Simplification of discriminant from scale-rotated-ellipse

Average Error: 41.4 → 6.3

Time: 1.2min

Precision: binary64

Cost: 1869

Math

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

↓

\[\begin{array}{l} t_0 := \frac{y-scale}{b} \cdot \frac{x-scale}{a}\\ \mathbf{if}\;\frac{angle}{180} \leq -5 \cdot 10^{-85}:\\ \;\;\;\;\frac{\frac{a}{y-scale} \cdot \left(-4 \cdot \frac{b}{x-scale}\right)}{\frac{y-scale}{a} \cdot \frac{x-scale}{b}}\\ \mathbf{elif}\;\frac{angle}{180} \leq 10^{+74} \lor \neg \left(\frac{angle}{180} \leq 2 \cdot 10^{+208}\right):\\ \;\;\;\;\frac{-4}{t_0 \cdot t_0}\\ \mathbf{else}:\\ \;\;\;\;\frac{-4 \cdot \left(b \cdot \frac{a}{y-scale \cdot x-scale}\right)}{y-scale \cdot x-scale} \cdot \left(a \cdot b\right)\\ \end{array} \]

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 (* (/ y-scale b) (/ x-scale a))))
   (if (<= (/ angle 180.0) -5e-85)
     (/
      (* (/ a y-scale) (* -4.0 (/ b x-scale)))
      (* (/ y-scale a) (/ x-scale b)))
     (if (or (<= (/ angle 180.0) 1e+74) (not (<= (/ angle 180.0) 2e+208)))
       (/ -4.0 (* t_0 t_0))
       (*
        (/ (* -4.0 (* b (/ a (* y-scale x-scale)))) (* y-scale x-scale))
        (* a b))))))

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 / b) * (x_45_scale / a);
	double tmp;
	if ((angle / 180.0) <= -5e-85) {
		tmp = ((a / y_45_scale) * (-4.0 * (b / x_45_scale))) / ((y_45_scale / a) * (x_45_scale / b));
	} else if (((angle / 180.0) <= 1e+74) || !((angle / 180.0) <= 2e+208)) {
		tmp = -4.0 / (t_0 * t_0);
	} else {
		tmp = ((-4.0 * (b * (a / (y_45_scale * x_45_scale)))) / (y_45_scale * x_45_scale)) * (a * b);
	}
	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 / b) * (x_45_scale / a);
	double tmp;
	if ((angle / 180.0) <= -5e-85) {
		tmp = ((a / y_45_scale) * (-4.0 * (b / x_45_scale))) / ((y_45_scale / a) * (x_45_scale / b));
	} else if (((angle / 180.0) <= 1e+74) || !((angle / 180.0) <= 2e+208)) {
		tmp = -4.0 / (t_0 * t_0);
	} else {
		tmp = ((-4.0 * (b * (a / (y_45_scale * x_45_scale)))) / (y_45_scale * x_45_scale)) * (a * b);
	}
	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 / b) * (x_45_scale / a)
	tmp = 0
	if (angle / 180.0) <= -5e-85:
		tmp = ((a / y_45_scale) * (-4.0 * (b / x_45_scale))) / ((y_45_scale / a) * (x_45_scale / b))
	elif ((angle / 180.0) <= 1e+74) or not ((angle / 180.0) <= 2e+208):
		tmp = -4.0 / (t_0 * t_0)
	else:
		tmp = ((-4.0 * (b * (a / (y_45_scale * x_45_scale)))) / (y_45_scale * x_45_scale)) * (a * b)
	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 / b) * Float64(x_45_scale / a))
	tmp = 0.0
	if (Float64(angle / 180.0) <= -5e-85)
		tmp = Float64(Float64(Float64(a / y_45_scale) * Float64(-4.0 * Float64(b / x_45_scale))) / Float64(Float64(y_45_scale / a) * Float64(x_45_scale / b)));
	elseif ((Float64(angle / 180.0) <= 1e+74) || !(Float64(angle / 180.0) <= 2e+208))
		tmp = Float64(-4.0 / Float64(t_0 * t_0));
	else
		tmp = Float64(Float64(Float64(-4.0 * Float64(b * Float64(a / Float64(y_45_scale * x_45_scale)))) / Float64(y_45_scale * x_45_scale)) * Float64(a * b));
	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 / b) * (x_45_scale / a);
	tmp = 0.0;
	if ((angle / 180.0) <= -5e-85)
		tmp = ((a / y_45_scale) * (-4.0 * (b / x_45_scale))) / ((y_45_scale / a) * (x_45_scale / b));
	elseif (((angle / 180.0) <= 1e+74) || ~(((angle / 180.0) <= 2e+208)))
		tmp = -4.0 / (t_0 * t_0);
	else
		tmp = ((-4.0 * (b * (a / (y_45_scale * x_45_scale)))) / (y_45_scale * x_45_scale)) * (a * b);
	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 / b), $MachinePrecision] * N[(x$45$scale / a), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[(angle / 180.0), $MachinePrecision], -5e-85], N[(N[(N[(a / y$45$scale), $MachinePrecision] * N[(-4.0 * N[(b / x$45$scale), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(N[(y$45$scale / a), $MachinePrecision] * N[(x$45$scale / b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[Or[LessEqual[N[(angle / 180.0), $MachinePrecision], 1e+74], N[Not[LessEqual[N[(angle / 180.0), $MachinePrecision], 2e+208]], $MachinePrecision]], N[(-4.0 / N[(t$95$0 * t$95$0), $MachinePrecision]), $MachinePrecision], N[(N[(N[(-4.0 * N[(b * N[(a / N[(y$45$scale * x$45$scale), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(y$45$scale * x$45$scale), $MachinePrecision]), $MachinePrecision] * N[(a * b), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if (/.f64 angle 180) < -5.0000000000000002e-85

