1 /* 2 * Double-precision SVE 10^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 "pl_sig.h" 10 #include "pl_test.h" 11 #include "poly_sve_f64.h" 12 13 #define SpecialBound 307.0 /* floor (log10 (2^1023)). */ 14 15 static const struct data 16 { 17 double poly[5]; 18 double shift, log10_2, log2_10_hi, log2_10_lo, scale_thres, special_bound; 19 } data = { 20 /* Coefficients generated using Remez algorithm. 21 rel error: 0x1.9fcb9b3p-60 22 abs error: 0x1.a20d9598p-60 in [ -log10(2)/128, log10(2)/128 ] 23 max ulp err 0.52 +0.5. */ 24 .poly = { 0x1.26bb1bbb55516p1, 0x1.53524c73cd32ap1, 0x1.0470591daeafbp1, 25 0x1.2bd77b1361ef6p0, 0x1.142b5d54e9621p-1 }, 26 /* 1.5*2^46+1023. This value is further explained below. */ 27 .shift = 0x1.800000000ffc0p+46, 28 .log10_2 = 0x1.a934f0979a371p1, /* 1/log2(10). */ 29 .log2_10_hi = 0x1.34413509f79ffp-2, /* log2(10). */ 30 .log2_10_lo = -0x1.9dc1da994fd21p-59, 31 .scale_thres = 1280.0, 32 .special_bound = SpecialBound, 33 }; 34 35 #define SpecialOffset 0x6000000000000000 /* 0x1p513. */ 36 /* SpecialBias1 + SpecialBias1 = asuint(1.0). */ 37 #define SpecialBias1 0x7000000000000000 /* 0x1p769. */ 38 #define SpecialBias2 0x3010000000000000 /* 0x1p-254. */ 39 40 /* Update of both special and non-special cases, if any special case is 41 detected. */ 42 static inline svfloat64_t 43 special_case (svbool_t pg, svfloat64_t s, svfloat64_t y, svfloat64_t n, 44 const struct data *d) 45 { 46 /* s=2^n may overflow, break it up into s=s1*s2, 47 such that exp = s + s*y can be computed as s1*(s2+s2*y) 48 and s1*s1 overflows only if n>0. */ 49 50 /* If n<=0 then set b to 0x6, 0 otherwise. */ 51 svbool_t p_sign = svcmple (pg, n, 0.0); /* n <= 0. */ 52 svuint64_t b = svdup_u64_z (p_sign, SpecialOffset); 53 54 /* Set s1 to generate overflow depending on sign of exponent n. */ 55 svfloat64_t s1 = svreinterpret_f64 (svsubr_x (pg, b, SpecialBias1)); 56 /* Offset s to avoid overflow in final result if n is below threshold. */ 57 svfloat64_t s2 = svreinterpret_f64 ( 58 svadd_x (pg, svsub_x (pg, svreinterpret_u64 (s), SpecialBias2), b)); 59 60 /* |n| > 1280 => 2^(n) overflows. */ 61 svbool_t p_cmp = svacgt (pg, n, d->scale_thres); 62 63 svfloat64_t r1 = svmul_x (pg, s1, s1); 64 svfloat64_t r2 = svmla_x (pg, s2, s2, y); 65 svfloat64_t r0 = svmul_x (pg, r2, s1); 66 67 return svsel (p_cmp, r1, r0); 68 } 69 70 /* Fast vector implementation of exp10 using FEXPA instruction. 71 Maximum measured error is 1.02 ulp. 72 SV_NAME_D1 (exp10)(-0x1.2862fec805e58p+2) got 0x1.885a89551d782p-16 73 want 0x1.885a89551d781p-16. */ 74 svfloat64_t SV_NAME_D1 (exp10) (svfloat64_t x, svbool_t pg) 75 { 76 const struct data *d = ptr_barrier (&data); 77 svbool_t no_big_scale = svacle (pg, x, d->special_bound); 78 svbool_t special = svnot_z (pg, no_big_scale); 79 80 /* n = round(x/(log10(2)/N)). */ 81 svfloat64_t shift = sv_f64 (d->shift); 82 svfloat64_t z = svmla_x (pg, shift, x, d->log10_2); 83 svfloat64_t n = svsub_x (pg, z, shift); 84 85 /* r = x - n*log10(2)/N. */ 86 svfloat64_t log2_10 = svld1rq (svptrue_b64 (), &d->log2_10_hi); 87 svfloat64_t r = x; 88 r = svmls_lane (r, n, log2_10, 0); 89 r = svmls_lane (r, n, log2_10, 1); 90 91 /* scale = 2^(n/N), computed using FEXPA. FEXPA does not propagate NaNs, so 92 for consistent NaN handling we have to manually propagate them. This 93 comes at significant performance cost. */ 94 svuint64_t u = svreinterpret_u64 (z); 95 svfloat64_t scale = svexpa (u); 96 97 /* Approximate exp10(r) using polynomial. */ 98 svfloat64_t r2 = svmul_x (pg, r, r); 99 svfloat64_t y = svmla_x (pg, svmul_x (pg, r, d->poly[0]), r2, 100 sv_pairwise_poly_3_f64_x (pg, r, r2, d->poly + 1)); 101 102 /* Assemble result as exp10(x) = 2^n * exp10(r). If |x| > SpecialBound 103 multiplication may overflow, so use special case routine. */ 104 if (unlikely (svptest_any (pg, special))) 105 { 106 /* FEXPA zeroes the sign bit, however the sign is meaningful to the 107 special case function so needs to be copied. 108 e = sign bit of u << 46. */ 109 svuint64_t e = svand_x (pg, svlsl_x (pg, u, 46), 0x8000000000000000); 110 /* Copy sign to scale. */ 111 scale = svreinterpret_f64 (svadd_x (pg, e, svreinterpret_u64 (scale))); 112 return special_case (pg, scale, y, n, d); 113 } 114 115 /* No special case. */ 116 return svmla_x (pg, scale, scale, y); 117 } 118 119 PL_SIG (SV, D, 1, exp10, -9.9, 9.9) 120 PL_TEST_ULP (SV_NAME_D1 (exp10), 0.52) 121 PL_TEST_SYM_INTERVAL (SV_NAME_D1 (exp10), 0, 307, 10000) 122 PL_TEST_SYM_INTERVAL (SV_NAME_D1 (exp10), 307, inf, 1000) 123