ab-angle->ABCF C

Percentage Accurate: 79.9% → 79.9%

Time: 1.0min

Alternatives: 16

Speedup: 1.0×

• 36.9% 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: 79.9% 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: 79.9% accurate, 0.3× speedup

Math

\[\begin{array}{l} angle_m = \left|angle\right| \\ \begin{array}{l} t_0 := \pi \cdot \left(angle\_m \cdot 0.005555555555555556\right)\\ t_1 := {\left(\sqrt[3]{t\_0}\right)}^{2}\\ t_2 := {t\_0}^{0.16666666666666666}\\ {\left(a \cdot \cos \left(t\_2 \cdot \left(t\_1 \cdot {\left(t\_2 \cdot \left(t\_2 \cdot t\_1\right)\right)}^{0.16666666666666666}\right)\right)\right)}^{2} + {\left(b \cdot \sin t\_0\right)}^{2} \end{array} \end{array} \]

FPCore

angle_m = (fabs.f64 angle)
(FPCore (a b angle_m)
 :precision binary64
 (let* ((t_0 (* PI (* angle_m 0.005555555555555556)))
        (t_1 (pow (cbrt t_0) 2.0))
        (t_2 (pow t_0 0.16666666666666666)))
   (+
    (pow
     (* a (cos (* t_2 (* t_1 (pow (* t_2 (* t_2 t_1)) 0.16666666666666666)))))
     2.0)
    (pow (* b (sin t_0)) 2.0))))

C

angle_m = fabs(angle);
double code(double a, double b, double angle_m) {
	double t_0 = ((double) M_PI) * (angle_m * 0.005555555555555556);
	double t_1 = pow(cbrt(t_0), 2.0);
	double t_2 = pow(t_0, 0.16666666666666666);
	return pow((a * cos((t_2 * (t_1 * pow((t_2 * (t_2 * t_1)), 0.16666666666666666))))), 2.0) + pow((b * sin(t_0)), 2.0);
}

Java

angle_m = Math.abs(angle);
public static double code(double a, double b, double angle_m) {
	double t_0 = Math.PI * (angle_m * 0.005555555555555556);
	double t_1 = Math.pow(Math.cbrt(t_0), 2.0);
	double t_2 = Math.pow(t_0, 0.16666666666666666);
	return Math.pow((a * Math.cos((t_2 * (t_1 * Math.pow((t_2 * (t_2 * t_1)), 0.16666666666666666))))), 2.0) + Math.pow((b * Math.sin(t_0)), 2.0);
}

Julia

angle_m = abs(angle)
function code(a, b, angle_m)
	t_0 = Float64(pi * Float64(angle_m * 0.005555555555555556))
	t_1 = cbrt(t_0) ^ 2.0
	t_2 = t_0 ^ 0.16666666666666666
	return Float64((Float64(a * cos(Float64(t_2 * Float64(t_1 * (Float64(t_2 * Float64(t_2 * t_1)) ^ 0.16666666666666666))))) ^ 2.0) + (Float64(b * sin(t_0)) ^ 2.0))
end

Wolfram

angle_m = N[Abs[angle], $MachinePrecision]
code[a_, b_, angle$95$m_] := Block[{t$95$0 = N[(Pi * N[(angle$95$m * 0.005555555555555556), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[Power[N[Power[t$95$0, 1/3], $MachinePrecision], 2.0], $MachinePrecision]}, Block[{t$95$2 = N[Power[t$95$0, 0.16666666666666666], $MachinePrecision]}, N[(N[Power[N[(a * N[Cos[N[(t$95$2 * N[(t$95$1 * N[Power[N[(t$95$2 * N[(t$95$2 * t$95$1), $MachinePrecision]), $MachinePrecision], 0.16666666666666666], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision] + N[Power[N[(b * N[Sin[t$95$0], $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Initial program 77.1%

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

3. Simplified 77.1%

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

   1. metadata-eval 77.1%

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

   2. div-inv 77.1%

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

   3. add-cube-cbrt 77.2%

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

   4. pow3 77.1%

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

   5. div-inv 77.2%

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

   6. metadata-eval 77.2%

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

6. Applied egg-rr 77.2%

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

   1. cube-mult 77.2%

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

   2. add-sqr-sqrt 42.1%

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

   3. associate-*l* 42.1%

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

   4. pow1/3 42.0%

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

   5. sqrt-pow1 42.0%

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

   6. metadata-eval 42.0%

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

   7. pow1/3 42.2%

      \[\leadsto {\left(a \cdot \cos \left({\left(\pi \cdot \left(angle \cdot 0.005555555555555556\right)\right)}^{0.16666666666666666} \cdot \left(\sqrt{\color{blue}{{\left(\pi \cdot \left(angle \cdot 0.005555555555555556\right)\right)}^{0.3333333333333333}}} \cdot \left(\sqrt[3]{\pi \cdot \left(angle \cdot 0.005555555555555556\right)} \cdot \sqrt[3]{\pi \cdot \left(angle \cdot 0.005555555555555556\right)}\right)\right)\right)\right)}^{2} + {\left(b \cdot \sin \left(\pi \cdot \left(angle \cdot 0.005555555555555556\right)\right)\right)}^{2} \]

   8. sqrt-pow1 42.2%

      \[\leadsto {\left(a \cdot \cos \left({\left(\pi \cdot \left(angle \cdot 0.005555555555555556\right)\right)}^{0.16666666666666666} \cdot \left(\color{blue}{{\left(\pi \cdot \left(angle \cdot 0.005555555555555556\right)\right)}^{\left(\frac{0.3333333333333333}{2}\right)}} \cdot \left(\sqrt[3]{\pi \cdot \left(angle \cdot 0.005555555555555556\right)} \cdot \sqrt[3]{\pi \cdot \left(angle \cdot 0.005555555555555556\right)}\right)\right)\right)\right)}^{2} + {\left(b \cdot \sin \left(\pi \cdot \left(angle \cdot 0.005555555555555556\right)\right)\right)}^{2} \]

   9. metadata-eval 42.2%

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

   10. pow2 42.2%

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

8. Applied egg-rr 42.2%

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

   1. metadata-eval 42.2%

      \[\leadsto {\left(a \cdot \cos \left({\left(\pi \cdot \left(angle \cdot 0.005555555555555556\right)\right)}^{0.16666666666666666} \cdot \left({\left(\pi \cdot \left(angle \cdot \color{blue}{\frac{1}{180}}\right)\right)}^{0.16666666666666666} \cdot {\left(\sqrt[3]{\pi \cdot \left(angle \cdot 0.005555555555555556\right)}\right)}^{2}\right)\right)\right)}^{2} + {\left(b \cdot \sin \left(\pi \cdot \left(angle \cdot 0.005555555555555556\right)\right)\right)}^{2} \]

