1*f3087befSAndrew Turner /* 2*f3087befSAndrew Turner * Helper for double-precision routines which calculate exp(x) - 1 and do not 3*f3087befSAndrew Turner * need special-case handling 4*f3087befSAndrew Turner * 5*f3087befSAndrew Turner * Copyright (c) 2022-2024, Arm Limited. 6*f3087befSAndrew Turner * SPDX-License-Identifier: MIT OR Apache-2.0 WITH LLVM-exception 7*f3087befSAndrew Turner */ 8*f3087befSAndrew Turner 9*f3087befSAndrew Turner #ifndef MATH_V_EXPM1_INLINE_H 10*f3087befSAndrew Turner #define MATH_V_EXPM1_INLINE_H 11*f3087befSAndrew Turner 12*f3087befSAndrew Turner #include "v_math.h" 13*f3087befSAndrew Turner 14*f3087befSAndrew Turner struct v_expm1_data 15*f3087befSAndrew Turner { 16*f3087befSAndrew Turner float64x2_t c2, c4, c6, c8; 17*f3087befSAndrew Turner float64x2_t invln2; 18*f3087befSAndrew Turner int64x2_t exponent_bias; 19*f3087befSAndrew Turner double c1, c3, c5, c7, c9, c10; 20*f3087befSAndrew Turner double ln2[2]; 21*f3087befSAndrew Turner }; 22*f3087befSAndrew Turner 23*f3087befSAndrew Turner /* Generated using fpminimax, with degree=12 in [log(2)/2, log(2)/2]. */ 24*f3087befSAndrew Turner #define V_EXPM1_DATA \ 25*f3087befSAndrew Turner { \ 26*f3087befSAndrew Turner .c1 = 0x1.5555555555559p-3, .c2 = V2 (0x1.555555555554bp-5), \ 27*f3087befSAndrew Turner .c3 = 0x1.111111110f663p-7, .c4 = V2 (0x1.6c16c16c1b5f3p-10), \ 28*f3087befSAndrew Turner .c5 = 0x1.a01a01affa35dp-13, .c6 = V2 (0x1.a01a018b4ecbbp-16), \ 29*f3087befSAndrew Turner .c7 = 0x1.71ddf82db5bb4p-19, .c8 = V2 (0x1.27e517fc0d54bp-22), \ 30*f3087befSAndrew Turner .c9 = 0x1.af5eedae67435p-26, .c10 = 0x1.1f143d060a28ap-29, \ 31*f3087befSAndrew Turner .ln2 = { 0x1.62e42fefa39efp-1, 0x1.abc9e3b39803fp-56 }, \ 32*f3087befSAndrew Turner .invln2 = V2 (0x1.71547652b82fep0), \ 33*f3087befSAndrew Turner .exponent_bias = V2 (0x3ff0000000000000), \ 34*f3087befSAndrew Turner } 35*f3087befSAndrew Turner 36*f3087befSAndrew Turner static inline float64x2_t 37*f3087befSAndrew Turner expm1_inline (float64x2_t x, const struct v_expm1_data *d) 38*f3087befSAndrew Turner { 39*f3087befSAndrew Turner /* Helper routine for calculating exp(x) - 1. */ 40*f3087befSAndrew Turner 41*f3087befSAndrew Turner float64x2_t ln2 = vld1q_f64 (&d->ln2[0]); 42*f3087befSAndrew Turner 43*f3087befSAndrew Turner /* Reduce argument to smaller range: 44*f3087befSAndrew Turner Let i = round(x / ln2) 45*f3087befSAndrew Turner and f = x - i * ln2, then f is in [-ln2/2, ln2/2]. 46*f3087befSAndrew Turner exp(x) - 1 = 2^i * (expm1(f) + 1) - 1 47*f3087befSAndrew Turner where 2^i is exact because i is an integer. */ 48*f3087befSAndrew Turner float64x2_t n = vrndaq_f64 (vmulq_f64 (x, d->invln2)); 49*f3087befSAndrew Turner int64x2_t i = vcvtq_s64_f64 (n); 50*f3087befSAndrew Turner float64x2_t f = vfmsq_laneq_f64 (x, n, ln2, 0); 51*f3087befSAndrew Turner f = vfmsq_laneq_f64 (f, n, ln2, 1); 52*f3087befSAndrew Turner 53*f3087befSAndrew Turner /* Approximate expm1(f) using polynomial. 54*f3087befSAndrew Turner Taylor expansion for expm1(x) has the form: 55*f3087befSAndrew Turner x + ax^2 + bx^3 + cx^4 .... 56*f3087befSAndrew Turner So we calculate the polynomial P(f) = a + bf + cf^2 + ... 57*f3087befSAndrew Turner and assemble the approximation expm1(f) ~= f + f^2 * P(f). */ 58*f3087befSAndrew Turner float64x2_t f2 = vmulq_f64 (f, f); 59*f3087befSAndrew Turner float64x2_t f4 = vmulq_f64 (f2, f2); 60*f3087befSAndrew Turner float64x2_t lane_consts_13 = vld1q_f64 (&d->c1); 61*f3087befSAndrew Turner float64x2_t lane_consts_57 = vld1q_f64 (&d->c5); 62*f3087befSAndrew Turner float64x2_t lane_consts_910 = vld1q_f64 (&d->c9); 63*f3087befSAndrew Turner float64x2_t p01 = vfmaq_laneq_f64 (v_f64 (0.5), f, lane_consts_13, 0); 64*f3087befSAndrew Turner float64x2_t p23 = vfmaq_laneq_f64 (d->c2, f, lane_consts_13, 1); 65*f3087befSAndrew Turner float64x2_t p45 = vfmaq_laneq_f64 (d->c4, f, lane_consts_57, 0); 66*f3087befSAndrew Turner float64x2_t p67 = vfmaq_laneq_f64 (d->c6, f, lane_consts_57, 1); 67*f3087befSAndrew Turner float64x2_t p03 = vfmaq_f64 (p01, f2, p23); 68*f3087befSAndrew Turner float64x2_t p47 = vfmaq_f64 (p45, f2, p67); 69*f3087befSAndrew Turner float64x2_t p89 = vfmaq_laneq_f64 (d->c8, f, lane_consts_910, 0); 70*f3087befSAndrew Turner float64x2_t p = vfmaq_laneq_f64 (p89, f2, lane_consts_910, 1); 71*f3087befSAndrew Turner p = vfmaq_f64 (p47, f4, p); 72*f3087befSAndrew Turner p = vfmaq_f64 (p03, f4, p); 73*f3087befSAndrew Turner 74*f3087befSAndrew Turner p = vfmaq_f64 (f, f2, p); 75*f3087befSAndrew Turner 76*f3087befSAndrew Turner /* Assemble the result. 77*f3087befSAndrew Turner expm1(x) ~= 2^i * (p + 1) - 1 78*f3087befSAndrew Turner Let t = 2^i. */ 79*f3087befSAndrew Turner int64x2_t u = vaddq_s64 (vshlq_n_s64 (i, 52), d->exponent_bias); 80*f3087befSAndrew Turner float64x2_t t = vreinterpretq_f64_s64 (u); 81*f3087befSAndrew Turner 82*f3087befSAndrew Turner /* expm1(x) ~= p * t + (t - 1). */ 83*f3087befSAndrew Turner return vfmaq_f64 (vsubq_f64 (t, v_f64 (1.0)), p, t); 84*f3087befSAndrew Turner } 85*f3087befSAndrew Turner 86*f3087befSAndrew Turner #endif // MATH_V_EXPM1_INLINE_H 87