1/* Single-precision pow function.
2 Copyright (C) 2017-2022 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-barriers.h>
21#include <math-narrow-eval.h>
22#include <stdint.h>
23#include <libm-alias-finite.h>
24#include <libm-alias-float.h>
25#include "math_config.h"
26
27/*
28POWF_LOG2_POLY_ORDER = 5
29EXP2F_TABLE_BITS = 5
30
31ULP error: 0.82 (~ 0.5 + relerr*2^24)
32relerr: 1.27 * 2^-26 (Relative error ~= 128*Ln2*relerr_log2 + relerr_exp2)
33relerr_log2: 1.83 * 2^-33 (Relative error of logx.)
34relerr_exp2: 1.69 * 2^-34 (Relative error of exp2(ylogx).)
35*/
36
37#define N (1 << POWF_LOG2_TABLE_BITS)
38#define T __powf_log2_data.tab
39#define A __powf_log2_data.poly
40#define OFF 0x3f330000
41
42/* Subnormal input is normalized so ix has negative biased exponent.
43 Output is multiplied by N (POWF_SCALE) if TOINT_INTRINICS is set. */
44static inline double_t
45log2_inline (uint32_t ix)
46{
47 /* double_t for better performance on targets with FLT_EVAL_METHOD==2. */
48 double_t z, r, r2, r4, p, q, y, y0, invc, logc;
49 uint32_t iz, top, tmp;
50 int k, i;
51
52 /* x = 2^k z; where z is in range [OFF,2*OFF] and exact.
53 The range is split into N subintervals.
54 The ith subinterval contains z and c is near its center. */
55 tmp = ix - OFF;
56 i = (tmp >> (23 - POWF_LOG2_TABLE_BITS)) % N;
57 top = tmp & 0xff800000;
58 iz = ix - top;
59 k = (int32_t) top >> (23 - POWF_SCALE_BITS); /* arithmetic shift */
60 invc = T[i].invc;
61 logc = T[i].logc;
62 z = (double_t) asfloat (iz);
63
64 /* log2(x) = log1p(z/c-1)/ln2 + log2(c) + k */
65 r = z * invc - 1;
66 y0 = logc + (double_t) k;
67
68 /* Pipelined polynomial evaluation to approximate log1p(r)/ln2. */
69 r2 = r * r;
70 y = A[0] * r + A[1];
71 p = A[2] * r + A[3];
72 r4 = r2 * r2;
73 q = A[4] * r + y0;
74 q = p * r2 + q;
75 y = y * r4 + q;
76 return y;
77}
78
79#undef N
80#undef T
81#define N (1 << EXP2F_TABLE_BITS)
82#define T __exp2f_data.tab
83#define SIGN_BIAS (1 << (EXP2F_TABLE_BITS + 11))
84
85/* The output of log2 and thus the input of exp2 is either scaled by N
86 (in case of fast toint intrinsics) or not. The unscaled xd must be
87 in [-1021,1023], sign_bias sets the sign of the result. */
88static inline double_t
89exp2_inline (double_t xd, uint32_t sign_bias)
90{
91 uint64_t ki, ski, t;
92 /* double_t for better performance on targets with FLT_EVAL_METHOD==2. */
93 double_t kd, z, r, r2, y, s;
94
95#if TOINT_INTRINSICS
96# define C __exp2f_data.poly_scaled
97 /* N*x = k + r with r in [-1/2, 1/2] */
98 kd = roundtoint (xd); /* k */
99 ki = converttoint (xd);
100#else
101# define C __exp2f_data.poly
102# define SHIFT __exp2f_data.shift_scaled
103 /* x = k/N + r with r in [-1/(2N), 1/(2N)] */
104 kd = (double) (xd + SHIFT); /* Rounding to double precision is required. */
105 ki = asuint64 (kd);
106 kd -= SHIFT; /* k/N */
107#endif
108 r = xd - kd;
109
110 /* exp2(x) = 2^(k/N) * 2^r ~= s * (C0*r^3 + C1*r^2 + C2*r + 1) */
111 t = T[ki % N];
112 ski = ki + sign_bias;
113 t += ski << (52 - EXP2F_TABLE_BITS);
114 s = asdouble (t);
115 z = C[0] * r + C[1];
116 r2 = r * r;
117 y = C[2] * r + 1;
118 y = z * r2 + y;
119 y = y * s;
120 return y;
121}
122
123/* Returns 0 if not int, 1 if odd int, 2 if even int. The argument is
