1*f3087befSAndrew Turner /* 2*f3087befSAndrew Turner * Double-precision vector log2 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 uint64x2_t off, sign_exp_mask, offset_lower_bound; 15*f3087befSAndrew Turner uint32x4_t special_bound; 16*f3087befSAndrew Turner float64x2_t c0, c2; 17*f3087befSAndrew Turner double c1, c3, invln2, c4; 18*f3087befSAndrew Turner } data = { 19*f3087befSAndrew Turner /* Each coefficient was generated to approximate log(r) for |r| < 0x1.fp-9 20*f3087befSAndrew Turner and N = 128, then scaled by log2(e) in extended precision and rounded back 21*f3087befSAndrew Turner to double precision. */ 22*f3087befSAndrew Turner .c0 = V2 (-0x1.71547652b8300p-1), 23*f3087befSAndrew Turner .c1 = 0x1.ec709dc340953p-2, 24*f3087befSAndrew Turner .c2 = V2 (-0x1.71547651c8f35p-2), 25*f3087befSAndrew Turner .c3 = 0x1.2777ebe12dda5p-2, 26*f3087befSAndrew Turner .c4 = -0x1.ec738d616fe26p-3, 27*f3087befSAndrew Turner .invln2 = 0x1.71547652b82fep0, 28*f3087befSAndrew Turner .off = V2 (0x3fe6900900000000), 29*f3087befSAndrew Turner .sign_exp_mask = V2 (0xfff0000000000000), 30*f3087befSAndrew Turner /* Lower bound is 0x0010000000000000. For 31*f3087befSAndrew Turner optimised register use subnormals are detected after offset has been 32*f3087befSAndrew Turner subtracted, so lower bound - offset (which wraps around). */ 33*f3087befSAndrew Turner .offset_lower_bound = V2 (0x0010000000000000 - 0x3fe6900900000000), 34*f3087befSAndrew Turner .special_bound = V4 (0x7fe00000), /* asuint64(inf) - asuint64(0x1p-1022). */ 35*f3087befSAndrew Turner }; 36*f3087befSAndrew Turner 37*f3087befSAndrew Turner #define N (1 << V_LOG2_TABLE_BITS) 38*f3087befSAndrew Turner #define IndexMask (N - 1) 39*f3087befSAndrew Turner 40*f3087befSAndrew Turner struct entry 41*f3087befSAndrew Turner { 42*f3087befSAndrew Turner float64x2_t invc; 43*f3087befSAndrew Turner float64x2_t log2c; 44*f3087befSAndrew Turner }; 45*f3087befSAndrew Turner 46*f3087befSAndrew Turner static inline struct entry 47*f3087befSAndrew Turner lookup (uint64x2_t i) 48*f3087befSAndrew Turner { 49*f3087befSAndrew Turner struct entry e; 50*f3087befSAndrew Turner uint64_t i0 51*f3087befSAndrew Turner = (vgetq_lane_u64 (i, 0) >> (52 - V_LOG2_TABLE_BITS)) & IndexMask; 52*f3087befSAndrew Turner uint64_t i1 53*f3087befSAndrew Turner = (vgetq_lane_u64 (i, 1) >> (52 - V_LOG2_TABLE_BITS)) & IndexMask; 54*f3087befSAndrew Turner float64x2_t e0 = vld1q_f64 (&__v_log2_data.table[i0].invc); 55*f3087befSAndrew Turner float64x2_t e1 = vld1q_f64 (&__v_log2_data.table[i1].invc); 56*f3087befSAndrew Turner e.invc = vuzp1q_f64 (e0, e1); 57*f3087befSAndrew Turner e.log2c = vuzp2q_f64 (e0, e1); 58*f3087befSAndrew Turner return e; 59*f3087befSAndrew Turner } 60*f3087befSAndrew Turner 61*f3087befSAndrew Turner static float64x2_t VPCS_ATTR NOINLINE 62*f3087befSAndrew Turner special_case (float64x2_t hi, uint64x2_t u_off, float64x2_t y, float64x2_t r2, 63*f3087befSAndrew Turner uint32x2_t special, const struct data *d) 64*f3087befSAndrew Turner { 65*f3087befSAndrew Turner float64x2_t x = vreinterpretq_f64_u64 (vaddq_u64 (u_off, d->off)); 66*f3087befSAndrew Turner return v_call_f64 (log2, x, vfmaq_f64 (hi, y, r2), vmovl_u32 (special)); 67*f3087befSAndrew Turner } 68*f3087befSAndrew Turner 69*f3087befSAndrew Turner /* Double-precision vector log2 routine. Implements the same algorithm as 70*f3087befSAndrew Turner vector log10, with coefficients and table entries