1*f3087befSAndrew Turner /* 2*f3087befSAndrew Turner * Double-precision atan(x) function. 3*f3087befSAndrew Turner * 4*f3087befSAndrew Turner * Copyright (c) 2022-2024, Arm Limited. 5*f3087befSAndrew Turner * SPDX-License-Identifier: MIT OR Apache-2.0 WITH LLVM-exception 6*f3087befSAndrew Turner */ 7*f3087befSAndrew Turner 8*f3087befSAndrew Turner #include "test_sig.h" 9*f3087befSAndrew Turner #include "test_defs.h" 10*f3087befSAndrew Turner #include "atan_common.h" 11*f3087befSAndrew Turner 12*f3087befSAndrew Turner #define AbsMask 0x7fffffffffffffff 13*f3087befSAndrew Turner #define PiOver2 0x1.921fb54442d18p+0 14*f3087befSAndrew Turner #define TinyBound 0x3e1 /* top12(asuint64(0x1p-30)). */ 15*f3087befSAndrew Turner #define BigBound 0x434 /* top12(asuint64(0x1p53)). */ 16*f3087befSAndrew Turner #define OneTop 0x3ff 17*f3087befSAndrew Turner 18*f3087befSAndrew Turner /* Fast implementation of double-precision atan. 19*f3087befSAndrew Turner Based on atan(x) ~ shift + z + z^3 * P(z^2) with reduction to [0,1] using 20*f3087befSAndrew Turner z=1/x and shift = pi/2. Maximum observed error is 2.27 ulps: 21*f3087befSAndrew Turner atan(0x1.0005af27c23e9p+0) got 0x1.9225645bdd7c1p-1 22*f3087befSAndrew Turner want 0x1.9225645bdd7c3p-1. */ 23*f3087befSAndrew Turner double 24*f3087befSAndrew Turner atan (double x) 25*f3087befSAndrew Turner { 26*f3087befSAndrew Turner uint64_t ix = asuint64 (x); 27*f3087befSAndrew Turner uint64_t sign = ix & ~AbsMask; 28*f3087befSAndrew Turner uint64_t ia = ix & AbsMask; 29*f3087befSAndrew Turner uint32_t ia12 = ia >> 52; 30*f3087befSAndrew Turner 31*f3087befSAndrew Turner if (unlikely (ia12 >= BigBound || ia12 < TinyBound)) 32*f3087befSAndrew Turner { 33*f3087befSAndrew Turner if (ia12 < TinyBound) 34*f3087befSAndrew Turner /* Avoid underflow by returning x. */ 35*f3087befSAndrew Turner return x; 36*f3087befSAndrew Turner if (ia > 0x7ff0000000000000) 37*f3087befSAndrew Turner /* Propagate NaN. */ 38*f3087befSAndrew Turner return __math_invalid (x); 39*f3087befSAndrew Turner /* atan(x) rounds to PiOver2 for large x. */ 40*f3087befSAndrew Turner return asdouble (asuint64 (PiOver2) ^ sign); 41*f3087befSAndrew Turner } 42*f3087befSAndrew Turner 43*f3087befSAndrew Turner double z, az, shift; 44*f3087befSAndrew Turner if (ia12 >= OneTop) 45*f3087befSAndrew Turner { 46*f3087befSAndrew Turner /* For x > 1, use atan(x) = pi / 2 + atan(-1 / x). */ 47*f3087befSAndrew Turner z = -1.0 / x; 48*f3087befSAndrew Turner shift = PiOver2; 49*f3087befSAndrew Turner /* Use absolute value only when needed (odd powers of z). */ 50*f3087befSAndrew Turner az = -fabs (z); 51*f3087befSAndrew Turner } 52*f3087befSAndrew Turner else 53*f3087befSAndrew Turner { 54*f3087befSAndrew Turner /* For x < 1, approximate atan(x) directly. */ 55*f3087befSAndrew Turner z = x; 56*f3087befSAndrew Turner shift = 0; 57*f3087befSAndrew Turner az = asdouble (ia); 58*f3087befSAndrew Turner } 59*f3087befSAndrew Turner 60*f3087befSAndrew Turner /* Calculate polynomial, shift + z + z^3 * P(z^2). */ 61*f3087befSAndrew Turner double y = eval_poly (z, az, shift); 62*f3087befSAndrew Turner /* Copy sign. */ 63*f3087befSAndrew Turner return asdouble (asuint64 (y) ^ sign); 64*f3087befSAndrew Turner } 65*f3087befSAndrew Turner 66*f3087befSAndrew Turner TEST_SIG (S, D, 1, atan, -10.0, 10.0) 67*f3087befSAndrew Turner TEST_ULP (atan, 1.78) 68*f3087befSAndrew Turner TEST_INTERVAL (atan, 0, 0x1p-30, 10000) 69*f3087befSAndrew Turner TEST_INTERVAL (atan, -0, -0x1p-30, 1000) 70*f3087befSAndrew Turner TEST_INTERVAL (atan, 0x1p-30, 0x1p53, 900000) 71*f3087befSAndrew Turner TEST_INTERVAL (atan, -0x1p-30, -0x1p53, 90000) 72*f3087befSAndrew Turner TEST_INTERVAL (atan, 0x1p53, inf, 10000) 73*f3087befSAndrew Turner TEST_INTERVAL (atan, -0x1p53, -inf, 1000) 74