ab-angle->ABCF C

Percentage Accurate: 79.8% → 79.7%

Time: 22.0s

Alternatives: 14

Speedup: 1.0×

• 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: 79.8% 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: 79.7% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \sqrt[3]{\pi \cdot angle}\\ {\left(a \cdot \cos \left(t_0 \cdot \left(0.005555555555555556 \cdot {\left({\left(\sqrt[3]{{\left(\sqrt[3]{t_0}\right)}^{3}}\right)}^{3}\right)}^{2}\right)\right)\right)}^{2} + {\left(b \cdot \sin \left(\pi \cdot \frac{angle}{180}\right)\right)}^{2} \end{array} \end{array} \]

FPCore

(FPCore (a b angle)
 :precision binary64
 (let* ((t_0 (cbrt (* PI angle))))
   (+
    (pow
     (*
      a
      (cos
       (*
        t_0
        (*
         0.005555555555555556
         (pow (pow (cbrt (pow (cbrt t_0) 3.0)) 3.0) 2.0)))))
     2.0)
    (pow (* b (sin (* PI (/ angle 180.0)))) 2.0))))

C

double code(double a, double b, double angle) {
	double t_0 = cbrt((((double) M_PI) * angle));
	return pow((a * cos((t_0 * (0.005555555555555556 * pow(pow(cbrt(pow(cbrt(t_0), 3.0)), 3.0), 2.0))))), 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.PI * angle));
	return Math.pow((a * Math.cos((t_0 * (0.005555555555555556 * Math.pow(Math.pow(Math.cbrt(Math.pow(Math.cbrt(t_0), 3.0)), 3.0), 2.0))))), 2.0) + Math.pow((b * Math.sin((Math.PI * (angle / 180.0)))), 2.0);
}

Julia

function code(a, b, angle)
	t_0 = cbrt(Float64(pi * angle))
	return Float64((Float64(a * cos(Float64(t_0 * Float64(0.005555555555555556 * ((cbrt((cbrt(t_0) ^ 3.0)) ^ 3.0) ^ 2.0))))) ^ 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[(Pi * angle), $MachinePrecision], 1/3], $MachinePrecision]}, N[(N[Power[N[(a * N[Cos[N[(t$95$0 * N[(0.005555555555555556 * N[Power[N[Power[N[Power[N[Power[N[Power[t$95$0, 1/3], $MachinePrecision], 3.0], $MachinePrecision], 1/3], $MachinePrecision], 3.0], $MachinePrecision], 2.0], $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 81.7%

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

   1. add-cube-cbrt 81.7%

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

   2. pow3 81.7%

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

   3. div-inv 81.8%

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

   4. metadata-eval 81.8%

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

3. Applied egg-rr 81.8%

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

   1. rem-cube-cbrt 81.7%

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

   2. associate-*r* 81.8%

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

   3. *-commutative 81.8%

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

   4. add-cube-cbrt 81.8%

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

   5. associate-*r* 81.8%

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

   6. pow2 81.8%

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

5. Applied egg-rr 81.8%

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

   1. add-cube-cbrt 81.8%

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

   2. pow3 81.9%

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

7. Applied egg-rr 81.9%

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

   1. add-cube-cbrt 81.8%

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

   2. pow3 81.9%

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

9. Applied egg-rr 81.9%

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

10. Final simplification 81.9%

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

Alternative 2: 79.7% accurate, 0.5× speedup

Math

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

FPCore

(FPCore (a b angle)
 :precision binary64
 (+
  (pow (* b (sin (* PI (/ angle 180.0)))) 2.0)
  (pow
   (*
    a
    (cos
     (*
      (cbrt (* PI angle))
      (*
       0.005555555555555556
       (pow (cbrt (* angle (cbrt (pow PI 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) * angle)) * (0.005555555555555556 * pow(cbrt((angle * cbrt(pow(((double) M_PI), 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 * angle)) * (0.005555555555555556 * Math.pow(Math.cbrt((angle * Math.cbrt(Math.pow(Math.PI, 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(cbrt(Float64(pi * angle)) * Float64(0.005555555555555556 * (cbrt(Float64(angle * cbrt((pi ^ 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[Power[N[(Pi * angle), $MachinePrecision], 1/3], $MachinePrecision] * N[(0.005555555555555556 * N[Power[N[Power[N[(angle * N[Power[N[Power[Pi, 3.0], $MachinePrecision], 1/3], $MachinePrecision]), $MachinePrecision], 1/3], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 81.7%