   1. Initial program 45.3

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

   2. Taylor expanded in angle around 0 39.9

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

   3. Simplified 19.9

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

      [Start] 39.9	\[ -4 \cdot \frac{{a}^{2} \cdot {b}^{2}}{{y-scale}^{2} \cdot {x-scale}^{2}} \]
      *-commutative [=>] 39.9	\[ \color{blue}{\frac{{a}^{2} \cdot {b}^{2}}{{y-scale}^{2} \cdot {x-scale}^{2}} \cdot -4} \]
      times-frac [=>] 39.9	\[ \color{blue}{\left(\frac{{a}^{2}}{{y-scale}^{2}} \cdot \frac{{b}^{2}}{{x-scale}^{2}}\right)} \cdot -4 \]
      associate-*l* [=>] 39.9	\[ \color{blue}{\frac{{a}^{2}}{{y-scale}^{2}} \cdot \left(\frac{{b}^{2}}{{x-scale}^{2}} \cdot -4\right)} \]
      unpow2 [=>] 39.9	\[ \frac{\color{blue}{a \cdot a}}{{y-scale}^{2}} \cdot \left(\frac{{b}^{2}}{{x-scale}^{2}} \cdot -4\right) \]
      unpow2 [=>] 39.9	\[ \frac{a \cdot a}{\color{blue}{y-scale \cdot y-scale}} \cdot \left(\frac{{b}^{2}}{{x-scale}^{2}} \cdot -4\right) \]
      times-frac [=>] 32.5	\[ \color{blue}{\left(\frac{a}{y-scale} \cdot \frac{a}{y-scale}\right)} \cdot \left(\frac{{b}^{2}}{{x-scale}^{2}} \cdot -4\right) \]
      unpow2 [=>] 32.5	\[ \left(\frac{a}{y-scale} \cdot \frac{a}{y-scale}\right) \cdot \left(\frac{\color{blue}{b \cdot b}}{{x-scale}^{2}} \cdot -4\right) \]
      unpow2 [=>] 32.5	\[ \left(\frac{a}{y-scale} \cdot \frac{a}{y-scale}\right) \cdot \left(\frac{b \cdot b}{\color{blue}{x-scale \cdot x-scale}} \cdot -4\right) \]
      times-frac [=>] 19.9	\[ \left(\frac{a}{y-scale} \cdot \frac{a}{y-scale}\right) \cdot \left(\color{blue}{\left(\frac{b}{x-scale} \cdot \frac{b}{x-scale}\right)} \cdot -4\right) \]

