Simplification of discriminant from scale-rotated-ellipse

Average Accuracy: 35.4% → 87.8%

Time: 1.5min

Precision: binary64

Cost: 7568

Math

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

↓

\[\begin{array}{l} t_0 := \frac{\frac{-4 \cdot \frac{b}{x-scale}}{\frac{y-scale}{a}}}{\frac{y-scale \cdot x-scale}{a \cdot b}}\\ t_1 := -4 \cdot {\left(\frac{b}{y-scale} \cdot \frac{a}{x-scale}\right)}^{2}\\ \mathbf{if}\;angle \leq -9.2 \cdot 10^{+91}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;angle \leq -1.92 \cdot 10^{-90}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;angle \leq 7.2 \cdot 10^{-271}:\\ \;\;\;\;-4 \cdot \left(\left(a \cdot \frac{\frac{b}{y-scale}}{x-scale}\right) \cdot \frac{a}{x-scale \cdot \frac{y-scale}{b}}\right)\\ \mathbf{elif}\;angle \leq 2.1 \cdot 10^{-27}:\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]

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
         (/
          (/ (* -4.0 (/ b x-scale)) (/ y-scale a))
          (/ (* y-scale x-scale) (* a b))))
        (t_1 (* -4.0 (pow (* (/ b y-scale) (/ a x-scale)) 2.0))))
   (if (<= angle -9.2e+91)
     t_1
     (if (<= angle -1.92e-90)
       t_0
       (if (<= angle 7.2e-271)
         (*
          -4.0
          (* (* a (/ (/ b y-scale) x-scale)) (/ a (* x-scale (/ y-scale b)))))
         (if (<= angle 2.1e-27) t_0 t_1))))))

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 = ((-4.0 * (b / x_45_scale)) / (y_45_scale / a)) / ((y_45_scale * x_45_scale) / (a * b));
	double t_1 = -4.0 * pow(((b / y_45_scale) * (a / x_45_scale)), 2.0);
	double tmp;
	if (angle <= -9.2e+91) {
		tmp = t_1;
	} else if (angle <= -1.92e-90) {
		tmp = t_0;
	} else if (angle <= 7.2e-271) {
		tmp = -4.0 * ((a * ((b / y_45_scale) / x_45_scale)) * (a / (x_45_scale * (y_45_scale / b))));
	} else if (angle <= 2.1e-27) {
		tmp = t_0;
	} else {
		tmp = t_1;
	}
	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 = ((-4.0 * (b / x_45_scale)) / (y_45_scale / a)) / ((y_45_scale * x_45_scale) / (a * b));
	double t_1 = -4.0 * Math.pow(((b / y_45_scale) * (a / x_45_scale)), 2.0);
	double tmp;
	if (angle <= -9.2e+91) {
		tmp = t_1;
	} else if (angle <= -1.92e-90) {
		tmp = t_0;
	} else if (angle <= 7.2e-271) {
		tmp = -4.0 * ((a * ((b / y_45_scale) / x_45_scale)) * (a / (x_45_scale * (y_45_scale / b))));
	} else if (angle <= 2.1e-27) {
		tmp = t_0;
	} else {
		tmp = t_1;
	}
	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 = ((-4.0 * (b / x_45_scale)) / (y_45_scale / a)) / ((y_45_scale * x_45_scale) / (a * b))
	t_1 = -4.0 * math.pow(((b / y_45_scale) * (a / x_45_scale)), 2.0)
	tmp = 0
	if angle <= -9.2e+91:
		tmp = t_1
	elif angle <= -1.92e-90:
		tmp = t_0
	elif angle <= 7.2e-271:
		tmp = -4.0 * ((a * ((b / y_45_scale) / x_45_scale)) * (a / (x_45_scale * (y_45_scale / b))))
	elif angle <= 2.1e-27:
		tmp = t_0
	else:
		tmp = t_1
	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(Float64(-4.0 * Float64(b / x_45_scale)) / Float64(y_45_scale / a)) / Float64(Float64(y_45_scale * x_45_scale) / Float64(a * b)))
	t_1 = Float64(-4.0 * (Float64(Float64(b / y_45_scale) * Float64(a / x_45_scale)) ^ 2.0))
	tmp = 0.0
	if (angle <= -9.2e+91)
		tmp = t_1;
	elseif (angle <= -1.92e-90)
		tmp = t_0;
	elseif (angle <= 7.2e-271)
		tmp = Float64(-4.0 * Float64(Float64(a * Float64(Float64(b / y_45_scale) / x_45_scale)) * Float64(a / Float64(x_45_scale * Float64(y_45_scale / b)))));
	elseif (angle <= 2.1e-27)
		tmp = t_0;
	else
		tmp = t_1;
	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 = ((-4.0 * (b / x_45_scale)) / (y_45_scale / a)) / ((y_45_scale * x_45_scale) / (a * b));
	t_1 = -4.0 * (((b / y_45_scale) * (a / x_45_scale)) ^ 2.0);
	tmp = 0.0;
	if (angle <= -9.2e+91)
		tmp = t_1;
	elseif (angle <= -1.92e-90)
		tmp = t_0;
	elseif (angle <= 7.2e-271)
		tmp = -4.0 * ((a * ((b / y_45_scale) / x_45_scale)) * (a / (x_45_scale * (y_45_scale / b))));
	elseif (angle <= 2.1e-27)
		tmp = t_0;
	else
		tmp = t_1;
	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[(N[(-4.0 * N[(b / x$45$scale), $MachinePrecision]), $MachinePrecision] / N[(y$45$scale / a), $MachinePrecision]), $MachinePrecision] / N[(N[(y$45$scale * x$45$scale), $MachinePrecision] / N[(a * b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(-4.0 * N[Power[N[(N[(b / y$45$scale), $MachinePrecision] * N[(a / x$45$scale), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[angle, -9.2e+91], t$95$1, If[LessEqual[angle, -1.92e-90], t$95$0, If[LessEqual[angle, 7.2e-271], N[(-4.0 * N[(N[(a * N[(N[(b / y$45$scale), $MachinePrecision] / x$45$scale), $MachinePrecision]), $MachinePrecision] * N[(a / N[(x$45$scale * N[(y$45$scale / b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[angle, 2.1e-27], t$95$0, t$95$1]]]]]]