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

   1. add-cube-cbrt 81.7%

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

   2. pow3 81.7%

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

   3. div-inv 81.8%

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

   4. metadata-eval 81.8%

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

3. Applied egg-rr 81.8%

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

   1. rem-cube-cbrt 81.7%

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

   2. associate-*r* 81.8%

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

   3. *-commutative 81.8%

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

   4. add-cube-cbrt 81.8%

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

   5. associate-*r* 81.8%

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

   6. pow2 81.8%

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

5. Applied egg-rr 81.8%

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

   1. rem-cbrt-cube 81.9%

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

7. Applied egg-rr 81.9%

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

8. Final simplification 81.9%

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

Alternative 3: 79.7% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \sqrt[3]{\pi \cdot angle}\\ {\left(b \cdot \sin \left(\pi \cdot \frac{angle}{180}\right)\right)}^{2} + {\left(a \cdot \cos \left(t_0 \cdot \left(0.005555555555555556 \cdot {\left({\left(\sqrt[3]{t_0}\right)}^{3}\right)}^{2}\right)\right)\right)}^{2} \end{array} \end{array} \]

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 (* t_0 (* 0.005555555555555556 (pow (pow (cbrt t_0) 3.0) 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((t_0 * (0.005555555555555556 * pow(pow(cbrt(t_0), 3.0), 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((t_0 * (0.005555555555555556 * Math.pow(Math.pow(Math.cbrt(t_0), 3.0), 2.0))))), 2.0);
}

Julia

function code(a, b, angle)
	t_0 = cbrt(Float64(pi * angle))
	return Float64((Float64(b * sin(Float64(pi * Float64(angle / 180.0)))) ^ 2.0) + (Float64(a * cos(Float64(t_0 * Float64(0.005555555555555556 * ((cbrt(t_0) ^ 3.0) ^ 2.0))))) ^ 2.0))
end

Wolfram

code[a_, b_, angle_] := Block[{t$95$0 = N[Power[N[(Pi * angle), $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[(t$95$0 * N[(0.005555555555555556 * N[Power[N[Power[N[Power[t$95$0, 1/3], $MachinePrecision], 3.0], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Initial program 81.7%

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

   1. add-cube-cbrt 81.7%

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

   2. pow3 81.7%

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

   3. div-inv 81.8%

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

   4. metadata-eval 81.8%

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

3. Applied egg-rr 81.8%

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

   1. rem-cube-cbrt 81.7%

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

   2. associate-*r* 81.8%

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

   3. *-commutative 81.8%

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

   4. add-cube-cbrt 81.8%

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

   5. associate-*r* 81.8%

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

   6. pow2 81.8%

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

5. Applied egg-rr 81.8%

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

   1. add-cube-cbrt 81.8%

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

   2. pow3 81.9%

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

7. Applied egg-rr 81.9%

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

8. Final simplification 81.9%

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

Alternative 4: 79.7% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \sqrt[3]{\pi \cdot angle}\\ {\left(b \cdot \sin \left(\pi \cdot \frac{angle}{180}\right)\right)}^{2} + {\left(a \cdot \cos \left(t_0 \cdot \left(0.005555555555555556 \cdot {t_0}^{2}\right)\right)\right)}^{2} \end{array} \end{array} \]

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 (* t_0 (* 0.005555555555555556 (pow 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((t_0 * (0.005555555555555556 * pow(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((t_0 * (0.005555555555555556 * Math.pow(t_0, 2.0))))), 2.0);
}

Julia

function code(a, b, angle)
	t_0 = cbrt(Float64(pi * angle))
	return Float64((Float64(b * sin(Float64(pi * Float64(angle / 180.0)))) ^ 2.0) + (Float64(a * cos(Float64(t_0 * Float64(0.005555555555555556 * (t_0 ^ 2.0))))) ^ 2.0))
end

Wolfram

code[a_, b_, angle_] := Block[{t$95$0 = N[Power[N[(Pi * angle), $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[(t$95$0 * N[(0.005555555555555556 * N[Power[t$95$0, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Initial program 81.7%

