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      <https://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^-114 */
      86    minarg = -7.9018778583833765273564461846232128760607E1L, big = 1e290L;
      87  
      88  
      89  long double
      90  __expm1l (long double x)
      91  {
      92    long double px, qx, xx;
      93    int32_t ix, lx, sign;
      94    int k;
      95    double xhi;
      96  
      97    /* Detect infinity and NaN.  */
      98    xhi = ldbl_high (x);
      99    EXTRACT_WORDS (ix, lx, xhi);
     100    sign = ix & 0x80000000;
     101    ix &= 0x7fffffff;
     102    if (!sign && ix >= 0x40600000)
     103      return __expl (x);
     104    if (ix >= 0x7ff00000)
     105      {
     106        /* Infinity (which must be negative infinity). */
     107        if (((ix - 0x7ff00000) | lx) == 0)
     108  	return -1.0L;
     109        /* NaN.  Invalid exception if signaling.  */
     110        return x + x;
     111      }
     112  
     113    /* expm1(+- 0) = +- 0.  */
     114    if ((ix | lx) == 0)
     115      return x;
     116  
     117    /* Minimum value.  */
     118    if (x < minarg)
     119      return (4.0/big - 1.0L);
     120  
     121    /* Express x = ln 2 (k + remainder), remainder not exceeding 1/2. */
     122    xx = C1 + C2;			/* ln 2. */
     123    px = floorl (0.5 + x / xx);
     124    k = px;
     125    /* remainder times ln 2 */
     126    x -= px * C1;
     127    x -= px * C2;
     128  
     129    /* Approximate exp(remainder ln 2).  */
     130    px = (((((((P7 * x
     131  	      + P6) * x
     132  	     + P5) * x + P4) * x + P3) * x + P2) * x + P1) * x + P0) * x;
     133  
     134    qx = (((((((x
     135  	      + Q7) * x
     136  	     + Q6) * x + Q5) * x + Q4) * x + Q3) * x + Q2) * x + Q1) * x + Q0;
     137  
     138    xx = x * x;
     139    qx = x + (0.5 * xx + xx * px / qx);
     140  
     141    /* exp(x) = exp(k ln 2) exp(remainder ln 2) = 2^k exp(remainder ln 2).
     142  
     143    We have qx = exp(remainder ln 2) - 1, so
     144    exp(x) - 1 = 2^k (qx + 1) - 1
     145               = 2^k qx + 2^k - 1.  */
     146  
     147    px = __ldexpl (1.0L, k);
     148    x = px * qx + (px - 1.0);
     149    return x;
     150  }
     151  libm_hidden_def (__expm1l)
     152  long_double_symbol (libm, __expm1l, expm1l);