Simplification of discriminant from scale-rotated-ellipse

Average Accuracy: 36.7% → 89.2%

Time: 1.5min

Precision: binary64

Cost: 1748

Math

\[\frac{\frac{\left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right) \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)}{x-scale}}{y-scale} \cdot \frac{\frac{\left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right) \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)}{x-scale}}{y-scale} - \left(4 \cdot \frac{\frac{{\left(a \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} + {\left(b \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2}}{x-scale}}{x-scale}\right) \cdot \frac{\frac{{\left(a \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} + {\left(b \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2}}{y-scale}}{y-scale} \]

↓

\[\begin{array}{l} t_0 := \frac{b \cdot a}{x-scale \cdot y-scale}\\ t_1 := \frac{a}{x-scale} \cdot \frac{b}{y-scale}\\ t_2 := -4 \cdot \left(t_1 \cdot t_1\right)\\ \mathbf{if}\;b \leq -2.15 \cdot 10^{-182}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;b \leq 3.5 \cdot 10^{-180}:\\ \;\;\;\;-4 \cdot \left(t_0 \cdot t_0\right)\\ \mathbf{elif}\;b \leq 6.8 \cdot 10^{+133}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;b \leq 9.5 \cdot 10^{+213}:\\ \;\;\;\;-4 \cdot \left(\frac{a}{x-scale \cdot y-scale} \cdot \left(b \cdot \left(\frac{a}{y-scale} \cdot \frac{b}{x-scale}\right)\right)\right)\\ \mathbf{elif}\;b \leq 3.9 \cdot 10^{+238}:\\ \;\;\;\;-4 \cdot \left(\left(\frac{b}{y-scale} \cdot \frac{b}{y-scale}\right) \cdot \frac{\frac{a \cdot a}{x-scale}}{x-scale}\right)\\ \mathbf{else}:\\ \;\;\;\;-4 \cdot \left(b \cdot \frac{t_1}{y-scale \cdot \frac{x-scale}{a}}\right)\\ \end{array} \]

FPCore

(FPCore (a b angle x-scale y-scale)
 :precision binary64
 (-
  (*
   (/
    (/
     (*
      (* (* 2.0 (- (pow b 2.0) (pow a 2.0))) (sin (* (/ angle 180.0) PI)))
      (cos (* (/ angle 180.0) PI)))
     x-scale)
    y-scale)
   (/
    (/
     (*
      (* (* 2.0 (- (pow b 2.0) (pow a 2.0))) (sin (* (/ angle 180.0) PI)))
      (cos (* (/ angle 180.0) PI)))
     x-scale)
    y-scale))
  (*
   (*
    4.0
    (/
     (/
      (+
       (pow (* a (sin (* (/ angle 180.0) PI))) 2.0)
       (pow (* b (cos (* (/ angle 180.0) PI))) 2.0))
      x-scale)
     x-scale))
   (/
    (/
     (+
      (pow (* a (cos (* (/ angle 180.0) PI))) 2.0)
      (pow (* b (sin (* (/ angle 180.0) PI))) 2.0))
     y-scale)
    y-scale))))

↓

(FPCore (a b angle x-scale y-scale)
 :precision binary64
 (let* ((t_0 (/ (* b a) (* x-scale y-scale)))
        (t_1 (* (/ a x-scale) (/ b y-scale)))
        (t_2 (* -4.0 (* t_1 t_1))))
   (if (<= b -2.15e-182)
     t_2
     (if (<= b 3.5e-180)
       (* -4.0 (* t_0 t_0))
       (if (<= b 6.8e+133)
         t_2
         (if (<= b 9.5e+213)
           (*
            -4.0
            (*
             (/ a (* x-scale y-scale))
             (* b (* (/ a y-scale) (/ b x-scale)))))
           (if (<= b 3.9e+238)
             (*
              -4.0
              (*
               (* (/ b y-scale) (/ b y-scale))
               (/ (/ (* a a) x-scale) x-scale)))
             (* -4.0 (* b (/ t_1 (* y-scale (/ x-scale a))))))))))))

C

double code(double a, double b, double angle, double x_45_scale, double y_45_scale) {
	return ((((((2.0 * (pow(b, 2.0) - pow(a, 2.0))) * sin(((angle / 180.0) * ((double) M_PI)))) * cos(((angle / 180.0) * ((double) M_PI)))) / x_45_scale) / y_45_scale) * (((((2.0 * (pow(b, 2.0) - pow(a, 2.0))) * sin(((angle / 180.0) * ((double) M_PI)))) * cos(((angle / 180.0) * ((double) M_PI)))) / x_45_scale) / y_45_scale)) - ((4.0 * (((pow((a * sin(((angle / 180.0) * ((double) M_PI)))), 2.0) + pow((b * cos(((angle / 180.0) * ((double) M_PI)))), 2.0)) / x_45_scale) / x_45_scale)) * (((pow((a * cos(((angle / 180.0) * ((double) M_PI)))), 2.0) + pow((b * sin(((angle / 180.0) * ((double) M_PI)))), 2.0)) / y_45_scale) / y_45_scale));
}

↓

double code(double a, double b, double angle, double x_45_scale, double y_45_scale) {
	double t_0 = (b * a) / (x_45_scale * y_45_scale);
	double t_1 = (a / x_45_scale) * (b / y_45_scale);
	double t_2 = -4.0 * (t_1 * t_1);
	double tmp;
	if (b <= -2.15e-182) {
		tmp = t_2;
	} else if (b <= 3.5e-180) {
		tmp = -4.0 * (t_0 * t_0);
	} else if (b <= 6.8e+133) {
		tmp = t_2;
	} else if (b <= 9.5e+213) {
		tmp = -4.0 * ((a / (x_45_scale * y_45_scale)) * (b * ((a / y_45_scale) * (b / x_45_scale))));
	} else if (b <= 3.9e+238) {
		tmp = -4.0 * (((b / y_45_scale) * (b / y_45_scale)) * (((a * a) / x_45_scale) / x_45_scale));
	} else {
		tmp = -4.0 * (b * (t_1 / (y_45_scale * (x_45_scale / a))));
	}
	return tmp;
}

Java

public static double code(double a, double b, double angle, double x_45_scale, double y_45_scale) {
	return ((((((2.0 * (Math.pow(b, 2.0) - Math.pow(a, 2.0))) * Math.sin(((angle / 180.0) * Math.PI))) * Math.cos(((angle / 180.0) * Math.PI))) / x_45_scale) / y_45_scale) * (((((2.0 * (Math.pow(b, 2.0) - Math.pow(a, 2.0))) * Math.sin(((angle / 180.0) * Math.PI))) * Math.cos(((angle / 180.0) * Math.PI))) / x_45_scale) / y_45_scale)) - ((4.0 * (((Math.pow((a * Math.sin(((angle / 180.0) * Math.PI))), 2.0) + Math.pow((b * Math.cos(((angle / 180.0) * Math.PI))), 2.0)) / x_45_scale) / x_45_scale)) * (((Math.pow((a * Math.cos(((angle / 180.0) * Math.PI))), 2.0) + Math.pow((b * Math.sin(((angle / 180.0) * Math.PI))), 2.0)) / y_45_scale) / y_45_scale));
}

↓

public static double code(double a, double b, double angle, double x_45_scale, double y_45_scale) {
	double t_0 = (b * a) / (x_45_scale * y_45_scale);
	double t_1 = (a / x_45_scale) * (b / y_45_scale);
	double t_2 = -4.0 * (t_1 * t_1);
	double tmp;
	if (b <= -2.15e-182) {
		tmp = t_2;
	} else if (b <= 3.5e-180) {
		tmp = -4.0 * (t_0 * t_0);
	} else if (b <= 6.8e+133) {
		tmp = t_2;
	} else if (b <= 9.5e+213) {
		tmp = -4.0 * ((a / (x_45_scale * y_45_scale)) * (b * ((a / y_45_scale) * (b / x_45_scale))));
	} else if (b <= 3.9e+238) {
		tmp = -4.0 * (((b / y_45_scale) * (b / y_45_scale)) * (((a * a) / x_45_scale) / x_45_scale));
	} else {
		tmp = -4.0 * (b * (t_1 / (y_45_scale * (x_45_scale / a))));
	}
	return tmp;
}

