1*412f47f9SXin Li /*
2*412f47f9SXin Li * Double-precision vector atan(x) function.
3*412f47f9SXin Li *
4*412f47f9SXin Li * Copyright (c) 2021-2023, Arm Limited.
5*412f47f9SXin Li * SPDX-License-Identifier: MIT OR Apache-2.0 WITH LLVM-exception
6*412f47f9SXin Li */
7*412f47f9SXin Li
8*412f47f9SXin Li #include "v_math.h"
9*412f47f9SXin Li #include "pl_sig.h"
10*412f47f9SXin Li #include "pl_test.h"
11*412f47f9SXin Li #include "poly_advsimd_f64.h"
12*412f47f9SXin Li
13*412f47f9SXin Li static const struct data
14*412f47f9SXin Li {
15*412f47f9SXin Li float64x2_t pi_over_2;
16*412f47f9SXin Li float64x2_t poly[20];
17*412f47f9SXin Li } data = {
18*412f47f9SXin Li /* Coefficients of polynomial P such that atan(x)~x+x*P(x^2) on
19*412f47f9SXin Li [2**-1022, 1.0]. */
20*412f47f9SXin Li .poly = { V2 (-0x1.5555555555555p-2), V2 (0x1.99999999996c1p-3),
21*412f47f9SXin Li V2 (-0x1.2492492478f88p-3), V2 (0x1.c71c71bc3951cp-4),
22*412f47f9SXin Li V2 (-0x1.745d160a7e368p-4), V2 (0x1.3b139b6a88ba1p-4),
23*412f47f9SXin Li V2 (-0x1.11100ee084227p-4), V2 (0x1.e1d0f9696f63bp-5),
24*412f47f9SXin Li V2 (-0x1.aebfe7b418581p-5), V2 (0x1.842dbe9b0d916p-5),
25*412f47f9SXin Li V2 (-0x1.5d30140ae5e99p-5), V2 (0x1.338e31eb2fbbcp-5),
26*412f47f9SXin Li V2 (-0x1.00e6eece7de8p-5), V2 (0x1.860897b29e5efp-6),
27*412f47f9SXin Li V2 (-0x1.0051381722a59p-6), V2 (0x1.14e9dc19a4a4ep-7),
28*412f47f9SXin Li V2 (-0x1.d0062b42fe3bfp-9), V2 (0x1.17739e210171ap-10),
29*412f47f9SXin Li V2 (-0x1.ab24da7be7402p-13), V2 (0x1.358851160a528p-16), },
30*412f47f9SXin Li .pi_over_2 = V2 (0x1.921fb54442d18p+0),
31*412f47f9SXin Li };
32*412f47f9SXin Li
33*412f47f9SXin Li #define SignMask v_u64 (0x8000000000000000)
34*412f47f9SXin Li #define TinyBound 0x3e10000000000000 /* asuint64(0x1p-30). */
35*412f47f9SXin Li #define BigBound 0x4340000000000000 /* asuint64(0x1p53). */
36*412f47f9SXin Li
37*412f47f9SXin Li /* Fast implementation of vector atan.
38*412f47f9SXin Li Based on atan(x) ~ shift + z + z^3 * P(z^2) with reduction to [0,1] using
39*412f47f9SXin Li z=1/x and shift = pi/2. Maximum observed error is 2.27 ulps:
40*412f47f9SXin Li _ZGVnN2v_atan (0x1.0005af27c23e9p+0) got 0x1.9225645bdd7c1p-1
41*412f47f9SXin Li want 0x1.9225645bdd7c3p-1. */
V_NAME_D1(atan)42*412f47f9SXin Li float64x2_t VPCS_ATTR V_NAME_D1 (atan) (float64x2_t x)
43*412f47f9SXin Li {
44*412f47f9SXin Li const struct data *d = ptr_barrier (&data);
45*412f47f9SXin Li
46*412f47f9SXin Li /* Small cases, infs and nans are supported by our approximation technique,
47*412f47f9SXin Li but do not set fenv flags correctly. Only trigger special case if we need
48*412f47f9SXin Li fenv. */
49*412f47f9SXin Li uint64x2_t ix = vreinterpretq_u64_f64 (x);
50*412f47f9SXin Li uint64x2_t sign = vandq_u64 (ix, SignMask);
51*412f47f9SXin Li
52*412f47f9SXin Li #if WANT_SIMD_EXCEPT
53*412f47f9SXin Li uint64x2_t ia12 = vandq_u64 (ix, v_u64 (0x7ff0000000000000));
54*412f47f9SXin Li uint64x2_t special = vcgtq_u64 (vsubq_u64 (ia12, v_u64 (TinyBound)),
