1*f3087befSAndrew Turner /*
2*f3087befSAndrew Turner * Single-precision inverse error function (AdvSIMD variant).
3*f3087befSAndrew Turner *
4*f3087befSAndrew Turner * Copyright (c) 2023-2024, Arm Limited.
5*f3087befSAndrew Turner * SPDX-License-Identifier: MIT OR Apache-2.0 WITH LLVM-exception
6*f3087befSAndrew Turner */
7*f3087befSAndrew Turner #include "v_math.h"
8*f3087befSAndrew Turner #include "test_sig.h"
9*f3087befSAndrew Turner #include "test_defs.h"
10*f3087befSAndrew Turner #include "v_poly_f32.h"
11*f3087befSAndrew Turner #include "v_logf_inline.h"
12*f3087befSAndrew Turner
13*f3087befSAndrew Turner const static struct data
14*f3087befSAndrew Turner {
15*f3087befSAndrew Turner /* We use P_N and Q_N to refer to arrays of coefficients, where P_N is the
16*f3087befSAndrew Turner coeffs of the numerator in table N of Blair et al, and Q_N is the coeffs
17*f3087befSAndrew Turner of the denominator. Coefficients are stored in various interleaved
18*f3087befSAndrew Turner formats to allow for table-based (vector-to-vector) lookup.
19*f3087befSAndrew Turner
20*f3087befSAndrew Turner Plo is first two coefficients of P_10 and P_29 interleaved.
21*f3087befSAndrew Turner PQ is third coeff of P_10 and first of Q_29 interleaved.
22*f3087befSAndrew Turner Qhi is second and third coeffs of Q_29 interleaved.
23*f3087befSAndrew Turner P29_3 is a homogenous vector with fourth coeff of P_29.
24*f3087befSAndrew Turner
25*f3087befSAndrew Turner P_10 and Q_10 are also stored in homogenous vectors to allow better
26*f3087befSAndrew Turner memory access when no lanes are in a tail region. */
27*f3087befSAndrew Turner float Plo[4], PQ[4], Qhi[4];
28*f3087befSAndrew Turner float32x4_t P29_3, tailshift;
29*f3087befSAndrew Turner float32x4_t P_50[6], Q_50[2];
30*f3087befSAndrew Turner float32x4_t P_10[3], Q_10[3];
31*f3087befSAndrew Turner uint8_t idxhi[16], idxlo[16];
32*f3087befSAndrew Turner struct v_logf_data logf_tbl;
33*f3087befSAndrew Turner } data = {
34*f3087befSAndrew Turner .idxlo = { 0, 1, 2, 3, 0, 1, 2, 3, 0, 1, 2, 3, 0, 1, 2, 3 },
35*f3087befSAndrew Turner .idxhi = { 8, 9, 10, 11, 8, 9, 10, 11, 8, 9, 10, 11, 8, 9, 10, 11 },
36*f3087befSAndrew Turner .P29_3 = V4 (0x1.b13626p-2),
37*f3087befSAndrew Turner .tailshift = V4 (-0.87890625),
38*f3087befSAndrew Turner .Plo = { -0x1.a31268p+3, -0x1.fc0252p-4, 0x1.ac9048p+4, 0x1.119d44p+0 },
39*f3087befSAndrew Turner .PQ = { -0x1.293ff6p+3, -0x1.f59ee2p+0, -0x1.8265eep+3, -0x1.69952p-4 },
40*f3087befSAndrew Turner .Qhi = { 0x1.ef5eaep+4, 0x1.c7b7d2p-1, -0x1.12665p+4, -0x1.167d7p+1 },
41*f3087befSAndrew Turner .P_50 = { V4 (0x1.3d8948p-3), V4 (0x1.61f9eap+0), V4 (0x1.61c6bcp-1),
42*f3087befSAndrew Turner V4 (-0x1.20c9f2p+0), V4 (0x1.5c704cp-1), V4 (-0x1.50c6bep-3) },
43*f3087befSAndrew Turner .Q_50 = { V4 (0x1.3d7dacp-3), V4 (0x1.629e5p+0) },
44*f3087befSAndrew Turner .P_10 = { V4 (-0x1.a31268p+3), V4 (0x1.ac9048p+4), V4 (-0x1.293ff6p+3) },
45*f3087befSAndrew Turner .Q_10 = { V4 (-0x1.8265eep+3), V4 (0x1.ef5eaep+4), V4 (-0x1.12665p+4) },
46*f3087befSAndrew Turner .logf_tbl = V_LOGF_CONSTANTS
47*f3087befSAndrew Turner };
48*f3087befSAndrew Turner
49*f3087befSAndrew Turner static inline float32x4_t
special(float32x4_t x,const struct data * d)50*f3087befSAndrew Turner special (float32x4_t x, const struct data *d)
51*f3087befSAndrew Turner {
52*f3087befSAndrew Turner /* Note erfinvf(inf) should return NaN, and erfinvf(1) should return Inf.
