1*f3087befSAndrew Turner /* 2*f3087befSAndrew Turner * Double-precision vector cosh(x) function. 3*f3087befSAndrew Turner * 4*f3087befSAndrew Turner * Copyright (c) 2022-2024, Arm Limited. 5*f3087befSAndrew Turner * SPDX-License-Identifier: MIT OR Apache-2.0 WITH LLVM-exception 6*f3087befSAndrew Turner */ 7*f3087befSAndrew Turner 8*f3087befSAndrew Turner #include "v_math.h" 9*f3087befSAndrew Turner #include "test_sig.h" 10*f3087befSAndrew Turner #include "test_defs.h" 11*f3087befSAndrew Turner 12*f3087befSAndrew Turner static const struct data 13*f3087befSAndrew Turner { 14*f3087befSAndrew Turner float64x2_t poly[3]; 15*f3087befSAndrew Turner float64x2_t inv_ln2; 16*f3087befSAndrew Turner double ln2[2]; 17*f3087befSAndrew Turner float64x2_t shift, thres; 18*f3087befSAndrew Turner uint64x2_t index_mask, special_bound; 19*f3087befSAndrew Turner } data = { 20*f3087befSAndrew Turner .poly = { V2 (0x1.fffffffffffd4p-2), V2 (0x1.5555571d6b68cp-3), 21*f3087befSAndrew Turner V2 (0x1.5555576a59599p-5), }, 22*f3087befSAndrew Turner 23*f3087befSAndrew Turner .inv_ln2 = V2 (0x1.71547652b82fep8), /* N/ln2. */ 24*f3087befSAndrew Turner /* -ln2/N. */ 25*f3087befSAndrew Turner .ln2 = {-0x1.62e42fefa39efp-9, -0x1.abc9e3b39803f3p-64}, 26*f3087befSAndrew Turner .shift = V2 (0x1.8p+52), 27*f3087befSAndrew Turner .thres = V2 (704.0), 28*f3087befSAndrew Turner 29*f3087befSAndrew Turner .index_mask = V2 (0xff), 30*f3087befSAndrew Turner /* 0x1.6p9, above which exp overflows. */ 31*f3087befSAndrew Turner .special_bound = V2 (0x4086000000000000), 32*f3087befSAndrew Turner }; 33*f3087befSAndrew Turner 34*f3087befSAndrew Turner static float64x2_t NOINLINE VPCS_ATTR 35*f3087befSAndrew Turner special_case (float64x2_t x, float64x2_t y, uint64x2_t special) 36*f3087befSAndrew Turner { 37*f3087befSAndrew Turner return v_call_f64 (cosh, x, y, special); 38*f3087befSAndrew Turner } 39*f3087befSAndrew Turner 40*f3087befSAndrew Turner /* Helper for approximating exp(x). Copied from v_exp_tail, with no 41*f3087befSAndrew Turner special-case handling or tail. */ 42*f3087befSAndrew Turner static inline float64x2_t 43*f3087befSAndrew Turner exp_inline (float64x2_t x) 44*f3087befSAndrew Turner { 45*f3087befSAndrew Turner const struct data *d = ptr_barrier (&data); 46*f3087befSAndrew Turner 47*f3087befSAndrew Turner /* n = round(x/(ln2/N)). */ 48*f3087befSAndrew Turner float64x2_t z = vfmaq_f64 (d->shift, x, d->inv_ln2); 49*f3087befSAndrew Turner uint64x2_t u = vreinterpretq_u64_f64 (z); 50*f3087befSAndrew Turner float64x2_t n = vsubq_f64 (z, d->shift); 51*f3087befSAndrew Turner 52*f3087befSAndrew Turner /* r = x - n*ln2/N. */ 53*f3087befSAndrew Turner float64x2_t ln2 = vld1q_f64 (d->ln2); 54*f3087befSAndrew Turner float64x2_t r = vfmaq_laneq_f64 (x, n, ln2, 0); 55*f3087befSAndrew Turner r = vfmaq_laneq_f64 (r, n, ln2, 1); 56*f3087befSAndrew Turner 57*f3087befSAndrew Turner uint64x2_t e = vshlq_n_u64 (u, 52 - V_EXP_TAIL_TABLE_BITS); 58*f3087befSAndrew Turner uint64x2_t i = vandq_u64 (u, d->index_mask); 