Simplification of discriminant from scale-rotated-ellipse

Average Error: 41.1 → 7.7

Time: 1.3min

Precision: binary64

Cost: 7304

Math

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

↓

\[\begin{array}{l} t_0 := \frac{y-scale \cdot x-scale}{b \cdot a}\\ \mathbf{if}\;a \leq 3 \cdot 10^{-271}:\\ \;\;\;\;-4 \cdot \frac{\frac{b}{x-scale}}{\frac{\frac{x-scale}{b}}{\frac{a}{y-scale}} \cdot \frac{y-scale}{a}}\\ \mathbf{elif}\;a \leq 9.5 \cdot 10^{-77}:\\ \;\;\;\;-4 \cdot {\left(\frac{b}{y-scale} \cdot \frac{a}{x-scale}\right)}^{2}\\ \mathbf{elif}\;a \leq 1.75 \cdot 10^{+81}:\\ \;\;\;\;-4 \cdot \left(\frac{b}{x-scale} \cdot \frac{a \cdot \left(\frac{a}{y-scale} \cdot \frac{b}{x-scale}\right)}{y-scale}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{-4}{t_0 \cdot t_0}\\ \end{array} \]

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 (/ (* y-scale x-scale) (* b a))))
   (if (<= a 3e-271)
     (*
      -4.0
      (/ (/ b x-scale) (* (/ (/ x-scale b) (/ a y-scale)) (/ y-scale a))))
     (if (<= a 9.5e-77)
       (* -4.0 (pow (* (/ b y-scale) (/ a x-scale)) 2.0))
       (if (<= a 1.75e+81)
         (*
          -4.0
          (* (/ b x-scale) (/ (* a (* (/ a y-scale) (/ b x-scale))) y-scale)))
         (/ -4.0 (* t_0 t_0)))))))

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 = (y_45_scale * x_45_scale) / (b * a);
	double tmp;
	if (a <= 3e-271) {
		tmp = -4.0 * ((b / x_45_scale) / (((x_45_scale / b) / (a / y_45_scale)) * (y_45_scale / a)));
	} else if (a <= 9.5e-77) {
		tmp = -4.0 * pow(((b / y_45_scale) * (a / x_45_scale)), 2.0);
	} else if (a <= 1.75e+81) {
		tmp = -4.0 * ((b / x_45_scale) * ((a * ((a / y_45_scale) * (b / x_45_scale))) / y_45_scale));
	} else {
		tmp = -4.0 / (t_0 * t_0);
	}
	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 = (y_45_scale * x_45_scale) / (b * a);
	double tmp;
	if (a <= 3e-271) {
		tmp = -4.0 * ((b / x_45_scale) / (((x_45_scale / b) / (a / y_45_scale)) * (y_45_scale / a)));
	} else if (a <= 9.5e-77) {
		tmp = -4.0 * Math.pow(((b / y_45_scale) * (a / x_45_scale)), 2.0);
	} else if (a <= 1.75e+81) {
		tmp = -4.0 * ((b / x_45_scale) * ((a * ((a / y_45_scale) * (b / x_45_scale))) / y_45_scale));
	} else {
		tmp = -4.0 / (t_0 * t_0);
	}
	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 = (y_45_scale * x_45_scale) / (b * a)
	tmp = 0
	if a <= 3e-271:
		tmp = -4.0 * ((b / x_45_scale) / (((x_45_scale / b) / (a / y_45_scale)) * (y_45_scale / a)))
	elif a <= 9.5e-77:
		tmp = -4.0 * math.pow(((b / y_45_scale) * (a / x_45_scale)), 2.0)
	elif a <= 1.75e+81:
		tmp = -4.0 * ((b / x_45_scale) * ((a * ((a / y_45_scale) * (b / x_45_scale))) / y_45_scale))
	else:
		tmp = -4.0 / (t_0 * t_0)
	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(y_45_scale * x_45_scale) / Float64(b * a))
	tmp = 0.0
	if (a <= 3e-271)
		tmp = Float64(-4.0 * Float64(Float64(b / x_45_scale) / Float64(Float64(Float64(x_45_scale / b) / Float64(a / y_45_scale)) * Float64(y_45_scale / a))));
	elseif (a <= 9.5e-77)
		tmp = Float64(-4.0 * (Float64(Float64(b / y_45_scale) * Float64(a / x_45_scale)) ^ 2.0));
	elseif (a <= 1.75e+81)
		tmp = Float64(-4.0 * Float64(Float64(b / x_45_scale) * Float64(Float64(a * Float64(Float64(a / y_45_scale) * Float64(b / x_45_scale))) / y_45_scale)));
	else
		tmp = Float64(-4.0 / Float64(t_0 * t_0));
	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 = (y_45_scale * x_45_scale) / (b * a);
	tmp = 0.0;
	if (a <= 3e-271)
		tmp = -4.0 * ((b / x_45_scale) / (((x_45_scale / b) / (a / y_45_scale)) * (y_45_scale / a)));
	elseif (a <= 9.5e-77)
		tmp = -4.0 * (((b / y_45_scale) * (a / x_45_scale)) ^ 2.0);
	elseif (a <= 1.75e+81)
		tmp = -4.0 * ((b / x_45_scale) * ((a * ((a / y_45_scale) * (b / x_45_scale))) / y_45_scale));
	else
		tmp = -4.0 / (t_0 * t_0);
	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[(y$45$scale * x$45$scale), $MachinePrecision] / N[(b * a), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[a, 3e-271], N[(-4.0 * N[(N[(b / x$45$scale), $MachinePrecision] / N[(N[(N[(x$45$scale / b), $MachinePrecision] / N[(a / y$45$scale), $MachinePrecision]), $MachinePrecision] * N[(y$45$scale / a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[a, 9.5e-77], 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[a, 1.75e+81], N[(-4.0 * N[(N[(b / x$45$scale), $MachinePrecision] * N[(N[(a * N[(N[(a / y$45$scale), $MachinePrecision] * N[(b / x$45$scale), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / y$45$scale), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(-4.0 / N[(t$95$0 * t$95$0), $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 4 regimes

