Simplification of discriminant from scale-rotated-ellipse

Percentage Accurate: 24.7% → 83.2%

Time: 1.5min

Alternatives: 6

Speedup: 2485.0×

• 33.0% of points produce a very large (infinite) output. You may want to add a precondition.

Specification

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{angle}{180} \cdot \pi\\ t_1 := \sin t_0\\ t_2 := \cos t_0\\ t_3 := \frac{\frac{\left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot t_1\right) \cdot t_2}{x-scale}}{y-scale}\\ t_3 \cdot t_3 - \left(4 \cdot \frac{\frac{{\left(a \cdot t_1\right)}^{2} + {\left(b \cdot t_2\right)}^{2}}{x-scale}}{x-scale}\right) \cdot \frac{\frac{{\left(a \cdot t_2\right)}^{2} + {\left(b \cdot t_1\right)}^{2}}{y-scale}}{y-scale} \end{array} \end{array} \]

FPCore

(FPCore (a b angle x-scale y-scale)
 :precision binary64
 (let* ((t_0 (* (/ angle 180.0) PI))
        (t_1 (sin t_0))
        (t_2 (cos t_0))
        (t_3
         (/
          (/ (* (* (* 2.0 (- (pow b 2.0) (pow a 2.0))) t_1) t_2) x-scale)
          y-scale)))
   (-
    (* t_3 t_3)
    (*
     (*
      4.0
      (/ (/ (+ (pow (* a t_1) 2.0) (pow (* b t_2) 2.0)) x-scale) x-scale))
     (/ (/ (+ (pow (* a t_2) 2.0) (pow (* b t_1) 2.0)) y-scale) y-scale)))))

C

double code(double a, double b, double angle, double x_45_scale, double y_45_scale) {
	double t_0 = (angle / 180.0) * ((double) M_PI);
	double t_1 = sin(t_0);
	double t_2 = cos(t_0);
	double t_3 = ((((2.0 * (pow(b, 2.0) - pow(a, 2.0))) * t_1) * t_2) / x_45_scale) / y_45_scale;
	return (t_3 * t_3) - ((4.0 * (((pow((a * t_1), 2.0) + pow((b * t_2), 2.0)) / x_45_scale) / x_45_scale)) * (((pow((a * t_2), 2.0) + pow((b * t_1), 2.0)) / y_45_scale) / y_45_scale));
}

Java

public static double code(double a, double b, double angle, double x_45_scale, double y_45_scale) {
	double t_0 = (angle / 180.0) * Math.PI;
	double t_1 = Math.sin(t_0);
	double t_2 = Math.cos(t_0);
	double t_3 = ((((2.0 * (Math.pow(b, 2.0) - Math.pow(a, 2.0))) * t_1) * t_2) / x_45_scale) / y_45_scale;
	return (t_3 * t_3) - ((4.0 * (((Math.pow((a * t_1), 2.0) + Math.pow((b * t_2), 2.0)) / x_45_scale) / x_45_scale)) * (((Math.pow((a * t_2), 2.0) + Math.pow((b * t_1), 2.0)) / y_45_scale) / y_45_scale));
}

Python

def code(a, b, angle, x_45_scale, y_45_scale):
	t_0 = (angle / 180.0) * math.pi
	t_1 = math.sin(t_0)
	t_2 = math.cos(t_0)
	t_3 = ((((2.0 * (math.pow(b, 2.0) - math.pow(a, 2.0))) * t_1) * t_2) / x_45_scale) / y_45_scale
	return (t_3 * t_3) - ((4.0 * (((math.pow((a * t_1), 2.0) + math.pow((b * t_2), 2.0)) / x_45_scale) / x_45_scale)) * (((math.pow((a * t_2), 2.0) + math.pow((b * t_1), 2.0)) / y_45_scale) / y_45_scale))

Julia

function code(a, b, angle, x_45_scale, y_45_scale)
	t_0 = Float64(Float64(angle / 180.0) * pi)
	t_1 = sin(t_0)
	t_2 = cos(t_0)
	t_3 = Float64(Float64(Float64(Float64(Float64(2.0 * Float64((b ^ 2.0) - (a ^ 2.0))) * t_1) * t_2) / x_45_scale) / y_45_scale)
	return Float64(Float64(t_3 * t_3) - Float64(Float64(4.0 * Float64(Float64(Float64((Float64(a * t_1) ^ 2.0) + (Float64(b * t_2) ^ 2.0)) / x_45_scale) / x_45_scale)) * Float64(Float64(Float64((Float64(a * t_2) ^ 2.0) + (Float64(b * t_1) ^ 2.0)) / y_45_scale) / y_45_scale)))
end

MATLAB

function tmp = code(a, b, angle, x_45_scale, y_45_scale)
	t_0 = (angle / 180.0) * pi;
	t_1 = sin(t_0);
	t_2 = cos(t_0);
	t_3 = ((((2.0 * ((b ^ 2.0) - (a ^ 2.0))) * t_1) * t_2) / x_45_scale) / y_45_scale;
	tmp = (t_3 * t_3) - ((4.0 * (((((a * t_1) ^ 2.0) + ((b * t_2) ^ 2.0)) / x_45_scale) / x_45_scale)) * (((((a * t_2) ^ 2.0) + ((b * t_1) ^ 2.0)) / y_45_scale) / y_45_scale));
end

Wolfram

code[a_, b_, angle_, x$45$scale_, y$45$scale_] := Block[{t$95$0 = N[(N[(angle / 180.0), $MachinePrecision] * Pi), $MachinePrecision]}, Block[{t$95$1 = N[Sin[t$95$0], $MachinePrecision]}, Block[{t$95$2 = N[Cos[t$95$0], $MachinePrecision]}, Block[{t$95$3 = N[(N[(N[(N[(N[(2.0 * N[(N[Power[b, 2.0], $MachinePrecision] - N[Power[a, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * t$95$1), $MachinePrecision] * t$95$2), $MachinePrecision] / x$45$scale), $MachinePrecision] / y$45$scale), $MachinePrecision]}, N[(N[(t$95$3 * t$95$3), $MachinePrecision] - N[(N[(4.0 * N[(N[(N[(N[Power[N[(a * t$95$1), $MachinePrecision], 2.0], $MachinePrecision] + N[Power[N[(b * t$95$2), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] / x$45$scale), $MachinePrecision] / x$45$scale), $MachinePrecision]), $MachinePrecision] * N[(N[(N[(N[Power[N[(a * t$95$2), $MachinePrecision], 2.0], $MachinePrecision] + N[Power[N[(b * t$95$1), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] / y$45$scale), $MachinePrecision] / y$45$scale), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]

