Simplification of discriminant from scale-rotated-ellipse

Average Accuracy: 36.3% → 90.6%

Time: 1.5min

Precision: binary64

Cost: 7300

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;\frac{angle}{180} \leq -1 \cdot 10^{-22}:\\ \;\;\;\;-4 \cdot {\left(\frac{\frac{a}{x-scale}}{\frac{y-scale}{b}}\right)}^{2}\\ \mathbf{elif}\;\frac{angle}{180} \leq -2 \cdot 10^{-258} \lor \neg \left(\frac{angle}{180} \leq 10^{-234}\right) \land \frac{angle}{180} \leq 5 \cdot 10^{-32}:\\ \;\;\;\;\frac{-4 \cdot \frac{a \cdot b}{x-scale \cdot y-scale}}{\frac{x-scale \cdot y-scale}{a \cdot b}}\\ \mathbf{else}:\\ \;\;\;\;-4 \cdot \frac{\frac{a}{x-scale} \cdot \frac{b}{y-scale}}{\frac{y-scale}{b} \cdot \frac{x-scale}{a}}\\ \end{array} \]

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
 (if (<= (/ angle 180.0) -1e-22)
   (* -4.0 (pow (/ (/ a x-scale) (/ y-scale b)) 2.0))
   (if (or (<= (/ angle 180.0) -2e-258)
           (and (not (<= (/ angle 180.0) 1e-234)) (<= (/ angle 180.0) 5e-32)))
     (/
      (* -4.0 (/ (* a b) (* x-scale y-scale)))
      (/ (* x-scale y-scale) (* a b)))
     (*
      -4.0
      (/ (* (/ a x-scale) (/ b y-scale)) (* (/ y-scale b) (/ 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 tmp;
	if ((angle / 180.0) <= -1e-22) {
		tmp = -4.0 * pow(((a / x_45_scale) / (y_45_scale / b)), 2.0);
	} else if (((angle / 180.0) <= -2e-258) || (!((angle / 180.0) <= 1e-234) && ((angle / 180.0) <= 5e-32))) {
		tmp = (-4.0 * ((a * b) / (x_45_scale * y_45_scale))) / ((x_45_scale * y_45_scale) / (a * b));
	} else {
		tmp = -4.0 * (((a / x_45_scale) * (b / y_45_scale)) / ((y_45_scale / b) * (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 tmp;
	if ((angle / 180.0) <= -1e-22) {
		tmp = -4.0 * Math.pow(((a / x_45_scale) / (y_45_scale / b)), 2.0);
	} else if (((angle / 180.0) <= -2e-258) || (!((angle / 180.0) <= 1e-234) && ((angle / 180.0) <= 5e-32))) {
		tmp = (-4.0 * ((a * b) / (x_45_scale * y_45_scale))) / ((x_45_scale * y_45_scale) / (a * b));
	} else {
		tmp = -4.0 * (((a / x_45_scale) * (b / y_45_scale)) / ((y_45_scale / b) * (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):
	tmp = 0
	if (angle / 180.0) <= -1e-22:
		tmp = -4.0 * math.pow(((a / x_45_scale) / (y_45_scale / b)), 2.0)
	elif ((angle / 180.0) <= -2e-258) or (not ((angle / 180.0) <= 1e-234) and ((angle / 180.0) <= 5e-32)):
		tmp = (-4.0 * ((a * b) / (x_45_scale * y_45_scale))) / ((x_45_scale * y_45_scale) / (a * b))
	else:
		tmp = -4.0 * (((a / x_45_scale) * (b / y_45_scale)) / ((y_45_scale / b) * (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)
	tmp = 0.0
	if (Float64(angle / 180.0) <= -1e-22)
		tmp = Float64(-4.0 * (Float64(Float64(a / x_45_scale) / Float64(y_45_scale / b)) ^ 2.0));
	elseif ((Float64(angle / 180.0) <= -2e-258) || (!(Float64(angle / 180.0) <= 1e-234) && (Float64(angle / 180.0) <= 5e-32)))
		tmp = Float64(Float64(-4.0 * Float64(Float64(a * b) / Float64(x_45_scale * y_45_scale))) / Float64(Float64(x_45_scale * y_45_scale) / Float64(a * b)));
	else
		tmp = Float64(-4.0 * Float64(Float64(Float64(a / x_45_scale) * Float64(b / y_45_scale)) / Float64(Float64(y_45_scale / b) * 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)
	tmp = 0.0;
	if ((angle / 180.0) <= -1e-22)
		tmp = -4.0 * (((a / x_45_scale) / (y_45_scale / b)) ^ 2.0);
	elseif (((angle / 180.0) <= -2e-258) || (~(((angle / 180.0) <= 1e-234)) && ((angle / 180.0) <= 5e-32)))
		tmp = (-4.0 * ((a * b) / (x_45_scale * y_45_scale))) / ((x_45_scale * y_45_scale) / (a * b));
	else
		tmp = -4.0 * (((a / x_45_scale) * (b / y_45_scale)) / ((y_45_scale / b) * (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_] := If[LessEqual[N[(angle / 180.0), $MachinePrecision], -1e-22], N[(-4.0 * N[Power[N[(N[(a / x$45$scale), $MachinePrecision] / N[(y$45$scale / b), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision], If[Or[LessEqual[N[(angle / 180.0), $MachinePrecision], -2e-258], And[N[Not[LessEqual[N[(angle / 180.0), $MachinePrecision], 1e-234]], $MachinePrecision], LessEqual[N[(angle / 180.0), $MachinePrecision], 5e-32]]], N[(N[(-4.0 * N[(N[(a * b), $MachinePrecision] / N[(x$45$scale * y$45$scale), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(N[(x$45$scale * y$45$scale), $MachinePrecision] / N[(a * b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(-4.0 * N[(N[(N[(a / x$45$scale), $MachinePrecision] * N[(b / y$45$scale), $MachinePrecision]), $MachinePrecision] / N[(N[(y$45$scale / b), $MachinePrecision] * N[(x$45$scale / a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if (/.f64 angle 180) < -1e-22

