1*f3087befSAndrew Turner /* 2*f3087befSAndrew Turner * Double-precision scalar atan2(x) function. 3*f3087befSAndrew Turner * 4*f3087befSAndrew Turner * Copyright (c) 2021-2024, 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 <stdbool.h> 9*f3087befSAndrew Turner 10*f3087befSAndrew Turner #include "atan_common.h" 11*f3087befSAndrew Turner #include "math_config.h" 12*f3087befSAndrew Turner #include "test_sig.h" 13*f3087befSAndrew Turner #include "test_defs.h" 14*f3087befSAndrew Turner 15*f3087befSAndrew Turner #define Pi (0x1.921fb54442d18p+1) 16*f3087befSAndrew Turner #define PiOver2 (0x1.921fb54442d18p+0) 17*f3087befSAndrew Turner #define PiOver4 (0x1.921fb54442d18p-1) 18*f3087befSAndrew Turner #define SignMask (0x8000000000000000) 19*f3087befSAndrew Turner #define ExpMask (0x7ff0000000000000) 20*f3087befSAndrew Turner 21*f3087befSAndrew Turner /* We calculate atan2 by P(n/d), where n and d are similar to the input 22*f3087befSAndrew Turner arguments, and P is a polynomial. Evaluating P(x) requires calculating x^8, 23*f3087befSAndrew Turner which may underflow if n and d have very different magnitude. 24*f3087befSAndrew Turner POW8_EXP_UFLOW_BOUND is the lower bound of the difference in exponents of n 25*f3087befSAndrew Turner and d for which P underflows, and is used to special-case such inputs. */ 26*f3087befSAndrew Turner #define POW8_EXP_UFLOW_BOUND 62 27*f3087befSAndrew Turner 28*f3087befSAndrew Turner static inline int64_t 29*f3087befSAndrew Turner biased_exponent (double f) 30*f3087befSAndrew Turner { 31*f3087befSAndrew Turner uint64_t fi = asuint64 (f); 32*f3087befSAndrew Turner return (fi & ExpMask) >> 52; 33*f3087befSAndrew Turner } 34*f3087befSAndrew Turner 35*f3087befSAndrew Turner /* Fast implementation of scalar atan2. Largest errors are when y and x are 36*f3087befSAndrew Turner close together. The greatest observed error is 2.28 ULP: 37*f3087befSAndrew Turner atan2(-0x1.5915b1498e82fp+732, 0x1.54d11ef838826p+732) 38*f3087befSAndrew Turner got -0x1.954f42f1fa841p-1 want -0x1.954f42f1fa843p-1. */ 39*f3087befSAndrew Turner double 40*f3087befSAndrew Turner atan2 (double y, double x) 41*f3087befSAndrew Turner { 42*f3087befSAndrew Turner uint64_t ix = asuint64 (x); 43*f3087befSAndrew Turner uint64_t iy = asuint64 (y); 44*f3087befSAndrew Turner 45*f3087befSAndrew Turner uint64_t sign_x = ix & SignMask; 46*f3087befSAndrew Turner uint64_t sign_y = iy & SignMask; 47*f3087befSAndrew Turner 48*f3087befSAndrew Turner uint64_t iax = ix & ~SignMask; 49*f3087befSAndrew Turner uint64_t iay = iy & ~SignMask; 50*f3087befSAndrew Turner 51*f3087befSAndrew Turner bool xisnan = isnan (x); 52*f3087befSAndrew Turner if (unlikely (isnan (y) && !xisnan)) 53*f3087befSAndrew Turner return __math_invalid (y); 54*f3087befSAndrew Turner if (unlikely (xisnan)) 55*f3087befSAndrew Turner return __math_invalid (x); 56*f3087befSAndrew Turner 57*f3087befSAndrew Turner /* m = 2 * sign(x) + sign(y). */ 58*f3087befSAndrew Turner uint32_t m = ((iy >> 63) & 1) | ((ix >> 62) & 2); 59*f3087befSAndrew Turner 60*f3087befSAndrew Turner int64_t exp_diff = biased_exponent (x) - biased_exponent (y); 61*f3087befSAndrew Turner 62*f3087befSAndrew Turner /* y = 0. */ 63*f3087befSAndrew Turner if (iay == 0) 64*f3087befSAndrew Turner { 65*f3087befSAndrew Turner switch (m) 66*f3087befSAndrew Turner { 67*f3087befSAndrew Turner case 0: 68*f3087befSAndrew Turner case 1: 69*f3087befSAndrew Turner return y; /* atan(+-0,+anything)=+-0. */ 70*f3087befSAndrew Turner case 2: 71*f3087befSAndrew Turner return Pi; /* atan(+0,-anything) = pi. */ 72*f3087befSAndrew Turner case 3: 73*f3087befSAndrew Turner return -Pi; /* atan(-0,-anything) =-pi. */ 74*f3087befSAndrew Turner } 75*f3087befSAndrew Turner } 76*f3087befSAndrew Turner /* Special case for (x, y) either on or very close to the y axis. Either x = 77*f3087befSAndrew Turner 0, or y is much larger than x (difference in exponents >= 78*f3087befSAndrew Turner POW8_EXP_UFLOW_BOUND). */ 79*f3087befSAndrew Turner