Initial Program: 24.7% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{angle}{180} \cdot \pi\\ t_1 := \sin t_0\\ t_2 := \cos t_0\\ t_3 := \frac{\frac{\left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot t_1\right) \cdot t_2}{x-scale}}{y-scale}\\ t_3 \cdot t_3 - \left(4 \cdot \frac{\frac{{\left(a \cdot t_1\right)}^{2} + {\left(b \cdot t_2\right)}^{2}}{x-scale}}{x-scale}\right) \cdot \frac{\frac{{\left(a \cdot t_2\right)}^{2} + {\left(b \cdot t_1\right)}^{2}}{y-scale}}{y-scale} \end{array} \end{array} \]

FPCore

(FPCore (a b angle x-scale y-scale)
 :precision binary64
 (let* ((t_0 (* (/ angle 180.0) PI))
        (t_1 (sin t_0))
        (t_2 (cos t_0))
        (t_3
         (/
          (/ (* (* (* 2.0 (- (pow b 2.0) (pow a 2.0))) t_1) t_2) x-scale)
          y-scale)))
   (-
    (* t_3 t_3)
    (*
     (*
      4.0
      (/ (/ (+ (pow (* a t_1) 2.0) (pow (* b t_2) 2.0)) x-scale) x-scale))
     (/ (/ (+ (pow (* a t_2) 2.0) (pow (* b t_1) 2.0)) y-scale) y-scale)))))

C

double code(double a, double b, double angle, double x_45_scale, double y_45_scale) {
	double t_0 = (angle / 180.0) * ((double) M_PI);
	double t_1 = sin(t_0);
	double t_2 = cos(t_0);
	double t_3 = ((((2.0 * (pow(b, 2.0) - pow(a, 2.0))) * t_1) * t_2) / x_45_scale) / y_45_scale;
	return (t_3 * t_3) - ((4.0 * (((pow((a * t_1), 2.0) + pow((b * t_2), 2.0)) / x_45_scale) / x_45_scale)) * (((pow((a * t_2), 2.0) + pow((b * t_1), 2.0)) / y_45_scale) / y_45_scale));
}

Java

public static double code(double a, double b, double angle, double x_45_scale, double y_45_scale) {
	double t_0 = (angle / 180.0) * Math.PI;
	double t_1 = Math.sin(t_0);
	double t_2 = Math.cos(t_0);
	double t_3 = ((((2.0 * (Math.pow(b, 2.0) - Math.pow(a, 2.0))) * t_1) * t_2) / x_45_scale) / y_45_scale;
	return (t_3 * t_3) - ((4.0 * (((Math.pow((a * t_1), 2.0) + Math.pow((b * t_2), 2.0)) / x_45_scale) / x_45_scale)) * (((Math.pow((a * t_2), 2.0) + Math.pow((b * t_1), 2.0)) / y_45_scale) / y_45_scale));
}

Python

def code(a, b, angle, x_45_scale, y_45_scale):
	t_0 = (angle / 180.0) * math.pi
	t_1 = math.sin(t_0)
	t_2 = math.cos(t_0)
	t_3 = ((((2.0 * (math.pow(b, 2.0) - math.pow(a, 2.0))) * t_1) * t_2) / x_45_scale) / y_45_scale
	return (t_3 * t_3) - ((4.0 * (((math.pow((a * t_1), 2.0) + math.pow((b * t_2), 2.0)) / x_45_scale) / x_45_scale)) * (((math.pow((a * t_2), 2.0) + math.pow((b * t_1), 2.0)) / y_45_scale) / y_45_scale))

Julia

function code(a, b, angle, x_45_scale, y_45_scale)
	t_0 = Float64(Float64(angle / 180.0) * pi)
	t_1 = sin(t_0)
	t_2 = cos(t_0)
	t_3 = Float64(Float64(Float64(Float64(Float64(2.0 * Float64((b ^ 2.0) - (a ^ 2.0))) * t_1) * t_2) / x_45_scale) / y_45_scale)
	return Float64(Float64(t_3 * t_3) - Float64(Float64(4.0 * Float64(Float64(Float64((Float64(a * t_1) ^ 2.0) + (Float64(b * t_2) ^ 2.0)) / x_45_scale) / x_45_scale)) * Float64(Float64(Float64((Float64(a * t_2) ^ 2.0) + (Float64(b * t_1) ^ 2.0)) / y_45_scale) / y_45_scale)))
end

MATLAB

function tmp = code(a, b, angle, x_45_scale, y_45_scale)
	t_0 = (angle / 180.0) * pi;
	t_1 = sin(t_0);
	t_2 = cos(t_0);
	t_3 = ((((2.0 * ((b ^ 2.0) - (a ^ 2.0))) * t_1) * t_2) / x_45_scale) / y_45_scale;
	tmp = (t_3 * t_3) - ((4.0 * (((((a * t_1) ^ 2.0) + ((b * t_2) ^ 2.0)) / x_45_scale) / x_45_scale)) * (((((a * t_2) ^ 2.0) + ((b * t_1) ^ 2.0)) / y_45_scale) / y_45_scale));
end

Wolfram

code[a_, b_, angle_, x$45$scale_, y$45$scale_] := Block[{t$95$0 = N[(N[(angle / 180.0), $MachinePrecision] * Pi), $MachinePrecision]}, Block[{t$95$1 = N[Sin[t$95$0], $MachinePrecision]}, Block[{t$95$2 = N[Cos[t$95$0], $MachinePrecision]}, Block[{t$95$3 = N[(N[(N[(N[(N[(2.0 * N[(N[Power[b, 2.0], $MachinePrecision] - N[Power[a, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * t$95$1), $MachinePrecision] * t$95$2), $MachinePrecision] / x$45$scale), $MachinePrecision] / y$45$scale), $MachinePrecision]}, N[(N[(t$95$3 * t$95$3), $MachinePrecision] - N[(N[(4.0 * N[(N[(N[(N[Power[N[(a * t$95$1), $MachinePrecision], 2.0], $MachinePrecision] + N[Power[N[(b * t$95$2), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] / x$45$scale), $MachinePrecision] / x$45$scale), $MachinePrecision]), $MachinePrecision] * N[(N[(N[(N[Power[N[(a * t$95$2), $MachinePrecision], 2.0], $MachinePrecision] + N[Power[N[(b * t$95$1), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] / y$45$scale), $MachinePrecision] / y$45$scale), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]