Python

def code(a, b, angle, x_45_scale, y_45_scale):
	return ((((((2.0 * (math.pow(b, 2.0) - math.pow(a, 2.0))) * math.sin(((angle / 180.0) * math.pi))) * math.cos(((angle / 180.0) * math.pi))) / x_45_scale) / y_45_scale) * (((((2.0 * (math.pow(b, 2.0) - math.pow(a, 2.0))) * math.sin(((angle / 180.0) * math.pi))) * math.cos(((angle / 180.0) * math.pi))) / x_45_scale) / y_45_scale)) - ((4.0 * (((math.pow((a * math.sin(((angle / 180.0) * math.pi))), 2.0) + math.pow((b * math.cos(((angle / 180.0) * math.pi))), 2.0)) / x_45_scale) / x_45_scale)) * (((math.pow((a * math.cos(((angle / 180.0) * math.pi))), 2.0) + math.pow((b * math.sin(((angle / 180.0) * math.pi))), 2.0)) / y_45_scale) / y_45_scale))

↓

def code(a, b, angle, x_45_scale, y_45_scale):
	t_0 = (b * a) / (x_45_scale * y_45_scale)
	t_1 = (a / x_45_scale) * (b / y_45_scale)
	t_2 = -4.0 * (t_1 * t_1)
	tmp = 0
	if b <= -2.15e-182:
		tmp = t_2
	elif b <= 3.5e-180:
		tmp = -4.0 * (t_0 * t_0)
	elif b <= 6.8e+133:
		tmp = t_2
	elif b <= 9.5e+213:
		tmp = -4.0 * ((a / (x_45_scale * y_45_scale)) * (b * ((a / y_45_scale) * (b / x_45_scale))))
	elif b <= 3.9e+238:
		tmp = -4.0 * (((b / y_45_scale) * (b / y_45_scale)) * (((a * a) / x_45_scale) / x_45_scale))
	else:
		tmp = -4.0 * (b * (t_1 / (y_45_scale * (x_45_scale / a))))
	return tmp

Julia

function code(a, b, angle, x_45_scale, y_45_scale)
	return Float64(Float64(Float64(Float64(Float64(Float64(Float64(2.0 * Float64((b ^ 2.0) - (a ^ 2.0))) * sin(Float64(Float64(angle / 180.0) * pi))) * cos(Float64(Float64(angle / 180.0) * pi))) / x_45_scale) / y_45_scale) * Float64(Float64(Float64(Float64(Float64(2.0 * Float64((b ^ 2.0) - (a ^ 2.0))) * sin(Float64(Float64(angle / 180.0) * pi))) * cos(Float64(Float64(angle / 180.0) * pi))) / x_45_scale) / y_45_scale)) - Float64(Float64(4.0 * Float64(Float64(Float64((Float64(a * sin(Float64(Float64(angle / 180.0) * pi))) ^ 2.0) + (Float64(b * cos(Float64(Float64(angle / 180.0) * pi))) ^ 2.0)) / x_45_scale) / x_45_scale)) * Float64(Float64(Float64((Float64(a * cos(Float64(Float64(angle / 180.0) * pi))) ^ 2.0) + (Float64(b * sin(Float64(Float64(angle / 180.0) * pi))) ^ 2.0)) / y_45_scale) / y_45_scale)))
end

↓

function code(a, b, angle, x_45_scale, y_45_scale)
	t_0 = Float64(Float64(b * a) / Float64(x_45_scale * y_45_scale))
	t_1 = Float64(Float64(a / x_45_scale) * Float64(b / y_45_scale))
	t_2 = Float64(-4.0 * Float64(t_1 * t_1))
	tmp = 0.0
	if (b <= -2.15e-182)
		tmp = t_2;
	elseif (b <= 3.5e-180)
		tmp = Float64(-4.0 * Float64(t_0 * t_0));
	elseif (b <= 6.8e+133)
		tmp = t_2;
	elseif (b <= 9.5e+213)
		tmp = Float64(-4.0 * Float64(Float64(a / Float64(x_45_scale * y_45_scale)) * Float64(b * Float64(Float64(a / y_45_scale) * Float64(b / x_45_scale)))));
	elseif (b <= 3.9e+238)
		tmp = Float64(-4.0 * Float64(Float64(Float64(b / y_45_scale) * Float64(b / y_45_scale)) * Float64(Float64(Float64(a * a) / x_45_scale) / x_45_scale)));
	else
		tmp = Float64(-4.0 * Float64(b * Float64(t_1 / Float64(y_45_scale * Float64(x_45_scale / a)))));
	end
	return tmp
end

MATLAB

function tmp = code(a, b, angle, x_45_scale, y_45_scale)
	tmp = ((((((2.0 * ((b ^ 2.0) - (a ^ 2.0))) * sin(((angle / 180.0) * pi))) * cos(((angle / 180.0) * pi))) / x_45_scale) / y_45_scale) * (((((2.0 * ((b ^ 2.0) - (a ^ 2.0))) * sin(((angle / 180.0) * pi))) * cos(((angle / 180.0) * pi))) / x_45_scale) / y_45_scale)) - ((4.0 * (((((a * sin(((angle / 180.0) * pi))) ^ 2.0) + ((b * cos(((angle / 180.0) * pi))) ^ 2.0)) / x_45_scale) / x_45_scale)) * (((((a * cos(((angle / 180.0) * pi))) ^ 2.0) + ((b * sin(((angle / 180.0) * pi))) ^ 2.0)) / y_45_scale) / y_45_scale));
end

↓

function tmp_2 = code(a, b, angle, x_45_scale, y_45_scale)
	t_0 = (b * a) / (x_45_scale * y_45_scale);
	t_1 = (a / x_45_scale) * (b / y_45_scale);
	t_2 = -4.0 * (t_1 * t_1);
	tmp = 0.0;
	if (b <= -2.15e-182)
		tmp = t_2;
	elseif (b <= 3.5e-180)
		tmp = -4.0 * (t_0 * t_0);
	elseif (b <= 6.8e+133)
		tmp = t_2;
	elseif (b <= 9.5e+213)
		tmp = -4.0 * ((a / (x_45_scale * y_45_scale)) * (b * ((a / y_45_scale) * (b / x_45_scale))));
	elseif (b <= 3.9e+238)
		tmp = -4.0 * (((b / y_45_scale) * (b / y_45_scale)) * (((a * a) / x_45_scale) / x_45_scale));
	else
		tmp = -4.0 * (b * (t_1 / (y_45_scale * (x_45_scale / a))));
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, angle_, x$45$scale_, y$45$scale_] := N[(N[(N[(N[(N[(N[(N[(2.0 * N[(N[Power[b, 2.0], $MachinePrecision] - N[Power[a, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[Sin[N[(N[(angle / 180.0), $MachinePrecision] * Pi), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[Cos[N[(N[(angle / 180.0), $MachinePrecision] * Pi), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / x$45$scale), $MachinePrecision] / y$45$scale), $MachinePrecision] * N[(N[(N[(N[(N[(2.0 * N[(N[Power[b, 2.0], $MachinePrecision] - N[Power[a, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[Sin[N[(N[(angle / 180.0), $MachinePrecision] * Pi), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[Cos[N[(N[(angle / 180.0), $MachinePrecision] * Pi), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / x$45$scale), $MachinePrecision] / y$45$scale), $MachinePrecision]), $MachinePrecision] - N[(N[(4.0 * N[(N[(N[(N[Power[N[(a * N[Sin[N[(N[(angle / 180.0), $MachinePrecision] * Pi), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision] + N[Power[N[(b * N[Cos[N[(N[(angle / 180.0), $MachinePrecision] * Pi), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] / x$45$scale), $MachinePrecision] / x$45$scale), $MachinePrecision]), $MachinePrecision] * N[(N[(N[(N[Power[N[(a * N[Cos[N[(N[(angle / 180.0), $MachinePrecision] * Pi), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision] + N[Power[N[(b * N[Sin[N[(N[(angle / 180.0), $MachinePrecision] * Pi), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] / y$45$scale), $MachinePrecision] / y$45$scale), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

