1 /* 2 * Double-precision SVE tan(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[9]; 16 double half_pi_hi, half_pi_lo, inv_half_pi, range_val, shift; 17 } data = { 18 /* Polynomial generated with FPMinimax. */ 19 .poly = { 0x1.5555555555556p-2, 0x1.1111111110a63p-3, 0x1.ba1ba1bb46414p-5, 20 0x1.664f47e5b5445p-6, 0x1.226e5e5ecdfa3p-7, 0x1.d6c7ddbf87047p-9, 21 0x1.7ea75d05b583ep-10, 0x1.289f22964a03cp-11, 22 0x1.4e4fd14147622p-12, }, 23 .half_pi_hi = 0x1.921fb54442d18p0, 24 .half_pi_lo = 0x1.1a62633145c07p-54, 25 .inv_half_pi = 0x1.45f306dc9c883p-1, 26 .range_val = 0x1p23, 27 .shift = 0x1.8p52, 28 }; 29 30 static svfloat64_t NOINLINE 31 special_case (svfloat64_t x, svfloat64_t y, svbool_t special) 32 { 33 return sv_call_f64 (tan, x, y, special); 34 } 35 36 /* Vector approximation for double-precision tan. 37 Maximum measured error is 3.48 ULP: 38 _ZGVsMxv_tan(0x1.4457047ef78d8p+20) got -0x1.f6ccd8ecf7dedp+37 39 want -0x1.f6ccd8ecf7deap+37. */ 40 svfloat64_t SV_NAME_D1 (tan) (svfloat64_t x, svbool_t pg) 41 { 42 const struct data *dat = ptr_barrier (&data); 43 44 /* Invert condition to catch NaNs and Infs as well as large values. */ 45 svbool_t special = svnot_z (pg, svaclt (pg, x, dat->range_val)); 46 47 /* q = nearest integer to 2 * x / pi. */ 48 svfloat64_t shift = sv_f64 (dat->shift); 49 svfloat64_t q = svmla_x (pg, shift, x, dat->inv_half_pi); 50 q = svsub_x (pg, q, shift); 51 svint64_t qi = svcvt_s64_x (pg, q); 52 53 /* Use q to reduce x to r in [-pi/4, pi/4], by: 54 r = x - q * pi/2, in extended precision. */ 55 svfloat64_t r = x; 56 svfloat64_t half_pi = svld1rq (svptrue_b64 (), &dat->half_pi_hi); 57 r = svmls_lane (r, q, half_pi, 0); 58 r = svmls_lane (r, q, half_pi, 1); 59 /* Further reduce r to [-pi/8, pi/8], to be reconstructed using double angle 60 formula. */ 61 r = svmul_x (pg, r, 0.5); 62 63 /* Approximate tan(r) using order 8 polynomial. 64 tan(x) is odd, so polynomial has the form: 65 tan(x) ~= x + C0 * x^3 + C1 * x^5 + C3 * x^7 + ... 66 Hence we first approximate P(r) = C1 + C2 * r^2 + C3 * r^4 + ... 67 Then compute the approximation by: 68 tan(r) ~= r + r^3 * (C0 + r^2 * P(r)). */ 69 svfloat64_t r2 = svmul_x (pg, r, r); 70 svfloat64_t r4 = svmul_x (pg, r2, r2); 71 svfloat64_t r8 = svmul_x (pg, r4, r4); 72 /* Use offset version coeff array by 1 to evaluate from C1 onwards. */ 73 svfloat64_t p = sv_estrin_7_f64_x (pg, r2, r4, r8, dat->poly + 1); 74 p = svmad_x (pg, p, r2, dat->poly[0]); 75 p = svmla_x (pg, r, r2, svmul_x (pg, p, r)); 76 77 /* Recombination uses double-angle formula: 78 tan(2x) = 2 * tan(x) / (1 - (tan(x))^2) 79 and reciprocity around pi/2: 80 tan(x) = 1 / (tan(pi/2 - x)) 81 to assemble result using change-of-sign and conditional selection of 82 numerator/denominator dependent on odd/even-ness of q (hence quadrant). */ 83 svbool_t use_recip 84 = svcmpeq (pg, svand_x (pg, svreinterpret_u64 (qi), 1), 0); 85 86 svfloat64_t n = svmad_x (pg, p, p, -1); 87 svfloat64_t d = svmul_x (pg, p, 2); 88 svfloat64_t swap = n; 89 n = svneg_m (n, use_recip, d); 90 d = svsel (use_recip, swap, d); 91 if (unlikely (svptest_any (pg, special))) 92 return special_case (x, svdiv_x (svnot_z (pg, special), n, d), special); 93 return svdiv_x (pg, n, d); 94 } 95 96 PL_SIG (SV, D, 1, tan, -3.1, 3.1) 97 PL_TEST_ULP (SV_NAME_D1 (tan), 2.99) 98 PL_TEST_SYM_INTERVAL (SV_NAME_D1 (tan), 0, 0x1p23, 500000) 99 PL_TEST_SYM_INTERVAL (SV_NAME_D1 (tan), 0x1p23, inf, 5000) 100