1 /*
2 * Single-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_f32.h"
12
13 static const struct data
14 {
15 float32x4_t poly[8];
16 float32x4_t pi_over_2;
17 } data = {
18 /* Coefficients of polynomial P such that atan(x)~x+x*P(x^2) on
19 [2**-128, 1.0].
20 Generated using fpminimax between FLT_MIN and 1. */
21 .poly = { V4 (-0x1.55555p-2f), V4 (0x1.99935ep-3f), V4 (-0x1.24051ep-3f),
22 V4 (0x1.bd7368p-4f), V4 (-0x1.491f0ep-4f), V4 (0x1.93a2c0p-5f),
23 V4 (-0x1.4c3c60p-6f), V4 (0x1.01fd88p-8f) },
24 .pi_over_2 = V4 (0x1.921fb6p+0f),
25 };
26
27 #define SignMask v_u32 (0x80000000)
28
29 #define P(i) d->poly[i]
30
31 #define TinyBound 0x30800000 /* asuint(0x1p-30). */
32 #define BigBound 0x4e800000 /* asuint(0x1p30). */
33
34 #if WANT_SIMD_EXCEPT
35 static float32x4_t VPCS_ATTR NOINLINE
special_case(float32x4_t x,float32x4_t y,uint32x4_t special)36 special_case (float32x4_t x, float32x4_t y, uint32x4_t special)
37 {
38 return v_call_f32 (atanf, x, y, special);
39 }
40 #endif
41
42 /* Fast implementation of vector atanf based on
43 atan(x) ~ shift + z + z^3 * P(z^2) with reduction to [0,1]
44 using z=-1/x and shift = pi/2. Maximum observed error is 2.9ulps:
45 _ZGVnN4v_atanf (0x1.0468f6p+0) got 0x1.967f06p-1 want 0x1.967fp-1. */
V_NAME_F1(atan)46 float32x4_t VPCS_ATTR V_NAME_F1 (atan) (float32x4_t x)
47 {
48 const struct data *d = ptr_barrier (&data);
49
50 /* Small cases, infs and nans are supported by our approximation technique,
51 but do not set fenv flags correctly. Only trigger special case if we need
52 fenv. */
53 uint32x4_t ix = vreinterpretq_u32_f32 (x);
54 uint32x4_t sign = vandq_u32 (ix, SignMask);
55
56 #if WANT_SIMD_EXCEPT
57 uint32x4_t ia = vandq_u32 (ix, v_u32 (0x7ff00000));
58 uint32x4_t special = vcgtq_u32 (vsubq_u32 (ia, v_u32 (TinyBound)),
59 v_u32 (BigBound - TinyBound));
60 /* If any lane is special, fall back to the scalar routine for all lanes. */
61 if (unlikely (v_any_u32 (special)))
62 return special_case (x, x, v_u32 (-1));
63 #endif
64
65 /* Argument reduction:
66 y := arctan(x) for x < 1
67 y := pi/2 + arctan(-1/x) for x > 1
68 Hence, use z=-1/a if x>=1, otherwise z=a. */
69 uint32x4_t red = vcagtq_f32 (x, v_f32 (1.0));
70 /* Avoid dependency in abs(x) in division (and comparison). */
71 float32x4_t z = vbslq_f32 (red, vdivq_f32 (v_f32 (1.0f), x), x);
72 float32x4_t shift = vreinterpretq_f32_u32 (
73 vandq_u32 (red, vreinterpretq_u32_f32 (d->pi_over_2)));
74 /* Use absolute value only when needed (odd powers of z). */
75 float32x4_t az = vbslq_f32 (
76 SignMask, vreinterpretq_f32_u32 (vandq_u32 (SignMask, red)), z);
77
78 /* Calculate the polynomial approximation.
79 Use 2-level Estrin scheme for P(z^2) with deg(P)=7. However,
80 a standard implementation using z8 creates spurious underflow
81 in the very last fma (when z^8 is small enough).
82 Therefore, we split the last fma into a mul and an fma.
83 Horner and single-level Estrin have higher errors that exceed
84 threshold. */
85 float32x4_t z2 = vmulq_f32 (z, z);
86 float32x4_t z4 = vmulq_f32 (z2, z2);
87
88 float32x4_t y = vfmaq_f32 (
89 v_pairwise_poly_3_f32 (z2, z4, d->poly), z4,
90 vmulq_f32 (z4, v_pairwise_poly_3_f32 (z2, z4, d->poly + 4)));
91
92 /* y = shift + z * P(z^2). */
93 y = vaddq_f32 (vfmaq_f32 (az, y, vmulq_f32 (z2, az)), shift);
94
95 /* y = atan(x) if x>0, -atan(-x) otherwise. */
96 y = vreinterpretq_f32_u32 (veorq_u32 (vreinterpretq_u32_f32 (y), sign));
97
98 return y;
99 }
100
101 PL_SIG (V, F, 1, atan, -10.0, 10.0)
102 PL_TEST_ULP (V_NAME_F1 (atan), 2.5)
103 PL_TEST_EXPECT_FENV (V_NAME_F1 (atan), WANT_SIMD_EXCEPT)
104 PL_TEST_SYM_INTERVAL (V_NAME_F1 (atan), 0, 0x1p-30, 5000)
105 PL_TEST_SYM_INTERVAL (V_NAME_F1 (atan), 0x1p-30, 1, 40000)
106 PL_TEST_SYM_INTERVAL (V_NAME_F1 (atan), 1, 0x1p30, 40000)
107 PL_TEST_SYM_INTERVAL (V_NAME_F1 (atan), 0x1p30, inf, 1000)
108