1 /*
2 * CDDL HEADER START
3 *
4 * The contents of this file are subject to the terms of the
5 * Common Development and Distribution License (the "License").
6 * You may not use this file except in compliance with the License.
7 *
8 * You can obtain a copy of the license at usr/src/OPENSOLARIS.LICENSE
9 * or http://www.opensolaris.org/os/licensing.
10 * See the License for the specific language governing permissions
11 * and limitations under the License.
12 *
13 * When distributing Covered Code, include this CDDL HEADER in each
14 * file and include the License file at usr/src/OPENSOLARIS.LICENSE.
15 * If applicable, add the following below this CDDL HEADER, with the
16 * fields enclosed by brackets "[]" replaced with your own identifying
17 * information: Portions Copyright [yyyy] [name of copyright owner]
18 *
19 * CDDL HEADER END
20 */
21
22 /*
23 * Copyright 2011 Nexenta Systems, Inc. All rights reserved.
24 */
25 /*
26 * Copyright 2006 Sun Microsystems, Inc. All rights reserved.
27 * Use is subject to license terms.
28 */
29
30 #if defined(ELFOBJ)
31 #pragma weak expm1l = __expm1l
32 #endif
33 #if !defined(__sparc)
34 #error Unsupported architecture
35 #endif
36
37 /*
38 * expm1l(x)
39 *
40 * Table driven method
41 * Written by K.C. Ng, June 1995.
42 * Algorithm :
43 * 1. expm1(x) = x if x<2**-114
44 * 2. if |x| <= 0.0625 = 1/16, use approximation
45 * expm1(x) = x + x*P/(2-P)
46 * where
47 * P = x - z*(P1+z*(P2+z*(P3+z*(P4+z*(P5+z*P6+z*P7))))), z = x*x;
48 * (this formula is derived from
49 * 2-P+x = R = x*(exp(x)+1)/(exp(x)-1) ~ 2 + x*x/6 - x^4/360 + ...)
50 *
51 * P1 = 1.66666666666666666666666666666638500528074603030e-0001
52 * P2 = -2.77777777777777777777777759668391122822266551158e-0003
53 * P3 = 6.61375661375661375657437408890138814721051293054e-0005
54 * P4 = -1.65343915343915303310185228411892601606669528828e-0006
55 * P5 = 4.17535139755122945763580609663414647067443411178e-0008
56 * P6 = -1.05683795988668526689182102605260986731620026832e-0009
57 * P7 = 2.67544168821852702827123344217198187229611470514e-0011
58 *
59 * Accuracy: |R-x*(exp(x)+1)/(exp(x)-1)|<=2**-119.13
60 *
61 * 3. For 1/16 < |x| < 1.125, choose x(+-i) ~ +-(i+4.5)/64, i=0,..,67
62 * since
63 * exp(x) = exp(xi+(x-xi))= exp(xi)*exp((x-xi))
64 * we have
65 * expm1(x) = expm1(xi)+(exp(xi))*(expm1(x-xi))
66 * where
67 * |s=x-xi| <= 1/128
68 * and
69 * expm1(s)=2s/(2-R), R= s-s^2*(T1+s^2*(T2+s^2*(T3+s^2*(T4+s^2*T5))))
70 *
71 * T1 = 1.666666666666666666666666666660876387437e-1L,
72 * T2 = -2.777777777777777777777707812093173478756e-3L,
73 * T3 = 6.613756613756613482074280932874221202424e-5L,
74 * T4 = -1.653439153392139954169609822742235851120e-6L,
75 * T5 = 4.175314851769539751387852116610973796053e-8L;
76 *
77 * 4. For |x| >= 1.125, return exp(x)-1.
78 * (see algorithm for exp)
79 *
80 * Special cases:
81 * expm1l(INF) is INF, expm1l(NaN) is NaN;
82 * expm1l(-INF)= -1;
83 * for finite argument, only expm1l(0)=0 is exact.
84 *
85 * Accuracy:
86 * according to an error analysis, the error is always less than
87 * 2 ulp (unit in the last place).
88 *
89 * Misc. info.
90 * For 113 bit long double
91 * if x > 1.135652340629414394949193107797076342845e+4
92 * then expm1l(x) overflow;
93 *
94 * Constants:
95 * Only decimal values are given. We assume that the compiler will convert
96 * from decimal to binary accurately enough to produce the correct
97 * hexadecimal values.
