1 /* 2 * Single-precision e^x function. 3 * 4 * Copyright (c) 2017-2018, Arm Limited. 5 * SPDX-License-Identifier: MIT 6 */ 7 8 #include <math.h> 9 #include <stdint.h> 10 #include "libm.h" 11 #include "exp2f_data.h" 12 13 /* 14 EXP2F_TABLE_BITS = 5 15 EXP2F_POLY_ORDER = 3 16 17 ULP error: 0.502 (nearest rounding.) 18 Relative error: 1.69 * 2^-34 in [-ln2/64, ln2/64] (before rounding.) 19 Wrong count: 170635 (all nearest rounding wrong results with fma.) 20 Non-nearest ULP error: 1 (rounded ULP error) 21 */ 22 23 #define N (1 << EXP2F_TABLE_BITS) 24 #define InvLn2N __exp2f_data.invln2_scaled 25 #define T __exp2f_data.tab 26 #define C __exp2f_data.poly_scaled 27 28 static inline uint32_t top12(float x) 29 { 30 return asuint(x) >> 20; 31 } 32 33 float expf(float x) 34 { 35 uint32_t abstop; 36 uint64_t ki, t; 37 double_t kd, xd, z, r, r2, y, s; 38 39 xd = (double_t)x; 40 abstop = top12(x) & 0x7ff; 41 if (predict_false(abstop >= top12(88.0f))) { 42 /* |x| >= 88 or x is nan. */ 43 if (asuint(x) == asuint(-INFINITY)) 44 return 0.0f; 45 if (abstop >= top12(INFINITY)) 46 return x + x; 47 if (x > 0x1.62e42ep6f) /* x > log(0x1p128) ~= 88.72 */ 48 return __math_oflowf(0); 49 if (x < -0x1.9fe368p6f) /* x < log(0x1p-150) ~= -103.97 */ 50 return __math_uflowf(0); 51 } 52 53 /* x*N/Ln2 = k + r with r in [-1/2, 1/2] and int k. */ 54 z = InvLn2N * xd; 55 56 /* Round and convert z to int, the result is in [-150*N, 128*N] and 57 ideally ties-to-even rule is used, otherwise the magnitude of r 58 can be bigger which gives larger approximation error. */ 59 #if TOINT_INTRINSICS 60 kd = roundtoint(z); 61 ki = converttoint(z); 62 #else 63 # define SHIFT __exp2f_data.shift 64 kd = eval_as_double(z + SHIFT); 65 ki = asuint64(kd); 66 kd -= SHIFT; 67 #endif 68 r = z - kd; 69 70 /* exp(x) = 2^(k/N) * 2^(r/N) ~= s * (C0*r^3 + C1*r^2 + C2*r + 1) */ 71 t = T[ki % N]; 72 t += ki << (52 - EXP2F_TABLE_BITS); 73 s = asdouble(t); 74 z = C[0] * r + C[1]; 75 r2 = r * r; 76 y = C[2] * r + 1; 77 y = z * r2 + y; 78 y = y * s; 79 return eval_as_float(y); 80 } 81