1 /* Quad-precision floating point e^x.
2 Copyright (C) 1999-2018 Free Software Foundation, Inc.
3 This file is part of the GNU C Library.
4 Contributed by Jakub Jelinek <jj@ultra.linux.cz>
5 Partly based on double-precision code
6 by Geoffrey Keating <geoffk@ozemail.com.au>
7
8 The GNU C Library is free software; you can redistribute it and/or
9 modify it under the terms of the GNU Lesser General Public
10 License as published by the Free Software Foundation; either
11 version 2.1 of the License, or (at your option) any later version.
12
13 The GNU C Library is distributed in the hope that it will be useful,
14 but WITHOUT ANY WARRANTY; without even the implied warranty of
15 MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the GNU
16 Lesser General Public License for more details.
17
18 You should have received a copy of the GNU Lesser General Public
19 License along with the GNU C Library; if not, see
20 <http://www.gnu.org/licenses/>. */
21
22 /* The basic design here is from
23 Abraham Ziv, "Fast Evaluation of Elementary Mathematical Functions with
24 Correctly Rounded Last Bit", ACM Trans. Math. Soft., 17 (3), September 1991,
25 pp. 410-423.
26
27 We work with number pairs where the first number is the high part and
28 the second one is the low part. Arithmetic with the high part numbers must
29 be exact, without any roundoff errors.
30
31 The input value, X, is written as
32 X = n * ln(2)_0 + arg1[t1]_0 + arg2[t2]_0 + x
33 - n * ln(2)_1 + arg1[t1]_1 + arg2[t2]_1 + xl
34
35 where:
36 - n is an integer, 16384 >= n >= -16495;
37 - ln(2)_0 is the first 93 bits of ln(2), and |ln(2)_0-ln(2)-ln(2)_1| < 2^-205
38 - t1 is an integer, 89 >= t1 >= -89
39 - t2 is an integer, 65 >= t2 >= -65
40 - |arg1[t1]-t1/256.0| < 2^-53
41 - |arg2[t2]-t2/32768.0| < 2^-53
42 - x + xl is whatever is left, |x + xl| < 2^-16 + 2^-53
43
44 Then e^x is approximated as
45
46 e^x = 2^n_1 ( 2^n_0 e^(arg1[t1]_0 + arg1[t1]_1) e^(arg2[t2]_0 + arg2[t2]_1)
47 + 2^n_0 e^(arg1[t1]_0 + arg1[t1]_1) e^(arg2[t2]_0 + arg2[t2]_1)
48 * p (x + xl + n * ln(2)_1))
49 where:
50 - p(x) is a polynomial approximating e(x)-1
51 - e^(arg1[t1]_0 + arg1[t1]_1) is obtained from a table
52 - e^(arg2[t2]_0 + arg2[t2]_1) likewise
53 - n_1 + n_0 = n, so that |n_0| < -FLT128_MIN_EXP-1.
54
55 If it happens that n_1 == 0 (this is the usual case), that multiplication
56 is omitted.
57 */
58
59 #ifndef _GNU_SOURCE
60 #define _GNU_SOURCE
61 #endif
62
63 #include "quadmath-imp.h"
64 #include "expq_table.h"
65
66 static const __float128 C[] = {
67 /* Smallest integer x for which e^x overflows. */
68 #define himark C[0]
69 11356.523406294143949491931077970765Q,
70
71 /* Largest integer x for which e^x underflows. */
72 #define lomark C[1]
73 -11433.4627433362978788372438434526231Q,
74
75 /* 3x2^96 */
76 #define THREEp96 C[2]
77 59421121885698253195157962752.0Q,
78
79 /* 3x2^103 */
80 #define THREEp103 C[3]
81 30423614405477505635920876929024.0Q,
82
83 /* 3x2^111 */
84 #define THREEp111 C[4]
85 7788445287802241442795744493830144.0Q,
86
87 /* 1/ln(2) */
88 #define M_1_LN2 C[5]
89 1.44269504088896340735992468100189204Q,
90
91 /* first 93 bits of ln(2) */
92 #define M_LN2_0 C[6]
93 0.693147180559945309417232121457981864Q,
94
95 /* ln2_0 - ln(2) */
96 #define M_LN2_1 C[7]
97 -1.94704509238074995158795957333327386E-31Q,
98
99 /* very small number */
100 #define TINY C[8]
101 1.0e-4900Q,
102
103 /* 2^16383 */
104 #define TWO16383 C[9]
105 5.94865747678615882542879663314003565E+4931Q,
106
107 /* 256 */
108 #define TWO8 C[10]
109 256,
110
111 /* 32768 */
112 #define TWO15 C[11]
113 32768,
114
