ab-angle->ABCF C

Percentage Accurate: 79.5% → 79.5%

Time: 48.0s

Alternatives: 13

Speedup: 1.3×

• 36.6% 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, 0.5× speedup

Math

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

FPCore

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

C

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

Java

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

Julia

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

Wolfram

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

Derivation

1. Initial program 84.3%

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

   1. associate-*r/ 84.2%

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

   2. metadata-eval 84.2%

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

   3. metadata-eval 84.2%

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

   4. distribute-neg-frac2 84.2%

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

   5. distribute-frac-neg 84.2%

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

   6. distribute-rgt-neg-out 84.2%

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

   7. associate-/l* 84.3%

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

   8. neg-mul-1 84.3%

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

   9. *-commutative 84.3%

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

   10. associate-/l* 84.3%

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

   11. metadata-eval 84.3%

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

   12. metadata-eval 84.3%

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

3. Simplified 84.5%

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

   1. metadata-eval 84.5%

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

   2. div-inv 84.5%

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

   3. add-cube-cbrt 84.3%

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

   4. pow3 84.6%

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

   5. div-inv 84.6%

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

   6. metadata-eval 84.6%

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

6. Applied egg-rr 84.6%

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

   1. add-sqr-sqrt 84.6%

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

   2. pow2 84.6%

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

8. Applied egg-rr 84.6%

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

9. Final simplification 84.6%

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

Alternative 2: 64.8% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ {\left(b \cdot \sin \left(\pi \cdot \left(angle \cdot 0.005555555555555556\right)\right)\right)}^{2} + {\left(a \cdot \cos \left({\left(\sqrt[3]{\pi \cdot \mathsf{expm1}\left(\mathsf{log1p}\left(angle \cdot 0.005555555555555556\right)\right)}\right)}^{3}\right)\right)}^{2} \end{array} \]

FPCore

(FPCore (a b angle)
 :precision binary64
 (+
  (pow (* b (sin (* PI (* angle 0.005555555555555556)))) 2.0)
  (pow
   (*
    a
    (cos
     (pow (cbrt (* PI (expm1 (log1p (* angle 0.005555555555555556))))) 3.0)))
   2.0)))

C

double code(double a, double b, double angle) {
	return pow((b * sin((((double) M_PI) * (angle * 0.005555555555555556)))), 2.0) + pow((a * cos(pow(cbrt((((double) M_PI) * expm1(log1p((angle * 0.005555555555555556))))), 3.0))), 2.0);
}

Java

public static double code(double a, double b, double angle) {
	return Math.pow((b * Math.sin((Math.PI * (angle * 0.005555555555555556)))), 2.0) + Math.pow((a * Math.cos(Math.pow(Math.cbrt((Math.PI * Math.expm1(Math.log1p((angle * 0.005555555555555556))))), 3.0))), 2.0);
}

Julia

function code(a, b, angle)
	return Float64((Float64(b * sin(Float64(pi * Float64(angle * 0.005555555555555556)))) ^ 2.0) + (Float64(a * cos((cbrt(Float64(pi * expm1(log1p(Float64(angle * 0.005555555555555556))))) ^ 3.0))) ^ 2.0))
end

Wolfram

code[a_, b_, angle_] := N[(N[Power[N[(b * N[Sin[N[(Pi * N[(angle * 0.005555555555555556), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision] + N[Power[N[(a * N[Cos[N[Power[N[Power[N[(Pi * N[(Exp[N[Log[1 + N[(angle * 0.005555555555555556), $MachinePrecision]], $MachinePrecision]] - 1), $MachinePrecision]), $MachinePrecision], 1/3], $MachinePrecision], 3.0], $MachinePrecision]], $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 84.3%

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

   1. associate-*r/ 84.2%

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

   2. metadata-eval 84.2%

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

   3. metadata-eval 84.2%

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

   4. distribute-neg-frac2 84.2%

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

   5. distribute-frac-neg 84.2%

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

   6. distribute-rgt-neg-out 84.2%

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

   7. associate-/l* 84.3%

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

   8. neg-mul-1 84.3%

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

   9. *-commutative 84.3%

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

   10. associate-/l* 84.3%

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

   11. metadata-eval 84.3%

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

   12. metadata-eval 84.3%

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

3. Simplified 84.5%

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

   1. metadata-eval 84.5%

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

   2. div-inv 84.5%

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

   3. add-cube-cbrt 84.3%

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

   4. pow3 84.6%

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

   5. div-inv 84.6%

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

   6. metadata-eval 84.6%

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

6. Applied egg-rr 84.6%

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

   1. expm1-log1p-u 66.9%

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

8. Applied egg-rr 66.8%

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

9. Final simplification 66.8%

   \[\leadsto {\left(b \cdot \sin \left(\pi \cdot \left(angle \cdot 0.005555555555555556\right)\right)\right)}^{2} + {\left(a \cdot \cos \left({\left(\sqrt[3]{\pi \cdot \mathsf{expm1}\left(\mathsf{log1p}\left(angle \cdot 0.005555555555555556\right)\right)}\right)}^{3}\right)\right)}^{2} \]
10. Add Preprocessing