   2. div-inv 42.2%

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

   3. add-cbrt-cube 31.1%

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

   4. pow1/3 31.0%

      \[\leadsto {\left(a \cdot \cos \left({\left(\pi \cdot \left(angle \cdot 0.005555555555555556\right)\right)}^{0.16666666666666666} \cdot \left({\color{blue}{\left({\left(\left(\left(\pi \cdot \frac{angle}{180}\right) \cdot \left(\pi \cdot \frac{angle}{180}\right)\right) \cdot \left(\pi \cdot \frac{angle}{180}\right)\right)}^{0.3333333333333333}\right)}}^{0.16666666666666666} \cdot {\left(\sqrt[3]{\pi \cdot \left(angle \cdot 0.005555555555555556\right)}\right)}^{2}\right)\right)\right)}^{2} + {\left(b \cdot \sin \left(\pi \cdot \left(angle \cdot 0.005555555555555556\right)\right)\right)}^{2} \]

   5. pow-to-exp 31.1%

      \[\leadsto {\left(a \cdot \cos \left({\left(\pi \cdot \left(angle \cdot 0.005555555555555556\right)\right)}^{0.16666666666666666} \cdot \left({\color{blue}{\left(e^{\log \left(\left(\left(\pi \cdot \frac{angle}{180}\right) \cdot \left(\pi \cdot \frac{angle}{180}\right)\right) \cdot \left(\pi \cdot \frac{angle}{180}\right)\right) \cdot 0.3333333333333333}\right)}}^{0.16666666666666666} \cdot {\left(\sqrt[3]{\pi \cdot \left(angle \cdot 0.005555555555555556\right)}\right)}^{2}\right)\right)\right)}^{2} + {\left(b \cdot \sin \left(\pi \cdot \left(angle \cdot 0.005555555555555556\right)\right)\right)}^{2} \]

   6. pow3 31.1%

      \[\leadsto {\left(a \cdot \cos \left({\left(\pi \cdot \left(angle \cdot 0.005555555555555556\right)\right)}^{0.16666666666666666} \cdot \left({\left(e^{\log \color{blue}{\left({\left(\pi \cdot \frac{angle}{180}\right)}^{3}\right)} \cdot 0.3333333333333333}\right)}^{0.16666666666666666} \cdot {\left(\sqrt[3]{\pi \cdot \left(angle \cdot 0.005555555555555556\right)}\right)}^{2}\right)\right)\right)}^{2} + {\left(b \cdot \sin \left(\pi \cdot \left(angle \cdot 0.005555555555555556\right)\right)\right)}^{2} \]

   7. log-pow 42.2%

      \[\leadsto {\left(a \cdot \cos \left({\left(\pi \cdot \left(angle \cdot 0.005555555555555556\right)\right)}^{0.16666666666666666} \cdot \left({\left(e^{\color{blue}{\left(3 \cdot \log \left(\pi \cdot \frac{angle}{180}\right)\right)} \cdot 0.3333333333333333}\right)}^{0.16666666666666666} \cdot {\left(\sqrt[3]{\pi \cdot \left(angle \cdot 0.005555555555555556\right)}\right)}^{2}\right)\right)\right)}^{2} + {\left(b \cdot \sin \left(\pi \cdot \left(angle \cdot 0.005555555555555556\right)\right)\right)}^{2} \]

   8. div-inv 42.2%

      \[\leadsto {\left(a \cdot \cos \left({\left(\pi \cdot \left(angle \cdot 0.005555555555555556\right)\right)}^{0.16666666666666666} \cdot \left({\left(e^{\left(3 \cdot \log \left(\pi \cdot \color{blue}{\left(angle \cdot \frac{1}{180}\right)}\right)\right) \cdot 0.3333333333333333}\right)}^{0.16666666666666666} \cdot {\left(\sqrt[3]{\pi \cdot \left(angle \cdot 0.005555555555555556\right)}\right)}^{2}\right)\right)\right)}^{2} + {\left(b \cdot \sin \left(\pi \cdot \left(angle \cdot 0.005555555555555556\right)\right)\right)}^{2} \]

   9. metadata-eval 42.2%

      \[\leadsto {\left(a \cdot \cos \left({\left(\pi \cdot \left(angle \cdot 0.005555555555555556\right)\right)}^{0.16666666666666666} \cdot \left({\left(e^{\left(3 \cdot \log \left(\pi \cdot \left(angle \cdot \color{blue}{0.005555555555555556}\right)\right)\right) \cdot 0.3333333333333333}\right)}^{0.16666666666666666} \cdot {\left(\sqrt[3]{\pi \cdot \left(angle \cdot 0.005555555555555556\right)}\right)}^{2}\right)\right)\right)}^{2} + {\left(b \cdot \sin \left(\pi \cdot \left(angle \cdot 0.005555555555555556\right)\right)\right)}^{2} \]

10. Applied egg-rr 42.2%

    \[\leadsto {\left(a \cdot \cos \left({\left(\pi \cdot \left(angle \cdot 0.005555555555555556\right)\right)}^{0.16666666666666666} \cdot \left({\color{blue}{\left(e^{\left(3 \cdot \log \left(\pi \cdot \left(angle \cdot 0.005555555555555556\right)\right)\right) \cdot 0.3333333333333333}\right)}}^{0.16666666666666666} \cdot {\left(\sqrt[3]{\pi \cdot \left(angle \cdot 0.005555555555555556\right)}\right)}^{2}\right)\right)\right)}^{2} + {\left(b \cdot \sin \left(\pi \cdot \left(angle \cdot 0.005555555555555556\right)\right)\right)}^{2} \]
11. Step-by-step derivation

    1. *-commutative 42.2%

       \[\leadsto {\left(a \cdot \cos \left({\left(\pi \cdot \left(angle \cdot 0.005555555555555556\right)\right)}^{0.16666666666666666} \cdot \left({\left(e^{\color{blue}{0.3333333333333333 \cdot \left(3 \cdot \log \left(\pi \cdot \left(angle \cdot 0.005555555555555556\right)\right)\right)}}\right)}^{0.16666666666666666} \cdot {\left(\sqrt[3]{\pi \cdot \left(angle \cdot 0.005555555555555556\right)}\right)}^{2}\right)\right)\right)}^{2} + {\left(b \cdot \sin \left(\pi \cdot \left(angle \cdot 0.005555555555555556\right)\right)\right)}^{2} \]

