[thirdparty/glibc.git] / sysdeps / ieee754 / ldbl-128 / e_expl.c

/* Quad-precision floating point e^x.
   Copyright (C) 1999-2015 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, see
   <http://www.gnu.org/licenses/>.  */

/* 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 <stdlib.h>
#include "t_expl.h"

static const long double C[] = {
/* Smallest integer x for which e^x overflows.  */
#define himark C[0]
 11356.523406294143949491931077970765L,

/* Largest integer x for which e^x underflows.  */
#define lomark C[1]
-11433.4627433362978788372438434526231L,

/* 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-4900L,

/* 2^16383 */
#define TWO16383 C[9]
 5.94865747678615882542879663314003565E+4931L,

/* 256 */
#define TWO8 C[10]
 256.0L,

/* 32768 */
#define TWO15 C[11]
 32768.0L,

/* Chebyshev polynom coefficients 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;
      long double x22, n, t, result, xl;
      union ieee854_long_double ex2_u, scale_u;
      fenv_t oldenv;

      feholdexcept (&oldenv);
#ifdef FE_TONEAREST
      fesetround (FE_TONEAREST);
#endif

      /* Calculate n.  */
      n = x * M_1_LN2 + THREEp111;
      n -= THREEp111;
      x = x - n * M_LN2_0;
      xl = n * M_LN2_1;

      /* Calculate t/256.  */
      t = x + THREEp103;
      t -= THREEp103;

      /* Compute tval1 = t.  */
      tval1 = (int) (t * TWO8);

      x -= __expl_table[T_EXPL_ARG1+2*tval1];
      xl -= __expl_table[T_EXPL_ARG1+2*tval1+1];

      /* Calculate t/32768.  */
      t = x + THREEp96;
      t -= THREEp96;

      /* Compute tval2 = t.  */
      tval2 = (int) (t * 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 = abs(n_i) >= 15000;
      ex2_u.ieee.exponent += n_i >> unsafe;

