Simplification of discriminant from scale-rotated-ellipse

Percentage Accurate: 24.8% → 94.6%

Time: 1.6min

Alternatives: 6

Speedup: 2485.0×

• 33.6% 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: 24.8% 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: 94.6% accurate, 4.1× speedup

Math

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

FPCore

(FPCore (a b angle x-scale y-scale)
 :precision binary64
 (*
  -4.0
  (pow
   (/ (pow (cbrt (* b a)) 2.0) (pow (* (cbrt y-scale) (cbrt x-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((b * a)), 2.0) / pow((cbrt(y_45_scale) * cbrt(x_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((b * a)), 2.0) / Math.pow((Math.cbrt(y_45_scale) * Math.cbrt(x_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(b * a)) ^ 2.0) / (Float64(cbrt(y_45_scale) * cbrt(x_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[(b * a), $MachinePrecision], 1/3], $MachinePrecision], 2.0], $MachinePrecision] / N[Power[N[(N[Power[y$45$scale, 1/3], $MachinePrecision] * N[Power[x$45$scale, 1/3], $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision], 3.0], $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 28.0%

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

3. Simplified 25.0%

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

5. Taylor expanded in angle around 0 49.6%

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

   1. *-commutative 49.6%

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

   2. unpow2 49.6%

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

   3. unpow2 49.6%

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

   4. swap-sqr 59.1%

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

   5. unpow2 59.1%

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

7. Simplified 59.1%

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

   1. *-commutative 59.1%

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

   2. unpow-prod-down 49.6%

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

   3. frac-times 49.4%

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

   4. add-cube-cbrt 49.4%

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

   5. pow3 49.4%

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

   6. frac-times 49.6%

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

   7. *-commutative 49.6%

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

   8. unpow-prod-down 59.0%

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

   9. div-inv 59.0%

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

   10. *-commutative 59.0%

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

   11. pow-prod-down 76.5%

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

   12. pow-flip 77.3%

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

   13. *-commutative 77.3%

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

   14. metadata-eval 77.3%

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

9. Applied egg-rr 77.3%

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

    1. *-commutative 77.3%

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

    2. metadata-eval 77.3%

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

    3. pow-flip 76.5%

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

    4. div-inv 76.9%

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

    5. cbrt-div 76.8%

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

    6. *-commutative 76.8%

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

    7. unpow2 76.8%

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

    8. cbrt-prod 81.1%

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

    9. pow2 81.1%

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

    10. unpow2 81.1%

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

    11. cbrt-prod 94.1%

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

    12. pow2 94.1%

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

11. Applied egg-rr 94.1%

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

    1. *-commutative 94.1%

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

    2. cbrt-prod 96.6%

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

13. Applied egg-rr 96.6%

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

14. Final simplification 96.6%

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

Alternative 2: 93.0% accurate, 4.9× speedup

Math

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

FPCore

(FPCore (a b angle x-scale y-scale)
 :precision binary64
 (*
  -4.0
  (pow (/ (pow (cbrt (* b a)) 2.0) (pow (cbrt (* y-scale x-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((b * a)), 2.0) / pow(cbrt((y_45_scale * x_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((b * a)), 2.0) / Math.pow(Math.cbrt((y_45_scale * x_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(b * a)) ^ 2.0) / (cbrt(Float64(y_45_scale * x_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[(b * a), $MachinePrecision], 1/3], $MachinePrecision], 2.0], $MachinePrecision] / N[Power[N[Power[N[(y$45$scale * x$45$scale), $MachinePrecision], 1/3], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision], 3.0], $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 28.0%

