1 /* 2 * Double-precision vector log(1+x) function. 3 * 4 * Copyright (c) 2022-2023, Arm Limited. 5 * SPDX-License-Identifier: MIT OR Apache-2.0 WITH LLVM-exception 6 */ 7 8 #include "v_math.h" 9 #include "poly_advsimd_f64.h" 10 #include "pl_sig.h" 11 #include "pl_test.h" 12 13 const static struct data 14 { 15 float64x2_t poly[19], ln2[2]; 16 uint64x2_t hf_rt2_top, one_m_hf_rt2_top, umask, inf, minus_one; 17 int64x2_t one_top; 18 } data = { 19 /* Generated using Remez, deg=20, in [sqrt(2)/2-1, sqrt(2)-1]. */ 20 .poly = { V2 (-0x1.ffffffffffffbp-2), V2 (0x1.55555555551a9p-2), 21 V2 (-0x1.00000000008e3p-2), V2 (0x1.9999999a32797p-3), 22 V2 (-0x1.555555552fecfp-3), V2 (0x1.249248e071e5ap-3), 23 V2 (-0x1.ffffff8bf8482p-4), V2 (0x1.c71c8f07da57ap-4), 24 V2 (-0x1.9999ca4ccb617p-4), V2 (0x1.7459ad2e1dfa3p-4), 25 V2 (-0x1.554d2680a3ff2p-4), V2 (0x1.3b4c54d487455p-4), 26 V2 (-0x1.2548a9ffe80e6p-4), V2 (0x1.0f389a24b2e07p-4), 27 V2 (-0x1.eee4db15db335p-5), V2 (0x1.e95b494d4a5ddp-5), 28 V2 (-0x1.15fdf07cb7c73p-4), V2 (0x1.0310b70800fcfp-4), 29 V2 (-0x1.cfa7385bdb37ep-6) }, 30 .ln2 = { V2 (0x1.62e42fefa3800p-1), V2 (0x1.ef35793c76730p-45) }, 31 /* top32(asuint64(sqrt(2)/2)) << 32. */ 32 .hf_rt2_top = V2 (0x3fe6a09e00000000), 33 /* (top32(asuint64(1)) - top32(asuint64(sqrt(2)/2))) << 32. */ 34 .one_m_hf_rt2_top = V2 (0x00095f6200000000), 35 .umask = V2 (0x000fffff00000000), 36 .one_top = V2 (0x3ff), 37 .inf = V2 (0x7ff0000000000000), 38 .minus_one = V2 (0xbff0000000000000) 39 }; 40 41 #define BottomMask v_u64 (0xffffffff) 42 43 static float64x2_t VPCS_ATTR NOINLINE 44 special_case (float64x2_t x, float64x2_t y, uint64x2_t special) 45 { 46 return v_call_f64 (log1p, x, y, special); 47 } 48 49 /* Vector log1p approximation using polynomial on reduced interval. Routine is 50 a modification of the algorithm used in scalar log1p, with no shortcut for 51 k=0 and no narrowing for f and k. Maximum observed error is 2.45 ULP: 52 _ZGVnN2v_log1p(0x1.658f7035c4014p+11) got 0x1.fd61d0727429dp+2 53 want 0x1.fd61d0727429fp+2 . */ 54 VPCS_ATTR float64x2_t V_NAME_D1 (log1p) (float64x2_t x) 55 { 56 const struct data *d = ptr_barrier (&data); 57 uint64x2_t ix = vreinterpretq_u64_f64 (x); 58 uint64x2_t ia = vreinterpretq_u64_f64 (vabsq_f64 (x)); 59 uint64x2_t special = vcgeq_u64 (ia, d->inf); 60 61 #if WANT_SIMD_EXCEPT 62 special = vorrq_u64 (special, 63 vcgeq_u64 (ix, vreinterpretq_u64_f64 (v_f64 (-1)))); 64 if (unlikely (v_any_u64 (special))) 65 x = v_zerofy_f64 (x, special); 66 #else 67 special = vorrq_u64 (special, vcleq_f64 (x, v_f64 (-1))); 68 #endif 69 70 /* With x + 1 = t * 2^k (where t = f + 1 and k is chosen such that f 71 is in [sqrt(2)/2, sqrt(2)]): 72 log1p(x) = k*log(2) + log1p(f). 