ab-angle->ABCF C

Percentage Accurate: 79.5% → 79.4%

Time: 37.3s

Alternatives: 14

Speedup: 1.3×

• 36.5% of points produce a very large (infinite) output. You may want to add a precondition.

Specification

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \pi \cdot \frac{angle}{180}\\ {\left(a \cdot \cos t\_0\right)}^{2} + {\left(b \cdot \sin t\_0\right)}^{2} \end{array} \end{array} \]

FPCore

(FPCore (a b angle)
 :precision binary64
 (let* ((t_0 (* PI (/ angle 180.0))))
   (+ (pow (* a (cos t_0)) 2.0) (pow (* b (sin t_0)) 2.0))))

C

double code(double a, double b, double angle) {
	double t_0 = ((double) M_PI) * (angle / 180.0);
	return pow((a * cos(t_0)), 2.0) + pow((b * sin(t_0)), 2.0);
}

Java

public static double code(double a, double b, double angle) {
	double t_0 = Math.PI * (angle / 180.0);
	return Math.pow((a * Math.cos(t_0)), 2.0) + Math.pow((b * Math.sin(t_0)), 2.0);
}

Python

def code(a, b, angle):
	t_0 = math.pi * (angle / 180.0)
	return math.pow((a * math.cos(t_0)), 2.0) + math.pow((b * math.sin(t_0)), 2.0)

Julia

function code(a, b, angle)
	t_0 = Float64(pi * Float64(angle / 180.0))
	return Float64((Float64(a * cos(t_0)) ^ 2.0) + (Float64(b * sin(t_0)) ^ 2.0))
end

MATLAB

function tmp = code(a, b, angle)
	t_0 = pi * (angle / 180.0);
	tmp = ((a * cos(t_0)) ^ 2.0) + ((b * sin(t_0)) ^ 2.0);
end

Wolfram

code[a_, b_, angle_] := Block[{t$95$0 = N[(Pi * N[(angle / 180.0), $MachinePrecision]), $MachinePrecision]}, N[(N[Power[N[(a * N[Cos[t$95$0], $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision] + N[Power[N[(b * N[Sin[t$95$0], $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]]

Initial Program: 79.5% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \pi \cdot \frac{angle}{180}\\ {\left(a \cdot \cos t\_0\right)}^{2} + {\left(b \cdot \sin t\_0\right)}^{2} \end{array} \end{array} \]

FPCore

(FPCore (a b angle)
 :precision binary64
 (let* ((t_0 (* PI (/ angle 180.0))))
   (+ (pow (* a (cos t_0)) 2.0) (pow (* b (sin t_0)) 2.0))))

C

double code(double a, double b, double angle) {
	double t_0 = ((double) M_PI) * (angle / 180.0);
	return pow((a * cos(t_0)), 2.0) + pow((b * sin(t_0)), 2.0);
}

Java

public static double code(double a, double b, double angle) {
	double t_0 = Math.PI * (angle / 180.0);
	return Math.pow((a * Math.cos(t_0)), 2.0) + Math.pow((b * Math.sin(t_0)), 2.0);
}

Python

def code(a, b, angle):
	t_0 = math.pi * (angle / 180.0)
	return math.pow((a * math.cos(t_0)), 2.0) + math.pow((b * math.sin(t_0)), 2.0)

Julia

function code(a, b, angle)
	t_0 = Float64(pi * Float64(angle / 180.0))
	return Float64((Float64(a * cos(t_0)) ^ 2.0) + (Float64(b * sin(t_0)) ^ 2.0))
end

MATLAB

function tmp = code(a, b, angle)
	t_0 = pi * (angle / 180.0);
	tmp = ((a * cos(t_0)) ^ 2.0) + ((b * sin(t_0)) ^ 2.0);
end

Wolfram

code[a_, b_, angle_] := Block[{t$95$0 = N[(Pi * N[(angle / 180.0), $MachinePrecision]), $MachinePrecision]}, N[(N[Power[N[(a * N[Cos[t$95$0], $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision] + N[Power[N[(b * N[Sin[t$95$0], $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]]

Alternative 1: 79.4% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \sqrt[3]{\sqrt{\pi}}\\ {\left(a \cdot \cos \left(\frac{t\_0 \cdot t\_0}{{\left({\left(\sqrt[3]{\sqrt[3]{\frac{\frac{-180}{\sqrt[3]{\pi}}}{angle}}}\right)}^{3}\right)}^{2}} \cdot \frac{\sqrt[3]{\pi}}{\sqrt[3]{\frac{-180}{\sqrt[3]{\pi} \cdot angle}}}\right)\right)}^{2} + {\left(b \cdot \sin \left(\pi \cdot \frac{angle}{180}\right)\right)}^{2} \end{array} \end{array} \]

FPCore

(FPCore (a b angle)
 :precision binary64
 (let* ((t_0 (cbrt (sqrt PI))))
   (+
    (pow
     (*
      a
      (cos
       (*
        (/
         (* t_0 t_0)
         (pow (pow (cbrt (cbrt (/ (/ -180.0 (cbrt PI)) angle))) 3.0) 2.0))
        (/ (cbrt PI) (cbrt (/ -180.0 (* (cbrt PI) angle)))))))
     2.0)
    (pow (* b (sin (* PI (/ angle 180.0)))) 2.0))))

C

double code(double a, double b, double angle) {
	double t_0 = cbrt(sqrt(((double) M_PI)));
	return pow((a * cos((((t_0 * t_0) / pow(pow(cbrt(cbrt(((-180.0 / cbrt(((double) M_PI))) / angle))), 3.0), 2.0)) * (cbrt(((double) M_PI)) / cbrt((-180.0 / (cbrt(((double) M_PI)) * angle))))))), 2.0) + pow((b * sin((((double) M_PI) * (angle / 180.0)))), 2.0);
}

Java

public static double code(double a, double b, double angle) {
	double t_0 = Math.cbrt(Math.sqrt(Math.PI));
	return Math.pow((a * Math.cos((((t_0 * t_0) / Math.pow(Math.pow(Math.cbrt(Math.cbrt(((-180.0 / Math.cbrt(Math.PI)) / angle))), 3.0), 2.0)) * (Math.cbrt(Math.PI) / Math.cbrt((-180.0 / (Math.cbrt(Math.PI) * angle))))))), 2.0) + Math.pow((b * Math.sin((Math.PI * (angle / 180.0)))), 2.0);
}

Julia

function code(a, b, angle)
	t_0 = cbrt(sqrt(pi))
	return Float64((Float64(a * cos(Float64(Float64(Float64(t_0 * t_0) / ((cbrt(cbrt(Float64(Float64(-180.0 / cbrt(pi)) / angle))) ^ 3.0) ^ 2.0)) * Float64(cbrt(pi) / cbrt(Float64(-180.0 / Float64(cbrt(pi) * angle))))))) ^ 2.0) + (Float64(b * sin(Float64(pi * Float64(angle / 180.0)))) ^ 2.0))
end

Wolfram

code[a_, b_, angle_] := Block[{t$95$0 = N[Power[N[Sqrt[Pi], $MachinePrecision], 1/3], $MachinePrecision]}, N[(N[Power[N[(a * N[Cos[N[(N[(N[(t$95$0 * t$95$0), $MachinePrecision] / N[Power[N[Power[N[Power[N[Power[N[(N[(-180.0 / N[Power[Pi, 1/3], $MachinePrecision]), $MachinePrecision] / angle), $MachinePrecision], 1/3], $MachinePrecision], 1/3], $MachinePrecision], 3.0], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] * N[(N[Power[Pi, 1/3], $MachinePrecision] / N[Power[N[(-180.0 / N[(N[Power[Pi, 1/3], $MachinePrecision] * angle), $MachinePrecision]), $MachinePrecision], 1/3], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision] + N[Power[N[(b * N[Sin[N[(Pi * N[(angle / 180.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Initial program 78.0%

   \[{\left(a \cdot \cos \left(\pi \cdot \frac{angle}{180}\right)\right)}^{2} + {\left(b \cdot \sin \left(\pi \cdot \frac{angle}{180}\right)\right)}^{2} \]
2. Add Preprocessing
3. Step-by-step derivation

   1. add-sqr-sqrt 36.8%

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

   2. sqrt-unprod 66.7%

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

   3. associate-*r/ 66.7%

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

   4. associate-*r/ 66.7%

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

   5. frac-times 66.8%

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

   6. *-commutative 66.8%

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

   7. *-commutative 66.8%

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

   8. metadata-eval 66.8%

      \[\leadsto {\left(a \cdot \cos \left(\sqrt{\frac{\left(angle \cdot \pi\right) \cdot \left(angle \cdot \pi\right)}{\color{blue}{32400}}}\right)\right)}^{2} + {\left(b \cdot \sin \left(\pi \cdot \frac{angle}{180}\right)\right)}^{2} \]

   9. metadata-eval 66.8%

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

   10. frac-times 66.7%

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

   11. associate-*r/ 66.7%

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

   12. associate-*r/ 66.7%

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

   13. sqrt-unprod 41.2%

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

   14. add-sqr-sqrt 78.0%

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

   15. *-commutative 78.0%

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

   16. associate-/r/ 77.9%

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

   17. add-cube-cbrt 78.0%

       \[\leadsto {\left(a \cdot \cos \left(\frac{\color{blue}{\left(\sqrt[3]{\pi} \cdot \sqrt[3]{\pi}\right) \cdot \sqrt[3]{\pi}}}{\frac{-180}{angle}}\right)\right)}^{2} + {\left(b \cdot \sin \left(\pi \cdot \frac{angle}{180}\right)\right)}^{2} \]

   18. associate-/l* 78.0%

       \[\leadsto {\left(a \cdot \cos \color{blue}{\left(\frac{\sqrt[3]{\pi} \cdot \sqrt[3]{\pi}}{\frac{\frac{-180}{angle}}{\sqrt[3]{\pi}}}\right)}\right)}^{2} + {\left(b \cdot \sin \left(\pi \cdot \frac{angle}{180}\right)\right)}^{2} \]

4. Applied egg-rr 78.0%

   \[\leadsto {\left(a \cdot \cos \color{blue}{\left(\frac{{\left(\sqrt[3]{\pi}\right)}^{2}}{\frac{\frac{-180}{angle}}{\sqrt[3]{\pi}}}\right)}\right)}^{2} + {\left(b \cdot \sin \left(\pi \cdot \frac{angle}{180}\right)\right)}^{2} \]
5. Step-by-step derivation

   1. unpow2 78.0%

      \[\leadsto {\left(a \cdot \cos \left(\frac{\color{blue}{\sqrt[3]{\pi} \cdot \sqrt[3]{\pi}}}{\frac{\frac{-180}{angle}}{\sqrt[3]{\pi}}}\right)\right)}^{2} + {\left(b \cdot \sin \left(\pi \cdot \frac{angle}{180}\right)\right)}^{2} \]

   2. add-cube-cbrt 78.0%

      \[\leadsto {\left(a \cdot \cos \left(\frac{\sqrt[3]{\pi} \cdot \sqrt[3]{\pi}}{\color{blue}{\left(\sqrt[3]{\frac{\frac{-180}{angle}}{\sqrt[3]{\pi}}} \cdot \sqrt[3]{\frac{\frac{-180}{angle}}{\sqrt[3]{\pi}}}\right) \cdot \sqrt[3]{\frac{\frac{-180}{angle}}{\sqrt[3]{\pi}}}}}\right)\right)}^{2} + {\left(b \cdot \sin \left(\pi \cdot \frac{angle}{180}\right)\right)}^{2} \]

   3. times-frac 78.1%

      \[\leadsto {\left(a \cdot \cos \color{blue}{\left(\frac{\sqrt[3]{\pi}}{\sqrt[3]{\frac{\frac{-180}{angle}}{\sqrt[3]{\pi}}} \cdot \sqrt[3]{\frac{\frac{-180}{angle}}{\sqrt[3]{\pi}}}} \cdot \frac{\sqrt[3]{\pi}}{\sqrt[3]{\frac{\frac{-180}{angle}}{\sqrt[3]{\pi}}}}\right)}\right)}^{2} + {\left(b \cdot \sin \left(\pi \cdot \frac{angle}{180}\right)\right)}^{2} \]

   4. pow2 78.1%

      \[\leadsto {\left(a \cdot \cos \left(\frac{\sqrt[3]{\pi}}{\color{blue}{{\left(\sqrt[3]{\frac{\frac{-180}{angle}}{\sqrt[3]{\pi}}}\right)}^{2}}} \cdot \frac{\sqrt[3]{\pi}}{\sqrt[3]{\frac{\frac{-180}{angle}}{\sqrt[3]{\pi}}}}\right)\right)}^{2} + {\left(b \cdot \sin \left(\pi \cdot \frac{angle}{180}\right)\right)}^{2} \]

   5. associate-/l/ 78.1%

      \[\leadsto {\left(a \cdot \cos \left(\frac{\sqrt[3]{\pi}}{{\left(\sqrt[3]{\color{blue}{\frac{-180}{\sqrt[3]{\pi} \cdot angle}}}\right)}^{2}} \cdot \frac{\sqrt[3]{\pi}}{\sqrt[3]{\frac{\frac{-180}{angle}}{\sqrt[3]{\pi}}}}\right)\right)}^{2} + {\left(b \cdot \sin \left(\pi \cdot \frac{angle}{180}\right)\right)}^{2} \]

   6. associate-/l/ 78.1%

      \[\leadsto {\left(a \cdot \cos \left(\frac{\sqrt[3]{\pi}}{{\left(\sqrt[3]{\frac{-180}{\sqrt[3]{\pi} \cdot angle}}\right)}^{2}} \cdot \frac{\sqrt[3]{\pi}}{\sqrt[3]{\color{blue}{\frac{-180}{\sqrt[3]{\pi} \cdot angle}}}}\right)\right)}^{2} + {\left(b \cdot \sin \left(\pi \cdot \frac{angle}{180}\right)\right)}^{2} \]

6. Applied egg-rr 78.1%

   \[\leadsto {\left(a \cdot \cos \color{blue}{\left(\frac{\sqrt[3]{\pi}}{{\left(\sqrt[3]{\frac{-180}{\sqrt[3]{\pi} \cdot angle}}\right)}^{2}} \cdot \frac{\sqrt[3]{\pi}}{\sqrt[3]{\frac{-180}{\sqrt[3]{\pi} \cdot angle}}}\right)}\right)}^{2} + {\left(b \cdot \sin \left(\pi \cdot \frac{angle}{180}\right)\right)}^{2} \]
7. Step-by-step derivation

   1. add-cube-cbrt 78.1%

      \[\leadsto {\left(a \cdot \cos \left(\frac{\sqrt[3]{\pi}}{{\color{blue}{\left(\left(\sqrt[3]{\sqrt[3]{\frac{-180}{\sqrt[3]{\pi} \cdot angle}}} \cdot \sqrt[3]{\sqrt[3]{\frac{-180}{\sqrt[3]{\pi} \cdot angle}}}\right) \cdot \sqrt[3]{\sqrt[3]{\frac{-180}{\sqrt[3]{\pi} \cdot angle}}}\right)}}^{2}} \cdot \frac{\sqrt[3]{\pi}}{\sqrt[3]{\frac{-180}{\sqrt[3]{\pi} \cdot angle}}}\right)\right)}^{2} + {\left(b \cdot \sin \left(\pi \cdot \frac{angle}{180}\right)\right)}^{2} \]