53*f3087befSAndrew Turner By using log here, instead of log1p, we return finite values for both
54*f3087befSAndrew Turner these inputs, and values outside [-1, 1]. This is non-compliant, but is an
55*f3087befSAndrew Turner acceptable optimisation at Ofast. To get correct behaviour for all finite
56*f3087befSAndrew Turner values use the log1pf_inline helper on -abs(x) - note that erfinvf(inf)
57*f3087befSAndrew Turner will still be finite. */
58*f3087befSAndrew Turner float32x4_t t = vdivq_f32 (
59*f3087befSAndrew Turner v_f32 (1), vsqrtq_f32 (vnegq_f32 (v_logf_inline (
60*f3087befSAndrew Turner vsubq_f32 (v_f32 (1), vabsq_f32 (x)), &d->logf_tbl))));
61*f3087befSAndrew Turner float32x4_t ts = vbslq_f32 (v_u32 (0x7fffffff), t, x);
62*f3087befSAndrew Turner float32x4_t q = vfmaq_f32 (d->Q_50[0], vaddq_f32 (t, d->Q_50[1]), t);
63*f3087befSAndrew Turner return vdivq_f32 (v_horner_5_f32 (t, d->P_50), vmulq_f32 (ts, q));
64*f3087befSAndrew Turner }
65*f3087befSAndrew Turner
66*f3087befSAndrew Turner static inline float32x4_t
notails(float32x4_t x,const struct data * d)67*f3087befSAndrew Turner notails (float32x4_t x, const struct data *d)
68*f3087befSAndrew Turner {
69*f3087befSAndrew Turner /* Shortcut when no input is in a tail region - no need to gather shift or
70*f3087befSAndrew Turner coefficients. */
71*f3087befSAndrew Turner float32x4_t t = vfmaq_f32 (v_f32 (-0.5625), x, x);
72*f3087befSAndrew Turner float32x4_t q = vaddq_f32 (t, d->Q_10[2]);
73*f3087befSAndrew Turner q = vfmaq_f32 (d->Q_10[1], t, q);
74*f3087befSAndrew Turner q = vfmaq_f32 (d->Q_10[0], t, q);
75*f3087befSAndrew Turner
76*f3087befSAndrew Turner return vdivq_f32 (vmulq_f32 (x, v_horner_2_f32 (t, d->P_10)), q);
77*f3087befSAndrew Turner }
78*f3087befSAndrew Turner
79*f3087befSAndrew Turner static inline float32x4_t
lookup(float32x4_t tbl,uint8x16_t idx)80*f3087befSAndrew Turner lookup (float32x4_t tbl, uint8x16_t idx)
81*f3087befSAndrew Turner {
82*f3087befSAndrew Turner return vreinterpretq_f32_u8 (vqtbl1q_u8 (vreinterpretq_u8_f32 (tbl), idx));
83*f3087befSAndrew Turner }
84*f3087befSAndrew Turner
85*f3087befSAndrew Turner /* Vector implementation of Blair et al's rational approximation to inverse
86*f3087befSAndrew Turner error function in single-precision. Worst-case error is 4.98 ULP, in the
87*f3087befSAndrew Turner tail region:
88*f3087befSAndrew Turner _ZGVnN4v_erfinvf(0x1.f7dbeep-1) got 0x1.b4793p+0
89*f3087befSAndrew Turner want 0x1.b4793ap+0 . */
V_NAME_F1(erfinv)90*f3087befSAndrew Turner float32x4_t VPCS_ATTR NOINLINE V_NAME_F1 (erfinv) (float32x4_t x)
91*f3087befSAndrew Turner {
92*f3087befSAndrew Turner const struct data *d = ptr_barrier (&data);
93*f3087befSAndrew Turner
94*f3087befSAndrew Turner /* Calculate inverse error using algorithm described in
95*f3087befSAndrew Turner J. M. Blair, C. A. Edwards, and J. H. Johnson,
96*f3087befSAndrew Turner "Rational Chebyshev approximations for the inverse of the error
97*f3087befSAndrew Turner function", Math. Comp. 30, pp. 827--830 (1976).