   4. Applied egg-rr 20.0

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

   5. Applied egg-rr 6.2

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

   if -5.0000000000000002e-85 < (/.f64 angle 180) < 9.99999999999999952e73 or 2e208 < (/.f64 angle 180)

   1. Initial program 39.0

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

   2. Simplified 46.6

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

      [Start] 39.0	\[ \frac{\frac{\left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right) \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)}{x-scale}}{y-scale} \cdot \frac{\frac{\left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right) \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)}{x-scale}}{y-scale} - \left(4 \cdot \frac{\frac{{\left(a \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} + {\left(b \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2}}{x-scale}}{x-scale}\right) \cdot \frac{\frac{{\left(a \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} + {\left(b \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2}}{y-scale}}{y-scale} \]
      sub-neg [=>] 39.0	\[ \color{blue}{\frac{\frac{\left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right) \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)}{x-scale}}{y-scale} \cdot \frac{\frac{\left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right) \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)}{x-scale}}{y-scale} + \left(-\left(4 \cdot \frac{\frac{{\left(a \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} + {\left(b \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2}}{x-scale}}{x-scale}\right) \cdot \frac{\frac{{\left(a \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} + {\left(b \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2}}{y-scale}}{y-scale}\right)} \]
      +-commutative [=>] 39.0	\[ \color{blue}{\left(-\left(4 \cdot \frac{\frac{{\left(a \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} + {\left(b \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2}}{x-scale}}{x-scale}\right) \cdot \frac{\frac{{\left(a \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} + {\left(b \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2}}{y-scale}}{y-scale}\right) + \frac{\frac{\left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right) \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)}{x-scale}}{y-scale} \cdot \frac{\frac{\left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right) \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)}{x-scale}}{y-scale}} \]

   3. Taylor expanded in angle around 0 39.5

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

   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] 39.5	\[ -4 \cdot \frac{{a}^{2} \cdot {b}^{2}}{{x-scale}^{2} \cdot {y-scale}^{2}} \]
      *-commutative [=>] 39.5	\[ -4 \cdot \frac{{a}^{2} \cdot {b}^{2}}{\color{blue}{{y-scale}^{2} \cdot {x-scale}^{2}}} \]
      associate-*r/ [=>] 39.5	\[ \color{blue}{\frac{-4 \cdot \left({a}^{2} \cdot {b}^{2}\right)}{{y-scale}^{2} \cdot {x-scale}^{2}}} \]
      associate-/l* [=>] 39.5	\[ \color{blue}{\frac{-4}{\frac{{y-scale}^{2} \cdot {x-scale}^{2}}{{a}^{2} \cdot {b}^{2}}}} \]
      unpow2 [=>] 39.5	\[ \frac{-4}{\frac{\color{blue}{\left(y-scale \cdot y-scale\right)} \cdot {x-scale}^{2}}{{a}^{2} \cdot {b}^{2}}} \]
      unpow2 [=>] 39.5	\[ \frac{-4}{\frac{\left(y-scale \cdot y-scale\right) \cdot \color{blue}{\left(x-scale \cdot x-scale\right)}}{{a}^{2} \cdot {b}^{2}}} \]
      unswap-sqr [=>] 31.6	\[ \frac{-4}{\frac{\color{blue}{\left(y-scale \cdot x-scale\right) \cdot \left(y-scale \cdot x-scale\right)}}{{a}^{2} \cdot {b}^{2}}} \]
      *-commutative [=>] 31.6	\[ \frac{-4}{\frac{\left(y-scale \cdot x-scale\right) \cdot \left(y-scale \cdot x-scale\right)}{\color{blue}{{b}^{2} \cdot {a}^{2}}}} \]
      unpow2 [=>] 31.6	\[ \frac{-4}{\frac{\left(y-scale \cdot x-scale\right) \cdot \left(y-scale \cdot x-scale\right)}{\color{blue}{\left(b \cdot b\right)} \cdot {a}^{2}}} \]
      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. Taylor expanded in y-scale around 0 39.5

