Simplification of discriminant from scale-rotated-ellipse

Average Accuracy: 24.0% → 59.9%

Time: 1.5min

Precision: binary64

Cost: 7172

Math

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

↓

\[\begin{array}{l} t_0 := \frac{b}{y-scale} \cdot \frac{a}{x-scale}\\ \mathbf{if}\;a \leq -4.3 \cdot 10^{-141}:\\ \;\;\;\;t_0 \cdot \left(t_0 \cdot -4\right)\\ \mathbf{else}:\\ \;\;\;\;-4 \cdot {\left(\frac{b}{x-scale} \cdot \frac{a}{y-scale}\right)}^{2}\\ \end{array} \]

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
 (let* ((t_0 (* (/ b y-scale) (/ a x-scale))))
   (if (<= a -4.3e-141)
     (* t_0 (* t_0 -4.0))
     (* -4.0 (pow (* (/ b x-scale) (/ a 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 t_0 = (b / y_45_scale) * (a / x_45_scale);
	double tmp;
	if (a <= -4.3e-141) {
		tmp = t_0 * (t_0 * -4.0);
	} else {
		tmp = -4.0 * pow(((b / x_45_scale) * (a / 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 t_0 = (b / y_45_scale) * (a / x_45_scale);
	double tmp;
	if (a <= -4.3e-141) {
		tmp = t_0 * (t_0 * -4.0);
	} else {
		tmp = -4.0 * Math.pow(((b / x_45_scale) * (a / 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):
	t_0 = (b / y_45_scale) * (a / x_45_scale)
	tmp = 0
	if a <= -4.3e-141:
		tmp = t_0 * (t_0 * -4.0)
	else:
		tmp = -4.0 * math.pow(((b / x_45_scale) * (a / 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)
	t_0 = Float64(Float64(b / y_45_scale) * Float64(a / x_45_scale))
	tmp = 0.0
	if (a <= -4.3e-141)
		tmp = Float64(t_0 * Float64(t_0 * -4.0));
	else
		tmp = Float64(-4.0 * (Float64(Float64(b / x_45_scale) * Float64(a / 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)
	t_0 = (b / y_45_scale) * (a / x_45_scale);
	tmp = 0.0;
	if (a <= -4.3e-141)
		tmp = t_0 * (t_0 * -4.0);
	else
		tmp = -4.0 * (((b / x_45_scale) * (a / 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_] := Block[{t$95$0 = N[(N[(b / y$45$scale), $MachinePrecision] * N[(a / x$45$scale), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[a, -4.3e-141], N[(t$95$0 * N[(t$95$0 * -4.0), $MachinePrecision]), $MachinePrecision], N[(-4.0 * N[Power[N[(N[(b / x$45$scale), $MachinePrecision] * N[(a / y$45$scale), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if a < -4.29999999999999974e-141

   1. Initial program 10.9%

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

   2. Taylor expanded in angle around 0 13.2%

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

   3. Simplified 28.6%

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

      [Start] 13.2	\[ -4 \cdot \frac{{a}^{2} \cdot {b}^{2}}{{y-scale}^{2} \cdot {x-scale}^{2}} \]
      *-commutative [=>] 13.2	\[ -4 \cdot \frac{\color{blue}{{b}^{2} \cdot {a}^{2}}}{{y-scale}^{2} \cdot {x-scale}^{2}} \]
      times-frac [=>] 13.2	\[ -4 \cdot \color{blue}{\left(\frac{{b}^{2}}{{y-scale}^{2}} \cdot \frac{{a}^{2}}{{x-scale}^{2}}\right)} \]
      unpow2 [=>] 13.2	\[ -4 \cdot \left(\frac{\color{blue}{b \cdot b}}{{y-scale}^{2}} \cdot \frac{{a}^{2}}{{x-scale}^{2}}\right) \]
      unpow2 [=>] 13.2	\[ -4 \cdot \left(\frac{b \cdot b}{\color{blue}{y-scale \cdot y-scale}} \cdot \frac{{a}^{2}}{{x-scale}^{2}}\right) \]
      times-frac [=>] 17.8	\[ -4 \cdot \left(\color{blue}{\left(\frac{b}{y-scale} \cdot \frac{b}{y-scale}\right)} \cdot \frac{{a}^{2}}{{x-scale}^{2}}\right) \]
      unpow2 [=>] 17.8	\[ -4 \cdot \left(\left(\frac{b}{y-scale} \cdot \frac{b}{y-scale}\right) \cdot \frac{\color{blue}{a \cdot a}}{{x-scale}^{2}}\right) \]
      unpow2 [=>] 17.8	\[ -4 \cdot \left(\left(\frac{b}{y-scale} \cdot \frac{b}{y-scale}\right) \cdot \frac{a \cdot a}{\color{blue}{x-scale \cdot x-scale}}\right) \]
      times-frac [=>] 28.6	\[ -4 \cdot \left(\left(\frac{b}{y-scale} \cdot \frac{b}{y-scale}\right) \cdot \color{blue}{\left(\frac{a}{x-scale} \cdot \frac{a}{x-scale}\right)}\right) \]

