ab-angle->ABCF C

Percentage Accurate: 79.1% → 79.1%

Time: 44.2s

Alternatives: 13

Speedup: 1.3×

• 36.0% of points produce a very large (infinite) output. You may want to add a precondition.

Specification

Math

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

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: 79.1% 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: 79.1% accurate, 0.5× speedup

Math

\[\begin{array}{l} angle_m = \left|angle\right| \\ {\left(a \cdot \cos \left(\left({angle\_m}^{0.16666666666666666} \cdot \sqrt{angle\_m}\right) \cdot \left(\sqrt[3]{angle\_m} \cdot \left(\sqrt[3]{{\pi}^{3}} \cdot -0.005555555555555556\right)\right)\right)\right)}^{2} + {\left(b \cdot \sin \left(\frac{\pi}{\frac{-180}{angle\_m}}\right)\right)}^{2} \end{array} \]

FPCore

angle_m = (fabs.f64 angle)
(FPCore (a b angle_m)
 :precision binary64
 (+
  (pow
   (*
    a
    (cos
     (*
      (* (pow angle_m 0.16666666666666666) (sqrt angle_m))
      (* (cbrt angle_m) (* (cbrt (pow PI 3.0)) -0.005555555555555556)))))
   2.0)
  (pow (* b (sin (/ PI (/ -180.0 angle_m)))) 2.0)))

C

angle_m = fabs(angle);
double code(double a, double b, double angle_m) {
	return pow((a * cos(((pow(angle_m, 0.16666666666666666) * sqrt(angle_m)) * (cbrt(angle_m) * (cbrt(pow(((double) M_PI), 3.0)) * -0.005555555555555556))))), 2.0) + pow((b * sin((((double) M_PI) / (-180.0 / angle_m)))), 2.0);
}

Java

angle_m = Math.abs(angle);
public static double code(double a, double b, double angle_m) {
	return Math.pow((a * Math.cos(((Math.pow(angle_m, 0.16666666666666666) * Math.sqrt(angle_m)) * (Math.cbrt(angle_m) * (Math.cbrt(Math.pow(Math.PI, 3.0)) * -0.005555555555555556))))), 2.0) + Math.pow((b * Math.sin((Math.PI / (-180.0 / angle_m)))), 2.0);
}

Julia

angle_m = abs(angle)
function code(a, b, angle_m)
	return Float64((Float64(a * cos(Float64(Float64((angle_m ^ 0.16666666666666666) * sqrt(angle_m)) * Float64(cbrt(angle_m) * Float64(cbrt((pi ^ 3.0)) * -0.005555555555555556))))) ^ 2.0) + (Float64(b * sin(Float64(pi / Float64(-180.0 / angle_m)))) ^ 2.0))
end

Wolfram

angle_m = N[Abs[angle], $MachinePrecision]
code[a_, b_, angle$95$m_] := N[(N[Power[N[(a * N[Cos[N[(N[(N[Power[angle$95$m, 0.16666666666666666], $MachinePrecision] * N[Sqrt[angle$95$m], $MachinePrecision]), $MachinePrecision] * N[(N[Power[angle$95$m, 1/3], $MachinePrecision] * N[(N[Power[N[Power[Pi, 3.0], $MachinePrecision], 1/3], $MachinePrecision] * -0.005555555555555556), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision] + N[Power[N[(b * N[Sin[N[(Pi / N[(-180.0 / angle$95$m), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 80.0%

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

3. Simplified 80.1%

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

   1. associate-/r/ 80.1%

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

   2. *-commutative 80.1%

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

   3. add-cube-cbrt 80.1%

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

   4. associate-*l* 80.1%

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

   5. pow2 80.1%

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

   6. div-inv 80.1%

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

   7. metadata-eval 80.1%

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

6. Applied egg-rr 80.1%

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

   1. unpow2 80.1%

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

   2. add-sqr-sqrt 37.6%

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

   3. associate-*l* 37.6%

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

   4. pow1/3 37.7%

      \[\leadsto {\left(a \cdot \cos \left(\left(\sqrt{\color{blue}{{angle}^{0.3333333333333333}}} \cdot \left(\sqrt{\sqrt[3]{angle}} \cdot \sqrt[3]{angle}\right)\right) \cdot \left(\sqrt[3]{angle} \cdot \left(\pi \cdot -0.005555555555555556\right)\right)\right)\right)}^{2} + {\left(b \cdot \sin \left(\frac{\pi}{\frac{-180}{angle}}\right)\right)}^{2} \]

   5. sqrt-pow1 37.7%

      \[\leadsto {\left(a \cdot \cos \left(\left(\color{blue}{{angle}^{\left(\frac{0.3333333333333333}{2}\right)}} \cdot \left(\sqrt{\sqrt[3]{angle}} \cdot \sqrt[3]{angle}\right)\right) \cdot \left(\sqrt[3]{angle} \cdot \left(\pi \cdot -0.005555555555555556\right)\right)\right)\right)}^{2} + {\left(b \cdot \sin \left(\frac{\pi}{\frac{-180}{angle}}\right)\right)}^{2} \]