Alternative 1: 83.2% accurate, 4.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y-scale \leq 4 \cdot 10^{-22}:\\ \;\;\;\;{\left({\left(\sqrt[3]{b} \cdot \sqrt[3]{a}\right)}^{2} \cdot \sqrt[3]{-4 \cdot {\left(y-scale \cdot x-scale\right)}^{-2}}\right)}^{3}\\ \mathbf{else}:\\ \;\;\;\;\frac{-4}{y-scale \cdot x-scale} \cdot \frac{{\left(b \cdot a\right)}^{2}}{y-scale \cdot x-scale}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b angle x-scale y-scale)
 :precision binary64
 (if (<= y-scale 4e-22)
   (pow
    (*
     (pow (* (cbrt b) (cbrt a)) 2.0)
     (cbrt (* -4.0 (pow (* y-scale x-scale) -2.0))))
    3.0)
   (* (/ -4.0 (* y-scale x-scale)) (/ (pow (* b a) 2.0) (* y-scale x-scale)))))

C

double code(double a, double b, double angle, double x_45_scale, double y_45_scale) {
	double tmp;
	if (y_45_scale <= 4e-22) {
		tmp = pow((pow((cbrt(b) * cbrt(a)), 2.0) * cbrt((-4.0 * pow((y_45_scale * x_45_scale), -2.0)))), 3.0);
	} else {
		tmp = (-4.0 / (y_45_scale * x_45_scale)) * (pow((b * a), 2.0) / (y_45_scale * x_45_scale));
	}
	return tmp;
}

Java

public static double code(double a, double b, double angle, double x_45_scale, double y_45_scale) {
	double tmp;
	if (y_45_scale <= 4e-22) {
		tmp = Math.pow((Math.pow((Math.cbrt(b) * Math.cbrt(a)), 2.0) * Math.cbrt((-4.0 * Math.pow((y_45_scale * x_45_scale), -2.0)))), 3.0);
	} else {
		tmp = (-4.0 / (y_45_scale * x_45_scale)) * (Math.pow((b * a), 2.0) / (y_45_scale * x_45_scale));
	}
	return tmp;
}

Julia

function code(a, b, angle, x_45_scale, y_45_scale)
	tmp = 0.0
	if (y_45_scale <= 4e-22)
		tmp = Float64((Float64(cbrt(b) * cbrt(a)) ^ 2.0) * cbrt(Float64(-4.0 * (Float64(y_45_scale * x_45_scale) ^ -2.0)))) ^ 3.0;
	else
		tmp = Float64(Float64(-4.0 / Float64(y_45_scale * x_45_scale)) * Float64((Float64(b * a) ^ 2.0) / Float64(y_45_scale * x_45_scale)));
	end
	return tmp
end

Wolfram

code[a_, b_, angle_, x$45$scale_, y$45$scale_] := If[LessEqual[y$45$scale, 4e-22], N[Power[N[(N[Power[N[(N[Power[b, 1/3], $MachinePrecision] * N[Power[a, 1/3], $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision] * N[Power[N[(-4.0 * N[Power[N[(y$45$scale * x$45$scale), $MachinePrecision], -2.0], $MachinePrecision]), $MachinePrecision], 1/3], $MachinePrecision]), $MachinePrecision], 3.0], $MachinePrecision], N[(N[(-4.0 / N[(y$45$scale * x$45$scale), $MachinePrecision]), $MachinePrecision] * N[(N[Power[N[(b * a), $MachinePrecision], 2.0], $MachinePrecision] / N[(y$45$scale * x$45$scale), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y-scale < 4.0000000000000002e-22

   1. Initial program 20.3%

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

   3. Simplified 16.0%

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

   4. Taylor expanded in angle around 0 45.5%

      \[\leadsto \color{blue}{-4 \cdot \frac{{a}^{2} \cdot {b}^{2}}{{x-scale}^{2} \cdot {y-scale}^{2}}} \]
   5. Step-by-step derivation

      1. associate-*r/ 45.5%

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

      2. *-commutative 45.5%

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

      3. unpow2 45.5%

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

      4. unpow2 45.5%

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

      5. swap-sqr 62.4%

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

      6. unpow2 62.4%

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

   6. Simplified 62.4%

      \[\leadsto \color{blue}{\frac{-4 \cdot \left({b}^{2} \cdot {a}^{2}\right)}{{\left(x-scale \cdot y-scale\right)}^{2}}} \]
   7. Step-by-step derivation

      1. pow-prod-down 45.5%

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

      2. add-cube-cbrt 45.5%

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

      3. pow3 45.5%

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

      4. div-inv 45.5%

         \[\leadsto {\left(\sqrt[3]{\color{blue}{\left(-4 \cdot \left({b}^{2} \cdot {a}^{2}\right)\right) \cdot \frac{1}{{x-scale}^{2} \cdot {y-scale}^{2}}}}\right)}^{3} \]

      5. *-commutative 45.5%

         \[\leadsto {\left(\sqrt[3]{\color{blue}{\left(\left({b}^{2} \cdot {a}^{2}\right) \cdot -4\right)} \cdot \frac{1}{{x-scale}^{2} \cdot {y-scale}^{2}}}\right)}^{3} \]

      6. pow-prod-down 58.1%

         \[\leadsto {\left(\sqrt[3]{\left(\color{blue}{{\left(b \cdot a\right)}^{2}} \cdot -4\right) \cdot \frac{1}{{x-scale}^{2} \cdot {y-scale}^{2}}}\right)}^{3} \]

      7. pow-prod-down 80.8%

         \[\leadsto {\left(\sqrt[3]{\left({\left(b \cdot a\right)}^{2} \cdot -4\right) \cdot \frac{1}{\color{blue}{{\left(x-scale \cdot y-scale\right)}^{2}}}}\right)}^{3} \]