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

   1. add-cube-cbrt 81.7%

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

   2. pow3 81.7%

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

   3. div-inv 81.8%

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

   4. metadata-eval 81.8%

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

3. Applied egg-rr 81.8%

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

   1. rem-cube-cbrt 81.7%

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

   2. associate-*r* 81.8%

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

   3. *-commutative 81.8%

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

   4. add-cube-cbrt 81.8%

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

   5. associate-*r* 81.8%

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

   6. pow2 81.8%

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

5. Applied egg-rr 81.8%

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

6. Final simplification 81.8%

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

Alternative 5: 79.8% accurate, 0.8× speedup

Math

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

FPCore

(FPCore (a b angle)
 :precision binary64
 (+
  (pow (* b (sin (* PI (/ angle 180.0)))) 2.0)
  (pow (* a (cos (pow (/ 1.0 (cbrt (/ (/ 180.0 angle) PI))) 3.0))) 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(pow((1.0 / cbrt(((180.0 / angle) / ((double) M_PI)))), 3.0))), 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.pow((1.0 / Math.cbrt(((180.0 / angle) / Math.PI))), 3.0))), 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(1.0 / cbrt(Float64(Float64(180.0 / angle) / pi))) ^ 3.0))) ^ 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[Power[N[(1.0 / N[Power[N[(N[(180.0 / angle), $MachinePrecision] / Pi), $MachinePrecision], 1/3], $MachinePrecision]), $MachinePrecision], 3.0], $MachinePrecision]], $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 81.7%

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

   1. add-cube-cbrt 81.7%

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

   2. pow3 81.7%

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

   3. div-inv 81.8%

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

   4. metadata-eval 81.8%

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

3. Applied egg-rr 81.8%

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

   1. metadata-eval 81.8%

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

   2. div-inv 81.7%

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

   3. associate-*r/ 81.8%

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

   4. clear-num 81.8%

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

   5. clear-num 81.8%

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

   6. cbrt-div 81.7%

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

   7. metadata-eval 81.7%

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

   8. /-rgt-identity 81.7%

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

   9. *-commutative 81.7%

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

   10. associate-/r* 81.8%

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

5. Applied egg-rr 81.8%

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

6. Final simplification 81.8%

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

Alternative 6: 79.7% accurate, 0.8× speedup

Math

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

FPCore

(FPCore (a b angle)
 :precision binary64
 (+
  (pow (* b (sin (* PI (/ angle 180.0)))) 2.0)
  (pow
   (* a (cos (pow (cbrt (* 0.005555555555555556 (* PI angle))) 3.0)))
   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(pow(cbrt((0.005555555555555556 * (((double) M_PI) * angle))), 3.0))), 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.pow(Math.cbrt((0.005555555555555556 * (Math.PI * angle))), 3.0))), 2.0);
}

Julia

function code(a, b, angle)
	return Float64((Float64(b * sin(Float64(pi * Float64(angle / 180.0)))) ^ 2.0) + (Float64(a * cos((cbrt(Float64(0.005555555555555556 * Float64(pi * angle))) ^ 3.0))) ^ 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[Power[N[Power[N[(0.005555555555555556 * N[(Pi * angle), $MachinePrecision]), $MachinePrecision], 1/3], $MachinePrecision], 3.0], $MachinePrecision]], $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 81.7%

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

   1. add-cube-cbrt 81.7%

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

   2. pow3 81.7%

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

   3. div-inv 81.8%

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

   4. metadata-eval 81.8%

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

3. Applied egg-rr 81.8%

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

   1. expm1-log1p-u 68.2%

      \[\leadsto {\left(a \cdot \cos \left({\color{blue}{\left(\mathsf{expm1}\left(\mathsf{log1p}\left(\sqrt[3]{\pi \cdot \left(angle \cdot 0.005555555555555556\right)}\right)\right)\right)}}^{3}\right)\right)}^{2} + {\left(b \cdot \sin \left(\pi \cdot \frac{angle}{180}\right)\right)}^{2} \]