   6. metadata-eval 37.7%

      \[\leadsto {\left(a \cdot \cos \left(\left({angle}^{\color{blue}{0.16666666666666666}} \cdot \left(\sqrt{\sqrt[3]{angle}} \cdot \sqrt[3]{angle}\right)\right) \cdot \left(\sqrt[3]{angle} \cdot \left(\pi \cdot -0.005555555555555556\right)\right)\right)\right)}^{2} + {\left(b \cdot \sin \left(\frac{\pi}{\frac{-180}{angle}}\right)\right)}^{2} \]

   7. add-cbrt-cube 37.7%

      \[\leadsto {\left(a \cdot \cos \left(\left({angle}^{0.16666666666666666} \cdot \left(\color{blue}{\sqrt[3]{\left(\sqrt{\sqrt[3]{angle}} \cdot \sqrt{\sqrt[3]{angle}}\right) \cdot \sqrt{\sqrt[3]{angle}}}} \cdot \sqrt[3]{angle}\right)\right) \cdot \left(\sqrt[3]{angle} \cdot \left(\pi \cdot -0.005555555555555556\right)\right)\right)\right)}^{2} + {\left(b \cdot \sin \left(\frac{\pi}{\frac{-180}{angle}}\right)\right)}^{2} \]

   8. sqrt-prod 37.7%

      \[\leadsto {\left(a \cdot \cos \left(\left({angle}^{0.16666666666666666} \cdot \left(\sqrt[3]{\color{blue}{\sqrt{\sqrt[3]{angle} \cdot \sqrt[3]{angle}}} \cdot \sqrt{\sqrt[3]{angle}}} \cdot \sqrt[3]{angle}\right)\right) \cdot \left(\sqrt[3]{angle} \cdot \left(\pi \cdot -0.005555555555555556\right)\right)\right)\right)}^{2} + {\left(b \cdot \sin \left(\frac{\pi}{\frac{-180}{angle}}\right)\right)}^{2} \]

   9. unpow2 37.7%

      \[\leadsto {\left(a \cdot \cos \left(\left({angle}^{0.16666666666666666} \cdot \left(\sqrt[3]{\sqrt{\color{blue}{{\left(\sqrt[3]{angle}\right)}^{2}}} \cdot \sqrt{\sqrt[3]{angle}}} \cdot \sqrt[3]{angle}\right)\right) \cdot \left(\sqrt[3]{angle} \cdot \left(\pi \cdot -0.005555555555555556\right)\right)\right)\right)}^{2} + {\left(b \cdot \sin \left(\frac{\pi}{\frac{-180}{angle}}\right)\right)}^{2} \]

   10. sqrt-prod 37.7%

       \[\leadsto {\left(a \cdot \cos \left(\left({angle}^{0.16666666666666666} \cdot \left(\sqrt[3]{\color{blue}{\sqrt{{\left(\sqrt[3]{angle}\right)}^{2} \cdot \sqrt[3]{angle}}}} \cdot \sqrt[3]{angle}\right)\right) \cdot \left(\sqrt[3]{angle} \cdot \left(\pi \cdot -0.005555555555555556\right)\right)\right)\right)}^{2} + {\left(b \cdot \sin \left(\frac{\pi}{\frac{-180}{angle}}\right)\right)}^{2} \]

   11. unpow2 37.7%

       \[\leadsto {\left(a \cdot \cos \left(\left({angle}^{0.16666666666666666} \cdot \left(\sqrt[3]{\sqrt{\color{blue}{\left(\sqrt[3]{angle} \cdot \sqrt[3]{angle}\right)} \cdot \sqrt[3]{angle}}} \cdot \sqrt[3]{angle}\right)\right) \cdot \left(\sqrt[3]{angle} \cdot \left(\pi \cdot -0.005555555555555556\right)\right)\right)\right)}^{2} + {\left(b \cdot \sin \left(\frac{\pi}{\frac{-180}{angle}}\right)\right)}^{2} \]

   12. add-cube-cbrt 37.7%

       \[\leadsto {\left(a \cdot \cos \left(\left({angle}^{0.16666666666666666} \cdot \left(\sqrt[3]{\sqrt{\color{blue}{angle}}} \cdot \sqrt[3]{angle}\right)\right) \cdot \left(\sqrt[3]{angle} \cdot \left(\pi \cdot -0.005555555555555556\right)\right)\right)\right)}^{2} + {\left(b \cdot \sin \left(\frac{\pi}{\frac{-180}{angle}}\right)\right)}^{2} \]

   13. add-sqr-sqrt 37.7%

       \[\leadsto {\left(a \cdot \cos \left(\left({angle}^{0.16666666666666666} \cdot \left(\sqrt[3]{\sqrt{angle}} \cdot \sqrt[3]{\color{blue}{\sqrt{angle} \cdot \sqrt{angle}}}\right)\right) \cdot \left(\sqrt[3]{angle} \cdot \left(\pi \cdot -0.005555555555555556\right)\right)\right)\right)}^{2} + {\left(b \cdot \sin \left(\frac{\pi}{\frac{-180}{angle}}\right)\right)}^{2} \]

   14. cbrt-prod 37.6%

       \[\leadsto {\left(a \cdot \cos \left(\left({angle}^{0.16666666666666666} \cdot \left(\sqrt[3]{\sqrt{angle}} \cdot \color{blue}{\left(\sqrt[3]{\sqrt{angle}} \cdot \sqrt[3]{\sqrt{angle}}\right)}\right)\right) \cdot \left(\sqrt[3]{angle} \cdot \left(\pi \cdot -0.005555555555555556\right)\right)\right)\right)}^{2} + {\left(b \cdot \sin \left(\frac{\pi}{\frac{-180}{angle}}\right)\right)}^{2} \]

