Simplification of discriminant from scale-rotated-ellipse

Percentage Accurate: 25.1% → 93.4%

Time: 39.4s

Alternatives: 6

Speedup: 1693.0×

• 33.4% 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: 25.1% 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: 93.4% accurate, 3.3× speedup

Math

\[\begin{array}{l} \\ -4 \cdot {\left(\frac{{\left(\sqrt[3]{a \cdot b}\right)}^{2}}{{\left(\sqrt[3]{x-scale \cdot y-scale}\right)}^{2}}\right)}^{3} \end{array} \]

FPCore

(FPCore (a b angle x-scale y-scale)
 :precision binary64
 (*
  -4.0
  (pow (/ (pow (cbrt (* a b)) 2.0) (pow (cbrt (* x-scale y-scale)) 2.0)) 3.0)))

C

double code(double a, double b, double angle, double x_45_scale, double y_45_scale) {
	return -4.0 * pow((pow(cbrt((a * b)), 2.0) / pow(cbrt((x_45_scale * y_45_scale)), 2.0)), 3.0);
}

Java

public static double code(double a, double b, double angle, double x_45_scale, double y_45_scale) {
	return -4.0 * Math.pow((Math.pow(Math.cbrt((a * b)), 2.0) / Math.pow(Math.cbrt((x_45_scale * y_45_scale)), 2.0)), 3.0);
}

Julia

function code(a, b, angle, x_45_scale, y_45_scale)
	return Float64(-4.0 * (Float64((cbrt(Float64(a * b)) ^ 2.0) / (cbrt(Float64(x_45_scale * y_45_scale)) ^ 2.0)) ^ 3.0))
end

Wolfram

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

Derivation

1. Initial program 24.3%

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

3. Taylor expanded in angle around 0 44.1%

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

   1. *-commutative 44.1%

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

   2. unpow2 44.1%

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

   3. unpow2 44.1%

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

   4. swap-sqr 53.4%

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

   5. unpow2 53.4%

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

5. Simplified 53.4%

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

   1. pow1 53.4%

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

   2. pow-prod-down 76.4%

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

7. Applied egg-rr 76.4%

   \[\leadsto -4 \cdot \frac{{\left(b \cdot a\right)}^{2}}{\color{blue}{{\left({\left(x-scale \cdot y-scale\right)}^{2}\right)}^{1}}} \]
8. Step-by-step derivation

   1. unpow1 76.4%

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

9. Simplified 76.4%

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

    1. add-cube-cbrt 76.3%

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

    2. pow3 76.3%

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

    3. add-cbrt-cube 64.5%

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

    4. unpow2 64.5%

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

    5. unpow2 64.5%

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

    6. add-cbrt-cube 76.3%

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

    7. div-inv 76.3%

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

    8. pow-flip 77.2%

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

    9. metadata-eval 77.2%

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

11. Applied egg-rr 77.2%

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

    1. metadata-eval 77.2%

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

    2. pow-flip 76.3%

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

    3. div-inv 76.3%

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

    4. add-cbrt-cube 64.5%

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

    5. unpow2 64.5%

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

    6. unpow2 64.5%

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

    7. add-cbrt-cube 76.3%

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

    8. cbrt-div 76.3%

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

13. Applied egg-rr 94.3%

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

Alternative 2: 83.5% accurate, 3.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;a \leq 3.8 \cdot 10^{+108}:\\ \;\;\;\;-4 \cdot {\left({\left(\sqrt[3]{a \cdot b}\right)}^{2} \cdot \sqrt[3]{{\left(x-scale \cdot y-scale\right)}^{-2}}\right)}^{3}\\ \mathbf{else}:\\ \;\;\;\;-4 \cdot {\left(\sqrt[3]{\frac{\frac{{\left(a \cdot b\right)}^{2}}{x-scale \cdot y-scale}}{x-scale \cdot y-scale}}\right)}^{3}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b angle x-scale y-scale)
 :precision binary64
 (if (<= a 3.8e+108)
   (*
    -4.0
    (pow
     (* (pow (cbrt (* a b)) 2.0) (cbrt (pow (* x-scale y-scale) -2.0)))
     3.0))
   (*
    -4.0
    (pow
     (cbrt (/ (/ (pow (* a b) 2.0) (* x-scale y-scale)) (* x-scale y-scale)))
     3.0))))

