223 lines · c
1/*2 * Single-precision pow function.3 *4 * Part of the LLVM Project, under the Apache License v2.0 with LLVM Exceptions.5 * See https://llvm.org/LICENSE.txt for license information.6 * SPDX-License-Identifier: Apache-2.0 WITH LLVM-exception7 */8 9#include <math.h>10#include <stdint.h>11#include "math_config.h"12 13/*14POWF_LOG2_POLY_ORDER = 515EXP2F_TABLE_BITS = 516 17ULP error: 0.82 (~ 0.5 + relerr*2^24)18relerr: 1.27 * 2^-26 (Relative error ~= 128*Ln2*relerr_log2 + relerr_exp2)19relerr_log2: 1.83 * 2^-33 (Relative error of logx.)20relerr_exp2: 1.69 * 2^-34 (Relative error of exp2(ylogx).)21*/22 23#define N (1 << POWF_LOG2_TABLE_BITS)24#define T __powf_log2_data.tab25#define A __powf_log2_data.poly26#define OFF 0x3f33000027 28/* Subnormal input is normalized so ix has negative biased exponent.29 Output is multiplied by N (POWF_SCALE) if TOINT_INTRINSICS is set. */30static inline double_t31log2_inline (uint32_t ix)32{33 /* double_t for better performance on targets with FLT_EVAL_METHOD==2. */34 double_t z, r, r2, r4, p, q, y, y0, invc, logc;35 uint32_t iz, top, tmp;36 int k, i;37 38 /* x = 2^k z; where z is in range [OFF,2*OFF] and exact.39 The range is split into N subintervals.40 The ith subinterval contains z and c is near its center. */41 tmp = ix - OFF;42 i = (tmp >> (23 - POWF_LOG2_TABLE_BITS)) % N;43 top = tmp & 0xff800000;44 iz = ix - top;45 k = (int32_t) top >> (23 - POWF_SCALE_BITS); /* arithmetic shift */46 invc = T[i].invc;47 logc = T[i].logc;48 z = (double_t) asfloat (iz);49 50 /* log2(x) = log1p(z/c-1)/ln2 + log2(c) + k */51 r = z * invc - 1;52 y0 = logc + (double_t) k;53 54 /* Pipelined polynomial evaluation to approximate log1p(r)/ln2. */55 r2 = r * r;56 y = A[0] * r + A[1];57 p = A[2] * r + A[3];58 r4 = r2 * r2;59 q = A[4] * r + y0;60 q = p * r2 + q;61 y = y * r4 + q;62 return y;63}64 65#undef N66#undef T67#define N (1 << EXP2F_TABLE_BITS)68#define T __exp2f_data.tab69#define SIGN_BIAS (1 << (EXP2F_TABLE_BITS + 11))70 71/* The output of log2 and thus the input of exp2 is either scaled by N72 (in case of fast toint intrinsics) or not. The unscaled xd must be73 in [-1021,1023], sign_bias sets the sign of the result. */74static inline float75exp2_inline (double_t xd, uint32_t sign_bias)76{77 uint64_t ki, ski, t;78 /* double_t for better performance on targets with FLT_EVAL_METHOD==2. */79 double_t kd, z, r, r2, y, s;80 81#if TOINT_INTRINSICS82# define C __exp2f_data.poly_scaled83 /* N*x = k + r with r in [-1/2, 1/2] */84 kd = roundtoint (xd); /* k */85 ki = converttoint (xd);86#else87# define C __exp2f_data.poly88# define SHIFT __exp2f_data.shift_scaled89 /* x = k/N + r with r in [-1/(2N), 1/(2N)] */90 kd = eval_as_double (xd + SHIFT);91 ki = asuint64 (kd);92 kd -= SHIFT; /* k/N */93#endif94 r = xd - kd;95 96 /* exp2(x) = 2^(k/N) * 2^r ~= s * (C0*r^3 + C1*r^2 + C2*r + 1) */97 t = T[ki % N];98 ski = ki + sign_bias;99 t += ski << (52 - EXP2F_TABLE_BITS);100 s = asdouble (t);101 z = C[0] * r + C[1];102 r2 = r * r;103 y = C[2] * r + 1;104 y = z * r2 + y;105 y = y * s;106 return eval_as_float (y);107}108 109/* Returns 0 if not int, 1 if odd int, 2 if