   15. associate-*l* 37.6%

       \[\leadsto {\left(a \cdot \cos \left(\left({angle}^{0.16666666666666666} \cdot \color{blue}{\left(\left(\sqrt[3]{\sqrt{angle}} \cdot \sqrt[3]{\sqrt{angle}}\right) \cdot \sqrt[3]{\sqrt{angle}}\right)}\right) \cdot \left(\sqrt[3]{angle} \cdot \left(\pi \cdot -0.005555555555555556\right)\right)\right)\right)}^{2} + {\left(b \cdot \sin \left(\frac{\pi}{\frac{-180}{angle}}\right)\right)}^{2} \]

   16. add-cube-cbrt 37.7%

       \[\leadsto {\left(a \cdot \cos \left(\left({angle}^{0.16666666666666666} \cdot \color{blue}{\sqrt{angle}}\right) \cdot \left(\sqrt[3]{angle} \cdot \left(\pi \cdot -0.005555555555555556\right)\right)\right)\right)}^{2} + {\left(b \cdot \sin \left(\frac{\pi}{\frac{-180}{angle}}\right)\right)}^{2} \]

8. Applied egg-rr 37.7%

   \[\leadsto {\left(a \cdot \cos \left(\color{blue}{\left({angle}^{0.16666666666666666} \cdot \sqrt{angle}\right)} \cdot \left(\sqrt[3]{angle} \cdot \left(\pi \cdot -0.005555555555555556\right)\right)\right)\right)}^{2} + {\left(b \cdot \sin \left(\frac{\pi}{\frac{-180}{angle}}\right)\right)}^{2} \]
9. Step-by-step derivation

   1. add-cbrt-cube 37.7%

      \[\leadsto {\left(a \cdot \cos \left(\left({angle}^{0.16666666666666666} \cdot \sqrt{angle}\right) \cdot \left(\sqrt[3]{angle} \cdot \left(\color{blue}{\sqrt[3]{\left(\pi \cdot \pi\right) \cdot \pi}} \cdot -0.005555555555555556\right)\right)\right)\right)}^{2} + {\left(b \cdot \sin \left(\frac{\pi}{\frac{-180}{angle}}\right)\right)}^{2} \]

   2. pow3 37.7%

      \[\leadsto {\left(a \cdot \cos \left(\left({angle}^{0.16666666666666666} \cdot \sqrt{angle}\right) \cdot \left(\sqrt[3]{angle} \cdot \left(\sqrt[3]{\color{blue}{{\pi}^{3}}} \cdot -0.005555555555555556\right)\right)\right)\right)}^{2} + {\left(b \cdot \sin \left(\frac{\pi}{\frac{-180}{angle}}\right)\right)}^{2} \]

10. Applied egg-rr 37.7%

    \[\leadsto {\left(a \cdot \cos \left(\left({angle}^{0.16666666666666666} \cdot \sqrt{angle}\right) \cdot \left(\sqrt[3]{angle} \cdot \left(\color{blue}{\sqrt[3]{{\pi}^{3}}} \cdot -0.005555555555555556\right)\right)\right)\right)}^{2} + {\left(b \cdot \sin \left(\frac{\pi}{\frac{-180}{angle}}\right)\right)}^{2} \]

11. Final simplification 37.7%

    \[\leadsto {\left(a \cdot \cos \left(\left({angle}^{0.16666666666666666} \cdot \sqrt{angle}\right) \cdot \left(\sqrt[3]{angle} \cdot \left(\sqrt[3]{{\pi}^{3}} \cdot -0.005555555555555556\right)\right)\right)\right)}^{2} + {\left(b \cdot \sin \left(\frac{\pi}{\frac{-180}{angle}}\right)\right)}^{2} \]
12. Add Preprocessing

Alternative 2: 79.1% accurate, 0.5× speedup

Math

\[\begin{array}{l} angle_m = \left|angle\right| \\ {\left(b \cdot \sin \left(\frac{\pi}{\frac{-180}{angle\_m}}\right)\right)}^{2} + {\left(a \cdot \cos \left({e}^{\left(2 \cdot \log \left(\sqrt[3]{angle\_m}\right)\right)} \cdot \left(\sqrt[3]{angle\_m} \cdot \left(\pi \cdot -0.005555555555555556\right)\right)\right)\right)}^{2} \end{array} \]

FPCore

angle_m = (fabs.f64 angle)
(FPCore (a b angle_m)
 :precision binary64
 (+
  (pow (* b (sin (/ PI (/ -180.0 angle_m)))) 2.0)
  (pow
   (*
    a
    (cos
     (*
      (pow E (* 2.0 (log (cbrt angle_m))))
      (* (cbrt angle_m) (* PI -0.005555555555555556)))))
   2.0)))

C

angle_m = fabs(angle);
double code(double a, double b, double angle_m) {
	return pow((b * sin((((double) M_PI) / (-180.0 / angle_m)))), 2.0) + pow((a * cos((pow(((double) M_E), (2.0 * log(cbrt(angle_m)))) * (cbrt(angle_m) * (((double) M_PI) * -0.005555555555555556))))), 2.0);
}

