Simplification of discriminant from scale-rotated-ellipse

Average Error: 41.2 → 5.9

Time: 1.3min

Precision: binary64

Cost: 1484

Math

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

↓

\[\begin{array}{l} t_0 := \frac{\left(-4 \cdot \frac{a}{x-scale}\right) \cdot \frac{b}{y-scale}}{\frac{y-scale}{b} \cdot \frac{x-scale}{a}}\\ t_1 := \frac{x-scale \cdot y-scale}{a \cdot b}\\ \mathbf{if}\;x-scale \leq -1.2 \cdot 10^{+117}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;x-scale \leq -7 \cdot 10^{-172}:\\ \;\;\;\;\frac{-4}{t_1 \cdot t_1}\\ \mathbf{elif}\;x-scale \leq 3.35 \cdot 10^{-202}:\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;-4 \cdot \left(\frac{\frac{b}{\frac{y-scale}{a}}}{x-scale} \cdot \frac{b \cdot \frac{a}{y-scale}}{x-scale}\right)\\ \end{array} \]

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
         (/
          (* (* -4.0 (/ a x-scale)) (/ b y-scale))
          (* (/ y-scale b) (/ x-scale a))))
        (t_1 (/ (* x-scale y-scale) (* a b))))
   (if (<= x-scale -1.2e+117)
     t_0
     (if (<= x-scale -7e-172)
       (/ -4.0 (* t_1 t_1))
       (if (<= x-scale 3.35e-202)
         t_0
         (*
          -4.0
          (*
           (/ (/ b (/ y-scale a)) x-scale)
           (/ (* b (/ a y-scale)) x-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 = ((-4.0 * (a / x_45_scale)) * (b / y_45_scale)) / ((y_45_scale / b) * (x_45_scale / a));
	double t_1 = (x_45_scale * y_45_scale) / (a * b);
	double tmp;
	if (x_45_scale <= -1.2e+117) {
		tmp = t_0;
	} else if (x_45_scale <= -7e-172) {
		tmp = -4.0 / (t_1 * t_1);
	} else if (x_45_scale <= 3.35e-202) {
		tmp = t_0;
	} else {
		tmp = -4.0 * (((b / (y_45_scale / a)) / x_45_scale) * ((b * (a / y_45_scale)) / x_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 = ((-4.0 * (a / x_45_scale)) * (b / y_45_scale)) / ((y_45_scale / b) * (x_45_scale / a));
	double t_1 = (x_45_scale * y_45_scale) / (a * b);
	double tmp;
	if (x_45_scale <= -1.2e+117) {
		tmp = t_0;
	} else if (x_45_scale <= -7e-172) {
		tmp = -4.0 / (t_1 * t_1);
	} else if (x_45_scale <= 3.35e-202) {
		tmp = t_0;
	} else {
		tmp = -4.0 * (((b / (y_45_scale / a)) / x_45_scale) * ((b * (a / y_45_scale)) / x_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 = ((-4.0 * (a / x_45_scale)) * (b / y_45_scale)) / ((y_45_scale / b) * (x_45_scale / a))
	t_1 = (x_45_scale * y_45_scale) / (a * b)
	tmp = 0
	if x_45_scale <= -1.2e+117:
		tmp = t_0
	elif x_45_scale <= -7e-172:
		tmp = -4.0 / (t_1 * t_1)
	elif x_45_scale <= 3.35e-202:
		tmp = t_0
	else:
		tmp = -4.0 * (((b / (y_45_scale / a)) / x_45_scale) * ((b * (a / y_45_scale)) / x_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(-4.0 * Float64(a / x_45_scale)) * Float64(b / y_45_scale)) / Float64(Float64(y_45_scale / b) * Float64(x_45_scale / a)))
	t_1 = Float64(Float64(x_45_scale * y_45_scale) / Float64(a * b))
	tmp = 0.0
	if (x_45_scale <= -1.2e+117)
		tmp = t_0;
	elseif (x_45_scale <= -7e-172)
		tmp = Float64(-4.0 / Float64(t_1 * t_1));
	elseif (x_45_scale <= 3.35e-202)
		tmp = t_0;
	else
		tmp = Float64(-4.0 * Float64(Float64(Float64(b / Float64(y_45_scale / a)) / x_45_scale) * Float64(Float64(b * Float64(a / y_45_scale)) / x_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 = ((-4.0 * (a / x_45_scale)) * (b / y_45_scale)) / ((y_45_scale / b) * (x_45_scale / a));
	t_1 = (x_45_scale * y_45_scale) / (a * b);
	tmp = 0.0;
	if (x_45_scale <= -1.2e+117)
		tmp = t_0;
	elseif (x_45_scale <= -7e-172)
		tmp = -4.0 / (t_1 * t_1);
	elseif (x_45_scale <= 3.35e-202)
		tmp = t_0;
	else
		tmp = -4.0 * (((b / (y_45_scale / a)) / x_45_scale) * ((b * (a / y_45_scale)) / x_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[(-4.0 * N[(a / x$45$scale), $MachinePrecision]), $MachinePrecision] * N[(b / y$45$scale), $MachinePrecision]), $MachinePrecision] / N[(N[(y$45$scale / b), $MachinePrecision] * N[(x$45$scale / a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[(x$45$scale * y$45$scale), $MachinePrecision] / N[(a * b), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[x$45$scale, -1.2e+117], t$95$0, If[LessEqual[x$45$scale, -7e-172], N[(-4.0 / N[(t$95$1 * t$95$1), $MachinePrecision]), $MachinePrecision], If[LessEqual[x$45$scale, 3.35e-202], t$95$0, N[(-4.0 * N[(N[(N[(b / N[(y$45$scale / a), $MachinePrecision]), $MachinePrecision] / x$45$scale), $MachinePrecision] * N[(N[(b * N[(a / y$45$scale), $MachinePrecision]), $MachinePrecision] / x$45$scale), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]]

