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.
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.
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.
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/>. */
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,
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.
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
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
41 Then e^x is approximated as
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))
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.
52 If it happens that n_1 == 0 (this is the usual case), that multiplication
64 #include <math-barriers.h>
65 #include <math_private.h>
66 #include <math-underflow.h>
69 #include <libm-alias-finite.h>
71 static const _Float128 C
[] = {
72 /* Smallest integer x for which e^x overflows. */
74 L(11356.523406294143949491931077970765),
76 /* Largest integer x for which e^x underflows. */
78 L(-11433.4627433362978788372438434526231),
82 L(59421121885698253195157962752.0),
85 #define THREEp103 C[3]
86 L(30423614405477505635920876929024.0),
89 #define THREEp111 C[4]
90 L(7788445287802241442795744493830144.0),
94 L(1.44269504088896340735992468100189204),
96 /* first 93 bits of ln(2) */
98 L(0.693147180559945309417232121457981864),
102 L(-1.94704509238074995158795957333327386E-31),
104 /* very small number */
109 #define TWO16383 C[9]
110 L(5.94865747678615882542879663314003565E+4931),
120 /* Chebyshev polynom coefficients for (exp(x)-1)/x */
128 L(1.66666666666666666666666666666666683E-01),
129 L(4.16666666666666666666654902320001674E-02),
130 L(8.33333333333333333333314659767198461E-03),
131 L(1.38888888889899438565058018857254025E-03),
132 L(1.98412698413981650382436541785404286E-04),
136 __ieee754_expl (_Float128 x
)
138 /* Check for usual case. */
139 if (isless (x
, himark
) && isgreater (x
, lomark
))
141 int tval1
, tval2
, unsafe
, n_i
;
142 _Float128 x22
, n
, t
, result
, xl
;
143 union ieee854_long_double ex2_u
, scale_u
;
146 feholdexcept (&oldenv
);
148 fesetround (FE_TONEAREST
);
152 n
= x
* M_1_LN2
+ THREEp111
;
157 /* Calculate t/256. */
161 /* Compute tval1 = t. */
162 tval1
= (int) (t
* TWO8
);
164 x
-= __expl_table
[T_EXPL_ARG1
+2*tval1
];
165 xl
-= __expl_table
[T_EXPL_ARG1
+2*tval1
+1];
167 /* Calculate t/32768. */
171 /* Compute tval2 = t. */
172 tval2
= (int) (t
* TWO15
);
174 x
-= __expl_table
[T_EXPL_ARG2
+2*tval2
];
175 xl
-= __expl_table
[T_EXPL_ARG2
+2*tval2
+1];
179 /* Compute ex2 = 2^n_0 e^(argtable[tval1]) e^(argtable[tval2]). */
180 ex2_u
.d
= __expl_table
[T_EXPL_RES1
+ tval1
]
181 * __expl_table
[T_EXPL_RES2
+ tval2
];
183 /* 'unsafe' is 1 iff n_1 != 0. */
184 unsafe
= abs(n_i
) >= 15000;
185 ex2_u
.ieee
.exponent
+= n_i
>> unsafe
;
187 /* Compute scale = 2^n_1. */
189 scale_u
.ieee
.exponent
+= n_i
- (n_i
>> unsafe
);
191 /* Approximate e^x2 - 1, using a seventh-degree polynomial,
192 with maximum error in [-2^-16-2^-53,2^-16+2^-53]
193 less than 4.8e-39. */
194 x22
= x
+ x
*x
*(P1
+x
*(P2
+x
*(P3
+x
*(P4
+x
*(P5
+x
*P6
)))));
195 math_force_eval (x22
);
200 result
= x22
* ex2_u
.d
+ ex2_u
.d
;
202 /* Now we can test whether the result is ultimate or if we are unsure.
203 In the later case we should probably call a mpn based routine to give
205 Empirically, this routine is already ultimate in about 99.9986% of
206 cases, the test below for the round to nearest case will be false
207 in ~ 99.9963% of cases.
208 Without proc2 routine maximum error which has been seen is
211 union ieee854_long_double ex3_u;
214 fesetround (FE_TONEAREST);
216 ex3_u.d = (result - ex2_u.d) - x22 * ex2_u.d;
218 ex3_u.ieee.exponent += LDBL_MANT_DIG + 15 + IEEE854_LONG_DOUBLE_BIAS
219 - ex2_u.ieee.exponent;
224 if (fegetround () == FE_TONEAREST)
228 return __ieee754_expl_proc2 (origx);
236 math_check_force_underflow_nonneg (result
);
240 /* Exceptional cases: */
241 else if (isless (x
, himark
))
244 /* e^-inf == 0, with no error. */
251 /* Return x, if x is a NaN or Inf; or overflow, otherwise. */
254 libm_alias_finite (__ieee754_expl
, __expl
)