↓

code[a_, b_, angle_, x$45$scale_, y$45$scale_] := Block[{t$95$0 = N[(N[(b * a), $MachinePrecision] / N[(x$45$scale * y$45$scale), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[(a / x$45$scale), $MachinePrecision] * N[(b / y$45$scale), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(-4.0 * N[(t$95$1 * t$95$1), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[b, -2.15e-182], t$95$2, If[LessEqual[b, 3.5e-180], N[(-4.0 * N[(t$95$0 * t$95$0), $MachinePrecision]), $MachinePrecision], If[LessEqual[b, 6.8e+133], t$95$2, If[LessEqual[b, 9.5e+213], N[(-4.0 * N[(N[(a / N[(x$45$scale * y$45$scale), $MachinePrecision]), $MachinePrecision] * N[(b * N[(N[(a / y$45$scale), $MachinePrecision] * N[(b / x$45$scale), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[b, 3.9e+238], N[(-4.0 * N[(N[(N[(b / y$45$scale), $MachinePrecision] * N[(b / y$45$scale), $MachinePrecision]), $MachinePrecision] * N[(N[(N[(a * a), $MachinePrecision] / x$45$scale), $MachinePrecision] / x$45$scale), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(-4.0 * N[(b * N[(t$95$1 / N[(y$45$scale * N[(x$45$scale / a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]]]]]

Derivation

1. Split input into 5 regimes

2. if b < -2.15e-182 or 3.5000000000000001e-180 < b < 6.79999999999999975e133

   1. Initial program 36.2%

      \[\frac{\frac{\left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right) \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)}{x-scale}}{y-scale} \cdot \frac{\frac{\left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right) \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)}{x-scale}}{y-scale} - \left(4 \cdot \frac{\frac{{\left(a \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} + {\left(b \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2}}{x-scale}}{x-scale}\right) \cdot \frac{\frac{{\left(a \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} + {\left(b \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2}}{y-scale}}{y-scale} \]

   2. Simplified 32.1%

      \[\leadsto \color{blue}{\frac{\left(\left(2 \cdot \left(b \cdot b - a \cdot a\right)\right) \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right) \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)}{y-scale \cdot x-scale} \cdot \frac{\left(\left(2 \cdot \left(b \cdot b - a \cdot a\right)\right) \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right) \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)}{y-scale \cdot x-scale} - \left(4 \cdot \frac{{\left(a \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} + {\left(b \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2}}{x-scale \cdot x-scale}\right) \cdot \frac{{\left(a \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} + {\left(b \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2}}{y-scale \cdot y-scale}} \]
      Proof

      [Start] 36.2

      \[ \frac{\frac{\left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right) \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)}{x-scale}}{y-scale} \cdot \frac{\frac{\left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right) \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)}{x-scale}}{y-scale} - \left(4 \cdot \frac{\frac{{\left(a \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} + {\left(b \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2}}{x-scale}}{x-scale}\right) \cdot \frac{\frac{{\left(a \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} + {\left(b \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2}}{y-scale}}{y-scale} \]

   3. Taylor expanded in angle around 0 41.8%

      \[\leadsto \color{blue}{-4 \cdot \frac{{a}^{2} \cdot {b}^{2}}{{y-scale}^{2} \cdot {x-scale}^{2}}} \]

   4. Simplified 55.8%

      \[\leadsto \color{blue}{-4 \cdot \left(\left(\frac{b}{y-scale} \cdot \frac{b}{y-scale}\right) \cdot \frac{\frac{a \cdot a}{x-scale}}{x-scale}\right)} \]
      Proof

      [Start] 41.8	\[ -4 \cdot \frac{{a}^{2} \cdot {b}^{2}}{{y-scale}^{2} \cdot {x-scale}^{2}} \]
      *-commutative [=>] 41.8	\[ -4 \cdot \frac{\color{blue}{{b}^{2} \cdot {a}^{2}}}{{y-scale}^{2} \cdot {x-scale}^{2}} \]
      times-frac [=>] 41.9	\[ -4 \cdot \color{blue}{\left(\frac{{b}^{2}}{{y-scale}^{2}} \cdot \frac{{a}^{2}}{{x-scale}^{2}}\right)} \]
      unpow2 [=>] 41.9	\[ -4 \cdot \left(\frac{\color{blue}{b \cdot b}}{{y-scale}^{2}} \cdot \frac{{a}^{2}}{{x-scale}^{2}}\right) \]
      unpow2 [=>] 41.9	\[ -4 \cdot \left(\frac{b \cdot b}{\color{blue}{y-scale \cdot y-scale}} \cdot \frac{{a}^{2}}{{x-scale}^{2}}\right) \]
      times-frac [=>] 50.1	\[ -4 \cdot \left(\color{blue}{\left(\frac{b}{y-scale} \cdot \frac{b}{y-scale}\right)} \cdot \frac{{a}^{2}}{{x-scale}^{2}}\right) \]
      unpow2 [=>] 50.1	\[ -4 \cdot \left(\left(\frac{b}{y-scale} \cdot \frac{b}{y-scale}\right) \cdot \frac{{a}^{2}}{\color{blue}{x-scale \cdot x-scale}}\right) \]
      associate-/r* [=>] 55.8	\[ -4 \cdot \left(\left(\frac{b}{y-scale} \cdot \frac{b}{y-scale}\right) \cdot \color{blue}{\frac{\frac{{a}^{2}}{x-scale}}{x-scale}}\right) \]
      unpow2 [=>] 55.8	\[ -4 \cdot \left(\left(\frac{b}{y-scale} \cdot \frac{b}{y-scale}\right) \cdot \frac{\frac{\color{blue}{a \cdot a}}{x-scale}}{x-scale}\right) \]

   5. Taylor expanded in b around 0 41.8%

      \[\leadsto -4 \cdot \color{blue}{\frac{{a}^{2} \cdot {b}^{2}}{{x-scale}^{2} \cdot {y-scale}^{2}}} \]

   6. Simplified 91.6%

      \[\leadsto -4 \cdot \color{blue}{{\left(\frac{b}{y-scale} \cdot \frac{a}{x-scale}\right)}^{2}} \]
      Proof