59*f3087befSAndrew Turner 60*f3087befSAndrew Turner /* y = tail + exp(r) - 1 ~= r + C1 r^2 + C2 r^3 + C3 r^4. */ 61*f3087befSAndrew Turner float64x2_t y = vfmaq_f64 (d->poly[1], d->poly[2], r); 62*f3087befSAndrew Turner y = vfmaq_f64 (d->poly[0], y, r); 63*f3087befSAndrew Turner y = vmulq_f64 (vfmaq_f64 (v_f64 (1), y, r), r); 64*f3087befSAndrew Turner 65*f3087befSAndrew Turner /* s = 2^(n/N). */ 66*f3087befSAndrew Turner u = v_lookup_u64 (__v_exp_tail_data, i); 67*f3087befSAndrew Turner float64x2_t s = vreinterpretq_f64_u64 (vaddq_u64 (u, e)); 68*f3087befSAndrew Turner 69*f3087befSAndrew Turner return vfmaq_f64 (s, y, s); 70*f3087befSAndrew Turner } 71*f3087befSAndrew Turner 72*f3087befSAndrew Turner /* Approximation for vector double-precision cosh(x) using exp_inline. 73*f3087befSAndrew Turner cosh(x) = (exp(x) + exp(-x)) / 2. 74*f3087befSAndrew Turner The greatest observed error is in the scalar fall-back region, so is the 75*f3087befSAndrew Turner same as the scalar routine, 1.93 ULP: 76*f3087befSAndrew Turner _ZGVnN2v_cosh (0x1.628af341989dap+9) got 0x1.fdf28623ef921p+1021 77*f3087befSAndrew Turner want 0x1.fdf28623ef923p+1021. 78*f3087befSAndrew Turner 79*f3087befSAndrew Turner The greatest observed error in the non-special region is 1.54 ULP: 80*f3087befSAndrew Turner _ZGVnN2v_cosh (0x1.8e205b6ecacf7p+2) got 0x1.f711dcb0c77afp+7 81*f3087befSAndrew Turner want 0x1.f711dcb0c77b1p+7. */ 82*f3087befSAndrew Turner float64x2_t VPCS_ATTR V_NAME_D1 (cosh) (float64x2_t x) 83*f3087befSAndrew Turner { 84*f3087befSAndrew Turner const struct data *d = ptr_barrier (&data); 85*f3087befSAndrew Turner 86*f3087befSAndrew Turner float64x2_t ax = vabsq_f64 (x); 87*f3087befSAndrew Turner uint64x2_t special 88*f3087befSAndrew Turner = vcgtq_u64 (vreinterpretq_u64_f64 (ax), d->special_bound); 89*f3087befSAndrew Turner 90*f3087befSAndrew Turner /* Up to the point that exp overflows, we can use it to calculate cosh by 91*f3087befSAndrew Turner exp(|x|) / 2 + 1 / (2 * exp(|x|)). */ 92*f3087befSAndrew Turner float64x2_t t = exp_inline (ax); 93*f3087befSAndrew Turner float64x2_t half_t = vmulq_n_f64 (t, 0.5); 94*f3087befSAndrew Turner float64x2_t half_over_t = vdivq_f64 (v_f64 (0.5), t); 95*f3087befSAndrew Turner 96*f3087befSAndrew Turner /* Fall back to scalar for any special cases. */ 97*f3087befSAndrew Turner if (unlikely (v_any_u64 (special))) 98*f3087befSAndrew Turner return special_case (x, vaddq_f64 (half_t, half_over_t), special); 99*f3087befSAndrew Turner 100*f3087befSAndrew Turner return vaddq_f64 (half_t, half_over_t); 101*f3087befSAndrew Turner } 102*f3087befSAndrew Turner 103*f3087befSAndrew Turner TEST_SIG (V, D, 1, cosh, -10.0, 10.0) 104*f3087befSAndrew Turner TEST_ULP (V_NAME_D1 (cosh), 1.43) 105*f3087befSAndrew Turner TEST_DISABLE_FENV_IF_NOT (V_NAME_D1 (cosh), WANT_SIMD_EXCEPT) 106*f3087befSAndrew Turner TEST_SYM_INTERVAL (V_NAME_D1 (cosh), 0, 0x1.6p9, 100000) 107*f3087befSAndrew Turner TEST_SYM_INTERVAL (V_NAME_D1 (cosh), 0x1.6p9, inf, 1000) 108