Derivation

1. Split input into 3 regimes

2. if angle < -9.19999999999999965e91 or 2.10000000000000015e-27 < angle

   1. Initial program 29.7%

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

   2. Simplified 25.3%

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

      [Start] 29.7

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

   3. Taylor expanded in angle around 0 37.9%

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

   4. Simplified 56.2%

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

      [Start] 37.9	\[ -4 \cdot \frac{{a}^{2} \cdot {b}^{2}}{{y-scale}^{2} \cdot {x-scale}^{2}} \]
      *-commutative [=>] 37.9	\[ -4 \cdot \frac{\color{blue}{{b}^{2} \cdot {a}^{2}}}{{y-scale}^{2} \cdot {x-scale}^{2}} \]
      times-frac [=>] 37.2	\[ -4 \cdot \color{blue}{\left(\frac{{b}^{2}}{{y-scale}^{2}} \cdot \frac{{a}^{2}}{{x-scale}^{2}}\right)} \]
      unpow2 [=>] 37.2	\[ -4 \cdot \left(\frac{\color{blue}{b \cdot b}}{{y-scale}^{2}} \cdot \frac{{a}^{2}}{{x-scale}^{2}}\right) \]
      unpow2 [=>] 37.2	\[ -4 \cdot \left(\frac{b \cdot b}{\color{blue}{y-scale \cdot y-scale}} \cdot \frac{{a}^{2}}{{x-scale}^{2}}\right) \]
      times-frac [=>] 49.8	\[ -4 \cdot \left(\color{blue}{\left(\frac{b}{y-scale} \cdot \frac{b}{y-scale}\right)} \cdot \frac{{a}^{2}}{{x-scale}^{2}}\right) \]
      unpow2 [=>] 49.8	\[ -4 \cdot \left(\left(\frac{b}{y-scale} \cdot \frac{b}{y-scale}\right) \cdot \frac{{a}^{2}}{\color{blue}{x-scale \cdot x-scale}}\right) \]
      associate-/r* [=>] 56.2	\[ -4 \cdot \left(\left(\frac{b}{y-scale} \cdot \frac{b}{y-scale}\right) \cdot \color{blue}{\frac{\frac{{a}^{2}}{x-scale}}{x-scale}}\right) \]
      unpow2 [=>] 56.2	\[ -4 \cdot \left(\left(\frac{b}{y-scale} \cdot \frac{b}{y-scale}\right) \cdot \frac{\frac{\color{blue}{a \cdot a}}{x-scale}}{x-scale}\right) \]

