1 /* 2 * Double-precision SVE cbrt(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 const static struct data 14 { 15 float64_t poly[4]; 16 float64_t table[5]; 17 float64_t one_third, two_thirds, shift; 18 int64_t exp_bias; 19 uint64_t tiny_bound, thresh; 20 } data = { 21 /* Generated with FPMinimax in [0.5, 1]. */ 22 .poly = { 0x1.c14e8ee44767p-2, 0x1.dd2d3f99e4c0ep-1, -0x1.08e83026b7e74p-1, 23 0x1.2c74eaa3ba428p-3, }, 24 /* table[i] = 2^((i - 2) / 3). */ 25 .table = { 0x1.428a2f98d728bp-1, 0x1.965fea53d6e3dp-1, 0x1p0, 26 0x1.428a2f98d728bp0, 0x1.965fea53d6e3dp0, }, 27 .one_third = 0x1.5555555555555p-2, 28 .two_thirds = 0x1.5555555555555p-1, 29 .shift = 0x1.8p52, 30 .exp_bias = 1022, 31 .tiny_bound = 0x0010000000000000, /* Smallest normal. */ 32 .thresh = 0x7fe0000000000000, /* asuint64 (infinity) - tiny_bound. */ 33 }; 34 35 #define MantissaMask 0x000fffffffffffff 36 #define HalfExp 0x3fe0000000000000 37 38 static svfloat64_t NOINLINE 39 special_case (svfloat64_t x, svfloat64_t y, svbool_t special) 40 { 41 return sv_call_f64 (cbrt, x, y, special); 42 } 43 44 static inline svfloat64_t 45 shifted_lookup (const svbool_t pg, const float64_t *table, svint64_t i) 46 { 47 return svld1_gather_index (pg, table, svadd_x (pg, i, 2)); 48 } 49 50 /* Approximation for double-precision vector cbrt(x), using low-order 51 polynomial and two Newton iterations. Greatest observed error is 1.79 ULP. 52 Errors repeat according to the exponent, for instance an error observed for 53 double value m * 2^e will be observed for any input m * 2^(e + 3*i), where i 54 is an integer. 55 _ZGVsMxv_cbrt (0x0.3fffb8d4413f3p-1022) got 0x1.965f53b0e5d97p-342 56 want 0x1.965f53b0e5d95p-342. */ 57 svfloat64_t SV_NAME_D1 (cbrt) (svfloat64_t x, const svbool_t pg) 58 { 59 const struct data *d = ptr_barrier (&data); 60 61 svfloat64_t ax = svabs_x (pg, x); 62 svuint64_t iax = svreinterpret_u64 (ax); 63 svuint64_t sign = sveor_x (pg, svreinterpret_u64 (x), iax); 64 65 /* Subnormal, +/-0 and special values. */ 66 svbool_t special = svcmpge (pg, svsub_x (pg, iax, d->tiny_bound), d->thresh); 67 68 /* Decompose |x| into m * 2^e, where m is in [0.5, 1.0]. This is a vector 69 version of frexp, which gets subnormal values wrong - these have to be 70 special-cased as a result. */ 71 svfloat64_t m = svreinterpret_f64 (svorr_x ( 72 pg, svand_x (pg, svreinterpret_u64 (x), MantissaMask), HalfExp)); 73 svint64_t e 74 = svsub_x (pg, svreinterpret_s64 (svlsr_x (pg, iax, 52)), d->exp_bias); 75 76 /* Calculate rough approximation for cbrt(m) in [0.5, 1.0], starting point 77 for Newton iterations. */ 78 svfloat64_t p 79 = sv_pairwise_poly_3_f64_x (pg, m, svmul_x (pg, m, m), d->poly); 80 81 /* Two iterations of Newton's method for iteratively approximating cbrt. */ 82 svfloat64_t m_by_3 = svmul_x (pg, m, d->one_third); 83 svfloat64_t a = svmla_x (pg, svdiv_x (pg, m_by_3, svmul_x (pg, p, p)), p, 84 d->two_thirds); 85 a = svmla_x (pg, svdiv_x (pg, m_by_3, svmul_x (pg, a, a)), a, d->two_thirds); 86 87 /* Assemble the result by the following: 88 89 cbrt(x) = cbrt(m) * 2 ^ (e / 3). 90 91 We can get 2 ^ round(e / 3) using ldexp and integer divide, but since e is 92 not necessarily a multiple of 3 we lose some information. 93 94 Let q = 2 ^ round(e / 3), then t = 2 ^ (e / 3) / q. 95 96 Then we know t = 2 ^ (i / 3), where i is the remainder from e / 3, which 97 is an integer in [-2, 2], and can be looked up in the table T. Hence the 98 result is assembled as: 99 100 cbrt(x) = cbrt(m) * t * 2 ^ round(e / 3) * sign. */ 101 svfloat64_t eb3f = svmul_x (pg, svcvt_f64_x (pg, e), d->one_third); 102 svint64_t ey = svcvt_s64_x (pg, eb3f); 103 svint64_t em3 = svmls_x (pg, e, ey, 3); 104 105 svfloat64_t my = shifted_lookup (pg, d->table, em3); 106 my = svmul_x (pg, my, a); 107 108 /* Vector version of ldexp. */ 109 svfloat64_t y = svscale_x (pg, my, ey); 110 111 if (unlikely (svptest_any (pg, special))) 112 return special_case ( 113 x, svreinterpret_f64 (svorr_x (pg, svreinterpret_u64 (y), sign)), 114 special); 115 116 /* Copy sign. */ 117 return svreinterpret_f64 (svorr_x (pg, svreinterpret_u64 (y), sign)); 118 } 119 120 PL_SIG (SV, D, 1, cbrt, -10.0, 10.0) 121 PL_TEST_ULP (SV_NAME_D1 (cbrt), 1.30) 122 PL_TEST_SYM_INTERVAL (SV_NAME_D1 (cbrt), 0, inf, 1000000) 123