1*f3087befSAndrew Turner /*
2*f3087befSAndrew Turner * Double-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_f64.h"
12*f3087befSAndrew Turner
13*f3087befSAndrew Turner const static struct data
14*f3087befSAndrew Turner {
15*f3087befSAndrew Turner float64x2_t poly[4], one_third, shift;
16*f3087befSAndrew Turner int64x2_t exp_bias;
17*f3087befSAndrew Turner uint64x2_t abs_mask, tiny_bound;
18*f3087befSAndrew Turner uint32x4_t thresh;
19*f3087befSAndrew Turner double table[5];
20*f3087befSAndrew Turner } data = {
21*f3087befSAndrew Turner .shift = V2 (0x1.8p52),
22*f3087befSAndrew Turner .poly = { /* Generated with fpminimax in [0.5, 1]. */
23*f3087befSAndrew Turner V2 (0x1.c14e8ee44767p-2), V2 (0x1.dd2d3f99e4c0ep-1),
24*f3087befSAndrew Turner V2 (-0x1.08e83026b7e74p-1), V2 (0x1.2c74eaa3ba428p-3) },
25*f3087befSAndrew Turner .exp_bias = V2 (1022),
26*f3087befSAndrew Turner .abs_mask = V2(0x7fffffffffffffff),
27*f3087befSAndrew Turner .tiny_bound = V2(0x0010000000000000), /* Smallest normal. */
28*f3087befSAndrew Turner .thresh = V4(0x7fe00000), /* asuint64 (infinity) - tiny_bound. */
29*f3087befSAndrew Turner .one_third = V2(0x1.5555555555555p-2),
30*f3087befSAndrew Turner .table = { /* table[i] = 2^((i - 2) / 3). */
31*f3087befSAndrew Turner 0x1.428a2f98d728bp-1, 0x1.965fea53d6e3dp-1, 0x1p0,
32*f3087befSAndrew Turner 0x1.428a2f98d728bp0, 0x1.965fea53d6e3dp0 }
33*f3087befSAndrew Turner };
34*f3087befSAndrew Turner
35*f3087befSAndrew Turner #define MantissaMask v_u64 (0x000fffffffffffff)
36*f3087befSAndrew Turner
37*f3087befSAndrew Turner static float64x2_t NOINLINE VPCS_ATTR
special_case(float64x2_t x,float64x2_t y,uint32x2_t special)38*f3087befSAndrew Turner special_case (float64x2_t x, float64x2_t y, uint32x2_t special)
39*f3087befSAndrew Turner {
40*f3087befSAndrew Turner return v_call_f64 (cbrt, x, y, vmovl_u32 (special));
41*f3087befSAndrew Turner }
42*f3087befSAndrew Turner
43*f3087befSAndrew Turner /* Approximation for double-precision vector cbrt(x), using low-order
44*f3087befSAndrew Turner polynomial and two Newton iterations.
45*f3087befSAndrew Turner
46*f3087befSAndrew Turner The vector version of frexp does not handle subnormals
47*f3087befSAndrew Turner correctly. As a result these need to be handled by the scalar
48*f3087befSAndrew Turner fallback, where accuracy may be worse than that of the vector code
49*f3087befSAndrew Turner path.
50*f3087befSAndrew Turner
51*f3087befSAndrew Turner Greatest observed error in the normal range is 1.79 ULP. Errors repeat
52*f3087befSAndrew Turner according to the exponent, for instance an error observed for double value
53*f3087befSAndrew Turner m * 2^e will be observed for any input m * 2^(e + 3*i), where i is an
54*f3087befSAndrew Turner integer.
55*f3087befSAndrew Turner _ZGVnN2v_cbrt (0x1.fffff403f0bc6p+1) got 0x1.965fe72821e9bp+0
56*f3087befSAndrew Turner want 0x1.965fe72821e99p+0. */
V_NAME_D1(cbrt)57*f3087befSAndrew Turner VPCS_ATTR float64x2_t V_NAME_D1 (cbrt) (float64x2_t x)
58*f3087befSAndrew Turner {
59*f3087befSAndrew Turner const struct data *d = ptr_barrier (&data);
60*f3087befSAndrew Turner uint64x2_t iax = vreinterpretq_u64_f64 (vabsq_f64 (x));
61*f3087befSAndrew Turner
62*f3087befSAndrew Turner /* Subnormal, +/-0 and special values. */
63*f3087befSAndrew Turner uint32x2_t special
64*f3087befSAndrew Turner = vcge_u32 (vsubhn_u64 (iax, d->tiny_bound), vget_low_u32 (d->thresh));
65*f3087befSAndrew Turner
66*f3087befSAndrew Turner /* Decompose |x| into m * 2^e, where m is in [0.5, 1.0]. This is a vector
67*f3087befSAndrew Turner version of frexp, which gets subnormal values wrong - these have to be
68*f3087befSAndrew Turner special-cased as a result. */
69*f3087befSAndrew Turner float64x2_t m = vbslq_f64 (MantissaMask, x, v_f64 (0.5));
