1*c9945492SAndroid Build Coastguard Worker /* origin: OpenBSD /usr/src/lib/libm/src/ld80/e_expl.c */
2*c9945492SAndroid Build Coastguard Worker /*
3*c9945492SAndroid Build Coastguard Worker * Copyright (c) 2008 Stephen L. Moshier <[email protected]>
4*c9945492SAndroid Build Coastguard Worker *
5*c9945492SAndroid Build Coastguard Worker * Permission to use, copy, modify, and distribute this software for any
6*c9945492SAndroid Build Coastguard Worker * purpose with or without fee is hereby granted, provided that the above
7*c9945492SAndroid Build Coastguard Worker * copyright notice and this permission notice appear in all copies.
8*c9945492SAndroid Build Coastguard Worker *
9*c9945492SAndroid Build Coastguard Worker * THE SOFTWARE IS PROVIDED "AS IS" AND THE AUTHOR DISCLAIMS ALL WARRANTIES
10*c9945492SAndroid Build Coastguard Worker * WITH REGARD TO THIS SOFTWARE INCLUDING ALL IMPLIED WARRANTIES OF
11*c9945492SAndroid Build Coastguard Worker * MERCHANTABILITY AND FITNESS. IN NO EVENT SHALL THE AUTHOR BE LIABLE FOR
12*c9945492SAndroid Build Coastguard Worker * ANY SPECIAL, DIRECT, INDIRECT, OR CONSEQUENTIAL DAMAGES OR ANY DAMAGES
13*c9945492SAndroid Build Coastguard Worker * WHATSOEVER RESULTING FROM LOSS OF USE, DATA OR PROFITS, WHETHER IN AN
14*c9945492SAndroid Build Coastguard Worker * ACTION OF CONTRACT, NEGLIGENCE OR OTHER TORTIOUS ACTION, ARISING OUT OF
15*c9945492SAndroid Build Coastguard Worker * OR IN CONNECTION WITH THE USE OR PERFORMANCE OF THIS SOFTWARE.
16*c9945492SAndroid Build Coastguard Worker */
17*c9945492SAndroid Build Coastguard Worker /*
18*c9945492SAndroid Build Coastguard Worker * Exponential function, long double precision
19*c9945492SAndroid Build Coastguard Worker *
20*c9945492SAndroid Build Coastguard Worker *
21*c9945492SAndroid Build Coastguard Worker * SYNOPSIS:
22*c9945492SAndroid Build Coastguard Worker *
23*c9945492SAndroid Build Coastguard Worker * long double x, y, expl();
24*c9945492SAndroid Build Coastguard Worker *
25*c9945492SAndroid Build Coastguard Worker * y = expl( x );
26*c9945492SAndroid Build Coastguard Worker *
27*c9945492SAndroid Build Coastguard Worker *
28*c9945492SAndroid Build Coastguard Worker * DESCRIPTION:
29*c9945492SAndroid Build Coastguard Worker *
30*c9945492SAndroid Build Coastguard Worker * Returns e (2.71828...) raised to the x power.
31*c9945492SAndroid Build Coastguard Worker *
32*c9945492SAndroid Build Coastguard Worker * Range reduction is accomplished by separating the argument
33*c9945492SAndroid Build Coastguard Worker * into an integer k and fraction f such that
34*c9945492SAndroid Build Coastguard Worker *
35*c9945492SAndroid Build Coastguard Worker * x k f
36*c9945492SAndroid Build Coastguard Worker * e = 2 e.
37*c9945492SAndroid Build Coastguard Worker *
38*c9945492SAndroid Build Coastguard Worker * A Pade' form of degree 5/6 is used to approximate exp(f) - 1
39*c9945492SAndroid Build Coastguard Worker * in the basic range [-0.5 ln 2, 0.5 ln 2].
40*c9945492SAndroid Build Coastguard Worker *
41*c9945492SAndroid Build Coastguard Worker *
42*c9945492SAndroid Build Coastguard Worker * ACCURACY:
43*c9945492SAndroid Build Coastguard Worker *
44*c9945492SAndroid Build Coastguard Worker * Relative error:
45*c9945492SAndroid Build Coastguard Worker * arithmetic domain # trials peak rms
46*c9945492SAndroid Build Coastguard Worker * IEEE +-10000 50000 1.12e-19 2.81e-20
47*c9945492SAndroid Build Coastguard Worker *
48*c9945492SAndroid Build Coastguard Worker *
49*c9945492SAndroid Build Coastguard Worker * Error amplification in the exponential function can be
50*c9945492SAndroid Build Coastguard Worker * a serious matter. The error propagation involves
51*c9945492SAndroid Build Coastguard Worker * exp( X(1+delta) ) = exp(X) ( 1 + X*delta + ... ),
52*c9945492SAndroid Build Coastguard Worker * which shows that a 1 lsb error in representing X produces
53*c9945492SAndroid Build Coastguard Worker * a relative error of X times 1 lsb in the function.
54*c9945492SAndroid Build Coastguard Worker * While the routine gives an accurate result for arguments
55*c9945492SAndroid Build Coastguard Worker * that are exactly represented by a long double precision
56*c9945492SAndroid Build Coastguard Worker * computer number, the result contains amplified roundoff
57*c9945492SAndroid Build Coastguard Worker * error for large arguments not exactly represented.