C

double code(double a, double b, double angle, double x_45_scale, double y_45_scale) {
	double tmp;
	if (a <= 3.8e+108) {
		tmp = -4.0 * pow((pow(cbrt((a * b)), 2.0) * cbrt(pow((x_45_scale * y_45_scale), -2.0))), 3.0);
	} else {
		tmp = -4.0 * pow(cbrt(((pow((a * b), 2.0) / (x_45_scale * y_45_scale)) / (x_45_scale * y_45_scale))), 3.0);
	}
	return tmp;
}

Java

public static double code(double a, double b, double angle, double x_45_scale, double y_45_scale) {
	double tmp;
	if (a <= 3.8e+108) {
		tmp = -4.0 * Math.pow((Math.pow(Math.cbrt((a * b)), 2.0) * Math.cbrt(Math.pow((x_45_scale * y_45_scale), -2.0))), 3.0);
	} else {
		tmp = -4.0 * Math.pow(Math.cbrt(((Math.pow((a * b), 2.0) / (x_45_scale * y_45_scale)) / (x_45_scale * y_45_scale))), 3.0);
	}
	return tmp;
}

Julia

function code(a, b, angle, x_45_scale, y_45_scale)
	tmp = 0.0
	if (a <= 3.8e+108)
		tmp = Float64(-4.0 * (Float64((cbrt(Float64(a * b)) ^ 2.0) * cbrt((Float64(x_45_scale * y_45_scale) ^ -2.0))) ^ 3.0));
	else
		tmp = Float64(-4.0 * (cbrt(Float64(Float64((Float64(a * b) ^ 2.0) / Float64(x_45_scale * y_45_scale)) / Float64(x_45_scale * y_45_scale))) ^ 3.0));
	end
	return tmp
end

Wolfram

code[a_, b_, angle_, x$45$scale_, y$45$scale_] := If[LessEqual[a, 3.8e+108], N[(-4.0 * N[Power[N[(N[Power[N[Power[N[(a * b), $MachinePrecision], 1/3], $MachinePrecision], 2.0], $MachinePrecision] * N[Power[N[Power[N[(x$45$scale * y$45$scale), $MachinePrecision], -2.0], $MachinePrecision], 1/3], $MachinePrecision]), $MachinePrecision], 3.0], $MachinePrecision]), $MachinePrecision], N[(-4.0 * N[Power[N[Power[N[(N[(N[Power[N[(a * b), $MachinePrecision], 2.0], $MachinePrecision] / N[(x$45$scale * y$45$scale), $MachinePrecision]), $MachinePrecision] / N[(x$45$scale * y$45$scale), $MachinePrecision]), $MachinePrecision], 1/3], $MachinePrecision], 3.0], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if a < 3.80000000000000008e108

   1. Initial program 28.2%

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

   3. Taylor expanded in angle around 0 45.1%

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

      1. *-commutative 45.1%

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

      2. unpow2 45.1%

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

      3. unpow2 45.1%

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

      4. swap-sqr 53.6%

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

      5. unpow2 53.6%

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

   5. Simplified 53.6%

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

      1. pow1 53.6%

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

      2. pow-prod-down 76.3%

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

   7. Applied egg-rr 76.3%

      \[\leadsto -4 \cdot \frac{{\left(b \cdot a\right)}^{2}}{\color{blue}{{\left({\left(x-scale \cdot y-scale\right)}^{2}\right)}^{1}}} \]
   8. Step-by-step derivation