55*412f47f9SXin Li v_u64 (BigBound - TinyBound));
56*412f47f9SXin Li /* If any lane is special, fall back to the scalar routine for all lanes. */
57*412f47f9SXin Li if (unlikely (v_any_u64 (special)))
58*412f47f9SXin Li return v_call_f64 (atan, x, v_f64 (0), v_u64 (-1));
59*412f47f9SXin Li #endif
60*412f47f9SXin Li
61*412f47f9SXin Li /* Argument reduction:
62*412f47f9SXin Li y := arctan(x) for x < 1
63*412f47f9SXin Li y := pi/2 + arctan(-1/x) for x > 1
64*412f47f9SXin Li Hence, use z=-1/a if x>=1, otherwise z=a. */
65*412f47f9SXin Li uint64x2_t red = vcagtq_f64 (x, v_f64 (1.0));
66*412f47f9SXin Li /* Avoid dependency in abs(x) in division (and comparison). */
67*412f47f9SXin Li float64x2_t z = vbslq_f64 (red, vdivq_f64 (v_f64 (1.0), x), x);
68*412f47f9SXin Li float64x2_t shift = vreinterpretq_f64_u64 (
69*412f47f9SXin Li vandq_u64 (red, vreinterpretq_u64_f64 (d->pi_over_2)));
70*412f47f9SXin Li /* Use absolute value only when needed (odd powers of z). */
71*412f47f9SXin Li float64x2_t az = vbslq_f64 (
72*412f47f9SXin Li SignMask, vreinterpretq_f64_u64 (vandq_u64 (SignMask, red)), z);
73*412f47f9SXin Li
74*412f47f9SXin Li /* Calculate the polynomial approximation.
75*412f47f9SXin Li Use split Estrin scheme for P(z^2) with deg(P)=19. Use split instead of
76*412f47f9SXin Li full scheme to avoid underflow in x^16.
77*412f47f9SXin Li The order 19 polynomial P approximates
78*412f47f9SXin Li (atan(sqrt(x))-sqrt(x))/x^(3/2). */
79*412f47f9SXin Li float64x2_t z2 = vmulq_f64 (z, z);
80*412f47f9SXin Li float64x2_t x2 = vmulq_f64 (z2, z2);
81*412f47f9SXin Li float64x2_t x4 = vmulq_f64 (x2, x2);
82*412f47f9SXin Li float64x2_t x8 = vmulq_f64 (x4, x4);
83*412f47f9SXin Li float64x2_t y
84*412f47f9SXin Li = vfmaq_f64 (v_estrin_7_f64 (z2, x2, x4, d->poly),
85*412f47f9SXin Li v_estrin_11_f64 (z2, x2, x4, x8, d->poly + 8), x8);
86*412f47f9SXin Li
87*412f47f9SXin Li /* Finalize. y = shift + z + z^3 * P(z^2). */
88*412f47f9SXin Li y = vfmaq_f64 (az, y, vmulq_f64 (z2, az));
89*412f47f9SXin Li y = vaddq_f64 (y, shift);
90*412f47f9SXin Li
91*412f47f9SXin Li /* y = atan(x) if x>0, -atan(-x) otherwise. */
92*412f47f9SXin Li y = vreinterpretq_f64_u64 (veorq_u64 (vreinterpretq_u64_f64 (y), sign));
93*412f47f9SXin Li return y;
94*412f47f9SXin Li }
95*412f47f9SXin Li
96*412f47f9SXin Li PL_SIG (V, D, 1, atan, -10.0, 10.0)
97*412f47f9SXin Li PL_TEST_ULP (V_NAME_D1 (atan), 1.78)
98*412f47f9SXin Li PL_TEST_EXPECT_FENV (V_NAME_D1 (atan), WANT_SIMD_EXCEPT)
99*412f47f9SXin Li PL_TEST_INTERVAL (V_NAME_D1 (atan), 0, 0x1p-30, 10000)
100*412f47f9SXin Li PL_TEST_INTERVAL (V_NAME_D1 (atan), -0, -0x1p-30, 1000)
101*412f47f9SXin Li PL_TEST_INTERVAL (V_NAME_D1 (atan), 0x1p-30, 0x1p53, 900000)
102*412f47f9SXin Li PL_TEST_INTERVAL (V_NAME_D1 (atan), -0x1p-30, -0x1p53, 90000)
103*412f47f9SXin Li PL_TEST_INTERVAL (V_NAME_D1 (atan), 0x1p53, inf, 10000)
104*412f47f9SXin Li PL_TEST_INTERVAL (V_NAME_D1 (atan), -0x1p53, -inf, 1000)
105