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4239f144 | 1 | /* expm1q.c |
1ec601bf FXC |
2 | * |
3 | * Exponential function, minus 1 | |
4239f144 | 4 | * 128-bit long double precision |
1ec601bf FXC |
5 | * |
6 | * | |
7 | * | |
8 | * SYNOPSIS: | |
9 | * | |
4239f144 | 10 | * long double x, y, expm1q(); |
1ec601bf | 11 | * |
4239f144 | 12 | * y = expm1q( x ); |
1ec601bf FXC |
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 | ||
1eba0867 | 38 | /* Copyright 2001 by Stephen L. Moshier |
1ec601bf FXC |
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 | |
4239f144 JM |
51 | License along with this library; if not, see |
52 | <http://www.gnu.org/licenses/>. */ | |
1ec601bf | 53 | |
1ec601bf FXC |
54 | #include "quadmath-imp.h" |
55 | ||
56 | /* exp(x) - 1 = x + 0.5 x^2 + x^3 P(x)/Q(x) | |
57 | -.5 ln 2 < x < .5 ln 2 | |
58 | Theoretical peak relative error = 8.1e-36 */ | |
59 | ||
60 | static const __float128 | |
61 | P0 = 2.943520915569954073888921213330863757240E8Q, | |
62 | P1 = -5.722847283900608941516165725053359168840E7Q, | |
63 | P2 = 8.944630806357575461578107295909719817253E6Q, | |
64 | P3 = -7.212432713558031519943281748462837065308E5Q, | |
65 | P4 = 4.578962475841642634225390068461943438441E4Q, | |
66 | P5 = -1.716772506388927649032068540558788106762E3Q, | |
67 | P6 = 4.401308817383362136048032038528753151144E1Q, | |
68 | P7 = -4.888737542888633647784737721812546636240E-1Q, | |
69 | Q0 = 1.766112549341972444333352727998584753865E9Q, | |
70 | Q1 = -7.848989743695296475743081255027098295771E8Q, | |
71 | Q2 = 1.615869009634292424463780387327037251069E8Q, | |
72 | Q3 = -2.019684072836541751428967854947019415698E7Q, | |
73 | Q4 = 1.682912729190313538934190635536631941751E6Q, | |
74 | Q5 = -9.615511549171441430850103489315371768998E4Q, | |
75 | Q6 = 3.697714952261803935521187272204485251835E3Q, | |
76 | Q7 = -8.802340681794263968892934703309274564037E1Q, | |
77 | /* Q8 = 1.000000000000000000000000000000000000000E0 */ | |
78 | /* C1 + C2 = ln 2 */ | |
79 | ||
80 | C1 = 6.93145751953125E-1Q, | |
81 | C2 = 1.428606820309417232121458176568075500134E-6Q, | |
1ec601bf | 82 | /* ln 2^-114 */ |
4239f144 | 83 | minarg = -7.9018778583833765273564461846232128760607E1Q, big = 1e4932Q; |
1ec601bf FXC |
84 | |
85 | ||
86 | __float128 | |
87 | expm1q (__float128 x) | |
88 | { | |
89 | __float128 px, qx, xx; | |
90 | int32_t ix, sign; | |
91 | ieee854_float128 u; | |
92 | int k; | |
93 | ||
94 | /* Detect infinity and NaN. */ | |
95 | u.value = x; | |
96 | ix = u.words32.w0; | |
97 | sign = ix & 0x80000000; | |
98 | ix &= 0x7fffffff; | |
737df6e6 TB |
99 | if (!sign && ix >= 0x40060000) |
100 | { | |
101 | /* If num is positive and exp >= 6 use plain exp. */ | |
102 | return expq (x); | |
103 | } | |
1ec601bf FXC |
104 | if (ix >= 0x7fff0000) |
105 | { | |
1eba0867 | 106 | /* Infinity (which must be negative infinity). */ |
1ec601bf | 107 | if (((ix & 0xffff) | u.words32.w1 | u.words32.w2 | u.words32.w3) == 0) |
4239f144 | 108 | return -1; |
1eba0867 JJ |
109 | /* NaN. Invalid exception if signaling. */ |
110 | return x + x; | |
1ec601bf FXC |
111 | } |
112 | ||
113 | /* expm1(+- 0) = +- 0. */ | |
114 | if ((ix == 0) && (u.words32.w1 | u.words32.w2 | u.words32.w3) == 0) | |
115 | return x; | |
116 | ||
1ec601bf FXC |
117 | /* Minimum value. */ |
118 | if (x < minarg) | |
4239f144 | 119 | return (4.0/big - 1); |
1ec601bf | 120 | |
1eba0867 JJ |
121 | /* Avoid internal underflow when result does not underflow, while |
122 | ensuring underflow (without returning a zero of the wrong sign) | |
123 | when the result does underflow. */ | |
124 | if (fabsq (x) < 0x1p-113Q) | |
125 | { | |
126 | math_check_force_underflow (x); | |
127 | return x; | |
128 | } | |
129 | ||
1ec601bf FXC |
130 | /* Express x = ln 2 (k + remainder), remainder not exceeding 1/2. */ |
131 | xx = C1 + C2; /* ln 2. */ | |
132 | px = floorq (0.5 + x / xx); | |
133 | k = px; | |
134 | /* remainder times ln 2 */ | |
135 | x -= px * C1; | |
136 | x -= px * C2; | |
137 | ||
138 | /* Approximate exp(remainder ln 2). */ | |
139 | px = (((((((P7 * x | |
140 | + P6) * x | |
141 | + P5) * x + P4) * x + P3) * x + P2) * x + P1) * x + P0) * x; | |
142 | ||
143 | qx = (((((((x | |
144 | + Q7) * x | |
145 | + Q6) * x + Q5) * x + Q4) * x + Q3) * x + Q2) * x + Q1) * x + Q0; | |
146 | ||
147 | xx = x * x; | |
148 | qx = x + (0.5 * xx + xx * px / qx); | |
149 | ||
150 | /* exp(x) = exp(k ln 2) exp(remainder ln 2) = 2^k exp(remainder ln 2). | |
151 | ||
152 | We have qx = exp(remainder ln 2) - 1, so | |
153 | exp(x) - 1 = 2^k (qx + 1) - 1 | |
154 | = 2^k qx + 2^k - 1. */ | |
155 | ||
4239f144 | 156 | px = ldexpq (1, k); |
1ec601bf FXC |
157 | x = px * qx + (px - 1.0); |
158 | return x; | |
159 | } |