    2. associate-*r* 42.2%

       \[\leadsto {\left(a \cdot \cos \left({\left(\pi \cdot \left(angle \cdot 0.005555555555555556\right)\right)}^{0.16666666666666666} \cdot \left({\left(e^{\color{blue}{\left(0.3333333333333333 \cdot 3\right) \cdot \log \left(\pi \cdot \left(angle \cdot 0.005555555555555556\right)\right)}}\right)}^{0.16666666666666666} \cdot {\left(\sqrt[3]{\pi \cdot \left(angle \cdot 0.005555555555555556\right)}\right)}^{2}\right)\right)\right)}^{2} + {\left(b \cdot \sin \left(\pi \cdot \left(angle \cdot 0.005555555555555556\right)\right)\right)}^{2} \]

    3. metadata-eval 42.2%

       \[\leadsto {\left(a \cdot \cos \left({\left(\pi \cdot \left(angle \cdot 0.005555555555555556\right)\right)}^{0.16666666666666666} \cdot \left({\left(e^{\color{blue}{1} \cdot \log \left(\pi \cdot \left(angle \cdot 0.005555555555555556\right)\right)}\right)}^{0.16666666666666666} \cdot {\left(\sqrt[3]{\pi \cdot \left(angle \cdot 0.005555555555555556\right)}\right)}^{2}\right)\right)\right)}^{2} + {\left(b \cdot \sin \left(\pi \cdot \left(angle \cdot 0.005555555555555556\right)\right)\right)}^{2} \]

    4. *-un-lft-identity 42.2%

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

    5. add-exp-log 42.2%

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

    6. metadata-eval 42.2%

       \[\leadsto {\left(a \cdot \cos \left({\left(\pi \cdot \left(angle \cdot 0.005555555555555556\right)\right)}^{0.16666666666666666} \cdot \left({\left(\pi \cdot \left(angle \cdot \color{blue}{\frac{1}{180}}\right)\right)}^{0.16666666666666666} \cdot {\left(\sqrt[3]{\pi \cdot \left(angle \cdot 0.005555555555555556\right)}\right)}^{2}\right)\right)\right)}^{2} + {\left(b \cdot \sin \left(\pi \cdot \left(angle \cdot 0.005555555555555556\right)\right)\right)}^{2} \]

    7. div-inv 42.2%

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

    8. clear-num 42.2%

       \[\leadsto {\left(a \cdot \cos \left({\left(\pi \cdot \left(angle \cdot 0.005555555555555556\right)\right)}^{0.16666666666666666} \cdot \left({\left(\pi \cdot \color{blue}{\frac{1}{\frac{180}{angle}}}\right)}^{0.16666666666666666} \cdot {\left(\sqrt[3]{\pi \cdot \left(angle \cdot 0.005555555555555556\right)}\right)}^{2}\right)\right)\right)}^{2} + {\left(b \cdot \sin \left(\pi \cdot \left(angle \cdot 0.005555555555555556\right)\right)\right)}^{2} \]

    9. div-inv 42.1%

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

    10. rem-3cbrt-rft 42.1%

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

12. Applied egg-rr 42.3%

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

13. Final simplification 42.3%

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

Alternative 2: 79.9% accurate, 0.4× speedup

Math

\[\begin{array}{l} angle_m = \left|angle\right| \\ \begin{array}{l} t_0 := \pi \cdot \left(angle\_m \cdot 0.005555555555555556\right)\\ {\left(b \cdot \sin t\_0\right)}^{2} + {\left(a \cdot \cos \left({t\_0}^{0.16666666666666666} \cdot \left({\left(\sqrt[3]{t\_0}\right)}^{2} \cdot {\left(e^{\left(3 \cdot \log t\_0\right) \cdot 0.3333333333333333}\right)}^{0.16666666666666666}\right)\right)\right)}^{2} \end{array} \end{array} \]

FPCore

angle_m = (fabs.f64 angle)
(FPCore (a b angle_m)
 :precision binary64
 (let* ((t_0 (* PI (* angle_m 0.005555555555555556))))
   (+
    (pow (* b (sin t_0)) 2.0)
    (pow
     (*
      a
      (cos
       (*
        (pow t_0 0.16666666666666666)
        (*
         (pow (cbrt t_0) 2.0)
         (pow
          (exp (* (* 3.0 (log t_0)) 0.3333333333333333))
          0.16666666666666666)))))
     2.0))))

C

angle_m = fabs(angle);
double code(double a, double b, double angle_m) {
	double t_0 = ((double) M_PI) * (angle_m * 0.005555555555555556);
	return pow((b * sin(t_0)), 2.0) + pow((a * cos((pow(t_0, 0.16666666666666666) * (pow(cbrt(t_0), 2.0) * pow(exp(((3.0 * log(t_0)) * 0.3333333333333333)), 0.16666666666666666))))), 2.0);
}

Java

angle_m = Math.abs(angle);
public static double code(double a, double b, double angle_m) {
	double t_0 = Math.PI * (angle_m * 0.005555555555555556);
	return Math.pow((b * Math.sin(t_0)), 2.0) + Math.pow((a * Math.cos((Math.pow(t_0, 0.16666666666666666) * (Math.pow(Math.cbrt(t_0), 2.0) * Math.pow(Math.exp(((3.0 * Math.log(t_0)) * 0.3333333333333333)), 0.16666666666666666))))), 2.0);
}

Julia

angle_m = abs(angle)
function code(a, b, angle_m)
	t_0 = Float64(pi * Float64(angle_m * 0.005555555555555556))
	return Float64((Float64(b * sin(t_0)) ^ 2.0) + (Float64(a * cos(Float64((t_0 ^ 0.16666666666666666) * Float64((cbrt(t_0) ^ 2.0) * (exp(Float64(Float64(3.0 * log(t_0)) * 0.3333333333333333)) ^ 0.16666666666666666))))) ^ 2.0))
end

Wolfram

angle_m = N[Abs[angle], $MachinePrecision]
code[a_, b_, angle$95$m_] := Block[{t$95$0 = N[(Pi * N[(angle$95$m * 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[(N[Power[t$95$0, 0.16666666666666666], $MachinePrecision] * N[(N[Power[N[Power[t$95$0, 1/3], $MachinePrecision], 2.0], $MachinePrecision] * N[Power[N[Exp[N[(N[(3.0 * N[Log[t$95$0], $MachinePrecision]), $MachinePrecision] * 0.3333333333333333), $MachinePrecision]], $MachinePrecision], 0.16666666666666666], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Initial program 77.1%

