Simplification of discriminant from scale-rotated-ellipse

Average Accuracy: 36.8% → 91.9%

Time: 1.3min

Precision: binary64

Cost: 7305

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;b \leq -2.4 \cdot 10^{+44} \lor \neg \left(b \leq 3.5 \cdot 10^{-281}\right):\\ \;\;\;\;-4 \cdot {\left(\frac{\frac{a}{x-scale}}{\frac{y-scale}{b}}\right)}^{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{a}{y-scale} \cdot \left(-4 \cdot \frac{b}{x-scale}\right)}{\frac{y-scale}{a} \cdot \frac{x-scale}{b}}\\ \end{array} \]

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 (or (<= b -2.4e+44) (not (<= b 3.5e-281)))
   (* -4.0 (pow (/ (/ a x-scale) (/ y-scale b)) 2.0))
   (/
    (* (/ a y-scale) (* -4.0 (/ b x-scale)))
    (* (/ y-scale a) (/ x-scale 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 tmp;
	if ((b <= -2.4e+44) || !(b <= 3.5e-281)) {
		tmp = -4.0 * pow(((a / x_45_scale) / (y_45_scale / b)), 2.0);
	} else {
		tmp = ((a / y_45_scale) * (-4.0 * (b / x_45_scale))) / ((y_45_scale / a) * (x_45_scale / 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 tmp;
	if ((b <= -2.4e+44) || !(b <= 3.5e-281)) {
		tmp = -4.0 * Math.pow(((a / x_45_scale) / (y_45_scale / b)), 2.0);
	} else {
		tmp = ((a / y_45_scale) * (-4.0 * (b / x_45_scale))) / ((y_45_scale / a) * (x_45_scale / 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):
	tmp = 0
	if (b <= -2.4e+44) or not (b <= 3.5e-281):
		tmp = -4.0 * math.pow(((a / x_45_scale) / (y_45_scale / b)), 2.0)
	else:
		tmp = ((a / y_45_scale) * (-4.0 * (b / x_45_scale))) / ((y_45_scale / a) * (x_45_scale / 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)
	tmp = 0.0
	if ((b <= -2.4e+44) || !(b <= 3.5e-281))
		tmp = Float64(-4.0 * (Float64(Float64(a / x_45_scale) / Float64(y_45_scale / b)) ^ 2.0));
	else
		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)));
	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 ((b <= -2.4e+44) || ~((b <= 3.5e-281)))
		tmp = -4.0 * (((a / x_45_scale) / (y_45_scale / b)) ^ 2.0);
	else
		tmp = ((a / y_45_scale) * (-4.0 * (b / x_45_scale))) / ((y_45_scale / a) * (x_45_scale / 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_] := If[Or[LessEqual[b, -2.4e+44], N[Not[LessEqual[b, 3.5e-281]], $MachinePrecision]], N[(-4.0 * N[Power[N[(N[(a / x$45$scale), $MachinePrecision] / N[(y$45$scale / b), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision], 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]]

Derivation

1. Split input into 2 regimes

2. if b < -2.40000000000000013e44 or 3.50000000000000022e-281 < b

   1. Initial program 30.4%

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

   2. Simplified 23.2%

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

      [Start] 30.4	\[ \frac{\frac{\left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right) \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)}{x-scale}}{y-scale} \cdot \frac{\frac{\left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right) \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)}{x-scale}}{y-scale} - \left(4 \cdot \frac{\frac{{\left(a \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} + {\left(b \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2}}{x-scale}}{x-scale}\right) \cdot \frac{\frac{{\left(a \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} + {\left(b \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2}}{y-scale}}{y-scale} \]
      sub-neg [=>] 30.4	\[ \color{blue}{\frac{\frac{\left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right) \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)}{x-scale}}{y-scale} \cdot \frac{\frac{\left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right) \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)}{x-scale}}{y-scale} + \left(-\left(4 \cdot \frac{\frac{{\left(a \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} + {\left(b \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2}}{x-scale}}{x-scale}\right) \cdot \frac{\frac{{\left(a \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} + {\left(b \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2}}{y-scale}}{y-scale}\right)} \]
      +-commutative [=>] 30.4	\[ \color{blue}{\left(-\left(4 \cdot \frac{\frac{{\left(a \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} + {\left(b \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2}}{x-scale}}{x-scale}\right) \cdot \frac{\frac{{\left(a \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} + {\left(b \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2}}{y-scale}}{y-scale}\right) + \frac{\frac{\left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right) \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)}{x-scale}}{y-scale} \cdot \frac{\frac{\left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right) \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)}{x-scale}}{y-scale}} \]