Java

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

Julia

angle_m = abs(angle)
function code(a, b, angle_m)
	return Float64((Float64(b * sin(Float64(pi / Float64(-180.0 / angle_m)))) ^ 2.0) + (Float64(a * cos(Float64((exp(1) ^ Float64(2.0 * log(cbrt(angle_m)))) * Float64(cbrt(angle_m) * Float64(pi * -0.005555555555555556))))) ^ 2.0))
end

Wolfram

angle_m = N[Abs[angle], $MachinePrecision]
code[a_, b_, angle$95$m_] := N[(N[Power[N[(b * N[Sin[N[(Pi / N[(-180.0 / angle$95$m), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision] + N[Power[N[(a * N[Cos[N[(N[Power[E, N[(2.0 * N[Log[N[Power[angle$95$m, 1/3], $MachinePrecision]], $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * N[(N[Power[angle$95$m, 1/3], $MachinePrecision] * N[(Pi * -0.005555555555555556), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 80.0%

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

3. Simplified 80.1%

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

   1. associate-/r/ 80.1%

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

   2. *-commutative 80.1%

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

   3. add-cube-cbrt 80.1%

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

   4. associate-*l* 80.1%

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

   5. pow2 80.1%

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

   6. div-inv 80.1%

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

   7. metadata-eval 80.1%

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

6. Applied egg-rr 80.1%

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

   1. add-exp-log 80.1%

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

   2. log-pow 37.6%

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

8. Applied egg-rr 37.6%

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

   1. *-un-lft-identity 37.6%

      \[\leadsto {\left(a \cdot \cos \left(e^{\color{blue}{1 \cdot \left(2 \cdot \log \left(\sqrt[3]{angle}\right)\right)}} \cdot \left(\sqrt[3]{angle} \cdot \left(\pi \cdot -0.005555555555555556\right)\right)\right)\right)}^{2} + {\left(b \cdot \sin \left(\frac{\pi}{\frac{-180}{angle}}\right)\right)}^{2} \]

   2. exp-prod 37.7%

      \[\leadsto {\left(a \cdot \cos \left(\color{blue}{{\left(e^{1}\right)}^{\left(2 \cdot \log \left(\sqrt[3]{angle}\right)\right)}} \cdot \left(\sqrt[3]{angle} \cdot \left(\pi \cdot -0.005555555555555556\right)\right)\right)\right)}^{2} + {\left(b \cdot \sin \left(\frac{\pi}{\frac{-180}{angle}}\right)\right)}^{2} \]

   3. *-commutative 37.7%

      \[\leadsto {\left(a \cdot \cos \left({\left(e^{1}\right)}^{\color{blue}{\left(\log \left(\sqrt[3]{angle}\right) \cdot 2\right)}} \cdot \left(\sqrt[3]{angle} \cdot \left(\pi \cdot -0.005555555555555556\right)\right)\right)\right)}^{2} + {\left(b \cdot \sin \left(\frac{\pi}{\frac{-180}{angle}}\right)\right)}^{2} \]

   4. add-log-exp 37.7%

      \[\leadsto {\left(a \cdot \cos \left({\left(e^{1}\right)}^{\color{blue}{\log \left(e^{\log \left(\sqrt[3]{angle}\right) \cdot 2}\right)}} \cdot \left(\sqrt[3]{angle} \cdot \left(\pi \cdot -0.005555555555555556\right)\right)\right)\right)}^{2} + {\left(b \cdot \sin \left(\frac{\pi}{\frac{-180}{angle}}\right)\right)}^{2} \]

   5. exp-to-pow 80.3%

      \[\leadsto {\left(a \cdot \cos \left({\left(e^{1}\right)}^{\log \color{blue}{\left({\left(\sqrt[3]{angle}\right)}^{2}\right)}} \cdot \left(\sqrt[3]{angle} \cdot \left(\pi \cdot -0.005555555555555556\right)\right)\right)\right)}^{2} + {\left(b \cdot \sin \left(\frac{\pi}{\frac{-180}{angle}}\right)\right)}^{2} \]

10. Applied egg-rr 80.3%

    \[\leadsto {\left(a \cdot \cos \left(\color{blue}{{\left(e^{1}\right)}^{\log \left({\left(\sqrt[3]{angle}\right)}^{2}\right)}} \cdot \left(\sqrt[3]{angle} \cdot \left(\pi \cdot -0.005555555555555556\right)\right)\right)\right)}^{2} + {\left(b \cdot \sin \left(\frac{\pi}{\frac{-180}{angle}}\right)\right)}^{2} \]
11. Step-by-step derivation

    1. exp-1-e 80.3%

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

    2. log-pow 37.7%

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

12. Simplified 37.7%

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

13. Final simplification 37.7%

    \[\leadsto {\left(b \cdot \sin \left(\frac{\pi}{\frac{-180}{angle}}\right)\right)}^{2} + {\left(a \cdot \cos \left({e}^{\left(2 \cdot \log \left(\sqrt[3]{angle}\right)\right)} \cdot \left(\sqrt[3]{angle} \cdot \left(\pi \cdot -0.005555555555555556\right)\right)\right)\right)}^{2} \]
14. Add Preprocessing

