1 /* 2 * Double-precision vector e^x function. 3 * 4 * Copyright (c) 2019, Arm Limited. 5 * SPDX-License-Identifier: MIT 6 */ 7 8 #include "mathlib.h" 9 #include "v_math.h" 10 #if V_SUPPORTED 11 #include "v_exp.h" 12 13 #if V_EXP_TABLE_BITS == 7 14 /* maxerr: 1.88 +0.5 ulp 15 rel error: 1.4337*2^-53 16 abs error: 1.4299*2^-53 in [ -ln2/256, ln2/256 ]. */ 17 #define C1 v_f64 (0x1.ffffffffffd43p-2) 18 #define C2 v_f64 (0x1.55555c75adbb2p-3) 19 #define C3 v_f64 (0x1.55555da646206p-5) 20 #define InvLn2 v_f64 (0x1.71547652b82fep7) /* N/ln2. */ 21 #define Ln2hi v_f64 (0x1.62e42fefa39efp-8) /* ln2/N. */ 22 #define Ln2lo v_f64 (0x1.abc9e3b39803f3p-63) 23 #elif V_EXP_TABLE_BITS == 8 24 /* maxerr: 0.54 +0.5 ulp 25 rel error: 1.4318*2^-58 26 abs error: 1.4299*2^-58 in [ -ln2/512, ln2/512 ]. */ 27 #define C1 v_f64 (0x1.fffffffffffd4p-2) 28 #define C2 v_f64 (0x1.5555571d6b68cp-3) 29 #define C3 v_f64 (0x1.5555576a59599p-5) 30 #define InvLn2 v_f64 (0x1.71547652b82fep8) 31 #define Ln2hi v_f64 (0x1.62e42fefa39efp-9) 32 #define Ln2lo v_f64 (0x1.abc9e3b39803f3p-64) 33 #endif 34 35 #define N (1 << V_EXP_TABLE_BITS) 36 #define Tab __v_exp_data 37 #define IndexMask v_u64 (N - 1) 38 #define Shift v_f64 (0x1.8p+52) 39 #define Thres v_f64 (704.0) 40 41 VPCS_ATTR 42 static v_f64_t 43 specialcase (v_f64_t s, v_f64_t y, v_f64_t n) 44 { 45 v_f64_t absn = v_abs_f64 (n); 46 47 /* 2^(n/N) may overflow, break it up into s1*s2. */ 48 v_u64_t b = v_cond_u64 (n <= v_f64 (0.0)) & v_u64 (0x6000000000000000); 49 v_f64_t s1 = v_as_f64_u64 (v_u64 (0x7000000000000000) - b); 50 v_f64_t s2 = v_as_f64_u64 (v_as_u64_f64 (s) - v_u64 (0x3010000000000000) + b); 51 v_u64_t cmp = v_cond_u64 (absn > v_f64 (1280.0 * N)); 52 v_f64_t r1 = s1 * s1; 53 v_f64_t r0 = v_fma_f64 (y, s2, s2) * s1; 54 return v_as_f64_u64 ((cmp & v_as_u64_f64 (r1)) | (~cmp & v_as_u64_f64 (r0))); 55 } 56 57 VPCS_ATTR 58 v_f64_t 59 V_NAME(exp) (v_f64_t x) 60 { 61 v_f64_t n, r, r2, s, y, z; 62 v_u64_t cmp, u, e, i; 63 64 cmp = v_cond_u64 (v_abs_f64 (x) > Thres); 65 66 /* n = round(x/(ln2/N)). */ 67 z = v_fma_f64 (x, InvLn2, Shift); 68 u = v_as_u64_f64 (z); 69 n = z - Shift; 70 71 /* r = x - n*ln2/N. */ 72 r = x; 73 r = v_fma_f64 (-Ln2hi, n, r); 74 r = v_fma_f64 (-Ln2lo, n, r); 75 76 e = u << (52 - V_EXP_TABLE_BITS); 77 i = u & IndexMask; 78 79 /* y = exp(r) - 1 ~= r + C1 r^2 + C2 r^3 + C3 r^4. */ 80 r2 = r * r; 81 y = v_fma_f64 (C2, r, C1); 82 y = v_fma_f64 (C3, r2, y); 83 y = v_fma_f64 (y, r2, r); 84 85 /* s = 2^(n/N). */ 86 u = v_lookup_u64 (Tab, i); 87 s = v_as_f64_u64 (u + e); 88 89 if (unlikely (v_any_u64 (cmp))) 90 return specialcase (s, y, n); 91 return v_fma_f64 (y, s, s); 92 } 93 VPCS_ALIAS 94 #endif 95