Derivation

1. Split input into 3 regimes

2. if x-scale < -1.1999999999999999e117 or -7.00000000000000057e-172 < x-scale < 3.35000000000000001e-202

   1. Initial program 42.4

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

   2. Simplified 49.3

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

      [Start] 42.4	\[ \frac{\frac{\left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right) \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)}{x-scale}}{y-scale} \cdot \frac{\frac{\left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right) \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)}{x-scale}}{y-scale} - \left(4 \cdot \frac{\frac{{\left(a \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} + {\left(b \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2}}{x-scale}}{x-scale}\right) \cdot \frac{\frac{{\left(a \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} + {\left(b \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2}}{y-scale}}{y-scale} \]
      sub-neg [=>] 42.4	\[ \color{blue}{\frac{\frac{\left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right) \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)}{x-scale}}{y-scale} \cdot \frac{\frac{\left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right) \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)}{x-scale}}{y-scale} + \left(-\left(4 \cdot \frac{\frac{{\left(a \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} + {\left(b \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2}}{x-scale}}{x-scale}\right) \cdot \frac{\frac{{\left(a \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} + {\left(b \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2}}{y-scale}}{y-scale}\right)} \]
      +-commutative [=>] 42.4	\[ \color{blue}{\left(-\left(4 \cdot \frac{\frac{{\left(a \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} + {\left(b \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2}}{x-scale}}{x-scale}\right) \cdot \frac{\frac{{\left(a \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} + {\left(b \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2}}{y-scale}}{y-scale}\right) + \frac{\frac{\left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right) \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)}{x-scale}}{y-scale} \cdot \frac{\frac{\left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right) \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)}{x-scale}}{y-scale}} \]