   1. Initial program 28.9%

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

   2. Simplified 23.1%

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

      [Start] 28.9	\[ \frac{\frac{\left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right) \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)}{x-scale}}{y-scale} \cdot \frac{\frac{\left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right) \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)}{x-scale}}{y-scale} - \left(4 \cdot \frac{\frac{{\left(a \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} + {\left(b \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2}}{x-scale}}{x-scale}\right) \cdot \frac{\frac{{\left(a \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} + {\left(b \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2}}{y-scale}}{y-scale} \]
      sub-neg [=>] 28.9	\[ \color{blue}{\frac{\frac{\left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right) \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)}{x-scale}}{y-scale} \cdot \frac{\frac{\left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right) \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)}{x-scale}}{y-scale} + \left(-\left(4 \cdot \frac{\frac{{\left(a \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} + {\left(b \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2}}{x-scale}}{x-scale}\right) \cdot \frac{\frac{{\left(a \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} + {\left(b \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2}}{y-scale}}{y-scale}\right)} \]
      +-commutative [=>] 28.9	\[ \color{blue}{\left(-\left(4 \cdot \frac{\frac{{\left(a \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} + {\left(b \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2}}{x-scale}}{x-scale}\right) \cdot \frac{\frac{{\left(a \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} + {\left(b \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2}}{y-scale}}{y-scale}\right) + \frac{\frac{\left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right) \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)}{x-scale}}{y-scale} \cdot \frac{\frac{\left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right) \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)}{x-scale}}{y-scale}} \]

   3. Taylor expanded in angle around 0 36.9%

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

   4. Simplified 67.5%

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

      [Start] 36.9	\[ -4 \cdot \frac{{a}^{2} \cdot {b}^{2}}{{x-scale}^{2} \cdot {y-scale}^{2}} \]
      times-frac [=>] 36.7	\[ -4 \cdot \color{blue}{\left(\frac{{a}^{2}}{{x-scale}^{2}} \cdot \frac{{b}^{2}}{{y-scale}^{2}}\right)} \]
      associate-*r* [=>] 36.7	\[ \color{blue}{\left(-4 \cdot \frac{{a}^{2}}{{x-scale}^{2}}\right) \cdot \frac{{b}^{2}}{{y-scale}^{2}}} \]
      unpow2 [=>] 36.7	\[ \left(-4 \cdot \frac{\color{blue}{a \cdot a}}{{x-scale}^{2}}\right) \cdot \frac{{b}^{2}}{{y-scale}^{2}} \]
      unpow2 [=>] 36.7	\[ \left(-4 \cdot \frac{a \cdot a}{\color{blue}{x-scale \cdot x-scale}}\right) \cdot \frac{{b}^{2}}{{y-scale}^{2}} \]
      times-frac [=>] 49.9	\[ \left(-4 \cdot \color{blue}{\left(\frac{a}{x-scale} \cdot \frac{a}{x-scale}\right)}\right) \cdot \frac{{b}^{2}}{{y-scale}^{2}} \]
      unpow2 [=>] 49.9	\[ \left(-4 \cdot \left(\frac{a}{x-scale} \cdot \frac{a}{x-scale}\right)\right) \cdot \frac{\color{blue}{b \cdot b}}{{y-scale}^{2}} \]
      unpow2 [=>] 49.9	\[ \left(-4 \cdot \left(\frac{a}{x-scale} \cdot \frac{a}{x-scale}\right)\right) \cdot \frac{b \cdot b}{\color{blue}{y-scale \cdot y-scale}} \]
      times-frac [=>] 67.5	\[ \left(-4 \cdot \left(\frac{a}{x-scale} \cdot \frac{a}{x-scale}\right)\right) \cdot \color{blue}{\left(\frac{b}{y-scale} \cdot \frac{b}{y-scale}\right)} \]

