1/*	$OpenBSD: s_expm1l.c,v 1.2 2016/09/12 19:47:02 guenther Exp $	*/
2
3/*
4 * Copyright (c) 2008 Stephen L. Moshier <steve@moshier.net>
5 *
6 * Permission to use, copy, modify, and distribute this software for any
7 * purpose with or without fee is hereby granted, provided that the above
8 * copyright notice and this permission notice appear in all copies.
9 *
10 * THE SOFTWARE IS PROVIDED "AS IS" AND THE AUTHOR DISCLAIMS ALL WARRANTIES
11 * WITH REGARD TO THIS SOFTWARE INCLUDING ALL IMPLIED WARRANTIES OF
12 * MERCHANTABILITY AND FITNESS. IN NO EVENT SHALL THE AUTHOR BE LIABLE FOR
13 * ANY SPECIAL, DIRECT, INDIRECT, OR CONSEQUENTIAL DAMAGES OR ANY DAMAGES
14 * WHATSOEVER RESULTING FROM LOSS OF USE, DATA OR PROFITS, WHETHER IN AN
15 * ACTION OF CONTRACT, NEGLIGENCE OR OTHER TORTIOUS ACTION, ARISING OUT OF
16 * OR IN CONNECTION WITH THE USE OR PERFORMANCE OF THIS SOFTWARE.
17 */
18
19/*							expm1l.c
20 *
21 *	Exponential function, minus 1
22 *      128-bit long double precision
23 *
24 *
25 *
26 * SYNOPSIS:
27 *
28 * long double x, y, expm1l();
29 *
30 * y = expm1l( x );
31 *
32 *
33 *
34 * DESCRIPTION:
35 *
36 * Returns e (2.71828...) raised to the x power, minus one.
37 *
38 * Range reduction is accomplished by separating the argument
39 * into an integer k and fraction f such that
40 *
41 *     x    k  f
42 *    e  = 2  e.
43 *
44 * An expansion x + .5 x^2 + x^3 R(x) approximates exp(f) - 1
45 * in the basic range [-0.5 ln 2, 0.5 ln 2].
46 *
47 *
48 * ACCURACY:
49 *
50 *                      Relative error:
51 * arithmetic   domain     # trials      peak         rms
52 *    IEEE    -79,+MAXLOG    100,000     1.7e-34     4.5e-35
53 *
54 */
55
56#include <errno.h>
57#include <math.h>
58
59#include "math_private.h"
60
61/* exp(x) - 1 = x + 0.5 x^2 + x^3 P(x)/Q(x)
62   -.5 ln 2  <  x  <  .5 ln 2
63   Theoretical peak relative error = 8.1e-36  */
64
65static const long double
66  P0 = 2.943520915569954073888921213330863757240E8L,
67  P1 = -5.722847283900608941516165725053359168840E7L,
68  P2 = 8.944630806357575461578107295909719817253E6L,
69  P3 = -7.212432713558031519943281748462837065308E5L,
70  P4 = 4.578962475841642634225390068461943438441E4L,
71  P5 = -1.716772506388927649032068540558788106762E3L,
72  P6 = 4.401308817383362136048032038528753151144E1L,
73  P7 = -4.888737542888633647784737721812546636240E-1L,
74  Q0 = 1.766112549341972444333352727998584753865E9L,
75  Q1 = -7.848989743695296475743081255027098295771E8L,
76  Q2 = 1.615869009634292424463780387327037251069E8L,
77  Q3 = -2.019684072836541751428967854947019415698E7L,
78  Q4 = 1.682912729190313538934190635536631941751E6L,
79  Q5 = -9.615511549171441430850103489315371768998E4L,
80  Q6 = 3.697714952261803935521187272204485251835E3L,
81  Q7 = -8.802340681794263968892934703309274564037E1L,
82  /* Q8 = 1.000000000000000000000000000000000000000E0 */
83/* C1 + C2 = ln 2 */
84
85  C1 = 6.93145751953125E-1L,
86  C2 = 1.428606820309417232121458176568075500134E-6L,
87/* ln (2^16384 * (1 - 2^-113)) */
88  maxlog = 1.1356523406294143949491931077970764891253E4L,
89/* ln 2^-114 */
90  minarg = -7.9018778583833765273564461846232128760607E1L, big = 1e4932L;
91
92
93long double
94expm1l(long double x)
95{
96  long double px, qx, xx;
97  int32_t ix, sign;
98  ieee_quad_shape_type u;
99  int k;
100
101  /* Detect infinity and NaN.  */
102  u.value = x;
103  ix = u.parts32.mswhi;
104  sign = ix & 0x80000000;
105  ix &= 0x7fffffff;
106  if (ix >= 0x7fff0000)
107    {
108      /* Infinity. */
109      if (((ix & 0xffff) | u.parts32.mswlo | u.parts32.lswhi |
110	u.parts32.lswlo) == 0)
111	{
112	  if (sign)
113	    return -1.0L;
114	  else
115	    return x;
116	}
117      /* NaN. No invalid exception. */
118      return x;
119    }
120
121  /* expm1(+- 0) = +- 0.  */
122  if ((ix == 0) && (u.parts32.mswlo | u.parts32.lswhi | u.parts32.lswlo) == 0)
123    return x;
124
125  /* Overflow.  */
126  if (x > maxlog)
127      return (big * big);
128
129  /* Minimum value.  */
130  if (x < minarg)
131    return (4.0/big - 1.0L);
132
133  /* Express x = ln 2 (k + remainder), remainder not exceeding 1/2. */
134  xx = C1 + C2;			/* ln 2. */
135  px = floorl (0.5 + x / xx);
136  k = px;
137  /* remainder times ln 2 */
138  x -= px * C1;
139  x -= px * C2;
140
141  /* Approximate exp(remainder ln 2).  */
142  px = (((((((P7 * x
143	      + P6) * x
144	     + P5) * x + P4) * x + P3) * x + P2) * x + P1) * x + P0) * x;
145
146  qx = (((((((x
147	      + Q7) * x
148	     + Q6) * x + Q5) * x + Q4) * x + Q3) * x + Q2) * x + Q1) * x + Q0;
149
150  xx = x * x;
151  qx = x + (0.5 * xx + xx * px / qx);
152
153  /* exp(x) = exp(k ln 2) exp(remainder ln 2) = 2^k exp(remainder ln 2).
154
155  We have qx = exp(remainder ln 2) - 1, so
156  exp(x) - 1 = 2^k (qx + 1) - 1
157	     = 2^k qx + 2^k - 1.  */
158
159  px = ldexpl (1.0L, k);
160  x = px * qx + (px - 1.0);
161  return x;
162}
163DEF_STD(expm1l);
164