1/* Single-precision pow function.
2 Copyright (C) 2017-2018 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 <http://www.gnu.org/licenses/>. */
18
19#include <math.h>
20#include <stdint.h>
21#include <shlib-compat.h>
22#include <libm-alias-float.h>
23#include "math_config.h"
24
25/*
26POWF_LOG2_POLY_ORDER = 5
27EXP2F_TABLE_BITS = 5
28
29ULP error: 0.82 (~ 0.5 + relerr*2^24)
30relerr: 1.27 * 2^-26 (Relative error ~= 128*Ln2*relerr_log2 + relerr_exp2)
31relerr_log2: 1.83 * 2^-33 (Relative error of logx.)
32relerr_exp2: 1.69 * 2^-34 (Relative error of exp2(ylogx).)
33*/
34
35#define N (1 << POWF_LOG2_TABLE_BITS)
36#define T __powf_log2_data.tab
37#define A __powf_log2_data.poly
38#define OFF 0x3f330000
39
40/* Subnormal input is normalized so ix has negative biased exponent.
41 Output is multiplied by N (POWF_SCALE) if TOINT_INTRINICS is set. */
42static inline double_t
43log2_inline (uint32_t ix)
44{
45 /* double_t for better performance on targets with FLT_EVAL_METHOD==2. */
46 double_t z, r, r2, r4, p, q, y, y0, invc, logc;
47 uint32_t iz, top, tmp;
48 int k, i;
49
50 /* x = 2^k z; where z is in range [OFF,2*OFF] and exact.
51 The range is split into N subintervals.
52 The ith subinterval contains z and c is near its center. */
53 tmp = ix - OFF;
54 i = (tmp >> (23 - POWF_LOG2_TABLE_BITS)) % N;
55 top = tmp & 0xff800000;
56 iz = ix - top;
57 k = (int32_t) top >> (23 - POWF_SCALE_BITS); /* arithmetic shift */
58 invc = T[i].invc;
59 logc = T[i].logc;
60 z = (double_t) asfloat (iz);
61
62 /* log2(x) = log1p(z/c-1)/ln2 + log2(c) + k */
63 r = z * invc - 1;
64 y0 = logc + (double_t) k;
65
66 /* Pipelined polynomial evaluation to approximate log1p(r)/ln2. */
67 r2 = r * r;
68 y = A[0] * r + A[1];
69 p = A[2] * r + A[3];
70 r4 = r2 * r2;
71 q = A[4] * r + y0;
72 q = p * r2 + q;
73 y = y * r4 + q;
74 return y;
75}
76
77#undef N
78#undef T
79#define N (1 << EXP2F_TABLE_BITS)
80#define T __exp2f_data.tab
81#define SIGN_BIAS (1 << (EXP2F_TABLE_BITS + 11))
82
83/* The output of log2 and thus the input of exp2 is either scaled by N
84 (in case of fast toint intrinsics) or not. The unscaled xd must be
85 in [-1021,1023], sign_bias sets the sign of the result. */
86static inline double_t
87exp2_inline (double_t xd, uint32_t sign_bias)
88{
89 uint64_t ki, ski, t;
90 /* double_t for better performance on targets with FLT_EVAL_METHOD==2. */
91 double_t kd, z, r, r2, y, s;
92
93#if TOINT_INTRINSICS
94# define C __exp2f_data.poly_scaled
95 /* N*x = k + r with r in [-1/2, 1/2] */
96 kd = roundtoint (xd); /* k */
97 ki = converttoint (xd);
98#else
99# define C __exp2f_data.poly
100# define SHIFT __exp2f_data.shift_scaled
101 /* x = k/N + r with r in [-1/(2N), 1/(2N)] */
102 kd = (double) (xd + SHIFT); /* Rounding to double precision is required. */
103 ki = asuint64 (kd);
104 kd -= SHIFT; /* k/N */
105#endif
106 r = xd - kd;
107
108 /* exp2(x) = 2^(k/N) * 2^r ~= s * (C0*r^3 + C1*r^2 + C2*r + 1) */
109 t = T[ki % N];
110 ski = ki + sign_bias;
