ab-angle->ABCF C

Percentage Accurate: 79.2% → 79.2%

Time: 15.7s

Alternatives: 10

Speedup: 1.3×

• 36.1% 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: 79.2% 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: 79.2% accurate, 1.3× speedup

Math

\[\begin{array}{l} \\ a \cdot \left(a \cdot \left(0.5 + 0.5 \cdot \cos \left(2 \cdot \left(angle \cdot \left(\pi \cdot 0.005555555555555556\right)\right)\right)\right)\right) + {\left(b \cdot \sin \left(\pi \cdot \left(angle \cdot 0.005555555555555556\right)\right)\right)}^{2} \end{array} \]

FPCore

(FPCore (a b angle)
 :precision binary64
 (+
  (*
   a
   (* a (+ 0.5 (* 0.5 (cos (* 2.0 (* angle (* PI 0.005555555555555556))))))))
  (pow (* b (sin (* PI (* angle 0.005555555555555556)))) 2.0)))

C

double code(double a, double b, double angle) {
	return (a * (a * (0.5 + (0.5 * cos((2.0 * (angle * (((double) M_PI) * 0.005555555555555556)))))))) + pow((b * sin((((double) M_PI) * (angle * 0.005555555555555556)))), 2.0);
}

Java

public static double code(double a, double b, double angle) {
	return (a * (a * (0.5 + (0.5 * Math.cos((2.0 * (angle * (Math.PI * 0.005555555555555556)))))))) + Math.pow((b * Math.sin((Math.PI * (angle * 0.005555555555555556)))), 2.0);
}

Python

def code(a, b, angle):
	return (a * (a * (0.5 + (0.5 * math.cos((2.0 * (angle * (math.pi * 0.005555555555555556)))))))) + math.pow((b * math.sin((math.pi * (angle * 0.005555555555555556)))), 2.0)

Julia

function code(a, b, angle)
	return Float64(Float64(a * Float64(a * Float64(0.5 + Float64(0.5 * cos(Float64(2.0 * Float64(angle * Float64(pi * 0.005555555555555556)))))))) + (Float64(b * sin(Float64(pi * Float64(angle * 0.005555555555555556)))) ^ 2.0))
end

MATLAB

function tmp = code(a, b, angle)
	tmp = (a * (a * (0.5 + (0.5 * cos((2.0 * (angle * (pi * 0.005555555555555556)))))))) + ((b * sin((pi * (angle * 0.005555555555555556)))) ^ 2.0);
end

Wolfram

code[a_, b_, angle_] := N[(N[(a * N[(a * N[(0.5 + N[(0.5 * N[Cos[N[(2.0 * N[(angle * N[(Pi * 0.005555555555555556), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[Power[N[(b * N[Sin[N[(Pi * N[(angle * 0.005555555555555556), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 78.3%

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

3. Simplified 78.3%

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

   1. metadata-eval 78.3%

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

   2. div-inv 78.3%

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

   3. add-cube-cbrt 77.7%

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

   4. pow3 77.7%

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

   5. *-commutative 77.7%

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

   6. *-commutative 77.7%

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

   7. div-inv 77.7%

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

   8. metadata-eval 77.7%

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

   9. associate-*l* 77.7%

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

6. Applied egg-rr 77.7%

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

   1. unpow3 77.7%

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

   2. add-cube-cbrt 78.3%

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

   3. *-commutative 78.3%

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

   4. unpow-prod-down 78.3%

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

   5. unpow2 78.3%

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

   6. associate-*r* 78.3%

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

   7. *-commutative 78.3%

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

   8. associate-*l* 78.3%

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

   9. *-commutative 78.3%

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

   10. associate-*r* 78.3%

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

8. Applied egg-rr 78.3%

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

   1. unpow2 78.3%

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

   2. sqr-cos-a 78.3%

      \[\leadsto a \cdot \left(a \cdot \color{blue}{\left(0.5 + 0.5 \cdot \cos \left(2 \cdot \left(angle \cdot \left(0.005555555555555556 \cdot \pi\right)\right)\right)\right)}\right) + {\left(b \cdot \sin \left(\pi \cdot \left(angle \cdot 0.005555555555555556\right)\right)\right)}^{2} \]

   3. associate-*r* 78.3%

      \[\leadsto a \cdot \left(a \cdot \left(0.5 + 0.5 \cdot \cos \left(2 \cdot \color{blue}{\left(\left(angle \cdot 0.005555555555555556\right) \cdot \pi\right)}\right)\right)\right) + {\left(b \cdot \sin \left(\pi \cdot \left(angle \cdot 0.005555555555555556\right)\right)\right)}^{2} \]

   4. *-commutative 78.3%

      \[\leadsto a \cdot \left(a \cdot \left(0.5 + 0.5 \cdot \cos \left(2 \cdot \color{blue}{\left(\pi \cdot \left(angle \cdot 0.005555555555555556\right)\right)}\right)\right)\right) + {\left(b \cdot \sin \left(\pi \cdot \left(angle \cdot 0.005555555555555556\right)\right)\right)}^{2} \]