98*f3087befSAndrew Turner https://doi.org/10.1090/S0025-5718-1976-0421040-7.
99*f3087befSAndrew Turner
100*f3087befSAndrew Turner Algorithm has 3 intervals:
101*f3087befSAndrew Turner - 'Normal' region [-0.75, 0.75]
102*f3087befSAndrew Turner - Tail region [0.75, 0.9375] U [-0.9375, -0.75]
103*f3087befSAndrew Turner - Extreme tail [-1, -0.9375] U [0.9375, 1]
104*f3087befSAndrew Turner Normal and tail are both rational approximation of similar order on
105*f3087befSAndrew Turner shifted input - these are typically performed in parallel using gather
106*f3087befSAndrew Turner loads to obtain correct coefficients depending on interval. */
107*f3087befSAndrew Turner uint32x4_t is_tail = vcageq_f32 (x, v_f32 (0.75));
108*f3087befSAndrew Turner uint32x4_t extreme_tail = vcageq_f32 (x, v_f32 (0.9375));
109*f3087befSAndrew Turner
110*f3087befSAndrew Turner if (unlikely (!v_any_u32 (is_tail)))
111*f3087befSAndrew Turner /* Shortcut for if all lanes are in [-0.75, 0.75] - can avoid having to
112*f3087befSAndrew Turner gather coefficients. If input is uniform in [-1, 1] then likelihood of
113*f3087befSAndrew Turner this is 0.75^4 ~= 0.31. */
114*f3087befSAndrew Turner return notails (x, d);
115*f3087befSAndrew Turner
116*f3087befSAndrew Turner /* Select requisite shift depending on interval: polynomial is evaluated on
117*f3087befSAndrew Turner x * x - shift.