   3. Taylor expanded in angle around 0 34.5%

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

   4. Simplified 67.9%

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

      [Start] 34.5	\[ -4 \cdot \frac{{a}^{2} \cdot {b}^{2}}{{x-scale}^{2} \cdot {y-scale}^{2}} \]
      times-frac [=>] 34.5	\[ -4 \cdot \color{blue}{\left(\frac{{a}^{2}}{{x-scale}^{2}} \cdot \frac{{b}^{2}}{{y-scale}^{2}}\right)} \]
      associate-*r* [=>] 34.5	\[ \color{blue}{\left(-4 \cdot \frac{{a}^{2}}{{x-scale}^{2}}\right) \cdot \frac{{b}^{2}}{{y-scale}^{2}}} \]
      unpow2 [=>] 34.5	\[ \left(-4 \cdot \frac{\color{blue}{a \cdot a}}{{x-scale}^{2}}\right) \cdot \frac{{b}^{2}}{{y-scale}^{2}} \]
      unpow2 [=>] 34.5	\[ \left(-4 \cdot \frac{a \cdot a}{\color{blue}{x-scale \cdot x-scale}}\right) \cdot \frac{{b}^{2}}{{y-scale}^{2}} \]
      times-frac [=>] 44.9	\[ \left(-4 \cdot \color{blue}{\left(\frac{a}{x-scale} \cdot \frac{a}{x-scale}\right)}\right) \cdot \frac{{b}^{2}}{{y-scale}^{2}} \]
      unpow2 [=>] 44.9	\[ \left(-4 \cdot \left(\frac{a}{x-scale} \cdot \frac{a}{x-scale}\right)\right) \cdot \frac{\color{blue}{b \cdot b}}{{y-scale}^{2}} \]
      unpow2 [=>] 44.9	\[ \left(-4 \cdot \left(\frac{a}{x-scale} \cdot \frac{a}{x-scale}\right)\right) \cdot \frac{b \cdot b}{\color{blue}{y-scale \cdot y-scale}} \]
      times-frac [=>] 67.9	\[ \left(-4 \cdot \left(\frac{a}{x-scale} \cdot \frac{a}{x-scale}\right)\right) \cdot \color{blue}{\left(\frac{b}{y-scale} \cdot \frac{b}{y-scale}\right)} \]

   5. Taylor expanded in a around 0 34.5%

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

   6. Simplified 91.2%

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

      [Start] 34.5	\[ -4 \cdot \frac{{a}^{2} \cdot {b}^{2}}{{y-scale}^{2} \cdot {x-scale}^{2}} \]
      *-commutative [=>] 34.5	\[ -4 \cdot \frac{{a}^{2} \cdot {b}^{2}}{\color{blue}{{x-scale}^{2} \cdot {y-scale}^{2}}} \]
      times-frac [=>] 34.5	\[ -4 \cdot \color{blue}{\left(\frac{{a}^{2}}{{x-scale}^{2}} \cdot \frac{{b}^{2}}{{y-scale}^{2}}\right)} \]
      unpow2 [=>] 34.5	\[ -4 \cdot \left(\frac{\color{blue}{a \cdot a}}{{x-scale}^{2}} \cdot \frac{{b}^{2}}{{y-scale}^{2}}\right) \]
      unpow2 [=>] 34.5	\[ -4 \cdot \left(\frac{a \cdot a}{\color{blue}{x-scale \cdot x-scale}} \cdot \frac{{b}^{2}}{{y-scale}^{2}}\right) \]
      times-frac [=>] 44.9	\[ -4 \cdot \left(\color{blue}{\left(\frac{a}{x-scale} \cdot \frac{a}{x-scale}\right)} \cdot \frac{{b}^{2}}{{y-scale}^{2}}\right) \]
      unpow2 [=>] 44.9	\[ -4 \cdot \left(\left(\frac{a}{x-scale} \cdot \frac{a}{x-scale}\right) \cdot \frac{\color{blue}{b \cdot b}}{{y-scale}^{2}}\right) \]
      unpow2 [=>] 44.9	\[ -4 \cdot \left(\left(\frac{a}{x-scale} \cdot \frac{a}{x-scale}\right) \cdot \frac{b \cdot b}{\color{blue}{y-scale \cdot y-scale}}\right) \]
      times-frac [=>] 67.9	\[ -4 \cdot \left(\left(\frac{a}{x-scale} \cdot \frac{a}{x-scale}\right) \cdot \color{blue}{\left(\frac{b}{y-scale} \cdot \frac{b}{y-scale}\right)}\right) \]
      swap-sqr [<=] 91.0	\[ -4 \cdot \color{blue}{\left(\left(\frac{a}{x-scale} \cdot \frac{b}{y-scale}\right) \cdot \left(\frac{a}{x-scale} \cdot \frac{b}{y-scale}\right)\right)} \]
      unpow2 [<=] 91.0	\[ -4 \cdot \color{blue}{{\left(\frac{a}{x-scale} \cdot \frac{b}{y-scale}\right)}^{2}} \]
      associate-*r/ [=>] 90.4	\[ -4 \cdot {\color{blue}{\left(\frac{\frac{a}{x-scale} \cdot b}{y-scale}\right)}}^{2} \]
      associate-/l* [=>] 91.2	\[ -4 \cdot {\color{blue}{\left(\frac{\frac{a}{x-scale}}{\frac{y-scale}{b}}\right)}}^{2} \]

