// Special Cases: // // atan2(y, nan) = nan // atan2(nan, x) = nan // atan2(+0, x>=0) = +0 // atan2(-0, x>=0) = -0 // atan2(+0, x<=-0) = +pi // atan2(-0, x<=-0) = -pi // atan2(y>0, 0) = +pi/2 // atan2(y<0, 0) = -pi/2 // atan2(+inf, +inf) = +pi/4 // atan2(-inf, +inf) = -pi/4 // atan2(+inf, -inf) = 3pi/4 // atan2(-inf, -inf) = -3pi/4 // atan2(y, +inf) = 0 // atan2(y>0, -inf) = +pi // atan2(y<0, -inf) = -pi // atan2(+inf, x) = +pi/2 // atan2(-inf, x) = -pi/2 const std = @import("../index.zig"); const math = std.math; const assert = std.debug.assert; fn atan2(comptime T: type, x: T, y: T) T { return switch (T) { f32 => atan2_32(x, y), f64 => atan2_64(x, y), else => @compileError("atan2 not implemented for " ++ @typeName(T)), }; } fn atan2_32(y: f32, x: f32) f32 { const pi: f32 = 3.1415927410e+00; const pi_lo: f32 = -8.7422776573e-08; if (math.isNan(x) or math.isNan(y)) { return x + y; } var ix = @bitCast(u32, x); var iy = @bitCast(u32, y); // x = 1.0 if (ix == 0x3F800000) { return math.atan(y); } // 2 * sign(x) + sign(y) const m = ((iy >> 31) & 1) | ((ix >> 30) & 2); ix &= 0x7FFFFFFF; iy &= 0x7FFFFFFF; if (iy == 0) { switch (m) { 0, 1 => return y, // atan(+-0, +...) 2 => return pi, // atan(+0, -...) 3 => return -pi, // atan(-0, -...) else => unreachable, } } if (ix == 0) { if (m & 1 != 0) { return -pi / 2; } else { return pi / 2; } } if (ix == 0x7F800000) { if (iy == 0x7F800000) { switch (m) { 0 => return pi / 4, // atan(+inf, +inf) 1 => return -pi / 4, // atan(-inf, +inf) 2 => return 3*pi / 4, // atan(+inf, -inf) 3 => return -3*pi / 4, // atan(-inf, -inf) else => unreachable, } } else { switch (m) { 0 => return 0.0, // atan(+..., +inf) 1 => return -0.0, // atan(-..., +inf) 2 => return pi, // atan(+..., -inf) 3 => return -pi, // atan(-...f, -inf) else => unreachable, } } } // |y / x| > 0x1p26 if (ix + (26 << 23) < iy or iy == 0x7F800000) { if (m & 1 != 0) { return -pi / 2; } else { return pi / 2; } } // z = atan(|y / x|) with correct underflow var z = z: { if ((m & 2) != 0 and iy + (26 << 23) < ix) { break :z 0.0; } else { break :z math.atan(math.fabs(y / x)); } }; switch (m) { 0 => return z, // atan(+, +) 1 => return -z, // atan(-, +) 2 => return pi - (z - pi_lo), // atan(+, -) 3 => return (z - pi_lo) - pi, // atan(-, -) else => unreachable, } } fn atan2_64(y: f64, x: f64) f64 { const pi: f64 = 3.1415926535897931160E+00; const pi_lo: f64 = 1.2246467991473531772E-16; if (math.isNan(x) or math.isNan(y)) { return x + y; } var ux = @bitCast(u64, x); var ix = u32(ux >> 32); var lx = u32(ux & 0xFFFFFFFF); var uy = @bitCast(u64, y); var iy = u32(uy >> 32); var ly = u32(uy & 0xFFFFFFFF); // x = 1.0 if ((ix -% 0x3FF00000) | lx == 0) { return math.atan(y); } // 2 * sign(x) + sign(y) const m = ((iy >> 31) & 1) | ((ix >> 30) & 2); ix &= 0x7FFFFFFF; iy &= 0x7FFFFFFF; if (iy | ly == 0) { switch (m) { 0, 1 => return y, // atan(+-0, +...) 