118*f3087befSAndrew Turner Normal shift = 0.5625
119*f3087befSAndrew Turner Tail shift = 0.87890625. */
120*f3087befSAndrew Turner float32x4_t t
121*f3087befSAndrew Turner = vfmaq_f32 (vbslq_f32 (is_tail, d->tailshift, v_f32 (-0.5625)), x, x);
122*f3087befSAndrew Turner
123*f3087befSAndrew Turner /* Calculate indexes for tbl: tbl is byte-wise, so:
124*f3087befSAndrew Turner [0, 1, 2, 3, 4, 5, 6, ....] copies the vector
125*f3087befSAndrew Turner Add 4 * i to a group of 4 lanes to copy 32-bit lane i. Each vector stores
126*f3087befSAndrew Turner two pairs of coeffs, so we need two idx vectors - one for each pair. */
127*f3087befSAndrew Turner uint8x16_t off = vandq_u8 (vreinterpretq_u8_u32 (is_tail), vdupq_n_u8 (4));
128*f3087befSAndrew Turner uint8x16_t idx_lo = vaddq_u8 (vld1q_u8 (d->idxlo), off);
129*f3087befSAndrew Turner uint8x16_t idx_hi = vaddq_u8 (vld1q_u8 (d->idxhi), off);
130*f3087befSAndrew Turner
131*f3087befSAndrew Turner /* Load the tables. */
132*f3087befSAndrew Turner float32x4_t plo = vld1q_f32 (d->Plo);
133*f3087befSAndrew Turner float32x4_t pq = vld1q_f32 (d->PQ);
134*f3087befSAndrew Turner float32x4_t qhi = vld1q_f32 (d->Qhi);
135*f3087befSAndrew Turner
136*f3087befSAndrew Turner /* Do the lookup (and calculate p3 by masking non-tail lanes). */
137*f3087befSAndrew Turner float32x4_t p3 = vreinterpretq_f32_u32 (
138*f3087befSAndrew Turner vandq_u32 (is_tail, vreinterpretq_u32_f32 (d->P29_3)));
139*f3087befSAndrew Turner float32x4_t p0 = lookup (plo, idx_lo), p1 = lookup (plo, idx_hi),
140*f3087befSAndrew Turner p2 = lookup (pq, idx_lo), q0 = lookup (pq, idx_hi),
141*f3087befSAndrew Turner q1 = lookup (qhi, idx_lo), q2 = lookup (qhi, idx_hi);
142*f3087befSAndrew Turner
143*f3087befSAndrew Turner float32x4_t p = vfmaq_f32 (p2, p3, t);
144*f3087befSAndrew Turner p = vfmaq_f32 (p1, p, t);
145*f3087befSAndrew Turner p = vfmaq_f32 (p0, p, t);
146*f3087befSAndrew Turner p = vmulq_f32 (x, p);
147*f3087befSAndrew Turner
148*f3087befSAndrew Turner float32x4_t q = vfmaq_f32 (q1, vaddq_f32 (q2, t), t);
149*f3087befSAndrew Turner q = vfmaq_f32 (q0, q, t);
150*f3087befSAndrew Turner
151*f3087befSAndrew Turner if (unlikely (v_any_u32 (extreme_tail)))
152*f3087befSAndrew Turner /* At least one lane is in the extreme tail - if input is uniform in
153*f3087befSAndrew Turner [-1, 1] the likelihood of this is ~0.23. */
154*f3087befSAndrew Turner return vbslq_f32 (extreme_tail, special (x, d), vdivq_f32 (p, q));
155*f3087befSAndrew Turner
156*f3087befSAndrew Turner return vdivq_f32 (p, q);
157*f3087befSAndrew Turner }
158*f3087befSAndrew Turner
159*f3087befSAndrew Turner HALF_WIDTH_ALIAS_F1 (erfinv)
160*f3087befSAndrew Turner
161*f3087befSAndrew Turner #if USE_MPFR
162*f3087befSAndrew Turner # warning Not generating tests for _ZGVnN4v_erfinvf, as MPFR has no suitable reference
163*f3087befSAndrew Turner #else
164*f3087befSAndrew Turner TEST_SIG (V, F, 1, erfinv, -0.99, 0.99)
165*f3087befSAndrew Turner TEST_DISABLE_FENV (V_NAME_F1 (erfinv))
166*f3087befSAndrew Turner TEST_ULP (V_NAME_F1 (erfinv), 4.49)
167*f3087befSAndrew Turner TEST_SYM_INTERVAL (V_NAME_F1 (erfinv), 0, 0x1.fffffep-1, 40000)
168*f3087befSAndrew Turner /* Test with control lane in each interval. */
169*f3087befSAndrew Turner TEST_CONTROL_VALUE (V_NAME_F1 (erfinv), 0.5)
170*f3087befSAndrew Turner TEST_CONTROL_VALUE (V_NAME_F1 (erfinv), 0.8)
171*f3087befSAndrew Turner TEST_CONTROL_VALUE (V_NAME_F1 (erfinv), 0.95)
172*f3087befSAndrew Turner #endif
173