]>
git.ipfire.org Git - thirdparty/glibc.git/blob - sysdeps/ieee754/ldbl-128ibm/s_expm1l.c
22abee4d9985320bfd3219a0139de26f71322f84
3 * Exponential function, minus 1
4 * 128-bit long double precision
10 * long double x, y, expm1l();
18 * Returns e (2.71828...) raised to the x power, minus one.
20 * Range reduction is accomplished by separating the argument
21 * into an integer k and fraction f such that
26 * An expansion x + .5 x^2 + x^3 R(x) approximates exp(f) - 1
27 * in the basic range [-0.5 ln 2, 0.5 ln 2].
33 * arithmetic domain # trials peak rms
34 * IEEE -79,+MAXLOG 100,000 1.7e-34 4.5e-35
38 /* Copyright 2001 by Stephen L. Moshier
40 This library is free software; you can redistribute it and/or
41 modify it under the terms of the GNU Lesser General Public
42 License as published by the Free Software Foundation; either
43 version 2.1 of the License, or (at your option) any later version.
45 This library is distributed in the hope that it will be useful,
46 but WITHOUT ANY WARRANTY; without even the implied warranty of
47 MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the GNU
48 Lesser General Public License for more details.
50 You should have received a copy of the GNU Lesser General Public
51 License along with this library; if not, see
52 <http://www.gnu.org/licenses/>. */
56 #include <math_private.h>
57 #include <math_ldbl_opt.h>
59 /* exp(x) - 1 = x + 0.5 x^2 + x^3 P(x)/Q(x)
60 -.5 ln 2 < x < .5 ln 2
61 Theoretical peak relative error = 8.1e-36 */
63 static const long double
64 P0
= 2.943520915569954073888921213330863757240E8L
,
65 P1
= -5.722847283900608941516165725053359168840E7L
,
66 P2
= 8.944630806357575461578107295909719817253E6L
,
67 P3
= -7.212432713558031519943281748462837065308E5L
,
68 P4
= 4.578962475841642634225390068461943438441E4L
,
69 P5
= -1.716772506388927649032068540558788106762E3L
,
70 P6
= 4.401308817383362136048032038528753151144E1L
,
71 P7
= -4.888737542888633647784737721812546636240E-1L,
72 Q0
= 1.766112549341972444333352727998584753865E9L
,
73 Q1
= -7.848989743695296475743081255027098295771E8L
,
74 Q2
= 1.615869009634292424463780387327037251069E8L
,
75 Q3
= -2.019684072836541751428967854947019415698E7L
,
76 Q4
= 1.682912729190313538934190635536631941751E6L
,
77 Q5
= -9.615511549171441430850103489315371768998E4L
,
78 Q6
= 3.697714952261803935521187272204485251835E3L
,
79 Q7
= -8.802340681794263968892934703309274564037E1L
,
80 /* Q8 = 1.000000000000000000000000000000000000000E0 */
83 C1
= 6.93145751953125E-1L,
84 C2
= 1.428606820309417232121458176568075500134E-6L,
86 minarg
= -7.9018778583833765273564461846232128760607E1L
, big
= 1e290L
;
90 __expm1l (long double x
)
92 long double px
, qx
, xx
;
97 /* Detect infinity and NaN. */
99 EXTRACT_WORDS (ix
, lx
, xhi
);
100 sign
= ix
& 0x80000000;
102 if (!sign
&& ix
>= 0x40600000)
104 if (ix
>= 0x7ff00000)
106 /* Infinity (which must be negative infinity). */
107 if (((ix
- 0x7ff00000) | lx
) == 0)
109 /* NaN. Invalid exception if signaling. */
113 /* expm1(+- 0) = +- 0. */
119 return (4.0/big
- 1.0L);
121 /* Express x = ln 2 (k + remainder), remainder not exceeding 1/2. */
122 xx
= C1
+ C2
; /* ln 2. */
123 px
= floorl (0.5 + x
/ xx
);
125 /* remainder times ln 2 */
129 /* Approximate exp(remainder ln 2). */
132 + P5
) * x
+ P4
) * x
+ P3
) * x
+ P2
) * x
+ P1
) * x
+ P0
) * x
;
136 + Q6
) * x
+ Q5
) * x
+ Q4
) * x
+ Q3
) * x
+ Q2
) * x
+ Q1
) * x
+ Q0
;
139 qx
= x
+ (0.5 * xx
+ xx
* px
/ qx
);
141 /* exp(x) = exp(k ln 2) exp(remainder ln 2) = 2^k exp(remainder ln 2).
143 We have qx = exp(remainder ln 2) - 1, so
144 exp(x) - 1 = 2^k (qx + 1) - 1
145 = 2^k qx + 2^k - 1. */
147 px
= __ldexpl (1.0L, k
);
148 x
= px
* qx
+ (px
- 1.0);
151 libm_hidden_def (__expm1l
)
152 long_double_symbol (libm
, __expm1l
, expm1l
);