2 => return pi, // atan(+0, -...) 3 => return -pi, // atan(-0, -...) else => unreachable, } } if (ix | lx == 0) { if (m & 1 != 0) { return -pi / 2; } else { return pi / 2; } } if (ix == 0x7FF00000) { if (iy == 0x7FF00000) { switch (m) { 0 => return pi / 4, // atan(+inf, +inf) 1 => return -pi / 4, // atan(-inf, +inf) 2 => return 3*pi / 4, // atan(+inf, -inf) 3 => return -3*pi / 4, // atan(-inf, -inf) else => unreachable, } } else { switch (m) { 0 => return 0.0, // atan(+..., +inf) 1 => return -0.0, // atan(-..., +inf) 2 => return pi, // atan(+..., -inf) 3 => return -pi, // atan(-...f, -inf) else => unreachable, } } } // |y / x| > 0x1p64 if (ix +% (64 << 20) < iy or iy == 0x7FF00000) { if (m & 1 != 0) { return -pi / 2; } else { return pi / 2; } } // z = atan(|y / x|) with correct underflow var z = z: { if ((m & 2) != 0 and iy +% (64 << 20) < ix) { break :z 0.0; } else { break :z math.atan(math.fabs(y / x)); } }; switch (m) { 0 => return z, // atan(+, +) 1 => return -z, // atan(-, +) 2 => return pi - (z - pi_lo), // atan(+, -) 3 => return (z - pi_lo) - pi, // atan(-, -) else => unreachable, } } test "math.atan2" { assert(atan2(f32, 0.2, 0.21) == atan2_32(0.2, 0.21)); assert(atan2(f64, 0.2, 0.21) == atan2_64(0.2, 0.21)); } test "math.atan2_32" { const epsilon = 0.000001; assert(math.approxEq(f32, atan2_32(0.0, 0.0), 0.0, epsilon)); assert(math.approxEq(f32, atan2_32(0.2, 0.2), 0.785398, epsilon)); assert(math.approxEq(f32, atan2_32(-0.2, 0.2), -0.785398, epsilon)); assert(math.approxEq(f32, atan2_32(0.2, -0.2), 2.356194, epsilon)); assert(math.approxEq(f32, atan2_32(-0.2, -0.2), -2.356194, epsilon)); assert(math.approxEq(f32, atan2_32(0.34, -0.4), 2.437099, epsilon)); assert(math.approxEq(f32, atan2_32(0.34, 1.243), 0.267001, epsilon)); } test "math.atan2_64" { const epsilon = 0.000001; assert(math.approxEq(f64, atan2_64(0.0, 0.0), 0.0, epsilon)); assert(math.approxEq(f64, atan2_64(0.2, 0.2), 0.785398, epsilon)); assert(math.approxEq(f64, atan2_64(-0.2, 0.2), -0.785398, epsilon)); assert(math.approxEq(f64, atan2_64(0.2, -0.2), 2.356194, epsilon)); assert(math.approxEq(f64, atan2_64(-0.2, -0.2), -2.356194, epsilon)); assert(math.approxEq(f64, atan2_64(0.34, -0.4), 2.437099, epsilon)); assert(math.approxEq(f64, atan2_64(0.34, 1.243), 0.267001, epsilon)); } test "math.atan2_32.special" { const epsilon = 0.000001; assert(math.isNan(atan2_32(1.0, math.nan(f32)))); assert(math.isNan(atan2_32(math.nan(f32), 1.0))); assert(atan2_32(0.0, 5.0) == 0.0); assert(atan2_32(-0.0, 