1 /* 2 * Double-precision SVE 2^x function. 3 * 4 * Copyright (c) 2023, Arm Limited. 5 * SPDX-License-Identifier: MIT OR Apache-2.0 WITH LLVM-exception 6 */ 7 8 #include "sv_math.h" 9 #include "poly_sve_f64.h" 10 #include "pl_sig.h" 11 #include "pl_test.h" 12 13 #define N (1 << V_EXP_TABLE_BITS) 14 15 #define BigBound 1022 16 #define UOFlowBound 1280 17 18 static const struct data 19 { 20 double poly[4]; 21 double shift, big_bound, uoflow_bound; 22 } data = { 23 /* Coefficients are computed using Remez algorithm with 24 minimisation of the absolute error. */ 25 .poly = { 0x1.62e42fefa3686p-1, 0x1.ebfbdff82c241p-3, 0x1.c6b09b16de99ap-5, 26 0x1.3b2abf5571ad8p-7 }, 27 .shift = 0x1.8p52 / N, 28 .uoflow_bound = UOFlowBound, 29 .big_bound = BigBound, 30 }; 31 32 #define SpecialOffset 0x6000000000000000 /* 0x1p513. */ 33 /* SpecialBias1 + SpecialBias1 = asuint(1.0). */ 34 #define SpecialBias1 0x7000000000000000 /* 0x1p769. */ 35 #define SpecialBias2 0x3010000000000000 /* 0x1p-254. */ 36 37 /* Update of both special and non-special cases, if any special case is 38 detected. */ 39 static inline svfloat64_t 40 special_case (svbool_t pg, svfloat64_t s, svfloat64_t y, svfloat64_t n, 41 const struct data *d) 42 { 43 /* s=2^n may overflow, break it up into s=s1*s2, 44 such that exp = s + s*y can be computed as s1*(s2+s2*y) 45 and s1*s1 overflows only if n>0. */ 46 47 /* If n<=0 then set b to 0x6, 0 otherwise. */ 48 svbool_t p_sign = svcmple (pg, n, 0.0); /* n <= 0. */ 49 svuint64_t b = svdup_u64_z (p_sign, SpecialOffset); 50 51 /* Set s1 to generate overflow depending on sign of exponent n. */ 52 svfloat64_t s1 = svreinterpret_f64 (svsubr_x (pg, b, SpecialBias1)); 53 /* Offset s to avoid overflow in final result if n is below threshold. */ 54 svfloat64_t s2 = svreinterpret_f64 ( 55 svadd_x (pg, svsub_x (pg, svreinterpret_u64 (s), SpecialBias2), b)); 56 57 /* |n| > 1280 => 2^(n) overflows. */ 58 svbool_t p_cmp = svacgt (pg, n, d->uoflow_bound); 59 60 svfloat64_t r1 = svmul_x (pg, s1, s1); 61 svfloat64_t r2 = svmla_x (pg, s2, s2, y); 62 svfloat64_t r0 = svmul_x (pg, r2, s1); 63 64 return svsel (p_cmp, r1, r0); 65 } 66 67 /* Fast vector implementation of exp2. 68 Maximum measured error is 1.65 ulp. 69 _ZGVsMxv_exp2(-0x1.4c264ab5b559bp-6) got 0x1.f8db0d4df721fp-1 70 want 0x1.f8db0d4df721dp-1. */ 71 svfloat64_t SV_NAME_D1 (exp2) (svfloat64_t x, svbool_t pg) 72 { 73 const struct data *d = ptr_barrier (&data); 74 svbool_t no_big_scale = svacle (pg, x, d->big_bound); 75 svbool_t special = svnot_z (pg, no_big_scale); 76 77 /* Reduce x to k/N + r, where k is integer and r in [-1/2N, 1/2N]. */ 78 svfloat64_t shift = sv_f64 (d->shift); 79 svfloat64_t kd = svadd_x (pg, x, shift); 80 svuint64_t ki = svreinterpret_u64 (kd); 81 /* kd = k/N. */ 82 kd = svsub_x (pg, kd, shift); 83 svfloat64_t r = svsub_x (pg, x, kd); 84 85 /* scale ~= 2^(k/N). */ 86 svuint64_t idx = svand_x (pg, ki, N - 1); 87 svuint64_t sbits = svld1_gather_index (pg, __v_exp_data, idx); 88 /* This is only a valid scale when -1023*N < k < 1024*N. */ 89 svuint64_t top = svlsl_x (pg, ki, 52 - V_EXP_TABLE_BITS); 90 svfloat64_t scale = svreinterpret_f64 (svadd_x (pg, sbits, top)); 91 92 /* Approximate exp2(r) using polynomial. */ 93 svfloat64_t r2 = svmul_x (pg, r, r); 94 svfloat64_t p = sv_pairwise_poly_3_f64_x (pg, r, r2, d->poly); 95 svfloat64_t y = svmul_x (pg, r, p); 96 97 /* Assemble exp2(x) = exp2(r) * scale. */ 98 if (unlikely (svptest_any (pg, special))) 99 return special_case (pg, scale, y, kd, d); 100 return svmla_x (pg, scale, scale, y); 101 } 102 103 PL_SIG (SV, D, 1, exp2, -9.9, 9.9) 104 PL_TEST_ULP (SV_NAME_D1 (exp2), 1.15) 105 PL_TEST_SYM_INTERVAL (SV_NAME_D1 (exp2), 0, BigBound, 1000) 106 PL_TEST_SYM_INTERVAL (SV_NAME_D1 (exp2), BigBound, UOFlowBound, 100000) 107 PL_TEST_SYM_INTERVAL (SV_NAME_D1 (exp2), UOFlowBound, inf, 1000) 108