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

     1  /* origin: OpenBSD /usr/src/lib/libm/src/ld80/e_expl.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, long double precision
    19   *
    20   *
    21   * SYNOPSIS:
    22   *
    23   * long double x, y, expl();
    24   *
    25   * y = expl( x );
    26   *
    27   *
    28   * DESCRIPTION:
    29   *
    30   * Returns e (2.71828...) raised to the x power.
    31   *
    32   * Range reduction is accomplished by separating the argument
    33   * into an integer k and fraction f such that
    34   *
    35   *     x    k  f
    36   *    e  = 2  e.
    37   *
    38   * A Pade' form of degree 5/6 is used to approximate exp(f) - 1
    39   * in the basic range [-0.5 ln 2, 0.5 ln 2].
    40   *
    41   *
    42   * ACCURACY:
    43   *
    44   *                      Relative error:
    45   * arithmetic   domain     # trials      peak         rms
    46   *    IEEE      +-10000     50000       1.12e-19    2.81e-20
    47   *
    48   *
    49   * Error amplification in the exponential function can be
    50   * a serious matter.  The error propagation involves
    51   * exp( X(1+delta) ) = exp(X) ( 1 + X*delta + ... ),
    52   * which shows that a 1 lsb error in representing X produces
    53   * a relative error of X times 1 lsb in the function.
    54   * While the routine gives an accurate result for arguments
    55   * that are exactly represented by a long double precision
    56   * computer number, the result contains amplified roundoff
    57   * error for large arguments not exactly represented.
    58   *
    59   *
    60   * ERROR MESSAGES:
    61   *
    62   *   message         condition      value returned
    63   * exp underflow    x < MINLOG         0.0
    64   * exp overflow     x > MAXLOG         MAXNUM
    65   *
    66   */
    67  
    68  #include "libm.h"
    69  
    70  #if LDBL_MANT_DIG == 53 && LDBL_MAX_EXP == 1024
    71  long double expl(long double x)
    72  {
    73  	return exp(x);
    74  }
    75  #elif LDBL_MANT_DIG == 64 && LDBL_MAX_EXP == 16384
    76  
    77  static const long double P[3] = {
    78   1.2617719307481059087798E-4L,
    79   3.0299440770744196129956E-2L,
    80   9.9999999999999999991025E-1L,
    81  };
    82  static const long double Q[4] = {
    83   3.0019850513866445504159E-6L,
    84   2.5244834034968410419224E-3L,
    85   2.2726554820815502876593E-1L,
    86   2.0000000000000000000897E0L,
    87  };
    88  static const long double
    89  LN2HI = 6.9314575195312500000000E-1L,
    90  LN2LO = 1.4286068203094172321215E-6L,
    91  LOG2E = 1.4426950408889634073599E0L;
    92  
    93  long double expl(long double x)
    94  {
    95  	long double px, xx;
    96  	int k;
    97  
    98  	if (isnan(x))
    99  		return x;
   100  	if (x > 11356.5234062941439488L) /* x > ln(2^16384 - 0.5) */
   101  		return x * 0x1p16383L;
   102  	if (x < -11399.4985314888605581L) /* x < ln(2^-16446) */
   103  		return -0x1p-16445L/x;
   104  
   105  	/* Express e**x = e**f 2**k
   106  	 *   = e**(f + k ln(2))
   107  	 */
   108  	px = floorl(LOG2E * x + 0.5);
   109  	k = px;
   110  	x -= px * LN2HI;
   111  	x -= px * LN2LO;
   112  
   113  	/* rational approximation of the fractional part:
   114  	 * e**x =  1 + 2x P(x**2)/(Q(x**2) - x P(x**2))
   115  	 */
   116  	xx = x * x;
   117  	px = x * __polevll(xx, P, 2);
   118  	x = px/(__polevll(xx, Q, 3) - px);
   119  	x = 1.0 + 2.0 * x;
   120  	return scalbnl(x, k);
   121  }
   122  #elif LDBL_MANT_DIG == 113 && LDBL_MAX_EXP == 16384
   123  // TODO: broken implementation to make things compile
   124  long double expl(long double x)
   125  {
   126  	return exp(x);
   127  }
   128  #endif