1*f3087befSAndrew Turner /*
2*f3087befSAndrew Turner * Single-precision SVE cbrt(x) function.
3*f3087befSAndrew Turner *
4*f3087befSAndrew Turner * Copyright (c) 2023-2024, 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 "sv_math.h"
9*f3087befSAndrew Turner #include "test_sig.h"
10*f3087befSAndrew Turner #include "test_defs.h"
11*f3087befSAndrew Turner #include "sv_poly_f32.h"
12*f3087befSAndrew Turner
13*f3087befSAndrew Turner const static struct data
14*f3087befSAndrew Turner {
15*f3087befSAndrew Turner float32_t poly[4];
16*f3087befSAndrew Turner float32_t table[5];
17*f3087befSAndrew Turner float32_t one_third, two_thirds;
18*f3087befSAndrew Turner } data = {
19*f3087befSAndrew Turner /* Very rough approximation of cbrt(x) in [0.5, 1], generated with FPMinimax.
20*f3087befSAndrew Turner */
21*f3087befSAndrew Turner .poly = { 0x1.c14e96p-2, 0x1.dd2d3p-1, -0x1.08e81ap-1,
22*f3087befSAndrew Turner 0x1.2c74c2p-3, },
23*f3087befSAndrew Turner /* table[i] = 2^((i - 2) / 3). */
24*f3087befSAndrew Turner .table = { 0x1.428a3p-1, 0x1.965feap-1, 0x1p0, 0x1.428a3p0, 0x1.965feap0 },
25*f3087befSAndrew Turner .one_third = 0x1.555556p-2f,
26*f3087befSAndrew Turner .two_thirds = 0x1.555556p-1f,
27*f3087befSAndrew Turner };
28*f3087befSAndrew Turner
29*f3087befSAndrew Turner #define SmallestNormal 0x00800000
30*f3087befSAndrew Turner #define Thresh 0x7f000000 /* asuint(INFINITY) - SmallestNormal. */
31*f3087befSAndrew Turner #define MantissaMask 0x007fffff
32*f3087befSAndrew Turner #define HalfExp 0x3f000000
33*f3087befSAndrew Turner
34*f3087befSAndrew Turner static svfloat32_t NOINLINE
special_case(svfloat32_t x,svfloat32_t y,svbool_t special)35*f3087befSAndrew Turner special_case (svfloat32_t x, svfloat32_t y, svbool_t special)
36*f3087befSAndrew Turner {
37*f3087befSAndrew Turner return sv_call_f32 (cbrtf, x, y, special);
38*f3087befSAndrew Turner }
39*f3087befSAndrew Turner
40*f3087befSAndrew Turner static inline svfloat32_t
shifted_lookup(const svbool_t pg,const float32_t * table,svint32_t i)41*f3087befSAndrew Turner shifted_lookup (const svbool_t pg, const float32_t *table, svint32_t i)
42*f3087befSAndrew Turner {
43*f3087befSAndrew Turner return svld1_gather_index (pg, table, svadd_x (pg, i, 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 _ZGVsMxv_cbrtf (0x1.85a2aap+3) got 0x1.267936p+1
51*f3087befSAndrew Turner want 0x1.267932p+1. */
SV_NAME_F1(cbrt)52*f3087befSAndrew Turner svfloat32_t SV_NAME_F1 (cbrt) (svfloat32_t x, const svbool_t pg)
53*f3087befSAndrew Turner {
54*f3087befSAndrew Turner const struct data *d = ptr_barrier (&data);
55*f3087befSAndrew Turner
56*f3087befSAndrew Turner svfloat32_t ax = svabs_x (pg, x);
57*f3087befSAndrew Turner svuint32_t iax = svreinterpret_u32 (ax);
58*f3087befSAndrew Turner svuint32_t sign = sveor_x (pg, svreinterpret_u32 (x), iax);
59*f3087befSAndrew Turner
60*f3087befSAndrew Turner /* Subnormal, +/-0 and special values. */
61*f3087befSAndrew Turner svbool_t special = svcmpge (pg, svsub_x (pg, iax, SmallestNormal), Thresh);
62*f3087befSAndrew Turner
63*f3087befSAndrew Turner /* Decompose |x| into m * 2^e, where m is in [0.5, 1.0]. This is a vector
64*f3087befSAndrew Turner version of frexpf, which gets subnormal values wrong - these have to be
65*f3087befSAndrew Turner special-cased as a result. */
