1 /* 2 * Single-precision pow function. 3 * 4 * Copyright (c) 2017-2024, 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 #include "test_defs.h" 12 13 /* 14 POWF_LOG2_POLY_ORDER = 5 15 EXP2F_TABLE_BITS = 5 16 17 ULP error: 0.82 (~ 0.5 + relerr*2^24) 18 relerr: 1.27 * 2^-26 (Relative error ~= 128*Ln2*relerr_log2 + relerr_exp2) 19 relerr_log2: 1.83 * 2^-33 (Relative error of logx.) 20 relerr_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.tab 25 #define A __powf_log2_data.poly 26 #define OFF 0x3f330000 27 28 /* Subnormal input is normalized so ix has negative biased exponent. 29 Output is multiplied by N (POWF_SCALE) if TOINT_INTRINICS is set. */ 30 static inline double_t 31 log2_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 N 66 #undef T 67 #define N (1 << EXP2F_TABLE_BITS) 68 #define T __exp2f_data.tab 69 #define SIGN_BIAS (1 << (EXP2F_TABLE_BITS + 11)) 70 71 /* The output of log2 and thus the input of exp2 is either scaled by N 72 (in case of fast toint intrinsics) or not. The unscaled xd must be 73 in [-1021,1023], sign_bias sets the sign of the result. */ 74 static inline float 75 exp2_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_INTRINSICS 82 # define C __exp2f_data.poly_scaled 83 /* N*x = k + r with r in [-1/2, 1/2] */ 84 kd = roundtoint (xd); /* k */ 85 ki = converttoint (xd); 86 #else 87 # define C __exp2f_data.poly 88 # define SHIFT __exp2f_data.shift_scaled 89 /* 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 #endif 94 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 is 110 the bit representation of a non-zero finite floating-point value. */ 111 static inline int 112 checkint (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 } 125 126 static inline int 127 zeroinfnan (uint32_t ix) 128 { 129 return 2 * ix - 1 >= 2u * 0x7f800000 - 1; 130 } 131 132 float 133 powf (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_ERRNO 166 if (2 * ix == 0 && iy & 0x80000000) 167 return __math_divzerof (sign_bias); 168 #endif 169 /* Without the barrier some versions of clang hoist the 1/x2 and 170 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_ERRNO 202 && ylogx > 0x1.fffffffa3aae2p+6 * POWF_SCALE) 203 /* |x^y| > 0x1.fffffep127, check if we round away from 0. */ 204 if ((!sign_bias 205 && eval_as_float (1.0f + opt_barrier_float (0x1p-25f)) != 1.0f) 206 || (sign_bias 207 && 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_UFLOW 213 if (ylogx < -149.0 * POWF_SCALE) 214 return __math_may_uflowf (sign_bias); 215 #endif 216 } 217 return exp2_inline (ylogx, sign_bias); 218 } 219 #if USE_GLIBC_ABI 220 strong_alias (powf, __powf_finite) 221 hidden_alias (powf, __ieee754_powf) 222 #endif 223 224 TEST_ULP (powf, 0.4) 225 TEST_ULP_NONNEAREST (powf, 0.5) 226 TEST_INTERVAL2 (powf, 0x1p-1, 0x1p1, 0x1p-7, 0x1p7, 50000) 227 TEST_INTERVAL2 (powf, 0x1p-1, 0x1p1, -0x1p-7, -0x1p7, 50000) 228 TEST_INTERVAL2 (powf, 0x1p-70, 0x1p70, 0x1p-1, 0x1p1, 50000) 229 TEST_INTERVAL2 (powf, 0x1p-70, 0x1p70, -0x1p-1, -0x1p1, 50000) 230 TEST_INTERVAL2 (powf, 0x1.ep-1, 0x1.1p0, 0x1p8, 0x1p14, 50000) 231 TEST_INTERVAL2 (powf, 0x1.ep-1, 0x1.1p0, -0x1p8, -0x1p14, 50000) 232