   if -2.40000000000000013e44 < b < 3.50000000000000022e-281

   1. Initial program 47.2%

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

   2. Taylor expanded in angle around 0 47.2%

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

   3. Simplified 76.3%

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

      [Start] 47.2	\[ -4 \cdot \frac{{a}^{2} \cdot {b}^{2}}{{y-scale}^{2} \cdot {x-scale}^{2}} \]
      *-commutative [=>] 47.2	\[ \color{blue}{\frac{{a}^{2} \cdot {b}^{2}}{{y-scale}^{2} \cdot {x-scale}^{2}} \cdot -4} \]
      times-frac [=>] 47.5	\[ \color{blue}{\left(\frac{{a}^{2}}{{y-scale}^{2}} \cdot \frac{{b}^{2}}{{x-scale}^{2}}\right)} \cdot -4 \]
      associate-*l* [=>] 47.5	\[ \color{blue}{\frac{{a}^{2}}{{y-scale}^{2}} \cdot \left(\frac{{b}^{2}}{{x-scale}^{2}} \cdot -4\right)} \]
      unpow2 [=>] 47.5	\[ \frac{\color{blue}{a \cdot a}}{{y-scale}^{2}} \cdot \left(\frac{{b}^{2}}{{x-scale}^{2}} \cdot -4\right) \]
      unpow2 [=>] 47.5	\[ \frac{a \cdot a}{\color{blue}{y-scale \cdot y-scale}} \cdot \left(\frac{{b}^{2}}{{x-scale}^{2}} \cdot -4\right) \]
      times-frac [=>] 61.9	\[ \color{blue}{\left(\frac{a}{y-scale} \cdot \frac{a}{y-scale}\right)} \cdot \left(\frac{{b}^{2}}{{x-scale}^{2}} \cdot -4\right) \]
      unpow2 [=>] 61.9	\[ \left(\frac{a}{y-scale} \cdot \frac{a}{y-scale}\right) \cdot \left(\frac{\color{blue}{b \cdot b}}{{x-scale}^{2}} \cdot -4\right) \]
      unpow2 [=>] 61.9	\[ \left(\frac{a}{y-scale} \cdot \frac{a}{y-scale}\right) \cdot \left(\frac{b \cdot b}{\color{blue}{x-scale \cdot x-scale}} \cdot -4\right) \]
      times-frac [=>] 76.3	\[ \left(\frac{a}{y-scale} \cdot \frac{a}{y-scale}\right) \cdot \left(\color{blue}{\left(\frac{b}{x-scale} \cdot \frac{b}{x-scale}\right)} \cdot -4\right) \]

   4. Applied egg-rr 76.1%

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

   5. Applied egg-rr 92.9%

      \[\leadsto \color{blue}{\frac{\frac{a}{y-scale} \cdot \left(-4 \cdot \frac{b}{x-scale}\right)}{\frac{y-scale}{a} \cdot \frac{x-scale}{b}}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 91.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -2.4 \cdot 10^{+44} \lor \neg \left(b \leq 3.5 \cdot 10^{-281}\right):\\ \;\;\;\;-4 \cdot {\left(\frac{\frac{a}{x-scale}}{\frac{y-scale}{b}}\right)}^{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{a}{y-scale} \cdot \left(-4 \cdot \frac{b}{x-scale}\right)}{\frac{y-scale}{a} \cdot \frac{x-scale}{b}}\\ \end{array} \]

