ab-angle->ABCF C

Percentage Accurate: 80.2% → 80.2%

Time: 1.1min

Alternatives: 17

Speedup: 1.0×

• 36.3% 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: 80.2% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 80.2% accurate, 0.7× speedup

Math

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

FPCore

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

C

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

Java

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

Julia

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

Wolfram

code[a_, b_, angle_] := Block[{t$95$0 = N[Power[N[(angle * 0.005555555555555556), $MachinePrecision], 1/3], $MachinePrecision]}, N[(N[Power[N[(a * N[Cos[N[(t$95$0 * N[(Pi * N[Power[t$95$0, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision] + N[Power[N[(b * N[Sin[N[(Pi * N[(angle / 180.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Initial program 79.6%

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

   1. add-sqr-sqrt 34.2%

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

   2. sqrt-unprod 66.7%

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

   3. associate-*r/ 66.7%

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

   4. associate-*r/ 66.7%

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

   5. frac-times 65.9%

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

   6. metadata-eval 65.9%

      \[\leadsto {\left(a \cdot \cos \left(\sqrt{\frac{\left(\pi \cdot angle\right) \cdot \left(\pi \cdot angle\right)}{\color{blue}{32400}}}\right)\right)}^{2} + {\left(b \cdot \sin \left(\pi \cdot \frac{angle}{180}\right)\right)}^{2} \]

   7. metadata-eval 65.9%

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

   8. frac-times 66.7%

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

   9. associate-*l/ 66.7%

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

   10. associate-*l/ 66.7%

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

   11. sqrt-unprod 45.4%

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

   12. add-sqr-sqrt 79.6%

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

   13. add-exp-log 45.5%

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

   14. add-sqr-sqrt 45.5%

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

   15. sqrt-unprod 66.7%

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

   16. associate-*l/ 66.7%

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

   17. associate-*l/ 66.7%

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

   18. frac-times 65.8%

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

3. Applied egg-rr 34.1%

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

   1. rem-exp-log 79.5%

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

   2. metadata-eval 79.5%

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

   3. div-inv 79.6%

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

   4. add-cube-cbrt 79.7%

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

   5. associate-*r* 79.7%

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

   6. pow2 79.7%

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

   7. div-inv 79.7%

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

   8. metadata-eval 79.7%

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

   9. div-inv 79.7%

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

   10. metadata-eval 79.7%

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

5. Applied egg-rr 79.7%

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

6. Final simplification 79.7%

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

Alternative 2: 80.2% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Python

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

Julia

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

MATLAB

function tmp = code(a, b, angle)
	t_0 = pi * (angle / 180.0);
	tmp = ((b * sin(t_0)) ^ 2.0) + ((a * cos(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[(b * N[Sin[t$95$0], $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision] + N[Power[N[(a * N[Cos[t$95$0], $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Initial program 79.6%

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

2. Final simplification 79.6%

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

Alternative 3: 80.1% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 79.6%

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

   1. clear-num 79.4%

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

   2. un-div-inv 79.4%

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

3. Applied egg-rr 79.6%

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

4. Final simplification 79.6%

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

Alternative 4: 80.2% accurate, 1.0× speedup

Math

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

FPCore

(FPCore (a b angle)
 :precision binary64
 (let* ((t_0 (* angle (/ PI -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 = angle * (((double) M_PI) / -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 = angle * (Math.PI / -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 = angle * (math.pi / -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(angle * Float64(pi / -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 = angle * (pi / -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[(angle * N[(Pi / -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]]

Derivation

1. Initial program 79.6%

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

3. Simplified 79.6%

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

4. Final simplification 79.6%

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

Alternative 5: 80.1% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

Wolfram

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

Derivation

1. Initial program 79.6%

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

2. Taylor expanded in angle around 0 79.4%

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

   1. add-sqr-sqrt 33.8%

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

   2. sqrt-unprod 61.2%

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

   3. associate-*r/ 61.2%

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

   4. associate-*r/ 61.3%

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

   5. frac-times 60.4%

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

   6. metadata-eval 60.4%

      \[\leadsto {\left(a \cdot 1\right)}^{2} + {\left(b \cdot \sin \left(\sqrt{\frac{\left(\pi \cdot angle\right) \cdot \left(\pi \cdot angle\right)}{\color{blue}{32400}}}\right)\right)}^{2} \]