5.0) == -0.0); assert(math.approxEq(f32, atan2_32(0.0, -5.0), math.pi, epsilon)); //assert(math.approxEq(f32, atan2_32(-0.0, -5.0), -math.pi, epsilon)); TODO support negative zero? assert(math.approxEq(f32, atan2_32(1.0, 0.0), math.pi / 2.0, epsilon)); assert(math.approxEq(f32, atan2_32(1.0, -0.0), math.pi / 2.0, epsilon)); assert(math.approxEq(f32, atan2_32(-1.0, 0.0), -math.pi / 2.0, epsilon)); assert(math.approxEq(f32, atan2_32(-1.0, -0.0), -math.pi / 2.0, epsilon)); assert(math.approxEq(f32, atan2_32(math.inf(f32), math.inf(f32)), math.pi / 4.0, epsilon)); assert(math.approxEq(f32, atan2_32(-math.inf(f32), math.inf(f32)), -math.pi / 4.0, epsilon)); assert(math.approxEq(f32, atan2_32(math.inf(f32), -math.inf(f32)), 3.0 * math.pi / 4.0, epsilon)); assert(math.approxEq(f32, atan2_32(-math.inf(f32), -math.inf(f32)), -3.0 * math.pi / 4.0, epsilon)); assert(atan2_32(1.0, math.inf(f32)) == 0.0); assert(math.approxEq(f32, atan2_32(1.0, -math.inf(f32)), math.pi, epsilon)); assert(math.approxEq(f32, atan2_32(-1.0, -math.inf(f32)), -math.pi, epsilon)); assert(math.approxEq(f32, atan2_32(math.inf(f32), 1.0), math.pi / 2.0, epsilon)); assert(math.approxEq(f32, atan2_32(-math.inf(f32), 1.0), -math.pi / 2.0, epsilon)); } test "math.atan2_64.special" { const epsilon = 0.000001; assert(math.isNan(atan2_64(1.0, math.nan(f64)))); assert(math.isNan(atan2_64(math.nan(f64), 1.0))); assert(atan2_64(0.0, 5.0) == 0.0); assert(atan2_64(-0.0, 5.0) == -0.0); assert(math.approxEq(f64, atan2_64(0.0, -5.0), math.pi, epsilon)); //assert(math.approxEq(f64, atan2_64(-0.0, -5.0), -math.pi, epsilon)); TODO support negative zero? assert(math.approxEq(f64, atan2_64(1.0, 0.0), math.pi / 2.0, epsilon)); assert(math.approxEq(f64, atan2_64(1.0, -0.0), math.pi / 2.0, epsilon)); assert(math.approxEq(f64, atan2_64(-1.0, 0.0), -math.pi / 2.0, epsilon)); assert(math.approxEq(f64, atan2_64(-1.0, -0.0), -math.pi / 2.0, epsilon)); assert(math.approxEq(f64, atan2_64(math.inf(f64), math.inf(f64)), math.pi / 4.0, epsilon)); assert(math.approxEq(f64, atan2_64(-math.inf(f64), math.inf(f64)), -math.pi / 4.0, epsilon)); assert(math.approxEq(f64, atan2_64(math.inf(f64), -math.inf(f64)), 3.0 * math.pi / 4.0, epsilon)); assert(math.approxEq(f64, atan2_64(-math.inf(f64), -math.inf(f64)), -3.0 * math.pi / 4.0, epsilon)); assert(atan2_64(1.0, math.inf(f64)) == 0.0); assert(math.approxEq(f64, atan2_64(1.0, -math.inf(f64)), math.pi, epsilon)); assert(math.approxEq(f64, atan2_64(-1.0, -math.inf(f64)), -math.pi, epsilon)); assert(math.approxEq(f64, atan2_64(math.inf(f64), 1.0), math.pi / 2.0, epsilon)); assert(math.approxEq(f64, atan2_64(-math.inf(f64), 1.0), -math.pi / 2.0, epsilon)); }