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
special_case(float32x4_t x,float32x4_t y,uint16x4_t special)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
shifted_lookup(const float * table,int32x4_t i)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. */
V_NAME_F1(cbrt)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