1 /* 2 * Double-precision vector exp(x) - 1 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 "poly_sve_f64.h" 10 #include "pl_sig.h" 11 #include "pl_test.h" 12 13 #define SpecialBound 0x1.62b7d369a5aa9p+9 14 #define ExponentBias 0x3ff0000000000000 15 16 static const struct data 17 { 18 double poly[11]; 19 double shift, inv_ln2, special_bound; 20 /* To be loaded in one quad-word. */ 21 double ln2_hi, ln2_lo; 22 } data = { 23 /* Generated using fpminimax. */ 24 .poly = { 0x1p-1, 0x1.5555555555559p-3, 0x1.555555555554bp-5, 25 0x1.111111110f663p-7, 0x1.6c16c16c1b5f3p-10, 0x1.a01a01affa35dp-13, 26 0x1.a01a018b4ecbbp-16, 0x1.71ddf82db5bb4p-19, 0x1.27e517fc0d54bp-22, 27 0x1.af5eedae67435p-26, 0x1.1f143d060a28ap-29, }, 28 29 .special_bound = SpecialBound, 30 .inv_ln2 = 0x1.71547652b82fep0, 31 .ln2_hi = 0x1.62e42fefa39efp-1, 32 .ln2_lo = 0x1.abc9e3b39803fp-56, 33 .shift = 0x1.8p52, 34 }; 35 36 static svfloat64_t NOINLINE 37 special_case (svfloat64_t x, svfloat64_t y, svbool_t pg) 38 { 39 return sv_call_f64 (expm1, x, y, pg); 40 } 41 42 /* Double-precision vector exp(x) - 1 function. 43 The maximum error observed error is 2.18 ULP: 44 _ZGVsMxv_expm1(0x1.634ba0c237d7bp-2) got 0x1.a8b9ea8d66e22p-2 45 want 0x1.a8b9ea8d66e2p-2. */ 46 svfloat64_t SV_NAME_D1 (expm1) (svfloat64_t x, svbool_t pg) 47 { 48 const struct data *d = ptr_barrier (&data); 49 50 /* Large, Nan/Inf. */ 51 svbool_t special = svnot_z (pg, svaclt (pg, x, d->special_bound)); 52 53 /* Reduce argument to smaller range: 54 Let i = round(x / ln2) 55 and f = x - i * ln2, then f is in [-ln2/2, ln2/2]. 56 exp(x) - 1 = 2^i * (expm1(f) + 1) - 1 57 where 2^i is exact because i is an integer. */ 58 svfloat64_t shift = sv_f64 (d->shift); 59 svfloat64_t n = svsub_x (pg, svmla_x (pg, shift, x, d->inv_ln2), shift); 60 svint64_t i = svcvt_s64_x (pg, n); 61 svfloat64_t ln2 = svld1rq (svptrue_b64 (), &d->ln2_hi); 62 svfloat64_t f = svmls_lane (x, n, ln2, 0); 63 f = svmls_lane (f, n, ln2, 1); 64 65 /* Approximate expm1(f) using polynomial. 66 Taylor expansion for expm1(x) has the form: 67 x + ax^2 + bx^3 + cx^4 .... 68 So we calculate the polynomial P(f) = a + bf + cf^2 + ... 69 and assemble the approximation expm1(f) ~= f + f^2 * P(f). */ 70 svfloat64_t f2 = svmul_x (pg, f, f); 71 svfloat64_t f4 = svmul_x (pg, f2, f2); 72 svfloat64_t f8 = svmul_x (pg, f4, f4); 73 svfloat64_t p 74 = svmla_x (pg, f, f2, sv_estrin_10_f64_x (pg, f, f2, f4, f8, d->poly)); 75 76 /* Assemble the result. 77 expm1(x) ~= 2^i * (p + 1) - 1 78 Let t = 2^i. */ 79 svint64_t u = svadd_x (pg, svlsl_x (pg, i, 52), ExponentBias); 80 svfloat64_t t = svreinterpret_f64 (u); 81 82 /* expm1(x) ~= p * t + (t - 1). */ 83 svfloat64_t y = svmla_x (pg, svsub_x (pg, t, 1), p, t); 84 85 if (unlikely (svptest_any (pg, special))) 86 return special_case (x, y, special); 87 88 return y; 89 } 90 91 PL_SIG (SV, D, 1, expm1, -9.9, 9.9) 92 PL_TEST_ULP (SV_NAME_D1 (expm1), 1.68) 93 PL_TEST_SYM_INTERVAL (SV_NAME_D1 (expm1), 0, 0x1p-23, 1000) 94 PL_TEST_SYM_INTERVAL (SV_NAME_D1 (expm1), 0x1p-23, SpecialBound, 200000) 95 PL_TEST_SYM_INTERVAL (SV_NAME_D1 (expm1), SpecialBound, inf, 1000) 96