58*c9945492SAndroid Build Coastguard Worker *
59*c9945492SAndroid Build Coastguard Worker *
60*c9945492SAndroid Build Coastguard Worker * ERROR MESSAGES:
61*c9945492SAndroid Build Coastguard Worker *
62*c9945492SAndroid Build Coastguard Worker * message condition value returned
63*c9945492SAndroid Build Coastguard Worker * exp underflow x < MINLOG 0.0
64*c9945492SAndroid Build Coastguard Worker * exp overflow x > MAXLOG MAXNUM
65*c9945492SAndroid Build Coastguard Worker *
66*c9945492SAndroid Build Coastguard Worker */
67*c9945492SAndroid Build Coastguard Worker
68*c9945492SAndroid Build Coastguard Worker #include "libm.h"
69*c9945492SAndroid Build Coastguard Worker
70*c9945492SAndroid Build Coastguard Worker #if LDBL_MANT_DIG == 53 && LDBL_MAX_EXP == 1024
expl(long double x)71*c9945492SAndroid Build Coastguard Worker long double expl(long double x)
72*c9945492SAndroid Build Coastguard Worker {
73*c9945492SAndroid Build Coastguard Worker return exp(x);
74*c9945492SAndroid Build Coastguard Worker }
75*c9945492SAndroid Build Coastguard Worker #elif LDBL_MANT_DIG == 64 && LDBL_MAX_EXP == 16384
76*c9945492SAndroid Build Coastguard Worker
77*c9945492SAndroid Build Coastguard Worker static const long double P[3] = {
78*c9945492SAndroid Build Coastguard Worker 1.2617719307481059087798E-4L,
79*c9945492SAndroid Build Coastguard Worker 3.0299440770744196129956E-2L,
80*c9945492SAndroid Build Coastguard Worker 9.9999999999999999991025E-1L,
81*c9945492SAndroid Build Coastguard Worker };
82*c9945492SAndroid Build Coastguard Worker static const long double Q[4] = {
83*c9945492SAndroid Build Coastguard Worker 3.0019850513866445504159E-6L,
84*c9945492SAndroid Build Coastguard Worker 2.5244834034968410419224E-3L,
85*c9945492SAndroid Build Coastguard Worker 2.2726554820815502876593E-1L,
86*c9945492SAndroid Build Coastguard Worker 2.0000000000000000000897E0L,
87*c9945492SAndroid Build Coastguard Worker };
88*c9945492SAndroid Build Coastguard Worker static const long double
89*c9945492SAndroid Build Coastguard Worker LN2HI = 6.9314575195312500000000E-1L,
90*c9945492SAndroid Build Coastguard Worker LN2LO = 1.4286068203094172321215E-6L,
91*c9945492SAndroid Build Coastguard Worker LOG2E = 1.4426950408889634073599E0L;
92*c9945492SAndroid Build Coastguard Worker
expl(long double x)93*c9945492SAndroid Build Coastguard Worker long double expl(long double x)
94*c9945492SAndroid Build Coastguard Worker {
95*c9945492SAndroid Build Coastguard Worker long double px, xx;
96*c9945492SAndroid Build Coastguard Worker int k;
97*c9945492SAndroid Build Coastguard Worker
98*c9945492SAndroid Build Coastguard Worker if (isnan(x))
99*c9945492SAndroid Build Coastguard Worker return x;
100*c9945492SAndroid Build Coastguard Worker if (x > 11356.5234062941439488L) /* x > ln(2^16384 - 0.5) */
101*c9945492SAndroid Build Coastguard Worker return x * 0x1p16383L;
102*c9945492SAndroid Build Coastguard Worker if (x < -11399.4985314888605581L) /* x < ln(2^-16446) */
103*c9945492SAndroid Build Coastguard Worker return -0x1p-16445L/x;
104*c9945492SAndroid Build Coastguard Worker
105*c9945492SAndroid Build Coastguard Worker /* Express e**x = e**f 2**k
106*c9945492SAndroid Build Coastguard Worker * = e**(f + k ln(2))
107*c9945492SAndroid Build Coastguard Worker */
108*c9945492SAndroid Build Coastguard Worker px = floorl(LOG2E * x + 0.5);
109*c9945492SAndroid Build Coastguard Worker k = px;
110*c9945492SAndroid Build Coastguard Worker x -= px * LN2HI;
111*c9945492SAndroid Build Coastguard Worker x -= px * LN2LO;
112*c9945492SAndroid Build Coastguard Worker
113*c9945492SAndroid Build Coastguard Worker /* rational approximation of the fractional part:
114*c9945492SAndroid Build Coastguard Worker * e**x = 1 + 2x P(x**2)/(Q(x**2) - x P(x**2))
115*c9945492SAndroid Build Coastguard Worker */
116*c9945492SAndroid Build Coastguard Worker xx = x * x;
117*c9945492SAndroid Build Coastguard Worker px = x * __polevll(xx, P, 2);
118*c9945492SAndroid Build Coastguard Worker x = px/(__polevll(xx, Q, 3) - px);
119*c9945492SAndroid Build Coastguard Worker x = 1.0 + 2.0 * x;
120*c9945492SAndroid Build Coastguard Worker return scalbnl(x, k);
121*c9945492SAndroid Build Coastguard Worker }
122*c9945492SAndroid Build Coastguard Worker #elif LDBL_MANT_DIG == 113 && LDBL_MAX_EXP == 16384
123*c9945492SAndroid Build Coastguard Worker // TODO: broken implementation to make things compile
expl(long double x)124*c9945492SAndroid Build Coastguard Worker long double expl(long double x)
125*c9945492SAndroid Build Coastguard Worker {
126*c9945492SAndroid Build Coastguard Worker return exp(x);
127*c9945492SAndroid Build Coastguard Worker }
128*c9945492SAndroid Build Coastguard Worker #endif
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