1 | /* Double-precision AdvSIMD inverse tan |
2 | |
3 | Copyright (C) 2023-2024 Free Software Foundation, Inc. |
4 | This file is part of the GNU C Library. |
5 | |
6 | The GNU C Library is free software; you can redistribute it and/or |
7 | modify it under the terms of the GNU Lesser General Public |
8 | License as published by the Free Software Foundation; either |
9 | version 2.1 of the License, or (at your option) any later version. |
10 | |
11 | The GNU C Library is distributed in the hope that it will be useful, |
12 | but WITHOUT ANY WARRANTY; without even the implied warranty of |
13 | MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the GNU |
14 | Lesser General Public License for more details. |
15 | |
16 | You should have received a copy of the GNU Lesser General Public |
17 | License along with the GNU C Library; if not, see |
18 | <https://www.gnu.org/licenses/>. */ |
19 | |
20 | #include "v_math.h" |
21 | #include "poly_advsimd_f64.h" |
22 | |
23 | static const struct data |
24 | { |
25 | float64x2_t pi_over_2; |
26 | float64x2_t poly[20]; |
27 | } data = { |
28 | /* Coefficients of polynomial P such that atan(x)~x+x*P(x^2) on |
29 | [2**-1022, 1.0]. */ |
30 | .poly = { V2 (-0x1.5555555555555p-2), V2 (0x1.99999999996c1p-3), |
31 | V2 (-0x1.2492492478f88p-3), V2 (0x1.c71c71bc3951cp-4), |
32 | V2 (-0x1.745d160a7e368p-4), V2 (0x1.3b139b6a88ba1p-4), |
33 | V2 (-0x1.11100ee084227p-4), V2 (0x1.e1d0f9696f63bp-5), |
34 | V2 (-0x1.aebfe7b418581p-5), V2 (0x1.842dbe9b0d916p-5), |
35 | V2 (-0x1.5d30140ae5e99p-5), V2 (0x1.338e31eb2fbbcp-5), |
36 | V2 (-0x1.00e6eece7de8p-5), V2 (0x1.860897b29e5efp-6), |
37 | V2 (-0x1.0051381722a59p-6), V2 (0x1.14e9dc19a4a4ep-7), |
38 | V2 (-0x1.d0062b42fe3bfp-9), V2 (0x1.17739e210171ap-10), |
39 | V2 (-0x1.ab24da7be7402p-13), V2 (0x1.358851160a528p-16), }, |
40 | .pi_over_2 = V2 (0x1.921fb54442d18p+0), |
41 | }; |
42 | |
43 | #define SignMask v_u64 (0x8000000000000000) |
44 | #define TinyBound 0x3e10000000000000 /* asuint64(0x1p-30). */ |
45 | #define BigBound 0x4340000000000000 /* asuint64(0x1p53). */ |
46 | |
47 | /* Fast implementation of vector atan. |
48 | Based on atan(x) ~ shift + z + z^3 * P(z^2) with reduction to [0,1] using |
49 | z=1/x and shift = pi/2. Maximum observed error is 2.27 ulps: |
50 | _ZGVnN2v_atan (0x1.0005af27c23e9p+0) got 0x1.9225645bdd7c1p-1 |
51 | want 0x1.9225645bdd7c3p-1. */ |
52 | float64x2_t VPCS_ATTR V_NAME_D1 (atan) (float64x2_t x) |
53 | { |
54 | const struct data *d = ptr_barrier (&data); |
55 | |
56 | /* Small cases, infs and nans are supported by our approximation technique, |
57 | but do not set fenv flags correctly. Only trigger special case if we need |
58 | fenv. */ |
59 | uint64x2_t ix = vreinterpretq_u64_f64 (x); |
60 | uint64x2_t sign = vandq_u64 (ix, SignMask); |
61 | |
62 | #if WANT_SIMD_EXCEPT |
63 | uint64x2_t ia12 = vandq_u64 (ix, v_u64 (0x7ff0000000000000)); |
64 | uint64x2_t special = vcgtq_u64 (vsubq_u64 (ia12, v_u64 (TinyBound)), |
65 | v_u64 (BigBound - TinyBound)); |
66 | /* If any lane is special, fall back to the scalar routine for all lanes. */ |
67 | if (__glibc_unlikely (v_any_u64 (special))) |
68 | return v_call_f64 (atan, x, v_f64 (0), v_u64 (-1)); |
69 | #endif |
70 | |
71 | /* Argument reduction: |
72 | y := arctan(x) for x < 1 |
73 | y := pi/2 + arctan(-1/x) for x > 1 |
74 | Hence, use z=-1/a if x>=1, otherwise z=a. */ |
75 | uint64x2_t red = vcagtq_f64 (x, v_f64 (1.0)); |
76 | /* Avoid dependency in abs(x) in division (and comparison). */ |
77 | float64x2_t z = vbslq_f64 (red, vdivq_f64 (v_f64 (1.0), x), x); |
78 | float64x2_t shift = vreinterpretq_f64_u64 ( |
79 | vandq_u64 (red, vreinterpretq_u64_f64 (d->pi_over_2))); |
80 | /* Use absolute value only when needed (odd powers of z). */ |
81 | float64x2_t az = vbslq_f64 ( |
82 | SignMask, vreinterpretq_f64_u64 (vandq_u64 (SignMask, red)), z); |
83 | |
84 | /* Calculate the polynomial approximation. |
85 | Use split Estrin scheme for P(z^2) with deg(P)=19. Use split instead of |
86 | full scheme to avoid underflow in x^16. |
87 | The order 19 polynomial P approximates |
88 | (atan(sqrt(x))-sqrt(x))/x^(3/2). */ |
89 | float64x2_t z2 = vmulq_f64 (z, z); |
90 | float64x2_t x2 = vmulq_f64 (z2, z2); |
91 | float64x2_t x4 = vmulq_f64 (x2, x2); |
92 | float64x2_t x8 = vmulq_f64 (x4, x4); |
93 | float64x2_t y |
94 | = vfmaq_f64 (v_estrin_7_f64 (z2, x2, x4, d->poly), |
95 | v_estrin_11_f64 (z2, x2, x4, x8, d->poly + 8), x8); |
96 | |
97 | /* Finalize. y = shift + z + z^3 * P(z^2). */ |
98 | y = vfmaq_f64 (az, y, vmulq_f64 (z2, az)); |
99 | y = vaddq_f64 (y, shift); |
100 | |
101 | /* y = atan(x) if x>0, -atan(-x) otherwise. */ |
102 | y = vreinterpretq_f64_u64 (veorq_u64 (vreinterpretq_u64_f64 (y), sign)); |
103 | return y; |
104 | } |
105 | |