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

3. Simplified 77.1%

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

   1. metadata-eval 77.1%

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

   2. div-inv 77.1%

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

   3. add-cube-cbrt 77.2%

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

   4. pow3 77.1%

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

   5. div-inv 77.2%

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

   6. metadata-eval 77.2%

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

6. Applied egg-rr 77.2%

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

   1. cube-mult 77.2%

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

   2. add-sqr-sqrt 42.1%

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

   3. associate-*l* 42.1%

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

   4. pow1/3 42.0%

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

   5. sqrt-pow1 42.0%

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

   6. metadata-eval 42.0%

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

   7. pow1/3 42.2%

      \[\leadsto {\left(a \cdot \cos \left({\left(\pi \cdot \left(angle \cdot 0.005555555555555556\right)\right)}^{0.16666666666666666} \cdot \left(\sqrt{\color{blue}{{\left(\pi \cdot \left(angle \cdot 0.005555555555555556\right)\right)}^{0.3333333333333333}}} \cdot \left(\sqrt[3]{\pi \cdot \left(angle \cdot 0.005555555555555556\right)} \cdot \sqrt[3]{\pi \cdot \left(angle \cdot 0.005555555555555556\right)}\right)\right)\right)\right)}^{2} + {\left(b \cdot \sin \left(\pi \cdot \left(angle \cdot 0.005555555555555556\right)\right)\right)}^{2} \]

   8. sqrt-pow1 42.2%

      \[\leadsto {\left(a \cdot \cos \left({\left(\pi \cdot \left(angle \cdot 0.005555555555555556\right)\right)}^{0.16666666666666666} \cdot \left(\color{blue}{{\left(\pi \cdot \left(angle \cdot 0.005555555555555556\right)\right)}^{\left(\frac{0.3333333333333333}{2}\right)}} \cdot \left(\sqrt[3]{\pi \cdot \left(angle \cdot 0.005555555555555556\right)} \cdot \sqrt[3]{\pi \cdot \left(angle \cdot 0.005555555555555556\right)}\right)\right)\right)\right)}^{2} + {\left(b \cdot \sin \left(\pi \cdot \left(angle \cdot 0.005555555555555556\right)\right)\right)}^{2} \]

   9. metadata-eval 42.2%

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

   10. pow2 42.2%

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

8. Applied egg-rr 42.2%

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

   1. metadata-eval 42.2%

      \[\leadsto {\left(a \cdot \cos \left({\left(\pi \cdot \left(angle \cdot 0.005555555555555556\right)\right)}^{0.16666666666666666} \cdot \left({\left(\pi \cdot \left(angle \cdot \color{blue}{\frac{1}{180}}\right)\right)}^{0.16666666666666666} \cdot {\left(\sqrt[3]{\pi \cdot \left(angle \cdot 0.005555555555555556\right)}\right)}^{2}\right)\right)\right)}^{2} + {\left(b \cdot \sin \left(\pi \cdot \left(angle \cdot 0.005555555555555556\right)\right)\right)}^{2} \]

   2. div-inv 42.2%

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

   3. add-cbrt-cube 31.1%

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

   4. pow1/3 31.0%

      \[\leadsto {\left(a \cdot \cos \left({\left(\pi \cdot \left(angle \cdot 0.005555555555555556\right)\right)}^{0.16666666666666666} \cdot \left({\color{blue}{\left({\left(\left(\left(\pi \cdot \frac{angle}{180}\right) \cdot \left(\pi \cdot \frac{angle}{180}\right)\right) \cdot \left(\pi \cdot \frac{angle}{180}\right)\right)}^{0.3333333333333333}\right)}}^{0.16666666666666666} \cdot {\left(\sqrt[3]{\pi \cdot \left(angle \cdot 0.005555555555555556\right)}\right)}^{2}\right)\right)\right)}^{2} + {\left(b \cdot \sin \left(\pi \cdot \left(angle \cdot 0.005555555555555556\right)\right)\right)}^{2} \]

   5. pow-to-exp 31.1%

      \[\leadsto {\left(a \cdot \cos \left({\left(\pi \cdot \left(angle \cdot 0.005555555555555556\right)\right)}^{0.16666666666666666} \cdot \left({\color{blue}{\left(e^{\log \left(\left(\left(\pi \cdot \frac{angle}{180}\right) \cdot \left(\pi \cdot \frac{angle}{180}\right)\right) \cdot \left(\pi \cdot \frac{angle}{180}\right)\right) \cdot 0.3333333333333333}\right)}}^{0.16666666666666666} \cdot {\left(\sqrt[3]{\pi \cdot \left(angle \cdot 0.005555555555555556\right)}\right)}^{2}\right)\right)\right)}^{2} + {\left(b \cdot \sin \left(\pi \cdot \left(angle \cdot 0.005555555555555556\right)\right)\right)}^{2} \]

   6. pow3 31.1%

      \[\leadsto {\left(a \cdot \cos \left({\left(\pi \cdot \left(angle \cdot 0.005555555555555556\right)\right)}^{0.16666666666666666} \cdot \left({\left(e^{\log \color{blue}{\left({\left(\pi \cdot \frac{angle}{180}\right)}^{3}\right)} \cdot 0.3333333333333333}\right)}^{0.16666666666666666} \cdot {\left(\sqrt[3]{\pi \cdot \left(angle \cdot 0.005555555555555556\right)}\right)}^{2}\right)\right)\right)}^{2} + {\left(b \cdot \sin \left(\pi \cdot \left(angle \cdot 0.005555555555555556\right)\right)\right)}^{2} \]

   7. log-pow 42.2%

      \[\leadsto {\left(a \cdot \cos \left({\left(\pi \cdot \left(angle \cdot 0.005555555555555556\right)\right)}^{0.16666666666666666} \cdot \left({\left(e^{\color{blue}{\left(3 \cdot \log \left(\pi \cdot \frac{angle}{180}\right)\right)} \cdot 0.3333333333333333}\right)}^{0.16666666666666666} \cdot {\left(\sqrt[3]{\pi \cdot \left(angle \cdot 0.005555555555555556\right)}\right)}^{2}\right)\right)\right)}^{2} + {\left(b \cdot \sin \left(\pi \cdot \left(angle \cdot 0.005555555555555556\right)\right)\right)}^{2} \]

