summaryrefslogtreecommitdiff
path: root/sysdeps/ieee754/ldbl-128/e_log2l.c
blob: cf4a380f16e5751a99adb6f9662295262ee5eb9b (plain)
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
22
23
24
25
26
27
28
29
30
31
32
33
34
35
36
37
38
39
40
41
42
43
44
45
46
47
48
49
50
51
52
53
54
55
56
57
58
59
60
61
62
63
64
65
66
67
68
69
70
71
72
73
74
75
76
77
78
79
80
81
82
83
84
85
86
87
88
89
90
91
92
93
94
95
96
97
98
99
100
101
102
103
104
105
106
107
108
109
110
111
112
113
114
115
116
117
118
119
120
121
122
123
124
125
126
127
128
129
130
131
132
133
134
135
136
137
138
139
140
141
142
143
144
145
146
147
148
149
150
151
152
153
154
155
156
157
158
159
160
161
162
163
164
165
166
167
168
169
170
171
172
173
174
175
176
177
178
179
180
181
182
183
184
185
186
187
188
189
190
191
192
193
194
195
196
197
198
199
200
201
202
203
204
205
206
207
208
209
210
211
212
213
214
215
216
217
218
219
220
221
222
223
224
225
226
227
228
229
230
231
232
233
234
235
236
237
238
239
240
241
242
243
244
245
246
247
248
249
250
251
252
/*                                                      log2l.c
 *      Base 2 logarithm, 128-bit long double precision
 *
 *
 *
 * SYNOPSIS:
 *
 * long double x, y, log2l();
 *
 * y = log2l( x );
 *
 *
 *
 * DESCRIPTION:
 *
 * Returns the base 2 logarithm of x.
 *
 * The argument is separated into its exponent and fractional
 * parts.  If the exponent is between -1 and +1, the (natural)
 * logarithm of the fraction is approximated by
 *
 *     log(1+x) = x - 0.5 x^2 + x^3 P(x)/Q(x).
 *
 * Otherwise, setting  z = 2(x-1)/x+1),
 *
 *     log(x) = z + z^3 P(z)/Q(z).
 *
 *
 *
 * ACCURACY:
 *
 *                      Relative error:
 * arithmetic   domain     # trials      peak         rms
 *    IEEE      0.5, 2.0     100,000    2.6e-34     4.9e-35
 *    IEEE     exp(+-10000)  100,000    9.6e-35     4.0e-35
 *
 * In the tests over the interval exp(+-10000), the logarithms
 * of the random arguments were uniformly distributed over
 * [-10000, +10000].
 *
 */

/*
   Cephes Math Library Release 2.2:  January, 1991
   Copyright 1984, 1991 by Stephen L. Moshier
   Adapted for glibc November, 2001

    This 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.

    This 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 this library; if not, see <http://www.gnu.org/licenses/>.
 */

#include <math.h>
#include <math_private.h>

/* Coefficients for ln(1+x) = x - x**2/2 + x**3 P(x)/Q(x)
 * 1/sqrt(2) <= x < sqrt(2)
 * Theoretical peak relative error = 5.3e-37,
 * relative peak error spread = 2.3e-14
 */
static const _Float128 P[13] =
{
  L(1.313572404063446165910279910527789794488E4),
  L(7.771154681358524243729929227226708890930E4),
  L(2.014652742082537582487669938141683759923E5),
  L(3.007007295140399532324943111654767187848E5),
  L(2.854829159639697837788887080758954924001E5),
  L(1.797628303815655343403735250238293741397E5),
  L(7.594356839258970405033155585486712125861E4),
  L(2.128857716871515081352991964243375186031E4),
  L(3.824952356185897735160588078446136783779E3),
  L(4.114517881637811823002128927449878962058E2),
  L(2.321125933898420063925789532045674660756E1),
  L(4.998469661968096229986658302195402690910E-1),
  L(1.538612243596254322971797716843006400388E-6)
};
static const _Float128 Q[12] =
{
  L(3.940717212190338497730839731583397586124E4),
  L(2.626900195321832660448791748036714883242E5),
  L(7.777690340007566932935753241556479363645E5),
  L(1.347518538384329112529391120390701166528E6),
  L(1.514882452993549494932585972882995548426E6),
  L(1.158019977462989115839826904108208787040E6),
  L(6.132189329546557743179177159925690841200E5),
  L(2.248234257620569139969141618556349415120E5),
  L(5.605842085972455027590989944010492125825E4),
  L(9.147150349299596453976674231612674085381E3),
  L(9.104928120962988414618126155557301584078E2),
  L(4.839208193348159620282142911143429644326E1)
/* 1.000000000000000000000000000000000000000E0L, */
};