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

   6. Simplified 5.9

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

      [Start] 39.5	\[ \frac{-4}{\frac{{y-scale}^{2} \cdot {x-scale}^{2}}{{a}^{2} \cdot {b}^{2}}} \]
      *-commutative [=>] 39.5	\[ \frac{-4}{\frac{\color{blue}{{x-scale}^{2} \cdot {y-scale}^{2}}}{{a}^{2} \cdot {b}^{2}}} \]
      times-frac [=>] 39.6	\[ \frac{-4}{\color{blue}{\frac{{x-scale}^{2}}{{a}^{2}} \cdot \frac{{y-scale}^{2}}{{b}^{2}}}} \]
      unpow2 [=>] 39.6	\[ \frac{-4}{\frac{\color{blue}{x-scale \cdot x-scale}}{{a}^{2}} \cdot \frac{{y-scale}^{2}}{{b}^{2}}} \]
      unpow2 [=>] 39.6	\[ \frac{-4}{\frac{x-scale \cdot x-scale}{\color{blue}{a \cdot a}} \cdot \frac{{y-scale}^{2}}{{b}^{2}}} \]
      unpow2 [=>] 39.6	\[ \frac{-4}{\frac{x-scale \cdot x-scale}{a \cdot a} \cdot \frac{\color{blue}{y-scale \cdot y-scale}}{{b}^{2}}} \]
      unpow2 [=>] 39.6	\[ \frac{-4}{\frac{x-scale \cdot x-scale}{a \cdot a} \cdot \frac{y-scale \cdot y-scale}{\color{blue}{b \cdot b}}} \]
      *-commutative [<=] 39.6	\[ \frac{-4}{\color{blue}{\frac{y-scale \cdot y-scale}{b \cdot b} \cdot \frac{x-scale \cdot x-scale}{a \cdot a}}} \]
      times-frac [=>] 30.9	\[ \frac{-4}{\color{blue}{\left(\frac{y-scale}{b} \cdot \frac{y-scale}{b}\right)} \cdot \frac{x-scale \cdot x-scale}{a \cdot a}} \]
      times-frac [=>] 18.6	\[ \frac{-4}{\left(\frac{y-scale}{b} \cdot \frac{y-scale}{b}\right) \cdot \color{blue}{\left(\frac{x-scale}{a} \cdot \frac{x-scale}{a}\right)}} \]
      swap-sqr [<=] 5.9	\[ \frac{-4}{\color{blue}{\left(\frac{y-scale}{b} \cdot \frac{x-scale}{a}\right) \cdot \left(\frac{y-scale}{b} \cdot \frac{x-scale}{a}\right)}} \]
      times-frac [<=] 10.8	\[ \frac{-4}{\left(\frac{y-scale}{b} \cdot \frac{x-scale}{a}\right) \cdot \color{blue}{\frac{y-scale \cdot x-scale}{b \cdot a}}} \]
      times-frac [<=] 5.8	\[ \frac{-4}{\color{blue}{\frac{y-scale \cdot x-scale}{b \cdot a}} \cdot \frac{y-scale \cdot x-scale}{b \cdot a}} \]
      unpow2 [<=] 5.8	\[ \frac{-4}{\color{blue}{{\left(\frac{y-scale \cdot x-scale}{b \cdot a}\right)}^{2}}} \]
      times-frac [=>] 5.9	\[ \frac{-4}{{\color{blue}{\left(\frac{y-scale}{b} \cdot \frac{x-scale}{a}\right)}}^{2}} \]