10. Applied egg-rr 78.3%

    \[\leadsto a \cdot \left(a \cdot \color{blue}{\left(0.5 + 0.5 \cdot \cos \left(2 \cdot \left(\pi \cdot \left(angle \cdot 0.005555555555555556\right)\right)\right)\right)}\right) + {\left(b \cdot \sin \left(\pi \cdot \left(angle \cdot 0.005555555555555556\right)\right)\right)}^{2} \]
11. Step-by-step derivation

    1. *-commutative 78.3%

       \[\leadsto a \cdot \left(a \cdot \left(0.5 + 0.5 \cdot \cos \left(2 \cdot \left(\pi \cdot \color{blue}{\left(0.005555555555555556 \cdot angle\right)}\right)\right)\right)\right) + {\left(b \cdot \sin \left(\pi \cdot \left(angle \cdot 0.005555555555555556\right)\right)\right)}^{2} \]

    2. associate-*l* 78.3%

       \[\leadsto a \cdot \left(a \cdot \left(0.5 + 0.5 \cdot \cos \left(2 \cdot \color{blue}{\left(\left(\pi \cdot 0.005555555555555556\right) \cdot angle\right)}\right)\right)\right) + {\left(b \cdot \sin \left(\pi \cdot \left(angle \cdot 0.005555555555555556\right)\right)\right)}^{2} \]

    3. *-commutative 78.3%

       \[\leadsto a \cdot \left(a \cdot \left(0.5 + 0.5 \cdot \cos \left(2 \cdot \color{blue}{\left(angle \cdot \left(\pi \cdot 0.005555555555555556\right)\right)}\right)\right)\right) + {\left(b \cdot \sin \left(\pi \cdot \left(angle \cdot 0.005555555555555556\right)\right)\right)}^{2} \]

12. Simplified 78.3%

    \[\leadsto a \cdot \left(a \cdot \color{blue}{\left(0.5 + 0.5 \cdot \cos \left(2 \cdot \left(angle \cdot \left(\pi \cdot 0.005555555555555556\right)\right)\right)\right)}\right) + {\left(b \cdot \sin \left(\pi \cdot \left(angle \cdot 0.005555555555555556\right)\right)\right)}^{2} \]
13. Add Preprocessing

Alternative 2: 79.2% accurate, 1.3× speedup

Math

\[\begin{array}{l} \\ {\left(b \cdot \sin \left(\pi \cdot \left(angle \cdot 0.005555555555555556\right)\right)\right)}^{2} + a \cdot \left(a \cdot \frac{1 + \cos \left(\pi \cdot \left(angle \cdot 0.011111111111111112\right)\right)}{2}\right) \end{array} \]

FPCore

(FPCore (a b angle)
 :precision binary64
 (+
  (pow (* b (sin (* PI (* angle 0.005555555555555556)))) 2.0)
  (* a (* a (/ (+ 1.0 (cos (* PI (* angle 0.011111111111111112)))) 2.0)))))

C

double code(double a, double b, double angle) {
	return pow((b * sin((((double) M_PI) * (angle * 0.005555555555555556)))), 2.0) + (a * (a * ((1.0 + cos((((double) M_PI) * (angle * 0.011111111111111112)))) / 2.0)));
}

Java

public static double code(double a, double b, double angle) {
	return Math.pow((b * Math.sin((Math.PI * (angle * 0.005555555555555556)))), 2.0) + (a * (a * ((1.0 + Math.cos((Math.PI * (angle * 0.011111111111111112)))) / 2.0)));
}

Python

def code(a, b, angle):
	return math.pow((b * math.sin((math.pi * (angle * 0.005555555555555556)))), 2.0) + (a * (a * ((1.0 + math.cos((math.pi * (angle * 0.011111111111111112)))) / 2.0)))

Julia

function code(a, b, angle)
	return Float64((Float64(b * sin(Float64(pi * Float64(angle * 0.005555555555555556)))) ^ 2.0) + Float64(a * Float64(a * Float64(Float64(1.0 + cos(Float64(pi * Float64(angle * 0.011111111111111112)))) / 2.0))))
end

MATLAB

function tmp = code(a, b, angle)
	tmp = ((b * sin((pi * (angle * 0.005555555555555556)))) ^ 2.0) + (a * (a * ((1.0 + cos((pi * (angle * 0.011111111111111112)))) / 2.0)));
end

Wolfram

code[a_, b_, angle_] := N[(N[Power[N[(b * N[Sin[N[(Pi * N[(angle * 0.005555555555555556), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision] + N[(a * N[(a * N[(N[(1.0 + N[Cos[N[(Pi * N[(angle * 0.011111111111111112), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 78.3%

