1 /* 2 * Single-precision vector erfc(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 uint32_t off_idx, off_arr; 15 float max, shift; 16 float third, two_thirds, two_over_fifteen, two_over_five, tenth; 17 } data = { 18 /* Set an offset so the range of the index used for lookup is 644, and it can 19 be clamped using a saturated add. */ 20 .off_idx = 0xb7fffd7b, /* 0xffffffff - asuint(shift) - 644. */ 21 .off_arr = 0xfffffd7b, /* 0xffffffff - 644. */ 22 .max = 10.0625f, /* 644/64. */ 23 .shift = 0x1p17f, 24 .third = 0x1.555556p-2f, 25 .two_thirds = 0x1.555556p-1f, 26 .two_over_fifteen = 0x1.111112p-3f, 27 .two_over_five = -0x1.99999ap-2f, 28 .tenth = -0x1.99999ap-4f, 29 }; 30 31 #define SignMask 0x80000000 32 #define TableScale 0x28000000 /* 0x1p-47. */ 33 34 /* Optimized single-precision vector erfcf(x). 35 Approximation based on series expansion near x rounded to 36 nearest multiple of 1/64. 37 Let d = x - r, and scale = 2 / sqrt(pi) * exp(-r^2). For x near r, 38 39 erfc(x) ~ erfc(r) - scale * d * poly(r, d), with 40 41 poly(r, d) = 1 - r d + (2/3 r^2 - 1/3) d^2 - r (1/3 r^2 - 1/2) d^3 42 + (2/15 r^4 - 2/5 r^2 + 1/10) d^4 43 44 Values of erfc(r) and scale are read from lookup tables. Stored values 45 are scaled to avoid hitting the subnormal range. 46 47 Note that for x < 0, erfc(x) = 2.0 - erfc(-x). 48 49 Maximum error: 1.63 ULP (~1.0 ULP for x < 0.0). 50 _ZGVsMxv_erfcf(0x1.1dbf7ap+3) got 0x1.f51212p-120 51 want 0x1.f51216p-120. */ 52 svfloat32_t SV_NAME_F1 (erfc) (svfloat32_t x, const svbool_t pg) 53 { 54 const struct data *dat = ptr_barrier (&data); 55 56 svfloat32_t a = svabs_x (pg, x); 57 58 /* Clamp input at |x| <= 10.0 + 4/64. */ 59 a = svmin_x (pg, a, dat->max); 60 61 /* Reduce x to the nearest multiple of 1/64. */ 62 svfloat32_t shift = sv_f32 (dat->shift); 63 svfloat32_t z = svadd_x (pg, a, shift); 64 65 /* Saturate index for the NaN case. */ 66 svuint32_t i = svqadd (svreinterpret_u32 (z), dat->off_idx); 67 68 /* Lookup erfc(r) and 2/sqrt(pi)*exp(-r^2) in tables. */ 69 i = svmul_x (pg, i, 2); 70 const float32_t *p = &__erfcf_data.tab[0].erfc - 2 * dat->off_arr; 71 svfloat32_t erfcr = svld1_gather_index (pg, p, i); 72 svfloat32_t scale = svld1_gather_index (pg, p + 1, i); 73 74 /* erfc(x) ~ erfc(r) - scale * d * poly(r, d). */ 75 svfloat32_t r = svsub_x (pg, z, shift); 76 svfloat32_t d = svsub_x (pg, a, r); 77 svfloat32_t d2 = svmul_x (pg, d, d); 78 svfloat32_t r2 = svmul_x (pg, r, r); 79 80 svfloat32_t coeffs = svld1rq (svptrue_b32 (), &dat->third); 81 svfloat32_t third = svdup_lane (coeffs, 0); 82 83 svfloat32_t p1 = r; 84 svfloat32_t p2 = svmls_lane (third, r2, coeffs, 1); 85 svfloat32_t p3 = svmul_x (pg, r, svmla_lane (sv_f32 (-0.5), r2, coeffs, 0)); 86 svfloat32_t p4 = svmla_lane (sv_f32 (dat->two_over_five), r2, coeffs, 2); 87 p4 = svmls_x (pg, sv_f32 (dat->tenth), r2, p4); 88 89 svfloat32_t y = svmla_x (pg, p3, d, p4); 90 y = svmla_x (pg, p2, d, y); 91 y = svmla_x (pg, p1, d, y); 92 93 /* Solves the |x| = inf/nan case. */ 94 y = svmls_x (pg, erfcr, scale, svmls_x (pg, d, d2, y)); 95 96 /* Offset equals 2.0f if sign, else 0.0f. */ 97 svuint32_t sign = svand_x (pg, svreinterpret_u32 (x), SignMask); 98 svfloat32_t off = svreinterpret_f32 (svlsr_x (pg, sign, 1)); 99 /* Handle sign and scale back in a single fma. */ 100 svfloat32_t fac = svreinterpret_f32 (svorr_x (pg, sign, TableScale)); 101 102 return svmla_x (pg, off, fac, y); 103 } 104 105 PL_SIG (SV, F, 1, erfc, -4.0, 10.0) 106 PL_TEST_ULP (SV_NAME_F1 (erfc), 1.14) 107 PL_TEST_SYM_INTERVAL (SV_NAME_F1 (erfc), 0.0, 0x1p-26, 40000) 108 PL_TEST_INTERVAL (SV_NAME_F1 (erfc), 0x1p-26, 10.0625, 40000) 109 PL_TEST_INTERVAL (SV_NAME_F1 (erfc), -0x1p-26, -4.0, 40000) 110 PL_TEST_INTERVAL (SV_NAME_F1 (erfc), 10.0625, inf, 40000) 111 PL_TEST_INTERVAL (SV_NAME_F1 (erfc), -4.0, -inf, 40000) 112