ab-angle->ABCF C

Percentage Accurate: 79.5% → 79.5%

Time: 14.3s

Alternatives: 12

Speedup: 1.0×

• 36.7% of points produce a very large (infinite) output. You may want to add a precondition.

Specification

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 79.5% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 79.5% accurate, 0.6× speedup

Math

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

FPCore

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

C

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

Java

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

Julia

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

Wolfram

code[a_, b_, angle_] := N[(N[Power[N[(a * N[Cos[N[(N[Power[N[Power[Pi, 1/3], $MachinePrecision], 2.0], $MachinePrecision] * N[(N[Power[Pi, 1/3], $MachinePrecision] * N[(0.005555555555555556 * angle), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision] + N[Power[N[(b * N[Sin[N[(Pi * N[(0.005555555555555556 * angle), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 81.0%

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

   1. associate-*r/ 81.0%

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

   2. metadata-eval 81.0%

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

   3. metadata-eval 81.0%

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

   4. distribute-neg-frac2 81.0%

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

   5. distribute-frac-neg 81.0%

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

   6. distribute-rgt-neg-out 81.0%

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

   7. associate-/l* 81.0%

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

   8. neg-mul-1 81.0%

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

   9. *-commutative 81.0%

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

   10. associate-/l* 81.1%

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

   11. metadata-eval 81.1%

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

   12. metadata-eval 81.1%

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

3. Simplified 81.2%

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

   1. metadata-eval 81.2%

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

   2. div-inv 81.1%

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

   3. add-sqr-sqrt 42.4%

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

   4. pow2 42.4%

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

   5. div-inv 42.5%

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

   6. metadata-eval 42.5%

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

   7. associate-*r* 42.5%

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

   8. *-commutative 42.5%

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

6. Applied egg-rr 42.5%

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

   1. unpow2 42.5%

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

   2. add-sqr-sqrt 81.1%

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

   3. *-commutative 81.1%

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

   4. associate-*r* 81.2%

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

   5. add-cube-cbrt 81.3%

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

   6. associate-*l* 81.3%

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

   7. pow2 81.3%

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

   8. *-commutative 81.3%

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

8. Applied egg-rr 81.3%

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

9. Final simplification 81.3%

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

Alternative 2: 79.5% accurate, 0.7× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 81.0%

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

   1. associate-*r/ 81.0%

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

   2. metadata-eval 81.0%

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

   3. metadata-eval 81.0%

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

   4. distribute-neg-frac2 81.0%

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

   5. distribute-frac-neg 81.0%

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

   6. distribute-rgt-neg-out 81.0%

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

   7. associate-/l* 81.0%

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

   8. neg-mul-1 81.0%

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

   9. *-commutative 81.0%

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

   10. associate-/l* 81.1%

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

   11. metadata-eval 81.1%

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

   12. metadata-eval 81.1%

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

3. Simplified 81.2%

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

   1. metadata-eval 81.2%

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

   2. div-inv 81.1%

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

   3. add-sqr-sqrt 42.4%

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

   4. pow2 42.4%

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

   5. div-inv 42.5%

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

   6. metadata-eval 42.5%

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

   7. associate-*r* 42.5%

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

   8. *-commutative 42.5%

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

6. Applied egg-rr 42.5%

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

   1. unpow2 42.5%

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

   2. add-sqr-sqrt 81.1%

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

   3. *-commutative 81.1%

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

   4. associate-*r* 81.2%

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

   5. *-commutative 81.2%

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

   6. add-sqr-sqrt 81.2%

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

   7. associate-*r* 81.3%

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

   8. *-commutative 81.3%

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

8. Applied egg-rr 81.3%

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

9. Final simplification 81.3%

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

Alternative 3: 79.4% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \pi \cdot \left(0.005555555555555556 \cdot angle\right)\\ {\left(b \cdot \sin t\_0\right)}^{2} + {\left(a \cdot \cos \left({\left(\sqrt[3]{t\_0}\right)}^{3}\right)\right)}^{2} \end{array} \end{array} \]