   2. expm1-udef 68.1%

      \[\leadsto {\left(a \cdot \cos \left({\color{blue}{\left(e^{\mathsf{log1p}\left(\sqrt[3]{\pi \cdot \left(angle \cdot 0.005555555555555556\right)}\right)} - 1\right)}}^{3}\right)\right)}^{2} + {\left(b \cdot \sin \left(\pi \cdot \frac{angle}{180}\right)\right)}^{2} \]

   3. metadata-eval 68.1%

      \[\leadsto {\left(a \cdot \cos \left({\left(e^{\mathsf{log1p}\left(\sqrt[3]{\pi \cdot \left(angle \cdot \color{blue}{\frac{1}{180}}\right)}\right)} - 1\right)}^{3}\right)\right)}^{2} + {\left(b \cdot \sin \left(\pi \cdot \frac{angle}{180}\right)\right)}^{2} \]

   4. div-inv 68.1%

      \[\leadsto {\left(a \cdot \cos \left({\left(e^{\mathsf{log1p}\left(\sqrt[3]{\pi \cdot \color{blue}{\frac{angle}{180}}}\right)} - 1\right)}^{3}\right)\right)}^{2} + {\left(b \cdot \sin \left(\pi \cdot \frac{angle}{180}\right)\right)}^{2} \]

5. Applied egg-rr 68.1%

   \[\leadsto {\left(a \cdot \cos \left({\color{blue}{\left(e^{\mathsf{log1p}\left(\sqrt[3]{\pi \cdot \frac{angle}{180}}\right)} - 1\right)}}^{3}\right)\right)}^{2} + {\left(b \cdot \sin \left(\pi \cdot \frac{angle}{180}\right)\right)}^{2} \]
6. Step-by-step derivation

   1. expm1-def 68.2%

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

   2. expm1-log1p 81.7%

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

   3. associate-*r/ 81.8%

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

   4. associate-/l* 81.7%

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

   5. *-lft-identity 81.7%

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

   6. associate-*l/ 81.7%

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

   7. associate-/r/ 81.8%

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

   8. metadata-eval 81.8%

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

   9. associate-*r* 81.8%

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

7. Simplified 81.8%

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

8. Final simplification 81.8%

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

Alternative 7: 79.7% accurate, 1.0× speedup

Math

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

FPCore

(FPCore (a b angle)
 :precision binary64
 (+
  (pow (* b (sin (* PI (/ angle 180.0)))) 2.0)
  (pow (* a (cos (* 0.005555555555555556 (* PI angle)))) 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((0.005555555555555556 * (((double) M_PI) * angle)))), 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((0.005555555555555556 * (Math.PI * angle)))), 2.0);
}

Python

def code(a, b, angle):
	return math.pow((b * math.sin((math.pi * (angle / 180.0)))), 2.0) + math.pow((a * math.cos((0.005555555555555556 * (math.pi * angle)))), 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(0.005555555555555556 * Float64(pi * angle)))) ^ 2.0))
end

MATLAB

function tmp = code(a, b, angle)
	tmp = ((b * sin((pi * (angle / 180.0)))) ^ 2.0) + ((a * cos((0.005555555555555556 * (pi * angle)))) ^ 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[(0.005555555555555556 * N[(Pi * angle), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 81.7%

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

2. Taylor expanded in angle around inf 81.8%

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

3. Final simplification 81.8%

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

Alternative 8: 79.8% accurate, 1.0× speedup

Math

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

FPCore

(FPCore (a b angle)
 :precision binary64
 (+
  (pow (* b (sin (* PI (/ angle 180.0)))) 2.0)
  (pow (* a (cos (* angle (/ PI 180.0)))) 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((angle * (((double) M_PI) / 180.0)))), 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((angle * (Math.PI / 180.0)))), 2.0);
}

Python

def code(a, b, angle):
	return math.pow((b * math.sin((math.pi * (angle / 180.0)))), 2.0) + math.pow((a * math.cos((angle * (math.pi / 180.0)))), 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(angle * Float64(pi / 180.0)))) ^ 2.0))
end

MATLAB

function tmp = code(a, b, angle)
	tmp = ((b * sin((pi * (angle / 180.0)))) ^ 2.0) + ((a * cos((angle * (pi / 180.0)))) ^ 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[(angle * N[(Pi / 180.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 81.7%