   5. Taylor expanded in b around 0 37.9%

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

   6. Simplified 90.6%

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

      [Start] 37.9	\[ -4 \cdot \frac{{a}^{2} \cdot {b}^{2}}{{x-scale}^{2} \cdot {y-scale}^{2}} \]
      *-commutative [<=] 37.9	\[ -4 \cdot \frac{{a}^{2} \cdot {b}^{2}}{\color{blue}{{y-scale}^{2} \cdot {x-scale}^{2}}} \]
      *-commutative [=>] 37.9	\[ -4 \cdot \frac{\color{blue}{{b}^{2} \cdot {a}^{2}}}{{y-scale}^{2} \cdot {x-scale}^{2}} \]
      unpow2 [=>] 37.9	\[ -4 \cdot \frac{\color{blue}{\left(b \cdot b\right)} \cdot {a}^{2}}{{y-scale}^{2} \cdot {x-scale}^{2}} \]
      unpow2 [=>] 37.9	\[ -4 \cdot \frac{\left(b \cdot b\right) \cdot \color{blue}{\left(a \cdot a\right)}}{{y-scale}^{2} \cdot {x-scale}^{2}} \]
      unpow2 [=>] 37.9	\[ -4 \cdot \frac{\left(b \cdot b\right) \cdot \left(a \cdot a\right)}{\color{blue}{\left(y-scale \cdot y-scale\right)} \cdot {x-scale}^{2}} \]
      unpow2 [=>] 37.9	\[ -4 \cdot \frac{\left(b \cdot b\right) \cdot \left(a \cdot a\right)}{\left(y-scale \cdot y-scale\right) \cdot \color{blue}{\left(x-scale \cdot x-scale\right)}} \]
      times-frac [=>] 37.2	\[ -4 \cdot \color{blue}{\left(\frac{b \cdot b}{y-scale \cdot y-scale} \cdot \frac{a \cdot a}{x-scale \cdot x-scale}\right)} \]
      associate-/l/ [<=] 41.7	\[ -4 \cdot \left(\frac{b \cdot b}{y-scale \cdot y-scale} \cdot \color{blue}{\frac{\frac{a \cdot a}{x-scale}}{x-scale}}\right) \]
      associate-*r/ [<=] 47.5	\[ -4 \cdot \left(\frac{b \cdot b}{y-scale \cdot y-scale} \cdot \frac{\color{blue}{a \cdot \frac{a}{x-scale}}}{x-scale}\right) \]
      times-frac [=>] 65.4	\[ -4 \cdot \left(\color{blue}{\left(\frac{b}{y-scale} \cdot \frac{b}{y-scale}\right)} \cdot \frac{a \cdot \frac{a}{x-scale}}{x-scale}\right) \]
      associate-*l/ [<=] 68.9	\[ -4 \cdot \left(\left(\frac{b}{y-scale} \cdot \frac{b}{y-scale}\right) \cdot \color{blue}{\left(\frac{a}{x-scale} \cdot \frac{a}{x-scale}\right)}\right) \]
      swap-sqr [<=] 90.6	\[ -4 \cdot \color{blue}{\left(\left(\frac{b}{y-scale} \cdot \frac{a}{x-scale}\right) \cdot \left(\frac{b}{y-scale} \cdot \frac{a}{x-scale}\right)\right)} \]
      unpow2 [<=] 90.6	\[ -4 \cdot \color{blue}{{\left(\frac{b}{y-scale} \cdot \frac{a}{x-scale}\right)}^{2}} \]

   if -9.19999999999999965e91 < angle < -1.92000000000000009e-90 or 7.1999999999999996e-271 < angle < 2.10000000000000015e-27

   1. Initial program 35.9%

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

   2. Taylor expanded in angle around 0 36.1%

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

   3. Simplified 69.2%

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

      [Start] 36.1	\[ -4 \cdot \frac{{a}^{2} \cdot {b}^{2}}{{y-scale}^{2} \cdot {x-scale}^{2}} \]
      *-commutative [=>] 36.1	\[ \color{blue}{\frac{{a}^{2} \cdot {b}^{2}}{{y-scale}^{2} \cdot {x-scale}^{2}} \cdot -4} \]
      times-frac [=>] 36.5	\[ \color{blue}{\left(\frac{{a}^{2}}{{y-scale}^{2}} \cdot \frac{{b}^{2}}{{x-scale}^{2}}\right)} \cdot -4 \]
      associate-*l* [=>] 36.5	\[ \color{blue}{\frac{{a}^{2}}{{y-scale}^{2}} \cdot \left(\frac{{b}^{2}}{{x-scale}^{2}} \cdot -4\right)} \]
      unpow2 [=>] 36.5	\[ \frac{\color{blue}{a \cdot a}}{{y-scale}^{2}} \cdot \left(\frac{{b}^{2}}{{x-scale}^{2}} \cdot -4\right) \]
      unpow2 [=>] 36.5	\[ \frac{a \cdot a}{\color{blue}{y-scale \cdot y-scale}} \cdot \left(\frac{{b}^{2}}{{x-scale}^{2}} \cdot -4\right) \]
      times-frac [=>] 49.9	\[ \color{blue}{\left(\frac{a}{y-scale} \cdot \frac{a}{y-scale}\right)} \cdot \left(\frac{{b}^{2}}{{x-scale}^{2}} \cdot -4\right) \]
      unpow2 [=>] 49.9	\[ \left(\frac{a}{y-scale} \cdot \frac{a}{y-scale}\right) \cdot \left(\frac{\color{blue}{b \cdot b}}{{x-scale}^{2}} \cdot -4\right) \]
      unpow2 [=>] 49.9	\[ \left(\frac{a}{y-scale} \cdot \frac{a}{y-scale}\right) \cdot \left(\frac{b \cdot b}{\color{blue}{x-scale \cdot x-scale}} \cdot -4\right) \]
      times-frac [=>] 69.2	\[ \left(\frac{a}{y-scale} \cdot \frac{a}{y-scale}\right) \cdot \left(\color{blue}{\left(\frac{b}{x-scale} \cdot \frac{b}{x-scale}\right)} \cdot -4\right) \]

