sysdeps/ieee754/ldbl-128ibm/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 #include <errno.h>
  55 #include <math.h>
  56 #include <math_private.h>
  57 #include <math_ldbl_opt.h>
  58
  59 /* exp(x) - 1 = x + 0.5 x^2 + x^3 P(x)/Q(x)
  60    -.5 ln 2  <  x  <  .5 ln 2
  61    Theoretical peak relative error = 8.1e-36  */
  62
  63 static const long double
  64   P0 = 2.943520915569954073888921213330863757240E8L,
  65   P1 = -5.722847283900608941516165725053359168840E7L,
  66   P2 = 8.944630806357575461578107295909719817253E6L,
  67   P3 = -7.212432713558031519943281748462837065308E5L,
  68   P4 = 4.578962475841642634225390068461943438441E4L,
  69   P5 = -1.716772506388927649032068540558788106762E3L,
  70   P6 = 4.401308817383362136048032038528753151144E1L,
  71   P7 = -4.888737542888633647784737721812546636240E-1L,
  72   Q0 = 1.766112549341972444333352727998584753865E9L,
  73   Q1 = -7.848989743695296475743081255027098295771E8L,
  74   Q2 = 1.615869009634292424463780387327037251069E8L,
  75   Q3 = -2.019684072836541751428967854947019415698E7L,
  76   Q4 = 1.682912729190313538934190635536631941751E6L,
  77   Q5 = -9.615511549171441430850103489315371768998E4L,
  78   Q6 = 3.697714952261803935521187272204485251835E3L,
  79   Q7 = -8.802340681794263968892934703309274564037E1L,
  80   /* Q8 = 1.000000000000000000000000000000000000000E0 */
  81 /* C1 + C2 = ln 2 */
  82
  83   C1 = 6.93145751953125E-1L,
  84   C2 = 1.428606820309417232121458176568075500134E-6L,
  85 /* ln (2^16384 * (1 - 2^-113)) */
  86   maxlog = 1.1356523406294143949491931077970764891253E4L,
  87 /* ln 2^-114 */
  88   minarg = -7.9018778583833765273564461846232128760607E1L, big = 1e290L;
  89
  90
  91 long double
  92 __expm1l (long double x)
  93 {
  94   long double px, qx, xx;
  95   int32_t ix, lx, sign;
  96   int k;
  97   double xhi;
  98
  99   /* Detect infinity and NaN.  */
 100   xhi = ldbl_high (x);
 101   EXTRACT_WORDS (ix, lx, xhi);
 102   sign = ix & 0x80000000;
 103   ix &= 0x7fffffff;
 104   if (!sign && ix >= 0x40600000)
 105     return __expl (x);
 106   if (ix >= 0x7ff00000)
 107     {
 108       /* Infinity. */
 109       if (((ix - 0x7ff00000) | lx) == 0)
 110         {
 111           if (sign)
 112             return -1.0L;
 113           else
 114             return x;
 115         }
 116       /* NaN. No invalid exception. */
 117       return x;
 118     }
 119
 120   /* expm1(+- 0) = +- 0.  */
 121   if ((ix | lx) == 0)
 122     return x;
 123
 124   /* Overflow.  */
 125   if (x > maxlog)
 126     {
 127       __set_errno (ERANGE);
 128       return (big * big);
 129     }
 130
 131   /* Minimum value.  */
 132   if (x < minarg)
 133     return (4.0/big - 1.0L);
 134
 135   /* Express x = ln 2 (k + remainder), remainder not exceeding 1/2. */
 136   xx = C1 + C2;                 /* ln 2. */
 137   px = __floorl (0.5 + x / xx);
 138   k = px;
 139   /* remainder times ln 2 */
 140   x -= px * C1;
 141   x -= px * C2;
 142
 143   /* Approximate exp(remainder ln 2).  */
 144   px = (((((((P7 * x
 145               + P6) * x
 146              + P5) * x + P4) * x + P3) * x + P2) * x + P1) * x + P0) * x;
 147
 148   qx = (((((((x
 149               + Q7) * x
 150              + Q6) * x + Q5) * x + Q4) * x + Q3) * x + Q2) * x + Q1) * x + Q0;
 151
 152   xx = x * x;
 153   qx = x + (0.5 * xx + xx * px / qx);
 154
 155   /* exp(x) = exp(k ln 2) exp(remainder ln 2) = 2^k exp(remainder ln 2).
 156
 157   We have qx = exp(remainder ln 2) - 1, so
 158   exp(x) - 1 = 2^k (qx + 1) - 1
 159              = 2^k qx + 2^k - 1.  */
 160
 161   px = __ldexpl (1.0L, k);
 162   x = px * qx + (px - 1.0);
 163   return x;
 164 }
 165 libm_hidden_def (__expm1l)
 166 long_double_symbol (libm, __expm1l, expm1l);