1 /* 2 * Copyright (c) 2017-2018, Arm Limited. 3 * SPDX-License-Identifier: MIT 4 */ 5 6 #include <math.h> 7 #include <stdint.h> 8 #include "libm.h" 9 #include "exp2f_data.h" 10 #include "powf_data.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 log2_inline(uint32_t ix) 30 { 31 double_t z, r, r2, r4, p, q, y, y0, invc, logc; 32 uint32_t iz, top, tmp; 33 int k, i; 34 35 /* x = 2^k z; where z is in range [OFF,2*OFF] and exact. 36 The range is split into N subintervals. 37 The ith subinterval contains z and c is near its center. */ 38 tmp = ix - OFF; 39 i = (tmp >> (23 - POWF_LOG2_TABLE_BITS)) % N; 40 top = tmp & 0xff800000; 41 iz = ix - top; 42 k = (int32_t)top >> (23 - POWF_SCALE_BITS); /* arithmetic shift */ 43 invc = T[i].invc; 44 logc = T[i].logc; 45 z = (double_t)asfloat(iz); 46 47 /* log2(x) = log1p(z/c-1)/ln2 + log2(c) + k */ 48 r = z * invc - 1; 49 y0 = logc + (double_t)k; 50 51 /* Pipelined polynomial evaluation to approximate log1p(r)/ln2. */ 52 r2 = r * r; 53 y = A[0] * r + A[1]; 54 p = A[2] * r + A[3]; 55 r4 = r2 * r2; 56 q = A[4] * r + y0; 57 q = p * r2 + q; 58 y = y * r4 + q; 59 return y; 60 } 61 62 #undef N 63 #undef T 64 #define N (1 << EXP2F_TABLE_BITS) 65 #define T __exp2f_data.tab 66 #define SIGN_BIAS (1 << (EXP2F_TABLE_BITS + 11)) 67 68 /* The output of log2 and thus the input of exp2 is either scaled by N 69 (in case of fast toint intrinsics) or not. The unscaled xd must be 70 in [-1021,1023], sign_bias sets the sign of the result. */ 71 static inline float exp2_inline(double_t xd, uint32_t sign_bias) 72 { 73 uint64_t ki, ski, t; 74 double_t kd, z, r, r2, y, s; 75 76 #if TOINT_INTRINSICS 77 #define C __exp2f_data.poly_scaled 78 /* N*x = k + r with r in [-1/2, 1/2] */ 79 kd = roundtoint(xd); /* k */ 80 ki = converttoint(xd); 81 #else 82 #define C __exp2f_data.poly 83 #define SHIFT __exp2f_data.shift_scaled 84 /* x = k/N + r with r in [-1/(2N), 1/(2N)] */ 85 kd = eval_as_double(xd + SHIFT); 86 ki = asuint64(kd); 87 kd -= SHIFT; /* k/N */ 88 #endif 89 r = xd - kd; 90 91 /* exp2(x) = 2^(k/N) * 2^r ~= s * (C0*r^3 + C1*r^2 + C2*r + 1) */ 92 t = T[ki % N]; 93 ski = ki + sign_bias; 94 t += ski << (52 - EXP2F_TABLE_BITS); 95 s = asdouble(t); 96 z = C[0] * r + C[1]; 97 r2 = r * r; 98 y = C[2] * r + 1; 99 y = z * r2 + y; 100 y = y * s; 101 return eval_as_float(y); 102 } 103 104 /* Returns 0 if not int, 1 if odd int, 2 if even int. The argument is 105 the bit representation of a non-zero finite floating-point value. */ 106 static inline int checkint(uint32_t iy) 107 { 108 int e = iy >> 23 & 0xff; 109 if (e < 0x7f) 110 return 0; 111 if (e > 0x7f + 23) 112 return 2; 113 if (iy & ((1 << (0x7f + 23 - e)) - 1)) 114 return 0; 115 if (iy & (1 << (0x7f + 23 - e))) 116 return 1; 117 return 2; 118 } 119 120 static inline int zeroinfnan(uint32_t ix) 121 { 122 return 2 * ix - 1 >= 2u * 0x7f800000 - 1; 123 } 124 125 float powf(float x, float y) 126 { 127 uint32_t sign_bias = 0; 128 uint32_t ix, iy; 129 130 ix = asuint(x); 131 iy = asuint(y); 132 if (predict_false(ix - 0x00800000 >= 0x7f800000 - 0x00800000 || 133 zeroinfnan(iy))) { 134 /* Either (x < 0x1p-126 or inf or nan) or (y is 0 or inf or nan). */ 135 if (predict_false(zeroinfnan(iy))) { 136 if (2 * iy == 0) 137 return issignalingf_inline(x) ? x + y : 1.0f; 138 if (ix == 0x3f800000) 139 return issignalingf_inline(y) ? x + y : 1.0f; 140 if (2 * ix > 2u * 0x7f800000 || 141 2 * iy > 2u * 0x7f800000) 142 return x + y; 143 if (2 * ix == 2 * 0x3f800000) 144 return 1.0f; 145 if ((2 * ix < 2 * 0x3f800000) == !(iy & 0x80000000)) 146 return 0.0f; /* |x|<1 && y==inf or |x|>1 && y==-inf. */ 147 return y * y; 148 } 149 if (predict_false(zeroinfnan(ix))) { 150 float_t x2 = x * x; 151 if (ix & 0x80000000 && checkint(iy) == 1) 152 x2 = -x2; 153 /* Without the barrier some versions of clang hoist the 1/x2 and 154 thus division by zero exception can be signaled spuriously. */ 155 return iy & 0x80000000 ? fp_barrierf(1 / x2) : x2; 156 } 157 /* x and y are non-zero finite. */ 158 if (ix & 0x80000000) { 159 /* Finite x < 0. */ 160 int yint = checkint(iy); 161 if (yint == 0) 162 return __math_invalidf(x); 163 if (yint == 1) 164 sign_bias = SIGN_BIAS; 165 ix &= 0x7fffffff; 166 } 167 if (ix < 0x00800000) { 168 /* Normalize subnormal x so exponent becomes negative. */ 169 ix = asuint(x * 0x1p23f); 170 ix &= 0x7fffffff; 171 ix -= 23 << 23; 172 } 173 } 174 double_t logx = log2_inline(ix); 175 double_t ylogx = y * logx; /* cannot overflow, y is single prec. */ 176 if (predict_false((asuint64(ylogx) >> 47 & 0xffff) >= 177 asuint64(126.0 * POWF_SCALE) >> 47)) { 178 /* |y*log(x)| >= 126. */ 179 if (ylogx > 0x1.fffffffd1d571p+6 * POWF_SCALE) 180 return __math_oflowf(sign_bias); 181 if (ylogx <= -150.0 * POWF_SCALE) 182 return __math_uflowf(sign_bias); 183 } 184 return exp2_inline(ylogx, sign_bias); 185 } 186