   4. Taylor expanded in a around 0 36.1%

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

   5. Simplified 90.6%

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

      [Start] 36.1	\[ -4 \cdot \frac{{a}^{2} \cdot {b}^{2}}{{x-scale}^{2} \cdot {y-scale}^{2}} \]
      *-commutative [=>] 36.1	\[ -4 \cdot \frac{{a}^{2} \cdot {b}^{2}}{\color{blue}{{y-scale}^{2} \cdot {x-scale}^{2}}} \]
      times-frac [=>] 36.5	\[ -4 \cdot \color{blue}{\left(\frac{{a}^{2}}{{y-scale}^{2}} \cdot \frac{{b}^{2}}{{x-scale}^{2}}\right)} \]
      unpow2 [=>] 36.5	\[ -4 \cdot \left(\frac{\color{blue}{a \cdot a}}{{y-scale}^{2}} \cdot \frac{{b}^{2}}{{x-scale}^{2}}\right) \]
      unpow2 [=>] 36.5	\[ -4 \cdot \left(\frac{a \cdot a}{\color{blue}{y-scale \cdot y-scale}} \cdot \frac{{b}^{2}}{{x-scale}^{2}}\right) \]
      times-frac [=>] 49.9	\[ -4 \cdot \left(\color{blue}{\left(\frac{a}{y-scale} \cdot \frac{a}{y-scale}\right)} \cdot \frac{{b}^{2}}{{x-scale}^{2}}\right) \]
      unpow2 [=>] 49.9	\[ -4 \cdot \left(\left(\frac{a}{y-scale} \cdot \frac{a}{y-scale}\right) \cdot \frac{\color{blue}{b \cdot b}}{{x-scale}^{2}}\right) \]
      unpow2 [=>] 49.9	\[ -4 \cdot \left(\left(\frac{a}{y-scale} \cdot \frac{a}{y-scale}\right) \cdot \frac{b \cdot b}{\color{blue}{x-scale \cdot x-scale}}\right) \]
      times-frac [=>] 69.2	\[ -4 \cdot \left(\left(\frac{a}{y-scale} \cdot \frac{a}{y-scale}\right) \cdot \color{blue}{\left(\frac{b}{x-scale} \cdot \frac{b}{x-scale}\right)}\right) \]
      swap-sqr [<=] 90.4	\[ -4 \cdot \color{blue}{\left(\left(\frac{a}{y-scale} \cdot \frac{b}{x-scale}\right) \cdot \left(\frac{a}{y-scale} \cdot \frac{b}{x-scale}\right)\right)} \]
      unpow2 [<=] 90.4	\[ -4 \cdot \color{blue}{{\left(\frac{a}{y-scale} \cdot \frac{b}{x-scale}\right)}^{2}} \]
      associate-*r/ [=>] 90.1	\[ -4 \cdot {\color{blue}{\left(\frac{\frac{a}{y-scale} \cdot b}{x-scale}\right)}}^{2} \]
      associate-/l* [=>] 90.6	\[ -4 \cdot {\color{blue}{\left(\frac{\frac{a}{y-scale}}{\frac{x-scale}{b}}\right)}}^{2} \]

   6. Applied egg-rr 86.5%

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

      [Start] 90.6	\[ -4 \cdot {\left(\frac{\frac{a}{y-scale}}{\frac{x-scale}{b}}\right)}^{2} \]
      unpow2 [=>] 90.6	\[ -4 \cdot \color{blue}{\left(\frac{\frac{a}{y-scale}}{\frac{x-scale}{b}} \cdot \frac{\frac{a}{y-scale}}{\frac{x-scale}{b}}\right)} \]
      clear-num [=>] 90.6	\[ -4 \cdot \left(\frac{\frac{a}{y-scale}}{\frac{x-scale}{b}} \cdot \color{blue}{\frac{1}{\frac{\frac{x-scale}{b}}{\frac{a}{y-scale}}}}\right) \]
      un-div-inv [=>] 90.6	\[ -4 \cdot \color{blue}{\frac{\frac{\frac{a}{y-scale}}{\frac{x-scale}{b}}}{\frac{\frac{x-scale}{b}}{\frac{a}{y-scale}}}} \]
      div-inv [=>] 90.4	\[ -4 \cdot \frac{\color{blue}{\frac{a}{y-scale} \cdot \frac{1}{\frac{x-scale}{b}}}}{\frac{\frac{x-scale}{b}}{\frac{a}{y-scale}}} \]
      clear-num [=>] 90.2	\[ -4 \cdot \frac{\color{blue}{\frac{1}{\frac{y-scale}{a}}} \cdot \frac{1}{\frac{x-scale}{b}}}{\frac{\frac{x-scale}{b}}{\frac{a}{y-scale}}} \]
      associate-*l/ [=>] 90.2	\[ -4 \cdot \frac{\color{blue}{\frac{1 \cdot \frac{1}{\frac{x-scale}{b}}}{\frac{y-scale}{a}}}}{\frac{\frac{x-scale}{b}}{\frac{a}{y-scale}}} \]
      *-un-lft-identity [<=] 90.2	\[ -4 \cdot \frac{\frac{\color{blue}{\frac{1}{\frac{x-scale}{b}}}}{\frac{y-scale}{a}}}{\frac{\frac{x-scale}{b}}{\frac{a}{y-scale}}} \]
      associate-/l/ [=>] 86.4	\[ -4 \cdot \color{blue}{\frac{\frac{1}{\frac{x-scale}{b}}}{\frac{\frac{x-scale}{b}}{\frac{a}{y-scale}} \cdot \frac{y-scale}{a}}} \]
      clear-num [<=] 86.5	\[ -4 \cdot \frac{\color{blue}{\frac{b}{x-scale}}}{\frac{\frac{x-scale}{b}}{\frac{a}{y-scale}} \cdot \frac{y-scale}{a}} \]

