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
129