1 /* 2 * Single-precision e^x function. 3 * 4 * Copyright (c) 2017-2023, 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 EXPF_TABLE_BITS = 5 14 EXPF_POLY_ORDER = 3 15 16 ULP error: 0.502 (nearest rounding.) 17 Relative error: 1.69 * 2^-34 in [-ln2/64, ln2/64] (before rounding.) 18 Wrong count: 170635 (all nearest rounding wrong results with fma.) 19 Non-nearest ULP error: 1 (rounded ULP error) 20 */ 21 22 #define N (1 << EXPF_TABLE_BITS) 23 #define InvLn2N __expf_data.invln2_scaled 24 #define T __expf_data.tab 25 #define C __expf_data.poly_scaled 26 27 static inline uint32_t 28 top12 (float x) 29 { 30 return asuint (x) >> 20; 31 } 32 33 float 34 optr_aor_exp_f32 (float x) 35 { 36 uint32_t abstop; 37 uint64_t ki, t; 38 /* double_t for better performance on targets with FLT_EVAL_METHOD==2. */ 39 double_t kd, xd, z, r, r2, y, s; 40 41 xd = (double_t) x; 42 abstop = top12 (x) & 0x7ff; 43 if (unlikely (abstop >= top12 (88.0f))) 44 { 45 /* |x| >= 88 or x is nan. */ 46 if (asuint (x) == asuint (-INFINITY)) 47 return 0.0f; 48 if (abstop >= top12 (INFINITY)) 49 return x + x; 50 if (x > 0x1.62e42ep6f) /* x > log(0x1p128) ~= 88.72 */ 51 return __math_oflowf (0); 52 if (x < -0x1.9fe368p6f) /* x < log(0x1p-150) ~= -103.97 */ 53 return __math_uflowf (0); 54 } 55 56 /* x*N/Ln2 = k + r with r in [-1/2, 1/2] and int k. */ 57 z = InvLn2N * xd; 58 59 /* Round and convert z to int, the result is in [-150*N, 128*N] and 60 ideally nearest int is used, otherwise the magnitude of r can be 61 bigger which gives larger approximation error. */ 62 kd = round (z); 63 ki = lround (z); 64 r = z - kd; 65 66 /* exp(x) = 2^(k/N) * 2^(r/N) ~= s * (C0*r^3 + C1*r^2 + C2*r + 1) */ 67 t = T[ki % N]; 68 t += ki << (52 - EXPF_TABLE_BITS); 69 s = asdouble (t); 70 z = C[0] * r + C[1]; 71 r2 = r * r; 72 y = C[2] * r + 1; 73 y = z * r2 + y; 74 y = y * s; 75 return eval_as_float (y); 76 } 77