   7. metadata-eval 60.4%

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

   8. frac-times 61.3%

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

   9. associate-*l/ 61.3%

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

   10. associate-*l/ 61.3%

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

   11. sqrt-unprod 45.2%

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

   12. add-sqr-sqrt 79.5%

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

   13. add-cbrt-cube 78.8%

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

   14. add-cbrt-cube 54.7%

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

   15. cbrt-unprod 54.8%

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

4. Applied egg-rr 54.7%

   \[\leadsto {\left(a \cdot 1\right)}^{2} + {\left(b \cdot \sin \color{blue}{\left(\sqrt[3]{{\left(\pi \cdot -0.005555555555555556\right)}^{3} \cdot {angle}^{3}}\right)}\right)}^{2} \]
5. Step-by-step derivation

   1. expm1-log1p-u 54.7%

      \[\leadsto {\left(a \cdot 1\right)}^{2} + {\left(b \cdot \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\sin \left(\sqrt[3]{{\left(\pi \cdot -0.005555555555555556\right)}^{3} \cdot {angle}^{3}}\right)\right)\right)}\right)}^{2} \]

   2. cbrt-prod 54.7%

      \[\leadsto {\left(a \cdot 1\right)}^{2} + {\left(b \cdot \mathsf{expm1}\left(\mathsf{log1p}\left(\sin \color{blue}{\left(\sqrt[3]{{\left(\pi \cdot -0.005555555555555556\right)}^{3}} \cdot \sqrt[3]{{angle}^{3}}\right)}\right)\right)\right)}^{2} \]

   3. rem-cbrt-cube 54.8%

      \[\leadsto {\left(a \cdot 1\right)}^{2} + {\left(b \cdot \mathsf{expm1}\left(\mathsf{log1p}\left(\sin \left(\color{blue}{\left(\pi \cdot -0.005555555555555556\right)} \cdot \sqrt[3]{{angle}^{3}}\right)\right)\right)\right)}^{2} \]

   4. rem-cbrt-cube 79.5%

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

   5. associate-*l* 79.4%

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

6. Applied egg-rr 79.4%

   \[\leadsto {\left(a \cdot 1\right)}^{2} + {\left(b \cdot \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\sin \left(\pi \cdot \left(-0.005555555555555556 \cdot angle\right)\right)\right)\right)}\right)}^{2} \]
7. Step-by-step derivation

   1. *-un-lft-identity 79.4%

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

   2. *-commutative 79.4%

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

   3. *-commutative 79.4%

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

   4. add-sqr-sqrt 45.5%

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

   5. sqrt-unprod 61.2%

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

   6. *-commutative 61.2%

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

   7. *-commutative 61.2%

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

   8. swap-sqr 60.8%

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

   9. metadata-eval 60.8%

      \[\leadsto {\left(a \cdot 1\right)}^{2} + {\left(b \cdot \mathsf{expm1}\left(\mathsf{log1p}\left(\sin \left(\sqrt{\left(angle \cdot angle\right) \cdot \color{blue}{3.08641975308642 \cdot 10^{-5}}} \cdot \pi\right) \cdot 1\right)\right)\right)}^{2} \]

   10. metadata-eval 60.8%

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

   11. swap-sqr 61.2%

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

   12. sqrt-unprod 33.8%

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

   13. add-sqr-sqrt 79.4%

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

   14. associate-*l* 79.5%

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

8. Applied egg-rr 79.5%

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

9. Final simplification 79.5%

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

Alternative 6: 80.0% accurate, 1.5× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 79.6%

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

2. Taylor expanded in angle around 0 79.4%

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

3. Taylor expanded in angle around inf 78.8%

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

4. Final simplification 78.8%

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

Alternative 7: 80.1% accurate, 1.5× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 79.6%

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

2. Taylor expanded in angle around 0 79.4%

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

3. Final simplification 79.4%

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

Alternative 8: 80.0% accurate, 1.5× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 79.6%

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

2. Taylor expanded in angle around 0 79.4%

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

   1. clear-num 79.4%

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

   2. un-div-inv 79.4%

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

4. Applied egg-rr 79.4%

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

5. Final simplification 79.4%

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

Alternative 9: 75.2% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;angle \leq 6.5 \cdot 10^{-99}:\\ \;\;\;\;{a}^{2} + {\left(b \cdot \left(0.005555555555555556 \cdot \left(\pi \cdot angle\right)\right)\right)}^{2}\\ \mathbf{else}:\\ \;\;\;\;{a}^{2} + \left(angle \cdot 0.005555555555555556\right) \cdot \left(\left(\pi \cdot b\right) \cdot \left(angle \cdot \left(0.005555555555555556 \cdot \left(\pi \cdot b\right)\right)\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (a b angle)
 :precision binary64
 (if (<= angle 6.5e-99)
   (+ (pow a 2.0) (pow (* b (* 0.005555555555555556 (* PI angle))) 2.0))
   (+
    (pow a 2.0)
    (*
     (* angle 0.005555555555555556)
     (* (* PI b) (* angle (* 0.005555555555555556 (* PI b))))))))

