1 /* Implement powl for x86 using extra-precision log.
2 Copyright (C) 2012-2023 Free Software Foundation, Inc.
3 This file is part of the GNU C Library.
4
5 The GNU C Library is free software; you can redistribute it and/or
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.
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
13 Lesser General Public License for more details.
14
15 You should have received a copy of the GNU Lesser General Public
16 License along with the GNU C Library; if not, see
17 <https://www.gnu.org/licenses/>. */
18
19 #include <math.h>
20 #include <math_private.h>
21 #include <math-underflow.h>
22 #include <stdbool.h>
23
24 /* High parts and low parts of -log (k/16), for integer k from 12 to
25 24. */
26
27 static const long double powl_log_table[] =
28 {
29 0x4.9a58844d36e49e1p-4L, -0x1.0522624fd558f574p-68L,
30 0x3.527da7915b3c6de4p-4L, 0x1.7d4ef4b901b99b9ep-68L,
31 0x2.22f1d044fc8f7bc8p-4L, -0x1.8e97c071a42fc388p-68L,
32 0x1.08598b59e3a0688ap-4L, 0x3.fd9bf503372c12fcp-72L,
33 -0x0p+0L, 0x0p+0L,
34 -0xf.85186008b15330cp-8L, 0x1.9b47488a6687672cp-72L,
35 -0x1.e27076e2af2e5e9ep-4L, -0xa.87ffe1fe9e155dcp-72L,
36 -0x2.bfe60e14f27a791p-4L, 0x1.83bebf1bdb88a032p-68L,
37 -0x3.91fef8f353443584p-4L, -0xb.b03de5ff734495cp-72L,
38 -0x4.59d72aeae98380e8p-4L, 0xc.e0aa3be4747dc1p-72L,
39 -0x5.1862f08717b09f4p-4L, -0x2.decdeccf1cd10578p-68L,
40 -0x5.ce75fdaef401a738p-4L, -0x9.314feb4fbde5aaep-72L,
41 -0x6.7cc8fb2fe612fcbp-4L, 0x2.5ca2642feb779f98p-68L,
42 };
43
44 /* High 32 bits of log2 (e), and remainder rounded to 64 bits. */
45 static const long double log2e_hi = 0x1.71547652p+0L;
46 static const long double log2e_lo = 0xb.82fe1777d0ffda1p-36L;
47
48 /* Given a number with high part HI and low part LO, add the number X
49 to it and store the result in *RHI and *RLO. It is given that
50 either |X| < |0.7 * HI|, or HI == LO == 0, and that the values are
51 small enough that no overflow occurs. The result does not need to
52 be exact to 128 bits; 78-bit accuracy of the final accumulated
53 result suffices. */
54
55 static inline void
56 acc_split (long double *rhi, long double *rlo, long double hi, long double lo,
57 long double x)
58 {
59 long double thi = hi + x;
60 long double tlo = (hi - thi) + x + lo;
61 *rhi = thi + tlo;
62 *rlo = (thi - *rhi) + tlo;
63 }
64
65 extern long double __powl_helper (long double x, long double y);
66 libm_hidden_proto (__powl_helper)
67
68 /* Given X a value that is finite and nonzero, or a NaN, and Y a
69 finite nonzero value with 0x1p-79 <= |Y| <= 0x1p78, compute X to
70 the power Y. */
71
72 long double
73 __powl_helper (long double x, long double y)
74 {
75 if (isnan (x))
76 return __ieee754_expl (y * __ieee754_logl (x));
77 bool negate;
78 if (x < 0)
79 {
80 long double absy = fabsl (y);
