1 /* 2 * Double-precision vector 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 "v_math.h" 9 #include "estrin.h" 10 #include "pl_sig.h" 11 #include "pl_test.h" 12 13 #if V_SUPPORTED 14 15 #define MHalfPiHi v_f64 (__v_tan_data.neg_half_pi_hi) 16 #define MHalfPiLo v_f64 (__v_tan_data.neg_half_pi_lo) 17 #define TwoOverPi v_f64 (0x1.45f306dc9c883p-1) 18 #define Shift v_f64 (0x1.8p52) 19 #define AbsMask 0x7fffffffffffffff 20 #define RangeVal 0x4160000000000000 /* asuint64(2^23). */ 21 #define TinyBound 0x3e50000000000000 /* asuint64(2^-26). */ 22 #define C(i) v_f64 (__v_tan_data.poly[i]) 23 24 /* Special cases (fall back to scalar calls). */ 25 VPCS_ATTR 26 NOINLINE static v_f64_t 27 specialcase (v_f64_t x) 28 { 29 return v_call_f64 (tan, x, x, v_u64 (-1)); 30 } 31 32 /* Vector approximation for double-precision tan. 33 Maximum measured error is 3.48 ULP: 34 __v_tan(0x1.4457047ef78d8p+20) got -0x1.f6ccd8ecf7dedp+37 35 want -0x1.f6ccd8ecf7deap+37. */ 36 VPCS_ATTR 37 v_f64_t V_NAME (tan) (v_f64_t x) 38 { 39 v_u64_t iax = v_as_u64_f64 (x) & AbsMask; 40 41 /* Our argument reduction cannot calculate q with sufficient accuracy for very 42 large inputs. Fall back to scalar routine for all lanes if any are too 43 large, or Inf/NaN. If fenv exceptions are expected, also fall back for tiny 44 input to avoid underflow. Note pl does not supply a scalar double-precision 45 tan, so the fallback will be statically linked from the system libm. */ 46 #if WANT_SIMD_EXCEPT 47 if (unlikely (v_any_u64 (iax - TinyBound > RangeVal - TinyBound))) 48 #else 49 if (unlikely (v_any_u64 (iax > RangeVal))) 50 #endif 51 return specialcase (x); 52 53 /* q = nearest integer to 2 * x / pi. */ 54 v_f64_t q = v_fma_f64 (x, TwoOverPi, Shift) - Shift; 55 v_s64_t qi = v_to_s64_f64 (q); 56 57 /* Use q to reduce x to r in [-pi/4, pi/4], by: 58 r = x - q * pi/2, in extended precision. */ 59 v_f64_t r = x; 60 r = v_fma_f64 (q, MHalfPiHi, r); 61 r = v_fma_f64 (q, MHalfPiLo, r); 62 /* Further reduce r to [-pi/8, pi/8], to be reconstructed using double angle 63 formula. */ 64 r = r * 0.5; 65 66 /* Approximate tan(r) using order 8 polynomial. 67 tan(x) is odd, so polynomial has the form: 68 tan(x) ~= x + C0 * x^3 + C1 * x^5 + C3 * x^7 + ... 69 Hence we first approximate P(r) = C1 + C2 * r^2 + C3 * r^4 + ... 70 Then compute the approximation by: 71 tan(r) ~= r + r^3 * (C0 + r^2 * P(r)). */ 72 v_f64_t r2 = r * r, r4 = r2 * r2, r8 = r4 * r4; 73 /* Use offset version of Estrin wrapper to evaluate from C1 onwards. */ 74 v_f64_t p = ESTRIN_7_ (r2, r4, r8, C, 1); 75 p = v_fma_f64 (p, r2, C (0)); 76 p = v_fma_f64 (r2, p * r, r); 77 78 /* Recombination uses double-angle formula: 79 tan(2x) = 2 * tan(x) / (1 - (tan(x))^2) 80 and reciprocity around pi/2: 81 tan(x) = 1 / (tan(pi/2 - x)) 82 to assemble result using change-of-sign and conditional selection of 83 numerator/denominator, dependent on odd/even-ness of q (hence quadrant). */ 84 v_f64_t n = v_fma_f64 (p, p, v_f64 (-1)); 85 v_f64_t d = p * 2; 86 87 v_u64_t use_recip = v_cond_u64 ((v_as_u64_s64 (qi) & 1) == 0); 88 89 return v_sel_f64 (use_recip, -d, n) / v_sel_f64 (use_recip, n, d); 90 } 91 VPCS_ALIAS 92 93 PL_SIG (V, D, 1, tan, -3.1, 3.1) 94 PL_TEST_ULP (V_NAME (tan), 2.99) 95 PL_TEST_EXPECT_FENV (V_NAME (tan), WANT_SIMD_EXCEPT) 96 PL_TEST_INTERVAL (V_NAME (tan), 0, TinyBound, 5000) 97 PL_TEST_INTERVAL (V_NAME (tan), TinyBound, RangeVal, 100000) 98 PL_TEST_INTERVAL (V_NAME (tan), RangeVal, inf, 5000) 99 PL_TEST_INTERVAL (V_NAME (tan), -0, -TinyBound, 5000) 100 PL_TEST_INTERVAL (V_NAME (tan), -TinyBound, -RangeVal, 100000) 101 PL_TEST_INTERVAL (V_NAME (tan), -RangeVal, -inf, 5000) 102 #endif 103