FPCore

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

C

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

Java

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

Julia

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

Wolfram

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

Derivation

1. Initial program 81.0%

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

   1. associate-*r/ 81.0%

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

   2. metadata-eval 81.0%

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

   3. metadata-eval 81.0%

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

   4. distribute-neg-frac2 81.0%

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

   5. distribute-frac-neg 81.0%

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

   6. distribute-rgt-neg-out 81.0%

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

   7. associate-/l* 81.0%

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

   8. neg-mul-1 81.0%

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

   9. *-commutative 81.0%

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

   10. associate-/l* 81.1%

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

   11. metadata-eval 81.1%

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

   12. metadata-eval 81.1%

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

3. Simplified 81.2%

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

   1. metadata-eval 81.2%

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

   2. div-inv 81.1%

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

   3. associate-*r/ 81.1%

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

   4. clear-num 81.1%

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

6. Applied egg-rr 81.1%

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

   1. associate-/r/ 81.1%

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

   2. metadata-eval 81.1%

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

   3. add-sqr-sqrt 42.5%

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

   4. unpow2 42.5%

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

   5. add-cube-cbrt 42.5%

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

   6. pow3 42.5%

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

   7. unpow2 42.5%

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

   8. add-sqr-sqrt 81.2%

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

   9. *-commutative 81.2%

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

   10. associate-*r* 81.2%

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

   11. *-commutative 81.2%

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

8. Applied egg-rr 81.2%

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

9. Final simplification 81.2%

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

Alternative 4: 79.5% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \pi \cdot \left(0.005555555555555556 \cdot angle\right)\\ {\left(\mathsf{hypot}\left(a \cdot \cos t\_0, b \cdot \sin t\_0\right)\right)}^{2} \end{array} \end{array} \]

FPCore

(FPCore (a b angle)
 :precision binary64
 (let* ((t_0 (* PI (* 0.005555555555555556 angle))))
   (pow (hypot (* a (cos t_0)) (* b (sin t_0))) 2.0)))

C

double code(double a, double b, double angle) {
	double t_0 = ((double) M_PI) * (0.005555555555555556 * angle);
	return pow(hypot((a * cos(t_0)), (b * sin(t_0))), 2.0);
}

Java

public static double code(double a, double b, double angle) {
	double t_0 = Math.PI * (0.005555555555555556 * angle);
	return Math.pow(Math.hypot((a * Math.cos(t_0)), (b * Math.sin(t_0))), 2.0);
}

Python

def code(a, b, angle):
	t_0 = math.pi * (0.005555555555555556 * angle)
	return math.pow(math.hypot((a * math.cos(t_0)), (b * math.sin(t_0))), 2.0)

Julia

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

MATLAB

function tmp = code(a, b, angle)
	t_0 = pi * (0.005555555555555556 * angle);
	tmp = hypot((a * cos(t_0)), (b * sin(t_0))) ^ 2.0;
end

Wolfram

code[a_, b_, angle_] := Block[{t$95$0 = N[(Pi * N[(0.005555555555555556 * angle), $MachinePrecision]), $MachinePrecision]}, N[Power[N[Sqrt[N[(a * N[Cos[t$95$0], $MachinePrecision]), $MachinePrecision] ^ 2 + N[(b * N[Sin[t$95$0], $MachinePrecision]), $MachinePrecision] ^ 2], $MachinePrecision], 2.0], $MachinePrecision]]