   5. Taylor expanded in a around 0 36.9%

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

   6. Simplified 91.0%

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

      [Start] 36.9	\[ -4 \cdot \frac{{a}^{2} \cdot {b}^{2}}{{y-scale}^{2} \cdot {x-scale}^{2}} \]
      *-commutative [=>] 36.9	\[ -4 \cdot \frac{{a}^{2} \cdot {b}^{2}}{\color{blue}{{x-scale}^{2} \cdot {y-scale}^{2}}} \]
      times-frac [=>] 36.7	\[ -4 \cdot \color{blue}{\left(\frac{{a}^{2}}{{x-scale}^{2}} \cdot \frac{{b}^{2}}{{y-scale}^{2}}\right)} \]
      unpow2 [=>] 36.7	\[ -4 \cdot \left(\frac{\color{blue}{a \cdot a}}{{x-scale}^{2}} \cdot \frac{{b}^{2}}{{y-scale}^{2}}\right) \]
      unpow2 [=>] 36.7	\[ -4 \cdot \left(\frac{a \cdot a}{\color{blue}{x-scale \cdot x-scale}} \cdot \frac{{b}^{2}}{{y-scale}^{2}}\right) \]
      times-frac [=>] 49.9	\[ -4 \cdot \left(\color{blue}{\left(\frac{a}{x-scale} \cdot \frac{a}{x-scale}\right)} \cdot \frac{{b}^{2}}{{y-scale}^{2}}\right) \]
      unpow2 [=>] 49.9	\[ -4 \cdot \left(\left(\frac{a}{x-scale} \cdot \frac{a}{x-scale}\right) \cdot \frac{\color{blue}{b \cdot b}}{{y-scale}^{2}}\right) \]
      unpow2 [=>] 49.9	\[ -4 \cdot \left(\left(\frac{a}{x-scale} \cdot \frac{a}{x-scale}\right) \cdot \frac{b \cdot b}{\color{blue}{y-scale \cdot y-scale}}\right) \]
      times-frac [=>] 67.5	\[ -4 \cdot \left(\left(\frac{a}{x-scale} \cdot \frac{a}{x-scale}\right) \cdot \color{blue}{\left(\frac{b}{y-scale} \cdot \frac{b}{y-scale}\right)}\right) \]
      swap-sqr [<=] 90.8	\[ -4 \cdot \color{blue}{\left(\left(\frac{a}{x-scale} \cdot \frac{b}{y-scale}\right) \cdot \left(\frac{a}{x-scale} \cdot \frac{b}{y-scale}\right)\right)} \]
      unpow2 [<=] 90.8	\[ -4 \cdot \color{blue}{{\left(\frac{a}{x-scale} \cdot \frac{b}{y-scale}\right)}^{2}} \]
      associate-*r/ [=>] 91.3	\[ -4 \cdot {\color{blue}{\left(\frac{\frac{a}{x-scale} \cdot b}{y-scale}\right)}}^{2} \]
      associate-/l* [=>] 91.0	\[ -4 \cdot {\color{blue}{\left(\frac{\frac{a}{x-scale}}{\frac{y-scale}{b}}\right)}}^{2} \]

   if -1e-22 < (/.f64 angle 180) < -1.99999999999999991e-258 or 9.9999999999999996e-235 < (/.f64 angle 180) < 5e-32

   1. Initial program 41.4%

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

   2. Simplified 32.7%

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

      [Start] 41.4

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

      fma-neg [=>] 40.5

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

   3. Taylor expanded in angle around 0 38.9%

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

   4. Simplified 50.9%

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

      [Start] 38.9	\[ -4 \cdot \frac{{a}^{2} \cdot {b}^{2}}{{y-scale}^{2} \cdot {x-scale}^{2}} \]
      associate-/l* [=>] 38.7	\[ -4 \cdot \color{blue}{\frac{{a}^{2}}{\frac{{y-scale}^{2} \cdot {x-scale}^{2}}{{b}^{2}}}} \]
      unpow2 [=>] 38.7	\[ -4 \cdot \frac{\color{blue}{a \cdot a}}{\frac{{y-scale}^{2} \cdot {x-scale}^{2}}{{b}^{2}}} \]
      *-commutative [=>] 38.7	\[ -4 \cdot \frac{a \cdot a}{\frac{\color{blue}{{x-scale}^{2} \cdot {y-scale}^{2}}}{{b}^{2}}} \]
      associate-/l* [=>] 38.5	\[ -4 \cdot \frac{a \cdot a}{\color{blue}{\frac{{x-scale}^{2}}{\frac{{b}^{2}}{{y-scale}^{2}}}}} \]
      unpow2 [=>] 38.5	\[ -4 \cdot \frac{a \cdot a}{\frac{\color{blue}{x-scale \cdot x-scale}}{\frac{{b}^{2}}{{y-scale}^{2}}}} \]
      unpow2 [=>] 38.5	\[ -4 \cdot \frac{a \cdot a}{\frac{x-scale \cdot x-scale}{\frac{\color{blue}{b \cdot b}}{{y-scale}^{2}}}} \]
      unpow2 [=>] 38.5	\[ -4 \cdot \frac{a \cdot a}{\frac{x-scale \cdot x-scale}{\frac{b \cdot b}{\color{blue}{y-scale \cdot y-scale}}}} \]
      times-frac [=>] 50.9	\[ -4 \cdot \frac{a \cdot a}{\frac{x-scale \cdot x-scale}{\color{blue}{\frac{b}{y-scale} \cdot \frac{b}{y-scale}}}} \]