2. if a < 3.00000000000000002e-271

   1. Initial program 41.2

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

   2. Taylor expanded in angle around 0 40.0

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

   3. Simplified 20.0

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

      [Start] 40.0	\[ -4 \cdot \frac{{a}^{2} \cdot {b}^{2}}{{y-scale}^{2} \cdot {x-scale}^{2}} \]
      *-commutative [=>] 40.0	\[ \color{blue}{\frac{{a}^{2} \cdot {b}^{2}}{{y-scale}^{2} \cdot {x-scale}^{2}} \cdot -4} \]
      times-frac [=>] 40.1	\[ \color{blue}{\left(\frac{{a}^{2}}{{y-scale}^{2}} \cdot \frac{{b}^{2}}{{x-scale}^{2}}\right)} \cdot -4 \]
      associate-*l* [=>] 40.2	\[ \color{blue}{\frac{{a}^{2}}{{y-scale}^{2}} \cdot \left(\frac{{b}^{2}}{{x-scale}^{2}} \cdot -4\right)} \]
      unpow2 [=>] 40.2	\[ \frac{\color{blue}{a \cdot a}}{{y-scale}^{2}} \cdot \left(\frac{{b}^{2}}{{x-scale}^{2}} \cdot -4\right) \]
      unpow2 [=>] 40.2	\[ \frac{a \cdot a}{\color{blue}{y-scale \cdot y-scale}} \cdot \left(\frac{{b}^{2}}{{x-scale}^{2}} \cdot -4\right) \]
      times-frac [=>] 31.7	\[ \color{blue}{\left(\frac{a}{y-scale} \cdot \frac{a}{y-scale}\right)} \cdot \left(\frac{{b}^{2}}{{x-scale}^{2}} \cdot -4\right) \]
      unpow2 [=>] 31.7	\[ \left(\frac{a}{y-scale} \cdot \frac{a}{y-scale}\right) \cdot \left(\frac{\color{blue}{b \cdot b}}{{x-scale}^{2}} \cdot -4\right) \]
      unpow2 [=>] 31.7	\[ \left(\frac{a}{y-scale} \cdot \frac{a}{y-scale}\right) \cdot \left(\frac{b \cdot b}{\color{blue}{x-scale \cdot x-scale}} \cdot -4\right) \]
      times-frac [=>] 20.0	\[ \left(\frac{a}{y-scale} \cdot \frac{a}{y-scale}\right) \cdot \left(\color{blue}{\left(\frac{b}{x-scale} \cdot \frac{b}{x-scale}\right)} \cdot -4\right) \]