      1. unpow1 76.3%

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

   9. Simplified 76.3%

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

       1. add-cube-cbrt 76.1%

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

       2. pow3 76.1%

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

       3. add-cbrt-cube 64.9%

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

       4. unpow2 64.9%

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

       5. unpow2 64.9%

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

       6. add-cbrt-cube 76.1%

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

       7. div-inv 76.1%

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

       8. pow-flip 76.7%

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

       9. metadata-eval 76.7%

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

   11. Applied egg-rr 76.7%

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

       1. cbrt-prod 76.6%

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

       2. unpow2 76.6%

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

       3. cbrt-prod 83.3%

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

       4. pow2 83.3%

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

       5. *-commutative 83.3%

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

   13. Applied egg-rr 83.3%

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

   if 3.80000000000000008e108 < a

   1. Initial program 2.6%

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

   3. Taylor expanded in angle around 0 38.9%

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

      1. *-commutative 38.9%

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

      2. unpow2 38.9%

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

      3. unpow2 38.9%

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

      4. swap-sqr 52.2%

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

      5. unpow2 52.2%

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

   5. Simplified 52.2%

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

      1. pow1 52.2%

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

      2. pow-prod-down 77.2%

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

   7. Applied egg-rr 77.2%

      \[\leadsto -4 \cdot \frac{{\left(b \cdot a\right)}^{2}}{\color{blue}{{\left({\left(x-scale \cdot y-scale\right)}^{2}\right)}^{1}}} \]
   8. Step-by-step derivation

      1. unpow1 77.2%

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

   9. Simplified 77.2%

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

       1. add-cube-cbrt 77.1%

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

       2. pow3 77.1%

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

       3. add-cbrt-cube 62.3%

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

       4. unpow2 62.3%

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

       5. unpow2 62.3%

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

       6. add-cbrt-cube 77.1%

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

       7. div-inv 77.1%

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

       8. pow-flip 79.8%

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

       9. metadata-eval 79.8%

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

   11. Applied egg-rr 79.8%

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

       1. metadata-eval 79.8%

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

       2. pow-flip 77.1%

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

       3. div-inv 77.1%

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

       4. pow2 77.1%

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

       5. associate-/r* 91.9%

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

       6. *-commutative 91.9%

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

   13. Applied egg-rr 91.9%

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

Alternative 3: 83.7% accurate, 5.4× speedup

Math

\[\begin{array}{l} \\ -4 \cdot {\left(\sqrt[3]{\frac{\frac{{\left(a \cdot b\right)}^{2}}{x-scale \cdot y-scale}}{x-scale \cdot y-scale}}\right)}^{3} \end{array} \]

FPCore

(FPCore (a b angle x-scale y-scale)
 :precision binary64
 (*
  -4.0
  (pow
   (cbrt (/ (/ (pow (* a b) 2.0) (* x-scale y-scale)) (* x-scale y-scale)))
   3.0)))

C

double code(double a, double b, double angle, double x_45_scale, double y_45_scale) {
	return -4.0 * pow(cbrt(((pow((a * b), 2.0) / (x_45_scale * y_45_scale)) / (x_45_scale * y_45_scale))), 3.0);
}

Java

public static double code(double a, double b, double angle, double x_45_scale, double y_45_scale) {
	return -4.0 * Math.pow(Math.cbrt(((Math.pow((a * b), 2.0) / (x_45_scale * y_45_scale)) / (x_45_scale * y_45_scale))), 3.0);
}

Julia

function code(a, b, angle, x_45_scale, y_45_scale)
	return Float64(-4.0 * (cbrt(Float64(Float64((Float64(a * b) ^ 2.0) / Float64(x_45_scale * y_45_scale)) / Float64(x_45_scale * y_45_scale))) ^ 3.0))
end

Wolfram

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

Derivation

1. Initial program 24.3%

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

3. Taylor expanded in angle around 0 44.1%

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

   1. *-commutative 44.1%

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

   2. unpow2 44.1%

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

   3. unpow2 44.1%

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

   4. swap-sqr 53.4%

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

   5. unpow2 53.4%

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

5. Simplified 53.4%

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

   1. pow1 53.4%

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

   2. pow-prod-down 76.4%

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

7. Applied egg-rr 76.4%

   \[\leadsto -4 \cdot \frac{{\left(b \cdot a\right)}^{2}}{\color{blue}{{\left({\left(x-scale \cdot y-scale\right)}^{2}\right)}^{1}}} \]
8. Step-by-step derivation

