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
  41 #include "math.h"
  42 #include "math_private.h"
  43
  44 /* exp(x) - 1 = x + 0.5 x^2 + x^3 P(x)/Q(x)
  45    -.5 ln 2  <  x  <  .5 ln 2
  46    Theoretical peak relative error = 8.1e-36  */
  47
  48 static long double
  49   P0 = 2.943520915569954073888921213330863757240E8L,
  50   P1 = -5.722847283900608941516165725053359168840E7L,
  51   P2 = 8.944630806357575461578107295909719817253E6L,
  52   P3 = -7.212432713558031519943281748462837065308E5L,
  53   P4 = 4.578962475841642634225390068461943438441E4L,
  54   P5 = -1.716772506388927649032068540558788106762E3L,
  55   P6 = 4.401308817383362136048032038528753151144E1L,
  56   P7 = -4.888737542888633647784737721812546636240E-1L,
  57   Q0 = 1.766112549341972444333352727998584753865E9L,
  58   Q1 = -7.848989743695296475743081255027098295771E8L,
  59   Q2 = 1.615869009634292424463780387327037251069E8L,
  60   Q3 = -2.019684072836541751428967854947019415698E7L,
  61   Q4 = 1.682912729190313538934190635536631941751E6L,
  62   Q5 = -9.615511549171441430850103489315371768998E4L,
  63   Q6 = 3.697714952261803935521187272204485251835E3L,
  64   Q7 = -8.802340681794263968892934703309274564037E1L,
  65   /* Q8 = 1.000000000000000000000000000000000000000E0 */
  66 /* C1 + C2 = ln 2 */
  67
  68   C1 = 6.93145751953125E-1L,
  69   C2 = 1.428606820309417232121458176568075500134E-6L,
  70 /* ln (2^16384 * (1 - 2^-113)) */
  71   maxlog = 1.1356523406294143949491931077970764891253E4L,
  72 /* ln 2^-114 */
  73   minarg = -7.9018778583833765273564461846232128760607E1L, big = 2e4932L;
  74
  75
  76 long double
  77 __expm1l (long double x)
  78 {
  79   long double px, qx, xx;
  80   int32_t ix, sign;
  81   ieee854_long_double_shape_type u;
  82   int k;
  83
  84   /* Detect infinity and NaN.  */
  85   u.value = x;
  86   ix = u.parts32.w0;
  87   sign = ix & 0x80000000;
  88   ix &= 0x7fffffff;
  89   if (ix >= 0x7fff0000)
  90     {
  91       /* Infinity. */
  92       if (((ix & 0xffff) | u.parts32.w1 | u.parts32.w2 | u.parts32.w3) == 0)
  93         {
  94           if (sign)
  95             return -1.0L;
  96           else
  97             return x;
  98         }
  99       /* NaN. No invalid exception. */
 100       return x;
 101     }
 102
 103   /* expm1(+- 0) = +- 0.  */
 104   if ((ix == 0) && (u.parts32.w1 | u.parts32.w2 | u.parts32.w3) == 0)
 105     return x;
 106
 107   /* Overflow.  */
 108   if (x > maxlog)
 109     return (big * big);
 110
 111   /* Minimum value.  */
 112   if (x < minarg)
 113     return (4.0/big - 1.0L);
 114
 115   /* Express x = ln 2 (k + remainder), remainder not exceeding 1/2. */
 116   xx = C1 + C2;                 /* ln 2. */
 117   px = __floorl (0.5 + x / xx);
 118   k = px;
 119   /* remainder times ln 2 */
 120   x -= px * C1;
 121   x -= px * C2;
 122
 123   /* Approximate exp(remainder ln 2).  */
 124   px = (((((((P7 * x
 125               + P6) * x
 126              + P5) * x + P4) * x + P3) * x + P2) * x + P1) * x + P0) * x;
 127
 128   qx = (((((((x
 129               + Q7) * x
 130              + Q6) * x + Q5) * x + Q4) * x + Q3) * x + Q2) * x + Q1) * x + Q0;
 131
 132   xx = x * x;
 133   qx = x + (0.5 * xx + xx * px / qx);
 134
 135   /* exp(x) = exp(k ln 2) exp(remainder ln 2) = 2^k exp(remainder ln 2).
 136
 137   We have qx = exp(remainder ln 2) - 1, so
 138   exp(x) - 1 = 2^k (qx + 1) - 1
 139              = 2^k qx + 2^k - 1.  */
 140
 141   px = ldexpl (1.0L, k);
 142   x = px * qx + (px - 1.0);
 143   return x;
 144 }
 145
 146 weak_alias (__expm1l, expm1l)
 147 #ifdef NO_LONG_DOUBLE
 148 strong_alias (__expm1, __expm1l) weak_alias (__expm1, expm1l)
 149 #endif