Simplification of discriminant from scale-rotated-ellipse

Average Accuracy: 35.0% → 90.9%

Time: 1.3min

Precision: binary64

Cost: 13572

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;a \leq -8 \cdot 10^{+138}:\\ \;\;\;\;-4 \cdot {\left(\frac{\frac{a}{x-scale}}{\frac{y-scale}{b}}\right)}^{2}\\ \mathbf{else}:\\ \;\;\;\;-4 \cdot {\left(\frac{a \cdot b}{\left|x-scale \cdot y-scale\right|}\right)}^{2}\\ \end{array} \]

FPCore

(FPCore (a b angle x-scale y-scale)
 :precision binary64
 (-
  (*
   (/
    (/
     (*
      (* (* 2.0 (- (pow b 2.0) (pow a 2.0))) (sin (* (/ angle 180.0) PI)))
      (cos (* (/ angle 180.0) PI)))
     x-scale)
    y-scale)
   (/
    (/
     (*
      (* (* 2.0 (- (pow b 2.0) (pow a 2.0))) (sin (* (/ angle 180.0) PI)))
      (cos (* (/ angle 180.0) PI)))
     x-scale)
    y-scale))
  (*
   (*
    4.0
    (/
     (/
      (+
       (pow (* a (sin (* (/ angle 180.0) PI))) 2.0)
       (pow (* b (cos (* (/ angle 180.0) PI))) 2.0))
      x-scale)
     x-scale))
   (/
    (/
     (+
      (pow (* a (cos (* (/ angle 180.0) PI))) 2.0)
      (pow (* b (sin (* (/ angle 180.0) PI))) 2.0))
     y-scale)
    y-scale))))

↓

(FPCore (a b angle x-scale y-scale)
 :precision binary64
 (if (<= a -8e+138)
   (* -4.0 (pow (/ (/ a x-scale) (/ y-scale b)) 2.0))
   (* -4.0 (pow (/ (* a b) (fabs (* x-scale y-scale))) 2.0))))

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 tmp;
	if (a <= -8e+138) {
		tmp = -4.0 * pow(((a / x_45_scale) / (y_45_scale / b)), 2.0);
	} else {
		tmp = -4.0 * pow(((a * b) / fabs((x_45_scale * y_45_scale))), 2.0);
	}
	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 tmp;
	if (a <= -8e+138) {
		tmp = -4.0 * Math.pow(((a / x_45_scale) / (y_45_scale / b)), 2.0);
	} else {
		tmp = -4.0 * Math.pow(((a * b) / Math.abs((x_45_scale * y_45_scale))), 2.0);
	}
	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):
	tmp = 0
	if a <= -8e+138:
		tmp = -4.0 * math.pow(((a / x_45_scale) / (y_45_scale / b)), 2.0)
	else:
		tmp = -4.0 * math.pow(((a * b) / math.fabs((x_45_scale * y_45_scale))), 2.0)
	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)
	tmp = 0.0
	if (a <= -8e+138)
		tmp = Float64(-4.0 * (Float64(Float64(a / x_45_scale) / Float64(y_45_scale / b)) ^ 2.0));
	else
		tmp = Float64(-4.0 * (Float64(Float64(a * b) / abs(Float64(x_45_scale * y_45_scale))) ^ 2.0));
	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)
	tmp = 0.0;
	if (a <= -8e+138)
		tmp = -4.0 * (((a / x_45_scale) / (y_45_scale / b)) ^ 2.0);
	else
		tmp = -4.0 * (((a * b) / abs((x_45_scale * y_45_scale))) ^ 2.0);
	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_] := If[LessEqual[a, -8e+138], N[(-4.0 * N[Power[N[(N[(a / x$45$scale), $MachinePrecision] / N[(y$45$scale / b), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision], N[(-4.0 * N[Power[N[(N[(a * b), $MachinePrecision] / N[Abs[N[(x$45$scale * y$45$scale), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if a < -8.0000000000000003e138