   7. Applied egg-rr 82.4%

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

      [Start] 86.5	\[ -4 \cdot \frac{\frac{b}{x-scale}}{\frac{\frac{x-scale}{b}}{\frac{a}{y-scale}} \cdot \frac{y-scale}{a}} \]
      associate-*r/ [=>] 86.5	\[ \color{blue}{\frac{-4 \cdot \frac{b}{x-scale}}{\frac{\frac{x-scale}{b}}{\frac{a}{y-scale}} \cdot \frac{y-scale}{a}}} \]
      *-commutative [=>] 86.5	\[ \frac{-4 \cdot \frac{b}{x-scale}}{\color{blue}{\frac{y-scale}{a} \cdot \frac{\frac{x-scale}{b}}{\frac{a}{y-scale}}}} \]
      associate-/r* [=>] 90.3	\[ \color{blue}{\frac{\frac{-4 \cdot \frac{b}{x-scale}}{\frac{y-scale}{a}}}{\frac{\frac{x-scale}{b}}{\frac{a}{y-scale}}}} \]
      *-commutative [=>] 90.3	\[ \frac{\frac{\color{blue}{\frac{b}{x-scale} \cdot -4}}{\frac{y-scale}{a}}}{\frac{\frac{x-scale}{b}}{\frac{a}{y-scale}}} \]
      div-inv [=>] 90.3	\[ \frac{\frac{\frac{b}{x-scale} \cdot -4}{\frac{y-scale}{a}}}{\color{blue}{\frac{x-scale}{b} \cdot \frac{1}{\frac{a}{y-scale}}}} \]
      clear-num [<=] 90.4	\[ \frac{\frac{\frac{b}{x-scale} \cdot -4}{\frac{y-scale}{a}}}{\frac{x-scale}{b} \cdot \color{blue}{\frac{y-scale}{a}}} \]
      frac-times [=>] 82.4	\[ \frac{\frac{\frac{b}{x-scale} \cdot -4}{\frac{y-scale}{a}}}{\color{blue}{\frac{x-scale \cdot y-scale}{b \cdot a}}} \]

   if -1.92000000000000009e-90 < angle < 7.1999999999999996e-271

   1. Initial program 47.4%

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

   2. Simplified 39.2%

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

      [Start] 47.4

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

   3. Taylor expanded in angle around 0 41.7%

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

   4. Simplified 58.1%

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

      [Start] 41.7	\[ -4 \cdot \frac{{a}^{2} \cdot {b}^{2}}{{y-scale}^{2} \cdot {x-scale}^{2}} \]
      *-commutative [=>] 41.7	\[ -4 \cdot \frac{\color{blue}{{b}^{2} \cdot {a}^{2}}}{{y-scale}^{2} \cdot {x-scale}^{2}} \]
      times-frac [=>] 41.4	\[ -4 \cdot \color{blue}{\left(\frac{{b}^{2}}{{y-scale}^{2}} \cdot \frac{{a}^{2}}{{x-scale}^{2}}\right)} \]
      unpow2 [=>] 41.4	\[ -4 \cdot \left(\frac{\color{blue}{b \cdot b}}{{y-scale}^{2}} \cdot \frac{{a}^{2}}{{x-scale}^{2}}\right) \]
      unpow2 [=>] 41.4	\[ -4 \cdot \left(\frac{b \cdot b}{\color{blue}{y-scale \cdot y-scale}} \cdot \frac{{a}^{2}}{{x-scale}^{2}}\right) \]
      times-frac [=>] 52.8	\[ -4 \cdot \left(\color{blue}{\left(\frac{b}{y-scale} \cdot \frac{b}{y-scale}\right)} \cdot \frac{{a}^{2}}{{x-scale}^{2}}\right) \]
      unpow2 [=>] 52.8	\[ -4 \cdot \left(\left(\frac{b}{y-scale} \cdot \frac{b}{y-scale}\right) \cdot \frac{{a}^{2}}{\color{blue}{x-scale \cdot x-scale}}\right) \]
      associate-/r* [=>] 58.1	\[ -4 \cdot \left(\left(\frac{b}{y-scale} \cdot \frac{b}{y-scale}\right) \cdot \color{blue}{\frac{\frac{{a}^{2}}{x-scale}}{x-scale}}\right) \]
      unpow2 [=>] 58.1	\[ -4 \cdot \left(\left(\frac{b}{y-scale} \cdot \frac{b}{y-scale}\right) \cdot \frac{\frac{\color{blue}{a \cdot a}}{x-scale}}{x-scale}\right) \]