   4. Taylor expanded in a around 0 40.0

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

   5. Simplified 6.2

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

      [Start] 40.0	\[ -4 \cdot \frac{{a}^{2} \cdot {b}^{2}}{{x-scale}^{2} \cdot {y-scale}^{2}} \]
      *-commutative [=>] 40.0	\[ -4 \cdot \frac{{a}^{2} \cdot {b}^{2}}{\color{blue}{{y-scale}^{2} \cdot {x-scale}^{2}}} \]
      times-frac [=>] 40.1	\[ -4 \cdot \color{blue}{\left(\frac{{a}^{2}}{{y-scale}^{2}} \cdot \frac{{b}^{2}}{{x-scale}^{2}}\right)} \]
      unpow2 [=>] 40.1	\[ -4 \cdot \left(\frac{\color{blue}{a \cdot a}}{{y-scale}^{2}} \cdot \frac{{b}^{2}}{{x-scale}^{2}}\right) \]
      unpow2 [=>] 40.1	\[ -4 \cdot \left(\frac{a \cdot a}{\color{blue}{y-scale \cdot y-scale}} \cdot \frac{{b}^{2}}{{x-scale}^{2}}\right) \]
      times-frac [=>] 31.7	\[ -4 \cdot \left(\color{blue}{\left(\frac{a}{y-scale} \cdot \frac{a}{y-scale}\right)} \cdot \frac{{b}^{2}}{{x-scale}^{2}}\right) \]
      unpow2 [=>] 31.7	\[ -4 \cdot \left(\left(\frac{a}{y-scale} \cdot \frac{a}{y-scale}\right) \cdot \frac{\color{blue}{b \cdot b}}{{x-scale}^{2}}\right) \]
      unpow2 [=>] 31.7	\[ -4 \cdot \left(\left(\frac{a}{y-scale} \cdot \frac{a}{y-scale}\right) \cdot \frac{b \cdot b}{\color{blue}{x-scale \cdot x-scale}}\right) \]
      times-frac [=>] 19.9	\[ -4 \cdot \left(\left(\frac{a}{y-scale} \cdot \frac{a}{y-scale}\right) \cdot \color{blue}{\left(\frac{b}{x-scale} \cdot \frac{b}{x-scale}\right)}\right) \]
      swap-sqr [<=] 6.4	\[ -4 \cdot \color{blue}{\left(\left(\frac{a}{y-scale} \cdot \frac{b}{x-scale}\right) \cdot \left(\frac{a}{y-scale} \cdot \frac{b}{x-scale}\right)\right)} \]
      unpow2 [<=] 6.4	\[ -4 \cdot \color{blue}{{\left(\frac{a}{y-scale} \cdot \frac{b}{x-scale}\right)}^{2}} \]
      associate-*r/ [=>] 6.2	\[ -4 \cdot {\color{blue}{\left(\frac{\frac{a}{y-scale} \cdot b}{x-scale}\right)}}^{2} \]
      associate-/l* [=>] 6.2	\[ -4 \cdot {\color{blue}{\left(\frac{\frac{a}{y-scale}}{\frac{x-scale}{b}}\right)}}^{2} \]

   6. Applied egg-rr 8.8

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

   if 3.00000000000000002e-271 < a < 9.5000000000000005e-77

   1. Initial program 32.7

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

   2. Simplified 39.4

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

      [Start] 32.7

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

   3. Taylor expanded in angle around 0 34.5

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

   4. Simplified 20.7

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

      [Start] 34.5	\[ -4 \cdot \frac{{a}^{2} \cdot {b}^{2}}{{y-scale}^{2} \cdot {x-scale}^{2}} \]
      *-commutative [=>] 34.5	\[ -4 \cdot \frac{\color{blue}{{b}^{2} \cdot {a}^{2}}}{{y-scale}^{2} \cdot {x-scale}^{2}} \]
      times-frac [=>] 34.6	\[ -4 \cdot \color{blue}{\left(\frac{{b}^{2}}{{y-scale}^{2}} \cdot \frac{{a}^{2}}{{x-scale}^{2}}\right)} \]
      unpow2 [=>] 34.6	\[ -4 \cdot \left(\frac{\color{blue}{b \cdot b}}{{y-scale}^{2}} \cdot \frac{{a}^{2}}{{x-scale}^{2}}\right) \]
      unpow2 [=>] 34.6	\[ -4 \cdot \left(\frac{b \cdot b}{\color{blue}{y-scale \cdot y-scale}} \cdot \frac{{a}^{2}}{{x-scale}^{2}}\right) \]
      times-frac [=>] 26.9	\[ -4 \cdot \left(\color{blue}{\left(\frac{b}{y-scale} \cdot \frac{b}{y-scale}\right)} \cdot \frac{{a}^{2}}{{x-scale}^{2}}\right) \]
      unpow2 [=>] 26.9	\[ -4 \cdot \left(\left(\frac{b}{y-scale} \cdot \frac{b}{y-scale}\right) \cdot \frac{{a}^{2}}{\color{blue}{x-scale \cdot x-scale}}\right) \]
      associate-/r* [=>] 20.7	\[ -4 \cdot \left(\left(\frac{b}{y-scale} \cdot \frac{b}{y-scale}\right) \cdot \color{blue}{\frac{\frac{{a}^{2}}{x-scale}}{x-scale}}\right) \]
      unpow2 [=>] 20.7	\[ -4 \cdot \left(\left(\frac{b}{y-scale} \cdot \frac{b}{y-scale}\right) \cdot \frac{\frac{\color{blue}{a \cdot a}}{x-scale}}{x-scale}\right) \]

