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f964490f | 1 | /* Quad-precision floating point e^x. |
581c785b | 2 | Copyright (C) 1999-2022 Free Software Foundation, Inc. |
f964490f | 3 | This file is part of the GNU C Library. |
f964490f RM |
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 | |
59ba27a6 | 16 | License along with the GNU C Library; if not, see |
5a82c748 | 17 | <https://www.gnu.org/licenses/>. */ |
f964490f RM |
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> | |
70e2ba33 | 65 | #include <fenv_private.h> |
220622dd | 66 | #include <libm-alias-finite.h> |
15089e04 | 67 | |
c5c2e667 | 68 | #include "t_expl.h" |
f964490f RM |
69 | |
70 | static const long double C[] = { | |
71 | /* Smallest integer x for which e^x overflows. */ | |
72 | #define himark C[0] | |
e42a38dd | 73 | 709.78271289338399678773454114191496482L, |
f964490f RM |
74 | |
75 | /* Largest integer x for which e^x underflows. */ | |
76 | #define lomark C[1] | |
e42a38dd | 77 | -744.44007192138126231410729844608163411L, |
f964490f RM |
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 | ||
382466e0 | 119 | /* Chebyshev polynom coefficients for (exp(x)-1)/x */ |
f964490f RM |
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 | ||
9755bc46 JM |
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 | ||
f964490f RM |
139 | long double |
140 | __ieee754_expl (long double x) | |
141 | { | |
41e8926a AZ |
142 | long double result, x22; |
143 | union ibm_extended_long_double ex2_u, scale_u; | |
144 | int unsafe; | |
145 | ||
f964490f RM |
146 | /* Check for usual case. */ |
147 | if (isless (x, himark) && isgreater (x, lomark)) | |
148 | { | |
41e8926a AZ |
149 | int tval1, tval2, n_i, exponent2; |
150 | long double n, xl; | |
151 | ||
152 | SET_RESTORE_ROUND (FE_TONEAREST); | |
f964490f | 153 | |
9755bc46 | 154 | n = roundl (x*M_1_LN2); |
f964490f RM |
155 | x = x-n*M_LN2_0; |
156 | xl = n*M_LN2_1; | |
157 | ||
9755bc46 | 158 | tval1 = roundl (x*TWO8); |
f964490f RM |
159 | x -= __expl_table[T_EXPL_ARG1+2*tval1]; |
160 | xl -= __expl_table[T_EXPL_ARG1+2*tval1+1]; | |
161 | ||
9755bc46 | 162 | tval2 = roundl (x*TWO15); |
f964490f RM |
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]). */ | |
9605ca6c AM |
169 | ex2_u.ld = (__expl_table[T_EXPL_RES1 + tval1] |
170 | * __expl_table[T_EXPL_RES2 + tval2]); | |
f964490f RM |
171 | n_i = (int)n; |
172 | /* 'unsafe' is 1 iff n_1 != 0. */ | |
d1cdd051 | 173 | unsafe = fabsl(n_i) >= -LDBL_MIN_EXP - 1; |
9605ca6c | 174 | ex2_u.d[0].ieee.exponent += n_i >> unsafe; |
f964490f RM |
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. */ | |
9605ca6c AM |
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) */; | |
f964490f | 181 | else if (exponent2 > 0) |
9605ca6c | 182 | ex2_u.d[1].ieee.exponent = exponent2; |
f964490f RM |
183 | else if (exponent2 <= -54) |
184 | { | |
9605ca6c AM |
185 | ex2_u.d[1].ieee.exponent = 0; |
186 | ex2_u.d[1].ieee.mantissa0 = 0; | |
187 | ex2_u.d[1].ieee.mantissa1 = 0; | |
f964490f RM |
188 | } |
189 | else | |
190 | { | |
191 | static const double | |
192 | two54 = 1.80143985094819840000e+16, /* 4350000000000000 */ | |
193 | twom54 = 5.55111512312578270212e-17; /* 3C90000000000000 */ | |
9605ca6c AM |
194 | ex2_u.d[1].d *= two54; |
195 | ex2_u.d[1].ieee.exponent += n_i >> unsafe; | |
196 | ex2_u.d[1].d *= twom54; | |
f964490f RM |
197 | } |
198 | ||
199 | /* Compute scale = 2^n_1. */ | |
9605ca6c AM |
200 | scale_u.ld = 1.0L; |
201 | scale_u.d[0].ieee.exponent += n_i - (n_i >> unsafe); | |
f964490f RM |
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 | ||
f964490f RM |
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 | |
350635a5 | 225 | - ex2_u.ieee.exponent; |
f964490f RM |
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 | */ | |
f964490f RM |
237 | } |
238 | /* Exceptional cases: */ | |
239 | else if (isless (x, himark)) | |
240 | { | |
d81f90cc | 241 | if (isinf (x)) |
f964490f RM |
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; | |
41e8926a AZ |
251 | |
252 | result = x22 * ex2_u.ld + ex2_u.ld; | |
253 | if (!unsafe) | |
254 | return result; | |
255 | return result * scale_u.ld; | |
f964490f | 256 | } |
220622dd | 257 | libm_alias_finite (__ieee754_expl, __expl) |