github.com/afumu/libc@v0.0.6/musl/src/math/expm1l.c (about)

     1  /* origin: OpenBSD /usr/src/lib/libm/src/ld80/e_expm1l.c */
     2  /*
     3   * Copyright (c) 2008 Stephen L. Moshier <steve@moshier.net>
     4   *
     5   * Permission to use, copy, modify, and distribute this software for any
     6   * purpose with or without fee is hereby granted, provided that the above
     7   * copyright notice and this permission notice appear in all copies.
     8   *
     9   * THE SOFTWARE IS PROVIDED "AS IS" AND THE AUTHOR DISCLAIMS ALL WARRANTIES
    10   * WITH REGARD TO THIS SOFTWARE INCLUDING ALL IMPLIED WARRANTIES OF
    11   * MERCHANTABILITY AND FITNESS. IN NO EVENT SHALL THE AUTHOR BE LIABLE FOR
    12   * ANY SPECIAL, DIRECT, INDIRECT, OR CONSEQUENTIAL DAMAGES OR ANY DAMAGES
    13   * WHATSOEVER RESULTING FROM LOSS OF USE, DATA OR PROFITS, WHETHER IN AN
    14   * ACTION OF CONTRACT, NEGLIGENCE OR OTHER TORTIOUS ACTION, ARISING OUT OF
    15   * OR IN CONNECTION WITH THE USE OR PERFORMANCE OF THIS SOFTWARE.
    16   */
    17  /*
    18   *      Exponential function, minus 1
    19   *      Long double precision
    20   *
    21   *
    22   * SYNOPSIS:
    23   *
    24   * long double x, y, expm1l();
    25   *
    26   * y = expm1l( x );
    27   *
    28   *
    29   * DESCRIPTION:
    30   *
    31   * Returns e (2.71828...) raised to the x power, minus 1.
    32   *
    33   * Range reduction is accomplished by separating the argument
    34   * into an integer k and fraction f such that
    35   *
    36   *     x    k  f
    37   *    e  = 2  e.
    38   *
    39   * An expansion x + .5 x^2 + x^3 R(x) approximates exp(f) - 1
    40   * in the basic range [-0.5 ln 2, 0.5 ln 2].
    41   *
    42   *
    43   * ACCURACY:
    44   *
    45   *                      Relative error:
    46   * arithmetic   domain     # trials      peak         rms
    47   *    IEEE    -45,+maxarg   200,000     1.2e-19     2.5e-20
    48   */
    49  
    50  #include "libm.h"
    51  
    52  #if LDBL_MANT_DIG == 53 && LDBL_MAX_EXP == 1024
    53  long double expm1l(long double x)
    54  {
    55  	return expm1(x);
    56  }
    57  #elif LDBL_MANT_DIG == 64 && LDBL_MAX_EXP == 16384
    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 = 3.4e-22  */
    62  static const long double
    63  P0 = -1.586135578666346600772998894928250240826E4L,
    64  P1 =  2.642771505685952966904660652518429479531E3L,
    65  P2 = -3.423199068835684263987132888286791620673E2L,
    66  P3 =  1.800826371455042224581246202420972737840E1L,
    67  P4 = -5.238523121205561042771939008061958820811E-1L,
    68  Q0 = -9.516813471998079611319047060563358064497E4L,
    69  Q1 =  3.964866271411091674556850458227710004570E4L,
    70  Q2 = -7.207678383830091850230366618190187434796E3L,
    71  Q3 =  7.206038318724600171970199625081491823079E2L,
    72  Q4 = -4.002027679107076077238836622982900945173E1L,
    73  /* Q5 = 1.000000000000000000000000000000000000000E0 */
    74  /* C1 + C2 = ln 2 */
    75  C1 = 6.93145751953125E-1L,
    76  C2 = 1.428606820309417232121458176568075500134E-6L,
    77  /* ln 2^-65 */
    78  minarg = -4.5054566736396445112120088E1L,
    79  /* ln 2^16384 */
    80  maxarg = 1.1356523406294143949492E4L;
    81  
    82  long double expm1l(long double x)
    83  {
    84  	long double px, qx, xx;
    85  	int k;
    86  
    87  	if (isnan(x))
    88  		return x;
    89  	if (x > maxarg)
    90  		return x*0x1p16383L; /* overflow, unless x==inf */
    91  	if (x == 0.0)
    92  		return x;
    93  	if (x < minarg)
    94  		return -1.0;
    95  
    96  	xx = C1 + C2;
    97  	/* Express x = ln 2 (k + remainder), remainder not exceeding 1/2. */
    98  	px = floorl(0.5 + x / xx);
    99  	k = px;
   100  	/* remainder times ln 2 */
   101  	x -= px * C1;
   102  	x -= px * C2;
   103  
   104  	/* Approximate exp(remainder ln 2).*/
   105  	px = (((( P4 * x + P3) * x + P2) * x + P1) * x + P0) * x;
   106  	qx = (((( x + Q4) * x + Q3) * x + Q2) * x + Q1) * x + Q0;
   107  	xx = x * x;
   108  	qx = x + (0.5 * xx + xx * px / qx);
   109  
   110  	/* exp(x) = exp(k ln 2) exp(remainder ln 2) = 2^k exp(remainder ln 2).
   111  	 We have qx = exp(remainder ln 2) - 1, so
   112  	 exp(x) - 1  =  2^k (qx + 1) - 1  =  2^k qx + 2^k - 1.  */
   113  	px = scalbnl(1.0, k);
   114  	x = px * qx + (px - 1.0);
   115  	return x;
   116  }
   117  #elif LDBL_MANT_DIG == 113 && LDBL_MAX_EXP == 16384
   118  // TODO: broken implementation to make things compile
   119  long double expm1l(long double x)
   120  {
   121  	return expm1(x);
   122  }
   123  #endif