   8. div-inv 42.2%

      \[\leadsto {\left(a \cdot \cos \left({\left(\pi \cdot \left(angle \cdot 0.005555555555555556\right)\right)}^{0.16666666666666666} \cdot \left({\left(e^{\left(3 \cdot \log \left(\pi \cdot \color{blue}{\left(angle \cdot \frac{1}{180}\right)}\right)\right) \cdot 0.3333333333333333}\right)}^{0.16666666666666666} \cdot {\left(\sqrt[3]{\pi \cdot \left(angle \cdot 0.005555555555555556\right)}\right)}^{2}\right)\right)\right)}^{2} + {\left(b \cdot \sin \left(\pi \cdot \left(angle \cdot 0.005555555555555556\right)\right)\right)}^{2} \]

   9. metadata-eval 42.2%

      \[\leadsto {\left(a \cdot \cos \left({\left(\pi \cdot \left(angle \cdot 0.005555555555555556\right)\right)}^{0.16666666666666666} \cdot \left({\left(e^{\left(3 \cdot \log \left(\pi \cdot \left(angle \cdot \color{blue}{0.005555555555555556}\right)\right)\right) \cdot 0.3333333333333333}\right)}^{0.16666666666666666} \cdot {\left(\sqrt[3]{\pi \cdot \left(angle \cdot 0.005555555555555556\right)}\right)}^{2}\right)\right)\right)}^{2} + {\left(b \cdot \sin \left(\pi \cdot \left(angle \cdot 0.005555555555555556\right)\right)\right)}^{2} \]

10. Applied egg-rr 42.2%

    \[\leadsto {\left(a \cdot \cos \left({\left(\pi \cdot \left(angle \cdot 0.005555555555555556\right)\right)}^{0.16666666666666666} \cdot \left({\color{blue}{\left(e^{\left(3 \cdot \log \left(\pi \cdot \left(angle \cdot 0.005555555555555556\right)\right)\right) \cdot 0.3333333333333333}\right)}}^{0.16666666666666666} \cdot {\left(\sqrt[3]{\pi \cdot \left(angle \cdot 0.005555555555555556\right)}\right)}^{2}\right)\right)\right)}^{2} + {\left(b \cdot \sin \left(\pi \cdot \left(angle \cdot 0.005555555555555556\right)\right)\right)}^{2} \]

11. Final simplification 42.2%

    \[\leadsto {\left(b \cdot \sin \left(\pi \cdot \left(angle \cdot 0.005555555555555556\right)\right)\right)}^{2} + {\left(a \cdot \cos \left({\left(\pi \cdot \left(angle \cdot 0.005555555555555556\right)\right)}^{0.16666666666666666} \cdot \left({\left(\sqrt[3]{\pi \cdot \left(angle \cdot 0.005555555555555556\right)}\right)}^{2} \cdot {\left(e^{\left(3 \cdot \log \left(\pi \cdot \left(angle \cdot 0.005555555555555556\right)\right)\right) \cdot 0.3333333333333333}\right)}^{0.16666666666666666}\right)\right)\right)}^{2} \]
12. Add Preprocessing

Alternative 3: 79.9% accurate, 0.4× speedup

Math

\[\begin{array}{l} angle_m = \left|angle\right| \\ \begin{array}{l} t_0 := \pi \cdot \left(angle\_m \cdot 0.005555555555555556\right)\\ {\left(b \cdot \sin t\_0\right)}^{2} + {\left(a \cdot \cos \left({t\_0}^{0.16666666666666666} \cdot \left({\left(\sqrt[3]{t\_0}\right)}^{2} \cdot {\left(\mathsf{expm1}\left(\mathsf{log1p}\left(t\_0\right)\right)\right)}^{0.16666666666666666}\right)\right)\right)}^{2} \end{array} \end{array} \]

FPCore

angle_m = (fabs.f64 angle)
(FPCore (a b angle_m)
 :precision binary64
 (let* ((t_0 (* PI (* angle_m 0.005555555555555556))))
   (+
    (pow (* b (sin t_0)) 2.0)
    (pow
     (*
      a
      (cos
       (*
        (pow t_0 0.16666666666666666)
        (*
         (pow (cbrt t_0) 2.0)
         (pow (expm1 (log1p t_0)) 0.16666666666666666)))))
     2.0))))

C

angle_m = fabs(angle);
double code(double a, double b, double angle_m) {
	double t_0 = ((double) M_PI) * (angle_m * 0.005555555555555556);
	return pow((b * sin(t_0)), 2.0) + pow((a * cos((pow(t_0, 0.16666666666666666) * (pow(cbrt(t_0), 2.0) * pow(expm1(log1p(t_0)), 0.16666666666666666))))), 2.0);
}

Java

angle_m = Math.abs(angle);
public static double code(double a, double b, double angle_m) {
	double t_0 = Math.PI * (angle_m * 0.005555555555555556);
	return Math.pow((b * Math.sin(t_0)), 2.0) + Math.pow((a * Math.cos((Math.pow(t_0, 0.16666666666666666) * (Math.pow(Math.cbrt(t_0), 2.0) * Math.pow(Math.expm1(Math.log1p(t_0)), 0.16666666666666666))))), 2.0);
}

Julia

angle_m = abs(angle)
function code(a, b, angle_m)
	t_0 = Float64(pi * Float64(angle_m * 0.005555555555555556))
	return Float64((Float64(b * sin(t_0)) ^ 2.0) + (Float64(a * cos(Float64((t_0 ^ 0.16666666666666666) * Float64((cbrt(t_0) ^ 2.0) * (expm1(log1p(t_0)) ^ 0.16666666666666666))))) ^ 2.0))
end

Wolfram

angle_m = N[Abs[angle], $MachinePrecision]
code[a_, b_, angle$95$m_] := Block[{t$95$0 = N[(Pi * N[(angle$95$m * 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[(N[Power[t$95$0, 0.16666666666666666], $MachinePrecision] * N[(N[Power[N[Power[t$95$0, 1/3], $MachinePrecision], 2.0], $MachinePrecision] * N[Power[N[(Exp[N[Log[1 + t$95$0], $MachinePrecision]] - 1), $MachinePrecision], 0.16666666666666666], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Initial program 77.1%

