libquadmath/math/expm1q.c

   1 /*                                                      expm1l.c
   2  *
   3  *      Exponential function, minus 1
   4  *      128-bit __float128 precision
   5  *
   6  *
   7  *
   8  * SYNOPSIS:
   9  *
  10  * __float128 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, write to the Free Software
  52     Foundation, Inc., 59 Temple Place, Suite 330, Boston, MA 02111-1307  USA */
  53
  54
  55
  56 #include <errno.h>
  57 #include "quadmath-imp.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 __float128
  64   P0 = 2.943520915569954073888921213330863757240E8Q,
  65   P1 = -5.722847283900608941516165725053359168840E7Q,
  66   P2 = 8.944630806357575461578107295909719817253E6Q,
  67   P3 = -7.212432713558031519943281748462837065308E5Q,
  68   P4 = 4.578962475841642634225390068461943438441E4Q,
  69   P5 = -1.716772506388927649032068540558788106762E3Q,
  70   P6 = 4.401308817383362136048032038528753151144E1Q,
  71   P7 = -4.888737542888633647784737721812546636240E-1Q,
  72   Q0 = 1.766112549341972444333352727998584753865E9Q,
  73   Q1 = -7.848989743695296475743081255027098295771E8Q,
  74   Q2 = 1.615869009634292424463780387327037251069E8Q,
  75   Q3 = -2.019684072836541751428967854947019415698E7Q,
  76   Q4 = 1.682912729190313538934190635536631941751E6Q,
  77   Q5 = -9.615511549171441430850103489315371768998E4Q,
  78   Q6 = 3.697714952261803935521187272204485251835E3Q,
  79   Q7 = -8.802340681794263968892934703309274564037E1Q,
  80   /* Q8 = 1.000000000000000000000000000000000000000E0 */
  81 /* C1 + C2 = ln 2 */
  82
  83   C1 = 6.93145751953125E-1Q,
  84   C2 = 1.428606820309417232121458176568075500134E-6Q,
  85 /* ln (2^16384 * (1 - 2^-113)) */
  86   maxlog = 1.1356523406294143949491931077970764891253E4Q,
  87 /* ln 2^-114 */
  88   minarg = -7.9018778583833765273564461846232128760607E1Q;
  89
  90
  91 __float128
  92 expm1q (__float128 x)
  93 {
  94   __float128 px, qx, xx;
  95   int32_t ix, sign;
  96   ieee854_float128 u;
  97   int k;
  98
  99   /* Detect infinity and NaN.  */
 100   u.value = x;
 101   ix = u.words32.w0;
 102   sign = ix & 0x80000000;
 103   ix &= 0x7fffffff;
 104   if (!sign && ix >= 0x40060000)
 105     {
 106       /* If num is positive and exp >= 6 use plain exp.  */
 107       return expq (x);
 108     }
 109   if (ix >= 0x7fff0000)
 110     {
 111       /* Infinity. */
 112       if (((ix & 0xffff) | u.words32.w1 | u.words32.w2 | u.words32.w3) == 0)
 113         {
 114           if (sign)
 115             return -1.0Q;
 116           else
 117             return x;
 118         }
 119       /* NaN. No invalid exception. */
 120       return x;
 121     }
 122
 123   /* expm1(+- 0) = +- 0.  */
 124   if ((ix == 0) && (u.words32.w1 | u.words32.w2 | u.words32.w3) == 0)
 125     return x;
 126
 127   /* Overflow.  */
 128   if (x > maxlog)
 129     {
 130       errno = ERANGE;
 131       return (HUGE_VALQ * HUGE_VALQ);
 132     }
 133
 134   /* Minimum value.  */
 135   if (x < minarg)
 136     return (4.0/HUGE_VALQ - 1.0Q);
 137
 138   /* Express x = ln 2 (k + remainder), remainder not exceeding 1/2. */
 139   xx = C1 + C2;                 /* ln 2. */
 140   px = floorq (0.5 + x / xx);
 141   k = px;
 142   /* remainder times ln 2 */
 143   x -= px * C1;
 144   x -= px * C2;
 145
 146   /* Approximate exp(remainder ln 2).  */
 147   px = (((((((P7 * x
 148               + P6) * x
 149              + P5) * x + P4) * x + P3) * x + P2) * x + P1) * x + P0) * x;
 150
 151   qx = (((((((x
 152               + Q7) * x
 153              + Q6) * x + Q5) * x + Q4) * x + Q3) * x + Q2) * x + Q1) * x + Q0;
 154
 155   xx = x * x;
 156   qx = x + (0.5 * xx + xx * px / qx);
 157
 158   /* exp(x) = exp(k ln 2) exp(remainder ln 2) = 2^k exp(remainder ln 2).
 159
 160   We have qx = exp(remainder ln 2) - 1, so
 161   exp(x) - 1 = 2^k (qx + 1) - 1
 162              = 2^k qx + 2^k - 1.  */
 163
 164   px = ldexpq (1.0Q, k);
 165   x = px * qx + (px - 1.0);
 166   return x;
 167 }