   2. pow3 78.1%

      \[\leadsto {\left(a \cdot \cos \left(\frac{\sqrt[3]{\pi}}{{\color{blue}{\left({\left(\sqrt[3]{\sqrt[3]{\frac{-180}{\sqrt[3]{\pi} \cdot angle}}}\right)}^{3}\right)}}^{2}} \cdot \frac{\sqrt[3]{\pi}}{\sqrt[3]{\frac{-180}{\sqrt[3]{\pi} \cdot angle}}}\right)\right)}^{2} + {\left(b \cdot \sin \left(\pi \cdot \frac{angle}{180}\right)\right)}^{2} \]

   3. associate-/r* 78.1%

      \[\leadsto {\left(a \cdot \cos \left(\frac{\sqrt[3]{\pi}}{{\left({\left(\sqrt[3]{\sqrt[3]{\color{blue}{\frac{\frac{-180}{\sqrt[3]{\pi}}}{angle}}}}\right)}^{3}\right)}^{2}} \cdot \frac{\sqrt[3]{\pi}}{\sqrt[3]{\frac{-180}{\sqrt[3]{\pi} \cdot angle}}}\right)\right)}^{2} + {\left(b \cdot \sin \left(\pi \cdot \frac{angle}{180}\right)\right)}^{2} \]

8. Applied egg-rr 78.1%

   \[\leadsto {\left(a \cdot \cos \left(\frac{\sqrt[3]{\pi}}{{\color{blue}{\left({\left(\sqrt[3]{\sqrt[3]{\frac{\frac{-180}{\sqrt[3]{\pi}}}{angle}}}\right)}^{3}\right)}}^{2}} \cdot \frac{\sqrt[3]{\pi}}{\sqrt[3]{\frac{-180}{\sqrt[3]{\pi} \cdot angle}}}\right)\right)}^{2} + {\left(b \cdot \sin \left(\pi \cdot \frac{angle}{180}\right)\right)}^{2} \]
9. Step-by-step derivation

   1. pow1/3 78.1%

      \[\leadsto {\left(a \cdot \cos \left(\frac{\color{blue}{{\pi}^{0.3333333333333333}}}{{\left({\left(\sqrt[3]{\sqrt[3]{\frac{\frac{-180}{\sqrt[3]{\pi}}}{angle}}}\right)}^{3}\right)}^{2}} \cdot \frac{\sqrt[3]{\pi}}{\sqrt[3]{\frac{-180}{\sqrt[3]{\pi} \cdot angle}}}\right)\right)}^{2} + {\left(b \cdot \sin \left(\pi \cdot \frac{angle}{180}\right)\right)}^{2} \]

   2. add-sqr-sqrt 78.1%

      \[\leadsto {\left(a \cdot \cos \left(\frac{{\color{blue}{\left(\sqrt{\pi} \cdot \sqrt{\pi}\right)}}^{0.3333333333333333}}{{\left({\left(\sqrt[3]{\sqrt[3]{\frac{\frac{-180}{\sqrt[3]{\pi}}}{angle}}}\right)}^{3}\right)}^{2}} \cdot \frac{\sqrt[3]{\pi}}{\sqrt[3]{\frac{-180}{\sqrt[3]{\pi} \cdot angle}}}\right)\right)}^{2} + {\left(b \cdot \sin \left(\pi \cdot \frac{angle}{180}\right)\right)}^{2} \]

   3. unpow-prod-down 78.2%

      \[\leadsto {\left(a \cdot \cos \left(\frac{\color{blue}{{\left(\sqrt{\pi}\right)}^{0.3333333333333333} \cdot {\left(\sqrt{\pi}\right)}^{0.3333333333333333}}}{{\left({\left(\sqrt[3]{\sqrt[3]{\frac{\frac{-180}{\sqrt[3]{\pi}}}{angle}}}\right)}^{3}\right)}^{2}} \cdot \frac{\sqrt[3]{\pi}}{\sqrt[3]{\frac{-180}{\sqrt[3]{\pi} \cdot angle}}}\right)\right)}^{2} + {\left(b \cdot \sin \left(\pi \cdot \frac{angle}{180}\right)\right)}^{2} \]

10. Applied egg-rr 78.2%

    \[\leadsto {\left(a \cdot \cos \left(\frac{\color{blue}{{\left(\sqrt{\pi}\right)}^{0.3333333333333333} \cdot {\left(\sqrt{\pi}\right)}^{0.3333333333333333}}}{{\left({\left(\sqrt[3]{\sqrt[3]{\frac{\frac{-180}{\sqrt[3]{\pi}}}{angle}}}\right)}^{3}\right)}^{2}} \cdot \frac{\sqrt[3]{\pi}}{\sqrt[3]{\frac{-180}{\sqrt[3]{\pi} \cdot angle}}}\right)\right)}^{2} + {\left(b \cdot \sin \left(\pi \cdot \frac{angle}{180}\right)\right)}^{2} \]
11. Step-by-step derivation

    1. unpow1/3 78.1%

       \[\leadsto {\left(a \cdot \cos \left(\frac{\color{blue}{\sqrt[3]{\sqrt{\pi}}} \cdot {\left(\sqrt{\pi}\right)}^{0.3333333333333333}}{{\left({\left(\sqrt[3]{\sqrt[3]{\frac{\frac{-180}{\sqrt[3]{\pi}}}{angle}}}\right)}^{3}\right)}^{2}} \cdot \frac{\sqrt[3]{\pi}}{\sqrt[3]{\frac{-180}{\sqrt[3]{\pi} \cdot angle}}}\right)\right)}^{2} + {\left(b \cdot \sin \left(\pi \cdot \frac{angle}{180}\right)\right)}^{2} \]

    2. unpow1/3 78.1%

       \[\leadsto {\left(a \cdot \cos \left(\frac{\sqrt[3]{\sqrt{\pi}} \cdot \color{blue}{\sqrt[3]{\sqrt{\pi}}}}{{\left({\left(\sqrt[3]{\sqrt[3]{\frac{\frac{-180}{\sqrt[3]{\pi}}}{angle}}}\right)}^{3}\right)}^{2}} \cdot \frac{\sqrt[3]{\pi}}{\sqrt[3]{\frac{-180}{\sqrt[3]{\pi} \cdot angle}}}\right)\right)}^{2} + {\left(b \cdot \sin \left(\pi \cdot \frac{angle}{180}\right)\right)}^{2} \]

12. Simplified 78.1%

    \[\leadsto {\left(a \cdot \cos \left(\frac{\color{blue}{\sqrt[3]{\sqrt{\pi}} \cdot \sqrt[3]{\sqrt{\pi}}}}{{\left({\left(\sqrt[3]{\sqrt[3]{\frac{\frac{-180}{\sqrt[3]{\pi}}}{angle}}}\right)}^{3}\right)}^{2}} \cdot \frac{\sqrt[3]{\pi}}{\sqrt[3]{\frac{-180}{\sqrt[3]{\pi} \cdot angle}}}\right)\right)}^{2} + {\left(b \cdot \sin \left(\pi \cdot \frac{angle}{180}\right)\right)}^{2} \]

13. Final simplification 78.1%

    \[\leadsto {\left(a \cdot \cos \left(\frac{\sqrt[3]{\sqrt{\pi}} \cdot \sqrt[3]{\sqrt{\pi}}}{{\left({\left(\sqrt[3]{\sqrt[3]{\frac{\frac{-180}{\sqrt[3]{\pi}}}{angle}}}\right)}^{3}\right)}^{2}} \cdot \frac{\sqrt[3]{\pi}}{\sqrt[3]{\frac{-180}{\sqrt[3]{\pi} \cdot angle}}}\right)\right)}^{2} + {\left(b \cdot \sin \left(\pi \cdot \frac{angle}{180}\right)\right)}^{2} \]
14. Add Preprocessing

Alternative 2: 79.4% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ {\left(b \cdot \sin \left(\pi \cdot \frac{angle}{180}\right)\right)}^{2} + {\left(a \cdot \cos \left(\frac{\sqrt[3]{\pi}}{\sqrt[3]{\frac{-180}{\sqrt[3]{\pi} \cdot angle}}} \cdot \frac{\sqrt[3]{\pi}}{{\left({\left(\sqrt[3]{\sqrt[3]{\frac{\frac{-180}{\sqrt[3]{\pi}}}{angle}}}\right)}^{3}\right)}^{2}}\right)\right)}^{2} \end{array} \]

FPCore

(FPCore (a b angle)
 :precision binary64
 (+
  (pow (* b (sin (* PI (/ angle 180.0)))) 2.0)
  (pow
   (*
    a
    (cos
     (*
      (/ (cbrt PI) (cbrt (/ -180.0 (* (cbrt PI) angle))))
      (/
       (cbrt PI)
       (pow (pow (cbrt (cbrt (/ (/ -180.0 (cbrt PI)) angle))) 3.0) 2.0)))))
   2.0)))

C

double code(double a, double b, double angle) {
	return pow((b * sin((((double) M_PI) * (angle / 180.0)))), 2.0) + pow((a * cos(((cbrt(((double) M_PI)) / cbrt((-180.0 / (cbrt(((double) M_PI)) * angle)))) * (cbrt(((double) M_PI)) / pow(pow(cbrt(cbrt(((-180.0 / cbrt(((double) M_PI))) / angle))), 3.0), 2.0))))), 2.0);
}

Java

public static double code(double a, double b, double angle) {
	return Math.pow((b * Math.sin((Math.PI * (angle / 180.0)))), 2.0) + Math.pow((a * Math.cos(((Math.cbrt(Math.PI) / Math.cbrt((-180.0 / (Math.cbrt(Math.PI) * angle)))) * (Math.cbrt(Math.PI) / Math.pow(Math.pow(Math.cbrt(Math.cbrt(((-180.0 / Math.cbrt(Math.PI)) / angle))), 3.0), 2.0))))), 2.0);
}

Julia

function code(a, b, angle)
	return Float64((Float64(b * sin(Float64(pi * Float64(angle / 180.0)))) ^ 2.0) + (Float64(a * cos(Float64(Float64(cbrt(pi) / cbrt(Float64(-180.0 / Float64(cbrt(pi) * angle)))) * Float64(cbrt(pi) / ((cbrt(cbrt(Float64(Float64(-180.0 / cbrt(pi)) / angle))) ^ 3.0) ^ 2.0))))) ^ 2.0))
end

Wolfram

code[a_, b_, angle_] := N[(N[Power[N[(b * N[Sin[N[(Pi * N[(angle / 180.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision] + N[Power[N[(a * N[Cos[N[(N[(N[Power[Pi, 1/3], $MachinePrecision] / N[Power[N[(-180.0 / N[(N[Power[Pi, 1/3], $MachinePrecision] * angle), $MachinePrecision]), $MachinePrecision], 1/3], $MachinePrecision]), $MachinePrecision] * N[(N[Power[Pi, 1/3], $MachinePrecision] / N[Power[N[Power[N[Power[N[Power[N[(N[(-180.0 / N[Power[Pi, 1/3], $MachinePrecision]), $MachinePrecision] / angle), $MachinePrecision], 1/3], $MachinePrecision], 1/3], $MachinePrecision], 3.0], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 78.0%

   \[{\left(a \cdot \cos \left(\pi \cdot \frac{angle}{180}\right)\right)}^{2} + {\left(b \cdot \sin \left(\pi \cdot \frac{angle}{180}\right)\right)}^{2} \]
2. Add Preprocessing
3. Step-by-step derivation

   1. add-sqr-sqrt 36.8%

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

   2. sqrt-unprod 66.7%

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

   3. associate-*r/ 66.7%

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

   4. associate-*r/ 66.7%

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

   5. frac-times 66.8%

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

   6. *-commutative 66.8%

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

   7. *-commutative 66.8%

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

   8. metadata-eval 66.8%

      \[\leadsto {\left(a \cdot \cos \left(\sqrt{\frac{\left(angle \cdot \pi\right) \cdot \left(angle \cdot \pi\right)}{\color{blue}{32400}}}\right)\right)}^{2} + {\left(b \cdot \sin \left(\pi \cdot \frac{angle}{180}\right)\right)}^{2} \]

   9. metadata-eval 66.8%

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

   10. frac-times 66.7%

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

   11. associate-*r/ 66.7%

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

   12. associate-*r/ 66.7%

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

   13. sqrt-unprod 41.2%

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

   14. add-sqr-sqrt 78.0%

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

   15. *-commutative 78.0%

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

   16. associate-/r/ 77.9%

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

   17. add-cube-cbrt 78.0%

       \[\leadsto {\left(a \cdot \cos \left(\frac{\color{blue}{\left(\sqrt[3]{\pi} \cdot \sqrt[3]{\pi}\right) \cdot \sqrt[3]{\pi}}}{\frac{-180}{angle}}\right)\right)}^{2} + {\left(b \cdot \sin \left(\pi \cdot \frac{angle}{180}\right)\right)}^{2} \]

   18. associate-/l* 78.0%

       \[\leadsto {\left(a \cdot \cos \color{blue}{\left(\frac{\sqrt[3]{\pi} \cdot \sqrt[3]{\pi}}{\frac{\frac{-180}{angle}}{\sqrt[3]{\pi}}}\right)}\right)}^{2} + {\left(b \cdot \sin \left(\pi \cdot \frac{angle}{180}\right)\right)}^{2} \]

4. Applied egg-rr 78.0%

   \[\leadsto {\left(a \cdot \cos \color{blue}{\left(\frac{{\left(\sqrt[3]{\pi}\right)}^{2}}{\frac{\frac{-180}{angle}}{\sqrt[3]{\pi}}}\right)}\right)}^{2} + {\left(b \cdot \sin \left(\pi \cdot \frac{angle}{180}\right)\right)}^{2} \]
5. Step-by-step derivation

   1. unpow2 78.0%

      \[\leadsto {\left(a \cdot \cos \left(\frac{\color{blue}{\sqrt[3]{\pi} \cdot \sqrt[3]{\pi}}}{\frac{\frac{-180}{angle}}{\sqrt[3]{\pi}}}\right)\right)}^{2} + {\left(b \cdot \sin \left(\pi \cdot \frac{angle}{180}\right)\right)}^{2} \]

   2. add-cube-cbrt 78.0%

      \[\leadsto {\left(a \cdot \cos \left(\frac{\sqrt[3]{\pi} \cdot \sqrt[3]{\pi}}{\color{blue}{\left(\sqrt[3]{\frac{\frac{-180}{angle}}{\sqrt[3]{\pi}}} \cdot \sqrt[3]{\frac{\frac{-180}{angle}}{\sqrt[3]{\pi}}}\right) \cdot \sqrt[3]{\frac{\frac{-180}{angle}}{\sqrt[3]{\pi}}}}}\right)\right)}^{2} + {\left(b \cdot \sin \left(\pi \cdot \frac{angle}{180}\right)\right)}^{2} \]