      8. pow-flip 80.8%

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

      9. metadata-eval 80.8%

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

   8. Applied egg-rr 80.8%

      \[\leadsto \color{blue}{{\left(\sqrt[3]{\left({\left(b \cdot a\right)}^{2} \cdot -4\right) \cdot {\left(x-scale \cdot y-scale\right)}^{-2}}\right)}^{3}} \]
   9. Step-by-step derivation

      1. associate-*l* 80.8%

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

      2. cbrt-prod 80.6%

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

      3. unpow2 80.6%

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

      4. cbrt-prod 88.4%

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

      5. pow2 88.4%

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

   10. Applied egg-rr 88.4%

       \[\leadsto {\color{blue}{\left({\left(\sqrt[3]{b \cdot a}\right)}^{2} \cdot \sqrt[3]{-4 \cdot {\left(x-scale \cdot y-scale\right)}^{-2}}\right)}}^{3} \]
   11. Step-by-step derivation

       1. cbrt-prod 88.8%

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

   12. Applied egg-rr 88.8%

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

   if 4.0000000000000002e-22 < y-scale

   1. Initial program 27.2%

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

   3. Simplified 27.0%

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

   4. Taylor expanded in angle around 0 44.1%

      \[\leadsto \color{blue}{-4 \cdot \frac{{a}^{2} \cdot {b}^{2}}{{x-scale}^{2} \cdot {y-scale}^{2}}} \]
   5. Step-by-step derivation

      1. associate-*r/ 44.1%

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

      2. *-commutative 44.1%

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

      3. unpow2 44.1%

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

      4. unpow2 44.1%

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

      5. swap-sqr 54.5%

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

      6. unpow2 54.5%

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

   6. Simplified 54.5%

      \[\leadsto \color{blue}{\frac{-4 \cdot \left({b}^{2} \cdot {a}^{2}\right)}{{\left(x-scale \cdot y-scale\right)}^{2}}} \]
   7. Step-by-step derivation

      1. pow-prod-down 81.2%

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

   8. Applied egg-rr 81.2%

      \[\leadsto \frac{-4 \cdot \color{blue}{{\left(b \cdot a\right)}^{2}}}{{\left(x-scale \cdot y-scale\right)}^{2}} \]
   9. Step-by-step derivation

      1. unpow2 81.2%

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

   10. Applied egg-rr 81.2%

       \[\leadsto \frac{-4 \cdot {\left(b \cdot a\right)}^{2}}{\color{blue}{\left(x-scale \cdot y-scale\right) \cdot \left(x-scale \cdot y-scale\right)}} \]
   11. Step-by-step derivation

       1. times-frac 91.0%

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

   12. Applied egg-rr 91.0%

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

4. Final simplification 89.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y-scale \leq 4 \cdot 10^{-22}:\\ \;\;\;\;{\left({\left(\sqrt[3]{b} \cdot \sqrt[3]{a}\right)}^{2} \cdot \sqrt[3]{-4 \cdot {\left(y-scale \cdot x-scale\right)}^{-2}}\right)}^{3}\\ \mathbf{else}:\\ \;\;\;\;\frac{-4}{y-scale \cdot x-scale} \cdot \frac{{\left(b \cdot a\right)}^{2}}{y-scale \cdot x-scale}\\ \end{array} \]

Alternative 2: 83.0% accurate, 4.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y-scale \leq 2.25 \cdot 10^{-23}:\\ \;\;\;\;{\left({\left(\sqrt[3]{b \cdot a}\right)}^{2} \cdot \left(\sqrt[3]{{\left(y-scale \cdot x-scale\right)}^{-2}} \cdot \sqrt[3]{-4}\right)\right)}^{3}\\ \mathbf{else}:\\ \;\;\;\;\frac{-4}{y-scale \cdot x-scale} \cdot \frac{{\left(b \cdot a\right)}^{2}}{y-scale \cdot x-scale}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b angle x-scale y-scale)
 :precision binary64
 (if (<= y-scale 2.25e-23)
   (pow
    (*
     (pow (cbrt (* b a)) 2.0)
     (* (cbrt (pow (* y-scale x-scale) -2.0)) (cbrt -4.0)))
    3.0)
   (* (/ -4.0 (* y-scale x-scale)) (/ (pow (* b a) 2.0) (* y-scale x-scale)))))

C

double code(double a, double b, double angle, double x_45_scale, double y_45_scale) {
	double tmp;
	if (y_45_scale <= 2.25e-23) {
		tmp = pow((pow(cbrt((b * a)), 2.0) * (cbrt(pow((y_45_scale * x_45_scale), -2.0)) * cbrt(-4.0))), 3.0);
	} else {
		tmp = (-4.0 / (y_45_scale * x_45_scale)) * (pow((b * a), 2.0) / (y_45_scale * x_45_scale));
	}
	return tmp;
}

Java

public static double code(double a, double b, double angle, double x_45_scale, double y_45_scale) {
	double tmp;
	if (y_45_scale <= 2.25e-23) {
		tmp = Math.pow((Math.pow(Math.cbrt((b * a)), 2.0) * (Math.cbrt(Math.pow((y_45_scale * x_45_scale), -2.0)) * Math.cbrt(-4.0))), 3.0);
	} else {
		tmp = (-4.0 / (y_45_scale * x_45_scale)) * (Math.pow((b * a), 2.0) / (y_45_scale * x_45_scale));
	}
	return tmp;
}

Julia

function code(a, b, angle, x_45_scale, y_45_scale)
	tmp = 0.0
	if (y_45_scale <= 2.25e-23)
		tmp = Float64((cbrt(Float64(b * a)) ^ 2.0) * Float64(cbrt((Float64(y_45_scale * x_45_scale) ^ -2.0)) * cbrt(-4.0))) ^ 3.0;
	else
		tmp = Float64(Float64(-4.0 / Float64(y_45_scale * x_45_scale)) * Float64((Float64(b * a) ^ 2.0) / Float64(y_45_scale * x_45_scale)));
	end
	return tmp
end