81 if (absy >= 0x1p64L)
82 negate = false;
83 else
84 {
85 unsigned long long yll = absy;
86 if (yll != absy)
87 return __ieee754_expl (y * __ieee754_logl (x));
88 negate = (yll & 1) != 0;
89 }
90 x = fabsl (x);
91 }
92 else
93 negate = false;
94
95 /* We need to compute Y * log2 (X) to at least 64 bits after the
96 point for normal results (that is, to at least 78 bits
97 precision). */
98 int x_int_exponent;
99 long double x_frac;
100 x_frac = __frexpl (x, &x_int_exponent);
101 if (x_frac <= 0x0.aaaaaaaaaaaaaaaap0L) /* 2.0L / 3.0L, rounded down */
102 {
103 x_frac *= 2.0;
104 x_int_exponent--;
105 }
106
107 long double log_x_frac_hi, log_x_frac_lo;
108 /* Determine an initial approximation to log (X_FRAC) using
109 POWL_LOG_TABLE, and multiply by a value K/16 to reduce to an
110 interval (24/25, 26/25). */
111 int k = (int) ((16.0L / x_frac) + 0.5L);
112 log_x_frac_hi = powl_log_table[2 * k - 24];
113 log_x_frac_lo = powl_log_table[2 * k - 23];
114 long double x_frac_low;
115 if (k == 16)
116 x_frac_low = 0.0L;
117 else
118 {
119 /* Mask off low 5 bits of X_FRAC so the multiplication by K/16
120 is exact. These bits are small enough that they can be
121 corrected for by adding log2 (e) * X_FRAC_LOW to the final
122 result. */
123 int32_t se;
124 uint32_t i0, i1;
125 GET_LDOUBLE_WORDS (se, i0, i1, x_frac);
126 x_frac_low = x_frac;
127 i1 &= 0xffffffe0;
128 SET_LDOUBLE_WORDS (x_frac, se, i0, i1);
129 x_frac_low -= x_frac;
130 x_frac_low /= x_frac;
131 x_frac *= k / 16.0L;
132 }
133
134 /* Now compute log (X_FRAC) for X_FRAC in (24/25, 26/25). Separate
135 W = X_FRAC - 1 into high 16 bits and remaining bits, so that
136 multiplications for low-order power series terms are exact. The
137 remaining bits are small enough that adding a 64-bit value of
138 log2 (1 + W_LO / (1 + W_HI)) will be a sufficient correction for
139 them. */
140 long double w = x_frac - 1;
141 long double w_hi, w_lo;
142 int32_t se;
143 uint32_t i0, i1;
144 GET_LDOUBLE_WORDS (se, i0, i1, w);
145 i0 &= 0xffff0000;
146 i1 = 0;
147 SET_LDOUBLE_WORDS (w_hi, se, i0, i1);
148 w_lo = w - w_hi;
149 long double wp = w_hi;
150 acc_split (&log_x_frac_hi, &log_x_frac_lo, log_x_frac_hi, log_x_frac_lo, wp);
151 wp *= -w_hi;
152 acc_split (&log_x_frac_hi, &log_x_frac_lo, log_x_frac_hi, log_x_frac_lo,
153 wp / 2.0L);
154 wp *= -w_hi;
155 acc_split (&log_x_frac_hi, &log_x_frac_lo, log_x_frac_hi, log_x_frac_lo,
156 wp * 0x0.5555p0L); /* -W_HI**3 / 3, high part. */
157 acc_split (&log_x_frac_hi, &log_x_frac_lo, log_x_frac_hi, log_x_frac_lo,
158 wp * 0x0.5555555555555555p-16L); /* -W_HI**3 / 3, low part. */
159 wp *= -w_hi;
160 acc_split (&log_x_frac_hi, &log_x_frac_lo, log_x_frac_hi, log_x_frac_lo,
161 wp / 4.0L);
162 /* Subsequent terms are small enough that they only need be computed
163 to 64 bits. */