Alternative 3: 79.1% accurate, 0.7× speedup

Math

\[\begin{array}{l} angle_m = \left|angle\right| \\ {\left(b \cdot \sin \left(\frac{\pi}{\frac{-180}{angle\_m}}\right)\right)}^{2} + {\left(a \cdot \cos \left(\left(\sqrt[3]{angle\_m} \cdot \left(\pi \cdot -0.005555555555555556\right)\right) \cdot {angle\_m}^{0.6666666666666666}\right)\right)}^{2} \end{array} \]

FPCore

angle_m = (fabs.f64 angle)
(FPCore (a b angle_m)
 :precision binary64
 (+
  (pow (* b (sin (/ PI (/ -180.0 angle_m)))) 2.0)
  (pow
   (*
    a
    (cos
     (*
      (* (cbrt angle_m) (* PI -0.005555555555555556))
      (pow angle_m 0.6666666666666666))))
   2.0)))

C

angle_m = fabs(angle);
double code(double a, double b, double angle_m) {
	return pow((b * sin((((double) M_PI) / (-180.0 / angle_m)))), 2.0) + pow((a * cos(((cbrt(angle_m) * (((double) M_PI) * -0.005555555555555556)) * pow(angle_m, 0.6666666666666666)))), 2.0);
}

Java

angle_m = Math.abs(angle);
public static double code(double a, double b, double angle_m) {
	return Math.pow((b * Math.sin((Math.PI / (-180.0 / angle_m)))), 2.0) + Math.pow((a * Math.cos(((Math.cbrt(angle_m) * (Math.PI * -0.005555555555555556)) * Math.pow(angle_m, 0.6666666666666666)))), 2.0);
}

Julia

angle_m = abs(angle)
function code(a, b, angle_m)
	return Float64((Float64(b * sin(Float64(pi / Float64(-180.0 / angle_m)))) ^ 2.0) + (Float64(a * cos(Float64(Float64(cbrt(angle_m) * Float64(pi * -0.005555555555555556)) * (angle_m ^ 0.6666666666666666)))) ^ 2.0))
end

Wolfram

angle_m = N[Abs[angle], $MachinePrecision]
code[a_, b_, angle$95$m_] := N[(N[Power[N[(b * N[Sin[N[(Pi / N[(-180.0 / angle$95$m), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision] + N[Power[N[(a * N[Cos[N[(N[(N[Power[angle$95$m, 1/3], $MachinePrecision] * N[(Pi * -0.005555555555555556), $MachinePrecision]), $MachinePrecision] * N[Power[angle$95$m, 0.6666666666666666], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 80.0%

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

3. Simplified 80.1%

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

   1. associate-/r/ 80.1%

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

   2. *-commutative 80.1%

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

   3. add-cube-cbrt 80.1%

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

   4. associate-*l* 80.1%

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

   5. pow2 80.1%

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

   6. div-inv 80.1%

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

   7. metadata-eval 80.1%

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

6. Applied egg-rr 80.1%

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

   1. add-exp-log 80.1%

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

   2. log-pow 37.6%

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

8. Applied egg-rr 37.6%

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

   1. *-commutative 37.6%

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

   2. exp-to-pow 80.1%

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

   3. pow2 80.1%

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

   4. pow1/3 37.6%

      \[\leadsto {\left(a \cdot \cos \left(\left(\color{blue}{{angle}^{0.3333333333333333}} \cdot \sqrt[3]{angle}\right) \cdot \left(\sqrt[3]{angle} \cdot \left(\pi \cdot -0.005555555555555556\right)\right)\right)\right)}^{2} + {\left(b \cdot \sin \left(\frac{\pi}{\frac{-180}{angle}}\right)\right)}^{2} \]

   5. pow1/3 37.7%

      \[\leadsto {\left(a \cdot \cos \left(\left({angle}^{0.3333333333333333} \cdot \color{blue}{{angle}^{0.3333333333333333}}\right) \cdot \left(\sqrt[3]{angle} \cdot \left(\pi \cdot -0.005555555555555556\right)\right)\right)\right)}^{2} + {\left(b \cdot \sin \left(\frac{\pi}{\frac{-180}{angle}}\right)\right)}^{2} \]

   6. pow-prod-up 37.7%

      \[\leadsto {\left(a \cdot \cos \left(\color{blue}{{angle}^{\left(0.3333333333333333 + 0.3333333333333333\right)}} \cdot \left(\sqrt[3]{angle} \cdot \left(\pi \cdot -0.005555555555555556\right)\right)\right)\right)}^{2} + {\left(b \cdot \sin \left(\frac{\pi}{\frac{-180}{angle}}\right)\right)}^{2} \]

   7. metadata-eval 37.7%

      \[\leadsto {\left(a \cdot \cos \left({angle}^{\color{blue}{0.6666666666666666}} \cdot \left(\sqrt[3]{angle} \cdot \left(\pi \cdot -0.005555555555555556\right)\right)\right)\right)}^{2} + {\left(b \cdot \sin \left(\frac{\pi}{\frac{-180}{angle}}\right)\right)}^{2} \]

