1 /*	$OpenBSD: s_expm1l.c,v 1.2 2011/07/20 21:02:51 martynas 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  *      Long double precision
23  *
24  *
25  * SYNOPSIS:
26  *
27  * long double x, y, expm1l();
28  *
29  * y = expm1l( x );
30  *
31  *
32  *
33  * DESCRIPTION:
34  *
35  * Returns e (2.71828...) raised to the x power, minus 1.
36  *
37  * Range reduction is accomplished by separating the argument
38  * into an integer k and fraction f such that
39  *
40  *     x    k  f
41  *    e  = 2  e.
42  *
43  * An expansion x + .5 x^2 + x^3 R(x) approximates exp(f) - 1
44  * in the basic range [-0.5 ln 2, 0.5 ln 2].
45  *
46  *
47  * ACCURACY:
48  *
49  *                      Relative error:
50  * arithmetic   domain     # trials      peak         rms
51  *    IEEE    -45,+MAXLOG   200,000     1.2e-19     2.5e-20
52  *
53  * ERROR MESSAGES:
54  *
55  *   message         condition      value returned
56  * expm1l overflow   x > MAXLOG         MAXNUM
57  *
58  */
59 
60 #include <math.h>
61 
62 static const long double MAXLOGL = 1.1356523406294143949492E4L;
63 
64 /* exp(x) - 1 = x + 0.5 x^2 + x^3 P(x)/Q(x)
65    -.5 ln 2  <  x  <  .5 ln 2
66    Theoretical peak relative error = 3.4e-22  */
67 
68 static const long double
69   P0 = -1.586135578666346600772998894928250240826E4L,
70   P1 =  2.642771505685952966904660652518429479531E3L,
71   P2 = -3.423199068835684263987132888286791620673E2L,
72   P3 =  1.800826371455042224581246202420972737840E1L,
73   P4 = -5.238523121205561042771939008061958820811E-1L,
74 
75   Q0 = -9.516813471998079611319047060563358064497E4L,
76   Q1 =  3.964866271411091674556850458227710004570E4L,
77   Q2 = -7.207678383830091850230366618190187434796E3L,
78   Q3 =  7.206038318724600171970199625081491823079E2L,
79   Q4 = -4.002027679107076077238836622982900945173E1L,
80   /* Q5 = 1.000000000000000000000000000000000000000E0 */
81 
82 /* C1 + C2 = ln 2 */
83 C1 = 6.93145751953125E-1L,
84 C2 = 1.428606820309417232121458176568075500134E-6L,
85 /* ln 2^-65 */
86 minarg = -4.5054566736396445112120088E1L;
87 static const long double huge = 0x1p10000L;
88 
89 long double
90 expm1l(long double x)
91 {
92 long double px, qx, xx;
93 int k;
94 
95 /* Overflow.  */
96 if (x > MAXLOGL)
97   return (huge*huge);	/* overflow */
98 
99 if (x == 0.0)
100   return x;
101 
102 /* Minimum value.  */
103 if (x < minarg)
104   return -1.0L;
105 
106 xx = C1 + C2;
107 
108 /* Express x = ln 2 (k + remainder), remainder not exceeding 1/2. */
109 px = floorl (0.5 + x / xx);
110 k = px;
111 /* remainder times ln 2 */
112 x -= px * C1;
113 x -= px * C2;
114 
115 /* Approximate exp(remainder ln 2).  */
116 px = (((( P4 * x
117 	 + P3) * x
118 	+ P2) * x
119        + P1) * x
120       + P0) * x;
121 
122 qx = (((( x
123 	 + Q4) * x
124 	+ Q3) * x
125        + Q2) * x
126       + Q1) * x
127      + Q0;
128 
129 xx = x * x;
130 qx = x + (0.5 * xx + xx * px / qx);
131 
132 /* exp(x) = exp(k ln 2) exp(remainder ln 2) = 2^k exp(remainder ln 2).
133    We have qx = exp(remainder ln 2) - 1, so
134    exp(x) - 1  =  2^k (qx + 1) - 1  =  2^k qx + 2^k - 1.  */
135 px = ldexpl(1.0L, k);
136 x = px * qx + (px - 1.0);
137 return x;
138 }
139