1 /* 2 * Double-precision vector erf(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 12 static const struct data 13 { 14 double third; 15 double tenth, two_over_five, two_over_fifteen; 16 double two_over_nine, two_over_fortyfive; 17 double max, shift; 18 } data = { 19 .third = 0x1.5555555555556p-2, /* used to compute 2/3 and 1/6 too. */ 20 .two_over_fifteen = 0x1.1111111111111p-3, 21 .tenth = -0x1.999999999999ap-4, 22 .two_over_five = -0x1.999999999999ap-2, 23 .two_over_nine = -0x1.c71c71c71c71cp-3, 24 .two_over_fortyfive = 0x1.6c16c16c16c17p-5, 25 .max = 5.9921875, /* 6 - 1/128. */ 26 .shift = 0x1p45, 27 }; 28 29 #define SignMask (0x8000000000000000) 30 31 /* Double-precision implementation of vector erf(x). 32 Approximation based on series expansion near x rounded to 33 nearest multiple of 1/128. 34 Let d = x - r, and scale = 2 / sqrt(pi) * exp(-r^2). For x near r, 35 erf(x) ~ erf(r) + scale * d * [ 36 + 1 37 - r d 38 + 1/3 (2 r^2 - 1) d^2 39 - 1/6 (r (2 r^2 - 3)) d^3 40 + 1/30 (4 r^4 - 12 r^2 + 3) d^4 41 - 1/90 (4 r^4 - 20 r^2 + 15) d^5 42 ] 43 44 Maximum measure error: 2.29 ULP 45 _ZGVsMxv_erf(-0x1.00003c924e5d1p-8) got -0x1.20dd59132ebadp-8 46 want -0x1.20dd59132ebafp-8. */ 47 svfloat64_t SV_NAME_D1 (erf) (svfloat64_t x, const svbool_t pg) 48 { 49 const struct data *dat = ptr_barrier (&data); 50 51 /* |x| >= 6.0 - 1/128. Opposite conditions except none of them catch NaNs so 52 they can be used in lookup and BSLs to yield the expected results. */ 53 svbool_t a_ge_max = svacge (pg, x, dat->max); 54 svbool_t a_lt_max = svaclt (pg, x, dat->max); 55 56 /* Set r to multiple of 1/128 nearest to |x|. */ 57 svfloat64_t a = svabs_x (pg, x); 58 svfloat64_t shift = sv_f64 (dat->shift); 59 svfloat64_t z = svadd_x (pg, a, shift); 60 svuint64_t i 61 = svsub_x (pg, svreinterpret_u64 (z), svreinterpret_u64 (shift)); 62 63 /* Lookup without shortcut for small values but with predicate to avoid 64 segfault for large values and NaNs. */ 65 svfloat64_t r = svsub_x (pg, z, shift); 66 svfloat64_t erfr = svld1_gather_index (a_lt_max, __sv_erf_data.erf, i); 67 svfloat64_t scale = svld1_gather_index (a_lt_max, __sv_erf_data.scale, i); 68 69 /* erf(x) ~ erf(r) + scale * d * poly (r, d). */ 70 svfloat64_t d = svsub_x (pg, a, r); 71 svfloat64_t d2 = svmul_x (pg, d, d); 72 svfloat64_t r2 = svmul_x (pg, r, r); 73 74 /* poly (d, r) = 1 + p1(r) * d + p2(r) * d^2 + ... + p5(r) * d^5. */ 75 svfloat64_t p1 = r; 76 svfloat64_t third = sv_f64 (dat->third); 77 svfloat64_t twothird = svmul_x (pg, third, 2.0); 78 svfloat64_t sixth = svmul_x (pg, third, 0.5); 79 svfloat64_t p2 = svmls_x (pg, third, r2, twothird); 80 svfloat64_t p3 = svmad_x (pg, r2, third, -0.5); 81 p3 = svmul_x (pg, r, p3); 82 svfloat64_t p4 83 = svmla_x (pg, sv_f64 (dat->two_over_five), r2, dat->two_over_fifteen); 84 p4 = svmls_x (pg, sv_f64 (dat->tenth), r2, p4); 85 svfloat64_t p5 86 = svmla_x (pg, sv_f64 (dat->two_over_nine), r2, dat->two_over_fortyfive); 87 p5 = svmla_x (pg, sixth, r2, p5); 88 p5 = svmul_x (pg, r, p5); 89 90 svfloat64_t p34 = svmla_x (pg, p3, d, p4); 91 svfloat64_t p12 = svmla_x (pg, p1, d, p2); 92 svfloat64_t y = svmla_x (pg, p34, d2, p5); 93 y = svmla_x (pg, p12, d2, y); 94 95 y = svmla_x (pg, erfr, scale, svmls_x (pg, d, d2, y)); 96 97 /* Solves the |x| = inf and NaN cases. */ 98 y = svsel (a_ge_max, sv_f64 (1.0), y); 99 100 /* Copy sign. */ 101 svuint64_t ix = svreinterpret_u64 (x); 102 svuint64_t iy = svreinterpret_u64 (y); 103 svuint64_t sign = svand_x (pg, ix, SignMask); 104 return svreinterpret_f64 (svorr_x (pg, sign, iy)); 105 } 106 107 PL_SIG (SV, D, 1, erf, -6.0, 6.0) 108 PL_TEST_ULP (SV_NAME_D1 (erf), 1.79) 109 PL_TEST_SYM_INTERVAL (SV_NAME_D1 (erf), 0, 5.9921875, 40000) 110 PL_TEST_SYM_INTERVAL (SV_NAME_D1 (erf), 5.9921875, inf, 40000) 111 PL_TEST_SYM_INTERVAL (SV_NAME_D1 (erf), 0, inf, 4000) 112