sysdeps/ieee754/ldbl-128/s_expm1l.c

   1 /*                                                      expm1l.c
   2  *
   3  *      Exponential function, minus 1
   4  *      128-bit long double precision
   5  *
   6  *
   7  *
   8  * SYNOPSIS:
   9  *
  10  * long double x, y, expm1l();
  11  *
  12  * y = expm1l( x );
  13  *
  14  *
  15  *
  16  * DESCRIPTION:
  17  *
  18  * Returns e (2.71828...) raised to the x power, minus one.
  19  *
  20  * Range reduction is accomplished by separating the argument
  21  * into an integer k and fraction f such that
  22  *
  23  *     x    k  f
  24  *    e  = 2  e.
  25  *
  26  * An expansion x + .5 x^2 + x^3 R(x) approximates exp(f) - 1
  27  * in the basic range [-0.5 ln 2, 0.5 ln 2].
  28  *
  29  *
  30  * ACCURACY:
  31  *
  32  *                      Relative error:
  33  * arithmetic   domain     # trials      peak         rms
  34  *    IEEE    -79,+MAXLOG    100,000     1.7e-34     4.5e-35
  35  *
  36  */
  37
  38 /* Copyright 2001 by Stephen L. Moshier
  39
  40     This library is free software; you can redistribute it and/or
  41     modify it under the terms of the GNU Lesser General Public
  42     License as published by the Free Software Foundation; either
  43     version 2.1 of the License, or (at your option) any later version.
  44
  45     This library is distributed in the hope that it will be useful,
  46     but WITHOUT ANY WARRANTY; without even the implied warranty of
  47     MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE.  See the GNU
  48     Lesser General Public License for more details.
  49
  50     You should have received a copy of the GNU Lesser General Public
  51     License along with this library; if not, see
  52     <http://www.gnu.org/licenses/>.  */
  53
  54
  55
  56 #include <errno.h>
  57 #include "math.h"
  58 #include "math_private.h"
  59
  60 /* exp(x) - 1 = x + 0.5 x^2 + x^3 P(x)/Q(x)
  61    -.5 ln 2  <  x  <  .5 ln 2
  62    Theoretical peak relative error = 8.1e-36  */
  63
  64 static const long double
  65   P0 = 2.943520915569954073888921213330863757240E8L,
  66   P1 = -5.722847283900608941516165725053359168840E7L,
  67   P2 = 8.944630806357575461578107295909719817253E6L,
  68   P3 = -7.212432713558031519943281748462837065308E5L,
  69   P4 = 4.578962475841642634225390068461943438441E4L,
  70   P5 = -1.716772506388927649032068540558788106762E3L,
  71   P6 = 4.401308817383362136048032038528753151144E1L,
  72   P7 = -4.888737542888633647784737721812546636240E-1L,
  73   Q0 = 1.766112549341972444333352727998584753865E9L,
  74   Q1 = -7.848989743695296475743081255027098295771E8L,
  75   Q2 = 1.615869009634292424463780387327037251069E8L,
  76   Q3 = -2.019684072836541751428967854947019415698E7L,
  77   Q4 = 1.682912729190313538934190635536631941751E6L,
  78   Q5 = -9.615511549171441430850103489315371768998E4L,
  79   Q6 = 3.697714952261803935521187272204485251835E3L,
  80   Q7 = -8.802340681794263968892934703309274564037E1L,
  81   /* Q8 = 1.000000000000000000000000000000000000000E0 */
  82 /* C1 + C2 = ln 2 */
  83
  84   C1 = 6.93145751953125E-1L,
  85   C2 = 1.428606820309417232121458176568075500134E-6L,
  86 /* ln (2^16384 * (1 - 2^-113)) */
  87   maxlog = 1.1356523406294143949491931077970764891253E4L,
  88 /* ln 2^-114 */
  89   minarg = -7.9018778583833765273564461846232128760607E1L, big = 1e4932L;
  90
  91
  92 long double
  93 __expm1l (long double x)
  94 {
  95   long double px, qx, xx;
  96   int32_t ix, sign;
  97   ieee854_long_double_shape_type u;
  98   int k;
  99
 100   /* Detect infinity and NaN.  */
 101   u.value = x;
 102   ix = u.parts32.w0;
 103   sign = ix & 0x80000000;
 104   ix &= 0x7fffffff;
 105   if (ix >= 0x7fff0000)
 106     {
 107       /* Infinity. */
 108       if (((ix & 0xffff) | u.parts32.w1 | u.parts32.w2 | u.parts32.w3) == 0)
 109         {
 110           if (sign)
 111             return -1.0L;
 112           else
 113             return x;
 114         }
 115       /* NaN. No invalid exception. */
 116       return x;
 117     }
 118
 119   /* expm1(+- 0) = +- 0.  */
 120   if ((ix == 0) && (u.parts32.w1 | u.parts32.w2 | u.parts32.w3) == 0)
 121     return x;
 122
 123   /* Overflow.  */
 124   if (x > maxlog)
 125     {
 126       __set_errno (ERANGE);
 127       return (big * big);
 128     }
 129
 130   /* Minimum value.  */
 131   if (x < minarg)
 132     return (4.0/big - 1.0L);
 133
 134   /* Express x = ln 2 (k + remainder), remainder not exceeding 1/2. */
 135   xx = C1 + C2;                 /* ln 2. */
 136   px = __floorl (0.5 + x / xx);
 137   k = px;
 138   /* remainder times ln 2 */
 139   x -= px * C1;
 140   x -= px * C2;
 141
 142   /* Approximate exp(remainder ln 2).  */
 143   px = (((((((P7 * x
 144               + P6) * x
 145              + P5) * x + P4) * x + P3) * x + P2) * x + P1) * x + P0) * x;
 146
 147   qx = (((((((x
 148               + Q7) * x
 149              + Q6) * x + Q5) * x + Q4) * x + Q3) * x + Q2) * x + Q1) * x + Q0;
 150
 151   xx = x * x;
 152   qx = x + (0.5 * xx + xx * px / qx);
 153
 154   /* exp(x) = exp(k ln 2) exp(remainder ln 2) = 2^k exp(remainder ln 2).
 155
 156   We have qx = exp(remainder ln 2) - 1, so
 157   exp(x) - 1 = 2^k (qx + 1) - 1
 158              = 2^k qx + 2^k - 1.  */
 159
 160   px = __ldexpl (1.0L, k);
 161   x = px * qx + (px - 1.0);
 162   return x;
 163 }
 164 libm_hidden_def (__expm1l)
 165 weak_alias (__expm1l, expm1l)