   5. Applied egg-rr 74.5%

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

      [Start] 58.1	\[ -4 \cdot \left(\left(\frac{b}{y-scale} \cdot \frac{b}{y-scale}\right) \cdot \frac{\frac{a \cdot a}{x-scale}}{x-scale}\right) \]
      associate-*l* [=>] 63.4	\[ -4 \cdot \color{blue}{\left(\frac{b}{y-scale} \cdot \left(\frac{b}{y-scale} \cdot \frac{\frac{a \cdot a}{x-scale}}{x-scale}\right)\right)} \]
      clear-num [=>] 63.4	\[ -4 \cdot \left(\color{blue}{\frac{1}{\frac{y-scale}{b}}} \cdot \left(\frac{b}{y-scale} \cdot \frac{\frac{a \cdot a}{x-scale}}{x-scale}\right)\right) \]
      associate-*l/ [=>] 63.5	\[ -4 \cdot \color{blue}{\frac{1 \cdot \left(\frac{b}{y-scale} \cdot \frac{\frac{a \cdot a}{x-scale}}{x-scale}\right)}{\frac{y-scale}{b}}} \]
      *-un-lft-identity [<=] 63.5	\[ -4 \cdot \frac{\color{blue}{\frac{b}{y-scale} \cdot \frac{\frac{a \cdot a}{x-scale}}{x-scale}}}{\frac{y-scale}{b}} \]
      clear-num [=>] 63.4	\[ -4 \cdot \frac{\color{blue}{\frac{1}{\frac{y-scale}{b}}} \cdot \frac{\frac{a \cdot a}{x-scale}}{x-scale}}{\frac{y-scale}{b}} \]
      frac-times [=>] 68.5	\[ -4 \cdot \frac{\color{blue}{\frac{1 \cdot \frac{a \cdot a}{x-scale}}{\frac{y-scale}{b} \cdot x-scale}}}{\frac{y-scale}{b}} \]
      *-un-lft-identity [<=] 68.5	\[ -4 \cdot \frac{\frac{\color{blue}{\frac{a \cdot a}{x-scale}}}{\frac{y-scale}{b} \cdot x-scale}}{\frac{y-scale}{b}} \]
      associate-/l/ [=>] 66.7	\[ -4 \cdot \color{blue}{\frac{\frac{a \cdot a}{x-scale}}{\frac{y-scale}{b} \cdot \left(\frac{y-scale}{b} \cdot x-scale\right)}} \]

   6. Applied egg-rr 88.8%

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

      [Start] 74.5	\[ -4 \cdot \frac{a \cdot \frac{a}{x-scale}}{\frac{y-scale}{b} \cdot \left(\frac{y-scale}{b} \cdot x-scale\right)} \]
      *-commutative [=>] 74.5	\[ -4 \cdot \frac{\color{blue}{\frac{a}{x-scale} \cdot a}}{\frac{y-scale}{b} \cdot \left(\frac{y-scale}{b} \cdot x-scale\right)} \]
      times-frac [=>] 88.8	\[ -4 \cdot \color{blue}{\left(\frac{\frac{a}{x-scale}}{\frac{y-scale}{b}} \cdot \frac{a}{\frac{y-scale}{b} \cdot x-scale}\right)} \]
      *-commutative [=>] 88.8	\[ -4 \cdot \left(\frac{\frac{a}{x-scale}}{\frac{y-scale}{b}} \cdot \frac{a}{\color{blue}{x-scale \cdot \frac{y-scale}{b}}}\right) \]