/* Coefficients for log(x) = z + z^3 P(z^2)/Q(z^2),
 * where z = 2(x-1)/(x+1)
 * 1/sqrt(2) <= x < sqrt(2)
 * Theoretical peak relative error = 1.1e-35,
 * relative peak error spread 1.1e-9
 */
static const _Float128 R[6] =
{
  L(1.418134209872192732479751274970992665513E5),
 L(-8.977257995689735303686582344659576526998E4),
  L(2.048819892795278657810231591630928516206E4),
 L(-2.024301798136027039250415126250455056397E3),
  L(8.057002716646055371965756206836056074715E1),
 L(-8.828896441624934385266096344596648080902E-1)
};
static const _Float128 S[6] =
{
  L(1.701761051846631278975701529965589676574E6),
 L(-1.332535117259762928288745111081235577029E6),
  L(4.001557694070773974936904547424676279307E5),
 L(-5.748542087379434595104154610899551484314E4),
  L(3.998526750980007367835804959888064681098E3),
 L(-1.186359407982897997337150403816839480438E2)
/* 1.000000000000000000000000000000000000000E0L, */
};

static const _Float128
/* log2(e) - 1 */
LOG2EA = L(4.4269504088896340735992468100189213742664595E-1),
/* sqrt(2)/2 */
SQRTH = L(7.071067811865475244008443621048490392848359E-1);


/* Evaluate P[n] x^n  +  P[n-1] x^(n-1)  +  ...  +  P[0] */

static _Float128
neval (_Float128 x, const _Float128 *p, int n)
{
  _Float128 y;

  p += n;
  y = *p--;
  do
    {
      y = y * x + *p--;
    }
  while (--n > 0);
  return y;
}


/* Evaluate x^n+1  +  P[n] x^(n)  +  P[n-1] x^(n-1)  +  ...  +  P[0] */

static _Float128
deval (_Float128 x, const _Float128 *p, int n)
{
  _Float128 y;

  p += n;
  y = x + *p--;
  do
    {
      y = y * x + *p--;
    }
  while (--n > 0);
  return y;
}



_Float128
__ieee754_log2l (_Float128 x)
{
  _Float128 z;
  _Float128 y;
  int e;
  int64_t hx, lx;

/* Test for domain */
  GET_LDOUBLE_WORDS64 (hx, lx, x);
  if (((hx & 0x7fffffffffffffffLL) | lx) == 0)
    return (-1 / __fabsl (x));		/* log2l(+-0)=-inf  */
  if (hx < 0)
    return (x - x) / (x - x);
  if (hx >= 0x7fff000000000000LL)
    return (x + x);

  if (x == 1)
    return 0;

/* separate mantissa from exponent */

/* Note, frexp is used so that denormal numbers
 * will be handled properly.
 */
  x = __frexpl (x, &e);


/* logarithm using log(x) = z + z**3 P(z)/Q(z),
 * where z = 2(x-1)/x+1)
 */
  if ((e > 2) || (e < -2))
    {
      if (x < SQRTH)
	{			/* 2( 2x-1 )/( 2x+1 ) */
	  e -= 1;
	  z = x - L(0.5);
	  y = L(0.5) * z + L(0.5);
	}
      else
	{			/*  2 (x-1)/(x+1)   */
	  z = x - L(0.5);
	  z -= L(0.5);
	  y = L(0.5) * x + L(0.5);
	}
      x = z / y;
      z = x * x;
      y = x * (z * neval (z, R, 5) / deval (z, S, 5));
      goto done;
    }


/* logarithm using log(1+x) = x - .5x**2 + x**3 P(x)/Q(x) */

  if (x < SQRTH)
    {
      e -= 1;
      x = 2.0 * x - 1;	/*  2x - 1  */
    }
  else
    {
      x = x - 1;
    }
  z = x * x;
  y = x * (z * neval (x, P, 12) / deval (x, Q, 11));
  y = y - 0.5 * z;

done:

/* Multiply log of fraction by log2(e)
 * and base 2 exponent by 1
 */
  z = y * LOG2EA;
  z += x * LOG2EA;
  z += y;
  z += x;
  z += e;
  return (z);
}
strong_alias (__ieee754_log2l, __log2l_finite)