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+/* Quad-precision floating point e^x.
+ Copyright (C) 1999,2004,2006 Free Software Foundation, Inc.
+ This file is part of the GNU C Library.
+ Contributed by Jakub Jelinek <jj@ultra.linux.cz>
+ Partly based on double-precision code
+ by Geoffrey Keating <geoffk@ozemail.com.au>
+
+ The GNU C Library is free software; you can redistribute it and/or
+ modify it under the terms of the GNU Lesser General Public
+ License as published by the Free Software Foundation; either
+ version 2.1 of the License, or (at your option) any later version.
+
+ The GNU C Library is distributed in the hope that it will be useful,
+ but WITHOUT ANY WARRANTY; without even the implied warranty of
+ MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the GNU
+ Lesser General Public License for more details.
+
+ You should have received a copy of the GNU Lesser General Public
+ License along with the GNU C Library; if not, write to the Free
+ Software Foundation, Inc., 59 Temple Place, Suite 330, Boston, MA
+ 02111-1307 USA. */
+
+/* The basic design here is from
+ Abraham Ziv, "Fast Evaluation of Elementary Mathematical Functions with
+ Correctly Rounded Last Bit", ACM Trans. Math. Soft., 17 (3), September 1991,
+ pp. 410-423.
+
+ We work with number pairs where the first number is the high part and
+ the second one is the low part. Arithmetic with the high part numbers must
+ be exact, without any roundoff errors.
+
+ The input value, X, is written as
+ X = n * ln(2)_0 + arg1[t1]_0 + arg2[t2]_0 + x
+ - n * ln(2)_1 + arg1[t1]_1 + arg2[t2]_1 + xl
+
+ where:
+ - n is an integer, 16384 >= n >= -16495;
+ - ln(2)_0 is the first 93 bits of ln(2), and |ln(2)_0-ln(2)-ln(2)_1| < 2^-205
+ - t1 is an integer, 89 >= t1 >= -89
+ - t2 is an integer, 65 >= t2 >= -65
+ - |arg1[t1]-t1/256.0| < 2^-53
+ - |arg2[t2]-t2/32768.0| < 2^-53
+ - x + xl is whatever is left, |x + xl| < 2^-16 + 2^-53
+
+ Then e^x is approximated as
+
+ e^x = 2^n_1 ( 2^n_0 e^(arg1[t1]_0 + arg1[t1]_1) e^(arg2[t2]_0 + arg2[t2]_1)
+ + 2^n_0 e^(arg1[t1]_0 + arg1[t1]_1) e^(arg2[t2]_0 + arg2[t2]_1)
+ * p (x + xl + n * ln(2)_1))
+ where:
+ - p(x) is a polynomial approximating e(x)-1
+ - e^(arg1[t1]_0 + arg1[t1]_1) is obtained from a table
+ - e^(arg2[t2]_0 + arg2[t2]_1) likewise
+ - n_1 + n_0 = n, so that |n_0| < -LDBL_MIN_EXP-1.
+
+ If it happens that n_1 == 0 (this is the usual case), that multiplication
+ is omitted.
+ */
+
+#ifndef _GNU_SOURCE
+#define _GNU_SOURCE
+#endif
+#include <float.h>
+#include <ieee754.h>
+#include <math.h>
+#include <fenv.h>
+#include <inttypes.h>
+#include <math_private.h>
+#include <sysdeps/ieee754/ldbl-128/t_expl.h>
+
+static const long double C[] = {
+/* Smallest integer x for which e^x overflows. */
+#define himark C[0]
+ 709.08956571282405153382846025171462914L,
+
+/* Largest integer x for which e^x underflows. */
+#define lomark C[1]
+-709.08956571282405153382846025171462914L,
+
+/* 3x2^96 */
+#define THREEp96 C[2]
+ 59421121885698253195157962752.0L,
+
+/* 3x2^103 */
+#define THREEp103 C[3]
+ 30423614405477505635920876929024.0L,
+
+/* 3x2^111 */
+#define THREEp111 C[4]
+ 7788445287802241442795744493830144.0L,
+
+/* 1/ln(2) */
+#define M_1_LN2 C[5]
+ 1.44269504088896340735992468100189204L,
+
+/* first 93 bits of ln(2) */
+#define M_LN2_0 C[6]
+ 0.693147180559945309417232121457981864L,
+
+/* ln2_0 - ln(2) */
+#define M_LN2_1 C[7]
+-1.94704509238074995158795957333327386E-31L,
+
+/* very small number */
+#define TINY C[8]
+ 1.0e-308L,
+
+/* 2^16383 */
+#define TWO1023 C[9]
+ 8.988465674311579538646525953945123668E+307L,
+
+/* 256 */
+#define TWO8 C[10]
+ 256.0L,
+
+/* 32768 */
+#define TWO15 C[11]
+ 32768.0L,
+
+/* Chebyshev polynom coeficients for (exp(x)-1)/x */
