1 /* Quad-precision floating point e^x.
2 Copyright (C) 1999-2023 Free Software Foundation, Inc.
3 This file is part of the GNU C Library.
4
5 The GNU C Library is free software; you can redistribute it and/or
6 modify it under the terms of the GNU Lesser General Public
7 License as published by the Free Software Foundation; either
8 version 2.1 of the License, or (at your option) any later version.
9
10 The GNU C Library is distributed in the hope that it will be useful,
11 but WITHOUT ANY WARRANTY; without even the implied warranty of
12 MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the GNU
13 Lesser General Public License for more details.
14
15 You should have received a copy of the GNU Lesser General Public
16 License along with the GNU C Library; if not, see
17 <https://www.gnu.org/licenses/>. */
18
19 /* The basic design here is from
20 Abraham Ziv, "Fast Evaluation of Elementary Mathematical Functions with
21 Correctly Rounded Last Bit", ACM Trans. Math. Soft., 17 (3), September 1991,
22 pp. 410-423.
23
24 We work with number pairs where the first number is the high part and
25 the second one is the low part. Arithmetic with the high part numbers must
26 be exact, without any roundoff errors.
27
28 The input value, X, is written as
29 X = n * ln(2)_0 + arg1[t1]_0 + arg2[t2]_0 + x
30 - n * ln(2)_1 + arg1[t1]_1 + arg2[t2]_1 + xl
31
32 where:
33 - n is an integer, 16384 >= n >= -16495;
34 - ln(2)_0 is the first 93 bits of ln(2), and |ln(2)_0-ln(2)-ln(2)_1| < 2^-205
35 - t1 is an integer, 89 >= t1 >= -89
36 - t2 is an integer, 65 >= t2 >= -65
37 - |arg1[t1]-t1/256.0| < 2^-53
38 - |arg2[t2]-t2/32768.0| < 2^-53
39 - x + xl is whatever is left, |x + xl| < 2^-16 + 2^-53
40
41 Then e^x is approximated as
42
43 e^x = 2^n_1 ( 2^n_0 e^(arg1[t1]_0 + arg1[t1]_1) e^(arg2[t2]_0 + arg2[t2]_1)
44 + 2^n_0 e^(arg1[t1]_0 + arg1[t1]_1) e^(arg2[t2]_0 + arg2[t2]_1)
45 * p (x + xl + n * ln(2)_1))
46 where:
47 - p(x) is a polynomial approximating e(x)-1
48 - e^(arg1[t1]_0 + arg1[t1]_1) is obtained from a table
49 - e^(arg2[t2]_0 + arg2[t2]_1) likewise
50 - n_1 + n_0 = n, so that |n_0| < -LDBL_MIN_EXP-1.
51
52 If it happens that n_1 == 0 (this is the usual case), that multiplication
53 is omitted.
54 */
55
56 #ifndef _GNU_SOURCE
57 #define _GNU_SOURCE
58 #endif
59 #include <float.h>
60 #include <ieee754.h>
61 #include <math.h>
62 #include <fenv.h>
63 #include <inttypes.h>
64 #include <math_private.h>
65 #include <fenv_private.h>
66 #include <libm-alias-finite.h>
67
68 #include "t_expl.h"
69
70 static const long double C[] = {
71 /* Smallest integer x for which e^x overflows. */
72 #define himark C[0]
73 709.78271289338399678773454114191496482L,
74
75 /* Largest integer x for which e^x underflows. */
76 #define lomark C[1]
77 -744.44007192138126231410729844608163411L,
78
79 /* 3x2^96 */
80 #define THREEp96 C[2]
81 59421121885698253195157962752.0L,
82
83 /* 3x2^103 */
84 #define THREEp103 C[3]
85 30423614405477505635920876929024.0L,
86
87 /* 3x2^111 */
88 #define THREEp111 C[4]
89 7788445287802241442795744493830144.0L,
90
91 /* 1/ln(2) */
92 #define M_1_LN2 C[5]
93 1.44269504088896340735992468100189204L,
94
95 /* first 93 bits of ln(2) */
96 #define M_LN2_0 C[6]
97 0.693147180559945309417232121457981864L,
98
99 /* ln2_0 - ln(2) */
100 #define M_LN2_1 C[7]
101 -1.94704509238074995158795957333327386E-31L,
102
103 /* very small number */
104 #define TINY C[8]
105 1.0e-308L,
106
107 /* 2^16383 */
108 #define TWO1023 C[9]
109 8.988465674311579538646525953945123668E+307L,
110
111 /* 256 */
112 #define TWO8 C[10]
113 256.0L,
114
115 /* 32768 */
116 #define TWO15 C[11]
117 32768.0L,
118
119 /* Chebyshev polynom coefficients for (exp(x)-1)/x */
120 #define P1 C[12]
121 #define P2 C[13]
122 #define P3 C[14]
123 #define P4 C[15]
124 #define P5 C[16]
125 #define P6 C[17]
126 0.5L,
127 1.66666666666666666666666666666666683E-01L,
128 4.16666666666666666666654902320001674E-02L,