   3. Taylor expanded in angle around 0 46.2

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

   4. Simplified 20.0

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

      [Start] 46.2	\[ -4 \cdot \frac{{a}^{2} \cdot {b}^{2}}{{x-scale}^{2} \cdot {y-scale}^{2}} \]
      times-frac [=>] 46.2	\[ -4 \cdot \color{blue}{\left(\frac{{a}^{2}}{{x-scale}^{2}} \cdot \frac{{b}^{2}}{{y-scale}^{2}}\right)} \]
      associate-*r* [=>] 46.2	\[ \color{blue}{\left(-4 \cdot \frac{{a}^{2}}{{x-scale}^{2}}\right) \cdot \frac{{b}^{2}}{{y-scale}^{2}}} \]
      unpow2 [=>] 46.2	\[ \left(-4 \cdot \frac{\color{blue}{a \cdot a}}{{x-scale}^{2}}\right) \cdot \frac{{b}^{2}}{{y-scale}^{2}} \]
      unpow2 [=>] 46.2	\[ \left(-4 \cdot \frac{a \cdot a}{\color{blue}{x-scale \cdot x-scale}}\right) \cdot \frac{{b}^{2}}{{y-scale}^{2}} \]
      times-frac [=>] 32.6	\[ \left(-4 \cdot \color{blue}{\left(\frac{a}{x-scale} \cdot \frac{a}{x-scale}\right)}\right) \cdot \frac{{b}^{2}}{{y-scale}^{2}} \]
      unpow2 [=>] 32.6	\[ \left(-4 \cdot \left(\frac{a}{x-scale} \cdot \frac{a}{x-scale}\right)\right) \cdot \frac{\color{blue}{b \cdot b}}{{y-scale}^{2}} \]
      unpow2 [=>] 32.6	\[ \left(-4 \cdot \left(\frac{a}{x-scale} \cdot \frac{a}{x-scale}\right)\right) \cdot \frac{b \cdot b}{\color{blue}{y-scale \cdot y-scale}} \]
      times-frac [=>] 20.0	\[ \left(-4 \cdot \left(\frac{a}{x-scale} \cdot \frac{a}{x-scale}\right)\right) \cdot \color{blue}{\left(\frac{b}{y-scale} \cdot \frac{b}{y-scale}\right)} \]

   5. Applied egg-rr 20.0

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

   6. Applied egg-rr 8.0

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

   if -1.1999999999999999e117 < x-scale < -7.00000000000000057e-172

   1. Initial program 42.4

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

   2. Simplified 47.0

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

      [Start] 42.4	\[ \frac{\frac{\left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right) \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)}{x-scale}}{y-scale} \cdot \frac{\frac{\left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right) \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)}{x-scale}}{y-scale} - \left(4 \cdot \frac{\frac{{\left(a \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} + {\left(b \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2}}{x-scale}}{x-scale}\right) \cdot \frac{\frac{{\left(a \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} + {\left(b \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2}}{y-scale}}{y-scale} \]
      sub-neg [=>] 42.4	\[ \color{blue}{\frac{\frac{\left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right) \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)}{x-scale}}{y-scale} \cdot \frac{\frac{\left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right) \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)}{x-scale}}{y-scale} + \left(-\left(4 \cdot \frac{\frac{{\left(a \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} + {\left(b \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2}}{x-scale}}{x-scale}\right) \cdot \frac{\frac{{\left(a \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} + {\left(b \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2}}{y-scale}}{y-scale}\right)} \]
      +-commutative [=>] 42.4	\[ \color{blue}{\left(-\left(4 \cdot \frac{\frac{{\left(a \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} + {\left(b \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2}}{x-scale}}{x-scale}\right) \cdot \frac{\frac{{\left(a \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} + {\left(b \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2}}{y-scale}}{y-scale}\right) + \frac{\frac{\left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right) \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)}{x-scale}}{y-scale} \cdot \frac{\frac{\left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right) \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)}{x-scale}}{y-scale}} \]

