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+/* origin: OpenBSD /usr/src/lib/libm/src/ld80/e_expl.c */
+/*
+ * Copyright (c) 2008 Stephen L. Moshier <steve@moshier.net>
+ *
+ * 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 long double P[3] = {
+ 1.2617719307481059087798E-4L,
+ 3.0299440770744196129956E-2L,
+ 9.9999999999999999991025E-1L,
+};
+static 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 0.0L;
+
+ /* Express e**x = e**g 2**n
+ * = e**g e**(n loge(2))
+ * = e**(g + n loge(2))
+ */
+ px = floorl(LOG2EL * x + 0.5L); /* 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 = 1.0L + ldexpl(x, 1);
+ x = ldexpl(x, n);
+ return x;
+}
+#endif