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

2. Taylor expanded in angle around inf 81.8%

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

   1. metadata-eval 81.8%

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

   2. *-commutative 81.8%

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

   3. associate-/r/ 81.7%

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

   4. associate-/l* 81.6%

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

   5. *-commutative 81.6%

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

   6. associate-/l* 81.6%

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

   7. metadata-eval 81.6%

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

   8. *-commutative 81.6%

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

   9. associate-*r/ 81.8%

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

4. Simplified 81.8%

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

5. Final simplification 81.8%

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

Alternative 9: 79.6% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ {a}^{2} + {\left(b \cdot \mathsf{expm1}\left(\mathsf{log1p}\left(\sin \left(\pi \cdot \left(0.005555555555555556 \cdot angle\right)\right)\right)\right)\right)}^{2} \end{array} \]

FPCore

(FPCore (a b angle)
 :precision binary64
 (+
  (pow a 2.0)
  (pow (* b (expm1 (log1p (sin (* PI (* 0.005555555555555556 angle)))))) 2.0)))

C

double code(double a, double b, double angle) {
	return pow(a, 2.0) + pow((b * expm1(log1p(sin((((double) M_PI) * (0.005555555555555556 * angle)))))), 2.0);
}

Java

public static double code(double a, double b, double angle) {
	return Math.pow(a, 2.0) + Math.pow((b * Math.expm1(Math.log1p(Math.sin((Math.PI * (0.005555555555555556 * angle)))))), 2.0);
}

Python

def code(a, b, angle):
	return math.pow(a, 2.0) + math.pow((b * math.expm1(math.log1p(math.sin((math.pi * (0.005555555555555556 * angle)))))), 2.0)

Julia

function code(a, b, angle)
	return Float64((a ^ 2.0) + (Float64(b * expm1(log1p(sin(Float64(pi * Float64(0.005555555555555556 * angle)))))) ^ 2.0))
end

Wolfram

code[a_, b_, angle_] := N[(N[Power[a, 2.0], $MachinePrecision] + N[Power[N[(b * N[(Exp[N[Log[1 + N[Sin[N[(Pi * N[(0.005555555555555556 * angle), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]], $MachinePrecision]] - 1), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 81.7%

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

2. Taylor expanded in angle around 0 81.6%

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

   1. expm1-log1p-u 81.6%

      \[\leadsto {\left(a \cdot 1\right)}^{2} + {\left(b \cdot \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\sin \left(\pi \cdot \frac{angle}{180}\right)\right)\right)}\right)}^{2} \]

   2. div-inv 81.7%

      \[\leadsto {\left(a \cdot 1\right)}^{2} + {\left(b \cdot \mathsf{expm1}\left(\mathsf{log1p}\left(\sin \left(\pi \cdot \color{blue}{\left(angle \cdot \frac{1}{180}\right)}\right)\right)\right)\right)}^{2} \]

   3. metadata-eval 81.7%

      \[\leadsto {\left(a \cdot 1\right)}^{2} + {\left(b \cdot \mathsf{expm1}\left(\mathsf{log1p}\left(\sin \left(\pi \cdot \left(angle \cdot \color{blue}{0.005555555555555556}\right)\right)\right)\right)\right)}^{2} \]

4. Applied egg-rr 81.7%

   \[\leadsto {\left(a \cdot 1\right)}^{2} + {\left(b \cdot \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\sin \left(\pi \cdot \left(angle \cdot 0.005555555555555556\right)\right)\right)\right)}\right)}^{2} \]

5. Final simplification 81.7%

   \[\leadsto {a}^{2} + {\left(b \cdot \mathsf{expm1}\left(\mathsf{log1p}\left(\sin \left(\pi \cdot \left(0.005555555555555556 \cdot angle\right)\right)\right)\right)\right)}^{2} \]

Alternative 10: 79.5% accurate, 1.5× speedup

Math

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

FPCore

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

C

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 81.7%

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

2. Taylor expanded in angle around 0 81.6%

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

3. Taylor expanded in b around 0 81.6%

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

4. Final simplification 81.6%

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

Alternative 11: 74.6% accurate, 1.9× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 81.7%

