1 /* 2 * Double-precision SVE log(1+x) function. 3 * 4 * Copyright (c) 2023, Arm Limited. 5 * SPDX-License-Identifier: MIT OR Apache-2.0 WITH LLVM-exception 6 */ 7 8 #include "sv_math.h" 9 #include "poly_sve_f64.h" 10 #include "pl_sig.h" 11 #include "pl_test.h" 12 13 static const struct data 14 { 15 double poly[19]; 16 double ln2_hi, ln2_lo; 17 uint64_t hfrt2_top, onemhfrt2_top, inf, mone; 18 } data = { 19 /* Generated using Remez in [ sqrt(2)/2 - 1, sqrt(2) - 1]. Order 20 20 polynomial, however first 2 coefficients are 0 and 1 so are not stored. */ 21 .poly = { -0x1.ffffffffffffbp-2, 0x1.55555555551a9p-2, -0x1.00000000008e3p-2, 22 0x1.9999999a32797p-3, -0x1.555555552fecfp-3, 0x1.249248e071e5ap-3, 23 -0x1.ffffff8bf8482p-4, 0x1.c71c8f07da57ap-4, -0x1.9999ca4ccb617p-4, 24 0x1.7459ad2e1dfa3p-4, -0x1.554d2680a3ff2p-4, 0x1.3b4c54d487455p-4, 25 -0x1.2548a9ffe80e6p-4, 0x1.0f389a24b2e07p-4, -0x1.eee4db15db335p-5, 26 0x1.e95b494d4a5ddp-5, -0x1.15fdf07cb7c73p-4, 0x1.0310b70800fcfp-4, 27 -0x1.cfa7385bdb37ep-6, }, 28 .ln2_hi = 0x1.62e42fefa3800p-1, 29 .ln2_lo = 0x1.ef35793c76730p-45, 30 /* top32(asuint64(sqrt(2)/2)) << 32. */ 31 .hfrt2_top = 0x3fe6a09e00000000, 32 /* (top32(asuint64(1)) - top32(asuint64(sqrt(2)/2))) << 32. */ 33 .onemhfrt2_top = 0x00095f6200000000, 34 .inf = 0x7ff0000000000000, 35 .mone = 0xbff0000000000000, 36 }; 37 38 #define AbsMask 0x7fffffffffffffff 39 #define BottomMask 0xffffffff 40 41 static svfloat64_t NOINLINE 42 special_case (svbool_t special, svfloat64_t x, svfloat64_t y) 43 { 44 return sv_call_f64 (log1p, x, y, special); 45 } 46 47 /* Vector approximation for log1p using polynomial on reduced interval. Maximum 48 observed error is 2.46 ULP: 49 _ZGVsMxv_log1p(0x1.654a1307242a4p+11) got 0x1.fd5565fb590f4p+2 50 want 0x1.fd5565fb590f6p+2. */ 51 svfloat64_t SV_NAME_D1 (log1p) (svfloat64_t x, svbool_t pg) 52 { 53 const struct data *d = ptr_barrier (&data); 54 svuint64_t ix = svreinterpret_u64 (x); 55 svuint64_t ax = svand_x (pg, ix, AbsMask); 56 svbool_t special 57 = svorr_z (pg, svcmpge (pg, ax, d->inf), svcmpge (pg, ix, d->mone)); 58 59 /* With x + 1 = t * 2^k (where t = f + 1 and k is chosen such that f 60 is in [sqrt(2)/2, sqrt(2)]): 61 log1p(x) = k*log(2) + log1p(f). 62 63 f may not be representable exactly, so we need a correction term: 64 let m = round(1 + x), c = (1 + x) - m. 65 c << m: at very small x, log1p(x) ~ x, hence: 66 log(1+x) - log(m) ~ c/m. 67 68 We therefore calculate log1p(x) by k*log2 + log1p(f) + c/m. */ 69 70 /* Obtain correctly scaled k by manipulation in the exponent. 71 The scalar algorithm casts down to 32-bit at this point to calculate k and 72 u_red. We stay in double-width to obtain f and k, using the same constants 73 as the scalar algorithm but shifted left by 32. */ 74 svfloat64_t m = svadd_x (pg, x, 1); 75 svuint64_t mi = svreinterpret_u64 (m); 76 svuint64_t u = svadd_x (pg, mi, d->onemhfrt2_top); 77 78 svint64_t ki = svsub_x (pg, svreinterpret_s64 (svlsr_x (pg, u, 52)), 0x3ff); 79 svfloat64_t k = svcvt_f64_x (pg, ki); 80 81 /* Reduce x to f in [sqrt(2)/2, sqrt(2)]. */ 82 svuint64_t utop 83 = svadd_x (pg, svand_x (pg, u, 0x000fffff00000000), d->hfrt2_top); 84 svuint64_t u_red = svorr_x (pg, utop, svand_x (pg, mi, BottomMask)); 85 svfloat64_t f = svsub_x (pg, svreinterpret_f64 (u_red), 1); 86 87 /* Correction term c/m. */ 88 svfloat64_t cm = svdiv_x (pg, svsub_x (pg, x, svsub_x (pg, m, 1)), m); 89 90 /* Approximate log1p(x) on the reduced input using a polynomial. Because 91 log1p(0)=0 we choose an approximation of the form: 92 x + C0*x^2 + C1*x^3 + C2x^4 + ... 93 Hence approximation has the form f + f^2 * P(f) 94 where P(x) = C0 + C1*x + C2x^2 + ... 95 Assembling this all correctly is dealt with at the final step. */ 96 svfloat64_t f2 = svmul_x (pg, f, f), f4 = svmul_x (pg, f2, f2), 97 f8 = svmul_x (pg, f4, f4), f16 = svmul_x (pg, f8, f8); 98 svfloat64_t p = sv_estrin_18_f64_x (pg, f, f2, f4, f8, f16, d->poly); 99 100 svfloat64_t ylo = svmla_x (pg, cm, k, d->ln2_lo); 101 svfloat64_t yhi = svmla_x (pg, f, k, d->ln2_hi); 102 svfloat64_t y = svmla_x (pg, svadd_x (pg, ylo, yhi), f2, p); 103 104 if (unlikely (svptest_any (pg, special))) 105 return special_case (special, x, y); 106 107 return y; 108 } 109 110 PL_SIG (SV, D, 1, log1p, -0.9, 10.0) 111 PL_TEST_ULP (SV_NAME_D1 (log1p), 1.97) 112 PL_TEST_SYM_INTERVAL (SV_NAME_D1 (log1p), 0.0, 0x1p-23, 50000) 113 PL_TEST_SYM_INTERVAL (SV_NAME_D1 (log1p), 0x1p-23, 0.001, 50000) 114 PL_TEST_SYM_INTERVAL (SV_NAME_D1 (log1p), 0.001, 1.0, 50000) 115 PL_TEST_INTERVAL (SV_NAME_D1 (log1p), 1, inf, 10000) 116 PL_TEST_INTERVAL (SV_NAME_D1 (log1p), -1, -inf, 10) 117