' Cubic Bézier interpolation for easing curves
' pA ... pD are Bézier curve points
' pX the value for which the corresponding curve Y value shall be found
' pPrecision defines up to which error in Y the result shall be
' Returns: curve Y value at pX
Function CubicBezierEase:Float(pA:Float, pB:Float, pC:Float, pD:Float, pX:Float, pPrecision:Float = 0.0001)
	' boundary conditions
	If pX <= 0 Then Return 0
	If pX >= 1 Then Return 1
	' polynomial factors, precalculated for performance
	Local pQuadr:Float = 3*pC - 6*pA
	Local pCubic:Float = 1 + 3*pA - 3*pC
	' A cubic bezier spline is a vector function
	' (x, y) = f(t)
	' The algorithm finds t for a given x
	' iteratively
	' and returns the corresponding y value.
	' start at t = pX
	Local t:Float = pX
	' old value of t, step of t, how much t actually changed and next best guess for t
	Local oldt:Float, tstep:Float, dt:Float, dttarget:Float
	' value of x, old value of x and how much x actually changed in last iteration
	Local x:Float, oldx:Float, dx:Float
	' first iteration is special (in respect to calculation of tstep)
	Local firstrun:Int = True
	' iterate
	Repeat
		' calculate x of cubic bezier spline
		x = 3*pA*t + pQuadr * t*t + pCubic * t*t*t
		' accurately found t?
		If Abs(x - pX) <= pPrecision Then
			Exit
		' no, move t
		Else
			' update tstep
			' in first iteration
			If firstrun Then
				firstrun = False
				' tstep by how much x differs from target x
				tstep = pX - x
			' after that
			Else
				' adapt tstep by how much the value over- or undershoot
				dx = x - oldx
				dt = t - oldt
				dttarget = (pX - oldx) / dx * dt
				tstep = -dt + dttarget
			EndIf
			' update t and clamp
			oldt = t
			t += tstep
			If t > 1 Then t=1
			If t < 0 Then t=0
		EndIf
		oldx = x
	Forever
	' calculate y of cubic bezier spline and return
	Return 3*pB*t + (3*pD - 6*pB) * t*t + (1 + 3*pB - 3*pD) * t*t*t
End Function

' Cubic Bézier interpolation for easing curves
' (Imagine a curve on the X-Y plane. Returns curve value at pX)
' pY0 and pY1 are curve Y values at X=0 and X=1 respectively
' pA ... pD are Bézier curve points
' pX the value for which the corresponding curve Y value shall be found
' pPrecision defines up to which error in Y the result shall be
' Returns: curve Y value at pX
Function CubicBezierEase:Float(pY0:Float, pY1:Float, pA:Float, pB:Float, pC:Float, pD:Float, pX:Float, pPrecision:Float = 0.0001)
	' boundary conditions
	If pX <= 0 Then Return pY0
	If pX >= 1 Then Return pY1
	' polynomial factors, precalculated for performance
	Local pLin:Float = 3*pA
	Local pQuadr:Float = 3*pC - 6*pA
	Local pCubic:Float = 1 + 3*pA - 3*pC
	' A cubic bezier spline is a vector function
	' (x, y) = f(t)
	' The algorithm finds t for a given x
	' iteratively
	' and returns the corresponding y value.
	' start at t = pX
	Local t:Float = pX
	' old value of t, step of t, how much t actually changed and next best guess for t
	Local oldt:Float, tstep:Float, dt:Float, dttarget:Float
	' value of x, old value of x and how much x actually changed in last iteration
	Local x:Float, oldx:Float, dx:Float
	' first iteration is special (in respect to calculation of tstep)
	Local firstrun:Int = True
	' iterate
	Repeat
		' calculate x of cubic bezier spline
		x = pLin * t + pQuadr * t*t + pCubic * t*t*t
		' accurately found t?
		If Abs(x - pX) <= pPrecision Then
			Exit
		' no, move t
		Else
			' update tstep
			' in first iteration
			If firstrun Then
				firstrun = False
				' tstep by how much x differs from target x
				tstep = pX - x
			' after that
			Else
				' adapt tstep by how much the value over- or undershoot
				dx = x - oldx
				dt = t - oldt
				dttarget = (pX - oldx) / dx * dt
				tstep = -dt + dttarget
			EndIf
			' update t and clamp
			oldt = t
			t += tstep
			If t > 1 Then t=1
			If t < 0 Then t=0
		EndIf
		oldx = x
	Forever
	' calculate y of cubic bezier spline and return
	Return pY0 + (3*pB - 3*pY0) * t + (3*pY0 - 6*pB + 3*pD) * t*t + (pY1 + 3*pB - 3*pD - pY0) * t*t*t
End