Simplification of discriminant from scale-rotated-ellipse

Average Accuracy: 23.6% → 60.3%

Time: 55.9s

Precision: binary64

Cost: 7172

Math

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

↓

\[\begin{array}{l} t_0 := \frac{\frac{b \cdot a}{y-scale}}{x-scale}\\ \mathbf{if}\;x-scale \leq -1720:\\ \;\;\;\;-4 \cdot {\left(\frac{b}{y-scale} \cdot \frac{a}{x-scale}\right)}^{2}\\ \mathbf{elif}\;x-scale \leq 2.6 \cdot 10^{+180}:\\ \;\;\;\;-4 \cdot \left(t_0 \cdot t_0\right)\\ \mathbf{else}:\\ \;\;\;\;-4 \cdot \left(\left(\frac{b}{x-scale} \cdot \frac{a}{y-scale}\right) \cdot \left(a \cdot \frac{\frac{b}{x-scale}}{y-scale}\right)\right)\\ \end{array} \]

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) y-scale) x-scale)))
   (if (<= x-scale -1720.0)
     (* -4.0 (pow (* (/ b y-scale) (/ a x-scale)) 2.0))
     (if (<= x-scale 2.6e+180)
       (* -4.0 (* t_0 t_0))
       (*
        -4.0
        (*
         (* (/ b x-scale) (/ a y-scale))
         (* a (/ (/ b x-scale) y-scale))))))))

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) / y_45_scale) / x_45_scale;
	double tmp;
	if (x_45_scale <= -1720.0) {
		tmp = -4.0 * pow(((b / y_45_scale) * (a / x_45_scale)), 2.0);
	} else if (x_45_scale <= 2.6e+180) {
		tmp = -4.0 * (t_0 * t_0);
	} else {
		tmp = -4.0 * (((b / x_45_scale) * (a / y_45_scale)) * (a * ((b / x_45_scale) / y_45_scale)));
	}
	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) / y_45_scale) / x_45_scale;
	double tmp;
	if (x_45_scale <= -1720.0) {
		tmp = -4.0 * Math.pow(((b / y_45_scale) * (a / x_45_scale)), 2.0);
	} else if (x_45_scale <= 2.6e+180) {
		tmp = -4.0 * (t_0 * t_0);
	} else {
		tmp = -4.0 * (((b / x_45_scale) * (a / y_45_scale)) * (a * ((b / x_45_scale) / y_45_scale)));
	}
	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) / y_45_scale) / x_45_scale
	tmp = 0
	if x_45_scale <= -1720.0:
		tmp = -4.0 * math.pow(((b / y_45_scale) * (a / x_45_scale)), 2.0)
	elif x_45_scale <= 2.6e+180:
		tmp = -4.0 * (t_0 * t_0)
	else:
		tmp = -4.0 * (((b / x_45_scale) * (a / y_45_scale)) * (a * ((b / x_45_scale) / y_45_scale)))
	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(b * a) / y_45_scale) / x_45_scale)
	tmp = 0.0
	if (x_45_scale <= -1720.0)
		tmp = Float64(-4.0 * (Float64(Float64(b / y_45_scale) * Float64(a / x_45_scale)) ^ 2.0));
	elseif (x_45_scale <= 2.6e+180)
		tmp = Float64(-4.0 * Float64(t_0 * t_0));
	else
		tmp = Float64(-4.0 * Float64(Float64(Float64(b / x_45_scale) * Float64(a / y_45_scale)) * Float64(a * Float64(Float64(b / x_45_scale) / y_45_scale))));
	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) / y_45_scale) / x_45_scale;
	tmp = 0.0;
	if (x_45_scale <= -1720.0)
		tmp = -4.0 * (((b / y_45_scale) * (a / x_45_scale)) ^ 2.0);
	elseif (x_45_scale <= 2.6e+180)
		tmp = -4.0 * (t_0 * t_0);
	else
		tmp = -4.0 * (((b / x_45_scale) * (a / y_45_scale)) * (a * ((b / x_45_scale) / y_45_scale)));
	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[(b * a), $MachinePrecision] / y$45$scale), $MachinePrecision] / x$45$scale), $MachinePrecision]}, If[LessEqual[x$45$scale, -1720.0], 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[x$45$scale, 2.6e+180], N[(-4.0 * N[(t$95$0 * t$95$0), $MachinePrecision]), $MachinePrecision], N[(-4.0 * N[(N[(N[(b / x$45$scale), $MachinePrecision] * N[(a / y$45$scale), $MachinePrecision]), $MachinePrecision] * N[(a * N[(N[(b / x$45$scale), $MachinePrecision] / y$45$scale), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if x-scale < -1720

