1 /* 2 * Double-precision vector atan(x) function. 3 * 4 * Copyright (c) 2021-2023, Arm Limited. 5 * SPDX-License-Identifier: MIT OR Apache-2.0 WITH LLVM-exception 6 */ 7 8 #include "v_math.h" 9 #include "pl_sig.h" 10 #include "pl_test.h" 11 12 #if V_SUPPORTED 13 14 #include "atan_common.h" 15 16 #define PiOver2 v_f64 (0x1.921fb54442d18p+0) 17 #define AbsMask v_u64 (0x7fffffffffffffff) 18 #define TinyBound 0x3e1 /* top12(asuint64(0x1p-30)). */ 19 #define BigBound 0x434 /* top12(asuint64(0x1p53)). */ 20 21 /* Fast implementation of vector atan. 22 Based on atan(x) ~ shift + z + z^3 * P(z^2) with reduction to [0,1] using 23 z=1/x and shift = pi/2. Maximum observed error is 2.27 ulps: 24 __v_atan(0x1.0005af27c23e9p+0) got 0x1.9225645bdd7c1p-1 25 want 0x1.9225645bdd7c3p-1. */ 26 VPCS_ATTR 27 v_f64_t V_NAME (atan) (v_f64_t x) 28 { 29 /* Small cases, infs and nans are supported by our approximation technique, 30 but do not set fenv flags correctly. Only trigger special case if we need 31 fenv. */ 32 v_u64_t ix = v_as_u64_f64 (x); 33 v_u64_t sign = ix & ~AbsMask; 34 35 #if WANT_SIMD_EXCEPT 36 v_u64_t ia12 = (ix >> 52) & 0x7ff; 37 v_u64_t special = v_cond_u64 (ia12 - TinyBound > BigBound - TinyBound); 38 /* If any lane is special, fall back to the scalar routine for all lanes. */ 39 if (unlikely (v_any_u64 (special))) 40 return v_call_f64 (atan, x, v_f64 (0), v_u64 (-1)); 41 #endif 42 43 /* Argument reduction: 44 y := arctan(x) for x < 1 45 y := pi/2 + arctan(-1/x) for x > 1 46 Hence, use z=-1/a if x>=1, otherwise z=a. */ 47 v_u64_t red = v_cagt_f64 (x, v_f64 (1.0)); 48 /* Avoid dependency in abs(x) in division (and comparison). */ 49 v_f64_t z = v_sel_f64 (red, v_div_f64 (v_f64 (-1.0), x), x); 50 v_f64_t shift = v_sel_f64 (red, PiOver2, v_f64 (0.0)); 51 /* Use absolute value only when needed (odd powers of z). */ 52 v_f64_t az = v_abs_f64 (z); 53 az = v_sel_f64 (red, -az, az); 54 55 /* Calculate the polynomial approximation. */ 56 v_f64_t y = eval_poly (z, az, shift); 57 58 /* y = atan(x) if x>0, -atan(-x) otherwise. */ 59 y = v_as_f64_u64 (v_as_u64_f64 (y) ^ sign); 60 return y; 61 } 62 VPCS_ALIAS 63 64 PL_SIG (V, D, 1, atan, -10.0, 10.0) 65 PL_TEST_ULP (V_NAME (atan), 1.78) 66 PL_TEST_EXPECT_FENV (V_NAME (atan), WANT_SIMD_EXCEPT) 67 PL_TEST_INTERVAL (V_NAME (atan), 0, 0x1p-30, 10000) 68 PL_TEST_INTERVAL (V_NAME (atan), -0, -0x1p-30, 1000) 69 PL_TEST_INTERVAL (V_NAME (atan), 0x1p-30, 0x1p53, 900000) 70 PL_TEST_INTERVAL (V_NAME (atan), -0x1p-30, -0x1p53, 90000) 71 PL_TEST_INTERVAL (V_NAME (atan), 0x1p53, inf, 10000) 72 PL_TEST_INTERVAL (V_NAME (atan), -0x1p53, -inf, 1000) 73 74 #endif 75