   4. Applied egg-rr 50.6%

      \[\leadsto \color{blue}{{\left(\sqrt[3]{{\left(\frac{b}{y-scale} \cdot \frac{a}{x-scale}\right)}^{2} \cdot -4}\right)}^{3}} \]
      Proof

      [Start] 28.6	\[ -4 \cdot \left(\left(\frac{b}{y-scale} \cdot \frac{b}{y-scale}\right) \cdot \left(\frac{a}{x-scale} \cdot \frac{a}{x-scale}\right)\right) \]
      add-cube-cbrt [=>] 28.4	\[ \color{blue}{\left(\sqrt[3]{-4 \cdot \left(\left(\frac{b}{y-scale} \cdot \frac{b}{y-scale}\right) \cdot \left(\frac{a}{x-scale} \cdot \frac{a}{x-scale}\right)\right)} \cdot \sqrt[3]{-4 \cdot \left(\left(\frac{b}{y-scale} \cdot \frac{b}{y-scale}\right) \cdot \left(\frac{a}{x-scale} \cdot \frac{a}{x-scale}\right)\right)}\right) \cdot \sqrt[3]{-4 \cdot \left(\left(\frac{b}{y-scale} \cdot \frac{b}{y-scale}\right) \cdot \left(\frac{a}{x-scale} \cdot \frac{a}{x-scale}\right)\right)}} \]
      pow3 [=>] 28.4	\[ \color{blue}{{\left(\sqrt[3]{-4 \cdot \left(\left(\frac{b}{y-scale} \cdot \frac{b}{y-scale}\right) \cdot \left(\frac{a}{x-scale} \cdot \frac{a}{x-scale}\right)\right)}\right)}^{3}} \]
      *-commutative [=>] 28.4	\[ {\left(\sqrt[3]{\color{blue}{\left(\left(\frac{b}{y-scale} \cdot \frac{b}{y-scale}\right) \cdot \left(\frac{a}{x-scale} \cdot \frac{a}{x-scale}\right)\right) \cdot -4}}\right)}^{3} \]
      unswap-sqr [=>] 50.6	\[ {\left(\sqrt[3]{\color{blue}{\left(\left(\frac{b}{y-scale} \cdot \frac{a}{x-scale}\right) \cdot \left(\frac{b}{y-scale} \cdot \frac{a}{x-scale}\right)\right)} \cdot -4}\right)}^{3} \]
      pow2 [=>] 50.6	\[ {\left(\sqrt[3]{\color{blue}{{\left(\frac{b}{y-scale} \cdot \frac{a}{x-scale}\right)}^{2}} \cdot -4}\right)}^{3} \]

   5. Applied egg-rr 51.2%

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

      [Start] 50.6	\[ {\left(\sqrt[3]{{\left(\frac{b}{y-scale} \cdot \frac{a}{x-scale}\right)}^{2} \cdot -4}\right)}^{3} \]
      rem-cube-cbrt [=>] 51.1	\[ \color{blue}{{\left(\frac{b}{y-scale} \cdot \frac{a}{x-scale}\right)}^{2} \cdot -4} \]
      unpow2 [=>] 51.1	\[ \color{blue}{\left(\left(\frac{b}{y-scale} \cdot \frac{a}{x-scale}\right) \cdot \left(\frac{b}{y-scale} \cdot \frac{a}{x-scale}\right)\right)} \cdot -4 \]
      associate-*l* [=>] 51.2	\[ \color{blue}{\left(\frac{b}{y-scale} \cdot \frac{a}{x-scale}\right) \cdot \left(\left(\frac{b}{y-scale} \cdot \frac{a}{x-scale}\right) \cdot -4\right)} \]