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

3. Simplified 78.3%

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

   1. metadata-eval 78.3%

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

   2. div-inv 78.3%

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

   3. add-cube-cbrt 77.7%

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

   4. pow3 77.7%

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

   5. *-commutative 77.7%

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

   6. *-commutative 77.7%

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

   7. div-inv 77.7%

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

   8. metadata-eval 77.7%

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

   9. associate-*l* 77.7%

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

6. Applied egg-rr 77.7%

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

   1. unpow3 77.7%

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

   2. add-cube-cbrt 78.3%

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

   3. *-commutative 78.3%

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

   4. unpow-prod-down 78.3%

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

   5. unpow2 78.3%

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

   6. associate-*r* 78.3%

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

   7. *-commutative 78.3%

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

   8. associate-*l* 78.3%

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

   9. *-commutative 78.3%

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

   10. associate-*r* 78.3%

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

8. Applied egg-rr 78.3%

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

   1. unpow2 78.3%

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

   2. cos-mult 78.3%

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

   3. associate-*r* 78.3%

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

   4. *-commutative 78.3%

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

   5. associate-*r* 78.3%

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

   6. *-commutative 78.3%

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

   7. associate-*r* 77.1%

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

   8. *-commutative 77.1%

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

   9. associate-*r* 78.3%

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

   10. *-commutative 78.3%

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

10. Applied egg-rr 78.3%

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

    1. +-commutative 78.3%

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

    2. +-inverses 78.3%

       \[\leadsto a \cdot \left(a \cdot \frac{\cos \color{blue}{0} + \cos \left(\pi \cdot \left(angle \cdot 0.005555555555555556\right) + \pi \cdot \left(angle \cdot 0.005555555555555556\right)\right)}{2}\right) + {\left(b \cdot \sin \left(\pi \cdot \left(angle \cdot 0.005555555555555556\right)\right)\right)}^{2} \]

    3. cos-0 78.3%

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

    4. distribute-lft-out 78.3%

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

    5. distribute-lft-out 78.3%

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

    6. metadata-eval 78.3%

       \[\leadsto a \cdot \left(a \cdot \frac{1 + \cos \left(\pi \cdot \left(angle \cdot \color{blue}{0.011111111111111112}\right)\right)}{2}\right) + {\left(b \cdot \sin \left(\pi \cdot \left(angle \cdot 0.005555555555555556\right)\right)\right)}^{2} \]

12. Simplified 78.3%

    \[\leadsto a \cdot \left(a \cdot \color{blue}{\frac{1 + \cos \left(\pi \cdot \left(angle \cdot 0.011111111111111112\right)\right)}{2}}\right) + {\left(b \cdot \sin \left(\pi \cdot \left(angle \cdot 0.005555555555555556\right)\right)\right)}^{2} \]

13. Final simplification 78.3%

    \[\leadsto {\left(b \cdot \sin \left(\pi \cdot \left(angle \cdot 0.005555555555555556\right)\right)\right)}^{2} + a \cdot \left(a \cdot \frac{1 + \cos \left(\pi \cdot \left(angle \cdot 0.011111111111111112\right)\right)}{2}\right) \]
14. Add Preprocessing

Alternative 3: 79.1% accurate, 1.3× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 78.3%

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

3. Simplified 78.3%

   \[\leadsto \color{blue}{{\left(a \cdot \cos \left(\pi \cdot \left(angle \cdot 0.005555555555555556\right)\right)\right)}^{2} + {\left(b \cdot \sin \left(\pi \cdot \left(angle \cdot 0.005555555555555556\right)\right)\right)}^{2}} \]
4. Add Preprocessing

5. Taylor expanded in angle around 0 78.0%

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

6. Final simplification 78.0%

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

Alternative 4: 52.8% accurate, 2.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \pi \cdot \left(angle \cdot 0.005555555555555556\right)\\ \mathbf{if}\;a \leq 5.9 \cdot 10^{-96}:\\ \;\;\;\;{\left(b \cdot \sin t\_0\right)}^{2}\\ \mathbf{else}:\\ \;\;\;\;{\left(a \cdot \cos t\_0\right)}^{2}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b angle)
 :precision binary64
 (let* ((t_0 (* PI (* angle 0.005555555555555556))))
   (if (<= a 5.9e-96) (pow (* b (sin t_0)) 2.0) (pow (* a (cos t_0)) 2.0))))

C

double code(double a, double b, double angle) {
	double t_0 = ((double) M_PI) * (angle * 0.005555555555555556);
	double tmp;
	if (a <= 5.9e-96) {
		tmp = pow((b * sin(t_0)), 2.0);
	} else {
		tmp = pow((a * cos(t_0)), 2.0);
	}
	return tmp;
}

Java

public static double code(double a, double b, double angle) {
	double t_0 = Math.PI * (angle * 0.005555555555555556);
	double tmp;
	if (a <= 5.9e-96) {
		tmp = Math.pow((b * Math.sin(t_0)), 2.0);
	} else {
		tmp = Math.pow((a * Math.cos(t_0)), 2.0);
	}
	return tmp;
}

Python

def code(a, b, angle):
	t_0 = math.pi * (angle * 0.005555555555555556)
	tmp = 0
	if a <= 5.9e-96:
		tmp = math.pow((b * math.sin(t_0)), 2.0)
	else:
		tmp = math.pow((a * math.cos(t_0)), 2.0)
	return tmp