   3. times-frac 78.1%

      \[\leadsto {\left(a \cdot \cos \color{blue}{\left(\frac{\sqrt[3]{\pi}}{\sqrt[3]{\frac{\frac{-180}{angle}}{\sqrt[3]{\pi}}} \cdot \sqrt[3]{\frac{\frac{-180}{angle}}{\sqrt[3]{\pi}}}} \cdot \frac{\sqrt[3]{\pi}}{\sqrt[3]{\frac{\frac{-180}{angle}}{\sqrt[3]{\pi}}}}\right)}\right)}^{2} + {\left(b \cdot \sin \left(\pi \cdot \frac{angle}{180}\right)\right)}^{2} \]

   4. pow2 78.1%

      \[\leadsto {\left(a \cdot \cos \left(\frac{\sqrt[3]{\pi}}{\color{blue}{{\left(\sqrt[3]{\frac{\frac{-180}{angle}}{\sqrt[3]{\pi}}}\right)}^{2}}} \cdot \frac{\sqrt[3]{\pi}}{\sqrt[3]{\frac{\frac{-180}{angle}}{\sqrt[3]{\pi}}}}\right)\right)}^{2} + {\left(b \cdot \sin \left(\pi \cdot \frac{angle}{180}\right)\right)}^{2} \]

   5. associate-/l/ 78.1%

      \[\leadsto {\left(a \cdot \cos \left(\frac{\sqrt[3]{\pi}}{{\left(\sqrt[3]{\color{blue}{\frac{-180}{\sqrt[3]{\pi} \cdot angle}}}\right)}^{2}} \cdot \frac{\sqrt[3]{\pi}}{\sqrt[3]{\frac{\frac{-180}{angle}}{\sqrt[3]{\pi}}}}\right)\right)}^{2} + {\left(b \cdot \sin \left(\pi \cdot \frac{angle}{180}\right)\right)}^{2} \]

   6. associate-/l/ 78.1%

      \[\leadsto {\left(a \cdot \cos \left(\frac{\sqrt[3]{\pi}}{{\left(\sqrt[3]{\frac{-180}{\sqrt[3]{\pi} \cdot angle}}\right)}^{2}} \cdot \frac{\sqrt[3]{\pi}}{\sqrt[3]{\color{blue}{\frac{-180}{\sqrt[3]{\pi} \cdot angle}}}}\right)\right)}^{2} + {\left(b \cdot \sin \left(\pi \cdot \frac{angle}{180}\right)\right)}^{2} \]

6. Applied egg-rr 78.1%

   \[\leadsto {\left(a \cdot \cos \color{blue}{\left(\frac{\sqrt[3]{\pi}}{{\left(\sqrt[3]{\frac{-180}{\sqrt[3]{\pi} \cdot angle}}\right)}^{2}} \cdot \frac{\sqrt[3]{\pi}}{\sqrt[3]{\frac{-180}{\sqrt[3]{\pi} \cdot angle}}}\right)}\right)}^{2} + {\left(b \cdot \sin \left(\pi \cdot \frac{angle}{180}\right)\right)}^{2} \]
7. Step-by-step derivation

   1. add-cube-cbrt 78.1%

      \[\leadsto {\left(a \cdot \cos \left(\frac{\sqrt[3]{\pi}}{{\color{blue}{\left(\left(\sqrt[3]{\sqrt[3]{\frac{-180}{\sqrt[3]{\pi} \cdot angle}}} \cdot \sqrt[3]{\sqrt[3]{\frac{-180}{\sqrt[3]{\pi} \cdot angle}}}\right) \cdot \sqrt[3]{\sqrt[3]{\frac{-180}{\sqrt[3]{\pi} \cdot angle}}}\right)}}^{2}} \cdot \frac{\sqrt[3]{\pi}}{\sqrt[3]{\frac{-180}{\sqrt[3]{\pi} \cdot angle}}}\right)\right)}^{2} + {\left(b \cdot \sin \left(\pi \cdot \frac{angle}{180}\right)\right)}^{2} \]

   2. pow3 78.1%

      \[\leadsto {\left(a \cdot \cos \left(\frac{\sqrt[3]{\pi}}{{\color{blue}{\left({\left(\sqrt[3]{\sqrt[3]{\frac{-180}{\sqrt[3]{\pi} \cdot angle}}}\right)}^{3}\right)}}^{2}} \cdot \frac{\sqrt[3]{\pi}}{\sqrt[3]{\frac{-180}{\sqrt[3]{\pi} \cdot angle}}}\right)\right)}^{2} + {\left(b \cdot \sin \left(\pi \cdot \frac{angle}{180}\right)\right)}^{2} \]

   3. associate-/r* 78.1%

      \[\leadsto {\left(a \cdot \cos \left(\frac{\sqrt[3]{\pi}}{{\left({\left(\sqrt[3]{\sqrt[3]{\color{blue}{\frac{\frac{-180}{\sqrt[3]{\pi}}}{angle}}}}\right)}^{3}\right)}^{2}} \cdot \frac{\sqrt[3]{\pi}}{\sqrt[3]{\frac{-180}{\sqrt[3]{\pi} \cdot angle}}}\right)\right)}^{2} + {\left(b \cdot \sin \left(\pi \cdot \frac{angle}{180}\right)\right)}^{2} \]

8. Applied egg-rr 78.1%

   \[\leadsto {\left(a \cdot \cos \left(\frac{\sqrt[3]{\pi}}{{\color{blue}{\left({\left(\sqrt[3]{\sqrt[3]{\frac{\frac{-180}{\sqrt[3]{\pi}}}{angle}}}\right)}^{3}\right)}}^{2}} \cdot \frac{\sqrt[3]{\pi}}{\sqrt[3]{\frac{-180}{\sqrt[3]{\pi} \cdot angle}}}\right)\right)}^{2} + {\left(b \cdot \sin \left(\pi \cdot \frac{angle}{180}\right)\right)}^{2} \]

9. Final simplification 78.1%

   \[\leadsto {\left(b \cdot \sin \left(\pi \cdot \frac{angle}{180}\right)\right)}^{2} + {\left(a \cdot \cos \left(\frac{\sqrt[3]{\pi}}{\sqrt[3]{\frac{-180}{\sqrt[3]{\pi} \cdot angle}}} \cdot \frac{\sqrt[3]{\pi}}{{\left({\left(\sqrt[3]{\sqrt[3]{\frac{\frac{-180}{\sqrt[3]{\pi}}}{angle}}}\right)}^{3}\right)}^{2}}\right)\right)}^{2} \]
10. Add Preprocessing

Alternative 3: 39.7% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \sqrt[3]{\pi} \cdot angle\\ {\left(b \cdot \sin \left(\pi \cdot \frac{angle}{180}\right)\right)}^{2} + {\left(a \cdot \cos \left(\frac{\sqrt[3]{\pi}}{\sqrt[3]{\frac{-180}{t\_0}}} \cdot \frac{\sqrt[3]{\pi}}{{\left(\sqrt[3]{-180} \cdot {\left(\frac{1}{t\_0}\right)}^{0.3333333333333333}\right)}^{2}}\right)\right)}^{2} \end{array} \end{array} \]

FPCore

(FPCore (a b angle)
 :precision binary64
 (let* ((t_0 (* (cbrt PI) angle)))
   (+
    (pow (* b (sin (* PI (/ angle 180.0)))) 2.0)
    (pow
     (*
      a
      (cos
       (*
        (/ (cbrt PI) (cbrt (/ -180.0 t_0)))
        (/
         (cbrt PI)
         (pow (* (cbrt -180.0) (pow (/ 1.0 t_0) 0.3333333333333333)) 2.0)))))
     2.0))))

C

double code(double a, double b, double angle) {
	double t_0 = cbrt(((double) M_PI)) * angle;
	return pow((b * sin((((double) M_PI) * (angle / 180.0)))), 2.0) + pow((a * cos(((cbrt(((double) M_PI)) / cbrt((-180.0 / t_0))) * (cbrt(((double) M_PI)) / pow((cbrt(-180.0) * pow((1.0 / t_0), 0.3333333333333333)), 2.0))))), 2.0);
}

Java

public static double code(double a, double b, double angle) {
	double t_0 = Math.cbrt(Math.PI) * angle;
	return Math.pow((b * Math.sin((Math.PI * (angle / 180.0)))), 2.0) + Math.pow((a * Math.cos(((Math.cbrt(Math.PI) / Math.cbrt((-180.0 / t_0))) * (Math.cbrt(Math.PI) / Math.pow((Math.cbrt(-180.0) * Math.pow((1.0 / t_0), 0.3333333333333333)), 2.0))))), 2.0);
}

Julia

function code(a, b, angle)
	t_0 = Float64(cbrt(pi) * angle)
	return Float64((Float64(b * sin(Float64(pi * Float64(angle / 180.0)))) ^ 2.0) + (Float64(a * cos(Float64(Float64(cbrt(pi) / cbrt(Float64(-180.0 / t_0))) * Float64(cbrt(pi) / (Float64(cbrt(-180.0) * (Float64(1.0 / t_0) ^ 0.3333333333333333)) ^ 2.0))))) ^ 2.0))
end

Wolfram

code[a_, b_, angle_] := Block[{t$95$0 = N[(N[Power[Pi, 1/3], $MachinePrecision] * angle), $MachinePrecision]}, N[(N[Power[N[(b * N[Sin[N[(Pi * N[(angle / 180.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision] + N[Power[N[(a * N[Cos[N[(N[(N[Power[Pi, 1/3], $MachinePrecision] / N[Power[N[(-180.0 / t$95$0), $MachinePrecision], 1/3], $MachinePrecision]), $MachinePrecision] * N[(N[Power[Pi, 1/3], $MachinePrecision] / N[Power[N[(N[Power[-180.0, 1/3], $MachinePrecision] * N[Power[N[(1.0 / t$95$0), $MachinePrecision], 0.3333333333333333], $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Initial program 78.0%

   \[{\left(a \cdot \cos \left(\pi \cdot \frac{angle}{180}\right)\right)}^{2} + {\left(b \cdot \sin \left(\pi \cdot \frac{angle}{180}\right)\right)}^{2} \]
2. Add Preprocessing
3. Step-by-step derivation

   1. add-sqr-sqrt 36.8%

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

   2. sqrt-unprod 66.7%

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

   3. associate-*r/ 66.7%

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

   4. associate-*r/ 66.7%

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

   5. frac-times 66.8%

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

   6. *-commutative 66.8%

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

   7. *-commutative 66.8%

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

   8. metadata-eval 66.8%

      \[\leadsto {\left(a \cdot \cos \left(\sqrt{\frac{\left(angle \cdot \pi\right) \cdot \left(angle \cdot \pi\right)}{\color{blue}{32400}}}\right)\right)}^{2} + {\left(b \cdot \sin \left(\pi \cdot \frac{angle}{180}\right)\right)}^{2} \]

   9. metadata-eval 66.8%

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

   10. frac-times 66.7%

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

   11. associate-*r/ 66.7%

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

   12. associate-*r/ 66.7%

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

   13. sqrt-unprod 41.2%

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

   14. add-sqr-sqrt 78.0%

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

   15. *-commutative 78.0%

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

   16. associate-/r/ 77.9%

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

   17. add-cube-cbrt 78.0%

       \[\leadsto {\left(a \cdot \cos \left(\frac{\color{blue}{\left(\sqrt[3]{\pi} \cdot \sqrt[3]{\pi}\right) \cdot \sqrt[3]{\pi}}}{\frac{-180}{angle}}\right)\right)}^{2} + {\left(b \cdot \sin \left(\pi \cdot \frac{angle}{180}\right)\right)}^{2} \]

   18. associate-/l* 78.0%

       \[\leadsto {\left(a \cdot \cos \color{blue}{\left(\frac{\sqrt[3]{\pi} \cdot \sqrt[3]{\pi}}{\frac{\frac{-180}{angle}}{\sqrt[3]{\pi}}}\right)}\right)}^{2} + {\left(b \cdot \sin \left(\pi \cdot \frac{angle}{180}\right)\right)}^{2} \]

4. Applied egg-rr 78.0%

   \[\leadsto {\left(a \cdot \cos \color{blue}{\left(\frac{{\left(\sqrt[3]{\pi}\right)}^{2}}{\frac{\frac{-180}{angle}}{\sqrt[3]{\pi}}}\right)}\right)}^{2} + {\left(b \cdot \sin \left(\pi \cdot \frac{angle}{180}\right)\right)}^{2} \]
5. Step-by-step derivation

   1. unpow2 78.0%

      \[\leadsto {\left(a \cdot \cos \left(\frac{\color{blue}{\sqrt[3]{\pi} \cdot \sqrt[3]{\pi}}}{\frac{\frac{-180}{angle}}{\sqrt[3]{\pi}}}\right)\right)}^{2} + {\left(b \cdot \sin \left(\pi \cdot \frac{angle}{180}\right)\right)}^{2} \]

   2. add-cube-cbrt 78.0%

      \[\leadsto {\left(a \cdot \cos \left(\frac{\sqrt[3]{\pi} \cdot \sqrt[3]{\pi}}{\color{blue}{\left(\sqrt[3]{\frac{\frac{-180}{angle}}{\sqrt[3]{\pi}}} \cdot \sqrt[3]{\frac{\frac{-180}{angle}}{\sqrt[3]{\pi}}}\right) \cdot \sqrt[3]{\frac{\frac{-180}{angle}}{\sqrt[3]{\pi}}}}}\right)\right)}^{2} + {\left(b \cdot \sin \left(\pi \cdot \frac{angle}{180}\right)\right)}^{2} \]

   3. times-frac 78.1%

      \[\leadsto {\left(a \cdot \cos \color{blue}{\left(\frac{\sqrt[3]{\pi}}{\sqrt[3]{\frac{\frac{-180}{angle}}{\sqrt[3]{\pi}}} \cdot \sqrt[3]{\frac{\frac{-180}{angle}}{\sqrt[3]{\pi}}}} \cdot \frac{\sqrt[3]{\pi}}{\sqrt[3]{\frac{\frac{-180}{angle}}{\sqrt[3]{\pi}}}}\right)}\right)}^{2} + {\left(b \cdot \sin \left(\pi \cdot \frac{angle}{180}\right)\right)}^{2} \]

   4. pow2 78.1%

      \[\leadsto {\left(a \cdot \cos \left(\frac{\sqrt[3]{\pi}}{\color{blue}{{\left(\sqrt[3]{\frac{\frac{-180}{angle}}{\sqrt[3]{\pi}}}\right)}^{2}}} \cdot \frac{\sqrt[3]{\pi}}{\sqrt[3]{\frac{\frac{-180}{angle}}{\sqrt[3]{\pi}}}}\right)\right)}^{2} + {\left(b \cdot \sin \left(\pi \cdot \frac{angle}{180}\right)\right)}^{2} \]

