waves

// ┐ ┌─┐ ┌─┐ ┌─┐ ┌─┐ ┌─┐ ┌─┐ ┌─┐
// └─┘ └─┘ └─┘ └─┘ └─┘ └─┘ └─┘ └
float square_wave(float x)
{
    return 4.0f * floor(x) - 2.0f * floor(2.0f * x) + 1.0f;
}

// ┐  ┌┐  ┌┐  ┌┐  ┌┐  ┌┐  ┌┐  ┌┐
// └──┘└──┘└──┘└──┘└──┘└──┘└──┘└
float pulse_wave(float x, float t)
{
    return 2.0f * static_cast<float>(signbit(x - floor(x) - t)) - 1.0f;
}

// ╱│ ╱│ ╱│ ╱│ ╱│ ╱│ ╱│ ╱│ ╱│ ╱│
//  │╱ │╱ │╱ │╱ │╱ │╱ │╱ │╱ │╱
float sawtooth_wave(float x)
{
    return 2.0f * (x - floor(0.5f + x));
}

// ╲  ╱╲  ╱╲  ╱╲  ╱╲  ╱╲  ╱╲  ╱╲
//  ╲╱  ╲╱  ╲╱  ╲╱  ╲╱  ╲╱  ╲╱
float triangle_wave(float x)
{
    return abs(4.0f * fmod(x, 1.0f) - 2.0f) - 1.0f;
}

float rectified_sin(float x)
{
    return 2.0f * abs(sin(x / 2.0f)) - 1.0f;
}

float cycloid(float x)
{
    // Use slightly curtate cycloid parameters, so there's no non-differentiable
    // cusps.
    const float a = 1.0f;
    const float b = 0.95f;
    // Approximates parameter t given the value x by applying the Newton-Raphson
    // method to the equation f(t) = at - bsin(t) - x.
    float t = x;
    for(int i = 0; i < 5; ++i)
    {
        float ft = (a * t) - (b * sin(t)) - x;
        float dft = a - (b * cos(t));
        t -= ft / dft;
    }
    // Now that t is known, insert it into the parametric equation to get y,
    // then remap y to the range [-1,1].
    float y = a - (b * cos(t));
    return y / a - 1.0f;
}