   if -4.29999999999999974e-141 < a

   1. Initial program 32.2%

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

   2. Simplified 27.4%

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

      [Start] 32.2

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

   3. Taylor expanded in angle around 0 30.1%

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

   4. Simplified 53.5%

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

      [Start] 30.1	\[ -4 \cdot \frac{{a}^{2} \cdot {b}^{2}}{{y-scale}^{2} \cdot {x-scale}^{2}} \]
      times-frac [=>] 30.1	\[ -4 \cdot \color{blue}{\left(\frac{{a}^{2}}{{y-scale}^{2}} \cdot \frac{{b}^{2}}{{x-scale}^{2}}\right)} \]
      *-commutative [=>] 30.1	\[ -4 \cdot \color{blue}{\left(\frac{{b}^{2}}{{x-scale}^{2}} \cdot \frac{{a}^{2}}{{y-scale}^{2}}\right)} \]
      unpow2 [=>] 30.1	\[ -4 \cdot \left(\frac{\color{blue}{b \cdot b}}{{x-scale}^{2}} \cdot \frac{{a}^{2}}{{y-scale}^{2}}\right) \]
      unpow2 [=>] 30.1	\[ -4 \cdot \left(\frac{b \cdot b}{\color{blue}{x-scale \cdot x-scale}} \cdot \frac{{a}^{2}}{{y-scale}^{2}}\right) \]
      times-frac [=>] 40.3	\[ -4 \cdot \left(\color{blue}{\left(\frac{b}{x-scale} \cdot \frac{b}{x-scale}\right)} \cdot \frac{{a}^{2}}{{y-scale}^{2}}\right) \]
      unpow2 [=>] 40.3	\[ -4 \cdot \left(\left(\frac{b}{x-scale} \cdot \frac{b}{x-scale}\right) \cdot \frac{\color{blue}{a \cdot a}}{{y-scale}^{2}}\right) \]
      unpow2 [=>] 40.3	\[ -4 \cdot \left(\left(\frac{b}{x-scale} \cdot \frac{b}{x-scale}\right) \cdot \frac{a \cdot a}{\color{blue}{y-scale \cdot y-scale}}\right) \]
      times-frac [=>] 53.5	\[ -4 \cdot \left(\left(\frac{b}{x-scale} \cdot \frac{b}{x-scale}\right) \cdot \color{blue}{\left(\frac{a}{y-scale} \cdot \frac{a}{y-scale}\right)}\right) \]

   5. Taylor expanded in b around 0 30.1%

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

   6. Simplified 69.2%

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

      [Start] 30.1	\[ -4 \cdot \frac{{a}^{2} \cdot {b}^{2}}{{y-scale}^{2} \cdot {x-scale}^{2}} \]
      unpow2 [=>] 30.1	\[ -4 \cdot \frac{{a}^{2} \cdot {b}^{2}}{\color{blue}{\left(y-scale \cdot y-scale\right)} \cdot {x-scale}^{2}} \]
      unpow2 [=>] 30.1	\[ -4 \cdot \frac{{a}^{2} \cdot {b}^{2}}{\left(y-scale \cdot y-scale\right) \cdot \color{blue}{\left(x-scale \cdot x-scale\right)}} \]
      unpow2 [=>] 30.1	\[ -4 \cdot \frac{\color{blue}{\left(a \cdot a\right)} \cdot {b}^{2}}{\left(y-scale \cdot y-scale\right) \cdot \left(x-scale \cdot x-scale\right)} \]
      unpow2 [=>] 30.1	\[ -4 \cdot \frac{\left(a \cdot a\right) \cdot \color{blue}{\left(b \cdot b\right)}}{\left(y-scale \cdot y-scale\right) \cdot \left(x-scale \cdot x-scale\right)} \]
      times-frac [=>] 30.1	\[ -4 \cdot \color{blue}{\left(\frac{a \cdot a}{y-scale \cdot y-scale} \cdot \frac{b \cdot b}{x-scale \cdot x-scale}\right)} \]
      times-frac [=>] 39.3	\[ -4 \cdot \left(\color{blue}{\left(\frac{a}{y-scale} \cdot \frac{a}{y-scale}\right)} \cdot \frac{b \cdot b}{x-scale \cdot x-scale}\right) \]
      times-frac [=>] 53.5	\[ -4 \cdot \left(\left(\frac{a}{y-scale} \cdot \frac{a}{y-scale}\right) \cdot \color{blue}{\left(\frac{b}{x-scale} \cdot \frac{b}{x-scale}\right)}\right) \]
      swap-sqr [<=] 69.2	\[ -4 \cdot \color{blue}{\left(\left(\frac{a}{y-scale} \cdot \frac{b}{x-scale}\right) \cdot \left(\frac{a}{y-scale} \cdot \frac{b}{x-scale}\right)\right)} \]
      *-commutative [<=] 69.2	\[ -4 \cdot \left(\color{blue}{\left(\frac{b}{x-scale} \cdot \frac{a}{y-scale}\right)} \cdot \left(\frac{a}{y-scale} \cdot \frac{b}{x-scale}\right)\right) \]
      *-commutative [<=] 69.2	\[ -4 \cdot \left(\left(\frac{b}{x-scale} \cdot \frac{a}{y-scale}\right) \cdot \color{blue}{\left(\frac{b}{x-scale} \cdot \frac{a}{y-scale}\right)}\right) \]
      unpow2 [<=] 69.2	\[ -4 \cdot \color{blue}{{\left(\frac{b}{x-scale} \cdot \frac{a}{y-scale}\right)}^{2}} \]