   1. Initial program 35.0%

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

   2. Simplified 30.5%

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

      [Start] 35.0

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

      fma-neg [=>] 33.9

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

   3. Taylor expanded in angle around 0 38.4%

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

   4. Simplified 49.0%

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

      [Start] 38.4	\[ -4 \cdot \frac{{a}^{2} \cdot {b}^{2}}{{y-scale}^{2} \cdot {x-scale}^{2}} \]
      *-commutative [=>] 38.4	\[ -4 \cdot \frac{{a}^{2} \cdot {b}^{2}}{\color{blue}{{x-scale}^{2} \cdot {y-scale}^{2}}} \]
      times-frac [=>] 38.4	\[ -4 \cdot \color{blue}{\left(\frac{{a}^{2}}{{x-scale}^{2}} \cdot \frac{{b}^{2}}{{y-scale}^{2}}\right)} \]
      unpow2 [=>] 38.4	\[ -4 \cdot \left(\frac{\color{blue}{a \cdot a}}{{x-scale}^{2}} \cdot \frac{{b}^{2}}{{y-scale}^{2}}\right) \]
      unpow2 [=>] 38.4	\[ -4 \cdot \left(\frac{a \cdot a}{\color{blue}{x-scale \cdot x-scale}} \cdot \frac{{b}^{2}}{{y-scale}^{2}}\right) \]
      unpow2 [=>] 38.4	\[ -4 \cdot \left(\frac{a \cdot a}{x-scale \cdot x-scale} \cdot \frac{\color{blue}{b \cdot b}}{{y-scale}^{2}}\right) \]
      unpow2 [=>] 38.4	\[ -4 \cdot \left(\frac{a \cdot a}{x-scale \cdot x-scale} \cdot \frac{b \cdot b}{\color{blue}{y-scale \cdot y-scale}}\right) \]
      times-frac [=>] 49.0	\[ -4 \cdot \left(\frac{a \cdot a}{x-scale \cdot x-scale} \cdot \color{blue}{\left(\frac{b}{y-scale} \cdot \frac{b}{y-scale}\right)}\right) \]