98 */
99
100 #include "libm.h"
101
102 extern const long double _TBL_expl_hi[], _TBL_expl_lo[];
103 extern const long double _TBL_expm1lx[], _TBL_expm1l[];
104
105 static const long double
106 zero = +0.0L,
107 one = +1.0L,
108 two = +2.0L,
109 ln2_64 = +1.083042469624914545964425189778400898568e-2L,
110 ovflthreshold = +1.135652340629414394949193107797076342845e+4L,
111 invln2_32 = +4.616624130844682903551758979206054839765e+1L,
112 ln2_32hi = +2.166084939249829091928849858592451515688e-2L,
113 ln2_32lo = +5.209643502595475652782654157501186731779e-27L,
114 huge = +1.0e4000L,
115 tiny = +1.0e-4000L,
116 P1 = +1.66666666666666666666666666666638500528074603030e-0001L,
117 P2 = -2.77777777777777777777777759668391122822266551158e-0003L,
118 P3 = +6.61375661375661375657437408890138814721051293054e-0005L,
119 P4 = -1.65343915343915303310185228411892601606669528828e-0006L,
120 P5 = +4.17535139755122945763580609663414647067443411178e-0008L,
121 P6 = -1.05683795988668526689182102605260986731620026832e-0009L,
122 P7 = +2.67544168821852702827123344217198187229611470514e-0011L,
123 /* rational approximation coeffs for [-(ln2)/64,(ln2)/64] */
124 T1 = +1.666666666666666666666666666660876387437e-1L,
125 T2 = -2.777777777777777777777707812093173478756e-3L,
126 T3 = +6.613756613756613482074280932874221202424e-5L,
127 T4 = -1.653439153392139954169609822742235851120e-6L,
128 T5 = +4.175314851769539751387852116610973796053e-8L;
129
130 long double
131 expm1l(long double x) {
132 int hx, ix, j, k, m;
133 long double t, r, s, w;
134
135 hx = ((int *) &x)[HIXWORD];
136 ix = hx & ~0x80000000;
137 if (ix >= 0x7fff0000) {
138 if (x != x)
139 return (x + x); /* NaN */
140 if (x < zero)
141 return (-one); /* -inf */
142 return (x); /* +inf */
143 }
144 if (ix < 0x3fff4000) { /* |x| < 1.25 */
145 if (ix < 0x3ffb0000) { /* |x| < 0.0625 */
146 if (ix < 0x3f8d0000) {
147 if ((int) x == 0)
148 return (x); /* |x|<2^-114 */
149 }
150 t = x * x;
151 r = (x - t * (P1 + t * (P2 + t * (P3 + t * (P4 + t *
152 (P5 + t * (P6 + t * P7)))))));
153 return (x + (x * r) / (two - r));
154 }
155 /* compute i = [64*x] */
156 m = 0x4009 - (ix >> 16);
157 j = ((ix & 0x0000ffff) | 0x10000) >> m; /* j=4,...,67 */
158 if (hx < 0)
159 j += 82; /* negative */
160 s = x - _TBL_expm1lx[j];
161 t = s * s;
162 r = s - t * (T1 + t * (T2 + t * (T3 + t * (T4 + t * T5))));
163 r = (s + s) / (two - r);
164 w = _TBL_expm1l[j];
165 return (w + (w + one) * r);
166 }
167 if (hx > 0) {
168 if (x > ovflthreshold)
169 return (huge * huge);
170 k = (int) (invln2_32 * (x + ln2_64));
171 } else {
172 if (x < -80.0)
173 return (tiny - x / x);
174 k = (int) (invln2_32 * (x - ln2_64));
175 }
176 j = k & 0x1f;
177 m = k >> 5;
178 t = (long double) k;
179 x = (x - t * ln2_32hi) - t * ln2_32lo;
180 t = x * x;
181 r = (x - t * (T1 + t * (T2 + t * (T3 + t * (T4 + t * T5))))) - two;
182 x = _TBL_expl_hi[j] - ((_TBL_expl_hi[j] * (x + x)) / r -
183 _TBL_expl_lo[j]);
184 return (scalbnl(x, m) - one);
185 }