   7. Applied egg-rr 5.9

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

   if 9.99999999999999952e73 < (/.f64 angle 180) < 2e208

   1. Initial program 43.1

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

   2. Simplified 48.1

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

      [Start] 43.1	\[ \frac{\frac{\left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right) \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)}{x-scale}}{y-scale} \cdot \frac{\frac{\left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right) \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)}{x-scale}}{y-scale} - \left(4 \cdot \frac{\frac{{\left(a \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} + {\left(b \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2}}{x-scale}}{x-scale}\right) \cdot \frac{\frac{{\left(a \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} + {\left(b \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2}}{y-scale}}{y-scale} \]
      sub-neg [=>] 43.1	\[ \color{blue}{\frac{\frac{\left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right) \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)}{x-scale}}{y-scale} \cdot \frac{\frac{\left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right) \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)}{x-scale}}{y-scale} + \left(-\left(4 \cdot \frac{\frac{{\left(a \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} + {\left(b \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2}}{x-scale}}{x-scale}\right) \cdot \frac{\frac{{\left(a \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} + {\left(b \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2}}{y-scale}}{y-scale}\right)} \]
      +-commutative [=>] 43.1	\[ \color{blue}{\left(-\left(4 \cdot \frac{\frac{{\left(a \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} + {\left(b \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2}}{x-scale}}{x-scale}\right) \cdot \frac{\frac{{\left(a \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} + {\left(b \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2}}{y-scale}}{y-scale}\right) + \frac{\frac{\left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right) \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)}{x-scale}}{y-scale} \cdot \frac{\frac{\left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right) \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)}{x-scale}}{y-scale}} \]

   3. Taylor expanded in angle around 0 37.5

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

   4. Simplified 25.4

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

      [Start] 37.5	\[ -4 \cdot \frac{{a}^{2} \cdot {b}^{2}}{{x-scale}^{2} \cdot {y-scale}^{2}} \]
      *-commutative [=>] 37.5	\[ -4 \cdot \frac{{a}^{2} \cdot {b}^{2}}{\color{blue}{{y-scale}^{2} \cdot {x-scale}^{2}}} \]
      associate-*r/ [=>] 37.5	\[ \color{blue}{\frac{-4 \cdot \left({a}^{2} \cdot {b}^{2}\right)}{{y-scale}^{2} \cdot {x-scale}^{2}}} \]
      associate-/l* [=>] 37.6	\[ \color{blue}{\frac{-4}{\frac{{y-scale}^{2} \cdot {x-scale}^{2}}{{a}^{2} \cdot {b}^{2}}}} \]
      unpow2 [=>] 37.6	\[ \frac{-4}{\frac{\color{blue}{\left(y-scale \cdot y-scale\right)} \cdot {x-scale}^{2}}{{a}^{2} \cdot {b}^{2}}} \]
      unpow2 [=>] 37.6	\[ \frac{-4}{\frac{\left(y-scale \cdot y-scale\right) \cdot \color{blue}{\left(x-scale \cdot x-scale\right)}}{{a}^{2} \cdot {b}^{2}}} \]
      unswap-sqr [=>] 29.2	\[ \frac{-4}{\frac{\color{blue}{\left(y-scale \cdot x-scale\right) \cdot \left(y-scale \cdot x-scale\right)}}{{a}^{2} \cdot {b}^{2}}} \]
      *-commutative [=>] 29.2	\[ \frac{-4}{\frac{\left(y-scale \cdot x-scale\right) \cdot \left(y-scale \cdot x-scale\right)}{\color{blue}{{b}^{2} \cdot {a}^{2}}}} \]
      unpow2 [=>] 29.2	\[ \frac{-4}{\frac{\left(y-scale \cdot x-scale\right) \cdot \left(y-scale \cdot x-scale\right)}{\color{blue}{\left(b \cdot b\right)} \cdot {a}^{2}}} \]
      associate-*l* [=>] 25.4	\[ \frac{-4}{\frac{\left(y-scale \cdot x-scale\right) \cdot \left(y-scale \cdot x-scale\right)}{\color{blue}{b \cdot \left(b \cdot {a}^{2}\right)}}} \]
      unpow2 [=>] 25.4	\[ \frac{-4}{\frac{\left(y-scale \cdot x-scale\right) \cdot \left(y-scale \cdot x-scale\right)}{b \cdot \left(b \cdot \color{blue}{\left(a \cdot a\right)}\right)}} \]