   5. Taylor expanded in b around 0 34.5

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

   6. Simplified 5.0

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

      [Start] 34.5	\[ -4 \cdot \frac{{a}^{2} \cdot {b}^{2}}{{x-scale}^{2} \cdot {y-scale}^{2}} \]
      *-commutative [<=] 34.5	\[ -4 \cdot \frac{{a}^{2} \cdot {b}^{2}}{\color{blue}{{y-scale}^{2} \cdot {x-scale}^{2}}} \]
      *-commutative [=>] 34.5	\[ -4 \cdot \frac{\color{blue}{{b}^{2} \cdot {a}^{2}}}{{y-scale}^{2} \cdot {x-scale}^{2}} \]
      unpow2 [=>] 34.5	\[ -4 \cdot \frac{\color{blue}{\left(b \cdot b\right)} \cdot {a}^{2}}{{y-scale}^{2} \cdot {x-scale}^{2}} \]
      unpow2 [=>] 34.5	\[ -4 \cdot \frac{\left(b \cdot b\right) \cdot \color{blue}{\left(a \cdot a\right)}}{{y-scale}^{2} \cdot {x-scale}^{2}} \]
      unpow2 [=>] 34.5	\[ -4 \cdot \frac{\left(b \cdot b\right) \cdot \left(a \cdot a\right)}{\color{blue}{\left(y-scale \cdot y-scale\right)} \cdot {x-scale}^{2}} \]
      unpow2 [=>] 34.5	\[ -4 \cdot \frac{\left(b \cdot b\right) \cdot \left(a \cdot a\right)}{\left(y-scale \cdot y-scale\right) \cdot \color{blue}{\left(x-scale \cdot x-scale\right)}} \]
      times-frac [=>] 34.6	\[ -4 \cdot \color{blue}{\left(\frac{b \cdot b}{y-scale \cdot y-scale} \cdot \frac{a \cdot a}{x-scale \cdot x-scale}\right)} \]
      associate-/l/ [<=] 29.7	\[ -4 \cdot \left(\frac{b \cdot b}{y-scale \cdot y-scale} \cdot \color{blue}{\frac{\frac{a \cdot a}{x-scale}}{x-scale}}\right) \]
      associate-*r/ [<=] 27.1	\[ -4 \cdot \left(\frac{b \cdot b}{y-scale \cdot y-scale} \cdot \frac{\color{blue}{a \cdot \frac{a}{x-scale}}}{x-scale}\right) \]
      times-frac [=>] 15.5	\[ -4 \cdot \left(\color{blue}{\left(\frac{b}{y-scale} \cdot \frac{b}{y-scale}\right)} \cdot \frac{a \cdot \frac{a}{x-scale}}{x-scale}\right) \]
      associate-*l/ [<=] 14.7	\[ -4 \cdot \left(\left(\frac{b}{y-scale} \cdot \frac{b}{y-scale}\right) \cdot \color{blue}{\left(\frac{a}{x-scale} \cdot \frac{a}{x-scale}\right)}\right) \]
      swap-sqr [<=] 5.0	\[ -4 \cdot \color{blue}{\left(\left(\frac{b}{y-scale} \cdot \frac{a}{x-scale}\right) \cdot \left(\frac{b}{y-scale} \cdot \frac{a}{x-scale}\right)\right)} \]
      unpow2 [<=] 5.0	\[ -4 \cdot \color{blue}{{\left(\frac{b}{y-scale} \cdot \frac{a}{x-scale}\right)}^{2}} \]