Alternative 3: 64.8% accurate, 0.5× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

Wolfram

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

Derivation

1. Initial program 84.3%

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

   1. associate-*r/ 84.2%

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

   2. metadata-eval 84.2%

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

   3. metadata-eval 84.2%

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

   4. distribute-neg-frac2 84.2%

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

   5. distribute-frac-neg 84.2%

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

   6. distribute-rgt-neg-out 84.2%

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

   7. associate-/l* 84.3%

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

   8. neg-mul-1 84.3%

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

   9. *-commutative 84.3%

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

   10. associate-/l* 84.3%

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

   11. metadata-eval 84.3%

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

   12. metadata-eval 84.3%

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

3. Simplified 84.5%

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

   1. expm1-log1p-u 84.5%

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

6. Applied egg-rr 84.5%

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

   1. expm1-log1p-u 66.9%

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

8. Applied egg-rr 66.9%

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

9. Final simplification 66.9%

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

Alternative 4: 79.5% accurate, 0.7× speedup

Math

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

FPCore

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

C

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

Java

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

Julia

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

Wolfram

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

Derivation

1. Initial program 84.3%

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

   1. associate-*r/ 84.2%

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

   2. metadata-eval 84.2%

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

   3. metadata-eval 84.2%

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

   4. distribute-neg-frac2 84.2%

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

   5. distribute-frac-neg 84.2%

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

   6. distribute-rgt-neg-out 84.2%

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

   7. associate-/l* 84.3%

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

   8. neg-mul-1 84.3%

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

   9. *-commutative 84.3%

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

   10. associate-/l* 84.3%

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

   11. metadata-eval 84.3%

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

   12. metadata-eval 84.3%

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

3. Simplified 84.5%

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

   1. add-cbrt-cube 84.5%

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

   2. pow3 84.5%

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

6. Applied egg-rr 84.5%

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

7. Final simplification 84.5%

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

Alternative 5: 79.5% accurate, 0.7× speedup

Math

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

FPCore

(FPCore (a b angle)
 :precision binary64
 (let* ((t_0 (* PI (* angle 0.005555555555555556))))
   (+ (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) * (angle * 0.005555555555555556);
	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 * (angle * 0.005555555555555556);
	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(angle * 0.005555555555555556))
	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[(angle * 0.005555555555555556), $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 84.3%

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

   1. associate-*r/ 84.2%

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

   2. metadata-eval 84.2%

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

   3. metadata-eval 84.2%

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

   4. distribute-neg-frac2 84.2%

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

   5. distribute-frac-neg 84.2%

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

   6. distribute-rgt-neg-out 84.2%

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

   7. associate-/l* 84.3%

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

   8. neg-mul-1 84.3%

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

   9. *-commutative 84.3%

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

   10. associate-/l* 84.3%

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

   11. metadata-eval 84.3%

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

   12. metadata-eval 84.3%

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

3. Simplified 84.5%

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

   1. metadata-eval 84.5%

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

   2. div-inv 84.5%

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

   3. add-cube-cbrt 84.3%

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

   4. pow3 84.6%

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

   5. div-inv 84.6%

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

   6. metadata-eval 84.6%

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

6. Applied egg-rr 84.6%

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

7. Final simplification 84.6%

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

Alternative 6: 64.7% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \pi \cdot \left(angle \cdot 0.005555555555555556\right)\\ {\left(b \cdot \sin t\_0\right)}^{2} + {\left(a \cdot \cos \left(\mathsf{expm1}\left(\mathsf{log1p}\left(t\_0\right)\right)\right)\right)}^{2} \end{array} \end{array} \]

FPCore

(FPCore (a b angle)
 :precision binary64
 (let* ((t_0 (* PI (* angle 0.005555555555555556))))
   (+ (pow (* b (sin t_0)) 2.0) (pow (* a (cos (expm1 (log1p t_0)))) 2.0))))