   5. Taylor expanded in y-scale around 0 37.6

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

   6. Simplified 5.5

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

      [Start] 37.6	\[ \frac{-4}{\frac{{y-scale}^{2} \cdot {x-scale}^{2}}{{a}^{2} \cdot {b}^{2}}} \]
      *-commutative [=>] 37.6	\[ \frac{-4}{\frac{\color{blue}{{x-scale}^{2} \cdot {y-scale}^{2}}}{{a}^{2} \cdot {b}^{2}}} \]
      times-frac [=>] 37.6	\[ \frac{-4}{\color{blue}{\frac{{x-scale}^{2}}{{a}^{2}} \cdot \frac{{y-scale}^{2}}{{b}^{2}}}} \]
      unpow2 [=>] 37.6	\[ \frac{-4}{\frac{\color{blue}{x-scale \cdot x-scale}}{{a}^{2}} \cdot \frac{{y-scale}^{2}}{{b}^{2}}} \]
      unpow2 [=>] 37.6	\[ \frac{-4}{\frac{x-scale \cdot x-scale}{\color{blue}{a \cdot a}} \cdot \frac{{y-scale}^{2}}{{b}^{2}}} \]
      unpow2 [=>] 37.6	\[ \frac{-4}{\frac{x-scale \cdot x-scale}{a \cdot a} \cdot \frac{\color{blue}{y-scale \cdot y-scale}}{{b}^{2}}} \]
      unpow2 [=>] 37.6	\[ \frac{-4}{\frac{x-scale \cdot x-scale}{a \cdot a} \cdot \frac{y-scale \cdot y-scale}{\color{blue}{b \cdot b}}} \]
      *-commutative [<=] 37.6	\[ \frac{-4}{\color{blue}{\frac{y-scale \cdot y-scale}{b \cdot b} \cdot \frac{x-scale \cdot x-scale}{a \cdot a}}} \]
      times-frac [=>] 31.6	\[ \frac{-4}{\color{blue}{\left(\frac{y-scale}{b} \cdot \frac{y-scale}{b}\right)} \cdot \frac{x-scale \cdot x-scale}{a \cdot a}} \]
      times-frac [=>] 18.8	\[ \frac{-4}{\left(\frac{y-scale}{b} \cdot \frac{y-scale}{b}\right) \cdot \color{blue}{\left(\frac{x-scale}{a} \cdot \frac{x-scale}{a}\right)}} \]
      swap-sqr [<=] 5.5	\[ \frac{-4}{\color{blue}{\left(\frac{y-scale}{b} \cdot \frac{x-scale}{a}\right) \cdot \left(\frac{y-scale}{b} \cdot \frac{x-scale}{a}\right)}} \]
      times-frac [<=] 10.6	\[ \frac{-4}{\left(\frac{y-scale}{b} \cdot \frac{x-scale}{a}\right) \cdot \color{blue}{\frac{y-scale \cdot x-scale}{b \cdot a}}} \]
      times-frac [<=] 5.9	\[ \frac{-4}{\color{blue}{\frac{y-scale \cdot x-scale}{b \cdot a}} \cdot \frac{y-scale \cdot x-scale}{b \cdot a}} \]
      unpow2 [<=] 5.9	\[ \frac{-4}{\color{blue}{{\left(\frac{y-scale \cdot x-scale}{b \cdot a}\right)}^{2}}} \]
      times-frac [=>] 5.5	\[ \frac{-4}{{\color{blue}{\left(\frac{y-scale}{b} \cdot \frac{x-scale}{a}\right)}}^{2}} \]