   if 9.5000000000000005e-77 < a < 1.75e81

   1. Initial program 39.7

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

   2. Taylor expanded in angle around 0 33.4

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

   3. Simplified 20.6

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

      [Start] 33.4	\[ -4 \cdot \frac{{a}^{2} \cdot {b}^{2}}{{y-scale}^{2} \cdot {x-scale}^{2}} \]
      *-commutative [=>] 33.4	\[ \color{blue}{\frac{{a}^{2} \cdot {b}^{2}}{{y-scale}^{2} \cdot {x-scale}^{2}} \cdot -4} \]
      times-frac [=>] 33.5	\[ \color{blue}{\left(\frac{{a}^{2}}{{y-scale}^{2}} \cdot \frac{{b}^{2}}{{x-scale}^{2}}\right)} \cdot -4 \]
      associate-*l* [=>] 33.5	\[ \color{blue}{\frac{{a}^{2}}{{y-scale}^{2}} \cdot \left(\frac{{b}^{2}}{{x-scale}^{2}} \cdot -4\right)} \]
      unpow2 [=>] 33.5	\[ \frac{\color{blue}{a \cdot a}}{{y-scale}^{2}} \cdot \left(\frac{{b}^{2}}{{x-scale}^{2}} \cdot -4\right) \]
      unpow2 [=>] 33.5	\[ \frac{a \cdot a}{\color{blue}{y-scale \cdot y-scale}} \cdot \left(\frac{{b}^{2}}{{x-scale}^{2}} \cdot -4\right) \]
      times-frac [=>] 32.7	\[ \color{blue}{\left(\frac{a}{y-scale} \cdot \frac{a}{y-scale}\right)} \cdot \left(\frac{{b}^{2}}{{x-scale}^{2}} \cdot -4\right) \]
      unpow2 [=>] 32.7	\[ \left(\frac{a}{y-scale} \cdot \frac{a}{y-scale}\right) \cdot \left(\frac{\color{blue}{b \cdot b}}{{x-scale}^{2}} \cdot -4\right) \]
      unpow2 [=>] 32.7	\[ \left(\frac{a}{y-scale} \cdot \frac{a}{y-scale}\right) \cdot \left(\frac{b \cdot b}{\color{blue}{x-scale \cdot x-scale}} \cdot -4\right) \]
      times-frac [=>] 20.6	\[ \left(\frac{a}{y-scale} \cdot \frac{a}{y-scale}\right) \cdot \left(\color{blue}{\left(\frac{b}{x-scale} \cdot \frac{b}{x-scale}\right)} \cdot -4\right) \]

   4. Taylor expanded in a around 0 33.4

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

   5. Simplified 4.0

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

      [Start] 33.4	\[ -4 \cdot \frac{{a}^{2} \cdot {b}^{2}}{{x-scale}^{2} \cdot {y-scale}^{2}} \]
      *-commutative [=>] 33.4	\[ -4 \cdot \frac{{a}^{2} \cdot {b}^{2}}{\color{blue}{{y-scale}^{2} \cdot {x-scale}^{2}}} \]
      times-frac [=>] 33.5	\[ -4 \cdot \color{blue}{\left(\frac{{a}^{2}}{{y-scale}^{2}} \cdot \frac{{b}^{2}}{{x-scale}^{2}}\right)} \]
      unpow2 [=>] 33.5	\[ -4 \cdot \left(\frac{\color{blue}{a \cdot a}}{{y-scale}^{2}} \cdot \frac{{b}^{2}}{{x-scale}^{2}}\right) \]
      unpow2 [=>] 33.5	\[ -4 \cdot \left(\frac{a \cdot a}{\color{blue}{y-scale \cdot y-scale}} \cdot \frac{{b}^{2}}{{x-scale}^{2}}\right) \]
      times-frac [=>] 32.7	\[ -4 \cdot \left(\color{blue}{\left(\frac{a}{y-scale} \cdot \frac{a}{y-scale}\right)} \cdot \frac{{b}^{2}}{{x-scale}^{2}}\right) \]
      unpow2 [=>] 32.7	\[ -4 \cdot \left(\left(\frac{a}{y-scale} \cdot \frac{a}{y-scale}\right) \cdot \frac{\color{blue}{b \cdot b}}{{x-scale}^{2}}\right) \]
      unpow2 [=>] 32.7	\[ -4 \cdot \left(\left(\frac{a}{y-scale} \cdot \frac{a}{y-scale}\right) \cdot \frac{b \cdot b}{\color{blue}{x-scale \cdot x-scale}}\right) \]
      times-frac [=>] 20.6	\[ -4 \cdot \left(\left(\frac{a}{y-scale} \cdot \frac{a}{y-scale}\right) \cdot \color{blue}{\left(\frac{b}{x-scale} \cdot \frac{b}{x-scale}\right)}\right) \]
      swap-sqr [<=] 4.0	\[ -4 \cdot \color{blue}{\left(\left(\frac{a}{y-scale} \cdot \frac{b}{x-scale}\right) \cdot \left(\frac{a}{y-scale} \cdot \frac{b}{x-scale}\right)\right)} \]
      unpow2 [<=] 4.0	\[ -4 \cdot \color{blue}{{\left(\frac{a}{y-scale} \cdot \frac{b}{x-scale}\right)}^{2}} \]
      associate-*r/ [=>] 3.5	\[ -4 \cdot {\color{blue}{\left(\frac{\frac{a}{y-scale} \cdot b}{x-scale}\right)}}^{2} \]
      associate-/l* [=>] 4.0	\[ -4 \cdot {\color{blue}{\left(\frac{\frac{a}{y-scale}}{\frac{x-scale}{b}}\right)}}^{2} \]