Julia

function code(a, b, angle)
	t_0 = Float64(pi * Float64(angle * 0.005555555555555556))
	tmp = 0.0
	if (a <= 5.9e-96)
		tmp = Float64(b * sin(t_0)) ^ 2.0;
	else
		tmp = Float64(a * cos(t_0)) ^ 2.0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, angle)
	t_0 = pi * (angle * 0.005555555555555556);
	tmp = 0.0;
	if (a <= 5.9e-96)
		tmp = (b * sin(t_0)) ^ 2.0;
	else
		tmp = (a * cos(t_0)) ^ 2.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, angle_] := Block[{t$95$0 = N[(Pi * N[(angle * 0.005555555555555556), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[a, 5.9e-96], N[Power[N[(b * N[Sin[t$95$0], $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision], N[Power[N[(a * N[Cos[t$95$0], $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if a < 5.89999999999999966e-96

   1. Initial program 76.9%

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

   3. Simplified 76.9%

      \[\leadsto \color{blue}{{\left(a \cdot \cos \left(\pi \cdot \left(angle \cdot 0.005555555555555556\right)\right)\right)}^{2} + {\left(b \cdot \sin \left(\pi \cdot \left(angle \cdot 0.005555555555555556\right)\right)\right)}^{2}} \]
   4. Add Preprocessing

   5. Taylor expanded in a around 0 42.5%

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

      1. associate-*r* 42.6%

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

      2. *-commutative 42.6%

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

      3. associate-*r* 42.6%

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

      4. unpow2 42.6%

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

      5. unpow2 42.6%

         \[\leadsto \color{blue}{\left(b \cdot b\right)} \cdot \left(\sin \left(angle \cdot \left(0.005555555555555556 \cdot \pi\right)\right) \cdot \sin \left(angle \cdot \left(0.005555555555555556 \cdot \pi\right)\right)\right) \]

      6. swap-sqr 47.9%

         \[\leadsto \color{blue}{\left(b \cdot \sin \left(angle \cdot \left(0.005555555555555556 \cdot \pi\right)\right)\right) \cdot \left(b \cdot \sin \left(angle \cdot \left(0.005555555555555556 \cdot \pi\right)\right)\right)} \]

      7. unpow2 47.9%

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

      8. associate-*r* 47.9%

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

      9. *-commutative 47.9%

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

      10. associate-*r* 47.8%

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

      11. associate-*r* 47.9%

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

      12. *-commutative 47.9%

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

      13. *-commutative 47.9%

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

      14. *-commutative 47.9%

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

   7. Simplified 47.9%

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

   if 5.89999999999999966e-96 < a

   1. Initial program 80.8%

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

   3. Simplified 80.8%

      \[\leadsto \color{blue}{{\left(a \cdot \cos \left(\pi \cdot \left(angle \cdot 0.005555555555555556\right)\right)\right)}^{2} + {\left(b \cdot \sin \left(\pi \cdot \left(angle \cdot 0.005555555555555556\right)\right)\right)}^{2}} \]
   4. Add Preprocessing

   5. Taylor expanded in a around inf 72.8%

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

      1. *-commutative 72.8%

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

      2. associate-*r* 73.0%

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

      3. *-commutative 73.0%

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

      4. associate-*r* 73.0%

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

      5. unpow2 73.0%

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

      6. unpow2 73.0%

         \[\leadsto \left(\cos \left(angle \cdot \left(0.005555555555555556 \cdot \pi\right)\right) \cdot \cos \left(angle \cdot \left(0.005555555555555556 \cdot \pi\right)\right)\right) \cdot \color{blue}{\left(a \cdot a\right)} \]

      7. swap-sqr 73.0%

         \[\leadsto \color{blue}{\left(\cos \left(angle \cdot \left(0.005555555555555556 \cdot \pi\right)\right) \cdot a\right) \cdot \left(\cos \left(angle \cdot \left(0.005555555555555556 \cdot \pi\right)\right) \cdot a\right)} \]

      8. unpow2 73.0%

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

      9. *-commutative 73.0%

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

      10. associate-*r* 73.0%

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

      11. *-commutative 73.0%

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

   7. Simplified 73.0%

      \[\leadsto \color{blue}{{\left(a \cdot \cos \left(\left(0.005555555555555556 \cdot angle\right) \cdot \pi\right)\right)}^{2}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 57.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \leq 5.9 \cdot 10^{-96}:\\ \;\;\;\;{\left(b \cdot \sin \left(\pi \cdot \left(angle \cdot 0.005555555555555556\right)\right)\right)}^{2}\\ \mathbf{else}:\\ \;\;\;\;{\left(a \cdot \cos \left(\pi \cdot \left(angle \cdot 0.005555555555555556\right)\right)\right)}^{2}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 52.8% accurate, 2.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;a \leq 3.6 \cdot 10^{-96}:\\ \;\;\;\;{\left(b \cdot \sin \left(angle \cdot \left(\pi \cdot 0.005555555555555556\right)\right)\right)}^{2}\\ \mathbf{else}:\\ \;\;\;\;{\left(a \cdot \cos \left(\pi \cdot \left(angle \cdot 0.005555555555555556\right)\right)\right)}^{2}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b angle)
 :precision binary64
 (if (<= a 3.6e-96)
   (pow (* b (sin (* angle (* PI 0.005555555555555556)))) 2.0)
   (pow (* a (cos (* PI (* angle 0.005555555555555556)))) 2.0)))

