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ef25b29e AJ |
1 | /* expm1l.c |
2 | * | |
3 | * Exponential function, minus 1 | |
4 | * 128-bit long double precision | |
5 | * | |
6 | * | |
7 | * | |
8 | * SYNOPSIS: | |
9 | * | |
10 | * long double x, y, expm1l(); | |
11 | * | |
12 | * y = expm1l( x ); | |
13 | * | |
14 | * | |
15 | * | |
16 | * DESCRIPTION: | |
17 | * | |
18 | * Returns e (2.71828...) raised to the x power, minus one. | |
19 | * | |
20 | * Range reduction is accomplished by separating the argument | |
21 | * into an integer k and fraction f such that | |
22 | * | |
23 | * x k f | |
24 | * e = 2 e. | |
25 | * | |
26 | * An expansion x + .5 x^2 + x^3 R(x) approximates exp(f) - 1 | |
27 | * in the basic range [-0.5 ln 2, 0.5 ln 2]. | |
28 | * | |
29 | * | |
30 | * ACCURACY: | |
31 | * | |
32 | * Relative error: | |
33 | * arithmetic domain # trials peak rms | |
34 | * IEEE -79,+MAXLOG 100,000 1.7e-34 4.5e-35 | |
35 | * | |
36 | */ | |
37 | ||
9c84384c | 38 | /* Copyright 2001 by Stephen L. Moshier |
cc7375ce RM |
39 | |
40 | This library is free software; you can redistribute it and/or | |
41 | modify it under the terms of the GNU Lesser General Public | |
42 | License as published by the Free Software Foundation; either | |
43 | version 2.1 of the License, or (at your option) any later version. | |
44 | ||
45 | This library is distributed in the hope that it will be useful, | |
46 | but WITHOUT ANY WARRANTY; without even the implied warranty of | |
47 | MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the GNU | |
48 | Lesser General Public License for more details. | |
49 | ||
50 | You should have received a copy of the GNU Lesser General Public | |
59ba27a6 PE |
51 | License along with this library; if not, see |
52 | <http://www.gnu.org/licenses/>. */ | |
cc7375ce | 53 | |
ef25b29e AJ |
54 | |
55 | ||
7f3394bd | 56 | #include <errno.h> |
ce9c5b3e | 57 | #include <float.h> |
1ed0291c RH |
58 | #include <math.h> |
59 | #include <math_private.h> | |
8f5b00d3 | 60 | #include <math-underflow.h> |
fd3b4e7c | 61 | #include <libm-alias-ldouble.h> |
ef25b29e AJ |
62 | |
63 | /* exp(x) - 1 = x + 0.5 x^2 + x^3 P(x)/Q(x) | |
64 | -.5 ln 2 < x < .5 ln 2 | |
65 | Theoretical peak relative error = 8.1e-36 */ | |
66 | ||
15089e04 | 67 | static const _Float128 |
02bbfb41 PM |
68 | P0 = L(2.943520915569954073888921213330863757240E8), |
69 | P1 = L(-5.722847283900608941516165725053359168840E7), | |
70 | P2 = L(8.944630806357575461578107295909719817253E6), | |
71 | P3 = L(-7.212432713558031519943281748462837065308E5), | |
72 | P4 = L(4.578962475841642634225390068461943438441E4), | |
73 | P5 = L(-1.716772506388927649032068540558788106762E3), | |
74 | P6 = L(4.401308817383362136048032038528753151144E1), | |
75 | P7 = L(-4.888737542888633647784737721812546636240E-1), | |
76 | Q0 = L(1.766112549341972444333352727998584753865E9), | |
77 | Q1 = L(-7.848989743695296475743081255027098295771E8), | |
78 | Q2 = L(1.615869009634292424463780387327037251069E8), | |
79 | Q3 = L(-2.019684072836541751428967854947019415698E7), | |
80 | Q4 = L(1.682912729190313538934190635536631941751E6), | |
81 | Q5 = L(-9.615511549171441430850103489315371768998E4), | |