   5. Applied egg-rr 77.3%

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

      [Start] 49.0	\[ -4 \cdot \left(\frac{a \cdot a}{x-scale \cdot x-scale} \cdot \left(\frac{b}{y-scale} \cdot \frac{b}{y-scale}\right)\right) \]
      add-sqr-sqrt [=>] 49.0	\[ -4 \cdot \color{blue}{\left(\sqrt{\frac{a \cdot a}{x-scale \cdot x-scale} \cdot \left(\frac{b}{y-scale} \cdot \frac{b}{y-scale}\right)} \cdot \sqrt{\frac{a \cdot a}{x-scale \cdot x-scale} \cdot \left(\frac{b}{y-scale} \cdot \frac{b}{y-scale}\right)}\right)} \]
      pow2 [=>] 49.0	\[ -4 \cdot \color{blue}{{\left(\sqrt{\frac{a \cdot a}{x-scale \cdot x-scale} \cdot \left(\frac{b}{y-scale} \cdot \frac{b}{y-scale}\right)}\right)}^{2}} \]
      *-commutative [=>] 49.0	\[ -4 \cdot {\left(\sqrt{\color{blue}{\left(\frac{b}{y-scale} \cdot \frac{b}{y-scale}\right) \cdot \frac{a \cdot a}{x-scale \cdot x-scale}}}\right)}^{2} \]
      sqrt-prod [=>] 49.0	\[ -4 \cdot {\color{blue}{\left(\sqrt{\frac{b}{y-scale} \cdot \frac{b}{y-scale}} \cdot \sqrt{\frac{a \cdot a}{x-scale \cdot x-scale}}\right)}}^{2} \]
      sqrt-prod [=>] 31.5	\[ -4 \cdot {\left(\color{blue}{\left(\sqrt{\frac{b}{y-scale}} \cdot \sqrt{\frac{b}{y-scale}}\right)} \cdot \sqrt{\frac{a \cdot a}{x-scale \cdot x-scale}}\right)}^{2} \]
      add-sqr-sqrt [<=] 52.8	\[ -4 \cdot {\left(\color{blue}{\frac{b}{y-scale}} \cdot \sqrt{\frac{a \cdot a}{x-scale \cdot x-scale}}\right)}^{2} \]
      times-frac [=>] 68.6	\[ -4 \cdot {\left(\frac{b}{y-scale} \cdot \sqrt{\color{blue}{\frac{a}{x-scale} \cdot \frac{a}{x-scale}}}\right)}^{2} \]
      sqrt-prod [=>] 49.6	\[ -4 \cdot {\left(\frac{b}{y-scale} \cdot \color{blue}{\left(\sqrt{\frac{a}{x-scale}} \cdot \sqrt{\frac{a}{x-scale}}\right)}\right)}^{2} \]
      add-sqr-sqrt [<=] 77.3	\[ -4 \cdot {\left(\frac{b}{y-scale} \cdot \color{blue}{\frac{a}{x-scale}}\right)}^{2} \]

   if -1720 < x-scale < 2.60000000000000021e180

   1. Initial program 17.1%

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

   2. Simplified 12.4%

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

      [Start] 17.1

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

      fma-neg [=>] 16.5

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

   3. Taylor expanded in angle around 0 18.3%

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

   4. Simplified 25.1%

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

      [Start] 18.3	\[ -4 \cdot \frac{{a}^{2} \cdot {b}^{2}}{{y-scale}^{2} \cdot {x-scale}^{2}} \]
      *-commutative [=>] 18.3	\[ -4 \cdot \frac{{a}^{2} \cdot {b}^{2}}{\color{blue}{{x-scale}^{2} \cdot {y-scale}^{2}}} \]
      times-frac [=>] 18.6	\[ -4 \cdot \color{blue}{\left(\frac{{a}^{2}}{{x-scale}^{2}} \cdot \frac{{b}^{2}}{{y-scale}^{2}}\right)} \]
      unpow2 [=>] 18.6	\[ -4 \cdot \left(\frac{\color{blue}{a \cdot a}}{{x-scale}^{2}} \cdot \frac{{b}^{2}}{{y-scale}^{2}}\right) \]
      unpow2 [=>] 18.6	\[ -4 \cdot \left(\frac{a \cdot a}{\color{blue}{x-scale \cdot x-scale}} \cdot \frac{{b}^{2}}{{y-scale}^{2}}\right) \]
      unpow2 [=>] 18.6	\[ -4 \cdot \left(\frac{a \cdot a}{x-scale \cdot x-scale} \cdot \frac{\color{blue}{b \cdot b}}{{y-scale}^{2}}\right) \]
      unpow2 [=>] 18.6	\[ -4 \cdot \left(\frac{a \cdot a}{x-scale \cdot x-scale} \cdot \frac{b \cdot b}{\color{blue}{y-scale \cdot y-scale}}\right) \]
      times-frac [=>] 25.1	\[ -4 \cdot \left(\frac{a \cdot a}{x-scale \cdot x-scale} \cdot \color{blue}{\left(\frac{b}{y-scale} \cdot \frac{b}{y-scale}\right)}\right) \]