   5. Applied egg-rr 77.8%

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

      [Start] 50.9	\[ -4 \cdot \frac{a \cdot a}{\frac{x-scale \cdot x-scale}{\frac{b}{y-scale} \cdot \frac{b}{y-scale}}} \]
      associate-/r/ [=>] 50.6	\[ -4 \cdot \color{blue}{\left(\frac{a \cdot a}{x-scale \cdot x-scale} \cdot \left(\frac{b}{y-scale} \cdot \frac{b}{y-scale}\right)\right)} \]
      associate-*r* [=>] 55.1	\[ -4 \cdot \color{blue}{\left(\left(\frac{a \cdot a}{x-scale \cdot x-scale} \cdot \frac{b}{y-scale}\right) \cdot \frac{b}{y-scale}\right)} \]
      times-frac [=>] 77.8	\[ -4 \cdot \left(\left(\color{blue}{\left(\frac{a}{x-scale} \cdot \frac{a}{x-scale}\right)} \cdot \frac{b}{y-scale}\right) \cdot \frac{b}{y-scale}\right) \]

   6. Applied egg-rr 90.7%

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

      [Start] 77.8	\[ -4 \cdot \left(\left(\left(\frac{a}{x-scale} \cdot \frac{a}{x-scale}\right) \cdot \frac{b}{y-scale}\right) \cdot \frac{b}{y-scale}\right) \]
      associate-*r/ [=>] 74.0	\[ -4 \cdot \color{blue}{\frac{\left(\left(\frac{a}{x-scale} \cdot \frac{a}{x-scale}\right) \cdot \frac{b}{y-scale}\right) \cdot b}{y-scale}} \]
      associate-/l* [=>] 77.8	\[ -4 \cdot \color{blue}{\frac{\left(\frac{a}{x-scale} \cdot \frac{a}{x-scale}\right) \cdot \frac{b}{y-scale}}{\frac{y-scale}{b}}} \]
      associate-*l* [=>] 86.6	\[ -4 \cdot \frac{\color{blue}{\frac{a}{x-scale} \cdot \left(\frac{a}{x-scale} \cdot \frac{b}{y-scale}\right)}}{\frac{y-scale}{b}} \]
      clear-num [=>] 86.6	\[ -4 \cdot \frac{\color{blue}{\frac{1}{\frac{x-scale}{a}}} \cdot \left(\frac{a}{x-scale} \cdot \frac{b}{y-scale}\right)}{\frac{y-scale}{b}} \]
      associate-*l/ [=>] 86.6	\[ -4 \cdot \frac{\color{blue}{\frac{1 \cdot \left(\frac{a}{x-scale} \cdot \frac{b}{y-scale}\right)}{\frac{x-scale}{a}}}}{\frac{y-scale}{b}} \]
      *-un-lft-identity [<=] 86.6	\[ -4 \cdot \frac{\frac{\color{blue}{\frac{a}{x-scale} \cdot \frac{b}{y-scale}}}{\frac{x-scale}{a}}}{\frac{y-scale}{b}} \]
      add-sqr-sqrt [=>] 51.1	\[ -4 \cdot \frac{\frac{\color{blue}{\left(\sqrt{\frac{a}{x-scale}} \cdot \sqrt{\frac{a}{x-scale}}\right)} \cdot \frac{b}{y-scale}}{\frac{x-scale}{a}}}{\frac{y-scale}{b}} \]
      sqrt-prod [<=] 64.5	\[ -4 \cdot \frac{\frac{\color{blue}{\sqrt{\frac{a}{x-scale} \cdot \frac{a}{x-scale}}} \cdot \frac{b}{y-scale}}{\frac{x-scale}{a}}}{\frac{y-scale}{b}} \]
      add-sqr-sqrt [=>] 38.4	\[ -4 \cdot \frac{\frac{\sqrt{\frac{a}{x-scale} \cdot \frac{a}{x-scale}} \cdot \color{blue}{\left(\sqrt{\frac{b}{y-scale}} \cdot \sqrt{\frac{b}{y-scale}}\right)}}{\frac{x-scale}{a}}}{\frac{y-scale}{b}} \]
      sqrt-unprod [=>] 59.8	\[ -4 \cdot \frac{\frac{\sqrt{\frac{a}{x-scale} \cdot \frac{a}{x-scale}} \cdot \color{blue}{\sqrt{\frac{b}{y-scale} \cdot \frac{b}{y-scale}}}}{\frac{x-scale}{a}}}{\frac{y-scale}{b}} \]
      sqrt-prod [<=] 59.8	\[ -4 \cdot \frac{\frac{\color{blue}{\sqrt{\left(\frac{a}{x-scale} \cdot \frac{a}{x-scale}\right) \cdot \left(\frac{b}{y-scale} \cdot \frac{b}{y-scale}\right)}}}{\frac{x-scale}{a}}}{\frac{y-scale}{b}} \]
      associate-*l* [<=] 64.6	\[ -4 \cdot \frac{\frac{\sqrt{\color{blue}{\left(\left(\frac{a}{x-scale} \cdot \frac{a}{x-scale}\right) \cdot \frac{b}{y-scale}\right) \cdot \frac{b}{y-scale}}}}{\frac{x-scale}{a}}}{\frac{y-scale}{b}} \]
      associate-/l/ [=>] 64.6	\[ -4 \cdot \color{blue}{\frac{\sqrt{\left(\left(\frac{a}{x-scale} \cdot \frac{a}{x-scale}\right) \cdot \frac{b}{y-scale}\right) \cdot \frac{b}{y-scale}}}{\frac{y-scale}{b} \cdot \frac{x-scale}{a}}} \]