10. Applied egg-rr 37.7%

    \[\leadsto {\left(a \cdot \cos \left(\color{blue}{{angle}^{0.6666666666666666}} \cdot \left(\sqrt[3]{angle} \cdot \left(\pi \cdot -0.005555555555555556\right)\right)\right)\right)}^{2} + {\left(b \cdot \sin \left(\frac{\pi}{\frac{-180}{angle}}\right)\right)}^{2} \]

11. Final simplification 37.7%

    \[\leadsto {\left(b \cdot \sin \left(\frac{\pi}{\frac{-180}{angle}}\right)\right)}^{2} + {\left(a \cdot \cos \left(\left(\sqrt[3]{angle} \cdot \left(\pi \cdot -0.005555555555555556\right)\right) \cdot {angle}^{0.6666666666666666}\right)\right)}^{2} \]
12. Add Preprocessing

Alternative 4: 79.1% accurate, 0.7× speedup

Math

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

FPCore

angle_m = (fabs.f64 angle)
(FPCore (a b angle_m)
 :precision binary64
 (+
  (pow (* b (sin (/ PI (/ -180.0 angle_m)))) 2.0)
  (pow
   (*
    a
    (cos (* (sqrt angle_m) (* (sqrt angle_m) (* PI -0.005555555555555556)))))
   2.0)))

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

angle_m = N[Abs[angle], $MachinePrecision]
code[a_, b_, angle$95$m_] := N[(N[Power[N[(b * N[Sin[N[(Pi / N[(-180.0 / angle$95$m), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision] + N[Power[N[(a * N[Cos[N[(N[Sqrt[angle$95$m], $MachinePrecision] * N[(N[Sqrt[angle$95$m], $MachinePrecision] * N[(Pi * -0.005555555555555556), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 80.0%

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

3. Simplified 80.1%

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

   1. associate-/r/ 80.1%

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

   2. add-sqr-sqrt 37.7%

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

   3. associate-*r* 37.6%

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

   4. div-inv 37.6%

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

   5. metadata-eval 37.6%

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

6. Applied egg-rr 37.6%

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

7. Final simplification 37.6%

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

Alternative 5: 79.1% accurate, 0.7× speedup

Math

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

FPCore

angle_m = (fabs.f64 angle)
(FPCore (a b angle_m)
 :precision binary64
 (+
  (pow (* b (sin (/ PI (/ -180.0 angle_m)))) 2.0)
  (pow (* a (log (exp (cos (* angle_m (* PI -0.005555555555555556)))))) 2.0)))

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

angle_m = N[Abs[angle], $MachinePrecision]
code[a_, b_, angle$95$m_] := N[(N[Power[N[(b * N[Sin[N[(Pi / N[(-180.0 / angle$95$m), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision] + N[Power[N[(a * N[Log[N[Exp[N[Cos[N[(angle$95$m * N[(Pi * -0.005555555555555556), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]], $MachinePrecision]], $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 80.0%

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

3. Simplified 80.1%

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

   1. associate-/r/ 80.1%

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

   2. add-sqr-sqrt 37.7%

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

   3. associate-*r* 37.6%

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

   4. div-inv 37.6%

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

   5. metadata-eval 37.6%

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

6. Applied egg-rr 37.6%

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

   1. log1p-expm1-u 37.6%

      \[\leadsto {\left(a \cdot \color{blue}{\mathsf{log1p}\left(\mathsf{expm1}\left(\cos \left(\left(\left(\pi \cdot -0.005555555555555556\right) \cdot \sqrt{angle}\right) \cdot \sqrt{angle}\right)\right)\right)}\right)}^{2} + {\left(b \cdot \sin \left(\frac{\pi}{\frac{-180}{angle}}\right)\right)}^{2} \]

   2. log1p-udef 37.6%

      \[\leadsto {\left(a \cdot \color{blue}{\log \left(1 + \mathsf{expm1}\left(\cos \left(\left(\left(\pi \cdot -0.005555555555555556\right) \cdot \sqrt{angle}\right) \cdot \sqrt{angle}\right)\right)\right)}\right)}^{2} + {\left(b \cdot \sin \left(\frac{\pi}{\frac{-180}{angle}}\right)\right)}^{2} \]

   3. associate-*l* 37.7%

      \[\leadsto {\left(a \cdot \log \left(1 + \mathsf{expm1}\left(\cos \color{blue}{\left(\left(\pi \cdot -0.005555555555555556\right) \cdot \left(\sqrt{angle} \cdot \sqrt{angle}\right)\right)}\right)\right)\right)}^{2} + {\left(b \cdot \sin \left(\frac{\pi}{\frac{-180}{angle}}\right)\right)}^{2} \]

   4. add-sqr-sqrt 80.1%

      \[\leadsto {\left(a \cdot \log \left(1 + \mathsf{expm1}\left(\cos \left(\left(\pi \cdot -0.005555555555555556\right) \cdot \color{blue}{angle}\right)\right)\right)\right)}^{2} + {\left(b \cdot \sin \left(\frac{\pi}{\frac{-180}{angle}}\right)\right)}^{2} \]

   5. *-commutative 80.1%

      \[\leadsto {\left(a \cdot \log \left(1 + \mathsf{expm1}\left(\cos \color{blue}{\left(angle \cdot \left(\pi \cdot -0.005555555555555556\right)\right)}\right)\right)\right)}^{2} + {\left(b \cdot \sin \left(\frac{\pi}{\frac{-180}{angle}}\right)\right)}^{2} \]