   5. Applied egg-rr 50.4%

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

      [Start] 25.1	\[ -4 \cdot \left(\frac{a \cdot a}{x-scale \cdot x-scale} \cdot \left(\frac{b}{y-scale} \cdot \frac{b}{y-scale}\right)\right) \]
      add-sqr-sqrt [=>] 25.1	\[ -4 \cdot \color{blue}{\left(\sqrt{\frac{a \cdot a}{x-scale \cdot x-scale} \cdot \left(\frac{b}{y-scale} \cdot \frac{b}{y-scale}\right)} \cdot \sqrt{\frac{a \cdot a}{x-scale \cdot x-scale} \cdot \left(\frac{b}{y-scale} \cdot \frac{b}{y-scale}\right)}\right)} \]
      pow2 [=>] 25.1	\[ -4 \cdot \color{blue}{{\left(\sqrt{\frac{a \cdot a}{x-scale \cdot x-scale} \cdot \left(\frac{b}{y-scale} \cdot \frac{b}{y-scale}\right)}\right)}^{2}} \]
      *-commutative [=>] 25.1	\[ -4 \cdot {\left(\sqrt{\color{blue}{\left(\frac{b}{y-scale} \cdot \frac{b}{y-scale}\right) \cdot \frac{a \cdot a}{x-scale \cdot x-scale}}}\right)}^{2} \]
      sqrt-prod [=>] 25.0	\[ -4 \cdot {\color{blue}{\left(\sqrt{\frac{b}{y-scale} \cdot \frac{b}{y-scale}} \cdot \sqrt{\frac{a \cdot a}{x-scale \cdot x-scale}}\right)}}^{2} \]
      sqrt-prod [=>] 16.5	\[ -4 \cdot {\left(\color{blue}{\left(\sqrt{\frac{b}{y-scale}} \cdot \sqrt{\frac{b}{y-scale}}\right)} \cdot \sqrt{\frac{a \cdot a}{x-scale \cdot x-scale}}\right)}^{2} \]
      add-sqr-sqrt [<=] 26.9	\[ -4 \cdot {\left(\color{blue}{\frac{b}{y-scale}} \cdot \sqrt{\frac{a \cdot a}{x-scale \cdot x-scale}}\right)}^{2} \]
      times-frac [=>] 40.3	\[ -4 \cdot {\left(\frac{b}{y-scale} \cdot \sqrt{\color{blue}{\frac{a}{x-scale} \cdot \frac{a}{x-scale}}}\right)}^{2} \]
      sqrt-prod [=>] 26.5	\[ -4 \cdot {\left(\frac{b}{y-scale} \cdot \color{blue}{\left(\sqrt{\frac{a}{x-scale}} \cdot \sqrt{\frac{a}{x-scale}}\right)}\right)}^{2} \]
      add-sqr-sqrt [<=] 50.4	\[ -4 \cdot {\left(\frac{b}{y-scale} \cdot \color{blue}{\frac{a}{x-scale}}\right)}^{2} \]

   6. Applied egg-rr 50.7%

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

      [Start] 50.4	\[ -4 \cdot {\left(\frac{b}{y-scale} \cdot \frac{a}{x-scale}\right)}^{2} \]
      unpow2 [=>] 50.4	\[ -4 \cdot \color{blue}{\left(\left(\frac{b}{y-scale} \cdot \frac{a}{x-scale}\right) \cdot \left(\frac{b}{y-scale} \cdot \frac{a}{x-scale}\right)\right)} \]
      associate-*r/ [=>] 49.1	\[ -4 \cdot \left(\color{blue}{\frac{\frac{b}{y-scale} \cdot a}{x-scale}} \cdot \left(\frac{b}{y-scale} \cdot \frac{a}{x-scale}\right)\right) \]
      associate-*r/ [=>] 50.9	\[ -4 \cdot \left(\frac{\frac{b}{y-scale} \cdot a}{x-scale} \cdot \color{blue}{\frac{\frac{b}{y-scale} \cdot a}{x-scale}}\right) \]
      associate-*l/ [=>] 47.8	\[ -4 \cdot \left(\frac{\color{blue}{\frac{b \cdot a}{y-scale}}}{x-scale} \cdot \frac{\frac{b}{y-scale} \cdot a}{x-scale}\right) \]
      associate-*l/ [=>] 50.7	\[ -4 \cdot \left(\frac{\frac{b \cdot a}{y-scale}}{x-scale} \cdot \frac{\color{blue}{\frac{b \cdot a}{y-scale}}}{x-scale}\right) \]