      [Start] 41.8	\[ -4 \cdot \frac{{a}^{2} \cdot {b}^{2}}{{x-scale}^{2} \cdot {y-scale}^{2}} \]
      *-commutative [<=] 41.8	\[ -4 \cdot \frac{{a}^{2} \cdot {b}^{2}}{\color{blue}{{y-scale}^{2} \cdot {x-scale}^{2}}} \]
      *-commutative [=>] 41.8	\[ -4 \cdot \frac{\color{blue}{{b}^{2} \cdot {a}^{2}}}{{y-scale}^{2} \cdot {x-scale}^{2}} \]
      unpow2 [=>] 41.8	\[ -4 \cdot \frac{\color{blue}{\left(b \cdot b\right)} \cdot {a}^{2}}{{y-scale}^{2} \cdot {x-scale}^{2}} \]
      unpow2 [=>] 41.8	\[ -4 \cdot \frac{\left(b \cdot b\right) \cdot \color{blue}{\left(a \cdot a\right)}}{{y-scale}^{2} \cdot {x-scale}^{2}} \]
      unpow2 [=>] 41.8	\[ -4 \cdot \frac{\left(b \cdot b\right) \cdot \left(a \cdot a\right)}{\color{blue}{\left(y-scale \cdot y-scale\right)} \cdot {x-scale}^{2}} \]
      unpow2 [=>] 41.8	\[ -4 \cdot \frac{\left(b \cdot b\right) \cdot \left(a \cdot a\right)}{\left(y-scale \cdot y-scale\right) \cdot \color{blue}{\left(x-scale \cdot x-scale\right)}} \]
      times-frac [=>] 41.9	\[ -4 \cdot \color{blue}{\left(\frac{b \cdot b}{y-scale \cdot y-scale} \cdot \frac{a \cdot a}{x-scale \cdot x-scale}\right)} \]
      associate-/l/ [<=] 46.6	\[ -4 \cdot \left(\frac{b \cdot b}{y-scale \cdot y-scale} \cdot \color{blue}{\frac{\frac{a \cdot a}{x-scale}}{x-scale}}\right) \]
      associate-*r/ [<=] 53.4	\[ -4 \cdot \left(\frac{b \cdot b}{y-scale \cdot y-scale} \cdot \frac{\color{blue}{a \cdot \frac{a}{x-scale}}}{x-scale}\right) \]
      times-frac [=>] 65.5	\[ -4 \cdot \left(\color{blue}{\left(\frac{b}{y-scale} \cdot \frac{b}{y-scale}\right)} \cdot \frac{a \cdot \frac{a}{x-scale}}{x-scale}\right) \]
      associate-*l/ [<=] 68.5	\[ -4 \cdot \left(\left(\frac{b}{y-scale} \cdot \frac{b}{y-scale}\right) \cdot \color{blue}{\left(\frac{a}{x-scale} \cdot \frac{a}{x-scale}\right)}\right) \]
      swap-sqr [<=] 91.6	\[ -4 \cdot \color{blue}{\left(\left(\frac{b}{y-scale} \cdot \frac{a}{x-scale}\right) \cdot \left(\frac{b}{y-scale} \cdot \frac{a}{x-scale}\right)\right)} \]
      unpow2 [<=] 91.6	\[ -4 \cdot \color{blue}{{\left(\frac{b}{y-scale} \cdot \frac{a}{x-scale}\right)}^{2}} \]

   7. Applied egg-rr 91.4%

      \[\leadsto -4 \cdot \color{blue}{\frac{\frac{b}{y-scale} \cdot \frac{a}{x-scale}}{\frac{y-scale}{b} \cdot \frac{x-scale}{a}}} \]
      Proof

      [Start] 91.6	\[ -4 \cdot {\left(\frac{b}{y-scale} \cdot \frac{a}{x-scale}\right)}^{2} \]
      unpow2 [=>] 91.6	\[ -4 \cdot \color{blue}{\left(\left(\frac{b}{y-scale} \cdot \frac{a}{x-scale}\right) \cdot \left(\frac{b}{y-scale} \cdot \frac{a}{x-scale}\right)\right)} \]
      associate-*l/ [=>] 89.5	\[ -4 \cdot \left(\left(\frac{b}{y-scale} \cdot \frac{a}{x-scale}\right) \cdot \color{blue}{\frac{b \cdot \frac{a}{x-scale}}{y-scale}}\right) \]
      associate-*r/ [=>] 85.8	\[ -4 \cdot \color{blue}{\frac{\left(\frac{b}{y-scale} \cdot \frac{a}{x-scale}\right) \cdot \left(b \cdot \frac{a}{x-scale}\right)}{y-scale}} \]
      associate-/l* [=>] 89.5	\[ -4 \cdot \color{blue}{\frac{\frac{b}{y-scale} \cdot \frac{a}{x-scale}}{\frac{y-scale}{b \cdot \frac{a}{x-scale}}}} \]
      associate-*r/ [=>] 85.9	\[ -4 \cdot \frac{\frac{b}{y-scale} \cdot \frac{a}{x-scale}}{\frac{y-scale}{\color{blue}{\frac{b \cdot a}{x-scale}}}} \]
      associate-/l* [=>] 89.4	\[ -4 \cdot \frac{\frac{b}{y-scale} \cdot \frac{a}{x-scale}}{\frac{y-scale}{\color{blue}{\frac{b}{\frac{x-scale}{a}}}}} \]
      associate-/r/ [=>] 91.4	\[ -4 \cdot \frac{\frac{b}{y-scale} \cdot \frac{a}{x-scale}}{\color{blue}{\frac{y-scale}{b} \cdot \frac{x-scale}{a}}} \]

   8. Applied egg-rr 91.6%

      \[\leadsto -4 \cdot \color{blue}{\left(\left(\frac{b}{y-scale} \cdot \frac{a}{x-scale}\right) \cdot \left(\frac{b}{y-scale} \cdot \frac{a}{x-scale}\right)\right)} \]
      Proof

      [Start] 91.4	\[ -4 \cdot \frac{\frac{b}{y-scale} \cdot \frac{a}{x-scale}}{\frac{y-scale}{b} \cdot \frac{x-scale}{a}} \]
      associate-/r* [=>] 87.2	\[ -4 \cdot \color{blue}{\frac{\frac{\frac{b}{y-scale} \cdot \frac{a}{x-scale}}{\frac{y-scale}{b}}}{\frac{x-scale}{a}}} \]
      div-inv [=>] 87.2	\[ -4 \cdot \color{blue}{\left(\frac{\frac{b}{y-scale} \cdot \frac{a}{x-scale}}{\frac{y-scale}{b}} \cdot \frac{1}{\frac{x-scale}{a}}\right)} \]
      div-inv [=>] 87.2	\[ -4 \cdot \left(\color{blue}{\left(\left(\frac{b}{y-scale} \cdot \frac{a}{x-scale}\right) \cdot \frac{1}{\frac{y-scale}{b}}\right)} \cdot \frac{1}{\frac{x-scale}{a}}\right) \]
      clear-num [<=] 87.1	\[ -4 \cdot \left(\left(\left(\frac{b}{y-scale} \cdot \frac{a}{x-scale}\right) \cdot \color{blue}{\frac{b}{y-scale}}\right) \cdot \frac{1}{\frac{x-scale}{a}}\right) \]
      clear-num [<=] 87.3	\[ -4 \cdot \left(\left(\left(\frac{b}{y-scale} \cdot \frac{a}{x-scale}\right) \cdot \frac{b}{y-scale}\right) \cdot \color{blue}{\frac{a}{x-scale}}\right) \]
      associate-*r* [<=] 91.6	\[ -4 \cdot \color{blue}{\left(\left(\frac{b}{y-scale} \cdot \frac{a}{x-scale}\right) \cdot \left(\frac{b}{y-scale} \cdot \frac{a}{x-scale}\right)\right)} \]

   if -2.15e-182 < b < 3.5000000000000001e-180

   1. Initial program 48.6%

      \[\frac{\frac{\left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right) \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)}{x-scale}}{y-scale} \cdot \frac{\frac{\left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right) \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)}{x-scale}}{y-scale} - \left(4 \cdot \frac{\frac{{\left(a \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} + {\left(b \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2}}{x-scale}}{x-scale}\right) \cdot \frac{\frac{{\left(a \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} + {\left(b \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2}}{y-scale}}{y-scale} \]

   2. Simplified 36.9%

      \[\leadsto \color{blue}{\frac{\left(\left(2 \cdot \left(b \cdot b - a \cdot a\right)\right) \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right) \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)}{y-scale \cdot x-scale} \cdot \frac{\left(\left(2 \cdot \left(b \cdot b - a \cdot a\right)\right) \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right) \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)}{y-scale \cdot x-scale} - \left(4 \cdot \frac{{\left(a \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} + {\left(b \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2}}{x-scale \cdot x-scale}\right) \cdot \frac{{\left(a \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} + {\left(b \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2}}{y-scale \cdot y-scale}} \]
      Proof