C

double code(double a, double b, double angle) {
	double tmp;
	if (a <= 3.6e-96) {
		tmp = pow((b * sin((angle * (((double) M_PI) * 0.005555555555555556)))), 2.0);
	} else {
		tmp = pow((a * cos((((double) M_PI) * (angle * 0.005555555555555556)))), 2.0);
	}
	return tmp;
}

Java

public static double code(double a, double b, double angle) {
	double tmp;
	if (a <= 3.6e-96) {
		tmp = Math.pow((b * Math.sin((angle * (Math.PI * 0.005555555555555556)))), 2.0);
	} else {
		tmp = Math.pow((a * Math.cos((Math.PI * (angle * 0.005555555555555556)))), 2.0);
	}
	return tmp;
}

Python

def code(a, b, angle):
	tmp = 0
	if a <= 3.6e-96:
		tmp = math.pow((b * math.sin((angle * (math.pi * 0.005555555555555556)))), 2.0)
	else:
		tmp = math.pow((a * math.cos((math.pi * (angle * 0.005555555555555556)))), 2.0)
	return tmp

Julia

function code(a, b, angle)
	tmp = 0.0
	if (a <= 3.6e-96)
		tmp = Float64(b * sin(Float64(angle * Float64(pi * 0.005555555555555556)))) ^ 2.0;
	else
		tmp = Float64(a * cos(Float64(pi * Float64(angle * 0.005555555555555556)))) ^ 2.0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, angle)
	tmp = 0.0;
	if (a <= 3.6e-96)
		tmp = (b * sin((angle * (pi * 0.005555555555555556)))) ^ 2.0;
	else
		tmp = (a * cos((pi * (angle * 0.005555555555555556)))) ^ 2.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, angle_] := If[LessEqual[a, 3.6e-96], N[Power[N[(b * N[Sin[N[(angle * N[(Pi * 0.005555555555555556), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision], N[Power[N[(a * N[Cos[N[(Pi * N[(angle * 0.005555555555555556), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if a < 3.60000000000000008e-96

   1. Initial program 76.9%

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

   3. Simplified 76.9%

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

      1. metadata-eval 76.9%

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

      2. div-inv 76.9%

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

      3. unpow2 76.9%

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

      4. associate-*r* 76.9%

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

      5. *-commutative 76.9%

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

      6. *-commutative 76.9%

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

      7. div-inv 74.6%

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

      8. metadata-eval 74.6%

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

      9. associate-*l* 75.5%

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

      10. *-commutative 75.5%

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

      11. div-inv 75.9%

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

      12. metadata-eval 75.9%

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

      13. associate-*l* 76.8%

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

   6. Applied egg-rr 76.8%

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

   7. Taylor expanded in a around 0 42.5%

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

      1. unpow2 42.5%

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

      2. *-commutative 42.5%

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

      3. associate-*l* 42.6%

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

      4. *-commutative 42.6%

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

      5. unpow2 42.6%

         \[\leadsto \left(b \cdot b\right) \cdot \color{blue}{\left(\sin \left(angle \cdot \left(0.005555555555555556 \cdot \pi\right)\right) \cdot \sin \left(angle \cdot \left(0.005555555555555556 \cdot \pi\right)\right)\right)} \]

      6. swap-sqr 47.9%

         \[\leadsto \color{blue}{\left(b \cdot \sin \left(angle \cdot \left(0.005555555555555556 \cdot \pi\right)\right)\right) \cdot \left(b \cdot \sin \left(angle \cdot \left(0.005555555555555556 \cdot \pi\right)\right)\right)} \]

      7. unpow2 47.9%

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

      8. *-commutative 47.9%

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

   9. Simplified 47.9%

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

   if 3.60000000000000008e-96 < a

   1. Initial program 80.8%

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

   3. Simplified 80.8%

      \[\leadsto \color{blue}{{\left(a \cdot \cos \left(\pi \cdot \left(angle \cdot 0.005555555555555556\right)\right)\right)}^{2} + {\left(b \cdot \sin \left(\pi \cdot \left(angle \cdot 0.005555555555555556\right)\right)\right)}^{2}} \]
   4. Add Preprocessing

   5. Taylor expanded in a around inf 72.8%

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

      1. *-commutative 72.8%

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

      2. associate-*r* 73.0%

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

      3. *-commutative 73.0%

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

      4. associate-*r* 73.0%

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

      5. unpow2 73.0%

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

      6. unpow2 73.0%

         \[\leadsto \left(\cos \left(angle \cdot \left(0.005555555555555556 \cdot \pi\right)\right) \cdot \cos \left(angle \cdot \left(0.005555555555555556 \cdot \pi\right)\right)\right) \cdot \color{blue}{\left(a \cdot a\right)} \]

      7. swap-sqr 73.0%

         \[\leadsto \color{blue}{\left(\cos \left(angle \cdot \left(0.005555555555555556 \cdot \pi\right)\right) \cdot a\right) \cdot \left(\cos \left(angle \cdot \left(0.005555555555555556 \cdot \pi\right)\right) \cdot a\right)} \]