   if 2.60000000000000021e180 < x-scale

   1. Initial program 37.3%

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

   2. Simplified 33.2%

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

      [Start] 37.3

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

   3. Taylor expanded in angle around 0 36.5%

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

   4. Simplified 69.5%

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

      [Start] 36.5	\[ -4 \cdot \frac{{a}^{2} \cdot {b}^{2}}{{y-scale}^{2} \cdot {x-scale}^{2}} \]
      times-frac [=>] 36.3	\[ -4 \cdot \color{blue}{\left(\frac{{a}^{2}}{{y-scale}^{2}} \cdot \frac{{b}^{2}}{{x-scale}^{2}}\right)} \]
      *-commutative [=>] 36.3	\[ -4 \cdot \color{blue}{\left(\frac{{b}^{2}}{{x-scale}^{2}} \cdot \frac{{a}^{2}}{{y-scale}^{2}}\right)} \]
      unpow2 [=>] 36.3	\[ -4 \cdot \left(\frac{\color{blue}{b \cdot b}}{{x-scale}^{2}} \cdot \frac{{a}^{2}}{{y-scale}^{2}}\right) \]
      unpow2 [=>] 36.3	\[ -4 \cdot \left(\frac{b \cdot b}{\color{blue}{x-scale \cdot x-scale}} \cdot \frac{{a}^{2}}{{y-scale}^{2}}\right) \]
      times-frac [=>] 50.0	\[ -4 \cdot \left(\color{blue}{\left(\frac{b}{x-scale} \cdot \frac{b}{x-scale}\right)} \cdot \frac{{a}^{2}}{{y-scale}^{2}}\right) \]
      unpow2 [=>] 50.0	\[ -4 \cdot \left(\left(\frac{b}{x-scale} \cdot \frac{b}{x-scale}\right) \cdot \frac{\color{blue}{a \cdot a}}{{y-scale}^{2}}\right) \]
      unpow2 [=>] 50.0	\[ -4 \cdot \left(\left(\frac{b}{x-scale} \cdot \frac{b}{x-scale}\right) \cdot \frac{a \cdot a}{\color{blue}{y-scale \cdot y-scale}}\right) \]
      times-frac [=>] 69.5	\[ -4 \cdot \left(\left(\frac{b}{x-scale} \cdot \frac{b}{x-scale}\right) \cdot \color{blue}{\left(\frac{a}{y-scale} \cdot \frac{a}{y-scale}\right)}\right) \]

   5. Applied egg-rr 65.8%

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

      [Start] 69.5	\[ -4 \cdot \left(\left(\frac{b}{x-scale} \cdot \frac{b}{x-scale}\right) \cdot \left(\frac{a}{y-scale} \cdot \frac{a}{y-scale}\right)\right) \]
      associate-*l/ [=>] 66.5	\[ -4 \cdot \left(\color{blue}{\frac{b \cdot \frac{b}{x-scale}}{x-scale}} \cdot \left(\frac{a}{y-scale} \cdot \frac{a}{y-scale}\right)\right) \]
      associate-*l/ [=>] 63.6	\[ -4 \cdot \left(\frac{b \cdot \frac{b}{x-scale}}{x-scale} \cdot \color{blue}{\frac{a \cdot \frac{a}{y-scale}}{y-scale}}\right) \]
      frac-times [=>] 65.8	\[ -4 \cdot \color{blue}{\frac{\left(b \cdot \frac{b}{x-scale}\right) \cdot \left(a \cdot \frac{a}{y-scale}\right)}{x-scale \cdot y-scale}} \]

