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

/* Quad-precision floating point e^x.
   Copyright (C) 1999-2021 Free Software Foundation, Inc.
   This file is part of the GNU C Library.

   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
   <https://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-barriers.h>
#include <math_private.h>
#include <math-underflow.h>
#include <stdlib.h>
#include "t_expl.h"
#include <libm-alias-finite.h>

static const _Float128 C[] = {
/* Smallest integer x for which e^x overflows.  */
#define himark C[0]
 L(11356.523406294143949491931077970765),

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

/* 3x2^96 */
#define THREEp96 C[2]
 L(59421121885698253195157962752.0),

/* 3x2^103 */
#define THREEp103 C[3]
 L(30423614405477505635920876929024.0),

/* 3x2^111 */
#define THREEp111 C[4]
 L(7788445287802241442795744493830144.0),

/* 1/ln(2) */
#define M_1_LN2 C[5]
 L(1.44269504088896340735992468100189204),

/* first 93 bits of ln(2) */
#define M_LN2_0 C[6]
 L(0.693147180559945309417232121457981864),

/* ln2_0 - ln(2) */
#define M_LN2_1 C[7]
L(-1.94704509238074995158795957333327386E-31),

/* very small number */
#define TINY C[8]
 L(1.0e-4900),

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

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

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

/* 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]
 L(0.5),
 L(1.66666666666666666666666666666666683E-01),
 L(4.16666666666666666666654902320001674E-02),
 L(8.33333333333333333333314659767198461E-03),
 L(1.38888888889899438565058018857254025E-03),
 L(1.98412698413981650382436541785404286E-04),
};

_Float128
__ieee754_expl (_Float128 x)
{
  /* Check for usual case.  */
  if (isless (x, himark) && isgreater (x, lomark))
    {
      int tval1, tval2, unsafe, n_i;
      _Float128 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;
      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)))));
      math_force_eval (x22);