C

double code(double a, double b, double angle) {
	double tmp;
	if (angle <= 6.5e-99) {
		tmp = pow(a, 2.0) + pow((b * (0.005555555555555556 * (((double) M_PI) * angle))), 2.0);
	} else {
		tmp = pow(a, 2.0) + ((angle * 0.005555555555555556) * ((((double) M_PI) * b) * (angle * (0.005555555555555556 * (((double) M_PI) * b)))));
	}
	return tmp;
}

Java

public static double code(double a, double b, double angle) {
	double tmp;
	if (angle <= 6.5e-99) {
		tmp = Math.pow(a, 2.0) + Math.pow((b * (0.005555555555555556 * (Math.PI * angle))), 2.0);
	} else {
		tmp = Math.pow(a, 2.0) + ((angle * 0.005555555555555556) * ((Math.PI * b) * (angle * (0.005555555555555556 * (Math.PI * b)))));
	}
	return tmp;
}

Python

def code(a, b, angle):
	tmp = 0
	if angle <= 6.5e-99:
		tmp = math.pow(a, 2.0) + math.pow((b * (0.005555555555555556 * (math.pi * angle))), 2.0)
	else:
		tmp = math.pow(a, 2.0) + ((angle * 0.005555555555555556) * ((math.pi * b) * (angle * (0.005555555555555556 * (math.pi * b)))))
	return tmp

Julia

function code(a, b, angle)
	tmp = 0.0
	if (angle <= 6.5e-99)
		tmp = Float64((a ^ 2.0) + (Float64(b * Float64(0.005555555555555556 * Float64(pi * angle))) ^ 2.0));
	else
		tmp = Float64((a ^ 2.0) + Float64(Float64(angle * 0.005555555555555556) * Float64(Float64(pi * b) * Float64(angle * Float64(0.005555555555555556 * Float64(pi * b))))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, angle)
	tmp = 0.0;
	if (angle <= 6.5e-99)
		tmp = (a ^ 2.0) + ((b * (0.005555555555555556 * (pi * angle))) ^ 2.0);
	else
		tmp = (a ^ 2.0) + ((angle * 0.005555555555555556) * ((pi * b) * (angle * (0.005555555555555556 * (pi * b)))));
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, angle_] := If[LessEqual[angle, 6.5e-99], N[(N[Power[a, 2.0], $MachinePrecision] + N[Power[N[(b * N[(0.005555555555555556 * N[(Pi * angle), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision], N[(N[Power[a, 2.0], $MachinePrecision] + N[(N[(angle * 0.005555555555555556), $MachinePrecision] * N[(N[(Pi * b), $MachinePrecision] * N[(angle * N[(0.005555555555555556 * N[(Pi * b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if angle < 6.50000000000000033e-99

   1. Initial program 85.2%

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

   2. Taylor expanded in angle around 0 85.3%

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

   3. Taylor expanded in angle around 0 81.8%

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

   if 6.50000000000000033e-99 < angle

   1. Initial program 66.0%

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

   2. Taylor expanded in angle around 0 65.0%

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

   3. Taylor expanded in angle around 0 55.1%

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

      1. *-commutative 55.1%

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

   5. Simplified 55.1%

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

      1. unpow2 55.1%

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

      2. associate-*r* 55.1%

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

      3. *-commutative 55.1%

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

      4. associate-*l* 60.1%

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

      5. associate-*r* 60.1%

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

      6. *-commutative 60.1%

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

      7. associate-*l* 60.1%

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

   7. Applied egg-rr 60.1%

      \[\leadsto {\left(a \cdot 1\right)}^{2} + \color{blue}{\left(angle \cdot 0.005555555555555556\right) \cdot \left(\left(\pi \cdot b\right) \cdot \left(angle \cdot \left(0.005555555555555556 \cdot \left(\pi \cdot b\right)\right)\right)\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 75.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;angle \leq 6.5 \cdot 10^{-99}:\\ \;\;\;\;{a}^{2} + {\left(b \cdot \left(0.005555555555555556 \cdot \left(\pi \cdot angle\right)\right)\right)}^{2}\\ \mathbf{else}:\\ \;\;\;\;{a}^{2} + \left(angle \cdot 0.005555555555555556\right) \cdot \left(\left(\pi \cdot b\right) \cdot \left(angle \cdot \left(0.005555555555555556 \cdot \left(\pi \cdot b\right)\right)\right)\right)\\ \end{array} \]