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

3. Simplified 25.0%

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

5. Taylor expanded in angle around 0 49.6%

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

   1. *-commutative 49.6%

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

   2. unpow2 49.6%

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

   3. unpow2 49.6%

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

   4. swap-sqr 59.1%

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

   5. unpow2 59.1%

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

7. Simplified 59.1%

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

   1. *-commutative 59.1%

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

   2. unpow-prod-down 49.6%

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

   3. frac-times 49.4%

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

   4. add-cube-cbrt 49.4%

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

   5. pow3 49.4%

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

   6. frac-times 49.6%

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

   7. *-commutative 49.6%

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

   8. unpow-prod-down 59.0%

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

   9. div-inv 59.0%

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

   10. *-commutative 59.0%

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

   11. pow-prod-down 76.5%

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

   12. pow-flip 77.3%

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

   13. *-commutative 77.3%

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

   14. metadata-eval 77.3%

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

9. Applied egg-rr 77.3%

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

    1. *-commutative 77.3%

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

    2. metadata-eval 77.3%

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

    3. pow-flip 76.5%

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

    4. div-inv 76.9%

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

    5. cbrt-div 76.8%

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

    6. *-commutative 76.8%

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

    7. unpow2 76.8%

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

    8. cbrt-prod 81.1%

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

    9. pow2 81.1%

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

    10. unpow2 81.1%

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

    11. cbrt-prod 94.1%

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

    12. pow2 94.1%

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

11. Applied egg-rr 94.1%

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

12. Final simplification 94.1%

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

Alternative 3: 93.8% accurate, 22.6× speedup

Math

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

FPCore

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

C

double code(double a, double b, double angle, double x_45_scale, double y_45_scale) {
	return -4.0 * pow((a * ((b / 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 = (-4.0d0) * ((a * ((b / 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 -4.0 * Math.pow((a * ((b / x_45_scale) / y_45_scale)), 2.0);
}

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 28.0%

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

3. Simplified 26.7%

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

6. Applied egg-rr 27.3%

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

7. Taylor expanded in angle around 0 49.6%

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

   1. *-commutative 49.6%

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

   2. unpow2 49.6%

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

   3. unpow2 49.6%

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

   4. swap-sqr 62.8%

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

   5. unpow2 62.8%

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

   6. unpow2 62.8%

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

   7. unpow2 62.8%

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

   8. swap-sqr 77.0%

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

   9. unpow2 77.0%

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

9. Simplified 77.0%

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

    1. *-commutative 77.0%

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

    2. unpow-prod-down 59.1%

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

    3. associate-*l/ 58.5%

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

    4. *-commutative 58.5%

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

    5. add-sqr-sqrt 58.5%

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

    6. sqrt-prod 58.5%

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

    7. unpow2 58.5%

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

    8. sqrt-prod 31.4%

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

    9. add-sqr-sqrt 44.8%

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

    10. sqrt-div 44.8%

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

    11. unpow2 44.8%

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

    12. sqrt-prod 22.8%

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

    13. add-sqr-sqrt 39.5%

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

    14. unpow2 39.5%

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

    15. sqrt-prod 20.9%

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

    16. add-sqr-sqrt 45.2%

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

    17. sqrt-prod 45.2%

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

11. Applied egg-rr 93.5%

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

12. Taylor expanded in a around 0 49.6%

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

14. Simplified 93.9%

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

15. Final simplification 93.9%

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

Alternative 4: 93.7% accurate, 146.2× speedup

Math

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

FPCore

(FPCore (a b angle x-scale y-scale)
 :precision binary64
 (let* ((t_0 (* a (/ b (* y-scale x-scale))))) (* -4.0 (* t_0 t_0))))

C

double code(double a, double b, double angle, double x_45_scale, double y_45_scale) {
	double t_0 = a * (b / (y_45_scale * x_45_scale));
	return -4.0 * (t_0 * t_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
    real(8) :: t_0
    t_0 = a * (b / (y_45scale * x_45scale))
    code = (-4.0d0) * (t_0 * t_0)
end function

Java

public static double code(double a, double b, double angle, double x_45_scale, double y_45_scale) {
	double t_0 = a * (b / (y_45_scale * x_45_scale));
	return -4.0 * (t_0 * t_0);
}

Python

def code(a, b, angle, x_45_scale, y_45_scale):
	t_0 = a * (b / (y_45_scale * x_45_scale))
	return -4.0 * (t_0 * t_0)

Julia

function code(a, b, angle, x_45_scale, y_45_scale)
	t_0 = Float64(a * Float64(b / Float64(y_45_scale * x_45_scale)))
	return Float64(-4.0 * Float64(t_0 * t_0))
end