Wolfram

code[a_, b_, angle_, x$45$scale_, y$45$scale_] := If[LessEqual[y$45$scale, 2.25e-23], N[Power[N[(N[Power[N[Power[N[(b * a), $MachinePrecision], 1/3], $MachinePrecision], 2.0], $MachinePrecision] * N[(N[Power[N[Power[N[(y$45$scale * x$45$scale), $MachinePrecision], -2.0], $MachinePrecision], 1/3], $MachinePrecision] * N[Power[-4.0, 1/3], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 3.0], $MachinePrecision], N[(N[(-4.0 / N[(y$45$scale * x$45$scale), $MachinePrecision]), $MachinePrecision] * N[(N[Power[N[(b * a), $MachinePrecision], 2.0], $MachinePrecision] / N[(y$45$scale * x$45$scale), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y-scale < 2.24999999999999987e-23

   1. Initial program 20.3%

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

   3. Simplified 16.0%

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

   4. Taylor expanded in angle around 0 45.5%

      \[\leadsto \color{blue}{-4 \cdot \frac{{a}^{2} \cdot {b}^{2}}{{x-scale}^{2} \cdot {y-scale}^{2}}} \]
   5. Step-by-step derivation

      1. associate-*r/ 45.5%

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

      2. *-commutative 45.5%

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

      3. unpow2 45.5%

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

      4. unpow2 45.5%

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

      5. swap-sqr 62.4%

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

      6. unpow2 62.4%

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

   6. Simplified 62.4%

      \[\leadsto \color{blue}{\frac{-4 \cdot \left({b}^{2} \cdot {a}^{2}\right)}{{\left(x-scale \cdot y-scale\right)}^{2}}} \]
   7. Step-by-step derivation

      1. pow-prod-down 45.5%

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

      2. add-cube-cbrt 45.5%

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

      3. pow3 45.5%

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

      4. div-inv 45.5%

         \[\leadsto {\left(\sqrt[3]{\color{blue}{\left(-4 \cdot \left({b}^{2} \cdot {a}^{2}\right)\right) \cdot \frac{1}{{x-scale}^{2} \cdot {y-scale}^{2}}}}\right)}^{3} \]

      5. *-commutative 45.5%

         \[\leadsto {\left(\sqrt[3]{\color{blue}{\left(\left({b}^{2} \cdot {a}^{2}\right) \cdot -4\right)} \cdot \frac{1}{{x-scale}^{2} \cdot {y-scale}^{2}}}\right)}^{3} \]

      6. pow-prod-down 58.1%

         \[\leadsto {\left(\sqrt[3]{\left(\color{blue}{{\left(b \cdot a\right)}^{2}} \cdot -4\right) \cdot \frac{1}{{x-scale}^{2} \cdot {y-scale}^{2}}}\right)}^{3} \]

      7. pow-prod-down 80.8%

         \[\leadsto {\left(\sqrt[3]{\left({\left(b \cdot a\right)}^{2} \cdot -4\right) \cdot \frac{1}{\color{blue}{{\left(x-scale \cdot y-scale\right)}^{2}}}}\right)}^{3} \]

      8. pow-flip 80.8%

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

      9. metadata-eval 80.8%

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

   8. Applied egg-rr 80.8%

      \[\leadsto \color{blue}{{\left(\sqrt[3]{\left({\left(b \cdot a\right)}^{2} \cdot -4\right) \cdot {\left(x-scale \cdot y-scale\right)}^{-2}}\right)}^{3}} \]
   9. Step-by-step derivation

      1. associate-*l* 80.8%

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

      2. cbrt-prod 80.6%

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

      3. unpow2 80.6%

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

      4. cbrt-prod 88.4%

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

      5. pow2 88.4%

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

   10. Applied egg-rr 88.4%

       \[\leadsto {\color{blue}{\left({\left(\sqrt[3]{b \cdot a}\right)}^{2} \cdot \sqrt[3]{-4 \cdot {\left(x-scale \cdot y-scale\right)}^{-2}}\right)}}^{3} \]
   11. Step-by-step derivation

       1. unpow-prod-down 63.2%

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

       2. *-commutative 63.2%

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

       3. *-commutative 63.2%

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

       4. cbrt-prod 63.2%

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

       5. *-commutative 63.2%

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

       6. unpow-prod-down 88.4%

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

   12. Applied egg-rr 88.4%

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

   if 2.24999999999999987e-23 < y-scale

   1. Initial program 27.2%

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

   3. Simplified 27.0%

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

   4. Taylor expanded in angle around 0 44.1%

      \[\leadsto \color{blue}{-4 \cdot \frac{{a}^{2} \cdot {b}^{2}}{{x-scale}^{2} \cdot {y-scale}^{2}}} \]
   5. Step-by-step derivation

      1. associate-*r/ 44.1%

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

      2. *-commutative 44.1%

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

      3. unpow2 44.1%

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

      4. unpow2 44.1%

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

      5. swap-sqr 54.5%

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

      6. unpow2 54.5%

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

   6. Simplified 54.5%

      \[\leadsto \color{blue}{\frac{-4 \cdot \left({b}^{2} \cdot {a}^{2}\right)}{{\left(x-scale \cdot y-scale\right)}^{2}}} \]
   7. Step-by-step derivation

      1. pow-prod-down 81.2%

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

   8. Applied egg-rr 81.2%

      \[\leadsto \frac{-4 \cdot \color{blue}{{\left(b \cdot a\right)}^{2}}}{{\left(x-scale \cdot y-scale\right)}^{2}} \]
   9. Step-by-step derivation

      1. unpow2 81.2%

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

   10. Applied egg-rr 81.2%

       \[\leadsto \frac{-4 \cdot {\left(b \cdot a\right)}^{2}}{\color{blue}{\left(x-scale \cdot y-scale\right) \cdot \left(x-scale \cdot y-scale\right)}} \]
   11. Step-by-step derivation

       1. times-frac 91.0%

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

   12. Applied egg-rr 91.0%

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

4. Final simplification 89.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y-scale \leq 2.25 \cdot 10^{-23}:\\ \;\;\;\;{\left({\left(\sqrt[3]{b \cdot a}\right)}^{2} \cdot \left(\sqrt[3]{{\left(y-scale \cdot x-scale\right)}^{-2}} \cdot \sqrt[3]{-4}\right)\right)}^{3}\\ \mathbf{else}:\\ \;\;\;\;\frac{-4}{y-scale \cdot x-scale} \cdot \frac{{\left(b \cdot a\right)}^{2}}{y-scale \cdot x-scale}\\ \end{array} \]