Alternatives

Alternative 1
Accuracy	90.4%
Cost	7172

\[\begin{array}{l} \mathbf{if}\;b \leq -3.1 \cdot 10^{+185}:\\ \;\;\;\;-4 \cdot {\left(\frac{b \cdot a}{x-scale \cdot y-scale}\right)}^{2}\\ \mathbf{elif}\;b \leq 3.2 \cdot 10^{-281} \lor \neg \left(b \leq 6.8 \cdot 10^{-208}\right):\\ \;\;\;\;\frac{\frac{a}{y-scale} \cdot \left(-4 \cdot \frac{b}{x-scale}\right)}{\frac{y-scale}{a} \cdot \frac{x-scale}{b}}\\ \mathbf{else}:\\ \;\;\;\;-4 \cdot \frac{\frac{b}{y-scale}}{\frac{\frac{y-scale}{b}}{\frac{a}{x-scale}} \cdot \frac{x-scale}{a}}\\ \end{array} \]

Alternative 2
Accuracy	86.3%
Cost	1616

\[\begin{array}{l} t_0 := \frac{-4 \cdot \frac{b}{x-scale}}{\frac{y-scale}{a} \cdot \left(\frac{y-scale}{a} \cdot \frac{x-scale}{b}\right)}\\ t_1 := \frac{a}{x-scale} \cdot \frac{b}{y-scale}\\ \mathbf{if}\;b \leq -1.22 \cdot 10^{-58}:\\ \;\;\;\;-4 \cdot \left(\frac{t_1}{y-scale} \cdot \left(b \cdot \frac{a}{x-scale}\right)\right)\\ \mathbf{elif}\;b \leq 3.4 \cdot 10^{-281}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;b \leq 9.5 \cdot 10^{-207}:\\ \;\;\;\;-4 \cdot \frac{\frac{b}{y-scale}}{\frac{\frac{y-scale}{b}}{\frac{a}{x-scale}} \cdot \frac{x-scale}{a}}\\ \mathbf{elif}\;b \leq 1.45 \cdot 10^{+54}:\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;-4 \cdot \frac{a \cdot t_1}{x-scale \cdot \frac{y-scale}{b}}\\ \end{array} \]

Alternative 3
Accuracy	64.1%
Cost	1485

\[\begin{array}{l} \mathbf{if}\;y-scale \leq -1.15 \cdot 10^{-160} \lor \neg \left(y-scale \leq 3.4 \cdot 10^{-156}\right) \land y-scale \leq 1.35 \cdot 10^{+142}:\\ \;\;\;\;-4 \cdot \left(\left(a \cdot \frac{\frac{b}{x-scale}}{x-scale}\right) \cdot \left(a \cdot \frac{b}{y-scale \cdot y-scale}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \]

Alternative 4
Accuracy	83.6%
Cost	1485

\[\begin{array}{l} \mathbf{if}\;a \leq -1.25 \cdot 10^{-102} \lor \neg \left(a \leq 1.3 \cdot 10^{-220}\right) \land a \leq 4.5 \cdot 10^{+148}:\\ \;\;\;\;-4 \cdot \frac{a}{\left(x-scale \cdot \frac{y-scale}{b}\right) \cdot \left(x-scale \cdot \frac{\frac{y-scale}{a}}{b}\right)}\\ \mathbf{else}:\\ \;\;\;\;-4 \cdot \left(\frac{\frac{a}{x-scale} \cdot \frac{b}{y-scale}}{y-scale} \cdot \left(b \cdot \frac{a}{x-scale}\right)\right)\\ \end{array} \]

Alternative 5
Accuracy	84.9%
Cost	1485

\[\begin{array}{l} t_0 := x-scale \cdot \frac{y-scale}{b}\\ \mathbf{if}\;a \leq -5 \cdot 10^{-179}:\\ \;\;\;\;-4 \cdot \frac{a}{\frac{\frac{y-scale}{b}}{\frac{a}{x-scale}} \cdot t_0}\\ \mathbf{elif}\;a \leq 1.7 \cdot 10^{-219} \lor \neg \left(a \leq 1.35 \cdot 10^{+149}\right):\\ \;\;\;\;-4 \cdot \left(\frac{\frac{a}{x-scale} \cdot \frac{b}{y-scale}}{y-scale} \cdot \left(b \cdot \frac{a}{x-scale}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;-4 \cdot \frac{a}{t_0 \cdot \left(x-scale \cdot \frac{\frac{y-scale}{a}}{b}\right)}\\ \end{array} \]