Derivation

1. Initial program 81.0%

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

   1. associate-*r/ 81.0%

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

   2. metadata-eval 81.0%

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

   3. metadata-eval 81.0%

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

   4. distribute-neg-frac2 81.0%

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

   5. distribute-frac-neg 81.0%

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

   6. distribute-rgt-neg-out 81.0%

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

   7. associate-/l* 81.0%

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

   8. neg-mul-1 81.0%

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

   9. *-commutative 81.0%

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

   10. associate-/l* 81.1%

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

   11. metadata-eval 81.1%

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

   12. metadata-eval 81.1%

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

3. Simplified 81.2%

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

   1. metadata-eval 81.2%

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

   2. div-inv 81.1%

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

   3. add-sqr-sqrt 42.4%

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

   4. pow2 42.4%

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

   5. div-inv 42.5%

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

   6. metadata-eval 42.5%

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

   7. associate-*r* 42.5%

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

   8. *-commutative 42.5%

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

6. Applied egg-rr 42.5%

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

8. Applied egg-rr 81.2%

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

Alternative 5: 79.3% accurate, 1.3× speedup

Math

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

FPCore

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

C

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 81.0%

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

   1. associate-*r/ 81.0%

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

   2. metadata-eval 81.0%

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

   3. metadata-eval 81.0%

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

   4. distribute-neg-frac2 81.0%

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

   5. distribute-frac-neg 81.0%

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

   6. distribute-rgt-neg-out 81.0%

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

   7. associate-/l* 81.0%

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

   8. neg-mul-1 81.0%

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

   9. *-commutative 81.0%

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

   10. associate-/l* 81.1%

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

   11. metadata-eval 81.1%

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

   12. metadata-eval 81.1%

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

3. Simplified 81.2%

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

5. Taylor expanded in angle around 0 81.0%

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

6. Final simplification 81.0%

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

Alternative 6: 52.6% accurate, 2.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \pi \cdot \left(0.005555555555555556 \cdot angle\right)\\ \mathbf{if}\;a \leq 1.4 \cdot 10^{-132}:\\ \;\;\;\;{\left(b \cdot \sin t\_0\right)}^{2}\\ \mathbf{else}:\\ \;\;\;\;{\left(a \cdot \cos t\_0\right)}^{2}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b angle)
 :precision binary64
 (let* ((t_0 (* PI (* 0.005555555555555556 angle))))
   (if (<= a 1.4e-132) (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) * (0.005555555555555556 * angle);
	double tmp;
	if (a <= 1.4e-132) {
		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 * (0.005555555555555556 * angle);
	double tmp;
	if (a <= 1.4e-132) {
		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 * (0.005555555555555556 * angle)
	tmp = 0
	if a <= 1.4e-132:
		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(0.005555555555555556 * angle))
	tmp = 0.0
	if (a <= 1.4e-132)
		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 * (0.005555555555555556 * angle);
	tmp = 0.0;
	if (a <= 1.4e-132)
		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[(0.005555555555555556 * angle), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[a, 1.4e-132], 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 < 1.40000000000000001e-132

   1. Initial program 78.9%

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

      1. associate-*r/ 78.9%

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

      2. metadata-eval 78.9%

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

      3. metadata-eval 78.9%

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

      4. distribute-neg-frac2 78.9%

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

      5. distribute-frac-neg 78.9%

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

      6. distribute-rgt-neg-out 78.9%

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

      7. associate-/l* 78.9%

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

      8. neg-mul-1 78.9%

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

      9. *-commutative 78.9%

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

      10. associate-/l* 78.9%

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

      11. metadata-eval 78.9%

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

      12. metadata-eval 78.9%

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

   3. Simplified 79.0%

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

      1. metadata-eval 79.0%

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

      2. div-inv 79.0%

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

      3. add-sqr-sqrt 44.0%

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

      4. pow2 44.0%

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

      5. div-inv 44.0%

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

      6. metadata-eval 44.0%

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

      7. associate-*r* 44.1%

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

      8. *-commutative 44.1%

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

   6. Applied egg-rr 44.1%

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

      1. unpow2 44.1%

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

      2. add-sqr-sqrt 79.0%

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

      3. metadata-eval 79.0%

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

      4. associate-/r/ 79.0%

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

      5. clear-num 79.0%

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

      6. expm1-log1p-u 67.4%

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

      7. div-inv 67.4%

         \[\leadsto {\left(a \cdot \cos \left(\mathsf{expm1}\left(\mathsf{log1p}\left(\color{blue}{\left(\pi \cdot angle\right) \cdot \frac{1}{180}}\right)\right)\right)\right)}^{2} + {\left(b \cdot \sin \left(\pi \cdot \left(angle \cdot 0.005555555555555556\right)\right)\right)}^{2} \]

      8. metadata-eval 67.4%

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

      9. associate-*r* 67.4%

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

      10. *-commutative 67.4%

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

   8. Applied egg-rr 67.4%

      \[\leadsto {\left(a \cdot \cos \color{blue}{\left(\mathsf{expm1}\left(\mathsf{log1p}\left(\pi \cdot \left(0.005555555555555556 \cdot angle\right)\right)\right)\right)}\right)}^{2} + {\left(b \cdot \sin \left(\pi \cdot \left(angle \cdot 0.005555555555555556\right)\right)\right)}^{2} \]
   9. Step-by-step derivation