   3. Taylor expanded in angle around 0 35.6

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

   4. Simplified 29.4

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

      [Start] 35.6	\[ -4 \cdot \frac{{a}^{2} \cdot {b}^{2}}{{x-scale}^{2} \cdot {y-scale}^{2}} \]
      *-commutative [=>] 35.6	\[ -4 \cdot \frac{{a}^{2} \cdot {b}^{2}}{\color{blue}{{y-scale}^{2} \cdot {x-scale}^{2}}} \]
      associate-*r/ [=>] 35.6	\[ \color{blue}{\frac{-4 \cdot \left({a}^{2} \cdot {b}^{2}\right)}{{y-scale}^{2} \cdot {x-scale}^{2}}} \]
      associate-/l* [=>] 35.7	\[ \color{blue}{\frac{-4}{\frac{{y-scale}^{2} \cdot {x-scale}^{2}}{{a}^{2} \cdot {b}^{2}}}} \]
      unpow2 [=>] 35.7	\[ \frac{-4}{\frac{\color{blue}{\left(y-scale \cdot y-scale\right)} \cdot {x-scale}^{2}}{{a}^{2} \cdot {b}^{2}}} \]
      unpow2 [=>] 35.7	\[ \frac{-4}{\frac{\left(y-scale \cdot y-scale\right) \cdot \color{blue}{\left(x-scale \cdot x-scale\right)}}{{a}^{2} \cdot {b}^{2}}} \]
      unswap-sqr [=>] 33.0	\[ \frac{-4}{\frac{\color{blue}{\left(y-scale \cdot x-scale\right) \cdot \left(y-scale \cdot x-scale\right)}}{{a}^{2} \cdot {b}^{2}}} \]
      *-commutative [=>] 33.0	\[ \frac{-4}{\frac{\left(y-scale \cdot x-scale\right) \cdot \left(y-scale \cdot x-scale\right)}{\color{blue}{{b}^{2} \cdot {a}^{2}}}} \]
      unpow2 [=>] 33.0	\[ \frac{-4}{\frac{\left(y-scale \cdot x-scale\right) \cdot \left(y-scale \cdot x-scale\right)}{\color{blue}{\left(b \cdot b\right)} \cdot {a}^{2}}} \]
      associate-*l* [=>] 29.4	\[ \frac{-4}{\frac{\left(y-scale \cdot x-scale\right) \cdot \left(y-scale \cdot x-scale\right)}{\color{blue}{b \cdot \left(b \cdot {a}^{2}\right)}}} \]
      unpow2 [=>] 29.4	\[ \frac{-4}{\frac{\left(y-scale \cdot x-scale\right) \cdot \left(y-scale \cdot x-scale\right)}{b \cdot \left(b \cdot \color{blue}{\left(a \cdot a\right)}\right)}} \]

   5. Applied egg-rr 4.8

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

   if 3.35000000000000001e-202 < x-scale

   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. Simplified 45.6

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

      [Start] 39.7	\[ \frac{\frac{\left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right) \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)}{x-scale}}{y-scale} \cdot \frac{\frac{\left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right) \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)}{x-scale}}{y-scale} - \left(4 \cdot \frac{\frac{{\left(a \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} + {\left(b \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2}}{x-scale}}{x-scale}\right) \cdot \frac{\frac{{\left(a \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} + {\left(b \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2}}{y-scale}}{y-scale} \]
      sub-neg [=>] 39.7	\[ \color{blue}{\frac{\frac{\left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right) \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)}{x-scale}}{y-scale} \cdot \frac{\frac{\left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right) \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)}{x-scale}}{y-scale} + \left(-\left(4 \cdot \frac{\frac{{\left(a \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} + {\left(b \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2}}{x-scale}}{x-scale}\right) \cdot \frac{\frac{{\left(a \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} + {\left(b \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2}}{y-scale}}{y-scale}\right)} \]
      +-commutative [=>] 39.7	\[ \color{blue}{\left(-\left(4 \cdot \frac{\frac{{\left(a \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} + {\left(b \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2}}{x-scale}}{x-scale}\right) \cdot \frac{\frac{{\left(a \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} + {\left(b \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2}}{y-scale}}{y-scale}\right) + \frac{\frac{\left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right) \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)}{x-scale}}{y-scale} \cdot \frac{\frac{\left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right) \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)}{x-scale}}{y-scale}} \]