82 | Q6 = L(3.697714952261803935521187272204485251835E3), | |
83 | Q7 = L(-8.802340681794263968892934703309274564037E1), | |
ef25b29e AJ |
84 | /* Q8 = 1.000000000000000000000000000000000000000E0 */ |
85 | /* C1 + C2 = ln 2 */ | |
86 | ||
02bbfb41 PM |
87 | C1 = L(6.93145751953125E-1), |
88 | C2 = L(1.428606820309417232121458176568075500134E-6), | |
ef25b29e | 89 | /* ln 2^-114 */ |
02bbfb41 | 90 | minarg = L(-7.9018778583833765273564461846232128760607E1), big = L(1e4932); |
ef25b29e AJ |
91 | |
92 | ||
15089e04 PM |
93 | _Float128 |
94 | __expm1l (_Float128 x) | |
ef25b29e | 95 | { |
15089e04 | 96 | _Float128 px, qx, xx; |
ef25b29e AJ |
97 | int32_t ix, sign; |
98 | ieee854_long_double_shape_type u; | |
99 | int k; | |
100 | ||
ef25b29e AJ |
101 | /* Detect infinity and NaN. */ |
102 | u.value = x; | |
103 | ix = u.parts32.w0; | |
104 | sign = ix & 0x80000000; | |
105 | ix &= 0x7fffffff; | |
2a8ab7f2 DM |
106 | if (!sign && ix >= 0x40060000) |
107 | { | |
108 | /* If num is positive and exp >= 6 use plain exp. */ | |
109 | return __expl (x); | |
110 | } | |
ef25b29e AJ |
111 | if (ix >= 0x7fff0000) |
112 | { | |
8124ac3e | 113 | /* Infinity (which must be negative infinity). */ |
ef25b29e | 114 | if (((ix & 0xffff) | u.parts32.w1 | u.parts32.w2 | u.parts32.w3) == 0) |
02bbfb41 | 115 | return -1; |
59e53a78 JM |
116 | /* NaN. Invalid exception if signaling. */ |
117 | return x + x; | |
ef25b29e AJ |
118 | } |
119 | ||
514abd20 AJ |
120 | /* expm1(+- 0) = +- 0. */ |
121 | if ((ix == 0) && (u.parts32.w1 | u.parts32.w2 | u.parts32.w3) == 0) | |
122 | return x; | |
1f5649f8 | 123 | |
514abd20 AJ |
124 | /* Minimum value. */ |
125 | if (x < minarg) | |
02bbfb41 | 126 | return (4.0/big - 1); |
514abd20 | 127 | |
a04bb330 JM |
128 | /* Avoid internal underflow when result does not underflow, while |
129 | ensuring underflow (without returning a zero of the wrong sign) | |
130 | when the result does underflow. */ | |
02bbfb41 | 131 | if (fabsl (x) < L(0x1p-113)) |
a04bb330 | 132 | { |
d96164c3 | 133 | math_check_force_underflow (x); |
a04bb330 JM |
134 | return x; |
135 | } | |
ce9c5b3e | 136 | |
ef25b29e AJ |
137 | /* Express x = ln 2 (k + remainder), remainder not exceeding 1/2. */ |
138 | xx = C1 + C2; /* ln 2. */ | |
139 | px = __floorl (0.5 + x / xx); | |
140 | k = px; | |
141 | /* remainder times ln 2 */ | |
142 | x -= px * C1; | |
143 | x -= px * C2; | |
144 | ||
145 | /* Approximate exp(remainder ln 2). */ | |
146 | px = (((((((P7 * x | |
147 | + P6) * x | |
148 | + P5) * x + P4) * x + P3) * x + P2) * x + P1) * x + P0) * x; | |
149 | ||
150 | qx = (((((((x | |
151 | + Q7) * x | |
152 | + Q6) * x + Q5) * x + Q4) * x + Q3) * x + Q2) * x + Q1) * x + Q0; | |
153 | ||
154 | xx = x * x; | |
155 | qx = x + (0.5 * xx + xx * px / qx); | |
156 | ||
157 | /* exp(x) = exp(k ln 2) exp(remainder ln 2) = 2^k exp(remainder ln 2). | |
158 | ||
159 | We have qx = exp(remainder ln 2) - 1, so | |
160 | exp(x) - 1 = 2^k (qx + 1) - 1 | |
161 | = 2^k qx + 2^k - 1. */ | |
162 | ||
02bbfb41 | 163 | px = __ldexpl (1, k); |
ef25b29e AJ |
164 | x = px * qx + (px - 1.0); |
165 | return x; | |
166 | } | |
76f2646f | 167 | libm_hidden_def (__expm1l) |
fd3b4e7c | 168 | libm_alias_ldouble (__expm1, expm1) |