      8. unpow2 73.0%

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

      9. *-commutative 73.0%

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

      10. associate-*r* 73.0%

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

      11. *-commutative 73.0%

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

   7. Simplified 73.0%

      \[\leadsto \color{blue}{{\left(a \cdot \cos \left(\left(0.005555555555555556 \cdot angle\right) \cdot \pi\right)\right)}^{2}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 57.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \leq 3.6 \cdot 10^{-96}:\\ \;\;\;\;{\left(b \cdot \sin \left(angle \cdot \left(\pi \cdot 0.005555555555555556\right)\right)\right)}^{2}\\ \mathbf{else}:\\ \;\;\;\;{\left(a \cdot \cos \left(\pi \cdot \left(angle \cdot 0.005555555555555556\right)\right)\right)}^{2}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 52.8% accurate, 2.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;a \leq 8.2 \cdot 10^{-98}:\\ \;\;\;\;{\left(b \cdot \sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)\right)}^{2}\\ \mathbf{else}:\\ \;\;\;\;{\left(a \cdot \cos \left(\pi \cdot \left(angle \cdot 0.005555555555555556\right)\right)\right)}^{2}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b angle)
 :precision binary64
 (if (<= a 8.2e-98)
   (pow (* b (sin (* 0.005555555555555556 (* angle PI)))) 2.0)
   (pow (* a (cos (* PI (* angle 0.005555555555555556)))) 2.0)))

C

double code(double a, double b, double angle) {
	double tmp;
	if (a <= 8.2e-98) {
		tmp = pow((b * sin((0.005555555555555556 * (angle * ((double) M_PI))))), 2.0);
	} else {
		tmp = pow((a * cos((((double) M_PI) * (angle * 0.005555555555555556)))), 2.0);
	}
	return tmp;
}

Java

public static double code(double a, double b, double angle) {
	double tmp;
	if (a <= 8.2e-98) {
		tmp = Math.pow((b * Math.sin((0.005555555555555556 * (angle * Math.PI)))), 2.0);
	} else {
		tmp = Math.pow((a * Math.cos((Math.PI * (angle * 0.005555555555555556)))), 2.0);
	}
	return tmp;
}

Python

def code(a, b, angle):
	tmp = 0
	if a <= 8.2e-98:
		tmp = math.pow((b * math.sin((0.005555555555555556 * (angle * math.pi)))), 2.0)
	else:
		tmp = math.pow((a * math.cos((math.pi * (angle * 0.005555555555555556)))), 2.0)
	return tmp

Julia

function code(a, b, angle)
	tmp = 0.0
	if (a <= 8.2e-98)
		tmp = Float64(b * sin(Float64(0.005555555555555556 * Float64(angle * pi)))) ^ 2.0;
	else
		tmp = Float64(a * cos(Float64(pi * Float64(angle * 0.005555555555555556)))) ^ 2.0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, angle)
	tmp = 0.0;
	if (a <= 8.2e-98)
		tmp = (b * sin((0.005555555555555556 * (angle * pi)))) ^ 2.0;
	else
		tmp = (a * cos((pi * (angle * 0.005555555555555556)))) ^ 2.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, angle_] := If[LessEqual[a, 8.2e-98], N[Power[N[(b * N[Sin[N[(0.005555555555555556 * N[(angle * Pi), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision], N[Power[N[(a * N[Cos[N[(Pi * N[(angle * 0.005555555555555556), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if a < 8.1999999999999996e-98

   1. Initial program 76.9%

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

   3. Simplified 76.9%

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

      1. metadata-eval 76.9%

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

      2. div-inv 76.9%

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

      3. unpow2 76.9%

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

      4. associate-*r* 76.9%

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

      5. *-commutative 76.9%

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

      6. *-commutative 76.9%

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

      7. div-inv 74.6%

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

      8. metadata-eval 74.6%

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

      9. associate-*l* 75.5%

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

      10. *-commutative 75.5%

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

      11. div-inv 75.9%

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

      12. metadata-eval 75.9%

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

      13. associate-*l* 76.8%

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

   6. Applied egg-rr 76.8%

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

   8. Applied egg-rr 76.9%

      \[\leadsto \color{blue}{{\left(\mathsf{hypot}\left(\cos \left(angle \cdot \left(0.005555555555555556 \cdot \pi\right)\right) \cdot a, b \cdot \sin \left(angle \cdot \left(0.005555555555555556 \cdot \pi\right)\right)\right)\right)}^{2}} \]

   9. Taylor expanded in a around 0 42.5%

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

       1. *-commutative 42.5%

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

       2. associate-*r* 42.6%

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

       3. *-commutative 42.6%

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

       4. associate-*r* 42.6%

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

       5. unpow2 42.6%

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

       6. unpow2 42.6%

          \[\leadsto \left(\sin \left(angle \cdot \left(0.005555555555555556 \cdot \pi\right)\right) \cdot \sin \left(angle \cdot \left(0.005555555555555556 \cdot \pi\right)\right)\right) \cdot \color{blue}{\left(b \cdot b\right)} \]

