1 /* 2 * Single-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 "pl_sig.h" 10 #include "pl_test.h" 11 12 /* Largest value of x for which expm1(x) should round to -1. */ 13 #define SpecialBound 0x1.5ebc4p+6f 14 15 static const struct data 16 { 17 /* These 4 are grouped together so they can be loaded as one quadword, then 18 used with _lane forms of svmla/svmls. */ 19 float c2, c4, ln2_hi, ln2_lo; 20 float c0, c1, c3, inv_ln2, special_bound, shift; 21 } data = { 22 /* Generated using fpminimax. */ 23 .c0 = 0x1.fffffep-2, .c1 = 0x1.5554aep-3, 24 .c2 = 0x1.555736p-5, .c3 = 0x1.12287cp-7, 25 .c4 = 0x1.6b55a2p-10, 26 27 .special_bound = SpecialBound, .shift = 0x1.8p23f, 28 .inv_ln2 = 0x1.715476p+0f, .ln2_hi = 0x1.62e4p-1f, 29 .ln2_lo = 0x1.7f7d1cp-20f, 30 }; 31 32 #define C(i) sv_f32 (d->c##i) 33 34 static svfloat32_t NOINLINE 35 special_case (svfloat32_t x, svbool_t pg) 36 { 37 return sv_call_f32 (expm1f, x, x, pg); 38 } 39 40 /* Single-precision SVE exp(x) - 1. Maximum error is 1.52 ULP: 41 _ZGVsMxv_expm1f(0x1.8f4ebcp-2) got 0x1.e859dp-2 42 want 0x1.e859d4p-2. */ 43 svfloat32_t SV_NAME_F1 (expm1) (svfloat32_t x, svbool_t pg) 44 { 45 const struct data *d = ptr_barrier (&data); 46 47 /* Large, NaN/Inf. */ 48 svbool_t special = svnot_z (pg, svaclt (pg, x, d->special_bound)); 49 50 if (unlikely (svptest_any (pg, special))) 51 return special_case (x, pg); 52 53 /* This vector is reliant on layout of data - it contains constants 54 that can be used with _lane forms of svmla/svmls. Values are: 55 [ coeff_2, coeff_4, ln2_hi, ln2_lo ]. */ 56 svfloat32_t lane_constants = svld1rq (svptrue_b32 (), &d->c2); 57 58 /* Reduce argument to smaller range: 59 Let i = round(x / ln2) 60 and f = x - i * ln2, then f is in [-ln2/2, ln2/2]. 61 exp(x) - 1 = 2^i * (expm1(f) + 1) - 1 62 where 2^i is exact because i is an integer. */ 63 svfloat32_t j = svmla_x (pg, sv_f32 (d->shift), x, d->inv_ln2); 64 j = svsub_x (pg, j, d->shift); 65 svint32_t i = svcvt_s32_x (pg, j); 66 67 svfloat32_t f = svmls_lane (x, j, lane_constants, 2); 68 f = svmls_lane (f, j, lane_constants, 3); 69 70 /* Approximate expm1(f) using polynomial. 71 Taylor expansion for expm1(x) has the form: 72 x + ax^2 + bx^3 + cx^4 .... 73 So we calculate the polynomial P(f) = a + bf + cf^2 + ... 74 and assemble the approximation expm1(f) ~= f + f^2 * P(f). */ 75 svfloat32_t p12 = svmla_lane (C (1), f, lane_constants, 0); 76 svfloat32_t p34 = svmla_lane (C (3), f, lane_constants, 1); 77 svfloat32_t f2 = svmul_x (pg, f, f); 78 svfloat32_t p = svmla_x (pg, p12, f2, p34); 79 p = svmla_x (pg, C (0), f, p); 80 p = svmla_x (pg, f, f2, p); 81 82 /* Assemble the result. 83 expm1(x) ~= 2^i * (p + 1) - 1 84 Let t = 2^i. */ 85 svfloat32_t t = svreinterpret_f32 ( 86 svadd_x (pg, svreinterpret_u32 (svlsl_x (pg, i, 23)), 0x3f800000)); 87 return svmla_x (pg, svsub_x (pg, t, 1), p, t); 88 } 89 90 PL_SIG (SV, F, 1, expm1, -9.9, 9.9) 91 PL_TEST_ULP (SV_NAME_F1 (expm1), 1.02) 92 PL_TEST_SYM_INTERVAL (SV_NAME_F1 (expm1), 0, SpecialBound, 100000) 93 PL_TEST_SYM_INTERVAL (SV_NAME_F1 (expm1), SpecialBound, inf, 1000) 94