1 /* 2 * Double-precision vector exp(x) - 1 function. 3 * 4 * Copyright (c) 2022-2023, Arm Limited. 5 * SPDX-License-Identifier: MIT OR Apache-2.0 WITH LLVM-exception 6 */ 7 8 #include "v_math.h" 9 #include "pl_sig.h" 10 #include "pl_test.h" 11 12 #if V_SUPPORTED 13 14 #define InvLn2 v_f64 (0x1.71547652b82fep0) 15 #define MLn2hi v_f64 (-0x1.62e42fefa39efp-1) 16 #define MLn2lo v_f64 (-0x1.abc9e3b39803fp-56) 17 #define Shift v_f64 (0x1.8p52) 18 #define TinyBound \ 19 0x3cc0000000000000 /* 0x1p-51, below which expm1(x) is within 2 ULP of x. */ 20 #define SpecialBound \ 21 0x40862b7d369a5aa9 /* 0x1.62b7d369a5aa9p+9. For |x| > SpecialBound, the \ 22 final stage of the algorithm overflows so fall back to \ 23 scalar. */ 24 #define AbsMask 0x7fffffffffffffff 25 #define One 0x3ff0000000000000 26 27 #define C(i) v_f64 (__expm1_poly[i]) 28 29 static inline v_f64_t 30 eval_poly (v_f64_t f, v_f64_t f2) 31 { 32 /* Evaluate custom polynomial using Estrin scheme. */ 33 v_f64_t p_01 = v_fma_f64 (f, C (1), C (0)); 34 v_f64_t p_23 = v_fma_f64 (f, C (3), C (2)); 35 v_f64_t p_45 = v_fma_f64 (f, C (5), C (4)); 36 v_f64_t p_67 = v_fma_f64 (f, C (7), C (6)); 37 v_f64_t p_89 = v_fma_f64 (f, C (9), C (8)); 38 39 v_f64_t p_03 = v_fma_f64 (f2, p_23, p_01); 40 v_f64_t p_47 = v_fma_f64 (f2, p_67, p_45); 41 v_f64_t p_8a = v_fma_f64 (f2, C (10), p_89); 42 43 v_f64_t f4 = f2 * f2; 44 v_f64_t p_07 = v_fma_f64 (f4, p_47, p_03); 45 return v_fma_f64 (f4 * f4, p_8a, p_07); 46 } 47 48 /* Double-precision vector exp(x) - 1 function. 49 The maximum error observed error is 2.18 ULP: 50 __v_expm1(0x1.634ba0c237d7bp-2) got 0x1.a8b9ea8d66e22p-2 51 want 0x1.a8b9ea8d66e2p-2. */ 52 VPCS_ATTR 53 v_f64_t V_NAME (expm1) (v_f64_t x) 54 { 55 v_u64_t ix = v_as_u64_f64 (x); 56 v_u64_t ax = ix & AbsMask; 57 58 #if WANT_SIMD_EXCEPT 59 /* If fp exceptions are to be triggered correctly, fall back to the scalar 60 variant for all lanes if any of them should trigger an exception. */ 61 v_u64_t special = v_cond_u64 ((ax >= SpecialBound) | (ax <= TinyBound)); 62 if (unlikely (v_any_u64 (special))) 63 return v_call_f64 (expm1, x, x, v_u64 (-1)); 64 #else 65 /* Large input, NaNs and Infs. */ 66 v_u64_t special 67 = v_cond_u64 ((ax >= SpecialBound) | (ix == 0x8000000000000000)); 68 #endif 69 70 /* Reduce argument to smaller range: 71 Let i = round(x / ln2) 72 and f = x - i * ln2, then f is in [-ln2/2, ln2/2]. 73 exp(x) - 1 = 2^i * (expm1(f) + 1) - 1 74 where 2^i is exact because i is an integer. */ 75 v_f64_t j = v_fma_f64 (InvLn2, x, Shift) - Shift; 76 v_s64_t i = v_to_s64_f64 (j); 77 v_f64_t f = v_fma_f64 (j, MLn2hi, x); 78 f = v_fma_f64 (j, MLn2lo, f); 79 80 /* Approximate expm1(f) using polynomial. 81 Taylor expansion for expm1(x) has the form: 82 x + ax^2 + bx^3 + cx^4 .... 83 So we calculate the polynomial P(f) = a + bf + cf^2 + ... 84 and assemble the approximation expm1(f) ~= f + f^2 * P(f). */ 85 v_f64_t f2 = f * f; 86 v_f64_t p = v_fma_f64 (f2, eval_poly (f, f2), f); 87 88 /* Assemble the result. 89 expm1(x) ~= 2^i * (p + 1) - 1 90 Let t = 2^i. */ 91 v_f64_t t = v_as_f64_u64 (v_as_u64_s64 (i << 52) + One); 92 /* expm1(x) ~= p * t + (t - 1). */ 93 v_f64_t y = v_fma_f64 (p, t, t - 1); 94 95 #if !WANT_SIMD_EXCEPT 96 if (unlikely (v_any_u64 (special))) 97 return v_call_f64 (expm1, x, y, special); 98 #endif 99 100 return y; 101 } 102 VPCS_ALIAS 103 104 PL_SIG (V, D, 1, expm1, -9.9, 9.9) 105 PL_TEST_ULP (V_NAME (expm1), 1.68) 106 PL_TEST_EXPECT_FENV (V_NAME (expm1), WANT_SIMD_EXCEPT) 107 PL_TEST_INTERVAL (V_NAME (expm1), 0, 0x1p-51, 1000) 108 PL_TEST_INTERVAL (V_NAME (expm1), -0, -0x1p-51, 1000) 109 PL_TEST_INTERVAL (V_NAME (expm1), 0x1p-51, 0x1.63108c75a1937p+9, 100000) 110 PL_TEST_INTERVAL (V_NAME (expm1), -0x1p-51, -0x1.740bf7c0d927dp+9, 100000) 111 PL_TEST_INTERVAL (V_NAME (expm1), 0x1.63108c75a1937p+9, inf, 100) 112 PL_TEST_INTERVAL (V_NAME (expm1), -0x1.740bf7c0d927dp+9, -inf, 100) 113 #endif 114