1 /* 2 * Single-precision pow function. 3 * 4 * Copyright (c) 2017-2019, Arm Limited. 5 * SPDX-License-Identifier: MIT OR Apache-2.0 WITH LLVM-exception 6 */ 7 8 #include <math.h> 9 #include <stdint.h> 10 #include "math_config.h" 11 12 /* 13 POWF_LOG2_POLY_ORDER = 5 14 EXP2F_TABLE_BITS = 5 15 16 ULP error: 0.82 (~ 0.5 + relerr*2^24) 17 relerr: 1.27 * 2^-26 (Relative error ~= 128*Ln2*relerr_log2 + relerr_exp2) 18 relerr_log2: 1.83 * 2^-33 (Relative error of logx.) 19 relerr_exp2: 1.69 * 2^-34 (Relative error of exp2(ylogx).) 20 */ 21 22 #define N (1 << POWF_LOG2_TABLE_BITS) 23 #define T __powf_log2_data.tab 24 #define A __powf_log2_data.poly 25 #define OFF 0x3f330000 26 27 /* Subnormal input is normalized so ix has negative biased exponent. 28 Output is multiplied by N (POWF_SCALE) if TOINT_INTRINICS is set. */ 29 static inline double_t 30 log2_inline (uint32_t ix) 31 { 32 /* double_t for better performance on targets with FLT_EVAL_METHOD==2. */ 33 double_t z, r, r2, r4, p, q, y, y0, invc, logc; 34 uint32_t iz, top, tmp; 35 int k, i; 36 37 /* x = 2^k z; where z is in range [OFF,2*OFF] and exact. 38 The range is split into N subintervals. 39 The ith subinterval contains z and c is near its center. */ 40 tmp = ix - OFF; 41 i = (tmp >> (23 - POWF_LOG2_TABLE_BITS)) % N; 42 top = tmp & 0xff800000; 43 iz = ix - top; 44 k = (int32_t) top >> (23 - POWF_SCALE_BITS); /* arithmetic shift */ 45 invc = T[i].invc; 46 logc = T[i].logc; 47 z = (double_t) asfloat (iz); 48 49 /* log2(x) = log1p(z/c-1)/ln2 + log2(c) + k */ 50 r = z * invc - 1; 51 y0 = logc + (double_t) k; 52 53 /* Pipelined polynomial evaluation to approximate log1p(r)/ln2. */ 54 r2 = r * r; 55 y = A[0] * r + A[1]; 56 p = A[2] * r + A[3]; 57 r4 = r2 * r2; 58 q = A[4] * r + y0; 59 q = p * r2 + q; 60 y = y * r4 + q; 61 return y; 62 } 63 64 #undef N 65 #undef T 66 #define N (1 << EXP2F_TABLE_BITS) 67 #define T __exp2f_data.tab 68 #define SIGN_BIAS (1 << (EXP2F_TABLE_BITS + 11)) 69 70 /* The output of log2 and thus the input of exp2 is either scaled by N 71 (in case of fast toint intrinsics) or not. The unscaled xd must be 72 in [-1021,1023], sign_bias sets the sign of the result. */ 73 static inline float 74 exp2_inline (double_t xd, uint32_t sign_bias) 75 { 76 uint64_t ki, ski, t; 77 /* double_t for better performance on targets with FLT_EVAL_METHOD==2. */ 78 double_t kd, z, r, r2, y, s; 79 80 #if TOINT_INTRINSICS 81 # define C __exp2f_data.poly_scaled 82 /* N*x = k + r with r in [-1/2, 1/2] */ 83 kd = roundtoint (xd); /* k */ 84 ki = converttoint (xd); 85 #else 86 # define C __exp2f_data.poly 87 # define SHIFT __exp2f_data.shift_scaled 88 /* x = k/N + r with r in [-1/(2N), 1/(2N)] */ 89 kd = eval_as_double (xd + SHIFT); 90 ki = asuint64 (kd); 91 kd -= SHIFT; /* k/N */ 92 #endif 93 r = xd - kd; 94 95 /* exp2(x) = 2^(k/N) * 2^r ~= s * (C0*r^3 + C1*r^2 + C2*r + 1) */ 96 t = T[ki % N]; 97 ski = ki + sign_bias; 98 t += ski << (52 - EXP2F_TABLE_BITS); 99 s = asdouble (t); 100 z = C[0] * r + C[1]; 101 r2 = r * r; 102 y = C[2] * r + 1; 103 y = z * r2 + y; 104 y = y * s; 105 return eval_as_float (y); 106 } 107 108 /* Returns 0 if not int, 1 if odd int, 2 if even int. The argument is 109 the bit representation