+#define P1 C[12]
+#define P2 C[13]
+#define P3 C[14]
+#define P4 C[15]
+#define P5 C[16]
+#define P6 C[17]
+ 0.5L,
+ 1.66666666666666666666666666666666683E-01L,
+ 4.16666666666666666666654902320001674E-02L,
+ 8.33333333333333333333314659767198461E-03L,
+ 1.38888888889899438565058018857254025E-03L,
+ 1.98412698413981650382436541785404286E-04L,
+};
+
+long double
+__ieee754_expl (long double x)
+{
+ /* Check for usual case. */
+ if (isless (x, himark) && isgreater (x, lomark))
+ {
+ int tval1, tval2, unsafe, n_i, exponent2;
+ long double x22, n, result, xl;
+ union ibm_extended_long_double ex2_u, scale_u;
+ fenv_t oldenv;
+
+ feholdexcept (&oldenv);
+#ifdef FE_TONEAREST
+ fesetround (FE_TONEAREST);
+#endif
+
+ n = roundl(x*M_1_LN2);
+ x = x-n*M_LN2_0;
+ xl = n*M_LN2_1;
+
+ tval1 = roundl(x*TWO8);
+ x -= __expl_table[T_EXPL_ARG1+2*tval1];
+ xl -= __expl_table[T_EXPL_ARG1+2*tval1+1];
+
+ tval2 = roundl(x*TWO15);
+ x -= __expl_table[T_EXPL_ARG2+2*tval2];
+ xl -= __expl_table[T_EXPL_ARG2+2*tval2+1];
+
+ x = x + xl;
+
+ /* Compute ex2 = 2^n_0 e^(argtable[tval1]) e^(argtable[tval2]). */
+ ex2_u.d = __expl_table[T_EXPL_RES1 + tval1]
+ * __expl_table[T_EXPL_RES2 + tval2];
+ n_i = (int)n;
+ /* 'unsafe' is 1 iff n_1 != 0. */
+ unsafe = fabsl(n_i) >= -LDBL_MIN_EXP - 1;
+ ex2_u.ieee.exponent += n_i >> unsafe;
+ /* Fortunately, there are no subnormal lowpart doubles in
+ __expl_table, only normal values and zeros.
+ But after scaling it can be subnormal. */
+ exponent2 = ex2_u.ieee.exponent2 + (n_i >> unsafe);
+ if (ex2_u.ieee.exponent2 == 0)
+ /* assert ((ex2_u.ieee.mantissa2|ex2_u.ieee.mantissa3) == 0) */;
+ else if (exponent2 > 0)
+ ex2_u.ieee.exponent2 = exponent2;
+ else if (exponent2 <= -54)
+ {
+ ex2_u.ieee.exponent2 = 0;
+ ex2_u.ieee.mantissa2 = 0;
+ ex2_u.ieee.mantissa3 = 0;
+ }
+ else
+ {
+ static const double
+ two54 = 1.80143985094819840000e+16, /* 4350000000000000 */
+ twom54 = 5.55111512312578270212e-17; /* 3C90000000000000 */
+ ex2_u.dd[1] *= two54;
+ ex2_u.ieee.exponent2 += n_i >> unsafe;
+ ex2_u.dd[1] *= twom54;
+ }
+
+ /* Compute scale = 2^n_1. */
+ scale_u.d = 1.0L;
+ scale_u.ieee.exponent += n_i - (n_i >> unsafe);
+
+ /* Approximate e^x2 - 1, using a seventh-degree polynomial,
+ with maximum error in [-2^-16-2^-53,2^-16+2^-53]
+ less than 4.8e-39. */
+ x22 = x + x*x*(P1+x*(P2+x*(P3+x*(P4+x*(P5+x*P6)))));
+
+ /* Return result. */
+ fesetenv (&oldenv);
+
+ result = x22 * ex2_u.d + ex2_u.d;
+
+ /* Now we can test whether the result is ultimate or if we are unsure.
+ In the later case we should probably call a mpn based routine to give
+ the ultimate result.
+ Empirically, this routine is already ultimate in about 99.9986% of
+ cases, the test below for the round to nearest case will be false
+ in ~ 99.9963% of cases.
+ Without proc2 routine maximum error which has been seen is
+ 0.5000262 ulp.
+
+ union ieee854_long_double ex3_u;
+
+ #ifdef FE_TONEAREST
+ fesetround (FE_TONEAREST);
+ #endif
+ ex3_u.d = (result - ex2_u.d) - x22 * ex2_u.d;
+ ex2_u.d = result;
+ ex3_u.ieee.exponent += LDBL_MANT_DIG + 15 + IEEE854_LONG_DOUBLE_BIAS
+ - ex2_u.ieee.exponent;
+ n_i = abs (ex3_u.d);
+ n_i = (n_i + 1) / 2;
+ fesetenv (&oldenv);
+ #ifdef FE_TONEAREST
+ if (fegetround () == FE_TONEAREST)
+ n_i -= 0x4000;
+ #endif
+ if (!n_i) {
+ return __ieee754_expl_proc2 (origx);
+ }
+ */
+ if (!unsafe)
+ return result;
+ else
+ return result * scale_u.d;
+ }
+ /* Exceptional cases: */
+ else if (isless (x, himark))
+ {
+ if (__isinfl (x))
+ /* e^-inf == 0, with no error. */
+ return 0;
+ else
+ /* Underflow */
+ return TINY * TINY;
+ }
+ else
+ /* Return x, if x is a NaN or Inf; or overflow, otherwise. */
+ return TWO1023*x;
+}
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