1 /* $OpenBSD: s_expm1l.c,v 1.1 2011/07/06 00:02:42 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 * 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
57
58 /* exp(x) - 1 = x + 0.5 x^2 + x^3 P(x)/Q(x)
59 -.5 ln 2 < x < .5 ln 2
60 Theoretical peak relative error = 8.1e-36 */
61
62 static const long double
63 P0 = 2.943520915569954073888921213330863757240E8L,
64 P1 = -5.722847283900608941516165725053359168840E7L,
65 P2 = 8.944630806357575461578107295909719817253E6L,
66 P3 = -7.212432713558031519943281748462837065308E5L,
67 P4 = 4.578962475841642634225390068461943438441E4L,
68 P5 = -1.716772506388927649032068540558788106762E3L,
69 P6 = 4.401308817383362136048032038528753151144E1L,
70 P7 = -4.888737542888633647784737721812546636240E-1L,
71 Q0 = 1.766112549341972444333352727998584753865E9L,
72 Q1 = -7.848989743695296475743081255027098295771E8L,
73 Q2 = 1.615869009634292424463780387327037251069E8L,
74 Q3 = -2.019684072836541751428967854947019415698E7L,
75 Q4 = 1.682912729190313538934190635536631941751E6L,
76 Q5 = -9.615511549171441430850103489315371768998E4L,
77 Q6 = 3.697714952261803935521187272204485251835E3L,
78 Q7 = -8.802340681794263968892934703309274564037E1L,
79 /* Q8 = 1.000000000000000000000000000000000000000E0 */
80 /* C1 + C2 = ln 2 */
81
82 C1 = 6.93145751953125E-1L,
83 C2 = 1.428606820309417232121458176568075500134E-6L,
84 /* ln (2^16384 * (1 - 2^-113)) */
85 maxlog = 1.1356523406294143949491931077970764891253E4L,
86 /* ln 2^-114 */
87 minarg = -7.9018778583833765273564461846232128760607E1L, big = 1e4932L;
88
89
90 long double
expm1l(long double x)91 expm1l(long double x)
92 {
93 long double px, qx, xx;
94 int32_t ix, sign;
95 ieee_quad_shape_type u;
96 int k;
97
98 /* Detect infinity and NaN. */
99 u.value = x;
100 ix = u.parts32.mswhi;
101 sign = ix & 0x80000000;
102 ix &= 0x7fffffff;
103 if (ix >= 0x7fff0000)
104 {
105 /* Infinity. */
106 if (((ix & 0xffff) | u.parts32.mswlo | u.parts32.lswhi |
107 u.parts32.lswlo) == 0)
108 {
109 if (sign)
110 return -1.0L;
111 else
112 return x;
113 }
114 /* NaN. No invalid exception. */
115 return x + x;
116 }
117
118 /* expm1(+- 0) = +- 0. */
119 if ((ix == 0) && (u.parts32.mswlo | u.parts32.lswhi | u.parts32.lswlo) == 0)
120 return x;
121
122 /* Overflow. */
123 if (x > maxlog)
124 return __math_oflowl(0);
125
126 /* Minimum value. */
127 if (x < minarg)
128 return __math_inexactl(-1.0L);
129
130 /* Express x = ln 2 (k + remainder), remainder not exceeding 1/2. */
131 xx = C1 + C2; /* ln 2. */
132 px = floorl (0.5L + x / xx);
133 k = px;
134 /* remainder times ln 2 */
135 x -= px * C1;
136 x -= px * C2;
137
138 /* Approximate exp(remainder ln 2). */
139 px = (((((((P7 * x
140 + P6) * x
141 + P5) * x + P4) * x + P3) * x + P2) * x + P1) * x + P0) * x;
142
143 qx = (((((((x
144 + Q7) * x
145 + Q6) * x + Q5) * x + Q4) * x + Q3) * x + Q2) * x + Q1) * x + Q0;
146
147 xx = x * x;
148 qx = x + (0.5L * xx + xx * px / qx);
149
150 /* exp(x) = exp(k ln 2) exp(remainder ln 2) = 2^k exp(remainder ln 2).
151
152 We have qx = exp(remainder ln 2) - 1, so
153 exp(x) - 1 = 2^k (qx + 1) - 1
154 = 2^k qx + 2^k - 1. */
155
156 if (k >= __LDBL_MAX_EXP__) {
157 /* Avoid overflow in ldexpl. */
158 x = ldexpl(qx + 1.0L, k) - 1.0L;
159 } else {
160 px = ldexpl (1.0L, k);
161 x = px * qx + (px - 1.0L);
162 }
163 return x;
164 }
165