164 for (int i = 5; i <= 17; i++)
165 {
166 wp *= -w_hi;
167 acc_split (&log_x_frac_hi, &log_x_frac_lo, log_x_frac_hi, log_x_frac_lo,
168 wp / i);
169 }
170
171 /* Convert LOG_X_FRAC_HI + LOG_X_FRAC_LO to a base-2 logarithm. */
172 long double log2_x_frac_hi, log2_x_frac_lo;
173 long double log_x_frac_hi32, log_x_frac_lo64;
174 GET_LDOUBLE_WORDS (se, i0, i1, log_x_frac_hi);
175 i1 = 0;
176 SET_LDOUBLE_WORDS (log_x_frac_hi32, se, i0, i1);
177 log_x_frac_lo64 = (log_x_frac_hi - log_x_frac_hi32) + log_x_frac_lo;
178 long double log2_x_frac_hi1 = log_x_frac_hi32 * log2e_hi;
179 long double log2_x_frac_lo1
180 = log_x_frac_lo64 * log2e_hi + log_x_frac_hi * log2e_lo;
181 log2_x_frac_hi = log2_x_frac_hi1 + log2_x_frac_lo1;
182 log2_x_frac_lo = (log2_x_frac_hi1 - log2_x_frac_hi) + log2_x_frac_lo1;
183
184 /* Correct for the masking off of W_LO. */
185 long double log2_1p_w_lo;
186 asm ("fyl2xp1"
187 : "=t" (log2_1p_w_lo)
188 : "0" (w_lo / (1.0L + w_hi)), "u" (1.0L)
189 : "st(1)");
190 acc_split (&log2_x_frac_hi, &log2_x_frac_lo, log2_x_frac_hi, log2_x_frac_lo,
191 log2_1p_w_lo);
192
193 /* Correct for the masking off of X_FRAC_LOW. */
194 acc_split (&log2_x_frac_hi, &log2_x_frac_lo, log2_x_frac_hi, log2_x_frac_lo,
195 x_frac_low * M_LOG2El);
196
197 /* Add the integer and fractional parts of the base-2 logarithm. */
198 long double log2_x_hi, log2_x_lo;
199 log2_x_hi = x_int_exponent + log2_x_frac_hi;
200 log2_x_lo = ((x_int_exponent - log2_x_hi) + log2_x_frac_hi) + log2_x_frac_lo;
201
202 /* Compute the base-2 logarithm of the result. */
203 long double log2_res_hi, log2_res_lo;
204 long double log2_x_hi32, log2_x_lo64;
205 GET_LDOUBLE_WORDS (se, i0, i1, log2_x_hi);
206 i1 = 0;
207 SET_LDOUBLE_WORDS (log2_x_hi32, se, i0, i1);
208 log2_x_lo64 = (log2_x_hi - log2_x_hi32) + log2_x_lo;
209 long double y_hi32, y_lo32;
210 GET_LDOUBLE_WORDS (se, i0, i1, y);
211 i1 = 0;
212 SET_LDOUBLE_WORDS (y_hi32, se, i0, i1);
213 y_lo32 = y - y_hi32;
214 log2_res_hi = log2_x_hi32 * y_hi32;
215 log2_res_lo = log2_x_hi32 * y_lo32 + log2_x_lo64 * y;
216
217 /* Split the base-2 logarithm of the result into integer and
218 fractional parts. */
219 long double log2_res_int = roundl (log2_res_hi);
220 long double log2_res_frac = log2_res_hi - log2_res_int + log2_res_lo;
221 /* If the integer part is very large, the computed fractional part
222 may be outside the valid range for f2xm1. */
223 if (fabsl (log2_res_int) > 16500)
224 log2_res_frac = 0;
225
226 /* Compute the final result. */
227 long double res;
228 asm ("f2xm1" : "=t" (res) : "0" (log2_res_frac));
229 res += 1.0L;
230 if (negate)
231 res = -res;
232 asm ("fscale" : "=t" (res) : "0" (res), "u" (log2_res_int));
233 math_check_force_underflow (res);
234 return res;
235 }
236
237 libm_hidden_def (__powl_helper)