   3. Taylor expanded in angle around 0 37.2

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

   4. Simplified 26.9

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

      [Start] 37.2	\[ -4 \cdot \frac{{a}^{2} \cdot {b}^{2}}{{x-scale}^{2} \cdot {y-scale}^{2}} \]
      *-commutative [=>] 37.2	\[ -4 \cdot \frac{{a}^{2} \cdot {b}^{2}}{\color{blue}{{y-scale}^{2} \cdot {x-scale}^{2}}} \]
      associate-*r/ [=>] 37.2	\[ \color{blue}{\frac{-4 \cdot \left({a}^{2} \cdot {b}^{2}\right)}{{y-scale}^{2} \cdot {x-scale}^{2}}} \]
      associate-/l* [=>] 37.2	\[ \color{blue}{\frac{-4}{\frac{{y-scale}^{2} \cdot {x-scale}^{2}}{{a}^{2} \cdot {b}^{2}}}} \]
      unpow2 [=>] 37.2	\[ \frac{-4}{\frac{\color{blue}{\left(y-scale \cdot y-scale\right)} \cdot {x-scale}^{2}}{{a}^{2} \cdot {b}^{2}}} \]
      unpow2 [=>] 37.2	\[ \frac{-4}{\frac{\left(y-scale \cdot y-scale\right) \cdot \color{blue}{\left(x-scale \cdot x-scale\right)}}{{a}^{2} \cdot {b}^{2}}} \]
      unswap-sqr [=>] 31.3	\[ \frac{-4}{\frac{\color{blue}{\left(y-scale \cdot x-scale\right) \cdot \left(y-scale \cdot x-scale\right)}}{{a}^{2} \cdot {b}^{2}}} \]
      *-commutative [=>] 31.3	\[ \frac{-4}{\frac{\left(y-scale \cdot x-scale\right) \cdot \left(y-scale \cdot x-scale\right)}{\color{blue}{{b}^{2} \cdot {a}^{2}}}} \]
      unpow2 [=>] 31.3	\[ \frac{-4}{\frac{\left(y-scale \cdot x-scale\right) \cdot \left(y-scale \cdot x-scale\right)}{\color{blue}{\left(b \cdot b\right)} \cdot {a}^{2}}} \]
      associate-*l* [=>] 26.9	\[ \frac{-4}{\frac{\left(y-scale \cdot x-scale\right) \cdot \left(y-scale \cdot x-scale\right)}{\color{blue}{b \cdot \left(b \cdot {a}^{2}\right)}}} \]
      unpow2 [=>] 26.9	\[ \frac{-4}{\frac{\left(y-scale \cdot x-scale\right) \cdot \left(y-scale \cdot x-scale\right)}{b \cdot \left(b \cdot \color{blue}{\left(a \cdot a\right)}\right)}} \]

   5. Applied egg-rr 5.4

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

   6. Applied egg-rr 11.5

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

   7. Simplified 9.3

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

      [Start] 11.5	\[ \left(-4 \cdot \frac{b}{y-scale \cdot x-scale}\right) \cdot \left(\left(\frac{a}{x-scale} \cdot \frac{b}{y-scale}\right) \cdot a\right) \]
      associate-*l* [=>] 11.5	\[ \color{blue}{-4 \cdot \left(\frac{b}{y-scale \cdot x-scale} \cdot \left(\left(\frac{a}{x-scale} \cdot \frac{b}{y-scale}\right) \cdot a\right)\right)} \]
      *-commutative [<=] 11.5	\[ -4 \cdot \color{blue}{\left(\left(\left(\frac{a}{x-scale} \cdot \frac{b}{y-scale}\right) \cdot a\right) \cdot \frac{b}{y-scale \cdot x-scale}\right)} \]
      associate-*l* [=>] 9.4	\[ -4 \cdot \color{blue}{\left(\left(\frac{a}{x-scale} \cdot \frac{b}{y-scale}\right) \cdot \left(a \cdot \frac{b}{y-scale \cdot x-scale}\right)\right)} \]
      associate-*l/ [=>] 8.8	\[ -4 \cdot \left(\color{blue}{\frac{a \cdot \frac{b}{y-scale}}{x-scale}} \cdot \left(a \cdot \frac{b}{y-scale \cdot x-scale}\right)\right) \]
      *-commutative [<=] 8.8	\[ -4 \cdot \left(\frac{\color{blue}{\frac{b}{y-scale} \cdot a}}{x-scale} \cdot \left(a \cdot \frac{b}{y-scale \cdot x-scale}\right)\right) \]
      associate-/r/ [<=] 9.3	\[ -4 \cdot \left(\frac{\color{blue}{\frac{b}{\frac{y-scale}{a}}}}{x-scale} \cdot \left(a \cdot \frac{b}{y-scale \cdot x-scale}\right)\right) \]

   8. Applied egg-rr 4.9

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

4. Final simplification 5.9

   \[\leadsto \begin{array}{l} \mathbf{if}\;x-scale \leq -1.2 \cdot 10^{+117}:\\ \;\;\;\;\frac{\left(-4 \cdot \frac{a}{x-scale}\right) \cdot \frac{b}{y-scale}}{\frac{y-scale}{b} \cdot \frac{x-scale}{a}}\\ \mathbf{elif}\;x-scale \leq -7 \cdot 10^{-172}:\\ \;\;\;\;\frac{-4}{\frac{x-scale \cdot y-scale}{a \cdot b} \cdot \frac{x-scale \cdot y-scale}{a \cdot b}}\\ \mathbf{elif}\;x-scale \leq 3.35 \cdot 10^{-202}:\\ \;\;\;\;\frac{\left(-4 \cdot \frac{a}{x-scale}\right) \cdot \frac{b}{y-scale}}{\frac{y-scale}{b} \cdot \frac{x-scale}{a}}\\ \mathbf{else}:\\ \;\;\;\;-4 \cdot \left(\frac{\frac{b}{\frac{y-scale}{a}}}{x-scale} \cdot \frac{b \cdot \frac{a}{y-scale}}{x-scale}\right)\\ \end{array} \]

