1 /* origin: OpenBSD /usr/src/lib/libm/src/ld80/e_expm1l.c */
3 * Copyright (c) 2008 Stephen L. Moshier <steve@moshier.net>
5 * Permission to use, copy, modify, and distribute this software for any
6 * purpose with or without fee is hereby granted, provided that the above
7 * copyright notice and this permission notice appear in all copies.
9 * THE SOFTWARE IS PROVIDED "AS IS" AND THE AUTHOR DISCLAIMS ALL WARRANTIES
10 * WITH REGARD TO THIS SOFTWARE INCLUDING ALL IMPLIED WARRANTIES OF
11 * MERCHANTABILITY AND FITNESS. IN NO EVENT SHALL THE AUTHOR BE LIABLE FOR
12 * ANY SPECIAL, DIRECT, INDIRECT, OR CONSEQUENTIAL DAMAGES OR ANY DAMAGES
13 * WHATSOEVER RESULTING FROM LOSS OF USE, DATA OR PROFITS, WHETHER IN AN
14 * ACTION OF CONTRACT, NEGLIGENCE OR OTHER TORTIOUS ACTION, ARISING OUT OF
15 * OR IN CONNECTION WITH THE USE OR PERFORMANCE OF THIS SOFTWARE.
18 * Exponential function, minus 1
19 * Long double precision
24 * long double x, y, expm1l();
31 * Returns e (2.71828...) raised to the x power, minus 1.
33 * Range reduction is accomplished by separating the argument
34 * into an integer k and fraction f such that
39 * An expansion x + .5 x^2 + x^3 R(x) approximates exp(f) - 1
40 * in the basic range [-0.5 ln 2, 0.5 ln 2].
46 * arithmetic domain # trials peak rms
47 * IEEE -45,+MAXLOG 200,000 1.2e-19 2.5e-20
51 * message condition value returned
52 * expm1l overflow x > MAXLOG MAXNUM
58 #if LDBL_MANT_DIG == 53 && LDBL_MAX_EXP == 1024
59 long double expm1l(long double x)
63 #elif LDBL_MANT_DIG == 64 && LDBL_MAX_EXP == 16384
64 static const long double MAXLOGL = 1.1356523406294143949492E4L;
66 /* exp(x) - 1 = x + 0.5 x^2 + x^3 P(x)/Q(x)
67 -.5 ln 2 < x < .5 ln 2
68 Theoretical peak relative error = 3.4e-22 */
69 static const long double
70 P0 = -1.586135578666346600772998894928250240826E4L,
71 P1 = 2.642771505685952966904660652518429479531E3L,
72 P2 = -3.423199068835684263987132888286791620673E2L,
73 P3 = 1.800826371455042224581246202420972737840E1L,
74 P4 = -5.238523121205561042771939008061958820811E-1L,
75 Q0 = -9.516813471998079611319047060563358064497E4L,
76 Q1 = 3.964866271411091674556850458227710004570E4L,
77 Q2 = -7.207678383830091850230366618190187434796E3L,
78 Q3 = 7.206038318724600171970199625081491823079E2L,
79 Q4 = -4.002027679107076077238836622982900945173E1L,
80 /* Q5 = 1.000000000000000000000000000000000000000E0 */
82 C1 = 6.93145751953125E-1L,
83 C2 = 1.428606820309417232121458176568075500134E-6L,
85 minarg = -4.5054566736396445112120088E1L,
88 long double expm1l(long double x)
90 long double px, qx, xx;
95 return huge*huge; /* overflow */
103 /* Express x = ln 2 (k + remainder), remainder not exceeding 1/2. */
104 px = floorl (0.5 + x / xx);
106 /* remainder times ln 2 */
110 /* Approximate exp(remainder ln 2).*/
111 px = (((( P4 * x + P3) * x + P2) * x + P1) * x + P0) * x;
112 qx = (((( x + Q4) * x + Q3) * x + Q2) * x + Q1) * x + Q0;
114 qx = x + (0.5 * xx + xx * px / qx);
116 /* exp(x) = exp(k ln 2) exp(remainder ln 2) = 2^k exp(remainder ln 2).
117 We have qx = exp(remainder ln 2) - 1, so
118 exp(x) - 1 = 2^k (qx + 1) - 1 = 2^k qx + 2^k - 1. */
119 px = ldexpl(1.0L, k);
120 x = px * qx + (px - 1.0);