   7. Applied egg-rr 8.7

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

4. Final simplification 6.3

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{angle}{180} \leq -5 \cdot 10^{-85}:\\ \;\;\;\;\frac{\frac{a}{y-scale} \cdot \left(-4 \cdot \frac{b}{x-scale}\right)}{\frac{y-scale}{a} \cdot \frac{x-scale}{b}}\\ \mathbf{elif}\;\frac{angle}{180} \leq 10^{+74} \lor \neg \left(\frac{angle}{180} \leq 2 \cdot 10^{+208}\right):\\ \;\;\;\;\frac{-4}{\left(\frac{y-scale}{b} \cdot \frac{x-scale}{a}\right) \cdot \left(\frac{y-scale}{b} \cdot \frac{x-scale}{a}\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{-4 \cdot \left(b \cdot \frac{a}{y-scale \cdot x-scale}\right)}{y-scale \cdot x-scale} \cdot \left(a \cdot b\right)\\ \end{array} \]

Alternatives

Alternative 1
Error	24.2
Cost	1617

\[\begin{array}{l} \mathbf{if}\;a \leq -2 \cdot 10^{+141}:\\ \;\;\;\;0\\ \mathbf{elif}\;a \leq -2.95 \cdot 10^{-142} \lor \neg \left(a \leq 1.55 \cdot 10^{-118}\right) \land a \leq 8 \cdot 10^{+132}:\\ \;\;\;\;-4 \cdot \left(\left(\frac{b}{y-scale} \cdot \frac{b}{y-scale}\right) \cdot \frac{\frac{a \cdot a}{x-scale}}{x-scale}\right)\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \]

Alternative 2
Error	7.5
Cost	1353

\[\begin{array}{l} \mathbf{if}\;a \leq -5 \cdot 10^{-202} \lor \neg \left(a \leq 2.1 \cdot 10^{+83}\right):\\ \;\;\;\;-4 \cdot \left(\frac{a}{x-scale \cdot \frac{y-scale}{b}} \cdot \left(a \cdot \frac{\frac{b}{y-scale}}{x-scale}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(a \cdot b\right) \cdot \frac{-4 \cdot \frac{b}{\frac{y-scale \cdot x-scale}{a}}}{y-scale \cdot x-scale}\\ \end{array} \]

Alternative 3
Error	7.7
Cost	1352

\[\begin{array}{l} t_0 := \frac{a}{x-scale \cdot \frac{y-scale}{b}}\\ \mathbf{if}\;a \leq -1.5 \cdot 10^{-204}:\\ \;\;\;\;-4 \cdot \left(t_0 \cdot \left(a \cdot \frac{\frac{b}{y-scale}}{x-scale}\right)\right)\\ \mathbf{elif}\;a \leq 3.05 \cdot 10^{+193}:\\ \;\;\;\;\frac{-4 \cdot \left(b \cdot \frac{a}{y-scale \cdot x-scale}\right)}{y-scale \cdot x-scale} \cdot \left(a \cdot b\right)\\ \mathbf{else}:\\ \;\;\;\;-4 \cdot \left(t_0 \cdot t_0\right)\\ \end{array} \]

Alternative 4
Error	6.9
Cost	1352

\[\begin{array}{l} t_0 := \frac{a}{x-scale \cdot \frac{y-scale}{b}}\\ \mathbf{if}\;b \leq -5.6 \cdot 10^{-104}:\\ \;\;\;\;-4 \cdot \left(t_0 \cdot t_0\right)\\ \mathbf{elif}\;b \leq 2.2 \cdot 10^{-237}:\\ \;\;\;\;\left(a \cdot b\right) \cdot \frac{-4 \cdot \frac{\frac{a \cdot b}{x-scale}}{y-scale}}{y-scale \cdot x-scale}\\ \mathbf{else}:\\ \;\;\;\;-4 \cdot \left(t_0 \cdot \left(a \cdot \frac{\frac{b}{y-scale}}{x-scale}\right)\right)\\ \end{array} \]