66*f3087befSAndrew Turner svfloat32_t m = svreinterpret_f32 (svorr_x (
67*f3087befSAndrew Turner pg, svand_x (pg, svreinterpret_u32 (x), MantissaMask), HalfExp));
68*f3087befSAndrew Turner svint32_t e = svsub_x (pg, svreinterpret_s32 (svlsr_x (pg, iax, 23)), 126);
69*f3087befSAndrew Turner
70*f3087befSAndrew Turner /* p is a rough approximation for cbrt(m) in [0.5, 1.0]. The better this is,
71*f3087befSAndrew Turner the less accurate the next stage of the algorithm needs to be. An order-4
72*f3087befSAndrew Turner polynomial is enough for one Newton iteration. */
73*f3087befSAndrew Turner svfloat32_t p
74*f3087befSAndrew Turner = sv_pairwise_poly_3_f32_x (pg, m, svmul_x (pg, m, m), d->poly);
75*f3087befSAndrew Turner
76*f3087befSAndrew Turner /* One iteration of Newton's method for iteratively approximating cbrt. */
77*f3087befSAndrew Turner svfloat32_t m_by_3 = svmul_x (pg, m, d->one_third);
78*f3087befSAndrew Turner svfloat32_t a = svmla_x (pg, svdiv_x (pg, m_by_3, svmul_x (pg, p, p)), p,
79*f3087befSAndrew Turner d->two_thirds);
80*f3087befSAndrew Turner
81*f3087befSAndrew Turner /* Assemble the result by the following:
82*f3087befSAndrew Turner
83*f3087befSAndrew Turner cbrt(x) = cbrt(m) * 2 ^ (e / 3).
84*f3087befSAndrew Turner
85*f3087befSAndrew Turner We can get 2 ^ round(e / 3) using ldexp and integer divide, but since e is
86*f3087befSAndrew Turner not necessarily a multiple of 3 we lose some information.
87*f3087befSAndrew Turner
88*f3087befSAndrew Turner Let q = 2 ^ round(e / 3), then t = 2 ^ (e / 3) / q.
89*f3087befSAndrew Turner
90*f3087befSAndrew Turner Then we know t = 2 ^ (i / 3), where i is the remainder from e / 3, which
91*f3087befSAndrew Turner is an integer in [-2, 2], and can be looked up in the table T. Hence the
92*f3087befSAndrew Turner result is assembled as:
93*f3087befSAndrew Turner
94*f3087befSAndrew Turner cbrt(x) = cbrt(m) * t * 2 ^ round(e / 3) * sign. */
95*f3087befSAndrew Turner svfloat32_t ef = svmul_x (pg, svcvt_f32_x (pg, e), d->one_third);
96*f3087befSAndrew Turner svint32_t ey = svcvt_s32_x (pg, ef);
97*f3087befSAndrew Turner svint32_t em3 = svmls_x (pg, e, ey, 3);
98*f3087befSAndrew Turner
99*f3087befSAndrew Turner svfloat32_t my = shifted_lookup (pg, d->table, em3);
100*f3087befSAndrew Turner my = svmul_x (pg, my, a);
101*f3087befSAndrew Turner
102*f3087befSAndrew Turner /* Vector version of ldexpf. */
103*f3087befSAndrew Turner svfloat32_t y = svscale_x (pg, my, ey);
104*f3087befSAndrew Turner
105*f3087befSAndrew Turner if (unlikely (svptest_any (pg, special)))
106*f3087befSAndrew Turner return special_case (
107*f3087befSAndrew Turner x, svreinterpret_f32 (svorr_x (pg, svreinterpret_u32 (y), sign)),
108*f3087befSAndrew Turner special);
109*f3087befSAndrew Turner
110*f3087befSAndrew Turner /* Copy sign. */
111*f3087befSAndrew Turner return svreinterpret_f32 (svorr_x (pg, svreinterpret_u32 (y), sign));
112*f3087befSAndrew Turner }
113*f3087befSAndrew Turner
114*f3087befSAndrew Turner TEST_SIG (SV, F, 1, cbrt, -10.0, 10.0)
115*f3087befSAndrew Turner TEST_ULP (SV_NAME_F1 (cbrt), 1.15)
116*f3087befSAndrew Turner TEST_DISABLE_FENV (SV_NAME_F1 (cbrt))
117*f3087befSAndrew Turner TEST_SYM_INTERVAL (SV_NAME_F1 (cbrt), 0, inf, 1000000)
118*f3087befSAndrew Turner CLOSE_SVE_ATTR
119