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

2. Taylor expanded in angle around 0 81.6%

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

3. Taylor expanded in angle around 0 75.8%

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

   1. metadata-eval 75.8%

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

   2. *-commutative 75.8%

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

   3. associate-/r/ 75.7%

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

   4. associate-/l* 75.8%

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

   5. *-commutative 75.8%

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

   6. associate-/l* 75.8%

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

   7. metadata-eval 75.8%

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

   8. *-commutative 75.8%

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

   9. associate-*r/ 75.8%

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

5. Simplified 75.8%

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

   1. unpow2 75.8%

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

   2. associate-*r* 74.0%

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

   3. *-commutative 74.0%

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

   4. associate-*r/ 74.0%

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

   5. div-inv 74.0%

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

   6. *-commutative 74.0%

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

   7. metadata-eval 74.0%

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

   8. associate-*r* 74.0%

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

   9. *-commutative 74.0%

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

   10. *-commutative 74.0%

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

   11. associate-*r* 74.0%

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

   12. associate-*r* 74.0%

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

   13. *-commutative 74.0%

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

   14. *-commutative 74.0%

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

   15. associate-*r/ 74.0%

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

   16. div-inv 74.0%

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

   17. metadata-eval 74.0%

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

   18. *-commutative 74.0%

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

   19. associate-*r* 74.0%

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

   20. *-commutative 74.0%

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

   21. *-commutative 74.0%

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

7. Applied egg-rr 74.0%

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

   1. associate-*l* 75.8%

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

9. Simplified 75.8%

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

10. Final simplification 75.8%

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

Alternative 12: 74.6% accurate, 2.0× speedup

Math

\[\begin{array}{l} \\ {a}^{2} + 3.08641975308642 \cdot 10^{-5} \cdot {\left(\pi \cdot \left(angle \cdot b\right)\right)}^{2} \end{array} \]

FPCore

(FPCore (a b angle)
 :precision binary64
 (+ (pow a 2.0) (* 3.08641975308642e-5 (pow (* PI (* angle b)) 2.0))))

C

double code(double a, double b, double angle) {
	return pow(a, 2.0) + (3.08641975308642e-5 * pow((((double) M_PI) * (angle * b)), 2.0));
}

Java

public static double code(double a, double b, double angle) {
	return Math.pow(a, 2.0) + (3.08641975308642e-5 * Math.pow((Math.PI * (angle * b)), 2.0));
}

Python

def code(a, b, angle):
	return math.pow(a, 2.0) + (3.08641975308642e-5 * math.pow((math.pi * (angle * b)), 2.0))

Julia

function code(a, b, angle)
	return Float64((a ^ 2.0) + Float64(3.08641975308642e-5 * (Float64(pi * Float64(angle * b)) ^ 2.0)))
end

MATLAB

function tmp = code(a, b, angle)
	tmp = (a ^ 2.0) + (3.08641975308642e-5 * ((pi * (angle * b)) ^ 2.0));
end

Wolfram

code[a_, b_, angle_] := N[(N[Power[a, 2.0], $MachinePrecision] + N[(3.08641975308642e-5 * N[Power[N[(Pi * N[(angle * b), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 81.7%

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

2. Taylor expanded in angle around 0 81.6%

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

3. Taylor expanded in angle around 0 75.8%

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

   1. metadata-eval 75.8%

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

   2. *-commutative 75.8%

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

   3. associate-/r/ 75.7%

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

   4. associate-/l* 75.8%

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

   5. *-commutative 75.8%

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

   6. associate-/l* 75.8%

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

   7. metadata-eval 75.8%

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

   8. *-commutative 75.8%

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

   9. associate-*r/ 75.8%

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

5. Simplified 75.8%

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

6. Taylor expanded in b around 0 60.9%

   \[\leadsto {\left(a \cdot 1\right)}^{2} + \color{blue}{3.08641975308642 \cdot 10^{-5} \cdot \left({angle}^{2} \cdot \left({b}^{2} \cdot {\pi}^{2}\right)\right)} \]
7. Step-by-step derivation