      /* 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 (isinf (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 TWO16383*x;
}
strong_alias (__ieee754_expl, __expl_finite)
Commit	Line	Data
652df710	1	/* Quad-precision floating point e^x.
b168057a	2	Copyright (C) 1999-2015 Free Software Foundation, Inc.
652df710 UD	3	This file is part of the GNU C Library.
	4	Contributed by Jakub Jelinek <jj@ultra.linux.cz>
	5	Partly based on double-precision code
	6	by Geoffrey Keating <geoffk@ozemail.com.au>
	7
	8	The GNU C Library is free software; you can redistribute it and/or
41bdb6e2 AJ	9	modify it under the terms of the GNU Lesser General Public
	10	License as published by the Free Software Foundation; either
	11	version 2.1 of the License, or (at your option) any later version.
652df710 UD	12
	13	The GNU C Library is distributed in the hope that it will be useful,
	14	but WITHOUT ANY WARRANTY; without even the implied warranty of
	15	MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the GNU
41bdb6e2	16	Lesser General Public License for more details.
652df710	17
41bdb6e2	18	You should have received a copy of the GNU Lesser General Public
59ba27a6 PE	19	License along with the GNU C Library; if not, see
59ba27a6 PE	20	<http://www.gnu.org/licenses/>. */
652df710 UD	21
	22	/* The basic design here is from
	23	Abraham Ziv, "Fast Evaluation of Elementary Mathematical Functions with
	24	Correctly Rounded Last Bit", ACM Trans. Math. Soft., 17 (3), September 1991,
	25	pp. 410-423.
	26
	27	We work with number pairs where the first number is the high part and
	28	the second one is the low part. Arithmetic with the high part numbers must
	29	be exact, without any roundoff errors.
	30
	31	The input value, X, is written as
	32	X = n * ln(2)_0 + arg1[t1]_0 + arg2[t2]_0 + x
	33	- n * ln(2)_1 + arg1[t1]_1 + arg2[t2]_1 + xl
	34
	35	where:
	36	- n is an integer, 16384 >= n >= -16495;
	37	- ln(2)_0 is the first 93 bits of ln(2), and \|ln(2)_0-ln(2)-ln(2)_1\| < 2^-205
	38	- t1 is an integer, 89 >= t1 >= -89
	39	- t2 is an integer, 65 >= t2 >= -65
	40	- \|arg1[t1]-t1/256.0\| < 2^-53
	41	- \|arg2[t2]-t2/32768.0\| < 2^-53
	42	- x + xl is whatever is left, \|x + xl\| < 2^-16 + 2^-53
	43
	44	Then e^x is approximated as
	45
	46	e^x = 2^n_1 ( 2^n_0 e^(arg1[t1]_0 + arg1[t1]_1) e^(arg2[t2]_0 + arg2[t2]_1)
	47	+ 2^n_0 e^(arg1[t1]_0 + arg1[t1]_1) e^(arg2[t2]_0 + arg2[t2]_1)
	48	* p (x + xl + n * ln(2)_1))
	49	where:
	50	- p(x) is a polynomial approximating e(x)-1
	51	- e^(arg1[t1]_0 + arg1[t1]_1) is obtained from a table
	52	- e^(arg2[t2]_0 + arg2[t2]_1) likewise
	53	- n_1 + n_0 = n, so that \|n_0\| < -LDBL_MIN_EXP-1.
	54
	55	If it happens that n_1 == 0 (this is the usual case), that multiplication
	56	is omitted.
	57	*/
	58
	59	#ifndef _GNU_SOURCE
	60	#define _GNU_SOURCE
	61	#endif
	62	#include <float.h>
	63	#include <ieee754.h>
	64	#include <math.h>
	65	#include <fenv.h>
	66	#include <inttypes.h>
	67	#include <math_private.h>
e853ea00	68	#include <stdlib.h>
652df710 UD	69	#include "t_expl.h"
	70
	71	static const long double C[] = {
	72	/* Smallest integer x for which e^x overflows. */
	73	#define himark C[0]
	74	11356.523406294143949491931077970765L,
bcf01e6d	75
652df710 UD	76	/* Largest integer x for which e^x underflows. */
	77	#define lomark C[1]
	78	-11433.4627433362978788372438434526231L,
	79
	80	/* 3x2^96 */
	81	#define THREEp96 C[2]
	82	59421121885698253195157962752.0L,
	83
	84	/* 3x2^103 */
	85	#define THREEp103 C[3]
	86	30423614405477505635920876929024.0L,
	87
	88	/* 3x2^111 */
	89	#define THREEp111 C[4]
	90	7788445287802241442795744493830144.0L,
	91
	92	/* 1/ln(2) */
	93	#define M_1_LN2 C[5]
	94	1.44269504088896340735992468100189204L,
	95
	96	/* first 93 bits of ln(2) */
	97	#define M_LN2_0 C[6]
	98	0.693147180559945309417232121457981864L,
	99
	100	/* ln2_0 - ln(2) */
	101	#define M_LN2_1 C[7]
	102	-1.94704509238074995158795957333327386E-31L,
	103
	104	/* very small number */
	105	#define TINY C[8]
	106	1.0e-4900L,
