1 /* 2 * Single-precision inverse error function (AdvSIMD variant). 3 * 4 * Copyright (c) 2023, Arm Limited. 5 * SPDX-License-Identifier: MIT OR Apache-2.0 WITH LLVM-exception 6 */ 7 #include "v_math.h" 8 #include "pl_sig.h" 9 #include "pl_test.h" 10 #include "poly_advsimd_f32.h" 11 #include "v_logf_inline.h" 12 13 const static struct data 14 { 15 /* We use P_N and Q_N to refer to arrays of coefficients, where P_N is the 16 coeffs of the numerator in table N of Blair et al, and Q_N is the coeffs 17 of the denominator. Coefficients are stored in various interleaved 18 formats to allow for table-based (vector-to-vector) lookup. 19 20 Plo is first two coefficients of P_10 and P_29 interleaved. 21 PQ is third coeff of P_10 and first of Q_29 interleaved. 22 Qhi is second and third coeffs of Q_29 interleaved. 23 P29_3 is a homogenous vector with fourth coeff of P_29. 24 25 P_10 and Q_10 are also stored in homogenous vectors to allow better 26 memory access when no lanes are in a tail region. */ 27 float32x4_t Plo, PQ, Qhi, P29_3, tailshift; 28 float32x4_t P_50[6], Q_50[2]; 29 float32x4_t P_10[3], Q_10[3]; 30 uint8x16_t idxhi, idxlo; 31 struct v_logf_data logf_tbl; 32 } data = { 33 .idxlo = { 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15 }, 34 .idxhi = { 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23 }, 35 .P29_3 = V4 (0x1.b13626p-2), 36 .tailshift = V4 (-0.87890625), 37 .Plo = { -0x1.a31268p+3, -0x1.fc0252p-4, 0x1.ac9048p+4, 0x1.119d44p+0 }, 38 .PQ = { -0x1.293ff6p+3, -0x1.f59ee2p+0, -0x1.8265eep+3, -0x1.69952p-4 }, 39 .Qhi = { 0x1.ef5eaep+4, 0x1.c7b7d2p-1, -0x1.12665p+4, -0x1.167d7p+1 }, 40 .P_50 = { V4 (0x1.3d8948p-3), V4 (0x1.61f9eap+0), V4 (0x1.61c6bcp-1), 41 V4 (-0x1.20c9f2p+0), V4 (0x1.5c704cp-1), V4 (-0x1.50c6bep-3) }, 42 .Q_50 = { V4 (0x1.3d7dacp-3), V4 (0x1.629e5p+0) }, 43 .P_10 = { V4 (-0x1.a31268p+3), V4 (0x1.ac9048p+4), V4 (-0x1.293ff6p+3) }, 44 .Q_10 = { V4 (-0x1.8265eep+3), V4 (0x1.ef5eaep+4), V4 (-0x1.12665p+4) }, 45 .logf_tbl = V_LOGF_CONSTANTS 46 }; 47 48 static inline float32x4_t 49 special (float32x4_t x, const struct data *d) 50 { 51 /* Note erfinvf(inf) should return NaN, and erfinvf(1) should return Inf. 52 By using log here, instead of log1p, we return finite values for both 53 these inputs, and values outside [-1, 1]. This is non-compliant, but is an 54 acceptable optimisation at Ofast. To get correct behaviour for all finite 55 values use the log1pf_inline helper on -abs(x) - note that erfinvf(inf) 56 will still be finite. */ 57 float32x4_t t = vdivq_f32 ( 58 v_f32 (1), vsqrtq_f32 (vnegq_f32 (v_logf_inline ( 59 vsubq_f32 (v_f32 (1), vabsq_f32 (x)), &d->logf_tbl)))); 60 float32x4_t ts = vbslq_f32 (v_u32 (0x7fffffff), t, x); 61 float32x4_t q = vfmaq_f32 (d->Q_50[0], vaddq_f32 (t, d->Q_50[1]), t); 62 return vdivq_f32 (v_horner_5_f32 (t, d->P_50), vmulq_f32 (ts, q)); 63 } 64 65 static inline float32x4_t 66 notails (float32x4_t x, const struct data *d) 67 { 68 /* Shortcut when no input is in a tail region - no need to gather shift or 69 coefficients. */ 70 float32x4_t t = vfmaq_f32 (v_f32 (-0.5625), x, x); 71 float32x4_t q = vaddq_f32 (t, d->Q_10[2]); 72 q = vfmaq_f32 (d->Q_10[1], t, q); 73 q = vfmaq_f32 (d->Q_10[0], t, q); 74 75 return vdivq_f32 (vmulq_f32 (x, v_horner_2_f32 (t, d->P_10)), q); 76 } 77 78 static inline float32x4_t 79 lookup (float32x4_t tbl, uint8x16_t idx) 80 { 81 return vreinterpretq_f32_u8 (vqtbl1q_u8 (vreinterpretq_u8_f32 (tbl), idx)); 82 } 83 84 /* Vector implementation of Blair et al's rational approximation to inverse 85 error function in single-precision. Worst-case error is 4.98 ULP, in the 86 tail region: 87 _ZGVnN4v_erfinvf(0x1.f7dbeep-1) got 0x1.b4793p+0 88 want 0x1.b4793ap+0 . */ 89 float32x4_t VPCS_ATTR V_NAME_F1 (erfinv) (float32x4_t x) 90 { 91 const struct data *d = ptr_barrier (&data); 92 93 /* Calculate inverse error using algorithm described in 94 J. M. Blair, C. A. Edwards, and J. H. Johnson, 95 "Rational Chebyshev approximations for the inverse of the error 96 function", Math. Comp. 30, pp. 827--830 (1976). 97 https://doi.org/10.1090/S0025-5718-1976-0421040-7. 98 99 Algorithm has 3 intervals: 100 - 'Normal' region [-0.75, 0.75] 101 - Tail region [0.75, 0.9375] U [-0.9375, -0.75] 102 - Extreme tail [-1, -0.9375] U [0.9375, 1] 103 Normal and tail are both rational approximation of similar order on 104 shifted input - these are typically performed in parallel using gather 105 loads to obtain correct coefficients depending on interval. */ 106 uint32x4_t is_tail = vcageq_f32 (x, v_f32 (0.75)); 107 uint32x4_t extreme_tail = vcageq_f32 (x, v_f32 (0.9375)); 108 109 if (unlikely (!v_any_u32 (is_tail))) 110 /* Shortcut for if all lanes are in [-0.75, 0.75] - can avoid having to 111 gather coefficients. If input is uniform in [-1, 1] then likelihood of 112 this is 0.75^4 ~= 0.31. */ 113 return notails (x, d); 114 115 /* Select requisite shift depending on interval: polynomial is evaluated on 116 x * x - shift. 117 Normal shift = 0.5625 118 Tail shift = 0.87890625. */ 119 float32x4_t t 120 = vfmaq_f32 (vbslq_f32 (is_tail, d->tailshift, v_f32 (-0.5625)), x, x); 121 122 /* Calculate indexes for tbl: tbl is byte-wise, so: 123 [0, 1, 2, 3, 4, 5, 6, ....] copies the vector 124 Add 4 * i to a group of 4 lanes to copy 32-bit lane i. Each vector stores 125 two pairs of coeffs, so we need two idx vectors - one for each pair. */ 126 uint8x16_t off = vandq_u8 (vreinterpretq_u8_u32 (is_tail), vdupq_n_u8 (4)); 127 uint8x16_t idx_lo = vaddq_u8 (d->idxlo, off); 128 uint8x16_t idx_hi = vaddq_u8 (d->idxhi, off); 129 130 /* Load the tables. */ 131 float32x4_t p_lo = d->Plo; 132 float32x4_t pq = d->PQ; 133 float32x4_t qhi = d->Qhi; 134 135 /* Do the lookup (and calculate p3 by masking non-tail lanes). */ 136 float32x4_t p3 = vreinterpretq_f32_u32 ( 137 vandq_u32 (is_tail, vreinterpretq_u32_f32 (d->P29_3))); 138 float32x4_t p0 = lookup (p_lo, idx_lo), p1 = lookup (p_lo, idx_hi), 139 p2 = lookup (pq, idx_lo), q0 = lookup (pq, idx_hi), 140 q1 = lookup (qhi, idx_lo), q2 = lookup (qhi, idx_hi); 141 142 float32x4_t p = vfmaq_f32 (p2, p3, t); 143 p = vfmaq_f32 (p1, p, t); 144 p = vfmaq_f32 (p0, p, t); 145 p = vmulq_f32 (x, p); 146 147 float32x4_t q = vfmaq_f32 (q1, vaddq_f32 (q2, t), t); 148 q = vfmaq_f32 (q0, q, t); 149 150 if (unlikely (v_any_u32 (extreme_tail))) 151 /* At least one lane is in the extreme tail - if input is uniform in 152 [-1, 1] the likelihood of this is ~0.23. */ 153 return vbslq_f32 (extreme_tail, special (x, d), vdivq_f32 (p, q)); 154 155 return vdivq_f32 (p, q); 156 } 157 158 PL_SIG (V, F, 1, erfinv, -0.99, 0.99) 159 PL_TEST_ULP (V_NAME_F1 (erfinv), 4.49) 160 /* Test with control lane in each interval. */ 161 PL_TEST_SYM_INTERVAL_C (V_NAME_F1 (erfinv), 0, 0x1.fffffep-1, 40000, 0.5) 162 PL_TEST_SYM_INTERVAL_C (V_NAME_F1 (erfinv), 0, 0x1.fffffep-1, 40000, 0.8) 163 PL_TEST_SYM_INTERVAL_C (V_NAME_F1 (erfinv), 0, 0x1.fffffep-1, 40000, 0.95) 164