      [Start] 48.6

      \[ \frac{\frac{\left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right) \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)}{x-scale}}{y-scale} \cdot \frac{\frac{\left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right) \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)}{x-scale}}{y-scale} - \left(4 \cdot \frac{\frac{{\left(a \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} + {\left(b \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2}}{x-scale}}{x-scale}\right) \cdot \frac{\frac{{\left(a \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} + {\left(b \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2}}{y-scale}}{y-scale} \]

   3. Taylor expanded in angle around 0 43.0%

      \[\leadsto \color{blue}{-4 \cdot \frac{{a}^{2} \cdot {b}^{2}}{{y-scale}^{2} \cdot {x-scale}^{2}}} \]

   4. Simplified 43.0%

      \[\leadsto \color{blue}{-4 \cdot \frac{\left(a \cdot a\right) \cdot \left(b \cdot b\right)}{\left(y-scale \cdot y-scale\right) \cdot \left(x-scale \cdot x-scale\right)}} \]
      Proof

      [Start] 43.0	\[ -4 \cdot \frac{{a}^{2} \cdot {b}^{2}}{{y-scale}^{2} \cdot {x-scale}^{2}} \]
      unpow2 [=>] 43.0	\[ -4 \cdot \frac{\color{blue}{\left(a \cdot a\right)} \cdot {b}^{2}}{{y-scale}^{2} \cdot {x-scale}^{2}} \]
      unpow2 [=>] 43.0	\[ -4 \cdot \frac{\left(a \cdot a\right) \cdot \color{blue}{\left(b \cdot b\right)}}{{y-scale}^{2} \cdot {x-scale}^{2}} \]
      unpow2 [=>] 43.0	\[ -4 \cdot \frac{\left(a \cdot a\right) \cdot \left(b \cdot b\right)}{\color{blue}{\left(y-scale \cdot y-scale\right)} \cdot {x-scale}^{2}} \]
      unpow2 [=>] 43.0	\[ -4 \cdot \frac{\left(a \cdot a\right) \cdot \left(b \cdot b\right)}{\left(y-scale \cdot y-scale\right) \cdot \color{blue}{\left(x-scale \cdot x-scale\right)}} \]

   5. Applied egg-rr 91.7%

      \[\leadsto -4 \cdot \color{blue}{\left(\frac{a \cdot b}{y-scale \cdot x-scale} \cdot \frac{a \cdot b}{y-scale \cdot x-scale}\right)} \]
      Proof

      [Start] 43.0	\[ -4 \cdot \frac{\left(a \cdot a\right) \cdot \left(b \cdot b\right)}{\left(y-scale \cdot y-scale\right) \cdot \left(x-scale \cdot x-scale\right)} \]
      unswap-sqr [=>] 58.5	\[ -4 \cdot \frac{\color{blue}{\left(a \cdot b\right) \cdot \left(a \cdot b\right)}}{\left(y-scale \cdot y-scale\right) \cdot \left(x-scale \cdot x-scale\right)} \]
      unswap-sqr [=>] 77.3	\[ -4 \cdot \frac{\left(a \cdot b\right) \cdot \left(a \cdot b\right)}{\color{blue}{\left(y-scale \cdot x-scale\right) \cdot \left(y-scale \cdot x-scale\right)}} \]
      times-frac [=>] 91.7	\[ -4 \cdot \color{blue}{\left(\frac{a \cdot b}{y-scale \cdot x-scale} \cdot \frac{a \cdot b}{y-scale \cdot x-scale}\right)} \]

   if 6.79999999999999975e133 < b < 9.49999999999999993e213

   1. Initial program 5.6%

      \[\frac{\frac{\left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right) \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)}{x-scale}}{y-scale} \cdot \frac{\frac{\left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right) \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)}{x-scale}}{y-scale} - \left(4 \cdot \frac{\frac{{\left(a \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} + {\left(b \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2}}{x-scale}}{x-scale}\right) \cdot \frac{\frac{{\left(a \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} + {\left(b \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2}}{y-scale}}{y-scale} \]

   2. Simplified 6.6%

      \[\leadsto \color{blue}{\frac{\left(\left(2 \cdot \left(b \cdot b - a \cdot a\right)\right) \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right) \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)}{y-scale \cdot x-scale} \cdot \frac{\left(\left(2 \cdot \left(b \cdot b - a \cdot a\right)\right) \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right) \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)}{y-scale \cdot x-scale} - \left(4 \cdot \frac{{\left(a \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} + {\left(b \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2}}{x-scale \cdot x-scale}\right) \cdot \frac{{\left(a \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} + {\left(b \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2}}{y-scale \cdot y-scale}} \]
      Proof

      [Start] 5.6

      \[ \frac{\frac{\left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right) \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)}{x-scale}}{y-scale} \cdot \frac{\frac{\left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right) \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)}{x-scale}}{y-scale} - \left(4 \cdot \frac{\frac{{\left(a \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} + {\left(b \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2}}{x-scale}}{x-scale}\right) \cdot \frac{\frac{{\left(a \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} + {\left(b \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2}}{y-scale}}{y-scale} \]

   3. Taylor expanded in angle around 0 10.3%

      \[\leadsto \color{blue}{-4 \cdot \frac{{a}^{2} \cdot {b}^{2}}{{y-scale}^{2} \cdot {x-scale}^{2}}} \]

   4. Simplified 10.3%

      \[\leadsto \color{blue}{-4 \cdot \frac{\left(a \cdot a\right) \cdot \left(b \cdot b\right)}{\left(y-scale \cdot y-scale\right) \cdot \left(x-scale \cdot x-scale\right)}} \]
      Proof

      [Start] 10.3	\[ -4 \cdot \frac{{a}^{2} \cdot {b}^{2}}{{y-scale}^{2} \cdot {x-scale}^{2}} \]
      unpow2 [=>] 10.3	\[ -4 \cdot \frac{\color{blue}{\left(a \cdot a\right)} \cdot {b}^{2}}{{y-scale}^{2} \cdot {x-scale}^{2}} \]
      unpow2 [=>] 10.3	\[ -4 \cdot \frac{\left(a \cdot a\right) \cdot \color{blue}{\left(b \cdot b\right)}}{{y-scale}^{2} \cdot {x-scale}^{2}} \]
      unpow2 [=>] 10.3	\[ -4 \cdot \frac{\left(a \cdot a\right) \cdot \left(b \cdot b\right)}{\color{blue}{\left(y-scale \cdot y-scale\right)} \cdot {x-scale}^{2}} \]
      unpow2 [=>] 10.3	\[ -4 \cdot \frac{\left(a \cdot a\right) \cdot \left(b \cdot b\right)}{\left(y-scale \cdot y-scale\right) \cdot \color{blue}{\left(x-scale \cdot x-scale\right)}} \]

   5. Applied egg-rr 18.2%

      \[\leadsto -4 \cdot \color{blue}{\left(\frac{a}{y-scale \cdot x-scale} \cdot \frac{a \cdot \left(b \cdot b\right)}{y-scale \cdot x-scale}\right)} \]
      Proof

      [Start] 10.3	\[ -4 \cdot \frac{\left(a \cdot a\right) \cdot \left(b \cdot b\right)}{\left(y-scale \cdot y-scale\right) \cdot \left(x-scale \cdot x-scale\right)} \]
      associate-*l* [=>] 12.7	\[ -4 \cdot \frac{\color{blue}{a \cdot \left(a \cdot \left(b \cdot b\right)\right)}}{\left(y-scale \cdot y-scale\right) \cdot \left(x-scale \cdot x-scale\right)} \]
      unswap-sqr [=>] 17.0	\[ -4 \cdot \frac{a \cdot \left(a \cdot \left(b \cdot b\right)\right)}{\color{blue}{\left(y-scale \cdot x-scale\right) \cdot \left(y-scale \cdot x-scale\right)}} \]
      times-frac [=>] 18.2	\[ -4 \cdot \color{blue}{\left(\frac{a}{y-scale \cdot x-scale} \cdot \frac{a \cdot \left(b \cdot b\right)}{y-scale \cdot x-scale}\right)} \]

