1*f3087befSAndrew Turner /*
2*f3087befSAndrew Turner * Single-precision vector atan(x) function.
3*f3087befSAndrew Turner *
4*f3087befSAndrew Turner * Copyright (c) 2021-2024, 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 "v_math.h"
9*f3087befSAndrew Turner #include "test_sig.h"
10*f3087befSAndrew Turner #include "test_defs.h"
11*f3087befSAndrew Turner #include "v_poly_f32.h"
12*f3087befSAndrew Turner
13*f3087befSAndrew Turner static const struct data
14*f3087befSAndrew Turner {
15*f3087befSAndrew Turner float32x4_t poly[8];
16*f3087befSAndrew Turner float32x4_t pi_over_2;
17*f3087befSAndrew Turner } data = {
18*f3087befSAndrew Turner /* Coefficients of polynomial P such that atan(x)~x+x*P(x^2) on
19*f3087befSAndrew Turner [2**-128, 1.0].
20*f3087befSAndrew Turner Generated using fpminimax between FLT_MIN and 1. */
21*f3087befSAndrew Turner .poly = { V4 (-0x1.55555p-2f), V4 (0x1.99935ep-3f), V4 (-0x1.24051ep-3f),
22*f3087befSAndrew Turner V4 (0x1.bd7368p-4f), V4 (-0x1.491f0ep-4f), V4 (0x1.93a2c0p-5f),
23*f3087befSAndrew Turner V4 (-0x1.4c3c60p-6f), V4 (0x1.01fd88p-8f) },
24*f3087befSAndrew Turner .pi_over_2 = V4 (0x1.921fb6p+0f),
25*f3087befSAndrew Turner };
26*f3087befSAndrew Turner
27*f3087befSAndrew Turner #define SignMask v_u32 (0x80000000)
28*f3087befSAndrew Turner
29*f3087befSAndrew Turner #define P(i) d->poly[i]
30*f3087befSAndrew Turner
31*f3087befSAndrew Turner #define TinyBound 0x30800000 /* asuint(0x1p-30). */
32*f3087befSAndrew Turner #define BigBound 0x4e800000 /* asuint(0x1p30). */
33*f3087befSAndrew Turner
34*f3087befSAndrew Turner #if WANT_SIMD_EXCEPT
35*f3087befSAndrew Turner static float32x4_t VPCS_ATTR NOINLINE
special_case(float32x4_t x,float32x4_t y,uint32x4_t special)36*f3087befSAndrew Turner special_case (float32x4_t x, float32x4_t y, uint32x4_t special)
37*f3087befSAndrew Turner {
38*f3087befSAndrew Turner return v_call_f32 (atanf, x, y, special);
39*f3087befSAndrew Turner }
40*f3087befSAndrew Turner #endif
41*f3087befSAndrew Turner
42*f3087befSAndrew Turner /* Fast implementation of vector atanf based on
43*f3087befSAndrew Turner atan(x) ~ shift + z + z^3 * P(z^2) with reduction to [0,1]
44*f3087befSAndrew Turner using z=-1/x and shift = pi/2. Maximum observed error is 2.9ulps:
45*f3087befSAndrew Turner _ZGVnN4v_atanf (0x1.0468f6p+0) got 0x1.967f06p-1 want 0x1.967fp-1. */
V_NAME_F1(atan)46*f3087befSAndrew Turner float32x4_t VPCS_ATTR NOINLINE V_NAME_F1 (atan) (float32x4_t x)
47*f3087befSAndrew Turner {
48*f3087befSAndrew Turner const struct data *d = ptr_barrier (&data);
49*f3087befSAndrew Turner
50*f3087befSAndrew Turner /* Small cases, infs and nans are supported by our approximation technique,
51*f3087befSAndrew Turner but do not set fenv flags correctly. Only trigger special case if we need
52*f3087befSAndrew Turner fenv. */
53*f3087befSAndrew Turner uint32x4_t ix = vreinterpretq_u32_f32 (x);
54*f3087befSAndrew Turner uint32x4_t sign = vandq_u32 (ix, SignMask);
55*f3087befSAndrew Turner
56*f3087befSAndrew Turner #if WANT_SIMD_EXCEPT
57*f3087befSAndrew Turner uint32x4_t ia = vandq_u32 (ix, v_u32 (0x7ff00000));
58*f3087befSAndrew Turner uint32x4_t special = vcgtq_u32 (vsubq_u32 (ia, v_u32 (TinyBound)),