   6. Applied egg-rr 7.1

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

   if 1.75e81 < a

   1. Initial program 56.2

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

   2. Simplified 64.0

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

      [Start] 56.2	\[ \frac{\frac{\left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right) \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)}{x-scale}}{y-scale} \cdot \frac{\frac{\left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right) \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)}{x-scale}}{y-scale} - \left(4 \cdot \frac{\frac{{\left(a \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} + {\left(b \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2}}{x-scale}}{x-scale}\right) \cdot \frac{\frac{{\left(a \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} + {\left(b \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2}}{y-scale}}{y-scale} \]
      sub-neg [=>] 56.2	\[ \color{blue}{\frac{\frac{\left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right) \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)}{x-scale}}{y-scale} \cdot \frac{\frac{\left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right) \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)}{x-scale}}{y-scale} + \left(-\left(4 \cdot \frac{\frac{{\left(a \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} + {\left(b \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2}}{x-scale}}{x-scale}\right) \cdot \frac{\frac{{\left(a \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} + {\left(b \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2}}{y-scale}}{y-scale}\right)} \]
      +-commutative [=>] 56.2	\[ \color{blue}{\left(-\left(4 \cdot \frac{\frac{{\left(a \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} + {\left(b \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2}}{x-scale}}{x-scale}\right) \cdot \frac{\frac{{\left(a \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} + {\left(b \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2}}{y-scale}}{y-scale}\right) + \frac{\frac{\left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right) \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)}{x-scale}}{y-scale} \cdot \frac{\frac{\left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right) \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)}{x-scale}}{y-scale}} \]

   3. Taylor expanded in angle around 0 52.6

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

   4. Simplified 46.5

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

      [Start] 52.6	\[ -4 \cdot \frac{{a}^{2} \cdot {b}^{2}}{{x-scale}^{2} \cdot {y-scale}^{2}} \]
      *-commutative [=>] 52.6	\[ -4 \cdot \frac{{a}^{2} \cdot {b}^{2}}{\color{blue}{{y-scale}^{2} \cdot {x-scale}^{2}}} \]
      associate-*r/ [=>] 52.6	\[ \color{blue}{\frac{-4 \cdot \left({a}^{2} \cdot {b}^{2}\right)}{{y-scale}^{2} \cdot {x-scale}^{2}}} \]
      associate-/l* [=>] 52.6	\[ \color{blue}{\frac{-4}{\frac{{y-scale}^{2} \cdot {x-scale}^{2}}{{a}^{2} \cdot {b}^{2}}}} \]
      unpow2 [=>] 52.6	\[ \frac{-4}{\frac{\color{blue}{\left(y-scale \cdot y-scale\right)} \cdot {x-scale}^{2}}{{a}^{2} \cdot {b}^{2}}} \]
      unpow2 [=>] 52.6	\[ \frac{-4}{\frac{\left(y-scale \cdot y-scale\right) \cdot \color{blue}{\left(x-scale \cdot x-scale\right)}}{{a}^{2} \cdot {b}^{2}}} \]
      unswap-sqr [=>] 49.3	\[ \frac{-4}{\frac{\color{blue}{\left(y-scale \cdot x-scale\right) \cdot \left(y-scale \cdot x-scale\right)}}{{a}^{2} \cdot {b}^{2}}} \]
      *-commutative [=>] 49.3	\[ \frac{-4}{\frac{\left(y-scale \cdot x-scale\right) \cdot \left(y-scale \cdot x-scale\right)}{\color{blue}{{b}^{2} \cdot {a}^{2}}}} \]
      unpow2 [=>] 49.3	\[ \frac{-4}{\frac{\left(y-scale \cdot x-scale\right) \cdot \left(y-scale \cdot x-scale\right)}{\color{blue}{\left(b \cdot b\right)} \cdot {a}^{2}}} \]
      associate-*l* [=>] 46.5	\[ \frac{-4}{\frac{\left(y-scale \cdot x-scale\right) \cdot \left(y-scale \cdot x-scale\right)}{\color{blue}{b \cdot \left(b \cdot {a}^{2}\right)}}} \]
      unpow2 [=>] 46.5	\[ \frac{-4}{\frac{\left(y-scale \cdot x-scale\right) \cdot \left(y-scale \cdot x-scale\right)}{b \cdot \left(b \cdot \color{blue}{\left(a \cdot a\right)}\right)}} \]

