1 /* Quad-precision floating point e^x.
2 Copyright (C) 1999-2020 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>
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.
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.
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 <https://www.gnu.org/licenses/>. */
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,
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.
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
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
44 Then e^x is approximated as
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))
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| < -LDBL_MIN_EXP-1.
55 If it happens that n_1 == 0 (this is the usual case), that multiplication
67 #include <math-barriers.h>
68 #include <math_private.h>
69 #include <math-underflow.h>
72 #include <libm-alias-finite.h>
74 static const _Float128 C
[] = {
75 /* Smallest integer x for which e^x overflows. */
77 L(11356.523406294143949491931077970765),
79 /* Largest integer x for which e^x underflows. */
81 L(-11433.4627433362978788372438434526231),
85 L(59421121885698253195157962752.0),
88 #define THREEp103 C[3]
89 L(30423614405477505635920876929024.0),
92 #define THREEp111 C[4]
93 L(7788445287802241442795744493830144.0),
97 L(1.44269504088896340735992468100189204),
99 /* first 93 bits of ln(2) */
101 L(0.693147180559945309417232121457981864),
105 L(-1.94704509238074995158795957333327386E-31),
107 /* very small number */
112 #define TWO16383 C[9]
113 L(5.94865747678615882542879663314003565E+4931),
123 /* Chebyshev polynom coefficients for (exp(x)-1)/x */
131 L(1.66666666666666666666666666666666683E-01),
132 L(4.16666666666666666666654902320001674E-02),
133 L(8.33333333333333333333314659767198461E-03),
134 L(1.38888888889899438565058018857254025E-03),
135 L(1.98412698413981650382436541785404286E-04),
139 __ieee754_expl (_Float128 x
)
141 /* Check for usual case. */
142 if (isless (x
, himark
) && isgreater (x
, lomark
))
144 int tval1
, tval2
, unsafe
, n_i
;
145 _Float128 x22
, n
, t
, result
, xl
;
146 union ieee854_long_double ex2_u
, scale_u
;
149 feholdexcept (&oldenv
);
151 fesetround (FE_TONEAREST
);
155 n
= x
* M_1_LN2
+ THREEp111
;
160 /* Calculate t/256. */
164 /* Compute tval1 = t. */
165 tval1
= (int) (t
* TWO8
);
167 x
-= __expl_table
[T_EXPL_ARG1
+2*tval1
];
168 xl
-= __expl_table
[T_EXPL_ARG1
+2*tval1
+1];
170 /* Calculate t/32768. */
174 /* Compute tval2 = t. */
175 tval2
= (int) (t
* TWO15
);
177 x
-= __expl_table
[T_EXPL_ARG2
+2*tval2
];
178 xl
-= __expl_table
[T_EXPL_ARG2
+2*tval2
+1];
182 /* Compute ex2 = 2^n_0 e^(argtable[tval1]) e^(argtable[tval2]). */
183 ex2_u
.d
= __expl_table
[T_EXPL_RES1
+ tval1
]
184 * __expl_table
[T_EXPL_RES2
+ tval2
];
186 /* 'unsafe' is 1 iff n_1 != 0. */
187 unsafe
= abs(n_i
) >= 15000;
188 ex2_u
.ieee
.exponent
+= n_i
>> unsafe
;
190 /* Compute scale = 2^n_1. */
192 scale_u
.ieee
.exponent
+= n_i
- (n_i
>> unsafe
);
194 /* Approximate e^x2 - 1, using a seventh-degree polynomial,
195 with maximum error in [-2^-16-2^-53,2^-16+2^-53]
196 less than 4.8e-39. */
197 x22
= x
+ x
*x
*(P1
+x
*(P2
+x
*(P3
+x
*(P4
+x
*(P5
+x
*P6
)))));
198 math_force_eval (x22
);
203 result
= x22
* ex2_u
.d
+ ex2_u
.d
;
205 /* Now we can test whether the result is ultimate or if we are unsure.
206 In the later case we should probably call a mpn based routine to give
208 Empirically, this routine is already ultimate in about 99.9986% of
209 cases, the test below for the round to nearest case will be false
210 in ~ 99.9963% of cases.
211 Without proc2 routine maximum error which has been seen is
214 union ieee854_long_double ex3_u;
217 fesetround (FE_TONEAREST);
219 ex3_u.d = (result - ex2_u.d) - x22 * ex2_u.d;
221 ex3_u.ieee.exponent += LDBL_MANT_DIG + 15 + IEEE854_LONG_DOUBLE_BIAS
222 - ex2_u.ieee.exponent;
227 if (fegetround () == FE_TONEAREST)
231 return __ieee754_expl_proc2 (origx);
239 math_check_force_underflow_nonneg (result
);
243 /* Exceptional cases: */
244 else if (isless (x
, himark
))
247 /* e^-inf == 0, with no error. */
254 /* Return x, if x is a NaN or Inf; or overflow, otherwise. */
257 libm_alias_finite (__ieee754_expl
, __expl
)