   1. metadata-eval 60.9%

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

   2. *-commutative 60.9%

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

   3. unpow2 60.9%

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

   4. unpow2 60.9%

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

   5. unswap-sqr 60.9%

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

   6. unpow2 60.9%

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

   7. swap-sqr 75.8%

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

   8. associate-*r* 75.8%

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

   9. associate-*r* 75.8%

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

   10. swap-sqr 75.8%

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

   11. unpow2 75.8%

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

   12. associate-*r* 75.8%

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

   13. *-commutative 75.8%

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

   14. associate-*r* 75.8%

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

   15. *-commutative 75.8%

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

   16. *-commutative 75.8%

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

   17. *-commutative 75.8%

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

8. Simplified 75.8%

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

9. Taylor expanded in angle around 0 60.9%

   \[\leadsto {\left(a \cdot 1\right)}^{2} + \color{blue}{3.08641975308642 \cdot 10^{-5} \cdot \left({angle}^{2} \cdot \left({b}^{2} \cdot {\pi}^{2}\right)\right)} \]
10. Step-by-step derivation

    1. unpow2 60.9%

       \[\leadsto {\left(a \cdot 1\right)}^{2} + 3.08641975308642 \cdot 10^{-5} \cdot \left({angle}^{2} \cdot \left(\color{blue}{\left(b \cdot b\right)} \cdot {\pi}^{2}\right)\right) \]

    2. unpow2 60.9%

       \[\leadsto {\left(a \cdot 1\right)}^{2} + 3.08641975308642 \cdot 10^{-5} \cdot \left({angle}^{2} \cdot \left(\left(b \cdot b\right) \cdot \color{blue}{\left(\pi \cdot \pi\right)}\right)\right) \]

    3. unswap-sqr 60.9%

       \[\leadsto {\left(a \cdot 1\right)}^{2} + 3.08641975308642 \cdot 10^{-5} \cdot \left({angle}^{2} \cdot \color{blue}{\left(\left(b \cdot \pi\right) \cdot \left(b \cdot \pi\right)\right)}\right) \]

    4. unpow2 60.9%

       \[\leadsto {\left(a \cdot 1\right)}^{2} + 3.08641975308642 \cdot 10^{-5} \cdot \left(\color{blue}{\left(angle \cdot angle\right)} \cdot \left(\left(b \cdot \pi\right) \cdot \left(b \cdot \pi\right)\right)\right) \]

    5. swap-sqr 75.8%

       \[\leadsto {\left(a \cdot 1\right)}^{2} + 3.08641975308642 \cdot 10^{-5} \cdot \color{blue}{\left(\left(angle \cdot \left(b \cdot \pi\right)\right) \cdot \left(angle \cdot \left(b \cdot \pi\right)\right)\right)} \]

    6. unpow2 75.8%

       \[\leadsto {\left(a \cdot 1\right)}^{2} + 3.08641975308642 \cdot 10^{-5} \cdot \color{blue}{{\left(angle \cdot \left(b \cdot \pi\right)\right)}^{2}} \]

    7. associate-*r* 75.8%

       \[\leadsto {\left(a \cdot 1\right)}^{2} + 3.08641975308642 \cdot 10^{-5} \cdot {\color{blue}{\left(\left(angle \cdot b\right) \cdot \pi\right)}}^{2} \]

    8. *-commutative 75.8%

       \[\leadsto {\left(a \cdot 1\right)}^{2} + 3.08641975308642 \cdot 10^{-5} \cdot {\color{blue}{\left(\pi \cdot \left(angle \cdot b\right)\right)}}^{2} \]

11. Simplified 75.8%

    \[\leadsto {\left(a \cdot 1\right)}^{2} + \color{blue}{3.08641975308642 \cdot 10^{-5} \cdot {\left(\pi \cdot \left(angle \cdot b\right)\right)}^{2}} \]

12. Final simplification 75.8%

    \[\leadsto {a}^{2} + 3.08641975308642 \cdot 10^{-5} \cdot {\left(\pi \cdot \left(angle \cdot b\right)\right)}^{2} \]

Alternative 13: 74.6% accurate, 2.0× speedup

Math

\[\begin{array}{l} \\ {a}^{2} + {\left(angle \cdot \left(\pi \cdot b\right)\right)}^{2} \cdot 3.08641975308642 \cdot 10^{-5} \end{array} \]

FPCore

(FPCore (a b angle)
 :precision binary64
 (+ (pow a 2.0) (* (pow (* angle (* PI b)) 2.0) 3.08641975308642e-5)))

C

double code(double a, double b, double angle) {
	return pow(a, 2.0) + (pow((angle * (((double) M_PI) * b)), 2.0) * 3.08641975308642e-5);
}