Alternative 3: 83.0% accurate, 4.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y-scale \leq 2.55 \cdot 10^{-23}:\\ \;\;\;\;{\left(\sqrt[3]{-4 \cdot {\left(y-scale \cdot x-scale\right)}^{-2}} \cdot {\left(\sqrt[3]{b \cdot a}\right)}^{2}\right)}^{3}\\ \mathbf{else}:\\ \;\;\;\;\frac{-4}{y-scale \cdot x-scale} \cdot \frac{{\left(b \cdot a\right)}^{2}}{y-scale \cdot x-scale}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b angle x-scale y-scale)
 :precision binary64
 (if (<= y-scale 2.55e-23)
   (pow
    (* (cbrt (* -4.0 (pow (* y-scale x-scale) -2.0))) (pow (cbrt (* b a)) 2.0))
    3.0)
   (* (/ -4.0 (* y-scale x-scale)) (/ (pow (* b a) 2.0) (* y-scale x-scale)))))

C

double code(double a, double b, double angle, double x_45_scale, double y_45_scale) {
	double tmp;
	if (y_45_scale <= 2.55e-23) {
		tmp = pow((cbrt((-4.0 * pow((y_45_scale * x_45_scale), -2.0))) * pow(cbrt((b * a)), 2.0)), 3.0);
	} else {
		tmp = (-4.0 / (y_45_scale * x_45_scale)) * (pow((b * a), 2.0) / (y_45_scale * x_45_scale));
	}
	return tmp;
}

Java

public static double code(double a, double b, double angle, double x_45_scale, double y_45_scale) {
	double tmp;
	if (y_45_scale <= 2.55e-23) {
		tmp = Math.pow((Math.cbrt((-4.0 * Math.pow((y_45_scale * x_45_scale), -2.0))) * Math.pow(Math.cbrt((b * a)), 2.0)), 3.0);
	} else {
		tmp = (-4.0 / (y_45_scale * x_45_scale)) * (Math.pow((b * a), 2.0) / (y_45_scale * x_45_scale));
	}
	return tmp;
}

Julia

function code(a, b, angle, x_45_scale, y_45_scale)
	tmp = 0.0
	if (y_45_scale <= 2.55e-23)
		tmp = Float64(cbrt(Float64(-4.0 * (Float64(y_45_scale * x_45_scale) ^ -2.0))) * (cbrt(Float64(b * a)) ^ 2.0)) ^ 3.0;
	else
		tmp = Float64(Float64(-4.0 / Float64(y_45_scale * x_45_scale)) * Float64((Float64(b * a) ^ 2.0) / Float64(y_45_scale * x_45_scale)));
	end
	return tmp
end

Wolfram

code[a_, b_, angle_, x$45$scale_, y$45$scale_] := If[LessEqual[y$45$scale, 2.55e-23], N[Power[N[(N[Power[N[(-4.0 * N[Power[N[(y$45$scale * x$45$scale), $MachinePrecision], -2.0], $MachinePrecision]), $MachinePrecision], 1/3], $MachinePrecision] * N[Power[N[Power[N[(b * a), $MachinePrecision], 1/3], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision], 3.0], $MachinePrecision], N[(N[(-4.0 / N[(y$45$scale * x$45$scale), $MachinePrecision]), $MachinePrecision] * N[(N[Power[N[(b * a), $MachinePrecision], 2.0], $MachinePrecision] / N[(y$45$scale * x$45$scale), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y-scale < 2.55000000000000005e-23

   1. Initial program 20.3%

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

   3. Simplified 16.0%

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

   4. Taylor expanded in angle around 0 45.5%

      \[\leadsto \color{blue}{-4 \cdot \frac{{a}^{2} \cdot {b}^{2}}{{x-scale}^{2} \cdot {y-scale}^{2}}} \]
   5. Step-by-step derivation

      1. associate-*r/ 45.5%

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

      2. *-commutative 45.5%

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

      3. unpow2 45.5%

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

      4. unpow2 45.5%

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

      5. swap-sqr 62.4%

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

      6. unpow2 62.4%

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

   6. Simplified 62.4%

      \[\leadsto \color{blue}{\frac{-4 \cdot \left({b}^{2} \cdot {a}^{2}\right)}{{\left(x-scale \cdot y-scale\right)}^{2}}} \]
   7. Step-by-step derivation

      1. pow-prod-down 45.5%

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

      2. add-cube-cbrt 45.5%

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

      3. pow3 45.5%

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

      4. div-inv 45.5%

         \[\leadsto {\left(\sqrt[3]{\color{blue}{\left(-4 \cdot \left({b}^{2} \cdot {a}^{2}\right)\right) \cdot \frac{1}{{x-scale}^{2} \cdot {y-scale}^{2}}}}\right)}^{3} \]

      5. *-commutative 45.5%

         \[\leadsto {\left(\sqrt[3]{\color{blue}{\left(\left({b}^{2} \cdot {a}^{2}\right) \cdot -4\right)} \cdot \frac{1}{{x-scale}^{2} \cdot {y-scale}^{2}}}\right)}^{3} \]

      6. pow-prod-down 58.1%

         \[\leadsto {\left(\sqrt[3]{\left(\color{blue}{{\left(b \cdot a\right)}^{2}} \cdot -4\right) \cdot \frac{1}{{x-scale}^{2} \cdot {y-scale}^{2}}}\right)}^{3} \]

      7. pow-prod-down 80.8%

         \[\leadsto {\left(\sqrt[3]{\left({\left(b \cdot a\right)}^{2} \cdot -4\right) \cdot \frac{1}{\color{blue}{{\left(x-scale \cdot y-scale\right)}^{2}}}}\right)}^{3} \]

      8. pow-flip 80.8%

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

      9. metadata-eval 80.8%

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

   8. Applied egg-rr 80.8%

      \[\leadsto \color{blue}{{\left(\sqrt[3]{\left({\left(b \cdot a\right)}^{2} \cdot -4\right) \cdot {\left(x-scale \cdot y-scale\right)}^{-2}}\right)}^{3}} \]
   9. Step-by-step derivation