   5. associate-/l/ 78.1%

      \[\leadsto {\left(a \cdot \cos \left(\frac{\sqrt[3]{\pi}}{{\left(\sqrt[3]{\color{blue}{\frac{-180}{\sqrt[3]{\pi} \cdot angle}}}\right)}^{2}} \cdot \frac{\sqrt[3]{\pi}}{\sqrt[3]{\frac{\frac{-180}{angle}}{\sqrt[3]{\pi}}}}\right)\right)}^{2} + {\left(b \cdot \sin \left(\pi \cdot \frac{angle}{180}\right)\right)}^{2} \]

   6. associate-/l/ 78.1%

      \[\leadsto {\left(a \cdot \cos \left(\frac{\sqrt[3]{\pi}}{{\left(\sqrt[3]{\frac{-180}{\sqrt[3]{\pi} \cdot angle}}\right)}^{2}} \cdot \frac{\sqrt[3]{\pi}}{\sqrt[3]{\color{blue}{\frac{-180}{\sqrt[3]{\pi} \cdot angle}}}}\right)\right)}^{2} + {\left(b \cdot \sin \left(\pi \cdot \frac{angle}{180}\right)\right)}^{2} \]

6. Applied egg-rr 78.1%

   \[\leadsto {\left(a \cdot \cos \color{blue}{\left(\frac{\sqrt[3]{\pi}}{{\left(\sqrt[3]{\frac{-180}{\sqrt[3]{\pi} \cdot angle}}\right)}^{2}} \cdot \frac{\sqrt[3]{\pi}}{\sqrt[3]{\frac{-180}{\sqrt[3]{\pi} \cdot angle}}}\right)}\right)}^{2} + {\left(b \cdot \sin \left(\pi \cdot \frac{angle}{180}\right)\right)}^{2} \]
7. Step-by-step derivation

   1. pow1/3 41.7%

      \[\leadsto {\left(a \cdot \cos \left(\frac{\sqrt[3]{\pi}}{{\color{blue}{\left({\left(\frac{-180}{\sqrt[3]{\pi} \cdot angle}\right)}^{0.3333333333333333}\right)}}^{2}} \cdot \frac{\sqrt[3]{\pi}}{\sqrt[3]{\frac{-180}{\sqrt[3]{\pi} \cdot angle}}}\right)\right)}^{2} + {\left(b \cdot \sin \left(\pi \cdot \frac{angle}{180}\right)\right)}^{2} \]

   2. div-inv 41.7%

      \[\leadsto {\left(a \cdot \cos \left(\frac{\sqrt[3]{\pi}}{{\left({\color{blue}{\left(-180 \cdot \frac{1}{\sqrt[3]{\pi} \cdot angle}\right)}}^{0.3333333333333333}\right)}^{2}} \cdot \frac{\sqrt[3]{\pi}}{\sqrt[3]{\frac{-180}{\sqrt[3]{\pi} \cdot angle}}}\right)\right)}^{2} + {\left(b \cdot \sin \left(\pi \cdot \frac{angle}{180}\right)\right)}^{2} \]

   3. unpow-prod-down 0.0%

      \[\leadsto {\left(a \cdot \cos \left(\frac{\sqrt[3]{\pi}}{{\color{blue}{\left({-180}^{0.3333333333333333} \cdot {\left(\frac{1}{\sqrt[3]{\pi} \cdot angle}\right)}^{0.3333333333333333}\right)}}^{2}} \cdot \frac{\sqrt[3]{\pi}}{\sqrt[3]{\frac{-180}{\sqrt[3]{\pi} \cdot angle}}}\right)\right)}^{2} + {\left(b \cdot \sin \left(\pi \cdot \frac{angle}{180}\right)\right)}^{2} \]

   4. pow1/3 36.8%

      \[\leadsto {\left(a \cdot \cos \left(\frac{\sqrt[3]{\pi}}{{\left(\color{blue}{\sqrt[3]{-180}} \cdot {\left(\frac{1}{\sqrt[3]{\pi} \cdot angle}\right)}^{0.3333333333333333}\right)}^{2}} \cdot \frac{\sqrt[3]{\pi}}{\sqrt[3]{\frac{-180}{\sqrt[3]{\pi} \cdot angle}}}\right)\right)}^{2} + {\left(b \cdot \sin \left(\pi \cdot \frac{angle}{180}\right)\right)}^{2} \]

8. Applied egg-rr 36.8%

   \[\leadsto {\left(a \cdot \cos \left(\frac{\sqrt[3]{\pi}}{{\color{blue}{\left(\sqrt[3]{-180} \cdot {\left(\frac{1}{\sqrt[3]{\pi} \cdot angle}\right)}^{0.3333333333333333}\right)}}^{2}} \cdot \frac{\sqrt[3]{\pi}}{\sqrt[3]{\frac{-180}{\sqrt[3]{\pi} \cdot angle}}}\right)\right)}^{2} + {\left(b \cdot \sin \left(\pi \cdot \frac{angle}{180}\right)\right)}^{2} \]

9. Final simplification 36.8%

   \[\leadsto {\left(b \cdot \sin \left(\pi \cdot \frac{angle}{180}\right)\right)}^{2} + {\left(a \cdot \cos \left(\frac{\sqrt[3]{\pi}}{\sqrt[3]{\frac{-180}{\sqrt[3]{\pi} \cdot angle}}} \cdot \frac{\sqrt[3]{\pi}}{{\left(\sqrt[3]{-180} \cdot {\left(\frac{1}{\sqrt[3]{\pi} \cdot angle}\right)}^{0.3333333333333333}\right)}^{2}}\right)\right)}^{2} \]
10. Add Preprocessing

Alternative 4: 79.4% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \sqrt[3]{\pi} \cdot angle\\ {\left(b \cdot \sin \left(\pi \cdot \frac{angle}{180}\right)\right)}^{2} + {\left(a \cdot \cos \left(\frac{\sqrt[3]{\pi}}{\sqrt[3]{\frac{-180}{t\_0}}} \cdot \frac{\sqrt[3]{\pi}}{{\left(\frac{\sqrt[3]{-180}}{\sqrt[3]{t\_0}}\right)}^{2}}\right)\right)}^{2} \end{array} \end{array} \]

FPCore

(FPCore (a b angle)
 :precision binary64
 (let* ((t_0 (* (cbrt PI) angle)))
   (+
    (pow (* b (sin (* PI (/ angle 180.0)))) 2.0)
    (pow
     (*
      a
      (cos
       (*
        (/ (cbrt PI) (cbrt (/ -180.0 t_0)))
        (/ (cbrt PI) (pow (/ (cbrt -180.0) (cbrt t_0)) 2.0)))))
     2.0))))

C

double code(double a, double b, double angle) {
	double t_0 = cbrt(((double) M_PI)) * angle;
	return pow((b * sin((((double) M_PI) * (angle / 180.0)))), 2.0) + pow((a * cos(((cbrt(((double) M_PI)) / cbrt((-180.0 / t_0))) * (cbrt(((double) M_PI)) / pow((cbrt(-180.0) / cbrt(t_0)), 2.0))))), 2.0);
}

Java

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

Julia

function code(a, b, angle)
	t_0 = Float64(cbrt(pi) * angle)
	return Float64((Float64(b * sin(Float64(pi * Float64(angle / 180.0)))) ^ 2.0) + (Float64(a * cos(Float64(Float64(cbrt(pi) / cbrt(Float64(-180.0 / t_0))) * Float64(cbrt(pi) / (Float64(cbrt(-180.0) / cbrt(t_0)) ^ 2.0))))) ^ 2.0))
end

Wolfram

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

Derivation

1. Initial program 78.0%

   \[{\left(a \cdot \cos \left(\pi \cdot \frac{angle}{180}\right)\right)}^{2} + {\left(b \cdot \sin \left(\pi \cdot \frac{angle}{180}\right)\right)}^{2} \]
2. Add Preprocessing
3. Step-by-step derivation

   1. add-sqr-sqrt 36.8%

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

   2. sqrt-unprod 66.7%

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

   3. associate-*r/ 66.7%

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

   4. associate-*r/ 66.7%

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

   5. frac-times 66.8%

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

   6. *-commutative 66.8%

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

   7. *-commutative 66.8%

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

   8. metadata-eval 66.8%

      \[\leadsto {\left(a \cdot \cos \left(\sqrt{\frac{\left(angle \cdot \pi\right) \cdot \left(angle \cdot \pi\right)}{\color{blue}{32400}}}\right)\right)}^{2} + {\left(b \cdot \sin \left(\pi \cdot \frac{angle}{180}\right)\right)}^{2} \]

   9. metadata-eval 66.8%

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

   10. frac-times 66.7%

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

   11. associate-*r/ 66.7%

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

   12. associate-*r/ 66.7%

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

   13. sqrt-unprod 41.2%

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

   14. add-sqr-sqrt 78.0%

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

   15. *-commutative 78.0%

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

   16. associate-/r/ 77.9%

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

   17. add-cube-cbrt 78.0%

       \[\leadsto {\left(a \cdot \cos \left(\frac{\color{blue}{\left(\sqrt[3]{\pi} \cdot \sqrt[3]{\pi}\right) \cdot \sqrt[3]{\pi}}}{\frac{-180}{angle}}\right)\right)}^{2} + {\left(b \cdot \sin \left(\pi \cdot \frac{angle}{180}\right)\right)}^{2} \]

   18. associate-/l* 78.0%

       \[\leadsto {\left(a \cdot \cos \color{blue}{\left(\frac{\sqrt[3]{\pi} \cdot \sqrt[3]{\pi}}{\frac{\frac{-180}{angle}}{\sqrt[3]{\pi}}}\right)}\right)}^{2} + {\left(b \cdot \sin \left(\pi \cdot \frac{angle}{180}\right)\right)}^{2} \]

4. Applied egg-rr 78.0%

   \[\leadsto {\left(a \cdot \cos \color{blue}{\left(\frac{{\left(\sqrt[3]{\pi}\right)}^{2}}{\frac{\frac{-180}{angle}}{\sqrt[3]{\pi}}}\right)}\right)}^{2} + {\left(b \cdot \sin \left(\pi \cdot \frac{angle}{180}\right)\right)}^{2} \]
5. Step-by-step derivation

   1. unpow2 78.0%

      \[\leadsto {\left(a \cdot \cos \left(\frac{\color{blue}{\sqrt[3]{\pi} \cdot \sqrt[3]{\pi}}}{\frac{\frac{-180}{angle}}{\sqrt[3]{\pi}}}\right)\right)}^{2} + {\left(b \cdot \sin \left(\pi \cdot \frac{angle}{180}\right)\right)}^{2} \]

   2. add-cube-cbrt 78.0%

      \[\leadsto {\left(a \cdot \cos \left(\frac{\sqrt[3]{\pi} \cdot \sqrt[3]{\pi}}{\color{blue}{\left(\sqrt[3]{\frac{\frac{-180}{angle}}{\sqrt[3]{\pi}}} \cdot \sqrt[3]{\frac{\frac{-180}{angle}}{\sqrt[3]{\pi}}}\right) \cdot \sqrt[3]{\frac{\frac{-180}{angle}}{\sqrt[3]{\pi}}}}}\right)\right)}^{2} + {\left(b \cdot \sin \left(\pi \cdot \frac{angle}{180}\right)\right)}^{2} \]

   3. times-frac 78.1%

      \[\leadsto {\left(a \cdot \cos \color{blue}{\left(\frac{\sqrt[3]{\pi}}{\sqrt[3]{\frac{\frac{-180}{angle}}{\sqrt[3]{\pi}}} \cdot \sqrt[3]{\frac{\frac{-180}{angle}}{\sqrt[3]{\pi}}}} \cdot \frac{\sqrt[3]{\pi}}{\sqrt[3]{\frac{\frac{-180}{angle}}{\sqrt[3]{\pi}}}}\right)}\right)}^{2} + {\left(b \cdot \sin \left(\pi \cdot \frac{angle}{180}\right)\right)}^{2} \]

   4. pow2 78.1%

      \[\leadsto {\left(a \cdot \cos \left(\frac{\sqrt[3]{\pi}}{\color{blue}{{\left(\sqrt[3]{\frac{\frac{-180}{angle}}{\sqrt[3]{\pi}}}\right)}^{2}}} \cdot \frac{\sqrt[3]{\pi}}{\sqrt[3]{\frac{\frac{-180}{angle}}{\sqrt[3]{\pi}}}}\right)\right)}^{2} + {\left(b \cdot \sin \left(\pi \cdot \frac{angle}{180}\right)\right)}^{2} \]

   5. associate-/l/ 78.1%

      \[\leadsto {\left(a \cdot \cos \left(\frac{\sqrt[3]{\pi}}{{\left(\sqrt[3]{\color{blue}{\frac{-180}{\sqrt[3]{\pi} \cdot angle}}}\right)}^{2}} \cdot \frac{\sqrt[3]{\pi}}{\sqrt[3]{\frac{\frac{-180}{angle}}{\sqrt[3]{\pi}}}}\right)\right)}^{2} + {\left(b \cdot \sin \left(\pi \cdot \frac{angle}{180}\right)\right)}^{2} \]

   6. associate-/l/ 78.1%

      \[\leadsto {\left(a \cdot \cos \left(\frac{\sqrt[3]{\pi}}{{\left(\sqrt[3]{\frac{-180}{\sqrt[3]{\pi} \cdot angle}}\right)}^{2}} \cdot \frac{\sqrt[3]{\pi}}{\sqrt[3]{\color{blue}{\frac{-180}{\sqrt[3]{\pi} \cdot angle}}}}\right)\right)}^{2} + {\left(b \cdot \sin \left(\pi \cdot \frac{angle}{180}\right)\right)}^{2} \]

6. Applied egg-rr 78.1%

   \[\leadsto {\left(a \cdot \cos \color{blue}{\left(\frac{\sqrt[3]{\pi}}{{\left(\sqrt[3]{\frac{-180}{\sqrt[3]{\pi} \cdot angle}}\right)}^{2}} \cdot \frac{\sqrt[3]{\pi}}{\sqrt[3]{\frac{-180}{\sqrt[3]{\pi} \cdot angle}}}\right)}\right)}^{2} + {\left(b \cdot \sin \left(\pi \cdot \frac{angle}{180}\right)\right)}^{2} \]
7. Step-by-step derivation

   1. cbrt-div 78.1%

      \[\leadsto {\left(a \cdot \cos \left(\frac{\sqrt[3]{\pi}}{{\color{blue}{\left(\frac{\sqrt[3]{-180}}{\sqrt[3]{\sqrt[3]{\pi} \cdot angle}}\right)}}^{2}} \cdot \frac{\sqrt[3]{\pi}}{\sqrt[3]{\frac{-180}{\sqrt[3]{\pi} \cdot angle}}}\right)\right)}^{2} + {\left(b \cdot \sin \left(\pi \cdot \frac{angle}{180}\right)\right)}^{2} \]

8. Applied egg-rr 78.1%

   \[\leadsto {\left(a \cdot \cos \left(\frac{\sqrt[3]{\pi}}{{\color{blue}{\left(\frac{\sqrt[3]{-180}}{\sqrt[3]{\sqrt[3]{\pi} \cdot angle}}\right)}}^{2}} \cdot \frac{\sqrt[3]{\pi}}{\sqrt[3]{\frac{-180}{\sqrt[3]{\pi} \cdot angle}}}\right)\right)}^{2} + {\left(b \cdot \sin \left(\pi \cdot \frac{angle}{180}\right)\right)}^{2} \]