   7. Applied egg-rr 90.2%

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

      [Start] 90.7	\[ -4 \cdot \frac{\frac{a}{x-scale} \cdot \frac{b}{y-scale}}{\frac{y-scale}{b} \cdot \frac{x-scale}{a}} \]
      associate-*r/ [=>] 90.7	\[ \color{blue}{\frac{-4 \cdot \left(\frac{a}{x-scale} \cdot \frac{b}{y-scale}\right)}{\frac{y-scale}{b} \cdot \frac{x-scale}{a}}} \]
      *-commutative [=>] 90.7	\[ \frac{\color{blue}{\left(\frac{a}{x-scale} \cdot \frac{b}{y-scale}\right) \cdot -4}}{\frac{y-scale}{b} \cdot \frac{x-scale}{a}} \]
      frac-times [=>] 82.2	\[ \frac{\color{blue}{\frac{a \cdot b}{x-scale \cdot y-scale}} \cdot -4}{\frac{y-scale}{b} \cdot \frac{x-scale}{a}} \]
      *-commutative [=>] 82.2	\[ \frac{\frac{a \cdot b}{x-scale \cdot y-scale} \cdot -4}{\color{blue}{\frac{x-scale}{a} \cdot \frac{y-scale}{b}}} \]
      frac-times [=>] 90.2	\[ \frac{\frac{a \cdot b}{x-scale \cdot y-scale} \cdot -4}{\color{blue}{\frac{x-scale \cdot y-scale}{a \cdot b}}} \]

   if -1.99999999999999991e-258 < (/.f64 angle 180) < 9.9999999999999996e-235 or 5e-32 < (/.f64 angle 180)

   1. Initial program 36.7%

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

   2. Simplified 30.4%

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

      [Start] 36.7

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

      fma-neg [=>] 35.5

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

   3. Taylor expanded in angle around 0 40.5%

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

   4. Simplified 52.8%

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

      [Start] 40.5	\[ -4 \cdot \frac{{a}^{2} \cdot {b}^{2}}{{y-scale}^{2} \cdot {x-scale}^{2}} \]
      associate-/l* [=>] 40.8	\[ -4 \cdot \color{blue}{\frac{{a}^{2}}{\frac{{y-scale}^{2} \cdot {x-scale}^{2}}{{b}^{2}}}} \]
      unpow2 [=>] 40.8	\[ -4 \cdot \frac{\color{blue}{a \cdot a}}{\frac{{y-scale}^{2} \cdot {x-scale}^{2}}{{b}^{2}}} \]
      *-commutative [=>] 40.8	\[ -4 \cdot \frac{a \cdot a}{\frac{\color{blue}{{x-scale}^{2} \cdot {y-scale}^{2}}}{{b}^{2}}} \]
      associate-/l* [=>] 40.8	\[ -4 \cdot \frac{a \cdot a}{\color{blue}{\frac{{x-scale}^{2}}{\frac{{b}^{2}}{{y-scale}^{2}}}}} \]
      unpow2 [=>] 40.8	\[ -4 \cdot \frac{a \cdot a}{\frac{\color{blue}{x-scale \cdot x-scale}}{\frac{{b}^{2}}{{y-scale}^{2}}}} \]
      unpow2 [=>] 40.8	\[ -4 \cdot \frac{a \cdot a}{\frac{x-scale \cdot x-scale}{\frac{\color{blue}{b \cdot b}}{{y-scale}^{2}}}} \]
      unpow2 [=>] 40.8	\[ -4 \cdot \frac{a \cdot a}{\frac{x-scale \cdot x-scale}{\frac{b \cdot b}{\color{blue}{y-scale \cdot y-scale}}}} \]
      times-frac [=>] 52.8	\[ -4 \cdot \frac{a \cdot a}{\frac{x-scale \cdot x-scale}{\color{blue}{\frac{b}{y-scale} \cdot \frac{b}{y-scale}}}} \]