      1. expm1-log1p-u 79.0%

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

      2. add-cube-cbrt 79.1%

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

      3. unpow2 79.1%

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

      4. associate-*r* 79.2%

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

      5. *-commutative 79.2%

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

      6. associate-*r* 79.1%

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

      7. associate-*l* 79.1%

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

   10. Applied egg-rr 79.1%

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

   11. Taylor expanded in a around 0 40.1%

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

       1. unpow2 40.1%

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

       2. associate-*r* 40.2%

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

       3. *-commutative 40.2%

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

       4. unpow2 40.2%

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

       5. swap-sqr 45.2%

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

       6. unpow2 45.2%

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

   13. Simplified 45.2%

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

   if 1.40000000000000001e-132 < a

   1. Initial program 84.9%

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

      1. associate-*r/ 85.0%

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

      2. metadata-eval 85.0%

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

      3. metadata-eval 85.0%

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

      4. distribute-neg-frac2 85.0%

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

      5. distribute-frac-neg 85.0%

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

      6. distribute-rgt-neg-out 85.0%

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

      7. associate-/l* 84.9%

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

      8. neg-mul-1 84.9%

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

      9. *-commutative 84.9%

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

      10. associate-/l* 85.1%

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

      11. metadata-eval 85.1%

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

      12. metadata-eval 85.1%

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

   3. Simplified 85.1%

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

      1. metadata-eval 85.1%

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

      2. div-inv 84.9%

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

      3. add-sqr-sqrt 39.6%

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

      4. pow2 39.6%

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

      5. div-inv 39.6%

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

      6. metadata-eval 39.6%

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

      7. associate-*r* 39.6%

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

      8. *-commutative 39.6%

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

   6. Applied egg-rr 39.6%

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

      1. unpow2 39.6%

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

      2. add-sqr-sqrt 84.9%

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

      3. metadata-eval 84.9%

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

      4. associate-/r/ 85.0%

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

      5. clear-num 85.0%

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

      6. expm1-log1p-u 68.4%

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

      7. div-inv 68.4%

         \[\leadsto {\left(a \cdot \cos \left(\mathsf{expm1}\left(\mathsf{log1p}\left(\color{blue}{\left(\pi \cdot angle\right) \cdot \frac{1}{180}}\right)\right)\right)\right)}^{2} + {\left(b \cdot \sin \left(\pi \cdot \left(angle \cdot 0.005555555555555556\right)\right)\right)}^{2} \]

      8. metadata-eval 68.4%

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

      9. associate-*r* 68.4%

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

      10. *-commutative 68.4%

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

   8. Applied egg-rr 68.4%

      \[\leadsto {\left(a \cdot \cos \color{blue}{\left(\mathsf{expm1}\left(\mathsf{log1p}\left(\pi \cdot \left(0.005555555555555556 \cdot angle\right)\right)\right)\right)}\right)}^{2} + {\left(b \cdot \sin \left(\pi \cdot \left(angle \cdot 0.005555555555555556\right)\right)\right)}^{2} \]
   9. Step-by-step derivation

      1. expm1-log1p-u 85.1%

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

      2. add-cube-cbrt 85.2%

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

      3. unpow2 85.2%

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

      4. associate-*r* 85.1%

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

      5. *-commutative 85.1%

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

      6. associate-*r* 85.1%

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

      7. associate-*l* 85.1%

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

   10. Applied egg-rr 85.1%

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

   11. Taylor expanded in a around inf 74.9%

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

       1. unpow2 74.9%

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

       2. associate-*r* 75.1%

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

       3. *-commutative 75.1%

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

       4. unpow2 75.1%

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

       5. swap-sqr 75.1%

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

       6. unpow2 75.1%

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

   13. Simplified 75.1%

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

Alternative 7: 52.7% accurate, 2.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;a \leq 9 \cdot 10^{-132}:\\ \;\;\;\;{\left(b \cdot \sin \left(angle \cdot \left(\pi \cdot 0.005555555555555556\right)\right)\right)}^{2}\\ \mathbf{else}:\\ \;\;\;\;{\left(a \cdot \cos \left(\pi \cdot \left(0.005555555555555556 \cdot angle\right)\right)\right)}^{2}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b angle)
 :precision binary64
 (if (<= a 9e-132)
   (pow (* b (sin (* angle (* PI 0.005555555555555556)))) 2.0)
   (pow (* a (cos (* PI (* 0.005555555555555556 angle)))) 2.0)))

C

double code(double a, double b, double angle) {
	double tmp;
	if (a <= 9e-132) {
		tmp = pow((b * sin((angle * (((double) M_PI) * 0.005555555555555556)))), 2.0);
	} else {
		tmp = pow((a * cos((((double) M_PI) * (0.005555555555555556 * angle)))), 2.0);
	}
	return tmp;
}

Java

public static double code(double a, double b, double angle) {
	double tmp;
	if (a <= 9e-132) {
		tmp = Math.pow((b * Math.sin((angle * (Math.PI * 0.005555555555555556)))), 2.0);
	} else {
		tmp = Math.pow((a * Math.cos((Math.PI * (0.005555555555555556 * angle)))), 2.0);
	}
	return tmp;
}

Python

def code(a, b, angle):
	tmp = 0
	if a <= 9e-132:
		tmp = math.pow((b * math.sin((angle * (math.pi * 0.005555555555555556)))), 2.0)
	else:
		tmp = math.pow((a * math.cos((math.pi * (0.005555555555555556 * angle)))), 2.0)
	return tmp