of a non-zero finite floating-point value. */ 110 static inline int 111 checkint (uint32_t iy) 112 { 113 int e = iy >> 23 & 0xff; 114 if (e < 0x7f) 115 return 0; 116 if (e > 0x7f + 23) 117 return 2; 118 if (iy & ((1 << (0x7f + 23 - e)) - 1)) 119 return 0; 120 if (iy & (1 << (0x7f + 23 - e))) 121 return 1; 122 return 2; 123 } 124 125 static inline int 126 zeroinfnan (uint32_t ix) 127 { 128 return 2 * ix - 1 >= 2u * 0x7f800000 - 1; 129 } 130 131 float 132 powf (float x, float y) 133 { 134 uint32_t sign_bias = 0; 135 uint32_t ix, iy; 136 137 ix = asuint (x); 138 iy = asuint (y); 139 if (unlikely (ix - 0x00800000 >= 0x7f800000 - 0x00800000 || zeroinfnan (iy))) 140 { 141 /* Either (x < 0x1p-126 or inf or nan) or (y is 0 or inf or nan). */ 142 if (unlikely (zeroinfnan (iy))) 143 { 144 if (2 * iy == 0) 145 return issignalingf_inline (x) ? x + y : 1.0f; 146 if (ix == 0x3f800000) 147 return issignalingf_inline (y) ? x + y : 1.0f; 148 if (2 * ix > 2u * 0x7f800000 || 2 * iy > 2u * 0x7f800000) 149 return x + y; 150 if (2 * ix == 2 * 0x3f800000) 151 return 1.0f; 152 if ((2 * ix < 2 * 0x3f800000) == !(iy & 0x80000000)) 153 return 0.0f; /* |x|<1 && y==inf or |x|>1 && y==-inf. */ 154 return y * y; 155 } 156 if (unlikely (zeroinfnan (ix))) 157 { 158 float_t x2 = x * x; 159 if (ix & 0x80000000 && checkint (iy) == 1) 160 { 161 x2 = -x2; 162 sign_bias = 1; 163 } 164 #if WANT_ERRNO 165 if (2 * ix == 0 && iy & 0x80000000) 166 return __math_divzerof (sign_bias); 167 #endif 168 /* Without the barrier some versions of clang hoist the 1/x2 and 169 thus division by zero exception can be signaled spuriously. */ 170 return iy & 0x80000000 ? opt_barrier_float (1 / x2) : x2; 171 } 172 /* x and y are non-zero finite. */ 173 if (ix & 0x80000000) 174 { 175 /* Finite x < 0. */ 176 int yint = checkint (iy); 177 if (yint == 0) 178 return __math_invalidf (x); 179 if (yint == 1) 180 sign_bias = SIGN_BIAS; 181 ix &= 0x7fffffff; 182 } 183 if (ix < 0x00800000) 184 { 185 /* Normalize subnormal x so exponent becomes negative. */ 186 ix = asuint (x * 0x1p23f); 187 ix &= 0x7fffffff; 188 ix -= 23 << 23; 189 } 190 } 191 double_t logx = log2_inline (ix); 192 double_t ylogx = y * logx; /* Note: cannot overflow, y is single prec. */ 193 if (unlikely ((asuint64 (ylogx) >> 47 & 0xffff) 194 >= asuint64 (126.0 * POWF_SCALE) >> 47)) 195 { 196 /* |y*log(x)| >= 126. */ 197 if (ylogx > 0x1.fffffffd1d571p+6 * POWF_SCALE) 198 /* |x^y| > 0x1.ffffffp127. */ 199 return __math_oflowf (sign_bias); 200 if (WANT_ROUNDING && WANT_ERRNO 201 && ylogx > 0x1.fffffffa3aae2p+6 * POWF_SCALE) 202 /* |x^y| > 0x1.fffffep127, check if we round away from 0. */ 203 if ((!sign_bias 204 && eval_as_float (1.0f + opt_barrier_float (0x1p-25f)) != 1.0f) 205 || (sign_bias 206 && eval_as_float (-1.0f - opt_barrier_float (0x1p-25f)) 207 != -1.0f)) 208 return __math_oflowf (sign_bias); 209 if (ylogx <= -150.0 * POWF_SCALE) 210 return __math_uflowf (sign_bias); 211 #if WANT_ERRNO_UFLOW 212 if (ylogx < -149.0 * POWF_SCALE) 213 return __math_may_uflowf (sign_bias); 214 #endif 215 } 216 return exp2_inline (ylogx, sign_bias); 217 } 218 #if USE_GLIBC_ABI 219 strong_alias (powf, __powf_finite) 220 hidden_alias (powf, __ieee754_powf) 221 #endif 222