   6. Taylor expanded in a around 0 18.2%

      \[\leadsto -4 \cdot \left(\frac{a}{y-scale \cdot x-scale} \cdot \color{blue}{\frac{a \cdot {b}^{2}}{y-scale \cdot x-scale}}\right) \]

   7. Simplified 71.2%

      \[\leadsto -4 \cdot \left(\frac{a}{y-scale \cdot x-scale} \cdot \color{blue}{\left(b \cdot \left(\frac{a}{y-scale} \cdot \frac{b}{x-scale}\right)\right)}\right) \]
      Proof

      [Start] 18.2	\[ -4 \cdot \left(\frac{a}{y-scale \cdot x-scale} \cdot \frac{a \cdot {b}^{2}}{y-scale \cdot x-scale}\right) \]
      *-commutative [=>] 18.2	\[ -4 \cdot \left(\frac{a}{y-scale \cdot x-scale} \cdot \frac{\color{blue}{{b}^{2} \cdot a}}{y-scale \cdot x-scale}\right) \]
      unpow2 [=>] 18.2	\[ -4 \cdot \left(\frac{a}{y-scale \cdot x-scale} \cdot \frac{\color{blue}{\left(b \cdot b\right)} \cdot a}{y-scale \cdot x-scale}\right) \]
      *-commutative [<=] 18.2	\[ -4 \cdot \left(\frac{a}{y-scale \cdot x-scale} \cdot \frac{\color{blue}{a \cdot \left(b \cdot b\right)}}{y-scale \cdot x-scale}\right) \]
      associate-*r* [=>] 60.8	\[ -4 \cdot \left(\frac{a}{y-scale \cdot x-scale} \cdot \frac{\color{blue}{\left(a \cdot b\right) \cdot b}}{y-scale \cdot x-scale}\right) \]
      associate-*l/ [<=] 80.8	\[ -4 \cdot \left(\frac{a}{y-scale \cdot x-scale} \cdot \color{blue}{\left(\frac{a \cdot b}{y-scale \cdot x-scale} \cdot b\right)}\right) \]
      *-commutative [=>] 80.8	\[ -4 \cdot \left(\frac{a}{y-scale \cdot x-scale} \cdot \color{blue}{\left(b \cdot \frac{a \cdot b}{y-scale \cdot x-scale}\right)}\right) \]
      times-frac [=>] 71.2	\[ -4 \cdot \left(\frac{a}{y-scale \cdot x-scale} \cdot \left(b \cdot \color{blue}{\left(\frac{a}{y-scale} \cdot \frac{b}{x-scale}\right)}\right)\right) \]

   if 9.49999999999999993e213 < b < 3.89999999999999993e238

   1. Initial program 0.0%

      \[\frac{\frac{\left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right) \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)}{x-scale}}{y-scale} \cdot \frac{\frac{\left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right) \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)}{x-scale}}{y-scale} - \left(4 \cdot \frac{\frac{{\left(a \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} + {\left(b \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2}}{x-scale}}{x-scale}\right) \cdot \frac{\frac{{\left(a \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} + {\left(b \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2}}{y-scale}}{y-scale} \]

   2. Simplified 0.0%

      \[\leadsto \color{blue}{\frac{\left(\left(2 \cdot \left(b \cdot b - a \cdot a\right)\right) \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right) \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)}{y-scale \cdot x-scale} \cdot \frac{\left(\left(2 \cdot \left(b \cdot b - a \cdot a\right)\right) \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right) \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)}{y-scale \cdot x-scale} - \left(4 \cdot \frac{{\left(a \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} + {\left(b \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2}}{x-scale \cdot x-scale}\right) \cdot \frac{{\left(a \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} + {\left(b \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2}}{y-scale \cdot y-scale}} \]
      Proof

      [Start] 0.0

      \[ \frac{\frac{\left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right) \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)}{x-scale}}{y-scale} \cdot \frac{\frac{\left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right) \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)}{x-scale}}{y-scale} - \left(4 \cdot \frac{\frac{{\left(a \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} + {\left(b \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2}}{x-scale}}{x-scale}\right) \cdot \frac{\frac{{\left(a \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} + {\left(b \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2}}{y-scale}}{y-scale} \]

   3. Taylor expanded in angle around 0 0.0%

      \[\leadsto \color{blue}{-4 \cdot \frac{{a}^{2} \cdot {b}^{2}}{{y-scale}^{2} \cdot {x-scale}^{2}}} \]

   4. Simplified 41.6%

      \[\leadsto \color{blue}{-4 \cdot \left(\left(\frac{b}{y-scale} \cdot \frac{b}{y-scale}\right) \cdot \frac{\frac{a \cdot a}{x-scale}}{x-scale}\right)} \]
      Proof

      [Start] 0.0	\[ -4 \cdot \frac{{a}^{2} \cdot {b}^{2}}{{y-scale}^{2} \cdot {x-scale}^{2}} \]
      *-commutative [=>] 0.0	\[ -4 \cdot \frac{\color{blue}{{b}^{2} \cdot {a}^{2}}}{{y-scale}^{2} \cdot {x-scale}^{2}} \]
      times-frac [=>] 0.0	\[ -4 \cdot \color{blue}{\left(\frac{{b}^{2}}{{y-scale}^{2}} \cdot \frac{{a}^{2}}{{x-scale}^{2}}\right)} \]
      unpow2 [=>] 0.0	\[ -4 \cdot \left(\frac{\color{blue}{b \cdot b}}{{y-scale}^{2}} \cdot \frac{{a}^{2}}{{x-scale}^{2}}\right) \]
      unpow2 [=>] 0.0	\[ -4 \cdot \left(\frac{b \cdot b}{\color{blue}{y-scale \cdot y-scale}} \cdot \frac{{a}^{2}}{{x-scale}^{2}}\right) \]
      times-frac [=>] 37.5	\[ -4 \cdot \left(\color{blue}{\left(\frac{b}{y-scale} \cdot \frac{b}{y-scale}\right)} \cdot \frac{{a}^{2}}{{x-scale}^{2}}\right) \]
      unpow2 [=>] 37.5	\[ -4 \cdot \left(\left(\frac{b}{y-scale} \cdot \frac{b}{y-scale}\right) \cdot \frac{{a}^{2}}{\color{blue}{x-scale \cdot x-scale}}\right) \]
      associate-/r* [=>] 41.6	\[ -4 \cdot \left(\left(\frac{b}{y-scale} \cdot \frac{b}{y-scale}\right) \cdot \color{blue}{\frac{\frac{{a}^{2}}{x-scale}}{x-scale}}\right) \]
      unpow2 [=>] 41.6	\[ -4 \cdot \left(\left(\frac{b}{y-scale} \cdot \frac{b}{y-scale}\right) \cdot \frac{\frac{\color{blue}{a \cdot a}}{x-scale}}{x-scale}\right) \]

   if 3.89999999999999993e238 < b

   1. Initial program 0.0%

      \[\frac{\frac{\left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right) \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)}{x-scale}}{y-scale} \cdot \frac{\frac{\left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right) \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)}{x-scale}}{y-scale} - \left(4 \cdot \frac{\frac{{\left(a \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} + {\left(b \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2}}{x-scale}}{x-scale}\right) \cdot \frac{\frac{{\left(a \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} + {\left(b \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2}}{y-scale}}{y-scale} \]

   2. Simplified 0.0%

      \[\leadsto \color{blue}{\frac{\left(\left(2 \cdot \left(b \cdot b - a \cdot a\right)\right) \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right) \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)}{y-scale \cdot x-scale} \cdot \frac{\left(\left(2 \cdot \left(b \cdot b - a \cdot a\right)\right) \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right) \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)}{y-scale \cdot x-scale} - \left(4 \cdot \frac{{\left(a \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} + {\left(b \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2}}{x-scale \cdot x-scale}\right) \cdot \frac{{\left(a \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} + {\left(b \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2}}{y-scale \cdot y-scale}} \]
      Proof