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

3. Simplified 77.1%

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

   1. metadata-eval 77.1%

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

   2. div-inv 77.1%

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

   3. add-cube-cbrt 77.2%

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

   4. pow3 77.1%

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

   5. div-inv 77.2%

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

   6. metadata-eval 77.2%

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

6. Applied egg-rr 77.2%

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

   1. cube-mult 77.2%

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

   2. add-sqr-sqrt 42.1%

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

   3. associate-*l* 42.1%

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

   4. pow1/3 42.0%

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

   5. sqrt-pow1 42.0%

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

   6. metadata-eval 42.0%

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

   7. pow1/3 42.2%

      \[\leadsto {\left(a \cdot \cos \left({\left(\pi \cdot \left(angle \cdot 0.005555555555555556\right)\right)}^{0.16666666666666666} \cdot \left(\sqrt{\color{blue}{{\left(\pi \cdot \left(angle \cdot 0.005555555555555556\right)\right)}^{0.3333333333333333}}} \cdot \left(\sqrt[3]{\pi \cdot \left(angle \cdot 0.005555555555555556\right)} \cdot \sqrt[3]{\pi \cdot \left(angle \cdot 0.005555555555555556\right)}\right)\right)\right)\right)}^{2} + {\left(b \cdot \sin \left(\pi \cdot \left(angle \cdot 0.005555555555555556\right)\right)\right)}^{2} \]

   8. sqrt-pow1 42.2%

      \[\leadsto {\left(a \cdot \cos \left({\left(\pi \cdot \left(angle \cdot 0.005555555555555556\right)\right)}^{0.16666666666666666} \cdot \left(\color{blue}{{\left(\pi \cdot \left(angle \cdot 0.005555555555555556\right)\right)}^{\left(\frac{0.3333333333333333}{2}\right)}} \cdot \left(\sqrt[3]{\pi \cdot \left(angle \cdot 0.005555555555555556\right)} \cdot \sqrt[3]{\pi \cdot \left(angle \cdot 0.005555555555555556\right)}\right)\right)\right)\right)}^{2} + {\left(b \cdot \sin \left(\pi \cdot \left(angle \cdot 0.005555555555555556\right)\right)\right)}^{2} \]

   9. metadata-eval 42.2%

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

   10. pow2 42.2%

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

8. Applied egg-rr 42.2%

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

   1. expm1-log1p-u 42.2%

      \[\leadsto {\left(a \cdot \cos \left({\left(\pi \cdot \left(angle \cdot 0.005555555555555556\right)\right)}^{0.16666666666666666} \cdot \left({\color{blue}{\left(\mathsf{expm1}\left(\mathsf{log1p}\left(\pi \cdot \left(angle \cdot 0.005555555555555556\right)\right)\right)\right)}}^{0.16666666666666666} \cdot {\left(\sqrt[3]{\pi \cdot \left(angle \cdot 0.005555555555555556\right)}\right)}^{2}\right)\right)\right)}^{2} + {\left(b \cdot \sin \left(\pi \cdot \left(angle \cdot 0.005555555555555556\right)\right)\right)}^{2} \]

10. Applied egg-rr 42.2%

    \[\leadsto {\left(a \cdot \cos \left({\left(\pi \cdot \left(angle \cdot 0.005555555555555556\right)\right)}^{0.16666666666666666} \cdot \left({\color{blue}{\left(\mathsf{expm1}\left(\mathsf{log1p}\left(\pi \cdot \left(angle \cdot 0.005555555555555556\right)\right)\right)\right)}}^{0.16666666666666666} \cdot {\left(\sqrt[3]{\pi \cdot \left(angle \cdot 0.005555555555555556\right)}\right)}^{2}\right)\right)\right)}^{2} + {\left(b \cdot \sin \left(\pi \cdot \left(angle \cdot 0.005555555555555556\right)\right)\right)}^{2} \]

11. Final simplification 42.2%

    \[\leadsto {\left(b \cdot \sin \left(\pi \cdot \left(angle \cdot 0.005555555555555556\right)\right)\right)}^{2} + {\left(a \cdot \cos \left({\left(\pi \cdot \left(angle \cdot 0.005555555555555556\right)\right)}^{0.16666666666666666} \cdot \left({\left(\sqrt[3]{\pi \cdot \left(angle \cdot 0.005555555555555556\right)}\right)}^{2} \cdot {\left(\mathsf{expm1}\left(\mathsf{log1p}\left(\pi \cdot \left(angle \cdot 0.005555555555555556\right)\right)\right)\right)}^{0.16666666666666666}\right)\right)\right)}^{2} \]
12. Add Preprocessing

Alternative 4: 79.9% accurate, 0.5× speedup

Math

\[\begin{array}{l} angle_m = \left|angle\right| \\ \begin{array}{l} t_0 := \pi \cdot \left(angle\_m \cdot 0.005555555555555556\right)\\ t_1 := {t\_0}^{0.16666666666666666}\\ {\left(b \cdot \sin t\_0\right)}^{2} + {\left(a \cdot \cos \left(t\_1 \cdot \left(t\_1 \cdot {\left(\sqrt[3]{t\_0}\right)}^{2}\right)\right)\right)}^{2} \end{array} \end{array} \]

FPCore

angle_m = (fabs.f64 angle)
(FPCore (a b angle_m)
 :precision binary64
 (let* ((t_0 (* PI (* angle_m 0.005555555555555556)))
        (t_1 (pow t_0 0.16666666666666666)))
   (+
    (pow (* b (sin t_0)) 2.0)
    (pow (* a (cos (* t_1 (* t_1 (pow (cbrt t_0) 2.0))))) 2.0))))

C

angle_m = fabs(angle);
double code(double a, double b, double angle_m) {
	double t_0 = ((double) M_PI) * (angle_m * 0.005555555555555556);
	double t_1 = pow(t_0, 0.16666666666666666);
	return pow((b * sin(t_0)), 2.0) + pow((a * cos((t_1 * (t_1 * pow(cbrt(t_0), 2.0))))), 2.0);
}

Java

angle_m = Math.abs(angle);
public static double code(double a, double b, double angle_m) {
	double t_0 = Math.PI * (angle_m * 0.005555555555555556);
	double t_1 = Math.pow(t_0, 0.16666666666666666);
	return Math.pow((b * Math.sin(t_0)), 2.0) + Math.pow((a * Math.cos((t_1 * (t_1 * Math.pow(Math.cbrt(t_0), 2.0))))), 2.0);
}

Julia

angle_m = abs(angle)
function code(a, b, angle_m)
	t_0 = Float64(pi * Float64(angle_m * 0.005555555555555556))
	t_1 = t_0 ^ 0.16666666666666666
	return Float64((Float64(b * sin(t_0)) ^ 2.0) + (Float64(a * cos(Float64(t_1 * Float64(t_1 * (cbrt(t_0) ^ 2.0))))) ^ 2.0))
end