   5. Applied egg-rr 76.8%

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

      [Start] 52.8	\[ -4 \cdot \frac{a \cdot a}{\frac{x-scale \cdot x-scale}{\frac{b}{y-scale} \cdot \frac{b}{y-scale}}} \]
      associate-/r/ [=>] 52.5	\[ -4 \cdot \color{blue}{\left(\frac{a \cdot a}{x-scale \cdot x-scale} \cdot \left(\frac{b}{y-scale} \cdot \frac{b}{y-scale}\right)\right)} \]
      associate-*r* [=>] 56.1	\[ -4 \cdot \color{blue}{\left(\left(\frac{a \cdot a}{x-scale \cdot x-scale} \cdot \frac{b}{y-scale}\right) \cdot \frac{b}{y-scale}\right)} \]
      times-frac [=>] 76.8	\[ -4 \cdot \left(\left(\color{blue}{\left(\frac{a}{x-scale} \cdot \frac{a}{x-scale}\right)} \cdot \frac{b}{y-scale}\right) \cdot \frac{b}{y-scale}\right) \]

   6. Applied egg-rr 90.7%

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

      [Start] 76.8	\[ -4 \cdot \left(\left(\left(\frac{a}{x-scale} \cdot \frac{a}{x-scale}\right) \cdot \frac{b}{y-scale}\right) \cdot \frac{b}{y-scale}\right) \]
      associate-*r/ [=>] 73.4	\[ -4 \cdot \color{blue}{\frac{\left(\left(\frac{a}{x-scale} \cdot \frac{a}{x-scale}\right) \cdot \frac{b}{y-scale}\right) \cdot b}{y-scale}} \]
      associate-/l* [=>] 76.8	\[ -4 \cdot \color{blue}{\frac{\left(\frac{a}{x-scale} \cdot \frac{a}{x-scale}\right) \cdot \frac{b}{y-scale}}{\frac{y-scale}{b}}} \]
      associate-*l* [=>] 86.9	\[ -4 \cdot \frac{\color{blue}{\frac{a}{x-scale} \cdot \left(\frac{a}{x-scale} \cdot \frac{b}{y-scale}\right)}}{\frac{y-scale}{b}} \]
      clear-num [=>] 87.0	\[ -4 \cdot \frac{\color{blue}{\frac{1}{\frac{x-scale}{a}}} \cdot \left(\frac{a}{x-scale} \cdot \frac{b}{y-scale}\right)}{\frac{y-scale}{b}} \]
      associate-*l/ [=>] 87.0	\[ -4 \cdot \frac{\color{blue}{\frac{1 \cdot \left(\frac{a}{x-scale} \cdot \frac{b}{y-scale}\right)}{\frac{x-scale}{a}}}}{\frac{y-scale}{b}} \]
      *-un-lft-identity [<=] 87.0	\[ -4 \cdot \frac{\frac{\color{blue}{\frac{a}{x-scale} \cdot \frac{b}{y-scale}}}{\frac{x-scale}{a}}}{\frac{y-scale}{b}} \]
      add-sqr-sqrt [=>] 51.4	\[ -4 \cdot \frac{\frac{\color{blue}{\left(\sqrt{\frac{a}{x-scale}} \cdot \sqrt{\frac{a}{x-scale}}\right)} \cdot \frac{b}{y-scale}}{\frac{x-scale}{a}}}{\frac{y-scale}{b}} \]
      sqrt-prod [<=] 64.3	\[ -4 \cdot \frac{\frac{\color{blue}{\sqrt{\frac{a}{x-scale} \cdot \frac{a}{x-scale}}} \cdot \frac{b}{y-scale}}{\frac{x-scale}{a}}}{\frac{y-scale}{b}} \]
      add-sqr-sqrt [=>] 38.0	\[ -4 \cdot \frac{\frac{\sqrt{\frac{a}{x-scale} \cdot \frac{a}{x-scale}} \cdot \color{blue}{\left(\sqrt{\frac{b}{y-scale}} \cdot \sqrt{\frac{b}{y-scale}}\right)}}{\frac{x-scale}{a}}}{\frac{y-scale}{b}} \]
      sqrt-unprod [=>] 60.3	\[ -4 \cdot \frac{\frac{\sqrt{\frac{a}{x-scale} \cdot \frac{a}{x-scale}} \cdot \color{blue}{\sqrt{\frac{b}{y-scale} \cdot \frac{b}{y-scale}}}}{\frac{x-scale}{a}}}{\frac{y-scale}{b}} \]
      sqrt-prod [<=] 60.3	\[ -4 \cdot \frac{\frac{\color{blue}{\sqrt{\left(\frac{a}{x-scale} \cdot \frac{a}{x-scale}\right) \cdot \left(\frac{b}{y-scale} \cdot \frac{b}{y-scale}\right)}}}{\frac{x-scale}{a}}}{\frac{y-scale}{b}} \]
      associate-*l* [<=] 64.7	\[ -4 \cdot \frac{\frac{\sqrt{\color{blue}{\left(\left(\frac{a}{x-scale} \cdot \frac{a}{x-scale}\right) \cdot \frac{b}{y-scale}\right) \cdot \frac{b}{y-scale}}}}{\frac{x-scale}{a}}}{\frac{y-scale}{b}} \]
      associate-/l/ [=>] 64.7	\[ -4 \cdot \color{blue}{\frac{\sqrt{\left(\left(\frac{a}{x-scale} \cdot \frac{a}{x-scale}\right) \cdot \frac{b}{y-scale}\right) \cdot \frac{b}{y-scale}}}{\frac{y-scale}{b} \cdot \frac{x-scale}{a}}} \]

