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- /* origin: OpenBSD /usr/src/lib/libm/src/ld80/e_expl.c */
- /*
- * Copyright (c) 2008 Stephen L. Moshier <[email protected]>
- *
- * Permission to use, copy, modify, and distribute this software for any
- * purpose with or without fee is hereby granted, provided that the above
- * copyright notice and this permission notice appear in all copies.
- *
- * THE SOFTWARE IS PROVIDED "AS IS" AND THE AUTHOR DISCLAIMS ALL WARRANTIES
- * WITH REGARD TO THIS SOFTWARE INCLUDING ALL IMPLIED WARRANTIES OF
- * MERCHANTABILITY AND FITNESS. IN NO EVENT SHALL THE AUTHOR BE LIABLE FOR
- * ANY SPECIAL, DIRECT, INDIRECT, OR CONSEQUENTIAL DAMAGES OR ANY DAMAGES
- * WHATSOEVER RESULTING FROM LOSS OF USE, DATA OR PROFITS, WHETHER IN AN
- * ACTION OF CONTRACT, NEGLIGENCE OR OTHER TORTIOUS ACTION, ARISING OUT OF
- * OR IN CONNECTION WITH THE USE OR PERFORMANCE OF THIS SOFTWARE.
- */
- /*
- * Exponential function, long double precision
- *
- *
- * SYNOPSIS:
- *
- * long double x, y, expl();
- *
- * y = expl( x );
- *
- *
- * DESCRIPTION:
- *
- * Returns e (2.71828...) raised to the x power.
- *
- * Range reduction is accomplished by separating the argument
- * into an integer k and fraction f such that
- *
- * x k f
- * e = 2 e.
- *
- * A Pade' form of degree 2/3 is used to approximate exp(f) - 1
- * in the basic range [-0.5 ln 2, 0.5 ln 2].
- *
- *
- * ACCURACY:
- *
- * Relative error:
- * arithmetic domain # trials peak rms
- * IEEE +-10000 50000 1.12e-19 2.81e-20
- *
- *
- * Error amplification in the exponential function can be
- * a serious matter. The error propagation involves
- * exp( X(1+delta) ) = exp(X) ( 1 + X*delta + ... ),
- * which shows that a 1 lsb error in representing X produces
- * a relative error of X times 1 lsb in the function.
- * While the routine gives an accurate result for arguments
- * that are exactly represented by a long double precision
- * computer number, the result contains amplified roundoff
- * error for large arguments not exactly represented.
- *
- *
- * ERROR MESSAGES:
- *
- * message condition value returned
- * exp underflow x < MINLOG 0.0
- * exp overflow x > MAXLOG MAXNUM
- *
- */
- #include "libm.h"
- #if LDBL_MANT_DIG == 53 && LDBL_MAX_EXP == 1024
- long double expl(long double x)
- {
- return exp(x);
- }
- #elif LDBL_MANT_DIG == 64 && LDBL_MAX_EXP == 16384
- static const long double P[3] = {
- 1.2617719307481059087798E-4L,
- 3.0299440770744196129956E-2L,
- 9.9999999999999999991025E-1L,
- };
- static const long double Q[4] = {
- 3.0019850513866445504159E-6L,
- 2.5244834034968410419224E-3L,
- 2.2726554820815502876593E-1L,
- 2.0000000000000000000897E0L,
- };
- static const long double
- C1 = 6.9314575195312500000000E-1L,
- C2 = 1.4286068203094172321215E-6L,
- MAXLOGL = 1.1356523406294143949492E4L,
- MINLOGL = -1.13994985314888605586758E4L,
- LOG2EL = 1.4426950408889634073599E0L;
- long double expl(long double x)
- {
- long double px, xx;
- int n;
- if (isnan(x))
- return x;
- if (x > MAXLOGL)
- return INFINITY;
- if (x < MINLOGL)
- return 0.0;
- /* Express e**x = e**g 2**n
- * = e**g e**(n loge(2))
- * = e**(g + n loge(2))
- */
- px = floorl(LOG2EL * x + 0.5); /* floor() truncates toward -infinity. */
- n = px;
- x -= px * C1;
- x -= px * C2;
- /* rational approximation for exponential
- * of the fractional part:
- * e**x = 1 + 2x P(x**2)/(Q(x**2) - P(x**2))
- */
- xx = x * x;
- px = x * __polevll(xx, P, 2);
- x = px/(__polevll(xx, Q, 3) - px);
- x = 1.0 + 2.0 * x;
- x = scalbnl(x, n);
- return x;
- }
- #endif
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