   5. Applied egg-rr 8.3

      \[\leadsto \frac{-4}{\color{blue}{\frac{y-scale \cdot x-scale}{b \cdot a} \cdot \frac{y-scale \cdot x-scale}{b \cdot a}}} \]
3. Recombined 4 regimes into one program.

4. Final simplification 7.7

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \leq 3 \cdot 10^{-271}:\\ \;\;\;\;-4 \cdot \frac{\frac{b}{x-scale}}{\frac{\frac{x-scale}{b}}{\frac{a}{y-scale}} \cdot \frac{y-scale}{a}}\\ \mathbf{elif}\;a \leq 9.5 \cdot 10^{-77}:\\ \;\;\;\;-4 \cdot {\left(\frac{b}{y-scale} \cdot \frac{a}{x-scale}\right)}^{2}\\ \mathbf{elif}\;a \leq 1.75 \cdot 10^{+81}:\\ \;\;\;\;-4 \cdot \left(\frac{b}{x-scale} \cdot \frac{a \cdot \left(\frac{a}{y-scale} \cdot \frac{b}{x-scale}\right)}{y-scale}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{-4}{\frac{y-scale \cdot x-scale}{b \cdot a} \cdot \frac{y-scale \cdot x-scale}{b \cdot a}}\\ \end{array} \]

Alternatives

Alternative 1
Error	5.8
Cost	7300

\[\begin{array}{l} t_0 := \frac{y-scale \cdot x-scale}{b \cdot a}\\ \mathbf{if}\;\frac{angle}{180} \leq -5000:\\ \;\;\;\;\frac{-4}{t_0 \cdot t_0}\\ \mathbf{else}:\\ \;\;\;\;-4 \cdot {\left(\frac{\frac{a}{y-scale}}{\frac{x-scale}{b}}\right)}^{2}\\ \end{array} \]

Alternative 2
Error	10.5
Cost	1617

\[\begin{array}{l} t_0 := -4 \cdot \left(\frac{b}{x-scale} \cdot \frac{a \cdot \left(\frac{a}{y-scale} \cdot \frac{b}{x-scale}\right)}{y-scale}\right)\\ \mathbf{if}\;angle \leq -2.05 \cdot 10^{+232}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;angle \leq -950:\\ \;\;\;\;-4 \cdot \left(\frac{\frac{b}{y-scale} \cdot \frac{a}{x-scale}}{x-scale} \cdot \frac{a}{\frac{y-scale}{b}}\right)\\ \mathbf{elif}\;angle \leq 7.5 \cdot 10^{-290} \lor \neg \left(angle \leq 1.15 \cdot 10^{-79}\right):\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;-4 \cdot \frac{b \cdot \frac{a}{y-scale}}{x-scale \cdot \frac{\frac{x-scale}{b}}{\frac{a}{y-scale}}}\\ \end{array} \]

Alternative 3
Error	26.7
Cost	1485

\[\begin{array}{l} \mathbf{if}\;a \leq -6.4 \cdot 10^{-142} \lor \neg \left(a \leq 5.8 \cdot 10^{-158}\right) \land a \leq 1.55 \cdot 10^{+31}:\\ \;\;\;\;-4 \cdot \left(\left(\frac{b}{y-scale} \cdot \frac{b}{y-scale}\right) \cdot \frac{\frac{a \cdot a}{x-scale}}{x-scale}\right)\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \]

