1*f3087befSAndrew Turner /* 2*f3087befSAndrew Turner * Single-precision vector cbrt(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 #include "v_poly_f32.h" 12*f3087befSAndrew Turner 13*f3087befSAndrew Turner const static struct data 14*f3087befSAndrew Turner { 15*f3087befSAndrew Turner float32x4_t poly[4], one_third; 16*f3087befSAndrew Turner float table[5]; 17*f3087befSAndrew Turner } data = { 18*f3087befSAndrew Turner .poly = { /* Very rough approximation of cbrt(x) in [0.5, 1], generated with 19*f3087befSAndrew Turner FPMinimax. */ 20*f3087befSAndrew Turner V4 (0x1.c14e96p-2), V4 (0x1.dd2d3p-1), V4 (-0x1.08e81ap-1), 21*f3087befSAndrew Turner V4 (0x1.2c74c2p-3) }, 22*f3087befSAndrew Turner .table = { /* table[i] = 2^((i - 2) / 3). */ 23*f3087befSAndrew Turner 0x1.428a3p-1, 0x1.965feap-1, 0x1p0, 0x1.428a3p0, 0x1.965feap0 }, 24*f3087befSAndrew Turner .one_third = V4 (0x1.555556p-2f), 25*f3087befSAndrew Turner }; 26*f3087befSAndrew Turner 27*f3087befSAndrew Turner #define SignMask v_u32 (0x80000000) 28*f3087befSAndrew Turner #define SmallestNormal v_u32 (0x00800000) 29*f3087befSAndrew Turner #define Thresh vdup_n_u16 (0x7f00) /* asuint(INFINITY) - SmallestNormal. */ 30*f3087befSAndrew Turner #define MantissaMask v_u32 (0x007fffff) 31*f3087befSAndrew Turner #define HalfExp v_u32 (0x3f000000) 32*f3087befSAndrew Turner 33*f3087befSAndrew Turner static float32x4_t VPCS_ATTR NOINLINE 34*f3087befSAndrew Turner special_case (float32x4_t x, float32x4_t y, uint16x4_t special) 35*f3087befSAndrew Turner { 36*f3087befSAndrew Turner return v_call_f32 (cbrtf, x, y, vmovl_u16 (special)); 37*f3087befSAndrew Turner } 38*f3087befSAndrew Turner 39*f3087befSAndrew Turner static inline float32x4_t 40*f3087befSAndrew Turner shifted_lookup (const float *table, int32x4_t i) 41*f3087befSAndrew Turner { 42*f3087befSAndrew Turner return (float32x4_t){ table[i[0] + 2], table[i[1] + 2], table[i[2] + 2], 43*f3087befSAndrew Turner table[i[3] + 2] }; 44*f3087befSAndrew Turner } 45*f3087befSAndrew Turner 46*f3087befSAndrew Turner /* Approximation for vector single-precision cbrt(x) using Newton iteration 47*f3087befSAndrew Turner with initial guess obtained by a low-order polynomial. Greatest error 48*f3087befSAndrew Turner is 1.64 ULP. This is observed for every value where the mantissa is 49*f3087befSAndrew Turner 0x1.85a2aa and the exponent is a multiple of 3, for example: 50*f3087befSAndrew Turner _ZGVnN4v_cbrtf(0x1.85a2aap+3) got 0x1.267936p+1 51*f3087befSAndrew Turner want 0x1.267932p+1. */ 52*f3087befSAndrew Turner float32x4_t VPCS_ATTR NOINLINE V_NAME_F1 (cbrt) (float32x4_t x) 53*f3087befSAndrew Turner { 54*f3087befSAndrew Turner const struct data *d = ptr_barrier (&data); 55*f3087befSAndrew Turner uint32x4_t iax = vreinterpretq_u32_f32 (vabsq_f32 (x)); 56*f3087befSAndrew Turner 57*f3087befSAndrew Turner /* Subnormal, +/-0 and special values. */ 58*f3087befSAndrew Turner uint16x4_t special = vcge_u16 (vsubhn_u32 (iax, SmallestNormal), Thresh); 59*f3087befSAndrew Turner 60*f3087befSAndrew Turner /* Decompose |x| into m * 2^e, where m is in [0.5, 1.0]. This is a vector 61*f3087befSAndrew Turner version of frexpf, which gets subnormal values wrong - these have to be 62*f3087befSAndrew Turner special-cased as a result. */ 63*f3087befSAndrew Turner