       7. swap-sqr 47.9%

          \[\leadsto \color{blue}{\left(\sin \left(angle \cdot \left(0.005555555555555556 \cdot \pi\right)\right) \cdot b\right) \cdot \left(\sin \left(angle \cdot \left(0.005555555555555556 \cdot \pi\right)\right) \cdot b\right)} \]

       8. unpow2 47.9%

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

       9. associate-*r* 47.9%

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

       10. *-commutative 47.9%

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

       11. associate-*r* 47.8%

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

       12. *-commutative 47.8%

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

   11. Simplified 47.8%

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

   if 8.1999999999999996e-98 < a

   1. Initial program 80.8%

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

   3. Simplified 80.8%

      \[\leadsto \color{blue}{{\left(a \cdot \cos \left(\pi \cdot \left(angle \cdot 0.005555555555555556\right)\right)\right)}^{2} + {\left(b \cdot \sin \left(\pi \cdot \left(angle \cdot 0.005555555555555556\right)\right)\right)}^{2}} \]
   4. Add Preprocessing

   5. Taylor expanded in a around inf 72.8%

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

      1. *-commutative 72.8%

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

      2. associate-*r* 73.0%

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

      3. *-commutative 73.0%

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

      4. associate-*r* 73.0%

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

      5. unpow2 73.0%

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

      6. unpow2 73.0%

         \[\leadsto \left(\cos \left(angle \cdot \left(0.005555555555555556 \cdot \pi\right)\right) \cdot \cos \left(angle \cdot \left(0.005555555555555556 \cdot \pi\right)\right)\right) \cdot \color{blue}{\left(a \cdot a\right)} \]

      7. swap-sqr 73.0%

         \[\leadsto \color{blue}{\left(\cos \left(angle \cdot \left(0.005555555555555556 \cdot \pi\right)\right) \cdot a\right) \cdot \left(\cos \left(angle \cdot \left(0.005555555555555556 \cdot \pi\right)\right) \cdot a\right)} \]

      8. unpow2 73.0%

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

      9. *-commutative 73.0%

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

      10. associate-*r* 73.0%

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

      11. *-commutative 73.0%

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

   7. Simplified 73.0%

      \[\leadsto \color{blue}{{\left(a \cdot \cos \left(\left(0.005555555555555556 \cdot angle\right) \cdot \pi\right)\right)}^{2}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 56.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \leq 8.2 \cdot 10^{-98}:\\ \;\;\;\;{\left(b \cdot \sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)\right)}^{2}\\ \mathbf{else}:\\ \;\;\;\;{\left(a \cdot \cos \left(\pi \cdot \left(angle \cdot 0.005555555555555556\right)\right)\right)}^{2}\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 55.8% accurate, 2.0× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 78.3%

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

3. Simplified 78.3%

   \[\leadsto \color{blue}{{\left(a \cdot \cos \left(\pi \cdot \left(angle \cdot 0.005555555555555556\right)\right)\right)}^{2} + {\left(b \cdot \sin \left(\pi \cdot \left(angle \cdot 0.005555555555555556\right)\right)\right)}^{2}} \]
4. Add Preprocessing

5. Taylor expanded in a around inf 57.6%

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

   1. *-commutative 57.6%

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

   2. associate-*r* 57.8%

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

   3. *-commutative 57.8%

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

   4. associate-*r* 57.8%

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

   5. unpow2 57.8%

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

   6. unpow2 57.8%

      \[\leadsto \left(\cos \left(angle \cdot \left(0.005555555555555556 \cdot \pi\right)\right) \cdot \cos \left(angle \cdot \left(0.005555555555555556 \cdot \pi\right)\right)\right) \cdot \color{blue}{\left(a \cdot a\right)} \]

   7. swap-sqr 57.8%

      \[\leadsto \color{blue}{\left(\cos \left(angle \cdot \left(0.005555555555555556 \cdot \pi\right)\right) \cdot a\right) \cdot \left(\cos \left(angle \cdot \left(0.005555555555555556 \cdot \pi\right)\right) \cdot a\right)} \]

   8. unpow2 57.8%

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

   9. *-commutative 57.8%

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

   10. associate-*r* 57.8%

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

   11. *-commutative 57.8%

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

7. Simplified 57.8%

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

8. Final simplification 57.8%

   \[\leadsto {\left(a \cdot \cos \left(\pi \cdot \left(angle \cdot 0.005555555555555556\right)\right)\right)}^{2} \]
9. Add Preprocessing

Alternative 8: 55.8% accurate, 2.0× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 78.3%

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

3. Simplified 78.3%

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

   1. metadata-eval 78.3%

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

   2. div-inv 78.3%

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

   3. unpow2 78.3%

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

   4. associate-*r* 78.3%

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

   5. *-commutative 78.3%

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

   6. *-commutative 78.3%

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

   7. div-inv 75.6%

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

   8. metadata-eval 75.6%

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

   9. associate-*l* 76.3%

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

   10. *-commutative 76.3%

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

   11. div-inv 77.6%

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

   12. metadata-eval 77.6%

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

   13. associate-*l* 78.3%

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

6. Applied egg-rr 78.3%

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

7. Taylor expanded in a around inf 57.6%

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

   1. *-commutative 57.6%

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

   2. *-commutative 57.6%

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

   3. associate-*l* 57.8%

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

   4. *-commutative 57.8%

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

   5. unpow2 57.8%

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

   6. unpow2 57.8%

      \[\leadsto \left(\cos \left(angle \cdot \left(0.005555555555555556 \cdot \pi\right)\right) \cdot \cos \left(angle \cdot \left(0.005555555555555556 \cdot \pi\right)\right)\right) \cdot \color{blue}{\left(a \cdot a\right)} \]