Julia

function code(a, b, angle)
	tmp = 0.0
	if (a <= 9e-132)
		tmp = Float64(b * sin(Float64(angle * Float64(pi * 0.005555555555555556)))) ^ 2.0;
	else
		tmp = Float64(a * cos(Float64(pi * Float64(0.005555555555555556 * angle)))) ^ 2.0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, angle)
	tmp = 0.0;
	if (a <= 9e-132)
		tmp = (b * sin((angle * (pi * 0.005555555555555556)))) ^ 2.0;
	else
		tmp = (a * cos((pi * (0.005555555555555556 * angle)))) ^ 2.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, angle_] := If[LessEqual[a, 9e-132], 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[(0.005555555555555556 * angle), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if a < 8.9999999999999999e-132

   1. Initial program 78.9%

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

      1. associate-*r/ 78.9%

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

      2. metadata-eval 78.9%

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

      3. metadata-eval 78.9%

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

      4. distribute-neg-frac2 78.9%

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

      5. distribute-frac-neg 78.9%

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

      6. distribute-rgt-neg-out 78.9%

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

      7. associate-/l* 78.9%

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

      8. neg-mul-1 78.9%

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

      9. *-commutative 78.9%

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

      10. associate-/l* 78.9%

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

      11. metadata-eval 78.9%

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

      12. metadata-eval 78.9%

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

   3. Simplified 79.0%

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

      1. metadata-eval 79.0%

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

      2. div-inv 79.0%

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

      3. add-sqr-sqrt 44.0%

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

      4. pow2 44.0%

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

      5. div-inv 44.0%

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

      6. metadata-eval 44.0%

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

      7. associate-*r* 44.1%

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

      8. *-commutative 44.1%

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

   6. Applied egg-rr 44.1%

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

   7. Taylor expanded in a around 0 40.1%

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

      1. unpow2 40.1%

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

      2. associate-*r* 40.2%

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

      3. *-commutative 40.2%

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

      4. unpow2 40.2%

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

      5. swap-sqr 45.2%

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

      6. unpow2 45.2%

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

      7. associate-*r* 45.2%

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

      8. *-commutative 45.2%

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

      9. *-commutative 45.2%

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

   9. Simplified 45.2%

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

   if 8.9999999999999999e-132 < a

   1. Initial program 84.9%

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

      1. associate-*r/ 85.0%

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

      2. metadata-eval 85.0%

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

      3. metadata-eval 85.0%

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

      4. distribute-neg-frac2 85.0%

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

      5. distribute-frac-neg 85.0%

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

      6. distribute-rgt-neg-out 85.0%

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

      7. associate-/l* 84.9%

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

      8. neg-mul-1 84.9%

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

      9. *-commutative 84.9%

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

      10. associate-/l* 85.1%

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

      11. metadata-eval 85.1%

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

      12. metadata-eval 85.1%

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

   3. Simplified 85.1%

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

      1. metadata-eval 85.1%

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

      2. div-inv 84.9%

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

      3. add-sqr-sqrt 39.6%

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

      4. pow2 39.6%

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

      5. div-inv 39.6%

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

      6. metadata-eval 39.6%

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

      7. associate-*r* 39.6%

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

      8. *-commutative 39.6%

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

   6. Applied egg-rr 39.6%

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

      1. unpow2 39.6%

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

      2. add-sqr-sqrt 84.9%

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

      3. metadata-eval 84.9%

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

      4. associate-/r/ 85.0%

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

      5. clear-num 85.0%

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

      6. expm1-log1p-u 68.4%

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

      7. div-inv 68.4%

         \[\leadsto {\left(a \cdot \cos \left(\mathsf{expm1}\left(\mathsf{log1p}\left(\color{blue}{\left(\pi \cdot angle\right) \cdot \frac{1}{180}}\right)\right)\right)\right)}^{2} + {\left(b \cdot \sin \left(\pi \cdot \left(angle \cdot 0.005555555555555556\right)\right)\right)}^{2} \]

      8. metadata-eval 68.4%

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

      9. associate-*r* 68.4%

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

      10. *-commutative 68.4%

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

   8. Applied egg-rr 68.4%

      \[\leadsto {\left(a \cdot \cos \color{blue}{\left(\mathsf{expm1}\left(\mathsf{log1p}\left(\pi \cdot \left(0.005555555555555556 \cdot angle\right)\right)\right)\right)}\right)}^{2} + {\left(b \cdot \sin \left(\pi \cdot \left(angle \cdot 0.005555555555555556\right)\right)\right)}^{2} \]
   9. Step-by-step derivation