float32x4_t m = vbslq_f32 (MantissaMask, x, v_f32 (0.5)); 64*f3087befSAndrew Turner int32x4_t e 65*f3087befSAndrew Turner = vsubq_s32 (vreinterpretq_s32_u32 (vshrq_n_u32 (iax, 23)), v_s32 (126)); 66*f3087befSAndrew Turner 67*f3087befSAndrew Turner /* p is a rough approximation for cbrt(m) in [0.5, 1.0]. The better this is, 68*f3087befSAndrew Turner the less accurate the next stage of the algorithm needs to be. An order-4 69*f3087befSAndrew Turner polynomial is enough for one Newton iteration. */ 70*f3087befSAndrew Turner float32x4_t p = v_pairwise_poly_3_f32 (m, vmulq_f32 (m, m), d->poly); 71*f3087befSAndrew Turner 72*f3087befSAndrew Turner float32x4_t one_third = d->one_third; 73*f3087befSAndrew Turner float32x4_t two_thirds = vaddq_f32 (one_third, one_third); 74*f3087befSAndrew Turner 75*f3087befSAndrew Turner /* One iteration of Newton's method for iteratively approximating cbrt. */ 76*f3087befSAndrew Turner float32x4_t m_by_3 = vmulq_f32 (m, one_third); 77*f3087befSAndrew Turner float32x4_t a 78*f3087befSAndrew Turner = vfmaq_f32 (vdivq_f32 (m_by_3, vmulq_f32 (p, p)), two_thirds, p); 79*f3087befSAndrew Turner 80*f3087befSAndrew Turner /* Assemble the result by the following: 81*f3087befSAndrew Turner 82*f3087befSAndrew Turner cbrt(x) = cbrt(m) * 2 ^ (e / 3). 83*f3087befSAndrew Turner 84*f3087befSAndrew Turner We can get 2 ^ round(e / 3) using ldexp and integer divide, but since e is 85*f3087befSAndrew Turner not necessarily a multiple of 3 we lose some information. 86*f3087befSAndrew Turner 87*f3087befSAndrew Turner Let q = 2 ^ round(e / 3), then t = 2 ^ (e / 3) / q. 88*f3087befSAndrew Turner 89*f3087befSAndrew Turner Then we know t = 2 ^ (i / 3), where i is the remainder from e / 3, which 90*f3087befSAndrew Turner is an integer in [-2, 2], and can be looked up in the table T. Hence the 91*f3087befSAndrew Turner result is assembled as: 92*f3087befSAndrew Turner 93*f3087befSAndrew Turner cbrt(x) = cbrt(m) * t * 2 ^ round(e / 3) * sign. */ 94*f3087befSAndrew Turner float32x4_t ef = vmulq_f32 (vcvtq_f32_s32 (e), one_third); 95*f3087befSAndrew Turner int32x4_t ey = vcvtq_s32_f32 (ef); 96*f3087befSAndrew Turner int32x4_t em3 = vsubq_s32 (e, vmulq_s32 (ey, v_s32 (3))); 97*f3087befSAndrew Turner 98*f3087befSAndrew Turner float32x4_t my = shifted_lookup (d->table, em3); 99*f3087befSAndrew Turner my = vmulq_f32 (my, a); 100*f3087befSAndrew Turner 101*f3087befSAndrew Turner /* Vector version of ldexpf. */ 102*f3087befSAndrew Turner float32x4_t y 103*f3087befSAndrew Turner = vreinterpretq_f32_s32 (vshlq_n_s32 (vaddq_s32 (ey, v_s32 (127)), 23)); 104*f3087befSAndrew Turner y = vmulq_f32 (y, my); 105*f3087befSAndrew Turner 106*f3087befSAndrew Turner if (unlikely (v_any_u16h (special))) 107*f3087befSAndrew Turner return special_case (x, vbslq_f32 (SignMask, x, y), special); 108*f3087befSAndrew Turner 109*f3087befSAndrew Turner /* Copy sign. */ 110*f3087befSAndrew Turner return vbslq_f32 (SignMask, x, y); 111*f3087befSAndrew Turner } 112*f3087befSAndrew Turner 113*f3087befSAndrew Turner HALF_WIDTH_ALIAS_F1 (cbrt) 114*f3087befSAndrew Turner 115*f3087befSAndrew Turner TEST_SIG (V, F, 1, cbrt, -10.0, 10.0) 116*f3087befSAndrew Turner TEST_ULP (V_NAME_F1 (cbrt), 1.15) 117*f3087befSAndrew Turner TEST_SYM_INTERVAL (V_NAME_F1 (cbrt), 0, inf, 1000000) 118