   1. Initial program 2.8%

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

   2. Simplified 0.9%

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

      [Start] 2.8	\[ \frac{\frac{\left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right) \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)}{x-scale}}{y-scale} \cdot \frac{\frac{\left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right) \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)}{x-scale}}{y-scale} - \left(4 \cdot \frac{\frac{{\left(a \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} + {\left(b \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2}}{x-scale}}{x-scale}\right) \cdot \frac{\frac{{\left(a \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} + {\left(b \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2}}{y-scale}}{y-scale} \]
      sub-neg [=>] 2.8	\[ \color{blue}{\frac{\frac{\left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right) \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)}{x-scale}}{y-scale} \cdot \frac{\frac{\left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right) \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)}{x-scale}}{y-scale} + \left(-\left(4 \cdot \frac{\frac{{\left(a \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} + {\left(b \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2}}{x-scale}}{x-scale}\right) \cdot \frac{\frac{{\left(a \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} + {\left(b \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2}}{y-scale}}{y-scale}\right)} \]
      +-commutative [=>] 2.8	\[ \color{blue}{\left(-\left(4 \cdot \frac{\frac{{\left(a \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} + {\left(b \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2}}{x-scale}}{x-scale}\right) \cdot \frac{\frac{{\left(a \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} + {\left(b \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2}}{y-scale}}{y-scale}\right) + \frac{\frac{\left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right) \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)}{x-scale}}{y-scale} \cdot \frac{\frac{\left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right) \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)}{x-scale}}{y-scale}} \]

   3. Taylor expanded in angle around 0 4.3%

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

   4. Simplified 53.5%

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

      [Start] 4.3	\[ -4 \cdot \frac{{a}^{2} \cdot {b}^{2}}{{x-scale}^{2} \cdot {y-scale}^{2}} \]
      *-commutative [=>] 4.3	\[ \color{blue}{\frac{{a}^{2} \cdot {b}^{2}}{{x-scale}^{2} \cdot {y-scale}^{2}} \cdot -4} \]
      times-frac [=>] 4.5	\[ \color{blue}{\left(\frac{{a}^{2}}{{x-scale}^{2}} \cdot \frac{{b}^{2}}{{y-scale}^{2}}\right)} \cdot -4 \]
      associate-*l* [=>] 4.5	\[ \color{blue}{\frac{{a}^{2}}{{x-scale}^{2}} \cdot \left(\frac{{b}^{2}}{{y-scale}^{2}} \cdot -4\right)} \]
      unpow2 [=>] 4.5	\[ \frac{\color{blue}{a \cdot a}}{{x-scale}^{2}} \cdot \left(\frac{{b}^{2}}{{y-scale}^{2}} \cdot -4\right) \]
      unpow2 [=>] 4.5	\[ \frac{a \cdot a}{\color{blue}{x-scale \cdot x-scale}} \cdot \left(\frac{{b}^{2}}{{y-scale}^{2}} \cdot -4\right) \]
      times-frac [=>] 36.2	\[ \color{blue}{\left(\frac{a}{x-scale} \cdot \frac{a}{x-scale}\right)} \cdot \left(\frac{{b}^{2}}{{y-scale}^{2}} \cdot -4\right) \]
      unpow2 [=>] 36.2	\[ \left(\frac{a}{x-scale} \cdot \frac{a}{x-scale}\right) \cdot \left(\frac{\color{blue}{b \cdot b}}{{y-scale}^{2}} \cdot -4\right) \]
      unpow2 [=>] 36.2	\[ \left(\frac{a}{x-scale} \cdot \frac{a}{x-scale}\right) \cdot \left(\frac{b \cdot b}{\color{blue}{y-scale \cdot y-scale}} \cdot -4\right) \]
      times-frac [=>] 53.5	\[ \left(\frac{a}{x-scale} \cdot \frac{a}{x-scale}\right) \cdot \left(\color{blue}{\left(\frac{b}{y-scale} \cdot \frac{b}{y-scale}\right)} \cdot -4\right) \]