9. Final simplification 78.1%

   \[\leadsto {\left(b \cdot \sin \left(\pi \cdot \frac{angle}{180}\right)\right)}^{2} + {\left(a \cdot \cos \left(\frac{\sqrt[3]{\pi}}{\sqrt[3]{\frac{-180}{\sqrt[3]{\pi} \cdot angle}}} \cdot \frac{\sqrt[3]{\pi}}{{\left(\frac{\sqrt[3]{-180}}{\sqrt[3]{\sqrt[3]{\pi} \cdot angle}}\right)}^{2}}\right)\right)}^{2} \]
10. Add Preprocessing

Alternative 5: 79.4% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \sqrt[3]{\frac{-180}{\sqrt[3]{\pi} \cdot angle}}\\ {\left(b \cdot \sin \left(\pi \cdot \frac{angle}{180}\right)\right)}^{2} + {\left(a \cdot \cos \left(\frac{\sqrt[3]{\pi}}{t\_0} \cdot \frac{\sqrt[3]{\pi}}{{t\_0}^{2}}\right)\right)}^{2} \end{array} \end{array} \]

FPCore

(FPCore (a b angle)
 :precision binary64
 (let* ((t_0 (cbrt (/ -180.0 (* (cbrt PI) angle)))))
   (+
    (pow (* b (sin (* PI (/ angle 180.0)))) 2.0)
    (pow (* a (cos (* (/ (cbrt PI) t_0) (/ (cbrt PI) (pow t_0 2.0))))) 2.0))))

C

double code(double a, double b, double angle) {
	double t_0 = cbrt((-180.0 / (cbrt(((double) M_PI)) * angle)));
	return pow((b * sin((((double) M_PI) * (angle / 180.0)))), 2.0) + pow((a * cos(((cbrt(((double) M_PI)) / t_0) * (cbrt(((double) M_PI)) / pow(t_0, 2.0))))), 2.0);
}

Java

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

Julia

function code(a, b, angle)
	t_0 = cbrt(Float64(-180.0 / Float64(cbrt(pi) * angle)))
	return Float64((Float64(b * sin(Float64(pi * Float64(angle / 180.0)))) ^ 2.0) + (Float64(a * cos(Float64(Float64(cbrt(pi) / t_0) * Float64(cbrt(pi) / (t_0 ^ 2.0))))) ^ 2.0))
end

Wolfram

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

Derivation

1. Initial program 78.0%

   \[{\left(a \cdot \cos \left(\pi \cdot \frac{angle}{180}\right)\right)}^{2} + {\left(b \cdot \sin \left(\pi \cdot \frac{angle}{180}\right)\right)}^{2} \]
2. Add Preprocessing
3. Step-by-step derivation

   1. add-sqr-sqrt 36.8%

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

   2. sqrt-unprod 66.7%

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

   3. associate-*r/ 66.7%

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

   4. associate-*r/ 66.7%

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

   5. frac-times 66.8%

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

   6. *-commutative 66.8%

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

   7. *-commutative 66.8%

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

   8. metadata-eval 66.8%

      \[\leadsto {\left(a \cdot \cos \left(\sqrt{\frac{\left(angle \cdot \pi\right) \cdot \left(angle \cdot \pi\right)}{\color{blue}{32400}}}\right)\right)}^{2} + {\left(b \cdot \sin \left(\pi \cdot \frac{angle}{180}\right)\right)}^{2} \]

   9. metadata-eval 66.8%

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

   10. frac-times 66.7%

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

   11. associate-*r/ 66.7%

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

   12. associate-*r/ 66.7%

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

   13. sqrt-unprod 41.2%

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

   14. add-sqr-sqrt 78.0%

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

   15. *-commutative 78.0%

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

   16. associate-/r/ 77.9%

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

   17. add-cube-cbrt 78.0%

       \[\leadsto {\left(a \cdot \cos \left(\frac{\color{blue}{\left(\sqrt[3]{\pi} \cdot \sqrt[3]{\pi}\right) \cdot \sqrt[3]{\pi}}}{\frac{-180}{angle}}\right)\right)}^{2} + {\left(b \cdot \sin \left(\pi \cdot \frac{angle}{180}\right)\right)}^{2} \]

   18. associate-/l* 78.0%

       \[\leadsto {\left(a \cdot \cos \color{blue}{\left(\frac{\sqrt[3]{\pi} \cdot \sqrt[3]{\pi}}{\frac{\frac{-180}{angle}}{\sqrt[3]{\pi}}}\right)}\right)}^{2} + {\left(b \cdot \sin \left(\pi \cdot \frac{angle}{180}\right)\right)}^{2} \]

4. Applied egg-rr 78.0%

   \[\leadsto {\left(a \cdot \cos \color{blue}{\left(\frac{{\left(\sqrt[3]{\pi}\right)}^{2}}{\frac{\frac{-180}{angle}}{\sqrt[3]{\pi}}}\right)}\right)}^{2} + {\left(b \cdot \sin \left(\pi \cdot \frac{angle}{180}\right)\right)}^{2} \]
5. Step-by-step derivation

   1. unpow2 78.0%

      \[\leadsto {\left(a \cdot \cos \left(\frac{\color{blue}{\sqrt[3]{\pi} \cdot \sqrt[3]{\pi}}}{\frac{\frac{-180}{angle}}{\sqrt[3]{\pi}}}\right)\right)}^{2} + {\left(b \cdot \sin \left(\pi \cdot \frac{angle}{180}\right)\right)}^{2} \]

   2. add-cube-cbrt 78.0%

      \[\leadsto {\left(a \cdot \cos \left(\frac{\sqrt[3]{\pi} \cdot \sqrt[3]{\pi}}{\color{blue}{\left(\sqrt[3]{\frac{\frac{-180}{angle}}{\sqrt[3]{\pi}}} \cdot \sqrt[3]{\frac{\frac{-180}{angle}}{\sqrt[3]{\pi}}}\right) \cdot \sqrt[3]{\frac{\frac{-180}{angle}}{\sqrt[3]{\pi}}}}}\right)\right)}^{2} + {\left(b \cdot \sin \left(\pi \cdot \frac{angle}{180}\right)\right)}^{2} \]

   3. times-frac 78.1%

      \[\leadsto {\left(a \cdot \cos \color{blue}{\left(\frac{\sqrt[3]{\pi}}{\sqrt[3]{\frac{\frac{-180}{angle}}{\sqrt[3]{\pi}}} \cdot \sqrt[3]{\frac{\frac{-180}{angle}}{\sqrt[3]{\pi}}}} \cdot \frac{\sqrt[3]{\pi}}{\sqrt[3]{\frac{\frac{-180}{angle}}{\sqrt[3]{\pi}}}}\right)}\right)}^{2} + {\left(b \cdot \sin \left(\pi \cdot \frac{angle}{180}\right)\right)}^{2} \]

   4. pow2 78.1%

      \[\leadsto {\left(a \cdot \cos \left(\frac{\sqrt[3]{\pi}}{\color{blue}{{\left(\sqrt[3]{\frac{\frac{-180}{angle}}{\sqrt[3]{\pi}}}\right)}^{2}}} \cdot \frac{\sqrt[3]{\pi}}{\sqrt[3]{\frac{\frac{-180}{angle}}{\sqrt[3]{\pi}}}}\right)\right)}^{2} + {\left(b \cdot \sin \left(\pi \cdot \frac{angle}{180}\right)\right)}^{2} \]

   5. associate-/l/ 78.1%

      \[\leadsto {\left(a \cdot \cos \left(\frac{\sqrt[3]{\pi}}{{\left(\sqrt[3]{\color{blue}{\frac{-180}{\sqrt[3]{\pi} \cdot angle}}}\right)}^{2}} \cdot \frac{\sqrt[3]{\pi}}{\sqrt[3]{\frac{\frac{-180}{angle}}{\sqrt[3]{\pi}}}}\right)\right)}^{2} + {\left(b \cdot \sin \left(\pi \cdot \frac{angle}{180}\right)\right)}^{2} \]

   6. associate-/l/ 78.1%

      \[\leadsto {\left(a \cdot \cos \left(\frac{\sqrt[3]{\pi}}{{\left(\sqrt[3]{\frac{-180}{\sqrt[3]{\pi} \cdot angle}}\right)}^{2}} \cdot \frac{\sqrt[3]{\pi}}{\sqrt[3]{\color{blue}{\frac{-180}{\sqrt[3]{\pi} \cdot angle}}}}\right)\right)}^{2} + {\left(b \cdot \sin \left(\pi \cdot \frac{angle}{180}\right)\right)}^{2} \]

6. Applied egg-rr 78.1%

   \[\leadsto {\left(a \cdot \cos \color{blue}{\left(\frac{\sqrt[3]{\pi}}{{\left(\sqrt[3]{\frac{-180}{\sqrt[3]{\pi} \cdot angle}}\right)}^{2}} \cdot \frac{\sqrt[3]{\pi}}{\sqrt[3]{\frac{-180}{\sqrt[3]{\pi} \cdot angle}}}\right)}\right)}^{2} + {\left(b \cdot \sin \left(\pi \cdot \frac{angle}{180}\right)\right)}^{2} \]

7. Final simplification 78.1%

   \[\leadsto {\left(b \cdot \sin \left(\pi \cdot \frac{angle}{180}\right)\right)}^{2} + {\left(a \cdot \cos \left(\frac{\sqrt[3]{\pi}}{\sqrt[3]{\frac{-180}{\sqrt[3]{\pi} \cdot angle}}} \cdot \frac{\sqrt[3]{\pi}}{{\left(\sqrt[3]{\frac{-180}{\sqrt[3]{\pi} \cdot angle}}\right)}^{2}}\right)\right)}^{2} \]
8. Add Preprocessing

Alternative 6: 79.5% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \sqrt[3]{\sqrt{\pi}}\\ {\left(b \cdot \sin \left(\pi \cdot \frac{angle}{180}\right)\right)}^{2} + {\left(a \cdot \cos \left(\frac{{\left(t\_0 \cdot t\_0\right)}^{2}}{\frac{\frac{-180}{angle}}{\sqrt[3]{\pi}}}\right)\right)}^{2} \end{array} \end{array} \]

FPCore

(FPCore (a b angle)
 :precision binary64
 (let* ((t_0 (cbrt (sqrt PI))))
   (+
    (pow (* b (sin (* PI (/ angle 180.0)))) 2.0)
    (pow
     (* a (cos (/ (pow (* t_0 t_0) 2.0) (/ (/ -180.0 angle) (cbrt PI)))))
     2.0))))

C

double code(double a, double b, double angle) {
	double t_0 = cbrt(sqrt(((double) M_PI)));
	return pow((b * sin((((double) M_PI) * (angle / 180.0)))), 2.0) + pow((a * cos((pow((t_0 * t_0), 2.0) / ((-180.0 / angle) / cbrt(((double) M_PI)))))), 2.0);
}

Java

public static double code(double a, double b, double angle) {
	double t_0 = Math.cbrt(Math.sqrt(Math.PI));
	return Math.pow((b * Math.sin((Math.PI * (angle / 180.0)))), 2.0) + Math.pow((a * Math.cos((Math.pow((t_0 * t_0), 2.0) / ((-180.0 / angle) / Math.cbrt(Math.PI))))), 2.0);
}

Julia

function code(a, b, angle)
	t_0 = cbrt(sqrt(pi))
	return Float64((Float64(b * sin(Float64(pi * Float64(angle / 180.0)))) ^ 2.0) + (Float64(a * cos(Float64((Float64(t_0 * t_0) ^ 2.0) / Float64(Float64(-180.0 / angle) / cbrt(pi))))) ^ 2.0))
end

Wolfram

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

Derivation

1. Initial program 78.0%

   \[{\left(a \cdot \cos \left(\pi \cdot \frac{angle}{180}\right)\right)}^{2} + {\left(b \cdot \sin \left(\pi \cdot \frac{angle}{180}\right)\right)}^{2} \]
2. Add Preprocessing
3. Step-by-step derivation

   1. add-sqr-sqrt 36.8%

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

   2. sqrt-unprod 66.7%

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

   3. associate-*r/ 66.7%

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

   4. associate-*r/ 66.7%

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

   5. frac-times 66.8%

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

   6. *-commutative 66.8%

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

   7. *-commutative 66.8%

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

   8. metadata-eval 66.8%

      \[\leadsto {\left(a \cdot \cos \left(\sqrt{\frac{\left(angle \cdot \pi\right) \cdot \left(angle \cdot \pi\right)}{\color{blue}{32400}}}\right)\right)}^{2} + {\left(b \cdot \sin \left(\pi \cdot \frac{angle}{180}\right)\right)}^{2} \]

   9. metadata-eval 66.8%

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

   10. frac-times 66.7%

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

   11. associate-*r/ 66.7%

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

   12. associate-*r/ 66.7%

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

   13. sqrt-unprod 41.2%

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

   14. add-sqr-sqrt 78.0%

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

   15. *-commutative 78.0%

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

   16. associate-/r/ 77.9%

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

   17. add-cube-cbrt 78.0%

       \[\leadsto {\left(a \cdot \cos \left(\frac{\color{blue}{\left(\sqrt[3]{\pi} \cdot \sqrt[3]{\pi}\right) \cdot \sqrt[3]{\pi}}}{\frac{-180}{angle}}\right)\right)}^{2} + {\left(b \cdot \sin \left(\pi \cdot \frac{angle}{180}\right)\right)}^{2} \]

   18. associate-/l* 78.0%

       \[\leadsto {\left(a \cdot \cos \color{blue}{\left(\frac{\sqrt[3]{\pi} \cdot \sqrt[3]{\pi}}{\frac{\frac{-180}{angle}}{\sqrt[3]{\pi}}}\right)}\right)}^{2} + {\left(b \cdot \sin \left(\pi \cdot \frac{angle}{180}\right)\right)}^{2} \]

4. Applied egg-rr 78.0%

   \[\leadsto {\left(a \cdot \cos \color{blue}{\left(\frac{{\left(\sqrt[3]{\pi}\right)}^{2}}{\frac{\frac{-180}{angle}}{\sqrt[3]{\pi}}}\right)}\right)}^{2} + {\left(b \cdot \sin \left(\pi \cdot \frac{angle}{180}\right)\right)}^{2} \]
5. Step-by-step derivation

   1. pow1/3 78.1%

      \[\leadsto {\left(a \cdot \cos \left(\frac{\color{blue}{{\pi}^{0.3333333333333333}}}{{\left({\left(\sqrt[3]{\sqrt[3]{\frac{\frac{-180}{\sqrt[3]{\pi}}}{angle}}}\right)}^{3}\right)}^{2}} \cdot \frac{\sqrt[3]{\pi}}{\sqrt[3]{\frac{-180}{\sqrt[3]{\pi} \cdot angle}}}\right)\right)}^{2} + {\left(b \cdot \sin \left(\pi \cdot \frac{angle}{180}\right)\right)}^{2} \]