   7. Applied egg-rr 90.6%

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

      [Start] 88.8	\[ -4 \cdot \left(\frac{\frac{a}{x-scale}}{\frac{y-scale}{b}} \cdot \frac{a}{x-scale \cdot \frac{y-scale}{b}}\right) \]
      associate-/r* [<=] 91.2	\[ -4 \cdot \left(\color{blue}{\frac{a}{x-scale \cdot \frac{y-scale}{b}}} \cdot \frac{a}{x-scale \cdot \frac{y-scale}{b}}\right) \]
      div-inv [=>] 91.1	\[ -4 \cdot \left(\color{blue}{\left(a \cdot \frac{1}{x-scale \cdot \frac{y-scale}{b}}\right)} \cdot \frac{a}{x-scale \cdot \frac{y-scale}{b}}\right) \]
      *-commutative [=>] 91.1	\[ -4 \cdot \left(\color{blue}{\left(\frac{1}{x-scale \cdot \frac{y-scale}{b}} \cdot a\right)} \cdot \frac{a}{x-scale \cdot \frac{y-scale}{b}}\right) \]
      *-commutative [=>] 91.1	\[ -4 \cdot \left(\left(\frac{1}{\color{blue}{\frac{y-scale}{b} \cdot x-scale}} \cdot a\right) \cdot \frac{a}{x-scale \cdot \frac{y-scale}{b}}\right) \]
      associate-/r* [=>] 90.6	\[ -4 \cdot \left(\left(\color{blue}{\frac{\frac{1}{\frac{y-scale}{b}}}{x-scale}} \cdot a\right) \cdot \frac{a}{x-scale \cdot \frac{y-scale}{b}}\right) \]
      clear-num [<=] 90.6	\[ -4 \cdot \left(\left(\frac{\color{blue}{\frac{b}{y-scale}}}{x-scale} \cdot a\right) \cdot \frac{a}{x-scale \cdot \frac{y-scale}{b}}\right) \]

3. Recombined 3 regimes into one program.

4. Final simplification 87.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;angle \leq -9.2 \cdot 10^{+91}:\\ \;\;\;\;-4 \cdot {\left(\frac{b}{y-scale} \cdot \frac{a}{x-scale}\right)}^{2}\\ \mathbf{elif}\;angle \leq -1.92 \cdot 10^{-90}:\\ \;\;\;\;\frac{\frac{-4 \cdot \frac{b}{x-scale}}{\frac{y-scale}{a}}}{\frac{y-scale \cdot x-scale}{a \cdot b}}\\ \mathbf{elif}\;angle \leq 7.2 \cdot 10^{-271}:\\ \;\;\;\;-4 \cdot \left(\left(a \cdot \frac{\frac{b}{y-scale}}{x-scale}\right) \cdot \frac{a}{x-scale \cdot \frac{y-scale}{b}}\right)\\ \mathbf{elif}\;angle \leq 2.1 \cdot 10^{-27}:\\ \;\;\;\;\frac{\frac{-4 \cdot \frac{b}{x-scale}}{\frac{y-scale}{a}}}{\frac{y-scale \cdot x-scale}{a \cdot b}}\\ \mathbf{else}:\\ \;\;\;\;-4 \cdot {\left(\frac{b}{y-scale} \cdot \frac{a}{x-scale}\right)}^{2}\\ \end{array} \]

Alternatives

Alternative 1
Accuracy	90.4%
Cost	20684

\[\begin{array}{l} \mathbf{if}\;\frac{angle}{180} \leq -2 \cdot 10^{-90}:\\ \;\;\;\;-4 \cdot {\left(\frac{\frac{a}{y-scale}}{\frac{x-scale}{b}}\right)}^{2}\\ \mathbf{elif}\;\frac{angle}{180} \leq 10^{-256}:\\ \;\;\;\;-4 \cdot \left(\left(a \cdot \frac{\frac{b}{y-scale}}{x-scale}\right) \cdot \frac{a}{x-scale \cdot \frac{y-scale}{b}}\right)\\ \mathbf{elif}\;\frac{angle}{180} \leq 10000000000:\\ \;\;\;\;{\left(\sqrt[3]{-4 \cdot {\left(\frac{a}{y-scale} \cdot \frac{b}{x-scale}\right)}^{2}}\right)}^{3}\\ \mathbf{else}:\\ \;\;\;\;-4 \cdot {\left(\frac{b}{y-scale} \cdot \frac{a}{x-scale}\right)}^{2}\\ \end{array} \]

Alternative 2
Accuracy	90.5%
Cost	7436

\[\begin{array}{l} t_0 := -4 \cdot {\left(\frac{\frac{a}{y-scale}}{\frac{x-scale}{b}}\right)}^{2}\\ \mathbf{if}\;angle \leq -1.45 \cdot 10^{-91}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;angle \leq 8.2 \cdot 10^{-271}:\\ \;\;\;\;-4 \cdot \left(\left(a \cdot \frac{\frac{b}{y-scale}}{x-scale}\right) \cdot \frac{a}{x-scale \cdot \frac{y-scale}{b}}\right)\\ \mathbf{elif}\;angle \leq 3.2 \cdot 10^{+20}:\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;-4 \cdot {\left(\frac{b}{y-scale} \cdot \frac{a}{x-scale}\right)}^{2}\\ \end{array} \]