   5. Taylor expanded in a around 0 4.3%

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

   6. Simplified 81.6%

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

      [Start] 4.3	\[ -4 \cdot \frac{{a}^{2} \cdot {b}^{2}}{{y-scale}^{2} \cdot {x-scale}^{2}} \]
      *-commutative [=>] 4.3	\[ -4 \cdot \frac{{a}^{2} \cdot {b}^{2}}{\color{blue}{{x-scale}^{2} \cdot {y-scale}^{2}}} \]
      times-frac [=>] 4.5	\[ -4 \cdot \color{blue}{\left(\frac{{a}^{2}}{{x-scale}^{2}} \cdot \frac{{b}^{2}}{{y-scale}^{2}}\right)} \]
      unpow2 [=>] 4.5	\[ -4 \cdot \left(\frac{\color{blue}{a \cdot a}}{{x-scale}^{2}} \cdot \frac{{b}^{2}}{{y-scale}^{2}}\right) \]
      unpow2 [=>] 4.5	\[ -4 \cdot \left(\frac{a \cdot a}{\color{blue}{x-scale \cdot x-scale}} \cdot \frac{{b}^{2}}{{y-scale}^{2}}\right) \]
      times-frac [=>] 36.2	\[ -4 \cdot \left(\color{blue}{\left(\frac{a}{x-scale} \cdot \frac{a}{x-scale}\right)} \cdot \frac{{b}^{2}}{{y-scale}^{2}}\right) \]
      unpow2 [=>] 36.2	\[ -4 \cdot \left(\left(\frac{a}{x-scale} \cdot \frac{a}{x-scale}\right) \cdot \frac{\color{blue}{b \cdot b}}{{y-scale}^{2}}\right) \]
      unpow2 [=>] 36.2	\[ -4 \cdot \left(\left(\frac{a}{x-scale} \cdot \frac{a}{x-scale}\right) \cdot \frac{b \cdot b}{\color{blue}{y-scale \cdot y-scale}}\right) \]
      times-frac [=>] 53.5	\[ -4 \cdot \left(\left(\frac{a}{x-scale} \cdot \frac{a}{x-scale}\right) \cdot \color{blue}{\left(\frac{b}{y-scale} \cdot \frac{b}{y-scale}\right)}\right) \]
      swap-sqr [<=] 81.8	\[ -4 \cdot \color{blue}{\left(\left(\frac{a}{x-scale} \cdot \frac{b}{y-scale}\right) \cdot \left(\frac{a}{x-scale} \cdot \frac{b}{y-scale}\right)\right)} \]
      unpow2 [<=] 81.8	\[ -4 \cdot \color{blue}{{\left(\frac{a}{x-scale} \cdot \frac{b}{y-scale}\right)}^{2}} \]
      associate-*r/ [=>] 81.1	\[ -4 \cdot {\color{blue}{\left(\frac{\frac{a}{x-scale} \cdot b}{y-scale}\right)}}^{2} \]
      associate-/l* [=>] 81.6	\[ -4 \cdot {\color{blue}{\left(\frac{\frac{a}{x-scale}}{\frac{y-scale}{b}}\right)}}^{2} \]

   if -8.0000000000000003e138 < a

   1. Initial program 38.2%

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

   2. Simplified 32.3%

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

      [Start] 38.2

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

   3. Taylor expanded in angle around 0 41.5%

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

   4. Simplified 49.0%

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

      [Start] 41.5	\[ -4 \cdot \frac{{a}^{2} \cdot {b}^{2}}{{y-scale}^{2} \cdot {x-scale}^{2}} \]
      associate-/l* [=>] 41.9	\[ -4 \cdot \color{blue}{\frac{{a}^{2}}{\frac{{y-scale}^{2} \cdot {x-scale}^{2}}{{b}^{2}}}} \]
      unpow2 [=>] 41.9	\[ -4 \cdot \frac{\color{blue}{a \cdot a}}{\frac{{y-scale}^{2} \cdot {x-scale}^{2}}{{b}^{2}}} \]
      *-commutative [=>] 41.9	\[ -4 \cdot \frac{a \cdot a}{\frac{\color{blue}{{x-scale}^{2} \cdot {y-scale}^{2}}}{{b}^{2}}} \]
      unpow2 [=>] 41.9	\[ -4 \cdot \frac{a \cdot a}{\frac{{x-scale}^{2} \cdot {y-scale}^{2}}{\color{blue}{b \cdot b}}} \]
      times-frac [=>] 49.0	\[ -4 \cdot \frac{a \cdot a}{\color{blue}{\frac{{x-scale}^{2}}{b} \cdot \frac{{y-scale}^{2}}{b}}} \]
      unpow2 [=>] 49.0	\[ -4 \cdot \frac{a \cdot a}{\frac{\color{blue}{x-scale \cdot x-scale}}{b} \cdot \frac{{y-scale}^{2}}{b}} \]
      unpow2 [=>] 49.0	\[ -4 \cdot \frac{a \cdot a}{\frac{x-scale \cdot x-scale}{b} \cdot \frac{\color{blue}{y-scale \cdot y-scale}}{b}} \]

