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