Missile targeting algorithm via numeric approximation

target.lua

local sqrt = math.sqrt
local cos, sin = math.cos, math.sin

local targetStep = 0.1 -- Determines targeting accuracy, lower is better (but slower)
local function targetAngle(missile, target)
    local px = missile.pos.x
    local py = missile.pos.y
    local vx = missile.vel.x
    local vy = missile.vel.y
    local am = missile.accel

    local tx = target.pos.x
    local ty = target.pos.y
    local tvx = 0
    local tvy = 0
    if target.vel then
        tvx = target.vel.x
        tvy = target.vel.y
    end

    local closestDist = math.huge
    local closestAngle = math.pi / 2

    for angle = 0, math.pi*2, targetStep do
        local cosAngle = cos(angle)
        local sinAngle = sin(angle)
        local secA = 1 / cosAngle

        local tFront = (tvx - vx) / (cosAngle * am)
        local tHalf = sqrt(
            (secA * (2 * am * (tx - px) + (tvx - vx)*(tvx - vx) * secA))
            /
            (am*am)
        )
        local t1 = tFront - tHalf
        local t2 = tFront + tHalf

        if t1 > 0 then
            local xToBe = px + t1 * vx + (am * t1*t1 * cosAngle)/2
            local yToBe = py + t1 * vy + (am * t1*t1 * sinAngle)/2

            local xWillBe = tx + tvx*t1
            local yWillBe = ty + tvy*t1

            local dist = (xWillBe - xToBe)*(xWillBe - xToBe) + (yWillBe - yToBe)*(yWillBe - yToBe)
            if dist < closestDist then
                closestDist = dist
                closestAngle = angle
            end
        end

        if t2 > 0 then
            local xToBe = px + t2 * vx + (am * t2*t2 * cosAngle)/2
            local yToBe = py + t2 * vy + (am * t2*t2 * sinAngle)/2

            local xWillBe = tx + tvx*t2
            local yWillBe = ty + tvy*t2

            local dist = (xWillBe - xToBe)*(xWillBe - xToBe) + (yWillBe - yToBe)*(yWillBe - yToBe)
            if dist < closestDist then
                closestDist = dist
                closestAngle = angle
            end
        end
    end

    return closestAngle
end