   5. Applied egg-rr 63.2%

      \[\leadsto -4 \cdot \color{blue}{\left(e^{\mathsf{log1p}\left({\left(\frac{a}{\frac{\sqrt{{\left(x-scale \cdot y-scale\right)}^{2}}}{b}}\right)}^{2}\right)} - 1\right)} \]

   6. Simplified 91.9%

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

      [Start] 63.2	\[ -4 \cdot \left(e^{\mathsf{log1p}\left({\left(\frac{a}{\frac{\sqrt{{\left(x-scale \cdot y-scale\right)}^{2}}}{b}}\right)}^{2}\right)} - 1\right) \]
      expm1-def [=>] 76.2	\[ -4 \cdot \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left({\left(\frac{a}{\frac{\sqrt{{\left(x-scale \cdot y-scale\right)}^{2}}}{b}}\right)}^{2}\right)\right)} \]
      expm1-log1p [=>] 77.2	\[ -4 \cdot \color{blue}{{\left(\frac{a}{\frac{\sqrt{{\left(x-scale \cdot y-scale\right)}^{2}}}{b}}\right)}^{2}} \]
      associate-/r/ [=>] 77.0	\[ -4 \cdot {\color{blue}{\left(\frac{a}{\sqrt{{\left(x-scale \cdot y-scale\right)}^{2}}} \cdot b\right)}}^{2} \]
      associate-*l/ [=>] 77.9	\[ -4 \cdot {\color{blue}{\left(\frac{a \cdot b}{\sqrt{{\left(x-scale \cdot y-scale\right)}^{2}}}\right)}}^{2} \]
      unpow2 [=>] 77.9	\[ -4 \cdot {\left(\frac{a \cdot b}{\sqrt{\color{blue}{\left(x-scale \cdot y-scale\right) \cdot \left(x-scale \cdot y-scale\right)}}}\right)}^{2} \]
      rem-sqrt-square [=>] 91.9	\[ -4 \cdot {\left(\frac{a \cdot b}{\color{blue}{\left|x-scale \cdot y-scale\right|}}\right)}^{2} \]

3. Recombined 2 regimes into one program.

4. Final simplification 90.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \leq -8 \cdot 10^{+138}:\\ \;\;\;\;-4 \cdot {\left(\frac{\frac{a}{x-scale}}{\frac{y-scale}{b}}\right)}^{2}\\ \mathbf{else}:\\ \;\;\;\;-4 \cdot {\left(\frac{a \cdot b}{\left|x-scale \cdot y-scale\right|}\right)}^{2}\\ \end{array} \]

Alternatives

Alternative 1
Accuracy	90.7%
Cost	7172

\[\begin{array}{l} t_0 := \frac{y-scale}{\frac{a \cdot b}{x-scale}}\\ \mathbf{if}\;a \leq -5.8 \cdot 10^{-70}:\\ \;\;\;\;-4 \cdot {\left(\frac{\frac{a}{x-scale}}{\frac{y-scale}{b}}\right)}^{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{-4}{t_0}}{t_0}\\ \end{array} \]