124 the bit representation of a non-zero finite floating-point value. */
125static inline int
126checkint (uint32_t iy)
127{
128 int e = iy >> 23 & 0xff;
129 if (e < 0x7f)
130 return 0;
131 if (e > 0x7f + 23)
132 return 2;
133 if (iy & ((1 << (0x7f + 23 - e)) - 1))
134 return 0;
135 if (iy & (1 << (0x7f + 23 - e)))
136 return 1;
137 return 2;
138}
139
140static inline int
141zeroinfnan (uint32_t ix)
142{
143 return 2 * ix - 1 >= 2u * 0x7f800000 - 1;
144}
145
146float
147__powf (float x, float y)
148{
149 uint32_t sign_bias = 0;
150 uint32_t ix, iy;
151
152 ix = asuint (x);
153 iy = asuint (y);
154 if (__glibc_unlikely (ix - 0x00800000 >= 0x7f800000 - 0x00800000
155 || zeroinfnan (iy)))
156 {
157 /* Either (x < 0x1p-126 or inf or nan) or (y is 0 or inf or nan). */
158 if (__glibc_unlikely (zeroinfnan (iy)))
159 {
160 if (2 * iy == 0)
161 return issignaling (x) ? x + y : 1.0f;
162 if (ix == 0x3f800000)
163 return issignaling (y) ? x + y : 1.0f;
164 if (2 * ix > 2u * 0x7f800000 || 2 * iy > 2u * 0x7f800000)
165 return x + y;
166 if (2 * ix == 2 * 0x3f800000)
167 return 1.0f;
168 if ((2 * ix < 2 * 0x3f800000) == !(iy & 0x80000000))
169 return 0.0f; /* |x|<1 && y==inf or |x|>1 && y==-inf. */
170 return y * y;
171 }
172 if (__glibc_unlikely (zeroinfnan (ix)))
173 {
174 float_t x2 = x * x;
175 if (ix & 0x80000000 && checkint (iy) == 1)
176 {
177 x2 = -x2;
178 sign_bias = 1;
179 }
180#if WANT_ERRNO
181 if (2 * ix == 0 && iy & 0x80000000)
182 return __math_divzerof (sign_bias);
183#endif
184 return iy & 0x80000000 ? 1 / x2 : x2;
185 }
186 /* x and y are non-zero finite. */
187 if (ix & 0x80000000)
188 {
189 /* Finite x < 0. */
190 int yint = checkint (iy);
191 if (yint == 0)
192 return __math_invalidf (x);
193 if (yint == 1)
194 sign_bias = SIGN_BIAS;
195 ix &= 0x7fffffff;
196 }
197 if (ix < 0x00800000)
198 {
199 /* Normalize subnormal x so exponent becomes negative. */
200 ix = asuint (x * 0x1p23f);
201 ix &= 0x7fffffff;
202 ix -= 23 << 23;
203 }
204 }
205 double_t logx = log2_inline (ix);
206 double_t ylogx = y * logx; /* Note: cannot overflow, y is single prec. */
207 if (__glibc_unlikely ((asuint64 (ylogx) >> 47 & 0xffff)
208 >= asuint64 (126.0 * POWF_SCALE) >> 47))
209 {
210 /* |y*log(x)| >= 126. */
211 if (ylogx > 0x1.fffffffd1d571p+6 * POWF_SCALE)
212 /* |x^y| > 0x1.ffffffp127. */
213 return __math_oflowf (sign_bias);
214 if (WANT_ROUNDING && WANT_ERRNO
215 && ylogx > 0x1.fffffffa3aae2p+6 * POWF_SCALE)
216 /* |x^y| > 0x1.fffffep127, check if we round away from 0. */
217 if ((!sign_bias
218 && math_narrow_eval (1.0f + math_opt_barrier (0x1p-25f)) != 1.0f)
219 || (sign_bias
220 && math_narrow_eval (-1.0f - math_opt_barrier (0x1p-25f))
221 != -1.0f))
222 return __math_oflowf (sign_bias);
223 if (ylogx <= -150.0 * POWF_SCALE)
224 return __math_uflowf (sign_bias);
225#if WANT_ERRNO_UFLOW
226 if (ylogx < -149.0 * POWF_SCALE)
227 return __math_may_uflowf (sign_bias);
228#endif
229 }
230 return (float) exp2_inline (ylogx, sign_bias);
231}
232#ifndef __powf
233strong_alias (__powf, __ieee754_powf)
234libm_alias_finite (__ieee754_powf, __powf)
235versioned_symbol (libm, __powf, powf, GLIBC_2_27);
236libm_alias_float_other (__pow, pow)
237#endif
238