scaled in extended 71*f3087befSAndrew Turner precision. The maximum observed error is 2.58 ULP: 72*f3087befSAndrew Turner _ZGVnN2v_log2(0x1.0b556b093869bp+0) got 0x1.fffb34198d9dap-5 73*f3087befSAndrew Turner want 0x1.fffb34198d9ddp-5. */ 74*f3087befSAndrew Turner float64x2_t VPCS_ATTR V_NAME_D1 (log2) (float64x2_t x) 75*f3087befSAndrew Turner { 76*f3087befSAndrew Turner const struct data *d = ptr_barrier (&data); 77*f3087befSAndrew Turner 78*f3087befSAndrew Turner /* To avoid having to mov x out of the way, keep u after offset has been 79*f3087befSAndrew Turner applied, and recover x by adding the offset back in the special-case 80*f3087befSAndrew Turner handler. */ 81*f3087befSAndrew Turner uint64x2_t u = vreinterpretq_u64_f64 (x); 82*f3087befSAndrew Turner uint64x2_t u_off = vsubq_u64 (u, d->off); 83*f3087befSAndrew Turner 84*f3087befSAndrew Turner /* x = 2^k z; where z is in range [Off,2*Off) and exact. 85*f3087befSAndrew Turner The range is split into N subintervals. 86*f3087befSAndrew Turner The ith subinterval contains z and c is near its center. */ 87*f3087befSAndrew Turner int64x2_t k = vshrq_n_s64 (vreinterpretq_s64_u64 (u_off), 52); 88*f3087befSAndrew Turner uint64x2_t iz = vsubq_u64 (u, vandq_u64 (u_off, d->sign_exp_mask)); 89*f3087befSAndrew Turner float64x2_t z = vreinterpretq_f64_u64 (iz); 90*f3087befSAndrew Turner 91*f3087befSAndrew Turner struct entry e = lookup (u_off); 92*f3087befSAndrew Turner 93*f3087befSAndrew Turner uint32x2_t special = vcge_u32 (vsubhn_u64 (u_off, d->offset_lower_bound), 94*f3087befSAndrew Turner vget_low_u32 (d->special_bound)); 95*f3087befSAndrew Turner 96*f3087befSAndrew Turner /* log2(x) = log1p(z/c-1)/log(2) + log2(c) + k. */ 97*f3087befSAndrew Turner float64x2_t r = vfmaq_f64 (v_f64 (-1.0), z, e.invc); 98*f3087befSAndrew Turner float64x2_t kd = vcvtq_f64_s64 (k); 99*f3087befSAndrew Turner 100*f3087befSAndrew Turner float64x2_t invln2_and_c4 = vld1q_f64 (&d->invln2); 101*f3087befSAndrew Turner float64x2_t hi 102*f3087befSAndrew Turner = vfmaq_laneq_f64 (vaddq_f64 (e.log2c, kd), r, invln2_and_c4, 0); 103*f3087befSAndrew Turner 104*f3087befSAndrew Turner float64x2_t r2 = vmulq_f64 (r, r); 105*f3087befSAndrew Turner float64x2_t odd_coeffs = vld1q_f64 (&d->c1); 106*f3087befSAndrew Turner float64x2_t y = vfmaq_laneq_f64 (d->c2, r, odd_coeffs, 1); 107*f3087befSAndrew Turner float64x2_t p = vfmaq_laneq_f64 (d->c0, r, odd_coeffs, 0); 108*f3087befSAndrew Turner y = vfmaq_laneq_f64 (y, r2, invln2_and_c4, 1); 109*f3087befSAndrew Turner y = vfmaq_f64 (p, r2, y); 110*f3087befSAndrew Turner 111*f3087befSAndrew Turner if (unlikely (v_any_u32h (special))) 112*f3087befSAndrew Turner return special_case (hi, u_off, y, r2, special, d); 113*f3087befSAndrew Turner return vfmaq_f64 (hi, y, r2); 114*f3087befSAndrew Turner } 115*f3087befSAndrew Turner 116*f3087befSAndrew Turner TEST_SIG (V, D, 1, log2, 0.01, 11.1) 117*f3087befSAndrew Turner TEST_ULP (V_NAME_D1 (log2), 2.09) 118*f3087befSAndrew Turner TEST_INTERVAL (V_NAME_D1 (log2), -0.0, -0x1p126, 100) 119*f3087befSAndrew Turner TEST_INTERVAL (V_NAME_D1 (log2), 0x1p-149, 0x1p-126, 4000) 120*f3087befSAndrew Turner TEST_INTERVAL (V_NAME_D1 (log2), 0x1p-126, 0x1p-23, 50000) 121*f3087befSAndrew Turner TEST_INTERVAL (V_NAME_D1 (log2), 0x1p-23, 1.0, 50000) 122*f3087befSAndrew Turner TEST_INTERVAL (V_NAME_D1 (log2), 1.0, 100, 50000) 123*f3087befSAndrew Turner TEST_INTERVAL (V_NAME_D1 (log2), 100, inf, 50000) 124