   6. Applied egg-rr 81.6%

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

      [Start] 65.8	\[ -4 \cdot \frac{\left(b \cdot \frac{b}{x-scale}\right) \cdot \left(a \cdot \frac{a}{y-scale}\right)}{x-scale \cdot y-scale} \]
      add-sqr-sqrt [=>] 65.8	\[ -4 \cdot \color{blue}{\left(\sqrt{\frac{\left(b \cdot \frac{b}{x-scale}\right) \cdot \left(a \cdot \frac{a}{y-scale}\right)}{x-scale \cdot y-scale}} \cdot \sqrt{\frac{\left(b \cdot \frac{b}{x-scale}\right) \cdot \left(a \cdot \frac{a}{y-scale}\right)}{x-scale \cdot y-scale}}\right)} \]
      times-frac [=>] 60.4	\[ -4 \cdot \left(\sqrt{\color{blue}{\frac{b \cdot \frac{b}{x-scale}}{x-scale} \cdot \frac{a \cdot \frac{a}{y-scale}}{y-scale}}} \cdot \sqrt{\frac{\left(b \cdot \frac{b}{x-scale}\right) \cdot \left(a \cdot \frac{a}{y-scale}\right)}{x-scale \cdot y-scale}}\right) \]
      sqrt-prod [=>] 60.4	\[ -4 \cdot \left(\color{blue}{\left(\sqrt{\frac{b \cdot \frac{b}{x-scale}}{x-scale}} \cdot \sqrt{\frac{a \cdot \frac{a}{y-scale}}{y-scale}}\right)} \cdot \sqrt{\frac{\left(b \cdot \frac{b}{x-scale}\right) \cdot \left(a \cdot \frac{a}{y-scale}\right)}{x-scale \cdot y-scale}}\right) \]
      associate-*l/ [<=] 60.4	\[ -4 \cdot \left(\left(\sqrt{\color{blue}{\frac{b}{x-scale} \cdot \frac{b}{x-scale}}} \cdot \sqrt{\frac{a \cdot \frac{a}{y-scale}}{y-scale}}\right) \cdot \sqrt{\frac{\left(b \cdot \frac{b}{x-scale}\right) \cdot \left(a \cdot \frac{a}{y-scale}\right)}{x-scale \cdot y-scale}}\right) \]
      sqrt-unprod [<=] 43.7	\[ -4 \cdot \left(\left(\color{blue}{\left(\sqrt{\frac{b}{x-scale}} \cdot \sqrt{\frac{b}{x-scale}}\right)} \cdot \sqrt{\frac{a \cdot \frac{a}{y-scale}}{y-scale}}\right) \cdot \sqrt{\frac{\left(b \cdot \frac{b}{x-scale}\right) \cdot \left(a \cdot \frac{a}{y-scale}\right)}{x-scale \cdot y-scale}}\right) \]
      add-sqr-sqrt [<=] 57.7	\[ -4 \cdot \left(\left(\color{blue}{\frac{b}{x-scale}} \cdot \sqrt{\frac{a \cdot \frac{a}{y-scale}}{y-scale}}\right) \cdot \sqrt{\frac{\left(b \cdot \frac{b}{x-scale}\right) \cdot \left(a \cdot \frac{a}{y-scale}\right)}{x-scale \cdot y-scale}}\right) \]
      associate-*l/ [<=] 57.7	\[ -4 \cdot \left(\left(\frac{b}{x-scale} \cdot \sqrt{\color{blue}{\frac{a}{y-scale} \cdot \frac{a}{y-scale}}}\right) \cdot \sqrt{\frac{\left(b \cdot \frac{b}{x-scale}\right) \cdot \left(a \cdot \frac{a}{y-scale}\right)}{x-scale \cdot y-scale}}\right) \]
      sqrt-unprod [<=] 37.9	\[ -4 \cdot \left(\left(\frac{b}{x-scale} \cdot \color{blue}{\left(\sqrt{\frac{a}{y-scale}} \cdot \sqrt{\frac{a}{y-scale}}\right)}\right) \cdot \sqrt{\frac{\left(b \cdot \frac{b}{x-scale}\right) \cdot \left(a \cdot \frac{a}{y-scale}\right)}{x-scale \cdot y-scale}}\right) \]
      add-sqr-sqrt [<=] 61.0	\[ -4 \cdot \left(\left(\frac{b}{x-scale} \cdot \color{blue}{\frac{a}{y-scale}}\right) \cdot \sqrt{\frac{\left(b \cdot \frac{b}{x-scale}\right) \cdot \left(a \cdot \frac{a}{y-scale}\right)}{x-scale \cdot y-scale}}\right) \]
      times-frac [=>] 59.1	\[ -4 \cdot \left(\left(\frac{b}{x-scale} \cdot \frac{a}{y-scale}\right) \cdot \sqrt{\color{blue}{\frac{b \cdot \frac{b}{x-scale}}{x-scale} \cdot \frac{a \cdot \frac{a}{y-scale}}{y-scale}}}\right) \]
      sqrt-prod [=>] 59.1	\[ -4 \cdot \left(\left(\frac{b}{x-scale} \cdot \frac{a}{y-scale}\right) \cdot \color{blue}{\left(\sqrt{\frac{b \cdot \frac{b}{x-scale}}{x-scale}} \cdot \sqrt{\frac{a \cdot \frac{a}{y-scale}}{y-scale}}\right)}\right) \]
      associate-*l/ [<=] 60.9	\[ -4 \cdot \left(\left(\frac{b}{x-scale} \cdot \frac{a}{y-scale}\right) \cdot \left(\sqrt{\color{blue}{\frac{b}{x-scale} \cdot \frac{b}{x-scale}}} \cdot \sqrt{\frac{a \cdot \frac{a}{y-scale}}{y-scale}}\right)\right) \]
      sqrt-unprod [<=] 43.8	\[ -4 \cdot \left(\left(\frac{b}{x-scale} \cdot \frac{a}{y-scale}\right) \cdot \left(\color{blue}{\left(\sqrt{\frac{b}{x-scale}} \cdot \sqrt{\frac{b}{x-scale}}\right)} \cdot \sqrt{\frac{a \cdot \frac{a}{y-scale}}{y-scale}}\right)\right) \]
      add-sqr-sqrt [<=] 61.3	\[ -4 \cdot \left(\left(\frac{b}{x-scale} \cdot \frac{a}{y-scale}\right) \cdot \left(\color{blue}{\frac{b}{x-scale}} \cdot \sqrt{\frac{a \cdot \frac{a}{y-scale}}{y-scale}}\right)\right) \]
      associate-*l/ [<=] 63.9	\[ -4 \cdot \left(\left(\frac{b}{x-scale} \cdot \frac{a}{y-scale}\right) \cdot \left(\frac{b}{x-scale} \cdot \sqrt{\color{blue}{\frac{a}{y-scale} \cdot \frac{a}{y-scale}}}\right)\right) \]