      1. expm1-log1p-u 85.1%

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

      2. add-cube-cbrt 85.2%

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

      3. unpow2 85.2%

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

      4. associate-*r* 85.1%

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

      5. *-commutative 85.1%

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

      6. associate-*r* 85.1%

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

      7. associate-*l* 85.1%

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

   10. Applied egg-rr 85.1%

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

   11. Taylor expanded in a around inf 74.9%

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

       1. unpow2 74.9%

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

       2. associate-*r* 75.1%

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

       3. *-commutative 75.1%

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

       4. unpow2 75.1%

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

       5. swap-sqr 75.1%

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

       6. unpow2 75.1%

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

   13. Simplified 75.1%

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

4. Final simplification 55.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \leq 9 \cdot 10^{-132}:\\ \;\;\;\;{\left(b \cdot \sin \left(angle \cdot \left(\pi \cdot 0.005555555555555556\right)\right)\right)}^{2}\\ \mathbf{else}:\\ \;\;\;\;{\left(a \cdot \cos \left(\pi \cdot \left(0.005555555555555556 \cdot angle\right)\right)\right)}^{2}\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 52.6% accurate, 2.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;a \leq 1.4 \cdot 10^{-132}:\\ \;\;\;\;{\left(b \cdot \sin \left(0.005555555555555556 \cdot \left(\pi \cdot angle\right)\right)\right)}^{2}\\ \mathbf{else}:\\ \;\;\;\;{\left(a \cdot \cos \left(\pi \cdot \left(0.005555555555555556 \cdot angle\right)\right)\right)}^{2}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b angle)
 :precision binary64
 (if (<= a 1.4e-132)
   (pow (* b (sin (* 0.005555555555555556 (* PI angle)))) 2.0)
   (pow (* a (cos (* PI (* 0.005555555555555556 angle)))) 2.0)))

C

double code(double a, double b, double angle) {
	double tmp;
	if (a <= 1.4e-132) {
		tmp = pow((b * sin((0.005555555555555556 * (((double) M_PI) * angle)))), 2.0);
	} else {
		tmp = pow((a * cos((((double) M_PI) * (0.005555555555555556 * angle)))), 2.0);
	}
	return tmp;
}

Java

public static double code(double a, double b, double angle) {
	double tmp;
	if (a <= 1.4e-132) {
		tmp = Math.pow((b * Math.sin((0.005555555555555556 * (Math.PI * angle)))), 2.0);
	} else {
		tmp = Math.pow((a * Math.cos((Math.PI * (0.005555555555555556 * angle)))), 2.0);
	}
	return tmp;
}

Python

def code(a, b, angle):
	tmp = 0
	if a <= 1.4e-132:
		tmp = math.pow((b * math.sin((0.005555555555555556 * (math.pi * angle)))), 2.0)
	else:
		tmp = math.pow((a * math.cos((math.pi * (0.005555555555555556 * angle)))), 2.0)
	return tmp

Julia

function code(a, b, angle)
	tmp = 0.0
	if (a <= 1.4e-132)
		tmp = Float64(b * sin(Float64(0.005555555555555556 * Float64(pi * angle)))) ^ 2.0;
	else
		tmp = Float64(a * cos(Float64(pi * Float64(0.005555555555555556 * angle)))) ^ 2.0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, angle)
	tmp = 0.0;
	if (a <= 1.4e-132)
		tmp = (b * sin((0.005555555555555556 * (pi * angle)))) ^ 2.0;
	else
		tmp = (a * cos((pi * (0.005555555555555556 * angle)))) ^ 2.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, angle_] := If[LessEqual[a, 1.4e-132], N[Power[N[(b * N[Sin[N[(0.005555555555555556 * N[(Pi * angle), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision], N[Power[N[(a * N[Cos[N[(Pi * N[(0.005555555555555556 * angle), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if a < 1.40000000000000001e-132