Alternatives

Alternative 1
Error	7.0
Cost	1616

\[\begin{array}{l} t_0 := -4 \cdot \left(\frac{\frac{b}{\frac{y-scale}{a}}}{x-scale} \cdot \frac{b \cdot \frac{a}{y-scale}}{x-scale}\right)\\ t_1 := \frac{-4}{\frac{y-scale}{b}} \cdot \frac{\frac{a}{x-scale} \cdot \frac{b}{y-scale}}{\frac{x-scale}{a}}\\ \mathbf{if}\;b \leq -1600000:\\ \;\;\;\;t_1\\ \mathbf{elif}\;b \leq -5.7 \cdot 10^{-182}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;b \leq -2 \cdot 10^{-251}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;b \leq 3.4 \cdot 10^{-274}:\\ \;\;\;\;a \cdot \frac{-4 \cdot \frac{b}{x-scale \cdot y-scale}}{\frac{x-scale \cdot y-scale}{a \cdot b}}\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \]

Alternative 2
Error	6.1
Cost	1485

\[\begin{array}{l} t_0 := \frac{x-scale \cdot y-scale}{a \cdot b}\\ \mathbf{if}\;y-scale \leq -1.65 \cdot 10^{+70} \lor \neg \left(y-scale \leq -2.25 \cdot 10^{-217}\right) \land y-scale \leq 2.05 \cdot 10^{+77}:\\ \;\;\;\;\frac{-4}{t_0 \cdot t_0}\\ \mathbf{else}:\\ \;\;\;\;-4 \cdot \left(\frac{\frac{b}{\frac{y-scale}{a}}}{x-scale} \cdot \frac{b \cdot \frac{a}{y-scale}}{x-scale}\right)\\ \end{array} \]

Alternative 3
Error	25.3
Cost	1352

\[\begin{array}{l} \mathbf{if}\;a \leq -7.2 \cdot 10^{+145}:\\ \;\;\;\;0\\ \mathbf{elif}\;a \leq 1.35 \cdot 10^{+154}:\\ \;\;\;\;-4 \cdot \left(\left(\frac{b}{y-scale} \cdot \frac{b}{y-scale}\right) \cdot \frac{\frac{a \cdot a}{x-scale}}{x-scale}\right)\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \]

Alternative 4
Error	7.7
Cost	1220

\[\begin{array}{l} \mathbf{if}\;b \leq -3.4 \cdot 10^{+123}:\\ \;\;\;\;\left(-4 \cdot \left(\frac{a}{x-scale} \cdot \frac{a}{x-scale}\right)\right) \cdot \left(\frac{b}{y-scale} \cdot \frac{b}{y-scale}\right)\\ \mathbf{else}:\\ \;\;\;\;-4 \cdot \left(\frac{\frac{b}{\frac{y-scale}{a}}}{x-scale} \cdot \frac{b \cdot \frac{a}{y-scale}}{x-scale}\right)\\ \end{array} \]

Alternative 5
Error	6.6
Cost	1220

\[\begin{array}{l} \mathbf{if}\;b \leq -1400000:\\ \;\;\;\;\frac{-4}{\frac{y-scale}{b}} \cdot \frac{\frac{a}{x-scale} \cdot \frac{b}{y-scale}}{\frac{x-scale}{a}}\\ \mathbf{else}:\\ \;\;\;\;-4 \cdot \left(\frac{\frac{b}{\frac{y-scale}{a}}}{x-scale} \cdot \frac{b \cdot \frac{a}{y-scale}}{x-scale}\right)\\ \end{array} \]

Alternative 6
Error	11.0
Cost	1088

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

Alternative 7
Error	6.2
Cost	1088

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

Alternative 8
Error	30.5
Cost	64

\[0 \]

Reproduce

herbie shell --seed 2023060 
(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))))