   2. add-sqr-sqrt 78.1%

      \[\leadsto {\left(a \cdot \cos \left(\frac{{\color{blue}{\left(\sqrt{\pi} \cdot \sqrt{\pi}\right)}}^{0.3333333333333333}}{{\left({\left(\sqrt[3]{\sqrt[3]{\frac{\frac{-180}{\sqrt[3]{\pi}}}{angle}}}\right)}^{3}\right)}^{2}} \cdot \frac{\sqrt[3]{\pi}}{\sqrt[3]{\frac{-180}{\sqrt[3]{\pi} \cdot angle}}}\right)\right)}^{2} + {\left(b \cdot \sin \left(\pi \cdot \frac{angle}{180}\right)\right)}^{2} \]

   3. unpow-prod-down 78.2%

      \[\leadsto {\left(a \cdot \cos \left(\frac{\color{blue}{{\left(\sqrt{\pi}\right)}^{0.3333333333333333} \cdot {\left(\sqrt{\pi}\right)}^{0.3333333333333333}}}{{\left({\left(\sqrt[3]{\sqrt[3]{\frac{\frac{-180}{\sqrt[3]{\pi}}}{angle}}}\right)}^{3}\right)}^{2}} \cdot \frac{\sqrt[3]{\pi}}{\sqrt[3]{\frac{-180}{\sqrt[3]{\pi} \cdot angle}}}\right)\right)}^{2} + {\left(b \cdot \sin \left(\pi \cdot \frac{angle}{180}\right)\right)}^{2} \]

6. Applied egg-rr 78.0%

   \[\leadsto {\left(a \cdot \cos \left(\frac{{\color{blue}{\left({\left(\sqrt{\pi}\right)}^{0.3333333333333333} \cdot {\left(\sqrt{\pi}\right)}^{0.3333333333333333}\right)}}^{2}}{\frac{\frac{-180}{angle}}{\sqrt[3]{\pi}}}\right)\right)}^{2} + {\left(b \cdot \sin \left(\pi \cdot \frac{angle}{180}\right)\right)}^{2} \]
7. Step-by-step derivation

   1. unpow1/3 78.1%

      \[\leadsto {\left(a \cdot \cos \left(\frac{\color{blue}{\sqrt[3]{\sqrt{\pi}}} \cdot {\left(\sqrt{\pi}\right)}^{0.3333333333333333}}{{\left({\left(\sqrt[3]{\sqrt[3]{\frac{\frac{-180}{\sqrt[3]{\pi}}}{angle}}}\right)}^{3}\right)}^{2}} \cdot \frac{\sqrt[3]{\pi}}{\sqrt[3]{\frac{-180}{\sqrt[3]{\pi} \cdot angle}}}\right)\right)}^{2} + {\left(b \cdot \sin \left(\pi \cdot \frac{angle}{180}\right)\right)}^{2} \]

   2. unpow1/3 78.1%

      \[\leadsto {\left(a \cdot \cos \left(\frac{\sqrt[3]{\sqrt{\pi}} \cdot \color{blue}{\sqrt[3]{\sqrt{\pi}}}}{{\left({\left(\sqrt[3]{\sqrt[3]{\frac{\frac{-180}{\sqrt[3]{\pi}}}{angle}}}\right)}^{3}\right)}^{2}} \cdot \frac{\sqrt[3]{\pi}}{\sqrt[3]{\frac{-180}{\sqrt[3]{\pi} \cdot angle}}}\right)\right)}^{2} + {\left(b \cdot \sin \left(\pi \cdot \frac{angle}{180}\right)\right)}^{2} \]

8. Simplified 78.0%

   \[\leadsto {\left(a \cdot \cos \left(\frac{{\color{blue}{\left(\sqrt[3]{\sqrt{\pi}} \cdot \sqrt[3]{\sqrt{\pi}}\right)}}^{2}}{\frac{\frac{-180}{angle}}{\sqrt[3]{\pi}}}\right)\right)}^{2} + {\left(b \cdot \sin \left(\pi \cdot \frac{angle}{180}\right)\right)}^{2} \]

9. Final simplification 78.0%

   \[\leadsto {\left(b \cdot \sin \left(\pi \cdot \frac{angle}{180}\right)\right)}^{2} + {\left(a \cdot \cos \left(\frac{{\left(\sqrt[3]{\sqrt{\pi}} \cdot \sqrt[3]{\sqrt{\pi}}\right)}^{2}}{\frac{\frac{-180}{angle}}{\sqrt[3]{\pi}}}\right)\right)}^{2} \]
10. Add Preprocessing

Alternative 7: 79.5% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ {\left(b \cdot \sin \left(\pi \cdot \frac{angle}{180}\right)\right)}^{2} + {\left(a \cdot \cos \left(\frac{{\left(\sqrt[3]{\pi}\right)}^{2}}{\frac{\frac{-180}{angle}}{\sqrt[3]{\pi}}}\right)\right)}^{2} \end{array} \]

FPCore

(FPCore (a b angle)
 :precision binary64
 (+
  (pow (* b (sin (* PI (/ angle 180.0)))) 2.0)
  (pow
   (* a (cos (/ (pow (cbrt PI) 2.0) (/ (/ -180.0 angle) (cbrt PI)))))
   2.0)))

C

double code(double a, double b, double angle) {
	return pow((b * sin((((double) M_PI) * (angle / 180.0)))), 2.0) + pow((a * cos((pow(cbrt(((double) M_PI)), 2.0) / ((-180.0 / angle) / cbrt(((double) M_PI)))))), 2.0);
}

Java

public static double code(double a, double b, double angle) {
	return Math.pow((b * Math.sin((Math.PI * (angle / 180.0)))), 2.0) + Math.pow((a * Math.cos((Math.pow(Math.cbrt(Math.PI), 2.0) / ((-180.0 / angle) / Math.cbrt(Math.PI))))), 2.0);
}

Julia

function code(a, b, angle)
	return Float64((Float64(b * sin(Float64(pi * Float64(angle / 180.0)))) ^ 2.0) + (Float64(a * cos(Float64((cbrt(pi) ^ 2.0) / Float64(Float64(-180.0 / angle) / cbrt(pi))))) ^ 2.0))
end

Wolfram

code[a_, b_, angle_] := N[(N[Power[N[(b * N[Sin[N[(Pi * N[(angle / 180.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision] + N[Power[N[(a * N[Cos[N[(N[Power[N[Power[Pi, 1/3], $MachinePrecision], 2.0], $MachinePrecision] / N[(N[(-180.0 / angle), $MachinePrecision] / N[Power[Pi, 1/3], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 78.0%

   \[{\left(a \cdot \cos \left(\pi \cdot \frac{angle}{180}\right)\right)}^{2} + {\left(b \cdot \sin \left(\pi \cdot \frac{angle}{180}\right)\right)}^{2} \]
2. Add Preprocessing
3. Step-by-step derivation

   1. add-sqr-sqrt 36.8%

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

   2. sqrt-unprod 66.7%

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

   3. associate-*r/ 66.7%

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

   4. associate-*r/ 66.7%

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

   5. frac-times 66.8%

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

   6. *-commutative 66.8%

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

   7. *-commutative 66.8%

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

   8. metadata-eval 66.8%

      \[\leadsto {\left(a \cdot \cos \left(\sqrt{\frac{\left(angle \cdot \pi\right) \cdot \left(angle \cdot \pi\right)}{\color{blue}{32400}}}\right)\right)}^{2} + {\left(b \cdot \sin \left(\pi \cdot \frac{angle}{180}\right)\right)}^{2} \]

   9. metadata-eval 66.8%

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

   10. frac-times 66.7%

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

   11. associate-*r/ 66.7%

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

   12. associate-*r/ 66.7%

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

   13. sqrt-unprod 41.2%

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

   14. add-sqr-sqrt 78.0%

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

   15. *-commutative 78.0%

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

   16. associate-/r/ 77.9%

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

   17. add-cube-cbrt 78.0%

       \[\leadsto {\left(a \cdot \cos \left(\frac{\color{blue}{\left(\sqrt[3]{\pi} \cdot \sqrt[3]{\pi}\right) \cdot \sqrt[3]{\pi}}}{\frac{-180}{angle}}\right)\right)}^{2} + {\left(b \cdot \sin \left(\pi \cdot \frac{angle}{180}\right)\right)}^{2} \]

   18. associate-/l* 78.0%

       \[\leadsto {\left(a \cdot \cos \color{blue}{\left(\frac{\sqrt[3]{\pi} \cdot \sqrt[3]{\pi}}{\frac{\frac{-180}{angle}}{\sqrt[3]{\pi}}}\right)}\right)}^{2} + {\left(b \cdot \sin \left(\pi \cdot \frac{angle}{180}\right)\right)}^{2} \]

4. Applied egg-rr 78.0%

   \[\leadsto {\left(a \cdot \cos \color{blue}{\left(\frac{{\left(\sqrt[3]{\pi}\right)}^{2}}{\frac{\frac{-180}{angle}}{\sqrt[3]{\pi}}}\right)}\right)}^{2} + {\left(b \cdot \sin \left(\pi \cdot \frac{angle}{180}\right)\right)}^{2} \]

5. Final simplification 78.0%

   \[\leadsto {\left(b \cdot \sin \left(\pi \cdot \frac{angle}{180}\right)\right)}^{2} + {\left(a \cdot \cos \left(\frac{{\left(\sqrt[3]{\pi}\right)}^{2}}{\frac{\frac{-180}{angle}}{\sqrt[3]{\pi}}}\right)\right)}^{2} \]
6. Add Preprocessing

Alternative 8: 64.6% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ {\left(b \cdot \sin \left(\pi \cdot \frac{angle}{180}\right)\right)}^{2} + {\left(a \cdot \cos \left(\mathsf{expm1}\left(\mathsf{log1p}\left(\left(\pi \cdot angle\right) \cdot \sqrt[3]{1.7146776406035666 \cdot 10^{-7}}\right)\right)\right)\right)}^{2} \end{array} \]

FPCore

(FPCore (a b angle)
 :precision binary64
 (+
  (pow (* b (sin (* PI (/ angle 180.0)))) 2.0)
  (pow
   (* a (cos (expm1 (log1p (* (* PI angle) (cbrt 1.7146776406035666e-7))))))
   2.0)))

C

double code(double a, double b, double angle) {
	return pow((b * sin((((double) M_PI) * (angle / 180.0)))), 2.0) + pow((a * cos(expm1(log1p(((((double) M_PI) * angle) * cbrt(1.7146776406035666e-7)))))), 2.0);
}

Java

public static double code(double a, double b, double angle) {
	return Math.pow((b * Math.sin((Math.PI * (angle / 180.0)))), 2.0) + Math.pow((a * Math.cos(Math.expm1(Math.log1p(((Math.PI * angle) * Math.cbrt(1.7146776406035666e-7)))))), 2.0);
}

Julia

function code(a, b, angle)
	return Float64((Float64(b * sin(Float64(pi * Float64(angle / 180.0)))) ^ 2.0) + (Float64(a * cos(expm1(log1p(Float64(Float64(pi * angle) * cbrt(1.7146776406035666e-7)))))) ^ 2.0))
end

Wolfram

code[a_, b_, angle_] := N[(N[Power[N[(b * N[Sin[N[(Pi * N[(angle / 180.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision] + N[Power[N[(a * N[Cos[N[(Exp[N[Log[1 + N[(N[(Pi * angle), $MachinePrecision] * N[Power[1.7146776406035666e-7, 1/3], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]] - 1), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 78.0%

   \[{\left(a \cdot \cos \left(\pi \cdot \frac{angle}{180}\right)\right)}^{2} + {\left(b \cdot \sin \left(\pi \cdot \frac{angle}{180}\right)\right)}^{2} \]
2. Add Preprocessing
3. Step-by-step derivation

   1. add-cbrt-cube 78.0%

      \[\leadsto {\left(a \cdot \cos \left(\color{blue}{\sqrt[3]{\left(\pi \cdot \pi\right) \cdot \pi}} \cdot \frac{angle}{180}\right)\right)}^{2} + {\left(b \cdot \sin \left(\pi \cdot \frac{angle}{180}\right)\right)}^{2} \]

   2. add-cbrt-cube 62.8%

      \[\leadsto {\left(a \cdot \cos \left(\sqrt[3]{\left(\pi \cdot \pi\right) \cdot \pi} \cdot \color{blue}{\sqrt[3]{\left(\frac{angle}{180} \cdot \frac{angle}{180}\right) \cdot \frac{angle}{180}}}\right)\right)}^{2} + {\left(b \cdot \sin \left(\pi \cdot \frac{angle}{180}\right)\right)}^{2} \]

   3. cbrt-unprod 62.9%

      \[\leadsto {\left(a \cdot \cos \color{blue}{\left(\sqrt[3]{\left(\left(\pi \cdot \pi\right) \cdot \pi\right) \cdot \left(\left(\frac{angle}{180} \cdot \frac{angle}{180}\right) \cdot \frac{angle}{180}\right)}\right)}\right)}^{2} + {\left(b \cdot \sin \left(\pi \cdot \frac{angle}{180}\right)\right)}^{2} \]

   4. pow3 62.9%

      \[\leadsto {\left(a \cdot \cos \left(\sqrt[3]{\color{blue}{{\pi}^{3}} \cdot \left(\left(\frac{angle}{180} \cdot \frac{angle}{180}\right) \cdot \frac{angle}{180}\right)}\right)\right)}^{2} + {\left(b \cdot \sin \left(\pi \cdot \frac{angle}{180}\right)\right)}^{2} \]

   5. pow3 62.9%

      \[\leadsto {\left(a \cdot \cos \left(\sqrt[3]{{\pi}^{3} \cdot \color{blue}{{\left(\frac{angle}{180}\right)}^{3}}}\right)\right)}^{2} + {\left(b \cdot \sin \left(\pi \cdot \frac{angle}{180}\right)\right)}^{2} \]

   6. div-inv 62.8%

      \[\leadsto {\left(a \cdot \cos \left(\sqrt[3]{{\pi}^{3} \cdot {\color{blue}{\left(angle \cdot \frac{1}{180}\right)}}^{3}}\right)\right)}^{2} + {\left(b \cdot \sin \left(\pi \cdot \frac{angle}{180}\right)\right)}^{2} \]

   7. metadata-eval 62.8%

      \[\leadsto {\left(a \cdot \cos \left(\sqrt[3]{{\pi}^{3} \cdot {\left(angle \cdot \color{blue}{0.005555555555555556}\right)}^{3}}\right)\right)}^{2} + {\left(b \cdot \sin \left(\pi \cdot \frac{angle}{180}\right)\right)}^{2} \]

4. Applied egg-rr 62.8%

   \[\leadsto {\left(a \cdot \cos \color{blue}{\left(\sqrt[3]{{\pi}^{3} \cdot {\left(angle \cdot 0.005555555555555556\right)}^{3}}\right)}\right)}^{2} + {\left(b \cdot \sin \left(\pi \cdot \frac{angle}{180}\right)\right)}^{2} \]
5. Step-by-step derivation

   1. cube-prod 62.8%

      \[\leadsto {\left(a \cdot \cos \left(\sqrt[3]{{\pi}^{3} \cdot \color{blue}{\left({angle}^{3} \cdot {0.005555555555555556}^{3}\right)}}\right)\right)}^{2} + {\left(b \cdot \sin \left(\pi \cdot \frac{angle}{180}\right)\right)}^{2} \]