111 t += ski << (52 - EXP2F_TABLE_BITS);
112 s = asdouble (t);
113 z = C[0] * r + C[1];
114 r2 = r * r;
115 y = C[2] * r + 1;
116 y = z * r2 + y;
117 y = y * s;
118 return y;
119}
120
121/* Returns 0 if not int, 1 if odd int, 2 if even int. */
122static inline int
123checkint (uint32_t iy)
124{
125 int e = iy >> 23 & 0xff;
126 if (e < 0x7f)
127 return 0;
128 if (e > 0x7f + 23)
129 return 2;
130 if (iy & ((1 << (0x7f + 23 - e)) - 1))
131 return 0;
132 if (iy & (1 << (0x7f + 23 - e)))
133 return 1;
134 return 2;
135}
136
137static inline int
138zeroinfnan (uint32_t ix)
139{
140 return 2 * ix - 1 >= 2u * 0x7f800000 - 1;
141}
142
143float
144__powf (float x, float y)
145{
146 uint32_t sign_bias = 0;
147 uint32_t ix, iy;
148
149 ix = asuint (x);
150 iy = asuint (y);
151 if (__glibc_unlikely (ix - 0x00800000 >= 0x7f800000 - 0x00800000
152 || zeroinfnan (iy)))
153 {
154 /* Either (x < 0x1p-126 or inf or nan) or (y is 0 or inf or nan). */
155 if (__glibc_unlikely (zeroinfnan (iy)))
156 {
157 if (2 * iy == 0)
158 return issignalingf_inline (x) ? x + y : 1.0f;
159 if (ix == 0x3f800000)
160 return issignalingf_inline (y) ? x + y : 1.0f;
161 if (2 * ix > 2u * 0x7f800000 || 2 * iy > 2u * 0x7f800000)
162 return x + y;
163 if (2 * ix == 2 * 0x3f800000)
164 return 1.0f;
165 if ((2 * ix < 2 * 0x3f800000) == !(iy & 0x80000000))
166 return 0.0f; /* |x|<1 && y==inf or |x|>1 && y==-inf. */
167 return y * y;
168 }
169 if (__glibc_unlikely (zeroinfnan (ix)))
170 {
171 float_t x2 = x * x;
172 if (ix & 0x80000000 && checkint (iy) == 1)
173 {
174 x2 = -x2;
175 sign_bias = 1;
176 }
177#if WANT_ERRNO
178 if (2 * ix == 0 && iy & 0x80000000)
179 return __math_divzerof (sign_bias);
180#endif
181 return iy & 0x80000000 ? 1 / x2 : x2;
182 }
183 /* x and y are non-zero finite. */
184 if (ix & 0x80000000)
185 {
186 /* Finite x < 0. */
187 int yint = checkint (iy);
188 if (yint == 0)
189 return __math_invalidf (x);
190 if (yint == 1)
191 sign_bias = SIGN_BIAS;
192 ix &= 0x7fffffff;
193 }
194 if (ix < 0x00800000)
195 {
196 /* Normalize subnormal x so exponent becomes negative. */
197 ix = asuint (x * 0x1p23f);
198 ix &= 0x7fffffff;
199 ix -= 23 << 23;
200 }
201 }
202 double_t logx = log2_inline (ix);
203 double_t ylogx = y * logx; /* Note: cannot overflow, y is single prec. */
204 if (__glibc_unlikely ((asuint64 (ylogx) >> 47 & 0xffff)
205 >= asuint64 (126.0 * POWF_SCALE) >> 47))
206 {
207 /* |y*log(x)| >= 126. */
208 if (ylogx > 0x1.fffffffd1d571p+6 * POWF_SCALE)
209 return __math_oflowf (sign_bias);
210 if (ylogx <= -150.0 * POWF_SCALE)
211 return __math_uflowf (sign_bias);
212#if WANT_ERRNO_UFLOW
213 if (ylogx < -149.0 * POWF_SCALE)
214 return __math_may_uflowf (sign_bias);
215#endif
216 }
217 return (float) exp2_inline (ylogx, sign_bias);
218}
219#ifndef __powf
220strong_alias (__powf, __ieee754_powf)
221strong_alias (__powf, __powf_finite)
222versioned_symbol (libm, __powf, powf, GLIBC_2_27);
223libm_alias_float_other (__pow, pow)
224#endif
225