Wolfram

angle_m = N[Abs[angle], $MachinePrecision]
code[a_, b_, angle$95$m_] := Block[{t$95$0 = N[(Pi * N[(angle$95$m * 0.005555555555555556), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[Power[t$95$0, 0.16666666666666666], $MachinePrecision]}, N[(N[Power[N[(b * N[Sin[t$95$0], $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision] + N[Power[N[(a * N[Cos[N[(t$95$1 * N[(t$95$1 * N[Power[N[Power[t$95$0, 1/3], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Initial program 77.1%

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

3. Simplified 77.1%

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

   1. metadata-eval 77.1%

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

   2. div-inv 77.1%

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

   3. add-cube-cbrt 77.2%

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

   4. pow3 77.1%

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

   5. div-inv 77.2%

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

   6. metadata-eval 77.2%

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

6. Applied egg-rr 77.2%

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

   1. cube-mult 77.2%

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

   2. add-sqr-sqrt 42.1%

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

   3. associate-*l* 42.1%

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

   4. pow1/3 42.0%

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

   5. sqrt-pow1 42.0%

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

   6. metadata-eval 42.0%

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

   7. pow1/3 42.2%

      \[\leadsto {\left(a \cdot \cos \left({\left(\pi \cdot \left(angle \cdot 0.005555555555555556\right)\right)}^{0.16666666666666666} \cdot \left(\sqrt{\color{blue}{{\left(\pi \cdot \left(angle \cdot 0.005555555555555556\right)\right)}^{0.3333333333333333}}} \cdot \left(\sqrt[3]{\pi \cdot \left(angle \cdot 0.005555555555555556\right)} \cdot \sqrt[3]{\pi \cdot \left(angle \cdot 0.005555555555555556\right)}\right)\right)\right)\right)}^{2} + {\left(b \cdot \sin \left(\pi \cdot \left(angle \cdot 0.005555555555555556\right)\right)\right)}^{2} \]

   8. sqrt-pow1 42.2%

      \[\leadsto {\left(a \cdot \cos \left({\left(\pi \cdot \left(angle \cdot 0.005555555555555556\right)\right)}^{0.16666666666666666} \cdot \left(\color{blue}{{\left(\pi \cdot \left(angle \cdot 0.005555555555555556\right)\right)}^{\left(\frac{0.3333333333333333}{2}\right)}} \cdot \left(\sqrt[3]{\pi \cdot \left(angle \cdot 0.005555555555555556\right)} \cdot \sqrt[3]{\pi \cdot \left(angle \cdot 0.005555555555555556\right)}\right)\right)\right)\right)}^{2} + {\left(b \cdot \sin \left(\pi \cdot \left(angle \cdot 0.005555555555555556\right)\right)\right)}^{2} \]

   9. metadata-eval 42.2%

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

   10. pow2 42.2%

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

8. Applied egg-rr 42.2%

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

9. Final simplification 42.2%

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

Alternative 5: 79.9% accurate, 0.7× speedup

Math

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

FPCore

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

C

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

Java

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

Julia

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

Wolfram

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

Derivation

1. Initial program 77.1%

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

3. Simplified 77.1%

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

   1. metadata-eval 77.1%

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

   2. div-inv 77.1%

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

   3. associate-*r/ 77.3%

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

6. Applied egg-rr 77.3%

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

   1. add-cbrt-cube 77.3%

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

   2. pow3 77.3%

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

8. Applied egg-rr 77.3%

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

9. Final simplification 77.3%

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

Alternative 6: 80.0% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 77.1%

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

3. Simplified 77.1%

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

5. Taylor expanded in angle around inf 77.3%

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

   1. associate-*r* 77.1%

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

   2. *-commutative 77.1%

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

   3. associate-*r* 77.2%

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

7. Simplified 77.2%

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

8. Final simplification 77.2%

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

Alternative 7: 79.9% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 77.1%

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

3. Simplified 77.1%

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

   1. metadata-eval 77.1%

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

   2. div-inv 77.1%

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

   3. clear-num 77.1%

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

   4. un-div-inv 77.2%

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

6. Applied egg-rr 77.2%

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

7. Final simplification 77.2%

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

Alternative 8: 79.8% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 77.1%

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

3. Simplified 77.1%

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

   1. metadata-eval 77.1%

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

   2. div-inv 77.1%

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

   3. associate-*r/ 77.3%

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

6. Applied egg-rr 77.3%

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

7. Final simplification 77.3%

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

Alternative 9: 79.8% accurate, 1.3× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 77.1%

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

3. Simplified 77.1%

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

5. Taylor expanded in angle around 0 76.8%

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

6. Taylor expanded in angle around inf 76.8%

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

7. Final simplification 76.8%

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

Alternative 10: 74.6% accurate, 2.0× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 77.1%

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

3. Simplified 77.1%

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

5. Taylor expanded in angle around 0 76.8%

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

6. Taylor expanded in angle around 0 71.2%

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

   1. unpow2 71.2%

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

   2. associate-*r* 71.2%

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

   3. associate-*l* 71.1%

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

   4. *-commutative 71.1%

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

   5. *-commutative 71.1%

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

   6. *-commutative 71.1%

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

   7. associate-*r* 71.1%

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

   8. *-commutative 71.1%

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

   9. metadata-eval 71.1%

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

   10. div-inv 71.1%

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

   11. *-commutative 71.1%

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

   12. associate-*l* 71.1%

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

   13. div-inv 71.1%

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

   14. metadata-eval 71.1%

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

8. Applied egg-rr 71.1%

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

   1. *-commutative 71.1%

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

   2. *-commutative 71.1%

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

   3. associate-*l* 71.3%

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

   4. associate-*r* 71.2%

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

   5. associate-*l* 71.2%

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

   6. pow2 71.2%

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

   7. associate-*l* 71.2%

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

   8. associate-*r* 71.2%

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

   9. *-commutative 71.2%

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

   10. associate-*l* 71.2%

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

10. Applied egg-rr 71.2%

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

11. Final simplification 71.2%

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

Alternative 11: 74.6% accurate, 2.0× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 77.1%

