Equation API, By Symmetryc & Wobbo

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-- API that holds miscellaneous functions for equations
-- @author Symmetryc, Wobbo
-- @module api
-- @alias M

local M = {}

--- Create a derivative for a function
-- This function takes two arguments, a function and a delta. The delta can be left unspecified. The function returned is an approximation for the derivative of the given function, and delta specifies the accuracy, where a smaller delta results in a higher accuracy.
-- @tparam func _f The function to derive from
-- @tparam number delta The accuracy of the derivation, defaults to 1e-6
-- @treturn func The derivative of _f
M.derive = function(_f, _delta)
    local delta = _delta or 1e-6
    return function(x)
        return (_f(x + delta) - _f(x)) / delta
    end
end

--- Creates an equation object
-- @param f The function that the equation will use to create values
-- @return The equation
M.eq = function(_f)
    return setmetatable({f = _f}, {
        __index = function(self, _key)
            self[_key] = self.f(_key)
            return self[_key]
        end;
    })
end

--- Finds the solution to an equation using Newton-Raphson method
-- @param api The api itself (call using `:` syntactic sugar)
-- @param eq The equation that you are solving
-- @param init The initial guess
-- @param loop How many loops the approximation should go through
-- @return The solution
M.nr = function(api, _eq, _init, _loop)
    local ans = _init or 0.1
    local fprime = api.derive(_eq)
    for i = 1, _loop or 5 do
        ans = ans - _eq(ans) / fprime(ans)
    end
    return ans
end

--- Funciton for implicit equations
-- @param api The api itself (call using `:` syntactic sugar)
-- @param str The string of the equation
-- @return The equation solved for x
-- @return The equation solved for y
M.impl = function(api, _str)
    local f = function(x, y)
        return setfenv(loadstring("return ".._str:gsub("(.-)=(.+)", "(%1)-(%2)")), setmetatable({x = x, y = y}, {__index = math}))()
    end
    local fx = function(x)
        return api:nr(function(y) return f(x, y) end, x)
    end
    local fy = function(y)
        return api:nr(function(x) return f(x, y) end, y)
    end
    return api.eq(fx), api.eq(fy)
end

return M