   1. Initial program 78.9%

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

      1. associate-*r/ 78.9%

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

      2. metadata-eval 78.9%

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

      3. metadata-eval 78.9%

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

      4. distribute-neg-frac2 78.9%

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

      5. distribute-frac-neg 78.9%

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

      6. distribute-rgt-neg-out 78.9%

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

      7. associate-/l* 78.9%

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

      8. neg-mul-1 78.9%

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

      9. *-commutative 78.9%

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

      10. associate-/l* 78.9%

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

      11. metadata-eval 78.9%

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

      12. metadata-eval 78.9%

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

   3. Simplified 79.0%

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

   5. Taylor expanded in a around 0 40.1%

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

      1. unpow2 40.1%

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

      2. *-commutative 40.1%

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

      3. unpow2 40.1%

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

      4. swap-sqr 45.1%

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

      5. unpow2 45.1%

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

      6. *-commutative 45.1%

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

   7. Simplified 45.1%

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

   if 1.40000000000000001e-132 < a

   1. Initial program 84.9%

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

      1. associate-*r/ 85.0%

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

      2. metadata-eval 85.0%

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

      3. metadata-eval 85.0%

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

      4. distribute-neg-frac2 85.0%

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

      5. distribute-frac-neg 85.0%

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

      6. distribute-rgt-neg-out 85.0%

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

      7. associate-/l* 84.9%

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

      8. neg-mul-1 84.9%

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

      9. *-commutative 84.9%

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

      10. associate-/l* 85.1%

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

      11. metadata-eval 85.1%

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

      12. metadata-eval 85.1%

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

   3. Simplified 85.1%

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

      1. metadata-eval 85.1%

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

      2. div-inv 84.9%

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

      3. add-sqr-sqrt 39.6%

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

      4. pow2 39.6%

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

      5. div-inv 39.6%

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

      6. metadata-eval 39.6%

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

      7. associate-*r* 39.6%

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

      8. *-commutative 39.6%

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

   6. Applied egg-rr 39.6%

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

      1. unpow2 39.6%

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

      2. add-sqr-sqrt 84.9%

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

      3. metadata-eval 84.9%

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

      4. associate-/r/ 85.0%

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

      5. clear-num 85.0%

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

      6. expm1-log1p-u 68.4%

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

      7. div-inv 68.4%

         \[\leadsto {\left(a \cdot \cos \left(\mathsf{expm1}\left(\mathsf{log1p}\left(\color{blue}{\left(\pi \cdot angle\right) \cdot \frac{1}{180}}\right)\right)\right)\right)}^{2} + {\left(b \cdot \sin \left(\pi \cdot \left(angle \cdot 0.005555555555555556\right)\right)\right)}^{2} \]

      8. metadata-eval 68.4%

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

      9. associate-*r* 68.4%

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

      10. *-commutative 68.4%

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

   8. Applied egg-rr 68.4%

      \[\leadsto {\left(a \cdot \cos \color{blue}{\left(\mathsf{expm1}\left(\mathsf{log1p}\left(\pi \cdot \left(0.005555555555555556 \cdot angle\right)\right)\right)\right)}\right)}^{2} + {\left(b \cdot \sin \left(\pi \cdot \left(angle \cdot 0.005555555555555556\right)\right)\right)}^{2} \]
   9. Step-by-step derivation

      1. expm1-log1p-u 85.1%

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

      2. add-cube-cbrt 85.2%

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

      3. unpow2 85.2%

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

      4. associate-*r* 85.1%

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

      5. *-commutative 85.1%

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

      6. associate-*r* 85.1%

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

      7. associate-*l* 85.1%

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

   10. Applied egg-rr 85.1%

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

   11. Taylor expanded in a around inf 74.9%

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

       1. unpow2 74.9%

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

       2. associate-*r* 75.1%

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

       3. *-commutative 75.1%

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

       4. unpow2 75.1%

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

       5. swap-sqr 75.1%

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

       6. unpow2 75.1%

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

   13. Simplified 75.1%

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

4. Final simplification 55.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \leq 1.4 \cdot 10^{-132}:\\ \;\;\;\;{\left(b \cdot \sin \left(0.005555555555555556 \cdot \left(\pi \cdot angle\right)\right)\right)}^{2}\\ \mathbf{else}:\\ \;\;\;\;{\left(a \cdot \cos \left(\pi \cdot \left(0.005555555555555556 \cdot angle\right)\right)\right)}^{2}\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 56.8% accurate, 2.0× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 81.0%

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

   1. associate-*r/ 81.0%

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

   2. metadata-eval 81.0%

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

   3. metadata-eval 81.0%

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

   4. distribute-neg-frac2 81.0%

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

   5. distribute-frac-neg 81.0%

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

   6. distribute-rgt-neg-out 81.0%

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

   7. associate-/l* 81.0%

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

   8. neg-mul-1 81.0%

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

   9. *-commutative 81.0%

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

   10. associate-/l* 81.1%

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

   11. metadata-eval 81.1%

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

   12. metadata-eval 81.1%

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

3. Simplified 81.2%

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

   1. metadata-eval 81.2%

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

   2. div-inv 81.1%

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

   3. add-sqr-sqrt 42.4%

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

   4. pow2 42.4%

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

   5. div-inv 42.5%

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

   6. metadata-eval 42.5%

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

   7. associate-*r* 42.5%

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

   8. *-commutative 42.5%

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

6. Applied egg-rr 42.5%

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

   1. unpow2 42.5%

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

   2. add-sqr-sqrt 81.1%

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

   3. metadata-eval 81.1%

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

   4. associate-/r/ 81.1%

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

   5. clear-num 81.1%

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

   6. expm1-log1p-u 67.8%

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

   7. div-inv 67.8%

      \[\leadsto {\left(a \cdot \cos \left(\mathsf{expm1}\left(\mathsf{log1p}\left(\color{blue}{\left(\pi \cdot angle\right) \cdot \frac{1}{180}}\right)\right)\right)\right)}^{2} + {\left(b \cdot \sin \left(\pi \cdot \left(angle \cdot 0.005555555555555556\right)\right)\right)}^{2} \]