   6. associate-*r* 80.1%

      \[\leadsto {\left(a \cdot \log \left(1 + \mathsf{expm1}\left(\cos \color{blue}{\left(\left(angle \cdot \pi\right) \cdot -0.005555555555555556\right)}\right)\right)\right)}^{2} + {\left(b \cdot \sin \left(\frac{\pi}{\frac{-180}{angle}}\right)\right)}^{2} \]

   7. *-commutative 80.1%

      \[\leadsto {\left(a \cdot \log \left(1 + \mathsf{expm1}\left(\cos \left(\color{blue}{\left(\pi \cdot angle\right)} \cdot -0.005555555555555556\right)\right)\right)\right)}^{2} + {\left(b \cdot \sin \left(\frac{\pi}{\frac{-180}{angle}}\right)\right)}^{2} \]

   8. associate-*l* 80.0%

      \[\leadsto {\left(a \cdot \log \left(1 + \mathsf{expm1}\left(\cos \color{blue}{\left(\pi \cdot \left(angle \cdot -0.005555555555555556\right)\right)}\right)\right)\right)}^{2} + {\left(b \cdot \sin \left(\frac{\pi}{\frac{-180}{angle}}\right)\right)}^{2} \]

8. Applied egg-rr 80.0%

   \[\leadsto {\left(a \cdot \color{blue}{\log \left(1 + \mathsf{expm1}\left(\cos \left(\pi \cdot \left(angle \cdot -0.005555555555555556\right)\right)\right)\right)}\right)}^{2} + {\left(b \cdot \sin \left(\frac{\pi}{\frac{-180}{angle}}\right)\right)}^{2} \]

9. Taylor expanded in angle around inf 80.1%

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

    1. *-commutative 80.1%

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

    2. associate-*r* 80.1%

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

11. Simplified 80.1%

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

12. Final simplification 80.1%

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

Alternative 6: 79.2% accurate, 1.0× speedup

Math

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

FPCore

angle_m = (fabs.f64 angle)
(FPCore (a b angle_m)
 :precision binary64
 (let* ((t_0 (* angle_m (/ PI -180.0))))
   (+ (pow (* a (cos t_0)) 2.0) (pow (* b (sin t_0)) 2.0))))

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

angle_m = N[Abs[angle], $MachinePrecision]
code[a_, b_, angle$95$m_] := Block[{t$95$0 = N[(angle$95$m * 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 80.0%

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

3. Simplified 80.1%

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

5. Final simplification 80.1%

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

Alternative 7: 79.1% accurate, 1.0× speedup

Math

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

FPCore

angle_m = (fabs.f64 angle)
(FPCore (a b angle_m)
 :precision binary64
 (+
  (pow (* b (sin (/ PI (/ -180.0 angle_m)))) 2.0)
  (pow (* a (cos (* -0.005555555555555556 (* angle_m PI)))) 2.0)))

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

angle_m = N[Abs[angle], $MachinePrecision]
code[a_, b_, angle$95$m_] := N[(N[Power[N[(b * N[Sin[N[(Pi / N[(-180.0 / angle$95$m), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision] + N[Power[N[(a * N[Cos[N[(-0.005555555555555556 * N[(angle$95$m * Pi), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 80.0%

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

3. Simplified 80.1%

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

   1. associate-/r/ 80.1%

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

   2. *-commutative 80.1%

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

   3. associate-*r/ 80.1%

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

   4. div-inv 80.1%

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

   5. metadata-eval 80.1%

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

6. Applied egg-rr 80.1%

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

7. Final simplification 80.1%

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

Alternative 8: 79.1% accurate, 1.3× speedup

Math

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

FPCore

angle_m = (fabs.f64 angle)
(FPCore (a b angle_m)
 :precision binary64
 (+
  (pow a 2.0)
  (pow (* b (sin (* -0.005555555555555556 (* angle_m PI)))) 2.0)))

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

angle_m = N[Abs[angle], $MachinePrecision]
code[a_, b_, angle$95$m_] := N[(N[Power[a, 2.0], $MachinePrecision] + N[Power[N[(b * N[Sin[N[(-0.005555555555555556 * N[(angle$95$m * Pi), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 80.0%

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

3. Simplified 80.1%

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

5. Taylor expanded in angle around 0 80.0%

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

6. Taylor expanded in angle around inf 79.9%

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

7. Final simplification 79.9%

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

Alternative 9: 79.2% accurate, 1.3× speedup

Math

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

FPCore

angle_m = (fabs.f64 angle)
(FPCore (a b angle_m)
 :precision binary64
 (+ (pow (* b (sin (* angle_m (/ PI -180.0)))) 2.0) (pow a 2.0)))

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

angle_m = N[Abs[angle], $MachinePrecision]
code[a_, b_, angle$95$m_] := N[(N[Power[N[(b * N[Sin[N[(angle$95$m * N[(Pi / -180.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision] + N[Power[a, 2.0], $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 80.0%