70*f3087befSAndrew Turner int64x2_t exp_bias = d->exp_bias;
71*f3087befSAndrew Turner uint64x2_t ia12 = vshrq_n_u64 (iax, 52);
72*f3087befSAndrew Turner int64x2_t e = vsubq_s64 (vreinterpretq_s64_u64 (ia12), exp_bias);
73*f3087befSAndrew Turner
74*f3087befSAndrew Turner /* Calculate rough approximation for cbrt(m) in [0.5, 1.0], starting point
75*f3087befSAndrew Turner for Newton iterations. */
76*f3087befSAndrew Turner float64x2_t p = v_pairwise_poly_3_f64 (m, vmulq_f64 (m, m), d->poly);
77*f3087befSAndrew Turner float64x2_t one_third = d->one_third;
78*f3087befSAndrew Turner /* Two iterations of Newton's method for iteratively approximating cbrt. */
79*f3087befSAndrew Turner float64x2_t m_by_3 = vmulq_f64 (m, one_third);
80*f3087befSAndrew Turner float64x2_t two_thirds = vaddq_f64 (one_third, one_third);
81*f3087befSAndrew Turner float64x2_t a
82*f3087befSAndrew Turner = vfmaq_f64 (vdivq_f64 (m_by_3, vmulq_f64 (p, p)), two_thirds, p);
83*f3087befSAndrew Turner a = vfmaq_f64 (vdivq_f64 (m_by_3, vmulq_f64 (a, a)), two_thirds, a);
84*f3087befSAndrew Turner
85*f3087befSAndrew Turner /* Assemble the result by the following:
86*f3087befSAndrew Turner
87*f3087befSAndrew Turner cbrt(x) = cbrt(m) * 2 ^ (e / 3).
88*f3087befSAndrew Turner
89*f3087befSAndrew Turner We can get 2 ^ round(e / 3) using ldexp and integer divide, but since e is
90*f3087befSAndrew Turner not necessarily a multiple of 3 we lose some information.
91*f3087befSAndrew Turner
92*f3087befSAndrew Turner Let q = 2 ^ round(e / 3), then t = 2 ^ (e / 3) / q.
93*f3087befSAndrew Turner
94*f3087befSAndrew Turner Then we know t = 2 ^ (i / 3), where i is the remainder from e / 3, which
95*f3087befSAndrew Turner is an integer in [-2, 2], and can be looked up in the table T. Hence the
96*f3087befSAndrew Turner result is assembled as:
97*f3087befSAndrew Turner
98*f3087befSAndrew Turner cbrt(x) = cbrt(m) * t * 2 ^ round(e / 3) * sign. */
99*f3087befSAndrew Turner
100*f3087befSAndrew Turner float64x2_t ef = vcvtq_f64_s64 (e);
101*f3087befSAndrew Turner float64x2_t eb3f = vrndnq_f64 (vmulq_f64 (ef, one_third));
102*f3087befSAndrew Turner int64x2_t em3 = vcvtq_s64_f64 (vfmsq_f64 (ef, eb3f, v_f64 (3)));
103*f3087befSAndrew Turner int64x2_t ey = vcvtq_s64_f64 (eb3f);
104*f3087befSAndrew Turner
105*f3087befSAndrew Turner float64x2_t my = (float64x2_t){ d->table[em3[0] + 2], d->table[em3[1] + 2] };
106*f3087befSAndrew Turner my = vmulq_f64 (my, a);
107*f3087befSAndrew Turner
108*f3087befSAndrew Turner /* Vector version of ldexp. */
109*f3087befSAndrew Turner float64x2_t y = vreinterpretq_f64_s64 (
110*f3087befSAndrew Turner vshlq_n_s64 (vaddq_s64 (ey, vaddq_s64 (exp_bias, v_s64 (1))), 52));
111*f3087befSAndrew Turner y = vmulq_f64 (y, my);
112*f3087befSAndrew Turner
113*f3087befSAndrew Turner if (unlikely (v_any_u32h (special)))
114*f3087befSAndrew Turner return special_case (x, vbslq_f64 (d->abs_mask, y, x), special);
115*f3087befSAndrew Turner
116*f3087befSAndrew Turner /* Copy sign. */
117*f3087befSAndrew Turner return vbslq_f64 (d->abs_mask, y, x);
118*f3087befSAndrew Turner }
119*f3087befSAndrew Turner
120*f3087befSAndrew Turner /* Worse-case ULP error assumes that scalar fallback is GLIBC 2.40 cbrt, which
121*f3087befSAndrew Turner has ULP error of 3.67 at 0x1.7a337e1ba1ec2p-257 [1]. Largest observed error
122*f3087befSAndrew Turner in the vector path is 1.79 ULP.
123*f3087befSAndrew Turner [1] Innocente, V., & Zimmermann, P. (2024). Accuracy of Mathematical
124*f3087befSAndrew Turner Functions in Single, Double, Double Extended, and Quadruple Precision. */
125*f3087befSAndrew Turner TEST_ULP (V_NAME_D1 (cbrt), 3.17)
126*f3087befSAndrew Turner TEST_SIG (V, D, 1, cbrt, -10.0, 10.0)
127*f3087befSAndrew Turner TEST_SYM_INTERVAL (V_NAME_D1 (cbrt), 0, inf, 1000000)
128