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

3. Simplified 77.1%

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

5. Taylor expanded in angle around 0 76.8%

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

6. Taylor expanded in angle around 0 71.2%

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

7. Taylor expanded in b around 0 61.2%

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

   1. associate-*r* 61.1%

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

   2. *-commutative 61.1%

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

   3. unpow2 61.1%

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

   4. metadata-eval 61.1%

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

   5. swap-sqr 61.5%

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

   6. associate-*r* 61.5%

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

   7. unpow2 61.5%

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

   8. swap-sqr 71.3%

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

   9. *-commutative 71.3%

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

   10. unpow2 71.3%

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

   11. swap-sqr 71.2%

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

   12. unpow2 71.2%

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

9. Simplified 71.2%

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

10. Final simplification 71.2%

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

Alternative 12: 74.6% accurate, 2.0× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 77.1%

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

3. Simplified 77.1%

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

5. Taylor expanded in angle around 0 76.8%

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

6. Taylor expanded in angle around 0 71.2%

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

   1. associate-*r* 71.2%

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

   2. *-commutative 71.2%

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

   3. associate-*r* 71.3%

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

8. Simplified 71.3%

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

9. Final simplification 71.3%

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

Alternative 13: 73.7% accurate, 3.5× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 77.1%

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

3. Simplified 77.1%

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

5. Taylor expanded in angle around 0 76.8%

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

6. Taylor expanded in angle around 0 71.2%

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

   1. unpow2 71.2%

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

   2. associate-*r* 71.2%

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

   3. associate-*l* 71.1%

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

   4. *-commutative 71.1%

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

   5. *-commutative 71.1%

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

   6. *-commutative 71.1%

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

   7. associate-*r* 71.1%

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

   8. *-commutative 71.1%

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

   9. metadata-eval 71.1%

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

   10. div-inv 71.1%

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

   11. *-commutative 71.1%

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

   12. associate-*l* 71.1%

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

   13. div-inv 71.1%

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

   14. metadata-eval 71.1%

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

8. Applied egg-rr 71.1%

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

9. Taylor expanded in angle around 0 71.1%

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

    1. *-commutative 71.1%

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

11. Simplified 71.1%

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

12. Final simplification 71.1%

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

Alternative 14: 73.7% accurate, 3.5× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 77.1%

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

3. Simplified 77.1%

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

5. Taylor expanded in angle around 0 76.8%

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

6. Taylor expanded in angle around 0 71.2%

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

   1. unpow2 71.2%

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

   2. associate-*r* 71.2%

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

   3. associate-*l* 71.1%

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

   4. *-commutative 71.1%

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

   5. *-commutative 71.1%

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

   6. *-commutative 71.1%

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

   7. associate-*r* 71.1%

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

   8. *-commutative 71.1%

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

   9. metadata-eval 71.1%

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

   10. div-inv 71.1%

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

   11. *-commutative 71.1%

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

   12. associate-*l* 71.1%

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

   13. div-inv 71.1%

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

   14. metadata-eval 71.1%

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

8. Applied egg-rr 71.1%

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

9. Taylor expanded in angle around 0 71.1%

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

    1. *-commutative 71.1%

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

    2. associate-*r* 71.1%

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

    3. *-commutative 71.1%

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

    4. associate-*l* 71.1%

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

    5. *-commutative 71.1%

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

    6. metadata-eval 71.1%

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

    7. /-rgt-identity 71.1%

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

    8. associate-/r/ 71.1%

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

    9. times-frac 71.0%

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

    10. *-lft-identity 71.0%

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

    11. associate-/l/ 71.1%

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

    12. associate-/r/ 71.1%

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

    13. /-rgt-identity 71.1%

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

    14. associate-/l* 71.0%

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

    15. associate-*r* 71.1%

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

    16. *-commutative 71.1%

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

    17. associate-*r* 71.1%

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

    18. associate-/l* 71.1%

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

11. Simplified 71.1%

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

12. Final simplification 71.1%

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

Alternative 15: 73.7% accurate, 3.5× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 77.1%

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

3. Simplified 77.1%

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

5. Taylor expanded in angle around 0 76.8%

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

6. Taylor expanded in angle around 0 71.2%

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

   1. unpow2 71.2%

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

   2. associate-*r* 71.2%

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

   3. associate-*l* 71.1%

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

   4. *-commutative 71.1%

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

   5. *-commutative 71.1%

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

   6. *-commutative 71.1%

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

   7. associate-*r* 71.1%

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

   8. *-commutative 71.1%

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

   9. metadata-eval 71.1%

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

   10. div-inv 71.1%

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

   11. *-commutative 71.1%

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

   12. associate-*l* 71.1%

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

   13. div-inv 71.1%

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

   14. metadata-eval 71.1%

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

8. Applied egg-rr 71.1%

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

   1. associate-*r* 71.1%

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

   2. metadata-eval 71.1%

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

   3. div-inv 71.1%

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

   4. clear-num 71.0%

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

   5. div-inv 71.1%

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

   6. clear-num 71.1%

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

   7. associate-*l/ 71.1%

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

   8. *-un-lft-identity 71.1%

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

   9. associate-/l/ 71.1%

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

10. Applied egg-rr 71.1%

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

11. Final simplification 71.1%

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

Alternative 16: 74.6% accurate, 3.5× speedup

Math

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

FPCore

angle_m = (fabs.f64 angle)
(FPCore (a b angle_m)
 :precision binary64
 (let* ((t_0 (* PI (* (* angle_m 0.005555555555555556) b))))
   (+ (pow a 2.0) (* t_0 t_0))))

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

angle_m = N[Abs[angle], $MachinePrecision]
code[a_, b_, angle$95$m_] := Block[{t$95$0 = N[(Pi * N[(N[(angle$95$m * 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 77.1%

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

3. Simplified 77.1%

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

5. Taylor expanded in angle around 0 76.8%

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

6. Taylor expanded in angle around 0 71.2%

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

   1. unpow2 71.2%

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

   2. *-commutative 71.2%

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

   3. associate-*r* 71.2%

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

   4. *-commutative 71.2%

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

   5. metadata-eval 71.2%

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

   6. div-inv 71.2%

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

   7. *-commutative 71.2%

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

   8. associate-*l* 71.3%

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

   9. div-inv 71.2%

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

   10. metadata-eval 71.2%

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

   11. *-commutative 71.2%

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

   12. associate-*r* 71.2%

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

   13. *-commutative 71.2%

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

   14. metadata-eval 71.2%

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

   15. div-inv 71.3%

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

   16. *-commutative 71.3%

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

   17. associate-*l* 71.2%

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

   18. div-inv 71.2%

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

   19. metadata-eval 71.2%

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

8. Applied egg-rr 71.2%

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

9. Final simplification 71.2%

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

Reproduce

herbie shell --seed 2024080 
(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)))