C

double code(double a, double b, double angle) {
	double t_0 = ((double) M_PI) * (angle * 0.005555555555555556);
	return pow((b * sin(t_0)), 2.0) + pow((a * cos(expm1(log1p(t_0)))), 2.0);
}

Java

public static double code(double a, double b, double angle) {
	double t_0 = Math.PI * (angle * 0.005555555555555556);
	return Math.pow((b * Math.sin(t_0)), 2.0) + Math.pow((a * Math.cos(Math.expm1(Math.log1p(t_0)))), 2.0);
}

Python

def code(a, b, angle):
	t_0 = math.pi * (angle * 0.005555555555555556)
	return math.pow((b * math.sin(t_0)), 2.0) + math.pow((a * math.cos(math.expm1(math.log1p(t_0)))), 2.0)

Julia

function code(a, b, angle)
	t_0 = Float64(pi * Float64(angle * 0.005555555555555556))
	return Float64((Float64(b * sin(t_0)) ^ 2.0) + (Float64(a * cos(expm1(log1p(t_0)))) ^ 2.0))
end

Wolfram

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

Derivation

1. Initial program 84.3%

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

   1. associate-*r/ 84.2%

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

   2. metadata-eval 84.2%

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

   3. metadata-eval 84.2%

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

   4. distribute-neg-frac2 84.2%

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

   5. distribute-frac-neg 84.2%

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

   6. distribute-rgt-neg-out 84.2%

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

   7. associate-/l* 84.3%

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

   8. neg-mul-1 84.3%

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

   9. *-commutative 84.3%

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

   10. associate-/l* 84.3%

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

   11. metadata-eval 84.3%

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

   12. metadata-eval 84.3%

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

3. Simplified 84.5%

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

   1. expm1-log1p-u 66.8%

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

6. Applied egg-rr 66.8%

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

7. Final simplification 66.8%

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

Alternative 7: 79.4% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 84.3%

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

   1. associate-*r/ 84.2%

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

   2. metadata-eval 84.2%

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

   3. metadata-eval 84.2%

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

   4. distribute-neg-frac2 84.2%

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

   5. distribute-frac-neg 84.2%

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

   6. distribute-rgt-neg-out 84.2%

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

   7. associate-/l* 84.3%

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

   8. neg-mul-1 84.3%

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

   9. *-commutative 84.3%

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

   10. associate-/l* 84.3%

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

   11. metadata-eval 84.3%

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

   12. metadata-eval 84.3%

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

3. Simplified 84.5%

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

5. Taylor expanded in angle around inf 84.4%

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

6. Final simplification 84.4%

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

Alternative 8: 79.5% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 84.3%

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

   1. associate-*r/ 84.2%

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

   2. metadata-eval 84.2%

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

   3. metadata-eval 84.2%

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

   4. distribute-neg-frac2 84.2%

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

   5. distribute-frac-neg 84.2%

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

   6. distribute-rgt-neg-out 84.2%

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

   7. associate-/l* 84.3%

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

   8. neg-mul-1 84.3%

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

   9. *-commutative 84.3%

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

   10. associate-/l* 84.3%

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

   11. metadata-eval 84.3%

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

   12. metadata-eval 84.3%

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

3. Simplified 84.5%

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

5. Taylor expanded in angle around inf 84.4%

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

   1. *-commutative 84.4%

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

   2. associate-*r* 84.5%

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

7. Simplified 84.5%

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

8. Final simplification 84.5%

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

Alternative 9: 79.5% accurate, 1.3× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 84.3%

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

   1. associate-*r/ 84.2%

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

   2. metadata-eval 84.2%

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

   3. metadata-eval 84.2%

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

   4. distribute-neg-frac2 84.2%

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

   5. distribute-frac-neg 84.2%

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

   6. distribute-rgt-neg-out 84.2%

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

   7. associate-/l* 84.3%

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

   8. neg-mul-1 84.3%

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

   9. *-commutative 84.3%

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

   10. associate-/l* 84.3%

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

   11. metadata-eval 84.3%

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

   12. metadata-eval 84.3%

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

3. Simplified 84.5%

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

5. Taylor expanded in angle around 0 83.9%

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

6. Taylor expanded in angle around inf 83.7%

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

   1. *-commutative 83.7%

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

   2. associate-*r* 83.8%

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

8. Simplified 83.8%

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

9. Final simplification 83.8%

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

Alternative 10: 79.5% accurate, 1.3× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 84.3%