   7. swap-sqr 57.8%

      \[\leadsto \color{blue}{\left(\cos \left(angle \cdot \left(0.005555555555555556 \cdot \pi\right)\right) \cdot a\right) \cdot \left(\cos \left(angle \cdot \left(0.005555555555555556 \cdot \pi\right)\right) \cdot a\right)} \]

   8. unpow2 57.8%

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

   9. *-commutative 57.8%

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

   10. associate-*l* 57.6%

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

   11. *-commutative 57.6%

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

   12. *-commutative 57.6%

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

   13. *-commutative 57.6%

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

   14. associate-*l* 57.8%

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

9. Simplified 57.8%

   \[\leadsto \color{blue}{{\left(a \cdot \cos \left(angle \cdot \left(\pi \cdot 0.005555555555555556\right)\right)\right)}^{2}} \]
10. Add Preprocessing

Alternative 9: 55.8% accurate, 2.0× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 78.3%

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

3. Simplified 78.3%

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

   1. metadata-eval 78.3%

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

   2. div-inv 78.3%

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

   3. unpow2 78.3%

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

   4. associate-*r* 78.3%

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

   5. *-commutative 78.3%

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

   6. *-commutative 78.3%

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

   7. div-inv 75.6%

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

   8. metadata-eval 75.6%

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

   9. associate-*l* 76.3%

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

   10. *-commutative 76.3%

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

   11. div-inv 77.6%

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

   12. metadata-eval 77.6%

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

   13. associate-*l* 78.3%

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

6. Applied egg-rr 78.3%

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

8. Applied egg-rr 78.3%

   \[\leadsto \color{blue}{{\left(\mathsf{hypot}\left(\cos \left(angle \cdot \left(0.005555555555555556 \cdot \pi\right)\right) \cdot a, b \cdot \sin \left(angle \cdot \left(0.005555555555555556 \cdot \pi\right)\right)\right)\right)}^{2}} \]

9. Taylor expanded in a around inf 57.6%

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

    1. unpow2 57.6%

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

    2. associate-*r* 57.8%

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

    3. *-commutative 57.8%

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

    4. associate-*r* 57.8%

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

    5. unpow2 57.8%

       \[\leadsto \left(a \cdot a\right) \cdot \color{blue}{\left(\cos \left(angle \cdot \left(0.005555555555555556 \cdot \pi\right)\right) \cdot \cos \left(angle \cdot \left(0.005555555555555556 \cdot \pi\right)\right)\right)} \]

    6. swap-sqr 57.8%

       \[\leadsto \color{blue}{\left(a \cdot \cos \left(angle \cdot \left(0.005555555555555556 \cdot \pi\right)\right)\right) \cdot \left(a \cdot \cos \left(angle \cdot \left(0.005555555555555556 \cdot \pi\right)\right)\right)} \]

    7. unpow2 57.8%

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

    8. associate-*r* 57.8%

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

    9. *-commutative 57.8%

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

    10. associate-*r* 57.6%

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

11. Simplified 57.6%

    \[\leadsto \color{blue}{{\left(a \cdot \cos \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)\right)}^{2}} \]
12. Add Preprocessing

Alternative 10: 55.9% accurate, 139.0× speedup

Math

\[\begin{array}{l} \\ a \cdot a \end{array} \]

FPCore

(FPCore (a b angle) :precision binary64 (* a a))

C

double code(double a, double b, double angle) {
	return a * a;
}

Fortran

real(8) function code(a, b, angle)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: angle
    code = a * a
end function

Java

public static double code(double a, double b, double angle) {
	return a * a;
}

Python

def code(a, b, angle):
	return a * a

Julia

function code(a, b, angle)
	return Float64(a * a)
end

MATLAB

function tmp = code(a, b, angle)
	tmp = a * a;
end

Wolfram

code[a_, b_, angle_] := N[(a * a), $MachinePrecision]

Derivation

1. Initial program 78.3%

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

3. Simplified 78.3%

   \[\leadsto \color{blue}{{\left(a \cdot \cos \left(\pi \cdot \left(angle \cdot 0.005555555555555556\right)\right)\right)}^{2} + {\left(b \cdot \sin \left(\pi \cdot \left(angle \cdot 0.005555555555555556\right)\right)\right)}^{2}} \]
4. Add Preprocessing

5. Taylor expanded in angle around 0 57.5%

   \[\leadsto \color{blue}{{a}^{2}} \]
6. Step-by-step derivation

   1. unpow2 57.5%

      \[\leadsto \color{blue}{a \cdot a} \]

7. Applied egg-rr 57.5%

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

Reproduce

herbie shell --seed 2024121 
(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)))