      /* 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
	{
	  result *= scale_u.d;
	  math_check_force_underflow_nonneg (result);
	  return result;
	}
    }
  /* 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;
}
libm_alias_finite (__ieee754_expl, __expl)
Commit	Line	Data
652df710	1	/* Quad-precision floating point e^x.
2b778ceb	2	Copyright (C) 1999-2021 Free Software Foundation, Inc.
652df710	3	This file is part of the GNU C Library.
652df710 UD	4
652df710 UD	5	The GNU C Library is free software; you can redistribute it and/or
41bdb6e2 AJ	6	modify it under the terms of the GNU Lesser General Public
	7	License as published by the Free Software Foundation; either
	8	version 2.1 of the License, or (at your option) any later version.
652df710 UD	9
	10	The GNU C Library is distributed in the hope that it will be useful,
	11	but WITHOUT ANY WARRANTY; without even the implied warranty of
	12	MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the GNU
41bdb6e2	13	Lesser General Public License for more details.
652df710	14
41bdb6e2	15	You should have received a copy of the GNU Lesser General Public
59ba27a6	16	License along with the GNU C Library; if not, see
5a82c748	17	<https://www.gnu.org/licenses/>. */
652df710 UD	18
	19	/* The basic design here is from
	20	Abraham Ziv, "Fast Evaluation of Elementary Mathematical Functions with
	21	Correctly Rounded Last Bit", ACM Trans. Math. Soft., 17 (3), September 1991,
	22	pp. 410-423.
	23
	24	We work with number pairs where the first number is the high part and
	25	the second one is the low part. Arithmetic with the high part numbers must
	26	be exact, without any roundoff errors.
	27
	28	The input value, X, is written as
	29	X = n * ln(2)_0 + arg1[t1]_0 + arg2[t2]_0 + x
	30	- n * ln(2)_1 + arg1[t1]_1 + arg2[t2]_1 + xl
	31
	32	where:
	33	- n is an integer, 16384 >= n >= -16495;
	34	- ln(2)_0 is the first 93 bits of ln(2), and \|ln(2)_0-ln(2)-ln(2)_1\| < 2^-205
	35	- t1 is an integer, 89 >= t1 >= -89
	36	- t2 is an integer, 65 >= t2 >= -65
	37	- \|arg1[t1]-t1/256.0\| < 2^-53
	38	- \|arg2[t2]-t2/32768.0\| < 2^-53
	39	- x + xl is whatever is left, \|x + xl\| < 2^-16 + 2^-53
	40
	41	Then e^x is approximated as
	42
	43	e^x = 2^n_1 ( 2^n_0 e^(arg1[t1]_0 + arg1[t1]_1) e^(arg2[t2]_0 + arg2[t2]_1)
	44	+ 2^n_0 e^(arg1[t1]_0 + arg1[t1]_1) e^(arg2[t2]_0 + arg2[t2]_1)
	45	* p (x + xl + n * ln(2)_1))
	46	where:
	47	- p(x) is a polynomial approximating e(x)-1
	48	- e^(arg1[t1]_0 + arg1[t1]_1) is obtained from a table
	49	- e^(arg2[t2]_0 + arg2[t2]_1) likewise
	50	- n_1 + n_0 = n, so that \|n_0\| < -LDBL_MIN_EXP-1.
	51
	52	If it happens that n_1 == 0 (this is the usual case), that multiplication
	53	is omitted.
	54	*/
	55
	56	#ifndef _GNU_SOURCE
	57	#define _GNU_SOURCE
	58	#endif
	59	#include <float.h>
	60	#include <ieee754.h>
	61	#include <math.h>
	62	#include <fenv.h>
	63	#include <inttypes.h>
b4d5b8b0	64	#include <math-barriers.h>
652df710	65	#include <math_private.h>
8f5b00d3	66	#include <math-underflow.h>
e853ea00	67	#include <stdlib.h>
652df710	68	#include "t_expl.h"
220622dd	69	#include <libm-alias-finite.h>
652df710	70
15089e04	71	static const _Float128 C[] = {
652df710 UD	72	/* Smallest integer x for which e^x overflows. */
652df710 UD	73	#define himark C[0]
02bbfb41	74	L(11356.523406294143949491931077970765),
bcf01e6d	75
652df710 UD	76	/* Largest integer x for which e^x underflows. */
652df710 UD	77	#define lomark C[1]
02bbfb41	78	L(-11433.4627433362978788372438434526231),
652df710 UD	79
	80	/* 3x2^96 */
	81	#define THREEp96 C[2]
02bbfb41	82	L(59421121885698253195157962752.0),
652df710 UD	83
	84	/* 3x2^103 */
	85	#define THREEp103 C[3]
02bbfb41	86	L(30423614405477505635920876929024.0),
652df710 UD	87
	88	/* 3x2^111 */
	89	#define THREEp111 C[4]
02bbfb41	90	L(7788445287802241442795744493830144.0),
652df710 UD	91
	92	/* 1/ln(2) */
	93	#define M_1_LN2 C[5]
02bbfb41	94	L(1.44269504088896340735992468100189204),
652df710 UD	95
	96	/* first 93 bits of ln(2) */
	97	#define M_LN2_0 C[6]
02bbfb41	98	L(0.693147180559945309417232121457981864),
652df710 UD	99
	100	/* ln2_0 - ln(2) */
	101	#define M_LN2_1 C[7]
02bbfb41	102	L(-1.94704509238074995158795957333327386E-31),
652df710 UD	103
	104	/* very small number */
	105	#define TINY C[8]
02bbfb41	106	L(1.0e-4900),
652df710 UD	107
	108	/* 2^16383 */
	109	#define TWO16383 C[9]
02bbfb41	110	L(5.94865747678615882542879663314003565E+4931),
652df710 UD	111
	112	/* 256 */
	113	#define TWO8 C[10]
02bbfb41	114	256,
652df710 UD	115
	116	/* 32768 */
	117	#define TWO15 C[11]
02bbfb41	118	32768,
652df710	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]
02bbfb41 PM	127	L(0.5),
	128	L(1.66666666666666666666666666666666683E-01),
	129	L(4.16666666666666666666654902320001674E-02),
	130	L(8.33333333333333333333314659767198461E-03),
	131	L(1.38888888889899438565058018857254025E-03),
	132	L(1.98412698413981650382436541785404286E-04),
652df710 UD	133	};
652df710 UD	134
15089e04 PM	135	_Float128
15089e04 PM	136	__ieee754_expl (_Float128 x)
652df710 UD	137	{
	138	/* Check for usual case. */
	139	if (isless (x, himark) && isgreater (x, lomark))
	140	{
	141	int tval1, tval2, unsafe, n_i;
15089e04	142	_Float128 x22, n, t, result, xl;
652df710 UD	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. */
02bbfb41	188	scale_u.d = 1;
652df710 UD	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)))));
a4c9be1b	195	math_force_eval (x22);
652df710 UD	196
	197	/* Return result. */
	198	fesetenv (&oldenv);
	199
	200	result = x22 * ex2_u.d + ex2_u.d;
	201
	202	/* Now we can test whether the result is ultimate or if we are unsure.
	203	In the later case we should probably call a mpn based routine to give
	204	the ultimate result.
	205	Empirically, this routine is already ultimate in about 99.9986% of
	206	cases, the test below for the round to nearest case will be false
	207	in ~ 99.9963% of cases.
	208	Without proc2 routine maximum error which has been seen is
	209	0.5000262 ulp.
	210
	211	union ieee854_long_double ex3_u;
	212
	213	#ifdef FE_TONEAREST
	214	fesetround (FE_TONEAREST);
	215	#endif
	216	ex3_u.d = (result - ex2_u.d) - x22 * ex2_u.d;
	217	ex2_u.d = result;
	218	ex3_u.ieee.exponent += LDBL_MANT_DIG + 15 + IEEE854_LONG_DOUBLE_BIAS
350635a5	219	- ex2_u.ieee.exponent;
652df710 UD	220	n_i = abs (ex3_u.d);
	221	n_i = (n_i + 1) / 2;
	222	fesetenv (&oldenv);
	223	#ifdef FE_TONEAREST
	224	if (fegetround () == FE_TONEAREST)
	225	n_i -= 0x4000;
	226	#endif
	227	if (!n_i) {
	228	return __ieee754_expl_proc2 (origx);
	229	}
	230	*/
	231	if (!unsafe)
	232	return result;
	233	else
8475ab16 JM	234	{
8475ab16 JM	235	result *= scale_u.d;
d96164c3	236	math_check_force_underflow_nonneg (result);
8475ab16 JM	237	return result;
8475ab16 JM	238	}
652df710 UD	239	}
	240	/* Exceptional cases: */
	241	else if (isless (x, himark))
	242	{
d81f90cc	243	if (isinf (x))
652df710 UD	244	/* e^-inf == 0, with no error. */
	245	return 0;
	246	else
	247	/* Underflow */
	248	return TINY * TINY;
	249	}
	250	else
	251	/* Return x, if x is a NaN or Inf; or overflow, otherwise. */
	252	return TWO16383*x;
	253	}
220622dd	254	libm_alias_finite (__ieee754_expl, __expl)