Alternative 3
Accuracy	87.5%
Cost	1617

\[\begin{array}{l} t_0 := -4 \cdot \left(\left(a \cdot \frac{\frac{b}{y-scale}}{x-scale}\right) \cdot \frac{a}{x-scale \cdot \frac{y-scale}{b}}\right)\\ \mathbf{if}\;angle \leq -5 \cdot 10^{+90}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;angle \leq -1.6 \cdot 10^{-175}:\\ \;\;\;\;-4 \cdot \frac{-1}{\frac{y-scale}{a} \cdot \left(\frac{x-scale}{-b} \cdot \left(y-scale \cdot \frac{\frac{x-scale}{b}}{a}\right)\right)}\\ \mathbf{elif}\;angle \leq 1.4 \cdot 10^{-270} \lor \neg \left(angle \leq 1.25 \cdot 10^{-40}\right):\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{-4 \cdot \frac{b}{x-scale}}{\frac{y-scale}{a}}}{\frac{y-scale \cdot x-scale}{a \cdot b}}\\ \end{array} \]

Alternative 4
Accuracy	88.0%
Cost	1616

\[\begin{array}{l} t_0 := \frac{a}{x-scale \cdot \frac{y-scale}{b}}\\ \mathbf{if}\;y-scale \leq -3.9 \cdot 10^{-119}:\\ \;\;\;\;-4 \cdot \frac{\frac{b}{x-scale}}{\frac{y-scale}{a} \cdot \frac{\frac{x-scale}{b}}{\frac{a}{y-scale}}}\\ \mathbf{elif}\;y-scale \leq -9.6 \cdot 10^{-141}:\\ \;\;\;\;-4 \cdot \frac{\left(a \cdot b\right) \cdot \frac{a \cdot b}{x-scale}}{y-scale \cdot \left(y-scale \cdot x-scale\right)}\\ \mathbf{elif}\;y-scale \leq -7.5 \cdot 10^{-200}:\\ \;\;\;\;-4 \cdot \frac{\frac{b}{x-scale}}{\frac{y-scale}{a} \cdot \left(y-scale \cdot \frac{\frac{x-scale}{b}}{a}\right)}\\ \mathbf{elif}\;y-scale \leq 4 \cdot 10^{-199}:\\ \;\;\;\;-4 \cdot \left(t_0 \cdot \frac{\frac{a}{x-scale}}{\frac{y-scale}{b}}\right)\\ \mathbf{else}:\\ \;\;\;\;-4 \cdot \left(\left(a \cdot \frac{\frac{b}{y-scale}}{x-scale}\right) \cdot t_0\right)\\ \end{array} \]

Alternative 5
Accuracy	59.7%
Cost	1485

\[\begin{array}{l} \mathbf{if}\;a \leq -1.65 \cdot 10^{+130}:\\ \;\;\;\;0\\ \mathbf{elif}\;a \leq -1.7 \cdot 10^{-156} \lor \neg \left(a \leq 4 \cdot 10^{-185}\right):\\ \;\;\;\;-4 \cdot \left(\left(\frac{b}{y-scale} \cdot \frac{b}{y-scale}\right) \cdot \frac{\frac{a \cdot a}{x-scale}}{x-scale}\right)\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \]

Alternative 6
Accuracy	87.1%
Cost	1484

\[\begin{array}{l} \mathbf{if}\;y-scale \leq -8 \cdot 10^{+116}:\\ \;\;\;\;-4 \cdot \left(\frac{a}{y-scale} \cdot \frac{\frac{a}{y-scale} \cdot \frac{b}{x-scale}}{\frac{x-scale}{b}}\right)\\ \mathbf{elif}\;y-scale \leq -2 \cdot 10^{-143}:\\ \;\;\;\;-4 \cdot \frac{\left(a \cdot b\right) \cdot \frac{a \cdot b}{x-scale}}{y-scale \cdot \left(y-scale \cdot x-scale\right)}\\ \mathbf{elif}\;y-scale \leq -4.5 \cdot 10^{-201}:\\ \;\;\;\;-4 \cdot \frac{\frac{b}{x-scale}}{\frac{y-scale}{a} \cdot \left(y-scale \cdot \frac{\frac{x-scale}{b}}{a}\right)}\\ \mathbf{else}:\\ \;\;\;\;-4 \cdot \left(\left(a \cdot \frac{\frac{b}{y-scale}}{x-scale}\right) \cdot \frac{a}{x-scale \cdot \frac{y-scale}{b}}\right)\\ \end{array} \]

Alternative 7
Accuracy	89.0%
Cost	1353

\[\begin{array}{l} \mathbf{if}\;angle \leq 2.4 \cdot 10^{-269} \lor \neg \left(angle \leq 3.1 \cdot 10^{+31}\right):\\ \;\;\;\;-4 \cdot \left(\left(a \cdot \frac{\frac{b}{y-scale}}{x-scale}\right) \cdot \frac{a}{x-scale \cdot \frac{y-scale}{b}}\right)\\ \mathbf{else}:\\ \;\;\;\;-4 \cdot \left(\frac{a}{y-scale} \cdot \frac{\frac{a}{y-scale} \cdot \frac{b}{x-scale}}{\frac{x-scale}{b}}\right)\\ \end{array} \]

Alternative 8
Accuracy	89.9%
Cost	1088

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

Alternative 9
Accuracy	52.4%
Cost	64

\[0 \]

Reproduce

herbie shell --seed 2023138 
(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))))