73 74 f may not be representable exactly, so we need a correction term: 75 let m = round(1 + x), c = (1 + x) - m. 76 c << m: at very small x, log1p(x) ~ x, hence: 77 log(1+x) - log(m) ~ c/m. 78 79 We therefore calculate log1p(x) by k*log2 + log1p(f) + c/m. */ 80 81 /* Obtain correctly scaled k by manipulation in the exponent. 82 The scalar algorithm casts down to 32-bit at this point to calculate k and 83 u_red. We stay in double-width to obtain f and k, using the same constants 84 as the scalar algorithm but shifted left by 32. */ 85 float64x2_t m = vaddq_f64 (x, v_f64 (1)); 86 uint64x2_t mi = vreinterpretq_u64_f64 (m); 87 uint64x2_t u = vaddq_u64 (mi, d->one_m_hf_rt2_top); 88 89 int64x2_t ki 90 = vsubq_s64 (vreinterpretq_s64_u64 (vshrq_n_u64 (u, 52)), d->one_top); 91 float64x2_t k = vcvtq_f64_s64 (ki); 92 93 /* Reduce x to f in [sqrt(2)/2, sqrt(2)]. */ 94 uint64x2_t utop = vaddq_u64 (vandq_u64 (u, d->umask), d->hf_rt2_top); 95 uint64x2_t u_red = vorrq_u64 (utop, vandq_u64 (mi, BottomMask)); 96 float64x2_t f = vsubq_f64 (vreinterpretq_f64_u64 (u_red), v_f64 (1)); 97 98 /* Correction term c/m. */ 99 float64x2_t cm = vdivq_f64 (vsubq_f64 (x, vsubq_f64 (m, v_f64 (1))), m); 100 101 /* Approximate log1p(x) on the reduced input using a polynomial. Because 102 log1p(0)=0 we choose an approximation of the form: 103 x + C0*x^2 + C1*x^3 + C2x^4 + ... 104 Hence approximation has the form f + f^2 * P(f) 105 where P(x) = C0 + C1*x + C2x^2 + ... 106 Assembling this all correctly is dealt with at the final step. */ 107 float64x2_t f2 = vmulq_f64 (f, f); 108 float64x2_t p = v_pw_horner_18_f64 (f, f2, d->poly); 109 110 float64x2_t ylo = vfmaq_f64 (cm, k, d->ln2[1]); 111 float64x2_t yhi = vfmaq_f64 (f, k, d->ln2[0]); 112 float64x2_t y = vaddq_f64 (ylo, yhi); 113 114 if (unlikely (v_any_u64 (special))) 115 return special_case (vreinterpretq_f64_u64 (ix), vfmaq_f64 (y, f2, p), 116 special); 117 118 return vfmaq_f64 (y, f2, p); 119 } 120 121 PL_SIG (V, D, 1, log1p, -0.9, 10.0) 122 PL_TEST_ULP (V_NAME_D1 (log1p), 1.97) 123 PL_TEST_EXPECT_FENV (V_NAME_D1 (log1p), WANT_SIMD_EXCEPT) 124 PL_TEST_SYM_INTERVAL (V_NAME_D1 (log1p), 0.0, 0x1p-23, 50000) 125 PL_TEST_SYM_INTERVAL (V_NAME_D1 (log1p), 0x1p-23, 0.001, 50000) 126 PL_TEST_SYM_INTERVAL (V_NAME_D1 (log1p), 0.001, 1.0, 50000) 127 PL_TEST_INTERVAL (V_NAME_D1 (log1p), 1, inf, 40000) 128 PL_TEST_INTERVAL (V_NAME_D1 (log1p), -1.0, -inf, 500) 129