	107
	108	/* 2^16383 */
	109	#define TWO16383 C[9]
	110	5.94865747678615882542879663314003565E+4931L,
	111
	112	/* 256 */
	113	#define TWO8 C[10]
	114	256.0L,
	115
	116	/* 32768 */
	117	#define TWO15 C[11]
	118	32768.0L,
	119
382466e0	120	/* Chebyshev polynom coefficients for (exp(x)-1)/x */
652df710 UD	121	#define P1 C[12]
	122	#define P2 C[13]
	123	#define P3 C[14]
	124	#define P4 C[15]
	125	#define P5 C[16]
	126	#define P6 C[17]
	127	0.5L,
	128	1.66666666666666666666666666666666683E-01L,
	129	4.16666666666666666666654902320001674E-02L,
	130	8.33333333333333333333314659767198461E-03L,
	131	1.38888888889899438565058018857254025E-03L,
	132	1.98412698413981650382436541785404286E-04L,
	133	};
	134
	135	long double
	136	__ieee754_expl (long double x)
	137	{
	138	/* Check for usual case. */
	139	if (isless (x, himark) && isgreater (x, lomark))
	140	{
	141	int tval1, tval2, unsafe, n_i;
	142	long double x22, n, t, result, xl;
	143	union ieee854_long_double ex2_u, scale_u;
	144	fenv_t oldenv;
	145
	146	feholdexcept (&oldenv);
	147	#ifdef FE_TONEAREST
	148	fesetround (FE_TONEAREST);
	149	#endif
	150
	151	/* Calculate n. */
	152	n = x * M_1_LN2 + THREEp111;
	153	n -= THREEp111;
	154	x = x - n * M_LN2_0;
	155	xl = n * M_LN2_1;
	156
	157	/* Calculate t/256. */
	158	t = x + THREEp103;
	159	t -= THREEp103;
	160
	161	/* Compute tval1 = t. */
	162	tval1 = (int) (t * TWO8);
	163
	164	x -= __expl_table[T_EXPL_ARG1+2*tval1];
	165	xl -= __expl_table[T_EXPL_ARG1+2*tval1+1];
	166
	167	/* Calculate t/32768. */
	168	t = x + THREEp96;
	169	t -= THREEp96;
	170
	171	/* Compute tval2 = t. */
	172	tval2 = (int) (t * TWO15);
	173
	174	x -= __expl_table[T_EXPL_ARG2+2*tval2];
	175	xl -= __expl_table[T_EXPL_ARG2+2*tval2+1];
	176
	177	x = x + xl;
	178
	179	/* Compute ex2 = 2^n_0 e^(argtable[tval1]) e^(argtable[tval2]). */
	180	ex2_u.d = __expl_table[T_EXPL_RES1 + tval1]
	181	* __expl_table[T_EXPL_RES2 + tval2];
	182	n_i = (int)n;
	183	/* 'unsafe' is 1 iff n_1 != 0. */
7e6424e3	184	unsafe = abs(n_i) >= 15000;
652df710 UD	185	ex2_u.ieee.exponent += n_i >> unsafe;
	186
	187	/* Compute scale = 2^n_1. */
	188	scale_u.d = 1.0L;
	189	scale_u.ieee.exponent += n_i - (n_i >> unsafe);
	190
	191	/* Approximate e^x2 - 1, using a seventh-degree polynomial,
	192	with maximum error in [-2^-16-2^-53,2^-16+2^-53]
	193	less than 4.8e-39. */
	194	x22 = x + xx(P1+x(P2+x(P3+x(P4+x(P5+x*P6)))));
	195
	196	/* Return result. */
	197	fesetenv (&oldenv);
	198
	199	result = x22 * ex2_u.d + ex2_u.d;
	200
	201	/* Now we can test whether the result is ultimate or if we are unsure.
	202	In the later case we should probably call a mpn based routine to give
	203	the ultimate result.
	204	Empirically, this routine is already ultimate in about 99.9986% of
	205	cases, the test below for the round to nearest case will be false
	206	in ~ 99.9963% of cases.
	207	Without proc2 routine maximum error which has been seen is
	208	0.5000262 ulp.
	209
	210	union ieee854_long_double ex3_u;
	211
	212	#ifdef FE_TONEAREST
	213	fesetround (FE_TONEAREST);
	214	#endif
	215	ex3_u.d = (result - ex2_u.d) - x22 * ex2_u.d;
	216	ex2_u.d = result;
	217	ex3_u.ieee.exponent += LDBL_MANT_DIG + 15 + IEEE854_LONG_DOUBLE_BIAS
350635a5	218	- ex2_u.ieee.exponent;
652df710 UD	219	n_i = abs (ex3_u.d);
	220	n_i = (n_i + 1) / 2;
	221	fesetenv (&oldenv);
	222	#ifdef FE_TONEAREST
	223	if (fegetround () == FE_TONEAREST)
	224	n_i -= 0x4000;
	225	#endif
	226	if (!n_i) {
	227	return __ieee754_expl_proc2 (origx);
	228	}
	229	*/
	230	if (!unsafe)
	231	return result;
	232	else
	233	return result * scale_u.d;
	234	}
	235	/* Exceptional cases: */
	236	else if (isless (x, himark))
	237	{
d81f90cc	238	if (isinf (x))
652df710 UD	239	/* e^-inf == 0, with no error. */
	240	return 0;
	241	else
	242	/* Underflow */
	243	return TINY * TINY;
	244	}
	245	else
	246	/* Return x, if x is a NaN or Inf; or overflow, otherwise. */
	247	return TWO16383*x;
	248	}
bcf01e6d	249	strong_alias (__ieee754_expl, __expl_finite)