   7. Taylor expanded in b around 0 74.5%

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

   8. Simplified 80.5%

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

      [Start] 74.5	\[ -4 \cdot \left(\left(\frac{b}{x-scale} \cdot \frac{a}{y-scale}\right) \cdot \frac{a \cdot b}{y-scale \cdot x-scale}\right) \]
      *-commutative [=>] 74.5	\[ -4 \cdot \left(\left(\frac{b}{x-scale} \cdot \frac{a}{y-scale}\right) \cdot \frac{a \cdot b}{\color{blue}{x-scale \cdot y-scale}}\right) \]
      *-commutative [<=] 74.5	\[ -4 \cdot \left(\left(\frac{b}{x-scale} \cdot \frac{a}{y-scale}\right) \cdot \frac{\color{blue}{b \cdot a}}{x-scale \cdot y-scale}\right) \]
      associate-*l/ [<=] 76.1	\[ -4 \cdot \left(\left(\frac{b}{x-scale} \cdot \frac{a}{y-scale}\right) \cdot \color{blue}{\left(\frac{b}{x-scale \cdot y-scale} \cdot a\right)}\right) \]
      *-commutative [=>] 76.1	\[ -4 \cdot \left(\left(\frac{b}{x-scale} \cdot \frac{a}{y-scale}\right) \cdot \color{blue}{\left(a \cdot \frac{b}{x-scale \cdot y-scale}\right)}\right) \]
      associate-/r* [=>] 80.5	\[ -4 \cdot \left(\left(\frac{b}{x-scale} \cdot \frac{a}{y-scale}\right) \cdot \left(a \cdot \color{blue}{\frac{\frac{b}{x-scale}}{y-scale}}\right)\right) \]