      [Start] 0.0

      \[ \frac{\frac{\left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right) \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)}{x-scale}}{y-scale} \cdot \frac{\frac{\left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right) \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)}{x-scale}}{y-scale} - \left(4 \cdot \frac{\frac{{\left(a \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} + {\left(b \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2}}{x-scale}}{x-scale}\right) \cdot \frac{\frac{{\left(a \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} + {\left(b \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2}}{y-scale}}{y-scale} \]

   3. Taylor expanded in angle around 0 0.0%

      \[\leadsto \color{blue}{-4 \cdot \frac{{a}^{2} \cdot {b}^{2}}{{y-scale}^{2} \cdot {x-scale}^{2}}} \]

   4. Simplified 34.7%

      \[\leadsto \color{blue}{-4 \cdot \left(\left(\frac{b}{y-scale} \cdot \frac{b}{y-scale}\right) \cdot \frac{\frac{a \cdot a}{x-scale}}{x-scale}\right)} \]
      Proof

      [Start] 0.0	\[ -4 \cdot \frac{{a}^{2} \cdot {b}^{2}}{{y-scale}^{2} \cdot {x-scale}^{2}} \]
      *-commutative [=>] 0.0	\[ -4 \cdot \frac{\color{blue}{{b}^{2} \cdot {a}^{2}}}{{y-scale}^{2} \cdot {x-scale}^{2}} \]
      times-frac [=>] 0.0	\[ -4 \cdot \color{blue}{\left(\frac{{b}^{2}}{{y-scale}^{2}} \cdot \frac{{a}^{2}}{{x-scale}^{2}}\right)} \]
      unpow2 [=>] 0.0	\[ -4 \cdot \left(\frac{\color{blue}{b \cdot b}}{{y-scale}^{2}} \cdot \frac{{a}^{2}}{{x-scale}^{2}}\right) \]
      unpow2 [=>] 0.0	\[ -4 \cdot \left(\frac{b \cdot b}{\color{blue}{y-scale \cdot y-scale}} \cdot \frac{{a}^{2}}{{x-scale}^{2}}\right) \]
      times-frac [=>] 30.0	\[ -4 \cdot \left(\color{blue}{\left(\frac{b}{y-scale} \cdot \frac{b}{y-scale}\right)} \cdot \frac{{a}^{2}}{{x-scale}^{2}}\right) \]
      unpow2 [=>] 30.0	\[ -4 \cdot \left(\left(\frac{b}{y-scale} \cdot \frac{b}{y-scale}\right) \cdot \frac{{a}^{2}}{\color{blue}{x-scale \cdot x-scale}}\right) \]
      associate-/r* [=>] 34.7	\[ -4 \cdot \left(\left(\frac{b}{y-scale} \cdot \frac{b}{y-scale}\right) \cdot \color{blue}{\frac{\frac{{a}^{2}}{x-scale}}{x-scale}}\right) \]
      unpow2 [=>] 34.7	\[ -4 \cdot \left(\left(\frac{b}{y-scale} \cdot \frac{b}{y-scale}\right) \cdot \frac{\frac{\color{blue}{a \cdot a}}{x-scale}}{x-scale}\right) \]

   5. Taylor expanded in b around 0 0.0%

      \[\leadsto -4 \cdot \color{blue}{\frac{{a}^{2} \cdot {b}^{2}}{{x-scale}^{2} \cdot {y-scale}^{2}}} \]

   6. Simplified 83.7%

      \[\leadsto -4 \cdot \color{blue}{{\left(\frac{b}{y-scale} \cdot \frac{a}{x-scale}\right)}^{2}} \]
      Proof

      [Start] 0.0	\[ -4 \cdot \frac{{a}^{2} \cdot {b}^{2}}{{x-scale}^{2} \cdot {y-scale}^{2}} \]
      *-commutative [<=] 0.0	\[ -4 \cdot \frac{{a}^{2} \cdot {b}^{2}}{\color{blue}{{y-scale}^{2} \cdot {x-scale}^{2}}} \]
      *-commutative [=>] 0.0	\[ -4 \cdot \frac{\color{blue}{{b}^{2} \cdot {a}^{2}}}{{y-scale}^{2} \cdot {x-scale}^{2}} \]
      unpow2 [=>] 0.0	\[ -4 \cdot \frac{\color{blue}{\left(b \cdot b\right)} \cdot {a}^{2}}{{y-scale}^{2} \cdot {x-scale}^{2}} \]
      unpow2 [=>] 0.0	\[ -4 \cdot \frac{\left(b \cdot b\right) \cdot \color{blue}{\left(a \cdot a\right)}}{{y-scale}^{2} \cdot {x-scale}^{2}} \]
      unpow2 [=>] 0.0	\[ -4 \cdot \frac{\left(b \cdot b\right) \cdot \left(a \cdot a\right)}{\color{blue}{\left(y-scale \cdot y-scale\right)} \cdot {x-scale}^{2}} \]
      unpow2 [=>] 0.0	\[ -4 \cdot \frac{\left(b \cdot b\right) \cdot \left(a \cdot a\right)}{\left(y-scale \cdot y-scale\right) \cdot \color{blue}{\left(x-scale \cdot x-scale\right)}} \]
      times-frac [=>] 0.0	\[ -4 \cdot \color{blue}{\left(\frac{b \cdot b}{y-scale \cdot y-scale} \cdot \frac{a \cdot a}{x-scale \cdot x-scale}\right)} \]
      associate-/l/ [<=] 0.0	\[ -4 \cdot \left(\frac{b \cdot b}{y-scale \cdot y-scale} \cdot \color{blue}{\frac{\frac{a \cdot a}{x-scale}}{x-scale}}\right) \]
      associate-*r/ [<=] 0.0	\[ -4 \cdot \left(\frac{b \cdot b}{y-scale \cdot y-scale} \cdot \frac{\color{blue}{a \cdot \frac{a}{x-scale}}}{x-scale}\right) \]
      times-frac [=>] 47.4	\[ -4 \cdot \left(\color{blue}{\left(\frac{b}{y-scale} \cdot \frac{b}{y-scale}\right)} \cdot \frac{a \cdot \frac{a}{x-scale}}{x-scale}\right) \]
      associate-*l/ [<=] 51.4	\[ -4 \cdot \left(\left(\frac{b}{y-scale} \cdot \frac{b}{y-scale}\right) \cdot \color{blue}{\left(\frac{a}{x-scale} \cdot \frac{a}{x-scale}\right)}\right) \]
      swap-sqr [<=] 83.7	\[ -4 \cdot \color{blue}{\left(\left(\frac{b}{y-scale} \cdot \frac{a}{x-scale}\right) \cdot \left(\frac{b}{y-scale} \cdot \frac{a}{x-scale}\right)\right)} \]
      unpow2 [<=] 83.7	\[ -4 \cdot \color{blue}{{\left(\frac{b}{y-scale} \cdot \frac{a}{x-scale}\right)}^{2}} \]

   7. Applied egg-rr 83.3%

      \[\leadsto -4 \cdot \color{blue}{\frac{\frac{b}{y-scale} \cdot \frac{a}{x-scale}}{\frac{y-scale}{b} \cdot \frac{x-scale}{a}}} \]
      Proof