if (unlikely (iax == 0 || exp_diff <= -POW8_EXP_UFLOW_BOUND)) 80*f3087befSAndrew Turner return sign_y ? -PiOver2 : PiOver2; 81*f3087befSAndrew Turner 82*f3087befSAndrew Turner /* Special case for either x is INF or (x, y) is very close to x axis and x 83*f3087befSAndrew Turner is negative. */ 84*f3087befSAndrew Turner if (unlikely (iax == 0x7ff0000000000000 85*f3087befSAndrew Turner || (exp_diff >= POW8_EXP_UFLOW_BOUND && m >= 2))) 86*f3087befSAndrew Turner { 87*f3087befSAndrew Turner if (iay == 0x7ff0000000000000) 88*f3087befSAndrew Turner { 89*f3087befSAndrew Turner switch (m) 90*f3087befSAndrew Turner { 91*f3087befSAndrew Turner case 0: 92*f3087befSAndrew Turner return PiOver4; /* atan(+INF,+INF). */ 93*f3087befSAndrew Turner case 1: 94*f3087befSAndrew Turner return -PiOver4; /* atan(-INF,+INF). */ 95*f3087befSAndrew Turner case 2: 96*f3087befSAndrew Turner return 3.0 * PiOver4; /* atan(+INF,-INF). */ 97*f3087befSAndrew Turner case 3: 98*f3087befSAndrew Turner return -3.0 * PiOver4; /* atan(-INF,-INF). */ 99*f3087befSAndrew Turner } 100*f3087befSAndrew Turner } 101*f3087befSAndrew Turner else 102*f3087befSAndrew Turner { 103*f3087befSAndrew Turner switch (m) 104*f3087befSAndrew Turner { 105*f3087befSAndrew Turner case 0: 106*f3087befSAndrew Turner return 0.0; /* atan(+...,+INF). */ 107*f3087befSAndrew Turner case 1: 108*f3087befSAndrew Turner return -0.0; /* atan(-...,+INF). */ 109*f3087befSAndrew Turner case 2: 110*f3087befSAndrew Turner return Pi; /* atan(+...,-INF). */ 111*f3087befSAndrew Turner case 3: 112*f3087befSAndrew Turner return -Pi; /* atan(-...,-INF). */ 113*f3087befSAndrew Turner } 114*f3087befSAndrew Turner } 115*f3087befSAndrew Turner } 116*f3087befSAndrew Turner /* y is INF. */ 117*f3087befSAndrew Turner if (iay == 0x7ff0000000000000) 118*f3087befSAndrew Turner return sign_y ? -PiOver2 : PiOver2; 119*f3087befSAndrew Turner 120*f3087befSAndrew Turner uint64_t sign_xy = sign_x ^ sign_y; 121*f3087befSAndrew Turner 122*f3087befSAndrew Turner double ax = asdouble (iax); 123*f3087befSAndrew Turner double ay = asdouble (iay); 124*f3087befSAndrew Turner uint64_t pred_aygtax = (ay > ax); 125*f3087befSAndrew Turner 126*f3087befSAndrew Turner /* Set up z for call to atan. */ 127*f3087befSAndrew Turner double n = pred_aygtax ? -ax : ay; 128*f3087befSAndrew Turner double d = pred_aygtax ? ay : ax; 129*f3087befSAndrew Turner double z = n / d; 130*f3087befSAndrew Turner 131*f3087befSAndrew Turner double ret; 132*f3087befSAndrew Turner if (unlikely (m < 2 && exp_diff >= POW8_EXP_UFLOW_BOUND)) 133*f3087befSAndrew Turner { 134*f3087befSAndrew Turner /* If (x, y) is very close to x axis and x is positive, the polynomial 135*f3087befSAndrew Turner will underflow and evaluate to z. */ 136*f3087befSAndrew Turner ret = z; 137*f3087befSAndrew Turner } 138*f3087befSAndrew Turner else 139*f3087befSAndrew Turner { 140*f3087befSAndrew Turner /* Work out the correct shift. */ 141*f3087befSAndrew Turner double shift = sign_x ? -2.0 : 0.0; 142*f3087befSAndrew Turner shift = pred_aygtax ? shift + 1.0 : shift; 143*f3087befSAndrew Turner shift *= PiOver2; 144*f3087befSAndrew Turner 145*f3087befSAndrew Turner ret = eval_poly (z, z, shift); 146*f3087befSAndrew Turner } 147*f3087befSAndrew Turner 148*f3087befSAndrew Turner /* Account for the sign of x and y. */ 149*f3087befSAndrew Turner return asdouble (asuint64 (ret) ^ sign_xy); 150*f3087befSAndrew Turner } 151*f3087befSAndrew Turner 152*f3087befSAndrew Turner /* Arity of 2 means no mathbench entry emitted. See test/mathbench_funcs.h. */ 153*f3087befSAndrew Turner TEST_SIG (S, D, 2, atan2) 154*f3087befSAndrew Turner TEST_ULP (atan2, 1.78) 155*f3087befSAndrew Turner TEST_INTERVAL (atan2, -10.0, 10.0, 50000) 156*f3087befSAndrew Turner TEST_INTERVAL (atan2, -1.0, 1.0, 40000) 157*f3087befSAndrew Turner TEST_INTERVAL (atan2, 0.0, 1.0, 40000) 158*f3087befSAndrew Turner TEST_INTERVAL (atan2, 1.0, 100.0, 40000) 159*f3087befSAndrew Turner TEST_INTERVAL (atan2, 1e6, 1e32, 40000) 160