      1. associate-*l* 80.8%

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

      2. cbrt-prod 80.6%

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

      3. unpow2 80.6%

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

      4. cbrt-prod 88.4%

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

      5. pow2 88.4%

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

   10. Applied egg-rr 88.4%

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

   if 2.55000000000000005e-23 < y-scale

   1. Initial program 27.2%

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

   3. Simplified 27.0%

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

   4. Taylor expanded in angle around 0 44.1%

      \[\leadsto \color{blue}{-4 \cdot \frac{{a}^{2} \cdot {b}^{2}}{{x-scale}^{2} \cdot {y-scale}^{2}}} \]
   5. Step-by-step derivation

      1. associate-*r/ 44.1%

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

      2. *-commutative 44.1%

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

      3. unpow2 44.1%

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

      4. unpow2 44.1%

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

      5. swap-sqr 54.5%

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

      6. unpow2 54.5%

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

   6. Simplified 54.5%

      \[\leadsto \color{blue}{\frac{-4 \cdot \left({b}^{2} \cdot {a}^{2}\right)}{{\left(x-scale \cdot y-scale\right)}^{2}}} \]
   7. Step-by-step derivation

      1. pow-prod-down 81.2%

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

   8. Applied egg-rr 81.2%

      \[\leadsto \frac{-4 \cdot \color{blue}{{\left(b \cdot a\right)}^{2}}}{{\left(x-scale \cdot y-scale\right)}^{2}} \]
   9. Step-by-step derivation

      1. unpow2 81.2%

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

   10. Applied egg-rr 81.2%

       \[\leadsto \frac{-4 \cdot {\left(b \cdot a\right)}^{2}}{\color{blue}{\left(x-scale \cdot y-scale\right) \cdot \left(x-scale \cdot y-scale\right)}} \]
   11. Step-by-step derivation

       1. times-frac 91.0%

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

   12. Applied egg-rr 91.0%

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

4. Final simplification 89.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y-scale \leq 2.55 \cdot 10^{-23}:\\ \;\;\;\;{\left(\sqrt[3]{-4 \cdot {\left(y-scale \cdot x-scale\right)}^{-2}} \cdot {\left(\sqrt[3]{b \cdot a}\right)}^{2}\right)}^{3}\\ \mathbf{else}:\\ \;\;\;\;\frac{-4}{y-scale \cdot x-scale} \cdot \frac{{\left(b \cdot a\right)}^{2}}{y-scale \cdot x-scale}\\ \end{array} \]

Alternative 4: 83.2% accurate, 21.8× speedup

Math

\[\begin{array}{l} \\ \frac{-4}{y-scale \cdot x-scale} \cdot \frac{{\left(b \cdot a\right)}^{2}}{y-scale \cdot x-scale} \end{array} \]

FPCore

(FPCore (a b angle x-scale y-scale)
 :precision binary64
 (* (/ -4.0 (* y-scale x-scale)) (/ (pow (* b a) 2.0) (* y-scale x-scale))))

C

double code(double a, double b, double angle, double x_45_scale, double y_45_scale) {
	return (-4.0 / (y_45_scale * x_45_scale)) * (pow((b * a), 2.0) / (y_45_scale * x_45_scale));
}

Fortran

real(8) function code(a, b, angle, x_45scale, y_45scale)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: angle
    real(8), intent (in) :: x_45scale
    real(8), intent (in) :: y_45scale
    code = ((-4.0d0) / (y_45scale * x_45scale)) * (((b * a) ** 2.0d0) / (y_45scale * x_45scale))
end function

Java

public static double code(double a, double b, double angle, double x_45_scale, double y_45_scale) {
	return (-4.0 / (y_45_scale * x_45_scale)) * (Math.pow((b * a), 2.0) / (y_45_scale * x_45_scale));
}

Python

def code(a, b, angle, x_45_scale, y_45_scale):
	return (-4.0 / (y_45_scale * x_45_scale)) * (math.pow((b * a), 2.0) / (y_45_scale * x_45_scale))

Julia

function code(a, b, angle, x_45_scale, y_45_scale)
	return Float64(Float64(-4.0 / Float64(y_45_scale * x_45_scale)) * Float64((Float64(b * a) ^ 2.0) / Float64(y_45_scale * x_45_scale)))
end

MATLAB

function tmp = code(a, b, angle, x_45_scale, y_45_scale)
	tmp = (-4.0 / (y_45_scale * x_45_scale)) * (((b * a) ^ 2.0) / (y_45_scale * x_45_scale));
end

Wolfram

code[a_, b_, angle_, x$45$scale_, y$45$scale_] := N[(N[(-4.0 / N[(y$45$scale * x$45$scale), $MachinePrecision]), $MachinePrecision] * N[(N[Power[N[(b * a), $MachinePrecision], 2.0], $MachinePrecision] / N[(y$45$scale * x$45$scale), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 21.9%

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

3. Simplified 18.5%

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

4. Taylor expanded in angle around 0 45.2%

   \[\leadsto \color{blue}{-4 \cdot \frac{{a}^{2} \cdot {b}^{2}}{{x-scale}^{2} \cdot {y-scale}^{2}}} \]
5. Step-by-step derivation

   1. associate-*r/ 45.2%

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

   2. *-commutative 45.2%

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

   3. unpow2 45.2%

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

   4. unpow2 45.2%

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

   5. swap-sqr 60.5%

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

   6. unpow2 60.5%

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

6. Simplified 60.5%

   \[\leadsto \color{blue}{\frac{-4 \cdot \left({b}^{2} \cdot {a}^{2}\right)}{{\left(x-scale \cdot y-scale\right)}^{2}}} \]
7. Step-by-step derivation

   1. pow-prod-down 81.0%

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

8. Applied egg-rr 81.0%

   \[\leadsto \frac{-4 \cdot \color{blue}{{\left(b \cdot a\right)}^{2}}}{{\left(x-scale \cdot y-scale\right)}^{2}} \]
9. Step-by-step derivation

   1. unpow2 81.0%

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

10. Applied egg-rr 81.0%

    \[\leadsto \frac{-4 \cdot {\left(b \cdot a\right)}^{2}}{\color{blue}{\left(x-scale \cdot y-scale\right) \cdot \left(x-scale \cdot y-scale\right)}} \]
11. Step-by-step derivation