      [Start] 83.7	\[ -4 \cdot {\left(\frac{b}{y-scale} \cdot \frac{a}{x-scale}\right)}^{2} \]
      unpow2 [=>] 83.7	\[ -4 \cdot \color{blue}{\left(\left(\frac{b}{y-scale} \cdot \frac{a}{x-scale}\right) \cdot \left(\frac{b}{y-scale} \cdot \frac{a}{x-scale}\right)\right)} \]
      associate-*l/ [=>] 76.1	\[ -4 \cdot \left(\left(\frac{b}{y-scale} \cdot \frac{a}{x-scale}\right) \cdot \color{blue}{\frac{b \cdot \frac{a}{x-scale}}{y-scale}}\right) \]
      associate-*r/ [=>] 66.4	\[ -4 \cdot \color{blue}{\frac{\left(\frac{b}{y-scale} \cdot \frac{a}{x-scale}\right) \cdot \left(b \cdot \frac{a}{x-scale}\right)}{y-scale}} \]
      associate-/l* [=>] 76.1	\[ -4 \cdot \color{blue}{\frac{\frac{b}{y-scale} \cdot \frac{a}{x-scale}}{\frac{y-scale}{b \cdot \frac{a}{x-scale}}}} \]
      associate-*r/ [=>] 57.1	\[ -4 \cdot \frac{\frac{b}{y-scale} \cdot \frac{a}{x-scale}}{\frac{y-scale}{\color{blue}{\frac{b \cdot a}{x-scale}}}} \]
      associate-/l* [=>] 75.6	\[ -4 \cdot \frac{\frac{b}{y-scale} \cdot \frac{a}{x-scale}}{\frac{y-scale}{\color{blue}{\frac{b}{\frac{x-scale}{a}}}}} \]
      associate-/r/ [=>] 83.3	\[ -4 \cdot \frac{\frac{b}{y-scale} \cdot \frac{a}{x-scale}}{\color{blue}{\frac{y-scale}{b} \cdot \frac{x-scale}{a}}} \]

   8. Applied egg-rr 64.0%

      \[\leadsto -4 \cdot \color{blue}{\left(\frac{\frac{b}{y-scale} \cdot \frac{a}{x-scale}}{y-scale \cdot \frac{x-scale}{a}} \cdot b\right)} \]
      Proof

      [Start] 83.3	\[ -4 \cdot \frac{\frac{b}{y-scale} \cdot \frac{a}{x-scale}}{\frac{y-scale}{b} \cdot \frac{x-scale}{a}} \]
      associate-*l/ [=>] 65.7	\[ -4 \cdot \frac{\frac{b}{y-scale} \cdot \frac{a}{x-scale}}{\color{blue}{\frac{y-scale \cdot \frac{x-scale}{a}}{b}}} \]
      associate-/r/ [=>] 64.0	\[ -4 \cdot \color{blue}{\left(\frac{\frac{b}{y-scale} \cdot \frac{a}{x-scale}}{y-scale \cdot \frac{x-scale}{a}} \cdot b\right)} \]

3. Recombined 5 regimes into one program.

4. Final simplification 89.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -2.15 \cdot 10^{-182}:\\ \;\;\;\;-4 \cdot \left(\left(\frac{a}{x-scale} \cdot \frac{b}{y-scale}\right) \cdot \left(\frac{a}{x-scale} \cdot \frac{b}{y-scale}\right)\right)\\ \mathbf{elif}\;b \leq 3.5 \cdot 10^{-180}:\\ \;\;\;\;-4 \cdot \left(\frac{b \cdot a}{x-scale \cdot y-scale} \cdot \frac{b \cdot a}{x-scale \cdot y-scale}\right)\\ \mathbf{elif}\;b \leq 6.8 \cdot 10^{+133}:\\ \;\;\;\;-4 \cdot \left(\left(\frac{a}{x-scale} \cdot \frac{b}{y-scale}\right) \cdot \left(\frac{a}{x-scale} \cdot \frac{b}{y-scale}\right)\right)\\ \mathbf{elif}\;b \leq 9.5 \cdot 10^{+213}:\\ \;\;\;\;-4 \cdot \left(\frac{a}{x-scale \cdot y-scale} \cdot \left(b \cdot \left(\frac{a}{y-scale} \cdot \frac{b}{x-scale}\right)\right)\right)\\ \mathbf{elif}\;b \leq 3.9 \cdot 10^{+238}:\\ \;\;\;\;-4 \cdot \left(\left(\frac{b}{y-scale} \cdot \frac{b}{y-scale}\right) \cdot \frac{\frac{a \cdot a}{x-scale}}{x-scale}\right)\\ \mathbf{else}:\\ \;\;\;\;-4 \cdot \left(b \cdot \frac{\frac{a}{x-scale} \cdot \frac{b}{y-scale}}{y-scale \cdot \frac{x-scale}{a}}\right)\\ \end{array} \]

Alternatives

Alternative 1
Accuracy	90.5%
Cost	1353

\[\begin{array}{l} t_0 := \frac{a}{x-scale} \cdot \frac{b}{y-scale}\\ \mathbf{if}\;b \leq -3.9 \cdot 10^{-184} \lor \neg \left(b \leq 10^{-187}\right):\\ \;\;\;\;-4 \cdot \left(t_0 \cdot t_0\right)\\ \mathbf{else}:\\ \;\;\;\;-4 \cdot \left(\frac{b \cdot a}{x-scale \cdot y-scale} \cdot \left(a \cdot \frac{b}{x-scale \cdot y-scale}\right)\right)\\ \end{array} \]

Alternative 2
Accuracy	91.1%
Cost	1353

\[\begin{array}{l} t_0 := \frac{a}{x-scale} \cdot \frac{b}{y-scale}\\ t_1 := \frac{b \cdot a}{x-scale \cdot y-scale}\\ \mathbf{if}\;b \leq -4 \cdot 10^{-185} \lor \neg \left(b \leq 4 \cdot 10^{-180}\right):\\ \;\;\;\;-4 \cdot \left(t_0 \cdot t_0\right)\\ \mathbf{else}:\\ \;\;\;\;-4 \cdot \left(t_1 \cdot t_1\right)\\ \end{array} \]

Alternative 3
Accuracy	89.9%
Cost	1353

\[\begin{array}{l} t_0 := \frac{b \cdot a}{x-scale \cdot y-scale}\\ \mathbf{if}\;b \leq -4.2 \cdot 10^{-184} \lor \neg \left(b \leq 9.8 \cdot 10^{-200}\right):\\ \;\;\;\;-4 \cdot \frac{b \cdot \frac{\frac{a}{x-scale}}{y-scale}}{\frac{y-scale}{b} \cdot \frac{x-scale}{a}}\\ \mathbf{else}:\\ \;\;\;\;-4 \cdot \left(t_0 \cdot t_0\right)\\ \end{array} \]

Alternative 4
Accuracy	91.3%
Cost	1088

\[\begin{array}{l} t_0 := \frac{a}{x-scale} \cdot \frac{b}{y-scale}\\ -4 \cdot \left(t_0 \cdot t_0\right) \end{array} \]

Alternative 5
Accuracy	53.6%
Cost	64

\[0 \]

Reproduce

herbie shell --seed 2023131 
(FPCore (a b angle x-scale y-scale)
  :name "Simplification of discriminant from scale-rotated-ellipse"
  :precision binary64
  (- (* (/ (/ (* (* (* 2.0 (- (pow b 2.0) (pow a 2.0))) (sin (* (/ angle 180.0) PI))) (cos (* (/ angle 180.0) PI))) x-scale) y-scale) (/ (/ (* (* (* 2.0 (- (pow b 2.0) (pow a 2.0))) (sin (* (/ angle 180.0) PI))) (cos (* (/ angle 180.0) PI))) x-scale) y-scale)) (* (* 4.0 (/ (/ (+ (pow (* a (sin (* (/ angle 180.0) PI))) 2.0) (pow (* b (cos (* (/ angle 180.0) PI))) 2.0)) x-scale) x-scale)) (/ (/ (+ (pow (* a (cos (* (/ angle 180.0) PI))) 2.0) (pow (* b (sin (* (/ angle 180.0) PI))) 2.0)) y-scale) y-scale))))