Java

public static double code(double a, double b, double angle) {
	return Math.pow(a, 2.0) + (Math.pow((angle * (Math.PI * b)), 2.0) * 3.08641975308642e-5);
}

Python

def code(a, b, angle):
	return math.pow(a, 2.0) + (math.pow((angle * (math.pi * b)), 2.0) * 3.08641975308642e-5)

Julia

function code(a, b, angle)
	return Float64((a ^ 2.0) + Float64((Float64(angle * Float64(pi * b)) ^ 2.0) * 3.08641975308642e-5))
end

MATLAB

function tmp = code(a, b, angle)
	tmp = (a ^ 2.0) + (((angle * (pi * b)) ^ 2.0) * 3.08641975308642e-5);
end

Wolfram

code[a_, b_, angle_] := N[(N[Power[a, 2.0], $MachinePrecision] + N[(N[Power[N[(angle * N[(Pi * b), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision] * 3.08641975308642e-5), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 81.7%

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

2. Taylor expanded in angle around 0 81.6%

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

3. Taylor expanded in angle around 0 75.8%

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

   1. metadata-eval 75.8%

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

   2. *-commutative 75.8%

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

   3. associate-/r/ 75.7%

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

   4. associate-/l* 75.8%

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

   5. *-commutative 75.8%

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

   6. associate-/l* 75.8%

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

   7. metadata-eval 75.8%

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

   8. *-commutative 75.8%

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

   9. associate-*r/ 75.8%

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

5. Simplified 75.8%

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

6. Taylor expanded in b around 0 60.9%

   \[\leadsto {\left(a \cdot 1\right)}^{2} + \color{blue}{3.08641975308642 \cdot 10^{-5} \cdot \left({angle}^{2} \cdot \left({b}^{2} \cdot {\pi}^{2}\right)\right)} \]
7. Step-by-step derivation

   1. metadata-eval 60.9%

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

   2. *-commutative 60.9%

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

   3. unpow2 60.9%

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

   4. unpow2 60.9%

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

   5. unswap-sqr 60.9%

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

   6. unpow2 60.9%

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

   7. swap-sqr 75.8%

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

   8. associate-*r* 75.8%

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

   9. associate-*r* 75.8%

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

   10. swap-sqr 75.8%

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

   11. unpow2 75.8%

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

   12. associate-*r* 75.8%

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

   13. *-commutative 75.8%

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

   14. associate-*r* 75.8%

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

   15. *-commutative 75.8%

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

   16. *-commutative 75.8%

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

   17. *-commutative 75.8%

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

8. Simplified 75.8%

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

   1. *-commutative 75.8%

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

   2. unpow-prod-down 75.8%

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

   3. *-commutative 75.8%

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

   4. associate-*l* 75.8%

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

   5. metadata-eval 75.8%

      \[\leadsto {\left(a \cdot 1\right)}^{2} + {\left(angle \cdot \left(b \cdot \pi\right)\right)}^{2} \cdot \color{blue}{3.08641975308642 \cdot 10^{-5}} \]

10. Applied egg-rr 75.8%

    \[\leadsto {\left(a \cdot 1\right)}^{2} + \color{blue}{{\left(angle \cdot \left(b \cdot \pi\right)\right)}^{2} \cdot 3.08641975308642 \cdot 10^{-5}} \]

11. Final simplification 75.8%

    \[\leadsto {a}^{2} + {\left(angle \cdot \left(\pi \cdot b\right)\right)}^{2} \cdot 3.08641975308642 \cdot 10^{-5} \]

Alternative 14: 74.6% accurate, 2.0× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 81.7%

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

2. Taylor expanded in angle around 0 81.6%

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

3. Taylor expanded in angle around 0 75.8%

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

   1. metadata-eval 75.8%

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

   2. *-commutative 75.8%

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

   3. associate-/r/ 75.7%

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

   4. associate-/l* 75.8%

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

   5. *-commutative 75.8%

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

   6. associate-/l* 75.8%

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

   7. metadata-eval 75.8%

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

   8. *-commutative 75.8%

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

   9. associate-*r/ 75.8%

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

5. Simplified 75.8%

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

6. Final simplification 75.8%

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

Reproduce

herbie shell --seed 2023196 
(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)))