3. Recombined 3 regimes into one program.

4. Final simplification 90.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{angle}{180} \leq -1 \cdot 10^{-22}:\\ \;\;\;\;-4 \cdot {\left(\frac{\frac{a}{x-scale}}{\frac{y-scale}{b}}\right)}^{2}\\ \mathbf{elif}\;\frac{angle}{180} \leq -2 \cdot 10^{-258} \lor \neg \left(\frac{angle}{180} \leq 10^{-234}\right) \land \frac{angle}{180} \leq 5 \cdot 10^{-32}:\\ \;\;\;\;\frac{-4 \cdot \frac{a \cdot b}{x-scale \cdot y-scale}}{\frac{x-scale \cdot y-scale}{a \cdot b}}\\ \mathbf{else}:\\ \;\;\;\;-4 \cdot \frac{\frac{a}{x-scale} \cdot \frac{b}{y-scale}}{\frac{y-scale}{b} \cdot \frac{x-scale}{a}}\\ \end{array} \]

Alternatives

Alternative 1
Accuracy	90.4%
Cost	1618

\[\begin{array}{l} t_0 := \frac{x-scale \cdot y-scale}{a \cdot b}\\ \mathbf{if}\;angle \leq -3.6 \cdot 10^{-23} \lor \neg \left(angle \leq -3.4 \cdot 10^{-256}\right) \land \left(angle \leq 3.2 \cdot 10^{-232} \lor \neg \left(angle \leq 2.3 \cdot 10^{-28}\right)\right):\\ \;\;\;\;-4 \cdot \frac{\frac{a}{x-scale} \cdot \frac{b}{y-scale}}{\frac{y-scale}{b} \cdot \frac{x-scale}{a}}\\ \mathbf{else}:\\ \;\;\;\;\frac{-4}{t_0 \cdot t_0}\\ \end{array} \]

Alternative 2
Accuracy	90.5%
Cost	1618

\[\begin{array}{l} \mathbf{if}\;angle \leq -8 \cdot 10^{-25} \lor \neg \left(angle \leq -1.6 \cdot 10^{-256} \lor \neg \left(angle \leq 1.7 \cdot 10^{-232}\right) \land angle \leq 3.75 \cdot 10^{-25}\right):\\ \;\;\;\;-4 \cdot \frac{\frac{a}{x-scale} \cdot \frac{b}{y-scale}}{\frac{y-scale}{b} \cdot \frac{x-scale}{a}}\\ \mathbf{else}:\\ \;\;\;\;\frac{-4 \cdot \frac{a \cdot b}{x-scale \cdot y-scale}}{\frac{x-scale \cdot y-scale}{a \cdot b}}\\ \end{array} \]

