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
expm1l(long double x)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
expm1l(long double x)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
expm1l(long double x)119 long double expm1l(long double x)
120 {
121 return expm1(x);
122 }
123 #endif
124