Alternative 6
Accuracy	90.1%
Cost	1485

\[\begin{array}{l} \mathbf{if}\;b \leq -5 \cdot 10^{+183}:\\ \;\;\;\;-4 \cdot \left(\frac{\frac{a}{x-scale} \cdot \frac{b}{y-scale}}{y-scale} \cdot \left(b \cdot \frac{a}{x-scale}\right)\right)\\ \mathbf{elif}\;b \leq 3.2 \cdot 10^{-281} \lor \neg \left(b \leq 2 \cdot 10^{-204}\right):\\ \;\;\;\;\frac{\frac{a}{y-scale} \cdot \left(-4 \cdot \frac{b}{x-scale}\right)}{\frac{y-scale}{a} \cdot \frac{x-scale}{b}}\\ \mathbf{else}:\\ \;\;\;\;-4 \cdot \frac{\frac{b}{y-scale}}{\frac{\frac{y-scale}{b}}{\frac{a}{x-scale}} \cdot \frac{x-scale}{a}}\\ \end{array} \]

Alternative 7
Accuracy	82.7%
Cost	1353

\[\begin{array}{l} t_0 := \frac{a}{x-scale} \cdot \frac{b}{y-scale}\\ \mathbf{if}\;y-scale \leq 4.3 \cdot 10^{-224} \lor \neg \left(y-scale \leq 10^{-60}\right):\\ \;\;\;\;-4 \cdot \left(b \cdot \frac{\frac{a}{x-scale} \cdot t_0}{y-scale}\right)\\ \mathbf{else}:\\ \;\;\;\;-4 \cdot \left(\frac{a}{x-scale \cdot y-scale} \cdot \left(b \cdot t_0\right)\right)\\ \end{array} \]

Alternative 8
Accuracy	85.9%
Cost	1353

\[\begin{array}{l} t_0 := \frac{a}{x-scale} \cdot \frac{b}{y-scale}\\ \mathbf{if}\;y-scale \leq -2.3 \cdot 10^{-230} \lor \neg \left(y-scale \leq 6.2 \cdot 10^{-160}\right):\\ \;\;\;\;-4 \cdot \left(\frac{t_0}{y-scale} \cdot \left(b \cdot \frac{a}{x-scale}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;-4 \cdot \left(b \cdot \frac{\frac{a}{x-scale} \cdot t_0}{y-scale}\right)\\ \end{array} \]

Alternative 9
Accuracy	86.5%
Cost	1352

\[\begin{array}{l} t_0 := x-scale \cdot \frac{y-scale}{b}\\ t_1 := \frac{a}{x-scale} \cdot \frac{b}{y-scale}\\ \mathbf{if}\;y-scale \leq 4.1 \cdot 10^{-185}:\\ \;\;\;\;-4 \cdot \frac{a \cdot t_1}{t_0}\\ \mathbf{elif}\;y-scale \leq 10^{+165}:\\ \;\;\;\;-4 \cdot \left(\frac{t_1}{y-scale} \cdot \left(b \cdot \frac{a}{x-scale}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;-4 \cdot \frac{a}{t_0 \cdot \left(x-scale \cdot \frac{\frac{y-scale}{a}}{b}\right)}\\ \end{array} \]

Alternative 10
Accuracy	85.8%
Cost	1352

\[\begin{array}{l} t_0 := x-scale \cdot \frac{y-scale}{b}\\ \mathbf{if}\;angle \leq -8.5 \cdot 10^{-126}:\\ \;\;\;\;-4 \cdot \frac{\frac{b}{y-scale}}{\frac{\frac{y-scale}{b}}{\frac{a}{x-scale}} \cdot \frac{x-scale}{a}}\\ \mathbf{elif}\;angle \leq -7.5 \cdot 10^{-197}:\\ \;\;\;\;-4 \cdot \frac{a}{t_0 \cdot \left(x-scale \cdot \frac{\frac{y-scale}{a}}{b}\right)}\\ \mathbf{else}:\\ \;\;\;\;-4 \cdot \frac{a \cdot \left(\frac{a}{x-scale} \cdot \frac{b}{y-scale}\right)}{t_0}\\ \end{array} \]

Alternative 11
Accuracy	80.6%
Cost	1088

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

Alternative 12
Accuracy	53.2%
Cost	64

\[0 \]

Reproduce

herbie shell --seed 2023129 
(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))))