3. Recombined 3 regimes into one program.

4. Final simplification 60.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x-scale \leq -1720:\\ \;\;\;\;-4 \cdot {\left(\frac{b}{y-scale} \cdot \frac{a}{x-scale}\right)}^{2}\\ \mathbf{elif}\;x-scale \leq 2.6 \cdot 10^{+180}:\\ \;\;\;\;-4 \cdot \left(\frac{\frac{b \cdot a}{y-scale}}{x-scale} \cdot \frac{\frac{b \cdot a}{y-scale}}{x-scale}\right)\\ \mathbf{else}:\\ \;\;\;\;-4 \cdot \left(\left(\frac{b}{x-scale} \cdot \frac{a}{y-scale}\right) \cdot \left(a \cdot \frac{\frac{b}{x-scale}}{y-scale}\right)\right)\\ \end{array} \]

Alternatives

Alternative 1
Accuracy	38.8%
Cost	1353

\[\begin{array}{l} \mathbf{if}\;b \leq -4.5 \cdot 10^{-20} \lor \neg \left(b \leq 9 \cdot 10^{-183}\right):\\ \;\;\;\;-4 \cdot \left(\left(\frac{a}{x-scale} \cdot \frac{a}{x-scale}\right) \cdot \frac{b}{\frac{y-scale \cdot y-scale}{b}}\right)\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \]

Alternative 2
Accuracy	58.1%
Cost	1352

\[\begin{array}{l} t_0 := \frac{b}{x-scale} \cdot \frac{a}{y-scale}\\ \mathbf{if}\;a \leq -3 \cdot 10^{-98}:\\ \;\;\;\;-4 \cdot \left(t_0 \cdot \left(a \cdot \frac{\frac{b}{x-scale}}{y-scale}\right)\right)\\ \mathbf{elif}\;a \leq -9.2 \cdot 10^{-241}:\\ \;\;\;\;-4 \cdot \left(\left(\frac{b}{y-scale} \cdot \frac{b}{y-scale}\right) \cdot \left(\frac{a}{x-scale} \cdot \frac{a}{x-scale}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;-4 \cdot \left(t_0 \cdot t_0\right)\\ \end{array} \]

Alternative 3
Accuracy	60.1%
Cost	1352

\[\begin{array}{l} t_0 := \frac{\frac{b \cdot a}{y-scale}}{x-scale}\\ t_1 := \frac{b}{x-scale} \cdot \frac{a}{y-scale}\\ \mathbf{if}\;x-scale \leq -2:\\ \;\;\;\;-4 \cdot \left(t_1 \cdot t_1\right)\\ \mathbf{elif}\;x-scale \leq 2.8 \cdot 10^{+180}:\\ \;\;\;\;-4 \cdot \left(t_0 \cdot t_0\right)\\ \mathbf{else}:\\ \;\;\;\;-4 \cdot \left(t_1 \cdot \left(a \cdot \frac{\frac{b}{x-scale}}{y-scale}\right)\right)\\ \end{array} \]

Alternative 4
Accuracy	58.1%
Cost	1088

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

Alternative 5
Accuracy	59.7%
Cost	1088

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

Alternative 6
Accuracy	35.1%
Cost	64

\[0 \]

Reproduce

herbie shell --seed 2023153 
(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))))