in the basic range [-0.5 ln 2, 0.5 ln 2].
  46  *
  47  *
  48  * ACCURACY:
  49  *
  50  *                      Relative error:
  51  * arithmetic   domain     # trials      peak         rms
  52  *    IEEE    -79,+MAXLOG    100,000     1.7e-34     4.5e-35
  53  *
  54  */
  55
  56 #include <errno.h>
  57 #include "math.h"
  58
  59 #include "math_private.h"
  60
  61 /* exp(x) - 1 = x + 0.5 x^2 + x^3 P(x)/Q(x)
  62    -.5 ln 2  <  x  <  .5 ln 2
  63    Theoretical peak relative error = 8.1e-36  */
  64
  65 static const long double
  66   P0 = 2.943520915569954073888921213330863757240E8L,
  67   P1 = -5.722847283900608941516165725053359168840E7L,
  68   P2 = 8.944630806357575461578107295909719817253E6L,
  69   P3 = -7.212432713558031519943281748462837065308E5L,
  70   P4 = 4.578962475841642634225390068461943438441E4L,
  71   P5 = -1.716772506388927649032068540558788106762E3L,
  72   P6 = 4.401308817383362136048032038528753151144E1L,
  73   P7 = -4.888737542888633647784737721812546636240E-1L,
  74   Q0 = 1.766112549341972444333352727998584753865E9L,
  75   Q1 = -7.848989743695296475743081255027098295771E8L,
  76   Q2 = 1.615869009634292424463780387327037251069E8L,
  77   Q3 = -2.019684072836541751428967854947019415698E7L,
  78   Q4 = 1.682912729190313538934190635536631941751E6L,
  79   Q5 = -9.615511549171441430850103489315371768998E4L,
  80   Q6 = 3.697714952261803935521187272204485251835E3L,
  81   Q7 = -8.802340681794263968892934703309274564037E1L,
  82   /* Q8 = 1.000000000000000000000000000000000000000E0 */
  83 /* C1 + C2 = ln 2 */
  84
  85   C1 = 6.93145751953125E-1L,
  86   C2 = 1.428606820309417232121458176568075500134E-6L,
  87 /* ln (2^16384 * (1 - 2^-113)) */
  88   maxlog = 1.1356523406294143949491931077970764891253E4L,
  89 /* ln 2^-114 */
  90   minarg = -7.9018778583833765273564461846232128760607E1L, big = 1e4932L;
  91
  92
  93 long double
  94 expm1l(long double x)
  95 {
  96   long double px, qx, xx;
  97   int32_t ix, sign;
  98   ieee_quad_shape_type u;
  99   int k;
 100
 101   /* Detect infinity and NaN.  */
 102   u.value = x;
 103   ix = u.parts32.mswhi;
 104   sign = ix & 0x80000000;
 105   ix &= 0x7fffffff;
 106   if (ix >= 0x7fff0000)
 107     {
 108       /* Infinity. */
 109       if (((ix & 0xffff) | u.parts32.mswlo | u.parts32.lswhi |
 110         u.parts32.lswlo) == 0)
 111         {
 112           if (sign)
 113             return -1.0L;
 114           else
 115             return x;
 116         }
 117       /* NaN. No invalid exception. */
 118       return x;
 119     }
 120
 121   /* expm1(+- 0) = +- 0.  */
 122   if ((ix == 0) && (u.parts32.mswlo | u.parts32.lswhi | u.parts32.lswlo) == 0)
 123     return x;
 124
 125   /* Overflow.  */
 126   if (x > maxlog)
 127       return (big * big);
 128
 129   /* Minimum value.  */
 130   if (x < minarg)
 131     return (4.0/big - 1.0L);
 132
 133   /* Express x = ln 2 (k + remainder), remainder not exceeding 1/2. */
 134   xx = C1 + C2;                 /* ln 2. */
 135   px = floorl (0.5 + x / xx);
 136   k = px;
 137   /* remainder times ln 2 */
 138   x -= px * C1;
 139   x -= px * C2;
 140
 141   /* Approximate exp(remainder ln 2).  */
 142   px = (((((((P7 * x
 143               + P6) * x
 144              + P5) * x + P4) * x + P3) * x + P2) * x + P1) * x + P0) * x;
 145
 146   qx = (((((((x
 147               + Q7) * x
 148              + Q6) * x + Q5) * x + Q4) * x + Q3) * x + Q2) * x + Q1) * x + Q0;
 149
 150   xx = x * x;
 151   qx = x + (0.5 * xx + xx * px / qx);
 152
 153   /* exp(x) = exp(k ln 2) exp(remainder ln 2) = 2^k exp(remainder ln 2).
 154
 155   We have qx = exp(remainder ln 2) - 1, so
 156   exp(x) - 1 = 2^k (qx + 1) - 1
 157              = 2^k qx + 2^k - 1.  */
 158
 159   px = ldexpl (1.0L, k);
 160   x = px * qx + (px - 1.0);
 161   return x;
 162 }