even int. The argument is110 the bit representation of a non-zero finite floating-point value. */111static inline int112checkint (uint32_t iy)113{114 int e = iy >> 23 & 0xff;115 if (e < 0x7f)116 return 0;117 if (e > 0x7f + 23)118 return 2;119 if (iy & ((1 << (0x7f + 23 - e)) - 1))120 return 0;121 if (iy & (1 << (0x7f + 23 - e)))122 return 1;123 return 2;124}125 126static inline int127zeroinfnan (uint32_t ix)128{129 return 2 * ix - 1 >= 2u * 0x7f800000 - 1;130}131 132float133powf (float x, float y)134{135 uint32_t sign_bias = 0;136 uint32_t ix, iy;137 138 ix = asuint (x);139 iy = asuint (y);140 if (unlikely (ix - 0x00800000 >= 0x7f800000 - 0x00800000 || zeroinfnan (iy)))141 {142 /* Either (x < 0x1p-126 or inf or nan) or (y is 0 or inf or nan). */143 if (unlikely (zeroinfnan (iy)))144 {145 if (2 * iy == 0)146 return issignalingf_inline (x) ? x + y : 1.0f;147 if (ix == 0x3f800000)148 return issignalingf_inline (y) ? x + y : 1.0f;149 if (2 * ix > 2u * 0x7f800000 || 2 * iy > 2u * 0x7f800000)150 return x + y;151 if (2 * ix == 2 * 0x3f800000)152 return 1.0f;153 if ((2 * ix < 2 * 0x3f800000) == !(iy & 0x80000000))154 return 0.0f; /* |x|<1 && y==inf or |x|>1 && y==-inf. */155 return y * y;156 }157 if (unlikely (zeroinfnan (ix)))158 {159 float_t x2 = x * x;160 if (ix & 0x80000000 && checkint (iy) == 1)161 {162 x2 = -x2;163 sign_bias = 1;164 }165#if WANT_ERRNO166 if (2 * ix == 0 && iy & 0x80000000)167 return __math_divzerof (sign_bias);168#endif169 /* Without the barrier some versions of clang hoist the 1/x2 and170 thus division by zero exception can be signaled spuriously. */171 return iy & 0x80000000 ? opt_barrier_float (1 / x2) : x2;172 }173 /* x and y are non-zero finite. */174 if (ix & 0x80000000)175 {176 /* Finite x < 0. */177 int yint = checkint (iy);178 if (yint == 0)179 return __math_invalidf (x);180 if (yint == 1)181 sign_bias = SIGN_BIAS;182 ix &= 0x7fffffff;183 }184 if (ix < 0x00800000)185 {186 /* Normalize subnormal x so exponent becomes negative. */187 ix = asuint (x * 0x1p23f);188 ix &= 0x7fffffff;189 ix -= 23 << 23;190 }191 }192 double_t logx = log2_inline (ix);193 double_t ylogx = y * logx; /* Note: cannot overflow, y is single prec. */194 if (unlikely ((asuint64 (ylogx) >> 47 & 0xffff)195 >= asuint64 (126.0 * POWF_SCALE) >> 47))196 {197 /* |y*log(x)| >= 126. */198 if (ylogx > 0x1.fffffffd1d571p+6 * POWF_SCALE)199 /* |x^y| > 0x1.ffffffp127. */200 return __math_oflowf (sign_bias);201 if (WANT_ROUNDING && WANT_ERRNO202 && ylogx > 0x1.fffffffa3aae2p+6 * POWF_SCALE)203 /* |x^y| > 0x1.fffffep127, check if we round away from 0. */204 if ((!sign_bias205 && eval_as_float (1.0f + opt_barrier_float (0x1p-25f)) != 1.0f)206 || (sign_bias207 && eval_as_float (-1.0f - opt_barrier_float (0x1p-25f))208 != -1.0f))209 return __math_oflowf (sign_bias);210 if (ylogx <= -150.0 * POWF_SCALE)211 return __math_uflowf (sign_bias);212#if WANT_ERRNO_UFLOW213 if (ylogx < -149.0 * POWF_SCALE)214 return __math_may_uflowf (sign_bias);215#endif216 }217 return exp2_inline (ylogx, sign_bias);218}219#if USE_GLIBC_ABI220strong_alias (powf, __powf_finite)221hidden_alias (powf, __ieee754_powf)222#endif223