   1. unpow1 76.4%

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

9. Simplified 76.4%

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

    1. add-cube-cbrt 76.3%

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

    2. pow3 76.3%

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

    3. add-cbrt-cube 64.5%

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

    4. unpow2 64.5%

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

    5. unpow2 64.5%

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

    6. add-cbrt-cube 76.3%

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

    7. div-inv 76.3%

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

    8. pow-flip 77.2%

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

    9. metadata-eval 77.2%

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

11. Applied egg-rr 77.2%

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

    1. metadata-eval 77.2%

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

    2. pow-flip 76.3%

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

    3. div-inv 76.3%

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

    4. pow2 76.3%

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

    5. associate-/r* 81.9%

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

    6. *-commutative 81.9%

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

13. Applied egg-rr 81.9%

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

Alternative 4: 78.2% accurate, 14.9× speedup

Math

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

FPCore

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

C

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

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 = ((a * b) * (a * b)) * ((-4.0d0) * ((x_45scale * y_45scale) ** (-2.0d0)))
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 24.3%

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

3. Taylor expanded in angle around 0 44.1%

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

   1. *-commutative 44.1%

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

   2. unpow2 44.1%

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

   3. unpow2 44.1%

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

   4. swap-sqr 53.4%

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

   5. unpow2 53.4%

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

5. Simplified 53.4%

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

   1. associate-*r/ 53.4%

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

   2. pow-prod-down 76.4%

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

7. Applied egg-rr 76.4%

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

   1. associate-*r/ 76.4%

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

   2. add-cbrt-cube 64.6%

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

   3. unpow2 64.6%

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

   4. unpow2 64.6%

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

   5. add-cbrt-cube 76.4%

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

   6. div-inv 76.4%

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

   7. pow-flip 77.4%

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

   8. metadata-eval 77.4%

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

9. Applied egg-rr 77.4%

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

    1. *-commutative 77.4%

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

    2. associate-*l* 77.4%

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

    3. *-commutative 77.4%

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

11. Simplified 77.4%

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

    1. unpow2 77.4%

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

13. Applied egg-rr 77.4%

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

14. Final simplification 77.4%

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

Alternative 5: 78.2% accurate, 99.6× speedup

Math

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

FPCore

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

C

double code(double a, double b, double angle, double x_45_scale, double y_45_scale) {
	return -4.0 * (((a * b) * (a * b)) / ((x_45_scale * y_45_scale) * (x_45_scale * y_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) * (((a * b) * (a * b)) / ((x_45scale * y_45scale) * (x_45scale * y_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 * (((a * b) * (a * b)) / ((x_45_scale * y_45_scale) * (x_45_scale * y_45_scale)));
}

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 24.3%

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

3. Taylor expanded in angle around 0 44.1%

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

   1. *-commutative 44.1%

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

   2. unpow2 44.1%

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

   3. unpow2 44.1%

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

   4. swap-sqr 53.4%

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

   5. unpow2 53.4%

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

5. Simplified 53.4%

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

   1. pow1 53.4%

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

   2. pow-prod-down 76.4%

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

7. Applied egg-rr 76.4%

   \[\leadsto -4 \cdot \frac{{\left(b \cdot a\right)}^{2}}{\color{blue}{{\left({\left(x-scale \cdot y-scale\right)}^{2}\right)}^{1}}} \]
8. Step-by-step derivation

   1. unpow1 76.4%

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

9. Simplified 76.4%

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

    1. unpow2 76.4%

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

11. Applied egg-rr 76.4%

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

    1. unpow2 76.4%

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

13. Applied egg-rr 76.4%

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

14. Final simplification 76.4%

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

Alternative 6: 35.3% accurate, 1693.0× 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 24.3%

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

3. Simplified 20.5%

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

5. Taylor expanded in a around 0 22.1%

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

   1. distribute-rgt-out 22.1%

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

   2. metadata-eval 22.1%

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

   3. mul0-rgt 34.6%

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

7. Simplified 34.6%

   \[\leadsto \color{blue}{0} \]
8. Add Preprocessing

Reproduce

herbie shell --seed 2024131 
(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))))