1 /* 2 * Double-precision SVE 10^x function. 3 * 4 * Copyright (c) 2023-2025, Arm Limited. 5 * SPDX-License-Identifier: MIT OR Apache-2.0 WITH LLVM-exception 6 */ 7 8 #include "sv_math.h" 9 #include "test_sig.h" 10 #include "test_defs.h" 11 12 #define SpecialBound 307.0 /* floor (log10 (2^1023)). */ 13 14 static const struct data 15 { 16 double c1, c3, c2, c4, c0; 17 double shift, log10_2, log2_10_hi, log2_10_lo, scale_thres, special_bound; 18 } data = { 19 /* Coefficients generated using Remez algorithm. 20 rel error: 0x1.9fcb9b3p-60 21 abs error: 0x1.a20d9598p-60 in [ -log10(2)/128, log10(2)/128 ] 22 max ulp err 0.52 +0.5. */ 23 .c0 = 0x1.26bb1bbb55516p1, 24 .c1 = 0x1.53524c73cd32ap1, 25 .c2 = 0x1.0470591daeafbp1, 26 .c3 = 0x1.2bd77b1361ef6p0, 27 .c4 = 0x1.142b5d54e9621p-1, 28 /* 1.5*2^46+1023. This value is further explained below. */ 29 .shift = 0x1.800000000ffc0p+46, 30 .log10_2 = 0x1.a934f0979a371p1, /* 1/log2(10). */ 31 .log2_10_hi = 0x1.34413509f79ffp-2, /* log2(10). */ 32 .log2_10_lo = -0x1.9dc1da994fd21p-59, 33 .scale_thres = 1280.0, 34 .special_bound = SpecialBound, 35 }; 36 37 #define SpecialOffset 0x6000000000000000 /* 0x1p513. */ 38 /* SpecialBias1 + SpecialBias1 = asuint(1.0). */ 39 #define SpecialBias1 0x7000000000000000 /* 0x1p769. */ 40 #define SpecialBias2 0x3010000000000000 /* 0x1p-254. */ 41 42 /* Update of both special and non-special cases, if any special case is 43 detected. */ 44 static inline svfloat64_t 45 special_case (svbool_t pg, svfloat64_t s, svfloat64_t y, svfloat64_t n, 46 const struct data *d) 47 { 48 /* s=2^n may overflow, break it up into s=s1*s2, 49 such that exp = s + s*y can be computed as s1*(s2+s2*y) 50 and s1*s1 overflows only if n>0. */ 51 52 /* If n<=0 then set b to 0x6, 0 otherwise. */ 53 svbool_t p_sign = svcmple (pg, n, 0.0); /* n <= 0. */ 54 svuint64_t b = svdup_u64_z (p_sign, SpecialOffset); 55 56 /* Set s1 to generate overflow depending on sign of exponent n. */ 57 svfloat64_t s1 = svreinterpret_f64 (svsubr_x (pg, b, SpecialBias1)); 58 /* Offset s to avoid overflow in final result if n is below threshold. */ 59 svfloat64_t s2 = svreinterpret_f64 ( 60 svadd_x (pg, svsub_x (pg, svreinterpret_u64 (s), SpecialBias2), b)); 61 62 /* |n| > 1280 => 2^(n) overflows. */ 63 svbool_t p_cmp = svacgt (pg, n, d->scale_thres); 64 65 svfloat64_t r1 = svmul_x (svptrue_b64 (), s1, s1); 66 svfloat64_t r2 = svmla_x (pg, s2, s2, y); 67 svfloat64_t r0 = svmul_x (svptrue_b64 (), r2, s1); 68 69 return svsel (p_cmp, r1, r0); 70 } 71 72 /* Fast vector implementation of exp10 using FEXPA instruction. 73 Maximum measured error is 1.02 ulp. 74 SV_NAME_D1 (exp10)(-0x1.2862fec805e58p+2) got 0x1.885a89551d782p-16 75 want 0x1.885a89551d781p-16. */ 76 svfloat64_t SV_NAME_D1 (exp10) (svfloat64_t x, svbool_t pg) 77 { 78 const struct data *d = ptr_barrier (&data); 79 svbool_t no_big_scale = svacle (pg, x, d->special_bound); 80 svbool_t special = svnot_z (pg, no_big_scale); 81 82 /* n = round(x/(log10(2)/N)). */ 83 svfloat64_t shift = sv_f64 (d->shift); 84 svfloat64_t z = svmla_x (pg, shift, x, d->log10_2); 85 svfloat64_t n = svsub_x (pg, z, shift); 86 87 /* r = x - n*log10(2)/N. */ 88 svfloat64_t log2_10 = svld1rq (svptrue_b64 (), &d->log2_10_hi); 89 svfloat64_t r = x; 90 r = svmls_lane (r, n, log2_10, 0); 91 r = svmls_lane (r, n, log2_10, 1); 92 93 /* scale = 2^(n/N), computed using FEXPA. FEXPA does not propagate NaNs, so 94 for consistent NaN handling we have to manually propagate them. This 95 comes at significant performance cost. */ 96 svuint64_t u = svreinterpret_u64 (z); 97 svfloat64_t scale = svexpa (u); 98 svfloat64_t c24 = svld1rq (svptrue_b64 (), &d->c2); 99 /* Approximate exp10(r) using polynomial. */ 100 svfloat64_t r2 = svmul_x (svptrue_b64 (), r, r); 101 svfloat64_t p12 = svmla_lane (sv_f64 (d->c1), r, c24, 0); 102 svfloat64_t p34 = svmla_lane (sv_f64 (d->c3), r, c24, 1); 103 svfloat64_t p14 = svmla_x (pg, p12, p34, r2); 104 105 svfloat64_t y = svmla_x (pg, svmul_x (svptrue_b64 (), r, d->c0), r2, p14); 106 107 /* Assemble result as exp10(x) = 2^n * exp10(r). If |x| > SpecialBound 108 multiplication may overflow, so use special case routine. */ 109 if (unlikely (svptest_any (pg, special))) 110 { 111 /* FEXPA zeroes the sign bit, however the sign is meaningful to the 112 special case function so needs to be copied. 113 e = sign bit of u << 46. */ 114 svuint64_t e = svand_x (pg, svlsl_x (pg, u, 46), 0x8000000000000000); 115 /* Copy sign to scale. */ 116 scale = svreinterpret_f64 (svadd_x (pg, e, svreinterpret_u64 (scale))); 117 return special_case (pg, scale, y, n, d); 118 } 119 120 /* No special case. */ 121 return svmla_x (pg, scale, scale, y); 122 } 123 124 #if WANT_EXP10_TESTS 125 TEST_SIG (SV, D, 1, exp10, -9.9, 9.9) 126 TEST_ULP (SV_NAME_D1 (exp10), 0.52) 127 TEST_DISABLE_FENV (SV_NAME_D1 (exp10)) 128 TEST_SYM_INTERVAL (SV_NAME_D1 (exp10), 0, SpecialBound, 10000) 129 TEST_SYM_INTERVAL (SV_NAME_D1 (exp10), SpecialBound, inf, 1000) 130 #endif 131 CLOSE_SVE_ATTR 132