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 #pragma weak expm1l = __expm1l
31 #if !defined(__sparc)
32 #error Unsupported architecture
33 #endif
34
35 /*
36 * expm1l(x)
37 *
38 * Table driven method
39 * Written by K.C. Ng, June 1995.
40 * Algorithm :
41 * 1. expm1(x) = x if x<2**-114
42 * 2. if |x| <= 0.0625 = 1/16, use approximation
43 * expm1(x) = x + x*P/(2-P)
44 * where
45 * P = x - z*(P1+z*(P2+z*(P3+z*(P4+z*(P5+z*P6+z*P7))))), z = x*x;
46 * (this formula is derived from
47 * 2-P+x = R = x*(exp(x)+1)/(exp(x)-1) ~ 2 + x*x/6 - x^4/360 + ...)
48 *
49 * P1 = 1.66666666666666666666666666666638500528074603030e-0001
50 * P2 = -2.77777777777777777777777759668391122822266551158e-0003
51 * P3 = 6.61375661375661375657437408890138814721051293054e-0005
52 * P4 = -1.65343915343915303310185228411892601606669528828e-0006
53 * P5 = 4.17535139755122945763580609663414647067443411178e-0008
54 * P6 = -1.05683795988668526689182102605260986731620026832e-0009
55 * P7 = 2.67544168821852702827123344217198187229611470514e-0011
56 *
57 * Accuracy: |R-x*(exp(x)+1)/(exp(x)-1)|<=2**-119.13
58 *
59 * 3. For 1/16 < |x| < 1.125, choose x(+-i) ~ +-(i+4.5)/64, i=0,..,67
60 * since
61 * exp(x) = exp(xi+(x-xi))= exp(xi)*exp((x-xi))
62 * we have
63 * expm1(x) = expm1(xi)+(exp(xi))*(expm1(x-xi))
64 * where
65 * |s=x-xi| <= 1/128
66 * and
67 * expm1(s)=2s/(2-R), R= s-s^2*(T1+s^2*(T2+s^2*(T3+s^2*(T4+s^2*T5))))
68 *
69 * T1 = 1.666666666666666666666666666660876387437e-1L,
70 * T2 = -2.777777777777777777777707812093173478756e-3L,
71 * T3 = 6.613756613756613482074280932874221202424e-5L,
72 * T4 = -1.653439153392139954169609822742235851120e-6L,
73 * T5 = 4.175314851769539751387852116610973796053e-8L;
74 *
75 * 4. For |x| >= 1.125, return exp(x)-1.
76 * (see algorithm for exp)
77 *
78 * Special cases:
79 * expm1l(INF) is INF, expm1l(NaN) is NaN;
80 * expm1l(-INF)= -1;
81 * for finite argument, only expm1l(0)=0 is exact.
82 *
83 * Accuracy:
84 * according to an error analysis, the error is always less than
85 * 2 ulp (unit in the last place).
86 *
87 * Misc. info.
88 * For 113 bit long double
89 * if x > 1.135652340629414394949193107797076342845e+4
90 * then expm1l(x) overflow;
91 *
92 * Constants:
93 * Only decimal values are given. We assume that the compiler will convert
94 * from decimal to binary accurately enough to produce the correct
95 * hexadecimal values.
96 */
97
98 #include "libm.h"
99
100 extern const long double _TBL_expl_hi[], _TBL_expl_lo[];
101 extern const long double _TBL_expm1lx[], _TBL_expm1l[];
102
103 static const long double
104 zero = +0.0L,
105 one = +1.0L,
106 two = +2.0L,
107 ln2_64 = +1.083042469624914545964425189778400898568e-2L,
108 ovflthreshold = +1.135652340629414394949193107797076342845e+4L,
109 invln2_32 = +4.616624130844682903551758979206054839765e+1L,
110 ln2_32hi = +2.166084939249829091928849858592451515688e-2L,
111 ln2_32lo = +5.209643502595475652782654157501186731779e-27L,
112 huge = +1.0e4000L,
113 tiny = +1.0e-4000L,
114 P1 = +1.66666666666666666666666666666638500528074603030e-0001L,
115 P2 = -2.77777777777777777777777759668391122822266551158e-0003L,
116 P3 = +6.61375661375661375657437408890138814721051293054e-0005L,
117 P4 = -1.65343915343915303310185228411892601606669528828e-0006L,
118 P5 = +4.17535139755122945763580609663414647067443411178e-0008L,
119 P6 = -1.05683795988668526689182102605260986731620026832e-0009L,
120 P7 = +2.67544168821852702827123344217198187229611470514e-0011L,
121 /* rational approximation coeffs for [-(ln2)/64,(ln2)/64] */
122 T1 = +1.666666666666666666666666666660876387437e-1L,
123 T2 = -2.777777777777777777777707812093173478756e-3L,
124 T3 = +6.613756613756613482074280932874221202424e-5L,
125 T4 = -1.653439153392139954169609822742235851120e-6L,
126 T5 = +4.175314851769539751387852116610973796053e-8L;
127
128 long double
129 expm1l(long double x) {
130 int hx, ix, j, k, m;
131 long double t, r, s, w;
132
133 hx = ((int *) &x)[HIXWORD];
134 ix = hx & ~0x80000000;
135 if (ix >= 0x7fff0000) {
136 if (x != x)
137 return (x + x); /* NaN */
138 if (x < zero)
139 return (-one); /* -inf */
140 return (x); /* +inf */
141 }
142 if (ix < 0x3fff4000) { /* |x| < 1.25 */
143 if (ix < 0x3ffb0000) { /* |x| < 0.0625 */
144 if (ix < 0x3f8d0000) {
145 if ((int) x == 0)
146 return (x); /* |x|<2^-114 */
147 }
148 t = x * x;
149 r = (x - t * (P1 + t * (P2 + t * (P3 + t * (P4 + t *
150 (P5 + t * (P6 + t * P7)))))));
151 return (x + (x * r) / (two - r));
152 }
153 /* compute i = [64*x] */
154 m = 0x4009 - (ix >> 16);
155 j = ((ix & 0x0000ffff) | 0x10000) >> m; /* j=4,...,67 */
156 if (hx < 0)
157 j += 82; /* negative */
158 s = x - _TBL_expm1lx[j];
159 t = s * s;
160 r = s - t * (T1 + t * (T2 + t * (T3 + t * (T4 + t * T5))));
161 r = (s + s) / (two - r);
162 w = _TBL_expm1l[j];
163 return (w + (w + one) * r);
164 }
165 if (hx > 0) {
166 if (x > ovflthreshold)
167 return (huge * huge);
168 k = (int) (invln2_32 * (x + ln2_64));
169 } else {
170 if (x < -80.0)
171 return (tiny - x / x);
172 k = (int) (invln2_32 * (x - ln2_64));
173 }
174 j = k & 0x1f;
175 m = k >> 5;
176 t = (long double) k;
177 x = (x - t * ln2_32hi) - t * ln2_32lo;
178 t = x * x;
179 r = (x - t * (T1 + t * (T2 + t * (T3 + t * (T4 + t * T5))))) - two;
180 x = _TBL_expl_hi[j] - ((_TBL_expl_hi[j] * (x + x)) / r -
181 _TBL_expl_lo[j]);
182 return (scalbnl(x, m) - one);
183 }