Alternative 5
Error	7.7
Cost	1352

\[\begin{array}{l} \mathbf{if}\;a \leq -5 \cdot 10^{-203}:\\ \;\;\;\;-4 \cdot \left(\frac{a}{x-scale \cdot \frac{y-scale}{b}} \cdot \left(a \cdot \frac{\frac{b}{y-scale}}{x-scale}\right)\right)\\ \mathbf{elif}\;a \leq 1.5 \cdot 10^{+193}:\\ \;\;\;\;\left(a \cdot b\right) \cdot \frac{-4 \cdot \frac{b}{\frac{y-scale \cdot x-scale}{a}}}{y-scale \cdot x-scale}\\ \mathbf{else}:\\ \;\;\;\;\left(-4 \cdot \frac{a}{x-scale}\right) \cdot \frac{\frac{b}{y-scale} \cdot \frac{a}{x-scale}}{\frac{y-scale}{b}}\\ \end{array} \]

Alternative 6
Error	7.1
Cost	1352

\[\begin{array}{l} \mathbf{if}\;a \leq -2.6 \cdot 10^{-203}:\\ \;\;\;\;-4 \cdot \left(\frac{a}{x-scale \cdot \frac{y-scale}{b}} \cdot \left(a \cdot \frac{\frac{b}{y-scale}}{x-scale}\right)\right)\\ \mathbf{elif}\;a \leq 3.6 \cdot 10^{+197}:\\ \;\;\;\;\left(a \cdot b\right) \cdot \frac{\frac{-4}{\frac{y-scale \cdot x-scale}{a \cdot b}}}{y-scale \cdot x-scale}\\ \mathbf{else}:\\ \;\;\;\;\left(-4 \cdot \frac{a}{x-scale}\right) \cdot \frac{\frac{b}{y-scale} \cdot \frac{a}{x-scale}}{\frac{y-scale}{b}}\\ \end{array} \]

Alternative 7
Error	7.0
Cost	1352

\[\begin{array}{l} t_0 := \frac{y-scale}{b} \cdot \frac{x-scale}{a}\\ \mathbf{if}\;a \leq -3 \cdot 10^{-204}:\\ \;\;\;\;-4 \cdot \left(\frac{a}{x-scale \cdot \frac{y-scale}{b}} \cdot \left(a \cdot \frac{\frac{b}{y-scale}}{x-scale}\right)\right)\\ \mathbf{elif}\;a \leq 2.3 \cdot 10^{+88}:\\ \;\;\;\;\left(a \cdot b\right) \cdot \frac{\frac{-4}{\frac{y-scale \cdot x-scale}{a \cdot b}}}{y-scale \cdot x-scale}\\ \mathbf{else}:\\ \;\;\;\;\frac{-4}{t_0 \cdot t_0}\\ \end{array} \]

Alternative 8
Error	7.2
Cost	1352

\[\begin{array}{l} \mathbf{if}\;a \leq -1.3 \cdot 10^{-202}:\\ \;\;\;\;-4 \cdot \left(\frac{a}{x-scale \cdot \frac{y-scale}{b}} \cdot \left(a \cdot \frac{\frac{b}{y-scale}}{x-scale}\right)\right)\\ \mathbf{elif}\;a \leq 6.2 \cdot 10^{+201}:\\ \;\;\;\;\left(a \cdot b\right) \cdot \frac{\frac{-4}{\frac{y-scale \cdot x-scale}{a \cdot b}}}{y-scale \cdot x-scale}\\ \mathbf{else}:\\ \;\;\;\;\frac{-4 \cdot \frac{a}{x-scale}}{\frac{y-scale}{b} \cdot \left(\frac{y-scale}{b} \cdot \frac{x-scale}{a}\right)}\\ \end{array} \]

Alternative 9
Error	7.5
Cost	1088

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

Alternative 10
Error	6.2
Cost	1088

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

Alternative 11
Error	6.0
Cost	1088

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

Alternative 12
Error	30.2
Cost	64

\[0 \]

Reproduce

herbie shell --seed 2023039 
(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))))