sin

import std.math;
// f(x) = sin(x), f'(x) = cos(x)
auto x = dual(0.0, 1.0);
auto result = sin(x);
assert(approxEqual(result.re, 0.0)); // sin(0) = 0
assert(approxEqual(result.du, 1.0)); // cos(0) = 1

x = dual(PI / 2.0, 1.0);
result = sin(x);
assert(approxEqual(result.re, 1.0)); // sin(π/2) = 1
assert(approxEqual(result.du, 0.0)); // cos(π/2) = 0

// f(x) = sin(3x), f'(x) = 3cos(3x)
x = dual(5.0, 1.0);
result = sin(3 * x);
assert(approxEqual(result.re, std.math.sin(15.0)));
assert(approxEqual(result.du, 3.0 * std.math.cos(15.0)));