Alternative 3
Accuracy	79.4%
Cost	1616

\[\begin{array}{l} t_0 := \frac{a}{x-scale \cdot y-scale}\\ t_1 := -4 \cdot \left(\frac{b}{y-scale} \cdot \left(\frac{b}{y-scale} \cdot \left(\frac{a}{x-scale} \cdot \frac{a}{x-scale}\right)\right)\right)\\ t_2 := -4 \cdot \left(b \cdot \left(\left(\frac{a}{x-scale} \cdot \frac{b}{y-scale}\right) \cdot t_0\right)\right)\\ \mathbf{if}\;y-scale \leq -8.5 \cdot 10^{-195}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;y-scale \leq 2.2 \cdot 10^{-241}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;y-scale \leq 2.5 \cdot 10^{+135}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;y-scale \leq 7.5 \cdot 10^{+183}:\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;-4 \cdot \left(\left(t_0 \cdot t_0\right) \cdot \left(b \cdot b\right)\right)\\ \end{array} \]

Alternative 4
Accuracy	80.7%
Cost	1485

\[\begin{array}{l} \mathbf{if}\;y-scale \leq -3.1 \cdot 10^{-196} \lor \neg \left(y-scale \leq 3.3 \cdot 10^{-241}\right) \land y-scale \leq 3.4 \cdot 10^{+136}:\\ \;\;\;\;-4 \cdot \left(b \cdot \left(\left(\frac{a}{x-scale} \cdot \frac{b}{y-scale}\right) \cdot \frac{a}{x-scale \cdot y-scale}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;-4 \cdot \left(\frac{b}{y-scale} \cdot \left(\frac{b}{y-scale} \cdot \left(\frac{a}{x-scale} \cdot \frac{a}{x-scale}\right)\right)\right)\\ \end{array} \]

Alternative 5
Accuracy	89.0%
Cost	1352

\[\begin{array}{l} t_0 := \frac{a}{x-scale \cdot \frac{y-scale}{b}}\\ \mathbf{if}\;y-scale \leq 1.45 \cdot 10^{-147}:\\ \;\;\;\;-4 \cdot \left(t_0 \cdot t_0\right)\\ \mathbf{elif}\;y-scale \leq 8.4 \cdot 10^{+183}:\\ \;\;\;\;-4 \cdot \frac{b \cdot \frac{\frac{a}{x-scale}}{y-scale}}{\frac{y-scale}{b} \cdot \frac{x-scale}{a}}\\ \mathbf{else}:\\ \;\;\;\;-4 \cdot \left(b \cdot \frac{\frac{a \cdot \left(a \cdot \frac{b}{x-scale \cdot y-scale}\right)}{y-scale}}{x-scale}\right)\\ \end{array} \]

Alternative 6
Accuracy	89.4%
Cost	1352

\[\begin{array}{l} t_0 := \frac{a}{x-scale \cdot \frac{y-scale}{b}}\\ \mathbf{if}\;y-scale \leq 4.8 \cdot 10^{-147}:\\ \;\;\;\;-4 \cdot \left(t_0 \cdot t_0\right)\\ \mathbf{elif}\;y-scale \leq 4.8 \cdot 10^{+183}:\\ \;\;\;\;-4 \cdot \frac{b \cdot \frac{\frac{a}{x-scale}}{y-scale}}{\frac{y-scale}{b} \cdot \frac{x-scale}{a}}\\ \mathbf{else}:\\ \;\;\;\;-4 \cdot \frac{b \cdot \left(a \cdot \left(a \cdot \frac{b}{x-scale \cdot y-scale}\right)\right)}{x-scale \cdot y-scale}\\ \end{array} \]

Alternative 7
Accuracy	90.0%
Cost	1220

\[\begin{array}{l} \mathbf{if}\;y-scale \leq 3 \cdot 10^{+183}:\\ \;\;\;\;-4 \cdot \frac{\frac{a}{x-scale} \cdot \frac{b}{y-scale}}{\frac{y-scale}{b} \cdot \frac{x-scale}{a}}\\ \mathbf{else}:\\ \;\;\;\;-4 \cdot \frac{b \cdot \left(a \cdot \left(a \cdot \frac{b}{x-scale \cdot y-scale}\right)\right)}{x-scale \cdot y-scale}\\ \end{array} \]

Alternative 8
Accuracy	90.2%
Cost	1220

\[\begin{array}{l} \mathbf{if}\;y-scale \leq 8.5 \cdot 10^{+183}:\\ \;\;\;\;-4 \cdot \frac{\frac{a}{x-scale} \cdot \frac{b}{y-scale}}{\frac{y-scale}{b} \cdot \frac{x-scale}{a}}\\ \mathbf{else}:\\ \;\;\;\;\left(a \cdot b\right) \cdot \frac{\frac{-4}{\frac{x-scale \cdot y-scale}{a \cdot b}}}{x-scale \cdot y-scale}\\ \end{array} \]

Alternative 9
Accuracy	76.6%
Cost	1088

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

Alternative 10
Accuracy	90.7%
Cost	1088

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

Alternative 11
Accuracy	53.1%
Cost	64

\[0 \]

Reproduce

herbie shell --seed 2023137 
(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))))