Alternative 2
Accuracy	89.0%
Cost	1616

\[\begin{array}{l} t_0 := \frac{a}{x-scale} \cdot \frac{b}{y-scale}\\ t_1 := -4 \cdot \left(t_0 \cdot t_0\right)\\ \mathbf{if}\;y-scale \leq -1.55 \cdot 10^{-130}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;y-scale \leq -3.15 \cdot 10^{-295}:\\ \;\;\;\;\frac{\left(a \cdot b\right) \cdot \frac{-4}{\frac{y-scale}{\frac{a \cdot b}{x-scale}}}}{x-scale \cdot y-scale}\\ \mathbf{elif}\;y-scale \leq 1.1 \cdot 10^{-290}:\\ \;\;\;\;-4 \cdot \frac{t_0 \cdot \left(\frac{a}{x-scale} \cdot b\right)}{y-scale}\\ \mathbf{elif}\;y-scale \leq 1.3 \cdot 10^{-121}:\\ \;\;\;\;-4 \cdot \frac{b}{\frac{\frac{x-scale \cdot y-scale}{a \cdot b}}{\frac{a}{x-scale \cdot y-scale}}}\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]

Alternative 3
Accuracy	87.9%
Cost	1616

\[\begin{array}{l} t_0 := \frac{a \cdot b}{x-scale}\\ t_1 := \frac{-4}{\frac{y-scale}{t_0}}\\ t_2 := \frac{a}{x-scale} \cdot \frac{b}{y-scale}\\ \mathbf{if}\;y-scale \leq -1.55 \cdot 10^{-136}:\\ \;\;\;\;\frac{t_0 \cdot t_1}{y-scale}\\ \mathbf{elif}\;y-scale \leq -3.15 \cdot 10^{-295}:\\ \;\;\;\;\frac{\left(a \cdot b\right) \cdot t_1}{x-scale \cdot y-scale}\\ \mathbf{elif}\;y-scale \leq 1.1 \cdot 10^{-290}:\\ \;\;\;\;-4 \cdot \frac{t_2 \cdot \left(\frac{a}{x-scale} \cdot b\right)}{y-scale}\\ \mathbf{elif}\;y-scale \leq 1.7 \cdot 10^{-151}:\\ \;\;\;\;-4 \cdot \frac{b}{\frac{\frac{x-scale \cdot y-scale}{a \cdot b}}{\frac{a}{x-scale \cdot y-scale}}}\\ \mathbf{else}:\\ \;\;\;\;-4 \cdot \left(t_2 \cdot t_2\right)\\ \end{array} \]

Alternative 4
Accuracy	88.9%
Cost	1353

\[\begin{array}{l} t_0 := \frac{a}{x-scale} \cdot \frac{b}{y-scale}\\ \mathbf{if}\;angle \leq 1.3 \cdot 10^{-226} \lor \neg \left(angle \leq 1.35 \cdot 10^{+31}\right):\\ \;\;\;\;-4 \cdot \left(t_0 \cdot t_0\right)\\ \mathbf{else}:\\ \;\;\;\;-4 \cdot \left(b \cdot \frac{\frac{a}{x-scale \cdot y-scale}}{x-scale \cdot \frac{y-scale}{a \cdot b}}\right)\\ \end{array} \]

Alternative 5
Accuracy	89.3%
Cost	1353

\[\begin{array}{l} t_0 := \frac{a}{x-scale} \cdot \frac{b}{y-scale}\\ \mathbf{if}\;angle \leq 2.2 \cdot 10^{-226} \lor \neg \left(angle \leq 5.6 \cdot 10^{+33}\right):\\ \;\;\;\;-4 \cdot \left(t_0 \cdot t_0\right)\\ \mathbf{else}:\\ \;\;\;\;-4 \cdot \frac{b}{\frac{\frac{x-scale \cdot y-scale}{a \cdot b}}{\frac{a}{x-scale \cdot y-scale}}}\\ \end{array} \]

Alternative 6
Accuracy	90.7%
Cost	1220

\[\begin{array}{l} t_0 := \frac{y-scale}{\frac{a \cdot b}{x-scale}}\\ t_1 := \frac{a}{x-scale} \cdot \frac{b}{y-scale}\\ \mathbf{if}\;a \leq -4.6 \cdot 10^{-70}:\\ \;\;\;\;-4 \cdot \left(t_1 \cdot t_1\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{-4}{t_0}}{t_0}\\ \end{array} \]

Alternative 7
Accuracy	81.6%
Cost	1088

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

Alternative 8
Accuracy	51.6%
Cost	64

\[0 \]

Reproduce

herbie shell --seed 2023141 
(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))))