   2. associate-*r* 62.8%

      \[\leadsto {\left(a \cdot \cos \left(\sqrt[3]{\color{blue}{\left({\pi}^{3} \cdot {angle}^{3}\right) \cdot {0.005555555555555556}^{3}}}\right)\right)}^{2} + {\left(b \cdot \sin \left(\pi \cdot \frac{angle}{180}\right)\right)}^{2} \]

   3. *-commutative 62.8%

      \[\leadsto {\left(a \cdot \cos \left(\sqrt[3]{\color{blue}{\left({angle}^{3} \cdot {\pi}^{3}\right)} \cdot {0.005555555555555556}^{3}}\right)\right)}^{2} + {\left(b \cdot \sin \left(\pi \cdot \frac{angle}{180}\right)\right)}^{2} \]

   4. cube-prod 62.8%

      \[\leadsto {\left(a \cdot \cos \left(\sqrt[3]{\color{blue}{{\left(angle \cdot \pi\right)}^{3}} \cdot {0.005555555555555556}^{3}}\right)\right)}^{2} + {\left(b \cdot \sin \left(\pi \cdot \frac{angle}{180}\right)\right)}^{2} \]

   5. metadata-eval 62.9%

      \[\leadsto {\left(a \cdot \cos \left(\sqrt[3]{{\left(angle \cdot \pi\right)}^{3} \cdot \color{blue}{1.7146776406035666 \cdot 10^{-7}}}\right)\right)}^{2} + {\left(b \cdot \sin \left(\pi \cdot \frac{angle}{180}\right)\right)}^{2} \]

6. Simplified 62.9%

   \[\leadsto {\left(a \cdot \cos \color{blue}{\left(\sqrt[3]{{\left(angle \cdot \pi\right)}^{3} \cdot 1.7146776406035666 \cdot 10^{-7}}\right)}\right)}^{2} + {\left(b \cdot \sin \left(\pi \cdot \frac{angle}{180}\right)\right)}^{2} \]
7. Step-by-step derivation

   1. expm1-log1p-u 58.6%

      \[\leadsto {\left(a \cdot \cos \color{blue}{\left(\mathsf{expm1}\left(\mathsf{log1p}\left(\sqrt[3]{{\left(angle \cdot \pi\right)}^{3} \cdot 1.7146776406035666 \cdot 10^{-7}}\right)\right)\right)}\right)}^{2} + {\left(b \cdot \sin \left(\pi \cdot \frac{angle}{180}\right)\right)}^{2} \]

   2. cbrt-prod 58.6%

      \[\leadsto {\left(a \cdot \cos \left(\mathsf{expm1}\left(\mathsf{log1p}\left(\color{blue}{\sqrt[3]{{\left(angle \cdot \pi\right)}^{3}} \cdot \sqrt[3]{1.7146776406035666 \cdot 10^{-7}}}\right)\right)\right)\right)}^{2} + {\left(b \cdot \sin \left(\pi \cdot \frac{angle}{180}\right)\right)}^{2} \]

   3. rem-cbrt-cube 64.9%

      \[\leadsto {\left(a \cdot \cos \left(\mathsf{expm1}\left(\mathsf{log1p}\left(\color{blue}{\left(angle \cdot \pi\right)} \cdot \sqrt[3]{1.7146776406035666 \cdot 10^{-7}}\right)\right)\right)\right)}^{2} + {\left(b \cdot \sin \left(\pi \cdot \frac{angle}{180}\right)\right)}^{2} \]

   4. *-commutative 64.9%

      \[\leadsto {\left(a \cdot \cos \left(\mathsf{expm1}\left(\mathsf{log1p}\left(\color{blue}{\left(\pi \cdot angle\right)} \cdot \sqrt[3]{1.7146776406035666 \cdot 10^{-7}}\right)\right)\right)\right)}^{2} + {\left(b \cdot \sin \left(\pi \cdot \frac{angle}{180}\right)\right)}^{2} \]

8. Applied egg-rr 64.9%

   \[\leadsto {\left(a \cdot \cos \color{blue}{\left(\mathsf{expm1}\left(\mathsf{log1p}\left(\left(\pi \cdot angle\right) \cdot \sqrt[3]{1.7146776406035666 \cdot 10^{-7}}\right)\right)\right)}\right)}^{2} + {\left(b \cdot \sin \left(\pi \cdot \frac{angle}{180}\right)\right)}^{2} \]

9. Final simplification 64.9%

   \[\leadsto {\left(b \cdot \sin \left(\pi \cdot \frac{angle}{180}\right)\right)}^{2} + {\left(a \cdot \cos \left(\mathsf{expm1}\left(\mathsf{log1p}\left(\left(\pi \cdot angle\right) \cdot \sqrt[3]{1.7146776406035666 \cdot 10^{-7}}\right)\right)\right)\right)}^{2} \]
10. Add Preprocessing

Alternative 9: 79.5% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := angle \cdot \frac{\pi}{-180}\\ {\left(a \cdot \cos t\_0\right)}^{2} + {\left(b \cdot \sin t\_0\right)}^{2} \end{array} \end{array} \]

FPCore

(FPCore (a b angle)
 :precision binary64
 (let* ((t_0 (* angle (/ PI -180.0))))
   (+ (pow (* a (cos t_0)) 2.0) (pow (* b (sin t_0)) 2.0))))

C

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

Java

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

Python

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

Julia

function code(a, b, angle)
	t_0 = Float64(angle * Float64(pi / -180.0))
	return Float64((Float64(a * cos(t_0)) ^ 2.0) + (Float64(b * sin(t_0)) ^ 2.0))
end

MATLAB

function tmp = code(a, b, angle)
	t_0 = angle * (pi / -180.0);
	tmp = ((a * cos(t_0)) ^ 2.0) + ((b * sin(t_0)) ^ 2.0);
end

Wolfram

code[a_, b_, angle_] := Block[{t$95$0 = N[(angle * N[(Pi / -180.0), $MachinePrecision]), $MachinePrecision]}, N[(N[Power[N[(a * N[Cos[t$95$0], $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision] + N[Power[N[(b * N[Sin[t$95$0], $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Initial program 78.0%

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

3. Simplified 77.9%

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

5. Final simplification 77.9%

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

Alternative 10: 79.5% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ {\left(a \cdot \cos \left(angle \cdot \frac{\pi}{-180}\right)\right)}^{2} + {\left(b \cdot \sin \left(\pi \cdot \left(angle \cdot -0.005555555555555556\right)\right)\right)}^{2} \end{array} \]

FPCore

(FPCore (a b angle)
 :precision binary64
 (+
  (pow (* a (cos (* angle (/ PI -180.0)))) 2.0)
  (pow (* b (sin (* PI (* angle -0.005555555555555556)))) 2.0)))

C

double code(double a, double b, double angle) {
	return pow((a * cos((angle * (((double) M_PI) / -180.0)))), 2.0) + pow((b * sin((((double) M_PI) * (angle * -0.005555555555555556)))), 2.0);
}

Java

public static double code(double a, double b, double angle) {
	return Math.pow((a * Math.cos((angle * (Math.PI / -180.0)))), 2.0) + Math.pow((b * Math.sin((Math.PI * (angle * -0.005555555555555556)))), 2.0);
}

Python

def code(a, b, angle):
	return math.pow((a * math.cos((angle * (math.pi / -180.0)))), 2.0) + math.pow((b * math.sin((math.pi * (angle * -0.005555555555555556)))), 2.0)

Julia

function code(a, b, angle)
	return Float64((Float64(a * cos(Float64(angle * Float64(pi / -180.0)))) ^ 2.0) + (Float64(b * sin(Float64(pi * Float64(angle * -0.005555555555555556)))) ^ 2.0))
end

MATLAB

function tmp = code(a, b, angle)
	tmp = ((a * cos((angle * (pi / -180.0)))) ^ 2.0) + ((b * sin((pi * (angle * -0.005555555555555556)))) ^ 2.0);
end

Wolfram

code[a_, b_, angle_] := N[(N[Power[N[(a * N[Cos[N[(angle * N[(Pi / -180.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision] + N[Power[N[(b * N[Sin[N[(Pi * N[(angle * -0.005555555555555556), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 78.0%

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

3. Simplified 77.9%

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

5. Taylor expanded in b around 0 64.4%

   \[\leadsto {\left(a \cdot \cos \left(angle \cdot \frac{\pi}{-180}\right)\right)}^{2} + \color{blue}{{b}^{2} \cdot {\sin \left(-0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{2}} \]
6. Step-by-step derivation

   1. *-commutative 64.4%

      \[\leadsto {\left(a \cdot \cos \left(angle \cdot \frac{\pi}{-180}\right)\right)}^{2} + \color{blue}{{\sin \left(-0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{2} \cdot {b}^{2}} \]

   2. pow-prod-down 77.9%

      \[\leadsto {\left(a \cdot \cos \left(angle \cdot \frac{\pi}{-180}\right)\right)}^{2} + \color{blue}{{\left(\sin \left(-0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right) \cdot b\right)}^{2}} \]

   3. *-commutative 77.9%

      \[\leadsto {\left(a \cdot \cos \left(angle \cdot \frac{\pi}{-180}\right)\right)}^{2} + {\left(\sin \color{blue}{\left(\left(angle \cdot \pi\right) \cdot -0.005555555555555556\right)} \cdot b\right)}^{2} \]

   4. *-commutative 77.9%

      \[\leadsto {\left(a \cdot \cos \left(angle \cdot \frac{\pi}{-180}\right)\right)}^{2} + {\left(\sin \left(\color{blue}{\left(\pi \cdot angle\right)} \cdot -0.005555555555555556\right) \cdot b\right)}^{2} \]

   5. associate-*l* 78.0%

      \[\leadsto {\left(a \cdot \cos \left(angle \cdot \frac{\pi}{-180}\right)\right)}^{2} + {\left(\sin \color{blue}{\left(\pi \cdot \left(angle \cdot -0.005555555555555556\right)\right)} \cdot b\right)}^{2} \]

7. Applied egg-rr 78.0%

   \[\leadsto {\left(a \cdot \cos \left(angle \cdot \frac{\pi}{-180}\right)\right)}^{2} + \color{blue}{{\left(\sin \left(\pi \cdot \left(angle \cdot -0.005555555555555556\right)\right) \cdot b\right)}^{2}} \]

8. Final simplification 78.0%

   \[\leadsto {\left(a \cdot \cos \left(angle \cdot \frac{\pi}{-180}\right)\right)}^{2} + {\left(b \cdot \sin \left(\pi \cdot \left(angle \cdot -0.005555555555555556\right)\right)\right)}^{2} \]
9. Add Preprocessing

Alternative 11: 79.6% accurate, 1.3× speedup

Math

\[\begin{array}{l} \\ {a}^{2} + {\left(b \cdot \sin \left(\frac{\pi \cdot -0.005555555555555556}{\frac{1}{angle}}\right)\right)}^{2} \end{array} \]

FPCore

(FPCore (a b angle)
 :precision binary64
 (+
  (pow a 2.0)
  (pow (* b (sin (/ (* PI -0.005555555555555556) (/ 1.0 angle)))) 2.0)))

C

double code(double a, double b, double angle) {
	return pow(a, 2.0) + pow((b * sin(((((double) M_PI) * -0.005555555555555556) / (1.0 / angle)))), 2.0);
}

Java

public static double code(double a, double b, double angle) {
	return Math.pow(a, 2.0) + Math.pow((b * Math.sin(((Math.PI * -0.005555555555555556) / (1.0 / angle)))), 2.0);
}

Python

def code(a, b, angle):
	return math.pow(a, 2.0) + math.pow((b * math.sin(((math.pi * -0.005555555555555556) / (1.0 / angle)))), 2.0)

Julia

function code(a, b, angle)
	return Float64((a ^ 2.0) + (Float64(b * sin(Float64(Float64(pi * -0.005555555555555556) / Float64(1.0 / angle)))) ^ 2.0))
end

MATLAB

function tmp = code(a, b, angle)
	tmp = (a ^ 2.0) + ((b * sin(((pi * -0.005555555555555556) / (1.0 / angle)))) ^ 2.0);
end

Wolfram

code[a_, b_, angle_] := N[(N[Power[a, 2.0], $MachinePrecision] + N[Power[N[(b * N[Sin[N[(N[(Pi * -0.005555555555555556), $MachinePrecision] / N[(1.0 / angle), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 78.0%

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

3. Simplified 77.9%

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

5. Taylor expanded in angle around 0 77.7%

   \[\leadsto {\left(a \cdot \color{blue}{1}\right)}^{2} + {\left(b \cdot \sin \left(angle \cdot \frac{\pi}{-180}\right)\right)}^{2} \]
6. Step-by-step derivation

   1. *-commutative 77.7%

      \[\leadsto {\left(a \cdot 1\right)}^{2} + {\left(b \cdot \sin \color{blue}{\left(\frac{\pi}{-180} \cdot angle\right)}\right)}^{2} \]

   2. associate-/r/ 77.4%

      \[\leadsto {\left(a \cdot 1\right)}^{2} + {\left(b \cdot \sin \color{blue}{\left(\frac{\pi}{\frac{-180}{angle}}\right)}\right)}^{2} \]

   3. div-inv 77.4%

      \[\leadsto {\left(a \cdot 1\right)}^{2} + {\left(b \cdot \sin \left(\frac{\pi}{\color{blue}{-180 \cdot \frac{1}{angle}}}\right)\right)}^{2} \]

   4. associate-/r* 77.8%

      \[\leadsto {\left(a \cdot 1\right)}^{2} + {\left(b \cdot \sin \color{blue}{\left(\frac{\frac{\pi}{-180}}{\frac{1}{angle}}\right)}\right)}^{2} \]

   5. div-inv 77.8%

      \[\leadsto {\left(a \cdot 1\right)}^{2} + {\left(b \cdot \sin \left(\frac{\color{blue}{\pi \cdot \frac{1}{-180}}}{\frac{1}{angle}}\right)\right)}^{2} \]

   6. metadata-eval 77.8%

      \[\leadsto {\left(a \cdot 1\right)}^{2} + {\left(b \cdot \sin \left(\frac{\pi \cdot \color{blue}{-0.005555555555555556}}{\frac{1}{angle}}\right)\right)}^{2} \]

7. Applied egg-rr 77.8%

   \[\leadsto {\left(a \cdot 1\right)}^{2} + {\left(b \cdot \sin \color{blue}{\left(\frac{\pi \cdot -0.005555555555555556}{\frac{1}{angle}}\right)}\right)}^{2} \]

8. Final simplification 77.8%

   \[\leadsto {a}^{2} + {\left(b \cdot \sin \left(\frac{\pi \cdot -0.005555555555555556}{\frac{1}{angle}}\right)\right)}^{2} \]
9. Add Preprocessing

Alternative 12: 79.6% accurate, 1.3× speedup

Math

\[\begin{array}{l} \\ {\left(b \cdot \sin \left(angle \cdot \frac{\pi}{-180}\right)\right)}^{2} + {a}^{2} \end{array} \]

