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