Alternative 4
Error	10.1
Cost	1485

\[\begin{array}{l} \mathbf{if}\;a \leq -2.4 \cdot 10^{-202} \lor \neg \left(a \leq 2 \cdot 10^{-227}\right) \land a \leq 4.8 \cdot 10^{+235}:\\ \;\;\;\;-4 \cdot \left(\frac{b}{x-scale} \cdot \frac{a \cdot \left(\frac{a}{y-scale} \cdot \frac{b}{x-scale}\right)}{y-scale}\right)\\ \mathbf{else}:\\ \;\;\;\;-4 \cdot \left(\frac{\frac{b}{y-scale} \cdot \frac{a}{x-scale}}{x-scale} \cdot \frac{a}{\frac{y-scale}{b}}\right)\\ \end{array} \]

Alternative 5
Error	9.0
Cost	1353

\[\begin{array}{l} \mathbf{if}\;angle \leq -1.75 \cdot 10^{+232} \lor \neg \left(angle \leq -0.00015\right):\\ \;\;\;\;-4 \cdot \frac{\frac{b}{x-scale}}{\frac{\frac{x-scale}{b}}{\frac{a}{y-scale}} \cdot \frac{y-scale}{a}}\\ \mathbf{else}:\\ \;\;\;\;-4 \cdot \left(\frac{\frac{b}{y-scale} \cdot \frac{a}{x-scale}}{x-scale} \cdot \frac{a}{\frac{y-scale}{b}}\right)\\ \end{array} \]

Alternative 6
Error	9.2
Cost	1352

\[\begin{array}{l} \mathbf{if}\;angle \leq -7.2 \cdot 10^{+231}:\\ \;\;\;\;-4 \cdot \frac{\frac{b}{x-scale}}{\frac{\frac{y-scale \cdot \frac{x-scale}{b}}{\frac{a}{y-scale}}}{a}}\\ \mathbf{elif}\;angle \leq -2.18 \cdot 10^{-7}:\\ \;\;\;\;-4 \cdot \left(\frac{\frac{b}{y-scale} \cdot \frac{a}{x-scale}}{x-scale} \cdot \frac{a}{\frac{y-scale}{b}}\right)\\ \mathbf{else}:\\ \;\;\;\;-4 \cdot \frac{\frac{b}{x-scale}}{\frac{\frac{x-scale}{b}}{\frac{a}{y-scale}} \cdot \frac{y-scale}{a}}\\ \end{array} \]

Alternative 7
Error	8.7
Cost	1220

\[\begin{array}{l} \mathbf{if}\;angle \leq -1700000:\\ \;\;\;\;a \cdot \frac{-4 \cdot \frac{b}{y-scale \cdot x-scale}}{\frac{y-scale \cdot x-scale}{b \cdot a}}\\ \mathbf{else}:\\ \;\;\;\;-4 \cdot \frac{\frac{b}{x-scale}}{\frac{\frac{x-scale}{b}}{\frac{a}{y-scale}} \cdot \frac{y-scale}{a}}\\ \end{array} \]

Alternative 8
Error	8.4
Cost	1220

\[\begin{array}{l} \mathbf{if}\;angle \leq -2800:\\ \;\;\;\;\left(b \cdot a\right) \cdot \frac{\frac{-4}{\frac{y-scale \cdot x-scale}{b \cdot a}}}{y-scale \cdot x-scale}\\ \mathbf{else}:\\ \;\;\;\;-4 \cdot \frac{\frac{b}{x-scale}}{\frac{\frac{x-scale}{b}}{\frac{a}{y-scale}} \cdot \frac{y-scale}{a}}\\ \end{array} \]

Alternative 9
Error	7.8
Cost	1220

\[\begin{array}{l} t_0 := \frac{y-scale \cdot x-scale}{b \cdot a}\\ \mathbf{if}\;angle \leq -370:\\ \;\;\;\;\frac{-4}{t_0 \cdot t_0}\\ \mathbf{else}:\\ \;\;\;\;-4 \cdot \frac{\frac{b}{x-scale}}{\frac{\frac{x-scale}{b}}{\frac{a}{y-scale}} \cdot \frac{y-scale}{a}}\\ \end{array} \]

Alternative 10
Error	10.4
Cost	1088

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

Alternative 11
Error	30.5
Cost	64

\[0 \]

Reproduce

herbie shell --seed 2023053 
(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))))