FPCore

(FPCore (a b angle)
 :precision binary64
 (+ (pow (* b (sin (* angle (/ PI -180.0)))) 2.0) (pow a 2.0)))

C

double code(double a, double b, double angle) {
	return pow((b * sin((angle * (((double) M_PI) / -180.0)))), 2.0) + pow(a, 2.0);
}

Java

public static double code(double a, double b, double angle) {
	return Math.pow((b * Math.sin((angle * (Math.PI / -180.0)))), 2.0) + Math.pow(a, 2.0);
}

Python

def code(a, b, angle):
	return math.pow((b * math.sin((angle * (math.pi / -180.0)))), 2.0) + math.pow(a, 2.0)

Julia

function code(a, b, angle)
	return Float64((Float64(b * sin(Float64(angle * Float64(pi / -180.0)))) ^ 2.0) + (a ^ 2.0))
end

MATLAB

function tmp = code(a, b, angle)
	tmp = ((b * sin((angle * (pi / -180.0)))) ^ 2.0) + (a ^ 2.0);
end

Wolfram

code[a_, b_, angle_] := N[(N[Power[N[(b * N[Sin[N[(angle * N[(Pi / -180.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision] + N[Power[a, 2.0], $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 78.0%

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

3. Simplified 77.9%

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

5. Taylor expanded in angle around 0 77.7%

   \[\leadsto {\left(a \cdot \color{blue}{1}\right)}^{2} + {\left(b \cdot \sin \left(angle \cdot \frac{\pi}{-180}\right)\right)}^{2} \]

6. Final simplification 77.7%

   \[\leadsto {\left(b \cdot \sin \left(angle \cdot \frac{\pi}{-180}\right)\right)}^{2} + {a}^{2} \]
7. Add Preprocessing

Alternative 13: 79.6% accurate, 1.3× speedup

Math

\[\begin{array}{l} \\ {\left(b \cdot \sin \left(\pi \cdot \left(angle \cdot -0.005555555555555556\right)\right)\right)}^{2} + {a}^{2} \end{array} \]

FPCore

(FPCore (a b angle)
 :precision binary64
 (+ (pow (* b (sin (* PI (* angle -0.005555555555555556)))) 2.0) (pow a 2.0)))

C

double code(double a, double b, double angle) {
	return pow((b * sin((((double) M_PI) * (angle * -0.005555555555555556)))), 2.0) + pow(a, 2.0);
}

Java

public static double code(double a, double b, double angle) {
	return Math.pow((b * Math.sin((Math.PI * (angle * -0.005555555555555556)))), 2.0) + Math.pow(a, 2.0);
}

Python

def code(a, b, angle):
	return math.pow((b * math.sin((math.pi * (angle * -0.005555555555555556)))), 2.0) + math.pow(a, 2.0)

Julia

function code(a, b, angle)
	return Float64((Float64(b * sin(Float64(pi * Float64(angle * -0.005555555555555556)))) ^ 2.0) + (a ^ 2.0))
end

MATLAB

function tmp = code(a, b, angle)
	tmp = ((b * sin((pi * (angle * -0.005555555555555556)))) ^ 2.0) + (a ^ 2.0);
end

Wolfram

code[a_, b_, angle_] := N[(N[Power[N[(b * N[Sin[N[(Pi * N[(angle * -0.005555555555555556), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision] + N[Power[a, 2.0], $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 78.0%

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

3. Simplified 77.9%

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

5. Taylor expanded in angle around 0 77.7%

   \[\leadsto {\left(a \cdot \color{blue}{1}\right)}^{2} + {\left(b \cdot \sin \left(angle \cdot \frac{\pi}{-180}\right)\right)}^{2} \]
6. Step-by-step derivation

   1. add-sqr-sqrt 43.3%

      \[\leadsto {\left(a \cdot 1\right)}^{2} + {\color{blue}{\left(\sqrt{b \cdot \sin \left(angle \cdot \frac{\pi}{-180}\right)} \cdot \sqrt{b \cdot \sin \left(angle \cdot \frac{\pi}{-180}\right)}\right)}}^{2} \]

   2. sqrt-prod 77.7%

      \[\leadsto {\left(a \cdot 1\right)}^{2} + {\color{blue}{\left(\sqrt{\left(b \cdot \sin \left(angle \cdot \frac{\pi}{-180}\right)\right) \cdot \left(b \cdot \sin \left(angle \cdot \frac{\pi}{-180}\right)\right)}\right)}}^{2} \]

   3. unpow2 77.7%

      \[\leadsto {\left(a \cdot 1\right)}^{2} + {\left(\sqrt{\color{blue}{{\left(b \cdot \sin \left(angle \cdot \frac{\pi}{-180}\right)\right)}^{2}}}\right)}^{2} \]

   4. add-exp-log 76.8%

      \[\leadsto {\left(a \cdot 1\right)}^{2} + {\color{blue}{\left(e^{\log \left(\sqrt{{\left(b \cdot \sin \left(angle \cdot \frac{\pi}{-180}\right)\right)}^{2}}\right)}\right)}}^{2} \]

   5. sqrt-pow1 42.9%

      \[\leadsto {\left(a \cdot 1\right)}^{2} + {\left(e^{\log \color{blue}{\left({\left(b \cdot \sin \left(angle \cdot \frac{\pi}{-180}\right)\right)}^{\left(\frac{2}{2}\right)}\right)}}\right)}^{2} \]

   6. *-commutative 42.9%

      \[\leadsto {\left(a \cdot 1\right)}^{2} + {\left(e^{\log \left({\color{blue}{\left(\sin \left(angle \cdot \frac{\pi}{-180}\right) \cdot b\right)}}^{\left(\frac{2}{2}\right)}\right)}\right)}^{2} \]

   7. metadata-eval 42.9%

      \[\leadsto {\left(a \cdot 1\right)}^{2} + {\left(e^{\log \left({\left(\sin \left(angle \cdot \frac{\pi}{-180}\right) \cdot b\right)}^{\color{blue}{1}}\right)}\right)}^{2} \]

   8. pow1 42.9%

      \[\leadsto {\left(a \cdot 1\right)}^{2} + {\left(e^{\log \color{blue}{\left(\sin \left(angle \cdot \frac{\pi}{-180}\right) \cdot b\right)}}\right)}^{2} \]

   9. div-inv 42.9%

      \[\leadsto {\left(a \cdot 1\right)}^{2} + {\left(e^{\log \left(\sin \left(angle \cdot \color{blue}{\left(\pi \cdot \frac{1}{-180}\right)}\right) \cdot b\right)}\right)}^{2} \]

   10. metadata-eval 42.9%

       \[\leadsto {\left(a \cdot 1\right)}^{2} + {\left(e^{\log \left(\sin \left(angle \cdot \left(\pi \cdot \color{blue}{-0.005555555555555556}\right)\right) \cdot b\right)}\right)}^{2} \]

7. Applied egg-rr 42.9%

   \[\leadsto {\left(a \cdot 1\right)}^{2} + {\color{blue}{\left(e^{\log \left(\sin \left(angle \cdot \left(\pi \cdot -0.005555555555555556\right)\right) \cdot b\right)}\right)}}^{2} \]

8. Taylor expanded in angle around inf 77.7%

   \[\leadsto {\left(a \cdot 1\right)}^{2} + {\color{blue}{\left(b \cdot \sin \left(-0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)\right)}}^{2} \]
9. Step-by-step derivation

   1. associate-*r* 77.8%

      \[\leadsto {\left(a \cdot 1\right)}^{2} + {\left(b \cdot \sin \color{blue}{\left(\left(-0.005555555555555556 \cdot angle\right) \cdot \pi\right)}\right)}^{2} \]

   2. *-commutative 77.8%

      \[\leadsto {\left(a \cdot 1\right)}^{2} + {\left(b \cdot \sin \left(\color{blue}{\left(angle \cdot -0.005555555555555556\right)} \cdot \pi\right)\right)}^{2} \]

   3. *-commutative 77.8%

      \[\leadsto {\left(a \cdot 1\right)}^{2} + {\left(b \cdot \sin \color{blue}{\left(\pi \cdot \left(angle \cdot -0.005555555555555556\right)\right)}\right)}^{2} \]

10. Simplified 77.8%

    \[\leadsto {\left(a \cdot 1\right)}^{2} + {\color{blue}{\left(b \cdot \sin \left(\pi \cdot \left(angle \cdot -0.005555555555555556\right)\right)\right)}}^{2} \]

11. Final simplification 77.8%

    \[\leadsto {\left(b \cdot \sin \left(\pi \cdot \left(angle \cdot -0.005555555555555556\right)\right)\right)}^{2} + {a}^{2} \]
12. Add Preprocessing

Alternative 14: 74.7% accurate, 3.5× speedup

Math

\[\begin{array}{l} \\ {a}^{2} + \left(\left(angle \cdot -0.005555555555555556\right) \cdot \left(\pi \cdot b\right)\right) \cdot \left(-0.005555555555555556 \cdot \left(angle \cdot \left(\pi \cdot b\right)\right)\right) \end{array} \]

FPCore

(FPCore (a b angle)
 :precision binary64
 (+
  (pow a 2.0)
  (*
   (* (* angle -0.005555555555555556) (* PI b))
   (* -0.005555555555555556 (* angle (* PI b))))))

C

double code(double a, double b, double angle) {
	return pow(a, 2.0) + (((angle * -0.005555555555555556) * (((double) M_PI) * b)) * (-0.005555555555555556 * (angle * (((double) M_PI) * b))));
}

Java

public static double code(double a, double b, double angle) {
	return Math.pow(a, 2.0) + (((angle * -0.005555555555555556) * (Math.PI * b)) * (-0.005555555555555556 * (angle * (Math.PI * b))));
}

Python

def code(a, b, angle):
	return math.pow(a, 2.0) + (((angle * -0.005555555555555556) * (math.pi * b)) * (-0.005555555555555556 * (angle * (math.pi * b))))

Julia

function code(a, b, angle)
	return Float64((a ^ 2.0) + Float64(Float64(Float64(angle * -0.005555555555555556) * Float64(pi * b)) * Float64(-0.005555555555555556 * Float64(angle * Float64(pi * b)))))
end

MATLAB

function tmp = code(a, b, angle)
	tmp = (a ^ 2.0) + (((angle * -0.005555555555555556) * (pi * b)) * (-0.005555555555555556 * (angle * (pi * b))));
end

Wolfram

code[a_, b_, angle_] := N[(N[Power[a, 2.0], $MachinePrecision] + N[(N[(N[(angle * -0.005555555555555556), $MachinePrecision] * N[(Pi * b), $MachinePrecision]), $MachinePrecision] * N[(-0.005555555555555556 * N[(angle * N[(Pi * b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 78.0%

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

3. Simplified 77.9%

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

5. Taylor expanded in angle around 0 77.7%

   \[\leadsto {\left(a \cdot \color{blue}{1}\right)}^{2} + {\left(b \cdot \sin \left(angle \cdot \frac{\pi}{-180}\right)\right)}^{2} \]

6. Taylor expanded in angle around 0 73.4%

   \[\leadsto {\left(a \cdot 1\right)}^{2} + {\color{blue}{\left(-0.005555555555555556 \cdot \left(angle \cdot \left(b \cdot \pi\right)\right)\right)}}^{2} \]
7. Step-by-step derivation

   1. associate-*r* 73.4%

      \[\leadsto {\left(a \cdot 1\right)}^{2} + {\color{blue}{\left(\left(-0.005555555555555556 \cdot angle\right) \cdot \left(b \cdot \pi\right)\right)}}^{2} \]

   2. *-commutative 73.4%

      \[\leadsto {\left(a \cdot 1\right)}^{2} + {\left(\color{blue}{\left(angle \cdot -0.005555555555555556\right)} \cdot \left(b \cdot \pi\right)\right)}^{2} \]

   3. associate-*l* 73.4%

      \[\leadsto {\left(a \cdot 1\right)}^{2} + {\color{blue}{\left(angle \cdot \left(-0.005555555555555556 \cdot \left(b \cdot \pi\right)\right)\right)}}^{2} \]

   4. *-commutative 73.4%

      \[\leadsto {\left(a \cdot 1\right)}^{2} + {\left(angle \cdot \left(-0.005555555555555556 \cdot \color{blue}{\left(\pi \cdot b\right)}\right)\right)}^{2} \]

8. Simplified 73.4%

   \[\leadsto {\left(a \cdot 1\right)}^{2} + {\color{blue}{\left(angle \cdot \left(-0.005555555555555556 \cdot \left(\pi \cdot b\right)\right)\right)}}^{2} \]
9. Step-by-step derivation

   1. unpow2 73.4%

      \[\leadsto {\left(a \cdot 1\right)}^{2} + \color{blue}{\left(angle \cdot \left(-0.005555555555555556 \cdot \left(\pi \cdot b\right)\right)\right) \cdot \left(angle \cdot \left(-0.005555555555555556 \cdot \left(\pi \cdot b\right)\right)\right)} \]

   2. associate-*r* 73.4%

      \[\leadsto {\left(a \cdot 1\right)}^{2} + \color{blue}{\left(\left(angle \cdot -0.005555555555555556\right) \cdot \left(\pi \cdot b\right)\right)} \cdot \left(angle \cdot \left(-0.005555555555555556 \cdot \left(\pi \cdot b\right)\right)\right) \]

   3. associate-*r* 73.4%

      \[\leadsto {\left(a \cdot 1\right)}^{2} + \left(\left(angle \cdot -0.005555555555555556\right) \cdot \left(\pi \cdot b\right)\right) \cdot \color{blue}{\left(\left(angle \cdot -0.005555555555555556\right) \cdot \left(\pi \cdot b\right)\right)} \]

10. Applied egg-rr 73.4%

    \[\leadsto {\left(a \cdot 1\right)}^{2} + \color{blue}{\left(\left(angle \cdot -0.005555555555555556\right) \cdot \left(\pi \cdot b\right)\right) \cdot \left(\left(angle \cdot -0.005555555555555556\right) \cdot \left(\pi \cdot b\right)\right)} \]

11. Taylor expanded in angle around 0 73.4%

    \[\leadsto {\left(a \cdot 1\right)}^{2} + \left(\left(angle \cdot -0.005555555555555556\right) \cdot \left(\pi \cdot b\right)\right) \cdot \color{blue}{\left(-0.005555555555555556 \cdot \left(angle \cdot \left(b \cdot \pi\right)\right)\right)} \]

12. Final simplification 73.4%

    \[\leadsto {a}^{2} + \left(\left(angle \cdot -0.005555555555555556\right) \cdot \left(\pi \cdot b\right)\right) \cdot \left(-0.005555555555555556 \cdot \left(angle \cdot \left(\pi \cdot b\right)\right)\right) \]
13. Add Preprocessing

Reproduce

herbie shell --seed 2024026 
(FPCore (a b angle)
  :name "ab-angle->ABCF C"
  :precision binary64
  (+ (pow (* a (cos (* PI (/ angle 180.0)))) 2.0) (pow (* b (sin (* PI (/ angle 180.0)))) 2.0)))