115 /* Chebyshev polynom coefficients for (exp(x)-1)/x */
116 #define P1 C[12]
117 #define P2 C[13]
118 #define P3 C[14]
119 #define P4 C[15]
120 #define P5 C[16]
121 #define P6 C[17]
122 0.5Q,
123 1.66666666666666666666666666666666683E-01Q,
124 4.16666666666666666666654902320001674E-02Q,
125 8.33333333333333333333314659767198461E-03Q,
126 1.38888888889899438565058018857254025E-03Q,
127 1.98412698413981650382436541785404286E-04Q,
128 };
129
130 __float128
131 expq (__float128 x)
132 {
133 /* Check for usual case. */
134 if (__builtin_isless (x, himark) && __builtin_isgreater (x, lomark))
135 {
136 int tval1, tval2, unsafe, n_i;
137 __float128 x22, n, t, result, xl;
138 ieee854_float128 ex2_u, scale_u;
139 fenv_t oldenv;
140
141 feholdexcept (&oldenv);
142 #ifdef FE_TONEAREST
143 fesetround (FE_TONEAREST);
144 #endif
145
146 /* Calculate n. */
147 n = x * M_1_LN2 + THREEp111;
148 n -= THREEp111;
149 x = x - n * M_LN2_0;
150 xl = n * M_LN2_1;
151
152 /* Calculate t/256. */
153 t = x + THREEp103;
154 t -= THREEp103;
155
156 /* Compute tval1 = t. */
157 tval1 = (int) (t * TWO8);
158
159 x -= __expq_table[T_EXPL_ARG1+2*tval1];
160 xl -= __expq_table[T_EXPL_ARG1+2*tval1+1];
161
162 /* Calculate t/32768. */
163 t = x + THREEp96;
164 t -= THREEp96;
165
166 /* Compute tval2 = t. */
167 tval2 = (int) (t * TWO15);
168
169 x -= __expq_table[T_EXPL_ARG2+2*tval2];
170 xl -= __expq_table[T_EXPL_ARG2+2*tval2+1];
171
172 x = x + xl;
173
174 /* Compute ex2 = 2^n_0 e^(argtable[tval1]) e^(argtable[tval2]). */
175 ex2_u.value = __expq_table[T_EXPL_RES1 + tval1]
176 * __expq_table[T_EXPL_RES2 + tval2];
177 n_i = (int)n;
178 /* 'unsafe' is 1 iff n_1 != 0. */
179 unsafe = abs(n_i) >= 15000;
180 ex2_u.ieee.exponent += n_i >> unsafe;
181
182 /* Compute scale = 2^n_1. */
183 scale_u.value = 1;
184 scale_u.ieee.exponent += n_i - (n_i >> unsafe);
185
186 /* Approximate e^x2 - 1, using a seventh-degree polynomial,
187 with maximum error in [-2^-16-2^-53,2^-16+2^-53]
188 less than 4.8e-39. */
189 x22 = x + x*x*(P1+x*(P2+x*(P3+x*(P4+x*(P5+x*P6)))));
190 math_force_eval (x22);
191
192 /* Return result. */
193 fesetenv (&oldenv);
194
195 result = x22 * ex2_u.value + ex2_u.value;
196
197 /* Now we can test whether the result is ultimate or if we are unsure.
198 In the later case we should probably call a mpn based routine to give
199 the ultimate result.
200 Empirically, this routine is already ultimate in about 99.9986% of
201 cases, the test below for the round to nearest case will be false
202 in ~ 99.9963% of cases.
203 Without proc2 routine maximum error which has been seen is
204 0.5000262 ulp.
205
206 ieee854_float128 ex3_u;
207
208 #ifdef FE_TONEAREST
209 fesetround (FE_TONEAREST);
210 #endif
211 ex3_u.value = (result - ex2_u.value) - x22 * ex2_u.value;
212 ex2_u.value = result;
213 ex3_u.ieee.exponent += FLT128_MANT_DIG + 15 + IEEE854_FLOAT128_BIAS
214 - ex2_u.ieee.exponent;
215 n_i = abs (ex3_u.value);
216 n_i = (n_i + 1) / 2;
217 fesetenv (&oldenv);
218 #ifdef FE_TONEAREST
219 if (fegetround () == FE_TONEAREST)
220 n_i -= 0x4000;
221 #endif
222 if (!n_i) {
223 return __ieee754_expl_proc2 (origx);
224 }
225 */
226 if (!unsafe)
227 return result;
228 else
229 {
230 result *= scale_u.value;
231 math_check_force_underflow_nonneg (result);
232 return result;
233 }
234 }
235 /* Exceptional cases: */
236 else if (__builtin_isless (x, himark))
237 {
238 if (isinfq (x))
239 /* e^-inf == 0, with no error. */
240 return 0;
241 else
242 /* Underflow */
243 return TINY * TINY;
244 }
245 else
246 /* Return x, if x is a NaN or Inf; or overflow, otherwise. */
247 return TWO16383*x;
248 }