3. Recombined 2 regimes into one program.

4. Final simplification 63.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \leq -4.3 \cdot 10^{-141}:\\ \;\;\;\;\left(\frac{b}{y-scale} \cdot \frac{a}{x-scale}\right) \cdot \left(\left(\frac{b}{y-scale} \cdot \frac{a}{x-scale}\right) \cdot -4\right)\\ \mathbf{else}:\\ \;\;\;\;-4 \cdot {\left(\frac{b}{x-scale} \cdot \frac{a}{y-scale}\right)}^{2}\\ \end{array} \]

Alternatives

Alternative 1
Accuracy	50.5%
Cost	1749

\[\begin{array}{l} t_0 := \frac{\frac{a}{x-scale}}{y-scale}\\ t_1 := -4 \cdot \left(\left(t_0 \cdot t_0\right) \cdot \left(b \cdot b\right)\right)\\ t_2 := \frac{b}{y-scale} \cdot \frac{b}{y-scale}\\ \mathbf{if}\;b \leq -1.3 \cdot 10^{+113}:\\ \;\;\;\;-4 \cdot \left(t_2 \cdot \frac{a \cdot \frac{a}{x-scale}}{x-scale}\right)\\ \mathbf{elif}\;b \leq -4.4 \cdot 10^{-65}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;b \leq -2.7 \cdot 10^{-287}:\\ \;\;\;\;-4 \cdot \left(t_2 \cdot \left(\frac{a}{x-scale} \cdot \frac{a}{x-scale}\right)\right)\\ \mathbf{elif}\;b \leq 1.65 \cdot 10^{-105} \lor \neg \left(b \leq 2.7 \cdot 10^{+135}\right):\\ \;\;\;\;-4 \cdot \left(\left(\frac{b}{x-scale} \cdot \frac{b}{x-scale}\right) \cdot \left(\frac{a}{y-scale} \cdot \frac{a}{y-scale}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]

Alternative 2
Accuracy	39.5%
Cost	1485

\[\begin{array}{l} \mathbf{if}\;b \leq -3.9 \cdot 10^{+115}:\\ \;\;\;\;0\\ \mathbf{elif}\;b \leq -2.5 \cdot 10^{-105} \lor \neg \left(b \leq 2.4 \cdot 10^{-46}\right):\\ \;\;\;\;-4 \cdot \left(\left(b \cdot b\right) \cdot \left(\frac{a}{x-scale} \cdot \frac{a}{y-scale \cdot \left(y-scale \cdot x-scale\right)}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \]

Alternative 3
Accuracy	46.9%
Cost	1485

\[\begin{array}{l} \mathbf{if}\;b \leq -7 \cdot 10^{-288}:\\ \;\;\;\;-4 \cdot \left(\left(\frac{b}{y-scale} \cdot \frac{b}{y-scale}\right) \cdot \left(\frac{a}{x-scale} \cdot \frac{a}{x-scale}\right)\right)\\ \mathbf{elif}\;b \leq 2.4 \cdot 10^{-46} \lor \neg \left(b \leq 2.2 \cdot 10^{+136}\right):\\ \;\;\;\;-4 \cdot \left(\left(\frac{b}{x-scale} \cdot \frac{b}{x-scale}\right) \cdot \left(\frac{a}{y-scale} \cdot \frac{a}{y-scale}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;-4 \cdot \left(\left(b \cdot b\right) \cdot \left(\frac{a}{x-scale} \cdot \frac{a}{y-scale \cdot \left(y-scale \cdot x-scale\right)}\right)\right)\\ \end{array} \]

