1 /* 2 * Single-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 Shift v_f32 (0x1.8p23f) 15 #define InvLn2 v_f32 (0x1.715476p+0f) 16 #define MLn2hi v_f32 (-0x1.62e4p-1f) 17 #define MLn2lo v_f32 (-0x1.7f7d1cp-20f) 18 #define AbsMask (0x7fffffff) 19 #define One (0x3f800000) 20 #define SpecialBound \ 21 (0x42af5e20) /* asuint(0x1.5ebc4p+6). Largest value of x for which expm1(x) \ 22 should round to -1. */ 23 #define TinyBound (0x34000000) /* asuint(0x1p-23). */ 24 25 #define C(i) v_f32 (__expm1f_poly[i]) 26 27 /* Single-precision vector exp(x) - 1 function. 28 The maximum error is 1.51 ULP: 29 expm1f(0x1.8baa96p-2) got 0x1.e2fb9p-2 30 want 0x1.e2fb94p-2. */ 31 VPCS_ATTR 32 v_f32_t V_NAME (expm1f) (v_f32_t x) 33 { 34 v_u32_t ix = v_as_u32_f32 (x); 35 v_u32_t ax = ix & AbsMask; 36 37 #if WANT_SIMD_EXCEPT 38 /* If fp exceptions are to be triggered correctly, fall back to the scalar 39 variant for all lanes if any of them should trigger an exception. */ 40 v_u32_t special 41 = v_cond_u32 ((ax >= SpecialBound) | (ix == 0x80000000) | (ax < TinyBound)); 42 if (unlikely (v_any_u32 (special))) 43 return v_call_f32 (expm1f, x, x, v_u32 (0xffffffff)); 44 #else 45 /* Handles very large values (+ve and -ve), +/-NaN, +/-Inf and -0. */ 46 v_u32_t special = v_cond_u32 ((ax >= SpecialBound) | (ix == 0x80000000)); 47 #endif 48 49 /* Reduce argument to smaller range: 50 Let i = round(x / ln2) 51 and f = x - i * ln2, then f is in [-ln2/2, ln2/2]. 52 exp(x) - 1 = 2^i * (expm1(f) + 1) - 1 53 where 2^i is exact because i is an integer. */ 54 v_f32_t j = v_fma_f32 (InvLn2, x, Shift) - Shift; 55 v_s32_t i = v_to_s32_f32 (j); 56 v_f32_t f = v_fma_f32 (j, MLn2hi, x); 57 f = v_fma_f32 (j, MLn2lo, f); 58 59 /* Approximate expm1(f) using polynomial. 60 Taylor expansion for expm1(x) has the form: 61 x + ax^2 + bx^3 + cx^4 .... 62 So we calculate the polynomial P(f) = a + bf + cf^2 + ... 63 and assemble the approximation expm1(f) ~= f + f^2 * P(f). */ 64 65 v_f32_t p = v_fma_f32 (C (4), f, C (3)); 66 p = v_fma_f32 (p, f, C (2)); 67 p = v_fma_f32 (p, f, C (1)); 68 p = v_fma_f32 (p, f, C (0)); 69 p = v_fma_f32 (f * f, p, f); 70 71 /* Assemble the result. 72 expm1(x) ~= 2^i * (p + 1) - 1 73 Let t = 2^i. */ 74 v_f32_t t = v_as_f32_u32 (v_as_u32_s32 (i << 23) + One); 75 /* expm1(x) ~= p * t + (t - 1). */ 76 v_f32_t y = v_fma_f32 (p, t, t - 1); 77 78 #if !WANT_SIMD_EXCEPT 79 if (unlikely (v_any_u32 (special))) 80 return v_call_f32 (expm1f, x, y, special); 81 #endif 82 83 return y; 84 } 85 VPCS_ALIAS 86 87 PL_SIG (V, F, 1, expm1, -9.9, 9.9) 88 PL_TEST_ULP (V_NAME (expm1f), 1.02) 89 PL_TEST_EXPECT_FENV (V_NAME (expm1f), WANT_SIMD_EXCEPT) 90 PL_TEST_INTERVAL (V_NAME (expm1f), 0, 0x1p-23, 1000) 91 PL_TEST_INTERVAL (V_NAME (expm1f), -0, -0x1p-23, 1000) 92 PL_TEST_INTERVAL (V_NAME (expm1f), 0x1p-23, 0x1.644716p6, 1000000) 93 PL_TEST_INTERVAL (V_NAME (expm1f), -0x1p-23, -0x1.9bbabcp+6, 1000000) 94 #endif 95