MATLAB

function tmp = code(a, b, angle, x_45_scale, y_45_scale)
	t_0 = a * (b / (y_45_scale * x_45_scale));
	tmp = -4.0 * (t_0 * t_0);
end

Wolfram

code[a_, b_, angle_, x$45$scale_, y$45$scale_] := Block[{t$95$0 = N[(a * N[(b / N[(y$45$scale * x$45$scale), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, N[(-4.0 * N[(t$95$0 * t$95$0), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Initial program 28.0%

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

3. Simplified 26.7%

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

6. Applied egg-rr 27.3%

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

7. Taylor expanded in angle around 0 49.6%

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

   1. *-commutative 49.6%

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

   2. unpow2 49.6%

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

   3. unpow2 49.6%

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

   4. swap-sqr 62.8%

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

   5. unpow2 62.8%

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

   6. unpow2 62.8%

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

   7. unpow2 62.8%

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

   8. swap-sqr 77.0%

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

   9. unpow2 77.0%

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

9. Simplified 77.0%

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

    1. *-commutative 77.0%

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

    2. unpow-prod-down 59.1%

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

    3. associate-*l/ 58.5%

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

    4. *-commutative 58.5%

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

    5. add-sqr-sqrt 58.5%

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

    6. sqrt-prod 58.5%

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

    7. unpow2 58.5%

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

    8. sqrt-prod 31.4%

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

    9. add-sqr-sqrt 44.8%

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

    10. sqrt-div 44.8%

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

    11. unpow2 44.8%

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

    12. sqrt-prod 22.8%

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

    13. add-sqr-sqrt 39.5%

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

    14. unpow2 39.5%

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

    15. sqrt-prod 20.9%

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

    16. add-sqr-sqrt 45.2%

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

    17. sqrt-prod 45.2%

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

11. Applied egg-rr 93.5%

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

12. Final simplification 93.5%

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

Alternative 5: 93.6% accurate, 146.2× speedup

Math

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

FPCore

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

C

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 28.0%

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

3. Simplified 26.7%

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

6. Applied egg-rr 27.3%

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

7. Taylor expanded in angle around 0 49.6%

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

   1. *-commutative 49.6%

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

   2. unpow2 49.6%

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

   3. unpow2 49.6%

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

   4. swap-sqr 62.8%

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

   5. unpow2 62.8%

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

   6. unpow2 62.8%

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

   7. unpow2 62.8%

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

   8. swap-sqr 77.0%

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

   9. unpow2 77.0%

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

9. Simplified 77.0%

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

    1. *-commutative 77.0%

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

    2. unpow-prod-down 59.1%

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

    3. associate-*l/ 58.5%

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

    4. *-commutative 58.5%

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

    5. add-sqr-sqrt 58.5%

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

    6. sqrt-prod 58.5%

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

    7. unpow2 58.5%

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

    8. sqrt-prod 31.4%

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

    9. add-sqr-sqrt 44.8%

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

    10. sqrt-div 44.8%

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

    11. unpow2 44.8%

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

    12. sqrt-prod 22.8%

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

    13. add-sqr-sqrt 39.5%

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

    14. unpow2 39.5%

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

    15. sqrt-prod 20.9%

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

    16. add-sqr-sqrt 45.2%

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

    17. sqrt-prod 45.2%

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

11. Applied egg-rr 93.5%

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

    1. clear-num 93.5%

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

    2. un-div-inv 93.5%

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

13. Applied egg-rr 93.5%

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

14. Final simplification 93.5%

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

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 28.0%

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

3. Simplified 25.9%

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

5. Taylor expanded in b around 0 23.9%

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

   1. distribute-rgt-out 23.9%

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

   2. metadata-eval 23.9%

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

   3. mul0-rgt 36.0%

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

7. Simplified 36.0%

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

8. Final simplification 36.0%

   \[\leadsto 0 \]
9. Add Preprocessing

Reproduce

herbie shell --seed 2024010 
(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))))