129 8.33333333333333333333314659767198461E-03L,
130 1.38888888889899438565058018857254025E-03L,
131 1.98412698413981650382436541785404286E-04L,
132 };
133
134 /* Avoid local PLT entry use from (int) roundl (...) being converted
135 to a call to lroundl in the case of 32-bit long and roundl not
136 inlined. */
137 long int lroundl (long double) asm ("__lroundl");
138
139 long double
140 __ieee754_expl (long double x)
141 {
142 long double result, x22;
143 union ibm_extended_long_double ex2_u, scale_u;
144 int unsafe;
145
146 /* Check for usual case. */
147 if (isless (x, himark) && isgreater (x, lomark))
148 {
149 int tval1, tval2, n_i, exponent2;
150 long double n, xl;
151
152 SET_RESTORE_ROUND (FE_TONEAREST);
153
154 n = roundl (x*M_1_LN2);
155 x = x-n*M_LN2_0;
156 xl = n*M_LN2_1;
157
158 tval1 = roundl (x*TWO8);
159 x -= __expl_table[T_EXPL_ARG1+2*tval1];
160 xl -= __expl_table[T_EXPL_ARG1+2*tval1+1];
161
162 tval2 = roundl (x*TWO15);
163 x -= __expl_table[T_EXPL_ARG2+2*tval2];
164 xl -= __expl_table[T_EXPL_ARG2+2*tval2+1];
165
166 x = x + xl;
167
168 /* Compute ex2 = 2^n_0 e^(argtable[tval1]) e^(argtable[tval2]). */
169 ex2_u.ld = (__expl_table[T_EXPL_RES1 + tval1]
170 * __expl_table[T_EXPL_RES2 + tval2]);
171 n_i = (int)n;
172 /* 'unsafe' is 1 iff n_1 != 0. */
173 unsafe = fabsl(n_i) >= -LDBL_MIN_EXP - 1;
174 ex2_u.d[0].ieee.exponent += n_i >> unsafe;
175 /* Fortunately, there are no subnormal lowpart doubles in
176 __expl_table, only normal values and zeros.
177 But after scaling it can be subnormal. */
178 exponent2 = ex2_u.d[1].ieee.exponent + (n_i >> unsafe);
179 if (ex2_u.d[1].ieee.exponent == 0)
180 /* assert ((ex2_u.d[1].ieee.mantissa0|ex2_u.d[1].ieee.mantissa1) == 0) */;
181 else if (exponent2 > 0)
182 ex2_u.d[1].ieee.exponent = exponent2;
183 else if (exponent2 <= -54)
184 {
185 ex2_u.d[1].ieee.exponent = 0;
186 ex2_u.d[1].ieee.mantissa0 = 0;
187 ex2_u.d[1].ieee.mantissa1 = 0;
188 }
189 else
190 {
191 static const double
192 two54 = 1.80143985094819840000e+16, /* 4350000000000000 */
193 twom54 = 5.55111512312578270212e-17; /* 3C90000000000000 */
194 ex2_u.d[1].d *= two54;
195 ex2_u.d[1].ieee.exponent += n_i >> unsafe;
196 ex2_u.d[1].d *= twom54;
197 }
198
199 /* Compute scale = 2^n_1. */
200 scale_u.ld = 1.0L;
201 scale_u.d[0].ieee.exponent += n_i - (n_i >> unsafe);
202
203 /* Approximate e^x2 - 1, using a seventh-degree polynomial,
204 with maximum error in [-2^-16-2^-53,2^-16+2^-53]
205 less than 4.8e-39. */
206 x22 = x + x*x*(P1+x*(P2+x*(P3+x*(P4+x*(P5+x*P6)))));
207
208 /* Now we can test whether the result is ultimate or if we are unsure.
209 In the later case we should probably call a mpn based routine to give
210 the ultimate result.
211 Empirically, this routine is already ultimate in about 99.9986% of
212 cases, the test below for the round to nearest case will be false
213 in ~ 99.9963% of cases.
214 Without proc2 routine maximum error which has been seen is
215 0.5000262 ulp.
216
217 union ieee854_long_double ex3_u;
218
219 #ifdef FE_TONEAREST
220 fesetround (FE_TONEAREST);
221 #endif
222 ex3_u.d = (result - ex2_u.d) - x22 * ex2_u.d;
223 ex2_u.d = result;
224 ex3_u.ieee.exponent += LDBL_MANT_DIG + 15 + IEEE854_LONG_DOUBLE_BIAS
225 - ex2_u.ieee.exponent;
226 n_i = abs (ex3_u.d);
227 n_i = (n_i + 1) / 2;
228 fesetenv (&oldenv);
229 #ifdef FE_TONEAREST
230 if (fegetround () == FE_TONEAREST)
231 n_i -= 0x4000;
232 #endif
233 if (!n_i) {
234 return __ieee754_expl_proc2 (origx);
235 }
236 */
237 }
238 /* Exceptional cases: */
239 else if (isless (x, himark))
240 {
241 if (isinf (x))
242 /* e^-inf == 0, with no error. */
243 return 0;
244 else
245 /* Underflow */
246 return TINY * TINY;
247 }
248 else
249 /* Return x, if x is a NaN or Inf; or overflow, otherwise. */
250 return TWO1023*x;
251
252 result = x22 * ex2_u.ld + ex2_u.ld;
253 if (!unsafe)
254 return result;
255 return result * scale_u.ld;
256 }
257 libm_alias_finite (__ieee754_expl, __expl)