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

3. Simplified 80.1%

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

5. Taylor expanded in angle around 0 80.0%

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

6. Final simplification 80.0%

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

Alternative 10: 79.1% accurate, 1.3× speedup

Math

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

FPCore

angle_m = (fabs.f64 angle)
(FPCore (a b angle_m)
 :precision binary64
 (+ (pow (* b (sin (/ PI (/ -180.0 angle_m)))) 2.0) (pow a 2.0)))

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

angle_m = N[Abs[angle], $MachinePrecision]
code[a_, b_, angle$95$m_] := N[(N[Power[N[(b * N[Sin[N[(Pi / N[(-180.0 / angle$95$m), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision] + N[Power[a, 2.0], $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 80.0%

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

3. Simplified 80.1%

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

   1. associate-/r/ 80.1%

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

   2. add-sqr-sqrt 37.7%

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

   3. associate-*r* 37.6%

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

   4. div-inv 37.6%

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

   5. metadata-eval 37.6%

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

6. Applied egg-rr 37.6%

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

7. Taylor expanded in angle around 0 80.0%

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

8. Final simplification 80.0%

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

Alternative 11: 72.6% accurate, 3.5× speedup

Math

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

FPCore

angle_m = (fabs.f64 angle)
(FPCore (a b angle_m)
 :precision binary64
 (+
  (pow a 2.0)
  (*
   (* angle_m -0.005555555555555556)
   (* (* PI b) (* -0.005555555555555556 (* angle_m (* PI b)))))))

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

angle_m = N[Abs[angle], $MachinePrecision]
code[a_, b_, angle$95$m_] := N[(N[Power[a, 2.0], $MachinePrecision] + N[(N[(angle$95$m * -0.005555555555555556), $MachinePrecision] * N[(N[(Pi * b), $MachinePrecision] * N[(-0.005555555555555556 * N[(angle$95$m * N[(Pi * b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 80.0%

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

3. Simplified 80.1%

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

5. Taylor expanded in angle around 0 80.0%

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

6. Taylor expanded in angle around 0 76.0%

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

   1. *-commutative 76.0%

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

8. Simplified 76.0%

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

   1. unpow2 76.0%

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

   2. associate-*r* 76.1%

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

   3. associate-*l* 73.8%

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

   4. *-commutative 73.8%

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

   5. *-commutative 73.8%

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

   6. *-commutative 73.8%

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

   7. associate-*l* 73.8%

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

10. Applied egg-rr 73.8%

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

11. Taylor expanded in b around 0 73.8%

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

12. Final simplification 73.8%

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

Alternative 12: 73.0% accurate, 3.5× speedup

Math

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

FPCore

angle_m = (fabs.f64 angle)
(FPCore (a b angle_m)
 :precision binary64
 (+
  (pow a 2.0)
  (*
   (* PI b)
   (*
    (* angle_m -0.005555555555555556)
    (* (* PI b) (* angle_m -0.005555555555555556))))))

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

angle_m = N[Abs[angle], $MachinePrecision]
code[a_, b_, angle$95$m_] := N[(N[Power[a, 2.0], $MachinePrecision] + N[(N[(Pi * b), $MachinePrecision] * N[(N[(angle$95$m * -0.005555555555555556), $MachinePrecision] * N[(N[(Pi * b), $MachinePrecision] * N[(angle$95$m * -0.005555555555555556), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 80.0%

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

3. Simplified 80.1%

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

5. Taylor expanded in angle around 0 80.0%

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

6. Taylor expanded in angle around 0 76.0%

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

   1. *-commutative 76.0%

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

8. Simplified 76.0%

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

   1. unpow2 76.0%

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

   2. associate-*r* 76.1%

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

   3. associate-*r* 75.7%

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

   4. *-commutative 75.7%

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

   5. *-commutative 75.7%

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

   6. associate-*l* 75.7%

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

   7. *-commutative 75.7%

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

10. Applied egg-rr 75.7%

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

11. Final simplification 75.7%

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

Alternative 13: 74.1% accurate, 3.5× speedup

Math

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

FPCore

angle_m = (fabs.f64 angle)
(FPCore (a b angle_m)
 :precision binary64
 (+
  (pow a 2.0)
  (*
   -0.005555555555555556
   (* (* (* PI b) (* angle_m -0.005555555555555556)) (* PI (* angle_m b))))))

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

angle_m = N[Abs[angle], $MachinePrecision]
code[a_, b_, angle$95$m_] := N[(N[Power[a, 2.0], $MachinePrecision] + N[(-0.005555555555555556 * N[(N[(N[(Pi * b), $MachinePrecision] * N[(angle$95$m * -0.005555555555555556), $MachinePrecision]), $MachinePrecision] * N[(Pi * N[(angle$95$m * b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 80.0%

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

3. Simplified 80.1%

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

5. Taylor expanded in angle around 0 80.0%

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

6. Taylor expanded in angle around 0 76.0%

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

   1. *-commutative 76.0%

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

8. Simplified 76.0%

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

   1. unpow2 76.0%

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

   2. *-commutative 76.0%

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

   3. associate-*r* 76.0%

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

   4. *-commutative 76.0%

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

   5. *-commutative 76.0%

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

   6. associate-*l* 76.1%

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

   7. *-commutative 76.1%

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

   8. associate-*l* 76.1%

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

10. Applied egg-rr 76.1%

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

11. Final simplification 76.1%

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

Reproduce

herbie shell --seed 2024033 
(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)))