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

   1. associate-*r/ 84.2%

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

   2. metadata-eval 84.2%

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

   3. metadata-eval 84.2%

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

   4. distribute-neg-frac2 84.2%

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

   5. distribute-frac-neg 84.2%

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

   6. distribute-rgt-neg-out 84.2%

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

   7. associate-/l* 84.3%

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

   8. neg-mul-1 84.3%

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

   9. *-commutative 84.3%

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

   10. associate-/l* 84.3%

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

   11. metadata-eval 84.3%

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

   12. metadata-eval 84.3%

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

3. Simplified 84.5%

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

5. Taylor expanded in angle around 0 83.9%

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

6. Final simplification 83.9%

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

Alternative 11: 73.2% accurate, 3.5× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 84.3%

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

   1. associate-*r/ 84.2%

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

   2. metadata-eval 84.2%

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

   3. metadata-eval 84.2%

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

   4. distribute-neg-frac2 84.2%

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

   5. distribute-frac-neg 84.2%

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

   6. distribute-rgt-neg-out 84.2%

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

   7. associate-/l* 84.3%

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

   8. neg-mul-1 84.3%

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

   9. *-commutative 84.3%

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

   10. associate-/l* 84.3%

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

   11. metadata-eval 84.3%

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

   12. metadata-eval 84.3%

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

3. Simplified 84.5%

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

5. Taylor expanded in angle around 0 83.9%

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

6. Taylor expanded in angle around 0 79.4%

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

   1. unpow2 79.4%

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

   2. associate-*r* 79.5%

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

   3. associate-*l* 78.4%

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

   4. *-commutative 78.4%

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

   5. *-commutative 78.4%

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

   6. *-commutative 78.4%

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

   7. associate-*r* 78.4%

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

   8. *-commutative 78.4%

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

   9. *-commutative 78.4%

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

   10. associate-*l* 78.4%

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

8. Applied egg-rr 78.4%

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

9. Taylor expanded in angle around 0 78.4%

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

    1. associate-*r* 78.4%

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

    2. *-commutative 78.4%

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

    3. associate-*l* 78.4%

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

11. Simplified 78.4%

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

12. Final simplification 78.4%

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

Alternative 12: 73.2% accurate, 3.5× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 84.3%

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

   1. associate-*r/ 84.2%

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

   2. metadata-eval 84.2%

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

   3. metadata-eval 84.2%

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

   4. distribute-neg-frac2 84.2%

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

   5. distribute-frac-neg 84.2%

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

   6. distribute-rgt-neg-out 84.2%

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

   7. associate-/l* 84.3%

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

   8. neg-mul-1 84.3%

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

   9. *-commutative 84.3%

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

   10. associate-/l* 84.3%

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

   11. metadata-eval 84.3%

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

   12. metadata-eval 84.3%

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

3. Simplified 84.5%

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

5. Taylor expanded in angle around 0 83.9%

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

6. Taylor expanded in angle around 0 79.4%

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

   1. unpow2 79.4%

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

   2. associate-*r* 79.5%

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

   3. associate-*l* 78.4%

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

   4. *-commutative 78.4%

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

   5. *-commutative 78.4%

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

   6. *-commutative 78.4%

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

   7. associate-*r* 78.4%

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

   8. *-commutative 78.4%

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

   9. *-commutative 78.4%

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

   10. associate-*l* 78.4%

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

8. Applied egg-rr 78.4%

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

9. Final simplification 78.4%

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

Alternative 13: 74.3% accurate, 3.5× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 84.3%

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

   1. associate-*r/ 84.2%

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

   2. metadata-eval 84.2%

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

   3. metadata-eval 84.2%

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

   4. distribute-neg-frac2 84.2%

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

   5. distribute-frac-neg 84.2%

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

   6. distribute-rgt-neg-out 84.2%

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

   7. associate-/l* 84.3%

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

   8. neg-mul-1 84.3%

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

   9. *-commutative 84.3%

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

   10. associate-/l* 84.3%

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

   11. metadata-eval 84.3%

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

   12. metadata-eval 84.3%

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

3. Simplified 84.5%

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

5. Taylor expanded in angle around 0 83.9%

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

6. Taylor expanded in angle around 0 79.4%

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

   1. unpow2 79.4%

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

   2. *-commutative 79.4%

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

   3. *-commutative 79.4%

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

   4. associate-*r* 79.4%

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

   5. *-commutative 79.4%

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

   6. *-commutative 79.4%

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

   7. associate-*l* 79.5%

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

   8. associate-*r* 79.5%

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

   9. *-commutative 79.5%

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

   10. *-commutative 79.5%

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

   11. associate-*l* 79.5%

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

8. Applied egg-rr 79.5%

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

9. Final simplification 79.5%

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

Reproduce

herbie shell --seed 2024100 
(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)))