Alternative 4
Accuracy	46.7%
Cost	1353

\[\begin{array}{l} \mathbf{if}\;b \leq 4.5 \cdot 10^{-46} \lor \neg \left(b \leq 6 \cdot 10^{+129}\right):\\ \;\;\;\;-4 \cdot \left(\left(\frac{b}{x-scale} \cdot \frac{b}{x-scale}\right) \cdot \left(\frac{a}{y-scale} \cdot \frac{a}{y-scale}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;-4 \cdot \left(\left(b \cdot b\right) \cdot \left(\frac{a}{x-scale} \cdot \frac{a}{y-scale \cdot \left(y-scale \cdot x-scale\right)}\right)\right)\\ \end{array} \]

Alternative 5
Accuracy	53.2%
Cost	1353

\[\begin{array}{l} t_0 := \frac{\frac{a}{x-scale}}{y-scale}\\ \mathbf{if}\;b \leq 1.2 \cdot 10^{-104} \lor \neg \left(b \leq 2.25 \cdot 10^{+93}\right):\\ \;\;\;\;-4 \cdot \left(\frac{a}{x-scale} \cdot \left(\frac{b}{y-scale} \cdot \frac{\frac{a \cdot b}{y-scale}}{x-scale}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;-4 \cdot \left(\left(t_0 \cdot t_0\right) \cdot \left(b \cdot b\right)\right)\\ \end{array} \]

Alternative 6
Accuracy	60.0%
Cost	1353

\[\begin{array}{l} t_0 := \frac{\frac{a \cdot b}{y-scale}}{x-scale}\\ t_1 := \frac{b}{y-scale} \cdot \frac{a}{x-scale}\\ \mathbf{if}\;x-scale \leq 2 \cdot 10^{-237} \lor \neg \left(x-scale \leq 10^{+161}\right):\\ \;\;\;\;t_1 \cdot \left(t_1 \cdot -4\right)\\ \mathbf{else}:\\ \;\;\;\;-4 \cdot \left(t_0 \cdot t_0\right)\\ \end{array} \]

Alternative 7
Accuracy	53.1%
Cost	1352

\[\begin{array}{l} t_0 := \frac{\frac{a \cdot b}{y-scale}}{x-scale}\\ t_1 := \frac{\frac{a}{x-scale}}{y-scale}\\ \mathbf{if}\;b \leq 5 \cdot 10^{-105}:\\ \;\;\;\;-4 \cdot \left(\frac{a}{x-scale} \cdot \left(\frac{b}{y-scale} \cdot t_0\right)\right)\\ \mathbf{elif}\;b \leq 1.52 \cdot 10^{+93}:\\ \;\;\;\;-4 \cdot \left(\left(t_1 \cdot t_1\right) \cdot \left(b \cdot b\right)\right)\\ \mathbf{else}:\\ \;\;\;\;-4 \cdot \left(\frac{b}{y-scale} \cdot \left(\frac{a}{x-scale} \cdot t_0\right)\right)\\ \end{array} \]

Alternative 8
Accuracy	58.9%
Cost	1352

\[\begin{array}{l} t_0 := \frac{\frac{a \cdot b}{y-scale}}{x-scale}\\ \mathbf{if}\;a \leq -2.4 \cdot 10^{+203}:\\ \;\;\;\;-4 \cdot \frac{t_0 \cdot \frac{a \cdot b}{x-scale}}{y-scale}\\ \mathbf{elif}\;a \leq -10500000000:\\ \;\;\;\;\left(\frac{b}{y-scale} \cdot \frac{a}{x-scale}\right) \cdot \left(-4 \cdot \left(a \cdot \frac{\frac{b}{x-scale}}{y-scale}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;-4 \cdot \left(t_0 \cdot t_0\right)\\ \end{array} \]

Alternative 9
Accuracy	59.6%
Cost	1088

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

Alternative 10
Accuracy	35.6%
Cost	64

\[0 \]

Reproduce

herbie shell --seed 2023157 
(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))))