59*f3087befSAndrew Turner v_u32 (BigBound - TinyBound));
60*f3087befSAndrew Turner /* If any lane is special, fall back to the scalar routine for all lanes. */
61*f3087befSAndrew Turner if (unlikely (v_any_u32 (special)))
62*f3087befSAndrew Turner return special_case (x, x, v_u32 (-1));
63*f3087befSAndrew Turner #endif
64*f3087befSAndrew Turner
65*f3087befSAndrew Turner /* Argument reduction:
66*f3087befSAndrew Turner y := arctan(x) for x < 1
67*f3087befSAndrew Turner y := pi/2 + arctan(-1/x) for x > 1
68*f3087befSAndrew Turner Hence, use z=-1/a if x>=1, otherwise z=a. */
69*f3087befSAndrew Turner uint32x4_t red = vcagtq_f32 (x, v_f32 (1.0));
70*f3087befSAndrew Turner /* Avoid dependency in abs(x) in division (and comparison). */
71*f3087befSAndrew Turner float32x4_t z = vbslq_f32 (red, vdivq_f32 (v_f32 (1.0f), x), x);
72*f3087befSAndrew Turner float32x4_t shift = vreinterpretq_f32_u32 (
73*f3087befSAndrew Turner vandq_u32 (red, vreinterpretq_u32_f32 (d->pi_over_2)));
74*f3087befSAndrew Turner /* Use absolute value only when needed (odd powers of z). */
75*f3087befSAndrew Turner float32x4_t az = vbslq_f32 (
76*f3087befSAndrew Turner SignMask, vreinterpretq_f32_u32 (vandq_u32 (SignMask, red)), z);
77*f3087befSAndrew Turner
78*f3087befSAndrew Turner /* Calculate the polynomial approximation.
79*f3087befSAndrew Turner Use 2-level Estrin scheme for P(z^2) with deg(P)=7. However,
80*f3087befSAndrew Turner a standard implementation using z8 creates spurious underflow
81*f3087befSAndrew Turner in the very last fma (when z^8 is small enough).
82*f3087befSAndrew Turner Therefore, we split the last fma into a mul and an fma.
83*f3087befSAndrew Turner Horner and single-level Estrin have higher errors that exceed
84*f3087befSAndrew Turner threshold. */
85*f3087befSAndrew Turner float32x4_t z2 = vmulq_f32 (z, z);
86*f3087befSAndrew Turner float32x4_t z4 = vmulq_f32 (z2, z2);
87*f3087befSAndrew Turner
88*f3087befSAndrew Turner float32x4_t y = vfmaq_f32 (
89*f3087befSAndrew Turner v_pairwise_poly_3_f32 (z2, z4, d->poly), z4,
90*f3087befSAndrew Turner vmulq_f32 (z4, v_pairwise_poly_3_f32 (z2, z4, d->poly + 4)));
91*f3087befSAndrew Turner
92*f3087befSAndrew Turner /* y = shift + z * P(z^2). */
93*f3087befSAndrew Turner y = vaddq_f32 (vfmaq_f32 (az, y, vmulq_f32 (z2, az)), shift);
94*f3087befSAndrew Turner
95*f3087befSAndrew Turner /* y = atan(x) if x>0, -atan(-x) otherwise. */
96*f3087befSAndrew Turner y = vreinterpretq_f32_u32 (veorq_u32 (vreinterpretq_u32_f32 (y), sign));
97*f3087befSAndrew Turner
98*f3087befSAndrew Turner return y;
99*f3087befSAndrew Turner }
100*f3087befSAndrew Turner
101*f3087befSAndrew Turner HALF_WIDTH_ALIAS_F1 (atan)
102*f3087befSAndrew Turner
103*f3087befSAndrew Turner TEST_SIG (V, F, 1, atan, -10.0, 10.0)
104*f3087befSAndrew Turner TEST_ULP (V_NAME_F1 (atan), 2.5)
105*f3087befSAndrew Turner TEST_DISABLE_FENV_IF_NOT (V_NAME_F1 (atan), WANT_SIMD_EXCEPT)
106*f3087befSAndrew Turner TEST_SYM_INTERVAL (V_NAME_F1 (atan), 0, 0x1p-30, 5000)
107*f3087befSAndrew Turner TEST_SYM_INTERVAL (V_NAME_F1 (atan), 0x1p-30, 1, 40000)
108*f3087befSAndrew Turner TEST_SYM_INTERVAL (V_NAME_F1 (atan), 1, 0x1p30, 40000)
109*f3087befSAndrew Turner TEST_SYM_INTERVAL (V_NAME_F1 (atan), 0x1p30, inf, 1000)
110