1 /* 2 * Single-precision vector tan(x) function. 3 * 4 * Copyright (c) 2020-2023, Arm Limited. 5 * SPDX-License-Identifier: MIT OR Apache-2.0 WITH LLVM-exception 6 */ 7 8 #include "sv_math.h" 9 #include "pl_sig.h" 10 #include "pl_test.h" 11 12 static const struct data 13 { 14 float pio2_1, pio2_2, pio2_3, invpio2; 15 float c1, c3, c5; 16 float c0, c2, c4, range_val, shift; 17 } data = { 18 /* Coefficients generated using: 19 poly = fpminimax((tan(sqrt(x))-sqrt(x))/x^(3/2), 20 deg, 21 [|single ...|], 22 [a*a;b*b]); 23 optimize relative error 24 final prec : 23 bits 25 deg : 5 26 a : 0x1p-126 ^ 2 27 b : ((pi) / 0x1p2) ^ 2 28 dirty rel error: 0x1.f7c2e4p-25 29 dirty abs error: 0x1.f7c2ecp-25. */ 30 .c0 = 0x1.55555p-2, .c1 = 0x1.11166p-3, 31 .c2 = 0x1.b88a78p-5, .c3 = 0x1.7b5756p-6, 32 .c4 = 0x1.4ef4cep-8, .c5 = 0x1.0e1e74p-7, 33 34 .pio2_1 = 0x1.921fb6p+0f, .pio2_2 = -0x1.777a5cp-25f, 35 .pio2_3 = -0x1.ee59dap-50f, .invpio2 = 0x1.45f306p-1f, 36 .range_val = 0x1p15f, .shift = 0x1.8p+23f 37 }; 38 39 static svfloat32_t NOINLINE 40 special_case (svfloat32_t x, svfloat32_t y, svbool_t cmp) 41 { 42 return sv_call_f32 (tanf, x, y, cmp); 43 } 44 45 /* Fast implementation of SVE tanf. 46 Maximum error is 3.45 ULP: 47 SV_NAME_F1 (tan)(-0x1.e5f0cap+13) got 0x1.ff9856p-1 48 want 0x1.ff9850p-1. */ 49 svfloat32_t SV_NAME_F1 (tan) (svfloat32_t x, const svbool_t pg) 50 { 51 const struct data *d = ptr_barrier (&data); 52 53 /* Determine whether input is too large to perform fast regression. */ 54 svbool_t cmp = svacge (pg, x, d->range_val); 55 56 svfloat32_t odd_coeffs = svld1rq (svptrue_b32 (), &d->c1); 57 svfloat32_t pi_vals = svld1rq (svptrue_b32 (), &d->pio2_1); 58 59 /* n = rint(x/(pi/2)). */ 60 svfloat32_t q = svmla_lane (sv_f32 (d->shift), x, pi_vals, 3); 61 svfloat32_t n = svsub_x (pg, q, d->shift); 62 /* n is already a signed integer, simply convert it. */ 63 svint32_t in = svcvt_s32_x (pg, n); 64 /* Determine if x lives in an interval, where |tan(x)| grows to infinity. */ 65 svint32_t alt = svand_x (pg, in, 1); 66 svbool_t pred_alt = svcmpne (pg, alt, 0); 67 68 /* r = x - n * (pi/2) (range reduction into 0 .. pi/4). */ 69 svfloat32_t r; 70 r = svmls_lane (x, n, pi_vals, 0); 71 r = svmls_lane (r, n, pi_vals, 1); 72 r = svmls_lane (r, n, pi_vals, 2); 73 74 /* If x lives in an interval, where |tan(x)| 75 - is finite, then use a polynomial approximation of the form 76 tan(r) ~ r + r^3 * P(r^2) = r + r * r^2 * P(r^2). 77 - grows to infinity then use symmetries of tangent and the identity 78 tan(r) = cotan(pi/2 - r) to express tan(x) as 1/tan(-r). Finally, use 79 the same polynomial approximation of tan as above. */ 80 81 /* Perform additional reduction if required. */ 82 svfloat32_t z = svneg_m (r, pred_alt, r); 83 84 /* Evaluate polynomial approximation of tangent on [-pi/4, pi/4], 85 using Estrin on z^2. */ 86 svfloat32_t z2 = svmul_x (pg, z, z); 87 svfloat32_t p01 = svmla_lane (sv_f32 (d->c0), z2, odd_coeffs, 0); 88 svfloat32_t p23 = svmla_lane (sv_f32 (d->c2), z2, odd_coeffs, 1); 89 svfloat32_t p45 = svmla_lane (sv_f32 (d->c4), z2, odd_coeffs, 2); 90 91 svfloat32_t z4 = svmul_x (pg, z2, z2); 92 svfloat32_t p = svmla_x (pg, p01, z4, p23); 93 94 svfloat32_t z8 = svmul_x (pg, z4, z4); 95 p = svmla_x (pg, p, z8, p45); 96 97 svfloat32_t y = svmla_x (pg, z, p, svmul_x (pg, z, z2)); 98 99 /* Transform result back, if necessary. */ 100 svfloat32_t inv_y = svdivr_x (pg, y, 1.0f); 101 102 /* No need to pass pg to specialcase here since cmp is a strict subset, 103 guaranteed by the cmpge above. */ 104 if (unlikely (svptest_any (pg, cmp))) 105 return special_case (x, svsel (pred_alt, inv_y, y), cmp); 106 107 return svsel (pred_alt, inv_y, y); 108 } 109 110 PL_SIG (SV, F, 1, tan, -3.1, 3.1) 111 PL_TEST_ULP (SV_NAME_F1 (tan), 2.96) 112 PL_TEST_INTERVAL (SV_NAME_F1 (tan), -0.0, -0x1p126, 100) 113 PL_TEST_INTERVAL (SV_NAME_F1 (tan), 0x1p-149, 0x1p-126, 4000) 114 PL_TEST_INTERVAL (SV_NAME_F1 (tan), 0x1p-126, 0x1p-23, 50000) 115 PL_TEST_INTERVAL (SV_NAME_F1 (tan), 0x1p-23, 0.7, 50000) 116 PL_TEST_INTERVAL (SV_NAME_F1 (tan), 0.7, 1.5, 50000) 117 PL_TEST_INTERVAL (SV_NAME_F1 (tan), 1.5, 100, 50000) 118 PL_TEST_INTERVAL (SV_NAME_F1 (tan), 100, 0x1p17, 50000) 119 PL_TEST_INTERVAL (SV_NAME_F1 (tan), 0x1p17, inf, 50000) 120