    1. times-frac 85.4%

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

12. Applied egg-rr 85.4%

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

13. Final simplification 85.4%

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

Alternative 5: 77.7% accurate, 146.2× speedup

Math

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

FPCore

(FPCore (a b angle x-scale y-scale)
 :precision binary64
 (/ (* -4.0 (* (* b a) (* b a))) (* (* y-scale x-scale) (* y-scale x-scale))))

C

double code(double a, double b, double angle, double x_45_scale, double y_45_scale) {
	return (-4.0 * ((b * a) * (b * a))) / ((y_45_scale * x_45_scale) * (y_45_scale * x_45_scale));
}

Fortran

real(8) function code(a, b, angle, x_45scale, y_45scale)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: angle
    real(8), intent (in) :: x_45scale
    real(8), intent (in) :: y_45scale
    code = ((-4.0d0) * ((b * a) * (b * a))) / ((y_45scale * x_45scale) * (y_45scale * x_45scale))
end function

Java

public static double code(double a, double b, double angle, double x_45_scale, double y_45_scale) {
	return (-4.0 * ((b * a) * (b * a))) / ((y_45_scale * x_45_scale) * (y_45_scale * x_45_scale));
}

Python

def code(a, b, angle, x_45_scale, y_45_scale):
	return (-4.0 * ((b * a) * (b * a))) / ((y_45_scale * x_45_scale) * (y_45_scale * x_45_scale))

Julia

function code(a, b, angle, x_45_scale, y_45_scale)
	return Float64(Float64(-4.0 * Float64(Float64(b * a) * Float64(b * a))) / Float64(Float64(y_45_scale * x_45_scale) * Float64(y_45_scale * x_45_scale)))
end

MATLAB

function tmp = code(a, b, angle, x_45_scale, y_45_scale)
	tmp = (-4.0 * ((b * a) * (b * a))) / ((y_45_scale * x_45_scale) * (y_45_scale * x_45_scale));
end

Wolfram

code[a_, b_, angle_, x$45$scale_, y$45$scale_] := N[(N[(-4.0 * N[(N[(b * a), $MachinePrecision] * N[(b * a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(N[(y$45$scale * x$45$scale), $MachinePrecision] * N[(y$45$scale * x$45$scale), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 21.9%

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

3. Simplified 18.5%

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

4. Taylor expanded in angle around 0 45.2%

   \[\leadsto \color{blue}{-4 \cdot \frac{{a}^{2} \cdot {b}^{2}}{{x-scale}^{2} \cdot {y-scale}^{2}}} \]
5. Step-by-step derivation

   1. associate-*r/ 45.2%

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

   2. *-commutative 45.2%

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

   3. unpow2 45.2%

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

   4. unpow2 45.2%

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

   5. swap-sqr 60.5%

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

   6. unpow2 60.5%

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

6. Simplified 60.5%

   \[\leadsto \color{blue}{\frac{-4 \cdot \left({b}^{2} \cdot {a}^{2}\right)}{{\left(x-scale \cdot y-scale\right)}^{2}}} \]
7. Step-by-step derivation

   1. pow-prod-down 81.0%

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

8. Applied egg-rr 81.0%

   \[\leadsto \frac{-4 \cdot \color{blue}{{\left(b \cdot a\right)}^{2}}}{{\left(x-scale \cdot y-scale\right)}^{2}} \]
9. Step-by-step derivation

   1. unpow2 81.0%

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

10. Applied egg-rr 81.0%

    \[\leadsto \frac{-4 \cdot {\left(b \cdot a\right)}^{2}}{\color{blue}{\left(x-scale \cdot y-scale\right) \cdot \left(x-scale \cdot y-scale\right)}} \]
11. Step-by-step derivation

    1. unpow2 81.0%

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

12. Applied egg-rr 81.0%

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

13. Final simplification 81.0%

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

Alternative 6: 35.4% accurate, 2485.0× speedup

Math

\[\begin{array}{l} \\ 0 \end{array} \]

FPCore

(FPCore (a b angle x-scale y-scale) :precision binary64 0.0)

C

double code(double a, double b, double angle, double x_45_scale, double y_45_scale) {
	return 0.0;
}

Fortran

real(8) function code(a, b, angle, x_45scale, y_45scale)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: angle
    real(8), intent (in) :: x_45scale
    real(8), intent (in) :: y_45scale
    code = 0.0d0
end function

Java

public static double code(double a, double b, double angle, double x_45_scale, double y_45_scale) {
	return 0.0;
}

Python

def code(a, b, angle, x_45_scale, y_45_scale):
	return 0.0

Julia

function code(a, b, angle, x_45_scale, y_45_scale)
	return 0.0
end

MATLAB

function tmp = code(a, b, angle, x_45_scale, y_45_scale)
	tmp = 0.0;
end

Wolfram

code[a_, b_, angle_, x$45$scale_, y$45$scale_] := 0.0

Derivation

1. Initial program 21.9%

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

3. Simplified 18.6%

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

4. Taylor expanded in b around 0 21.2%

   \[\leadsto \color{blue}{-4 \cdot \frac{{a}^{4} \cdot \left({\cos \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{2} \cdot {\sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{2}\right)}{{x-scale}^{2} \cdot {y-scale}^{2}} + 4 \cdot \frac{{a}^{4} \cdot \left({\cos \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{2} \cdot {\sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{2}\right)}{{x-scale}^{2} \cdot {y-scale}^{2}}} \]
5. Step-by-step derivation

   1. distribute-rgt-out 21.2%

      \[\leadsto \color{blue}{\frac{{a}^{4} \cdot \left({\cos \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{2} \cdot {\sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{2}\right)}{{x-scale}^{2} \cdot {y-scale}^{2}} \cdot \left(-4 + 4\right)} \]

   2. metadata-eval 21.2%

      \[\leadsto \frac{{a}^{4} \cdot \left({\cos \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{2} \cdot {\sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{2}\right)}{{x-scale}^{2} \cdot {y-scale}^{2}} \cdot \color{blue}{0} \]

   3. mul0-rgt 33.6%

      \[\leadsto \color{blue}{0} \]

6. Simplified 33.6%

   \[\leadsto \color{blue}{0} \]

7. Final simplification 33.6%

   \[\leadsto 0 \]

Reproduce

herbie shell --seed 2023322 
(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))))