1/* Quad-precision floating point e^x.
2   Copyright (C) 1999-2018 Free Software Foundation, Inc.
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
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.
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
16   Lesser General Public License for more details.
17
18   You should have received a copy of the GNU Lesser General Public
19   License along with the GNU C Library; if not, see
20   <http://www.gnu.org/licenses/>.  */
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| < -FLT128_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
63#include "quadmath-imp.h"
64#include "expq_table.h"
65
66static const __float128 C[] = {
67/* Smallest integer x for which e^x overflows.  */
68#define himark C[0]
69 11356.523406294143949491931077970765Q,
70
71/* Largest integer x for which e^x underflows.  */
72#define lomark C[1]
73-11433.4627433362978788372438434526231Q,
74
75/* 3x2^96 */
76#define THREEp96 C[2]
77 59421121885698253195157962752.0Q,
78
79/* 3x2^103 */
80#define THREEp103 C[3]
81 30423614405477505635920876929024.0Q,
82
83/* 3x2^111 */
84#define THREEp111 C[4]
85 7788445287802241442795744493830144.0Q,
86
87/* 1/ln(2) */
88#define M_1_LN2 C[5]
89 1.44269504088896340735992468100189204Q,
90
91/* first 93 bits of ln(2) */
92#define M_LN2_0 C[6]
93 0.693147180559945309417232121457981864Q,
94
95/* ln2_0 - ln(2) */
96#define M_LN2_1 C[7]
97-1.94704509238074995158795957333327386E-31Q,
98
99/* very small number */
100#define TINY C[8]
101 1.0e-4900Q,
102
103/* 2^16383 */
104#define TWO16383 C[9]
105 5.94865747678615882542879663314003565E+4931Q,
106
107/* 256 */
108#define TWO8 C[10]
109 256,
110
111/* 32768 */
112#define TWO15 C[11]
113 32768,
114
115/* Chebyshev polynom coefficients for (exp(x)-1)/x */
116#define P1 C[12]
117#define P2 C[13]
118#define P3 C[14]
119#define P4 C[15]
120#define P5 C[16]
121#define P6 C[17]
122 0.5Q,
123 1.66666666666666666666666666666666683E-01Q,
124 4.16666666666666666666654902320001674E-02Q,
125 8.33333333333333333333314659767198461E-03Q,
126 1.38888888889899438565058018857254025E-03Q,
127 1.98412698413981650382436541785404286E-04Q,
128};
129
130__float128
131expq (__float128 x)
132{
133  /* Check for usual case.  */
134  if (__builtin_isless (x, himark) && __builtin_isgreater (x, lomark))
135    {
136      int tval1, tval2, unsafe, n_i;
137      __float128 x22, n, t, result, xl;
138      ieee854_float128 ex2_u, scale_u;
139      fenv_t oldenv;
140
141      feholdexcept (&oldenv);
142#ifdef FE_TONEAREST
143      fesetround (FE_TONEAREST);
144#endif
145
146      /* Calculate n.  */
147      n = x * M_1_LN2 + THREEp111;
148      n -= THREEp111;
149      x = x - n * M_LN2_0;
150      xl = n * M_LN2_1;
151
152      /* Calculate t/256.  */
153      t = x + THREEp103;
154      t -= THREEp103;
155
156      /* Compute tval1 = t.  */
157      tval1 = (int) (t * TWO8);
158
159      x -= __expq_table[T_EXPL_ARG1+2*tval1];
160      xl -= __expq_table[T_EXPL_ARG1+2*tval1+1];
161
162      /* Calculate t/32768.  */
163      t = x + THREEp96;
164      t -= THREEp96;
165
166      /* Compute tval2 = t.  */
167      tval2 = (int) (t * TWO15);
168
169      x -= __expq_table[T_EXPL_ARG2+2*tval2];
170      xl -= __expq_table[T_EXPL_ARG2+2*tval2+1];
171
172      x = x + xl;
173
174      /* Compute ex2 = 2^n_0 e^(argtable[tval1]) e^(argtable[tval2]).  */
175      ex2_u.value = __expq_table[T_EXPL_RES1 + tval1]
176		* __expq_table[T_EXPL_RES2 + tval2];
177      n_i = (int)n;
178      /* 'unsafe' is 1 iff n_1 != 0.  */
179      unsafe = abs(n_i) >= 15000;
180      ex2_u.ieee.exponent += n_i >> unsafe;
181
182      /* Compute scale = 2^n_1.  */
183      scale_u.value = 1;
184      scale_u.ieee.exponent += n_i - (n_i >> unsafe);
185
186      /* Approximate e^x2 - 1, using a seventh-degree polynomial,
187	 with maximum error in [-2^-16-2^-53,2^-16+2^-53]
188	 less than 4.8e-39.  */
189      x22 = x + x*x*(P1+x*(P2+x*(P3+x*(P4+x*(P5+x*P6)))));
190      math_force_eval (x22);
191
192      /* Return result.  */
193      fesetenv (&oldenv);
194
195      result = x22 * ex2_u.value + ex2_u.value;
196
197      /* Now we can test whether the result is ultimate or if we are unsure.
198	 In the later case we should probably call a mpn based routine to give
199	 the ultimate result.
200	 Empirically, this routine is already ultimate in about 99.9986% of
201	 cases, the test below for the round to nearest case will be false
202	 in ~ 99.9963% of cases.
203	 Without proc2 routine maximum error which has been seen is
204	 0.5000262 ulp.
205
206	  ieee854_float128 ex3_u;
207
208	  #ifdef FE_TONEAREST
209	    fesetround (FE_TONEAREST);
210	  #endif
211	  ex3_u.value = (result - ex2_u.value) - x22 * ex2_u.value;
212	  ex2_u.value = result;
213	  ex3_u.ieee.exponent += FLT128_MANT_DIG + 15 + IEEE854_FLOAT128_BIAS
214				 - ex2_u.ieee.exponent;
215	  n_i = abs (ex3_u.value);
216	  n_i = (n_i + 1) / 2;
217	  fesetenv (&oldenv);
218	  #ifdef FE_TONEAREST
219	  if (fegetround () == FE_TONEAREST)
220	    n_i -= 0x4000;
221	  #endif
222	  if (!n_i) {
223	    return __ieee754_expl_proc2 (origx);
224	  }
225       */
226      if (!unsafe)
227	return result;
228      else
229	{
230	  result *= scale_u.value;
231	  math_check_force_underflow_nonneg (result);
232	  return result;
233	}
234    }
235  /* Exceptional cases:  */
236  else if (__builtin_isless (x, himark))
237    {
238      if (isinfq (x))
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}
249