   8. metadata-eval 67.8%

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

   9. associate-*r* 67.8%

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

   10. *-commutative 67.8%

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

8. Applied egg-rr 67.8%

   \[\leadsto {\left(a \cdot \cos \color{blue}{\left(\mathsf{expm1}\left(\mathsf{log1p}\left(\pi \cdot \left(0.005555555555555556 \cdot angle\right)\right)\right)\right)}\right)}^{2} + {\left(b \cdot \sin \left(\pi \cdot \left(angle \cdot 0.005555555555555556\right)\right)\right)}^{2} \]
9. Step-by-step derivation

   1. expm1-log1p-u 81.2%

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

   2. add-cube-cbrt 81.3%

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

   3. unpow2 81.3%

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

   4. associate-*r* 81.3%

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

   5. *-commutative 81.3%

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

   6. associate-*r* 81.2%

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

   7. associate-*l* 81.2%

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

10. Applied egg-rr 81.2%

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

11. Taylor expanded in a around inf 64.0%

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

    1. unpow2 64.0%

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

    2. associate-*r* 64.0%

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

    3. *-commutative 64.0%

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

    4. unpow2 64.0%

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

    5. swap-sqr 64.0%

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

    6. unpow2 64.0%

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

13. Simplified 64.0%

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

Alternative 10: 56.7% 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 81.0%

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

   1. associate-*r/ 81.0%

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

   2. metadata-eval 81.0%

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

   3. metadata-eval 81.0%

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

   4. distribute-neg-frac2 81.0%

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

   5. distribute-frac-neg 81.0%

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

   6. distribute-rgt-neg-out 81.0%

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

   7. associate-/l* 81.0%

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

   8. neg-mul-1 81.0%

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

   9. *-commutative 81.0%

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

   10. associate-/l* 81.1%

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

   11. metadata-eval 81.1%

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

   12. metadata-eval 81.1%

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

3. Simplified 81.2%

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

   1. metadata-eval 81.2%

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

   2. div-inv 81.1%

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

   3. add-sqr-sqrt 42.4%

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

   4. pow2 42.4%

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

   5. div-inv 42.5%

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

   6. metadata-eval 42.5%

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

   7. associate-*r* 42.5%

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

   8. *-commutative 42.5%

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

6. Applied egg-rr 42.5%

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

7. Taylor expanded in a around inf 64.0%

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

9. Simplified 64.0%

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

Alternative 11: 56.7% accurate, 2.0× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 81.0%

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

   1. associate-*r/ 81.0%

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

   2. metadata-eval 81.0%

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

   3. metadata-eval 81.0%

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

   4. distribute-neg-frac2 81.0%

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

   5. distribute-frac-neg 81.0%

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

   6. distribute-rgt-neg-out 81.0%

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

   7. associate-/l* 81.0%

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

   8. neg-mul-1 81.0%

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

   9. *-commutative 81.0%

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

   10. associate-/l* 81.1%

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

   11. metadata-eval 81.1%

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

   12. metadata-eval 81.1%

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

3. Simplified 81.2%

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

5. Taylor expanded in a around inf 64.0%

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

   1. *-commutative 64.0%

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

   2. unpow2 64.0%

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

   3. unpow2 64.0%

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

   4. swap-sqr 64.0%

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

   5. unpow2 64.0%

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

   6. *-commutative 64.0%

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

7. Simplified 64.0%

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

8. Final simplification 64.0%

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

Alternative 12: 56.8% 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 81.0%

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

   1. associate-*r/ 81.0%

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

   2. metadata-eval 81.0%

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

   3. metadata-eval 81.0%

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

   4. distribute-neg-frac2 81.0%

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

   5. distribute-frac-neg 81.0%

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

   6. distribute-rgt-neg-out 81.0%

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

   7. associate-/l* 81.0%

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

   8. neg-mul-1 81.0%

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

   9. *-commutative 81.0%

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

   10. associate-/l* 81.1%

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

   11. metadata-eval 81.1%

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

   12. metadata-eval 81.1%

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

3. Simplified 81.2%

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

5. Taylor expanded in angle around 0 63.9%

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

   1. unpow2 63.9%

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

7. Applied egg-rr 63.9%

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

Reproduce

herbie shell --seed 2024177 
(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)))