1 /* 2 * Double-precision vector 2^x function. 3 * 4 * Copyright (c) 2019-2023, Arm Limited. 5 * SPDX-License-Identifier: MIT OR Apache-2.0 WITH LLVM-exception 6 */ 7 8 #include "v_math.h" 9 #include "poly_advsimd_f64.h" 10 #include "pl_sig.h" 11 #include "pl_test.h" 12 13 #define N (1 << V_EXP_TABLE_BITS) 14 #define IndexMask (N - 1) 15 #define BigBound 1022.0 16 #define UOFlowBound 1280.0 17 18 static const struct data 19 { 20 float64x2_t poly[4]; 21 float64x2_t shift, scale_big_bound, scale_uoflow_bound; 22 } data = { 23 /* Coefficients are computed using Remez algorithm with 24 minimisation of the absolute error. */ 25 .poly = { V2 (0x1.62e42fefa3686p-1), V2 (0x1.ebfbdff82c241p-3), 26 V2 (0x1.c6b09b16de99ap-5), V2 (0x1.3b2abf5571ad8p-7) }, 27 .shift = V2 (0x1.8p52 / N), 28 .scale_big_bound = V2 (BigBound), 29 .scale_uoflow_bound = V2 (UOFlowBound), 30 }; 31 32 static inline uint64x2_t 33 lookup_sbits (uint64x2_t i) 34 { 35 return (uint64x2_t){ __v_exp_data[i[0] & IndexMask], 36 __v_exp_data[i[1] & IndexMask] }; 37 } 38 39 #if WANT_SIMD_EXCEPT 40 41 # define TinyBound 0x2000000000000000 /* asuint64(0x1p-511). */ 42 # define Thres 0x2080000000000000 /* asuint64(512.0) - TinyBound. */ 43 44 /* Call scalar exp2 as a fallback. */ 45 static float64x2_t VPCS_ATTR NOINLINE 46 special_case (float64x2_t x, float64x2_t y, uint64x2_t is_special) 47 { 48 return v_call_f64 (exp2, x, y, is_special); 49 } 50 51 #else 52 53 # define SpecialOffset 0x6000000000000000 /* 0x1p513. */ 54 /* SpecialBias1 + SpecialBias1 = asuint(1.0). */ 55 # define SpecialBias1 0x7000000000000000 /* 0x1p769. */ 56 # define SpecialBias2 0x3010000000000000 /* 0x1p-254. */ 57 58 static inline float64x2_t VPCS_ATTR 59 special_case (float64x2_t s, float64x2_t y, float64x2_t n, 60 const struct data *d) 61 { 62 /* 2^(n/N) may overflow, break it up into s1*s2. */ 63 uint64x2_t b = vandq_u64 (vclezq_f64 (n), v_u64 (SpecialOffset)); 64 float64x2_t s1 = vreinterpretq_f64_u64 (vsubq_u64 (v_u64 (SpecialBias1), b)); 65 float64x2_t s2 = vreinterpretq_f64_u64 ( 66 vaddq_u64 (vsubq_u64 (vreinterpretq_u64_f64 (s), v_u64 (SpecialBias2)), b)); 67 uint64x2_t cmp = vcagtq_f64 (n, d->scale_uoflow_bound); 68 float64x2_t r1 = vmulq_f64 (s1, s1); 69 float64x2_t r0 = vmulq_f64 (vfmaq_f64 (s2, s2, y), s1); 70 return vbslq_f64 (cmp, r1, r0); 71 } 72 73 #endif 74 75 /* Fast vector implementation of exp2. 76 Maximum measured error is 1.65 ulp. 77 _ZGVnN2v_exp2(-0x1.4c264ab5b559bp-6) got 0x1.f8db0d4df721fp-1 78 want 0x1.f8db0d4df721dp-1. */ 79 VPCS_ATTR 80 float64x2_t V_NAME_D1 (exp2) (float64x2_t x) 81 { 82 const struct data *d = ptr_barrier (&data); 83 uint64x2_t cmp; 84 #if WANT_SIMD_EXCEPT 85 uint64x2_t ia = vreinterpretq_u64_f64 (vabsq_f64 (x)); 86 cmp = vcgeq_u64 (vsubq_u64 (ia, v_u64 (TinyBound)), v_u64 (Thres)); 87 /* Mask special lanes and retain a copy of x for passing to special-case 88 handler. */ 89 float64x2_t xc = x; 90 x = v_zerofy_f64 (x, cmp); 91 #else 92 cmp = vcagtq_f64 (x, d->scale_big_bound); 93 #endif 94 95 /* n = round(x/N). */ 96 float64x2_t z = vaddq_f64 (d->shift, x); 97 uint64x2_t u = vreinterpretq_u64_f64 (z); 98 float64x2_t n = vsubq_f64 (z, d->shift); 99 100 /* r = x - n/N. */ 101 float64x2_t r = vsubq_f64 (x, n); 102 103 /* s = 2^(n/N). */ 104 uint64x2_t e = vshlq_n_u64 (u, 52 - V_EXP_TABLE_BITS); 105 u = lookup_sbits (u); 106 float64x2_t s = vreinterpretq_f64_u64 (vaddq_u64 (u, e)); 107 108 /* y ~ exp2(r) - 1. */ 109 float64x2_t r2 = vmulq_f64 (r, r); 110 float64x2_t y = v_pairwise_poly_3_f64 (r, r2, d->poly); 111 y = vmulq_f64 (r, y); 112 113 if (unlikely (v_any_u64 (cmp))) 114 #if !WANT_SIMD_EXCEPT 115 return special_case (s, y, n, d); 116 #else 117 return special_case (xc, vfmaq_f64 (s, s, y), cmp); 118 #endif 119 return vfmaq_f64 (s, s, y); 120 } 121 122 PL_SIG (V, D, 1, exp2, -9.9, 9.9) 123 PL_TEST_ULP (V_NAME_D1 (exp2), 1.15) 124 PL_TEST_EXPECT_FENV (V_NAME_D1 (exp2), WANT_SIMD_EXCEPT) 125 PL_TEST_SYM_INTERVAL (V_NAME_D1 (exp2), 0, TinyBound, 5000) 126 PL_TEST_SYM_INTERVAL (V_NAME_D1 (exp2), TinyBound, BigBound, 10000) 127 PL_TEST_SYM_INTERVAL (V_NAME_D1 (exp2), BigBound, UOFlowBound, 5000) 128 PL_TEST_SYM_INTERVAL (V_NAME_D1 (exp2), UOFlowBound, inf, 10000) 129