Alternative 10: 75.2% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := angle \cdot \left(0.005555555555555556 \cdot \left(\pi \cdot b\right)\right)\\ \mathbf{if}\;angle \leq 4 \cdot 10^{-98}:\\ \;\;\;\;{a}^{2} + \left(b \cdot \left(\pi \cdot angle\right)\right) \cdot \left(0.005555555555555556 \cdot t_0\right)\\ \mathbf{else}:\\ \;\;\;\;{a}^{2} + \left(angle \cdot 0.005555555555555556\right) \cdot \left(\left(\pi \cdot b\right) \cdot t_0\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (a b angle)
 :precision binary64
 (let* ((t_0 (* angle (* 0.005555555555555556 (* PI b)))))
   (if (<= angle 4e-98)
     (+ (pow a 2.0) (* (* b (* PI angle)) (* 0.005555555555555556 t_0)))
     (+ (pow a 2.0) (* (* angle 0.005555555555555556) (* (* PI b) t_0))))))

C

double code(double a, double b, double angle) {
	double t_0 = angle * (0.005555555555555556 * (((double) M_PI) * b));
	double tmp;
	if (angle <= 4e-98) {
		tmp = pow(a, 2.0) + ((b * (((double) M_PI) * angle)) * (0.005555555555555556 * t_0));
	} else {
		tmp = pow(a, 2.0) + ((angle * 0.005555555555555556) * ((((double) M_PI) * b) * t_0));
	}
	return tmp;
}

Java

public static double code(double a, double b, double angle) {
	double t_0 = angle * (0.005555555555555556 * (Math.PI * b));
	double tmp;
	if (angle <= 4e-98) {
		tmp = Math.pow(a, 2.0) + ((b * (Math.PI * angle)) * (0.005555555555555556 * t_0));
	} else {
		tmp = Math.pow(a, 2.0) + ((angle * 0.005555555555555556) * ((Math.PI * b) * t_0));
	}
	return tmp;
}

Python

def code(a, b, angle):
	t_0 = angle * (0.005555555555555556 * (math.pi * b))
	tmp = 0
	if angle <= 4e-98:
		tmp = math.pow(a, 2.0) + ((b * (math.pi * angle)) * (0.005555555555555556 * t_0))
	else:
		tmp = math.pow(a, 2.0) + ((angle * 0.005555555555555556) * ((math.pi * b) * t_0))
	return tmp

Julia

function code(a, b, angle)
	t_0 = Float64(angle * Float64(0.005555555555555556 * Float64(pi * b)))
	tmp = 0.0
	if (angle <= 4e-98)
		tmp = Float64((a ^ 2.0) + Float64(Float64(b * Float64(pi * angle)) * Float64(0.005555555555555556 * t_0)));
	else
		tmp = Float64((a ^ 2.0) + Float64(Float64(angle * 0.005555555555555556) * Float64(Float64(pi * b) * t_0)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, angle)
	t_0 = angle * (0.005555555555555556 * (pi * b));
	tmp = 0.0;
	if (angle <= 4e-98)
		tmp = (a ^ 2.0) + ((b * (pi * angle)) * (0.005555555555555556 * t_0));
	else
		tmp = (a ^ 2.0) + ((angle * 0.005555555555555556) * ((pi * b) * t_0));
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, angle_] := Block[{t$95$0 = N[(angle * N[(0.005555555555555556 * N[(Pi * b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[angle, 4e-98], N[(N[Power[a, 2.0], $MachinePrecision] + N[(N[(b * N[(Pi * angle), $MachinePrecision]), $MachinePrecision] * N[(0.005555555555555556 * t$95$0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[Power[a, 2.0], $MachinePrecision] + N[(N[(angle * 0.005555555555555556), $MachinePrecision] * N[(N[(Pi * b), $MachinePrecision] * t$95$0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if angle < 3.99999999999999976e-98

   1. Initial program 85.2%

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

   2. Taylor expanded in angle around 0 85.3%

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

   3. Taylor expanded in angle around 0 81.7%

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

      1. *-commutative 81.7%

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

   5. Simplified 81.7%

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

      1. unpow2 81.7%

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

      2. associate-*r* 81.7%

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

      3. associate-*r* 81.7%

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

      4. *-commutative 81.7%

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

      5. associate-*l* 81.7%

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

      6. *-commutative 81.7%

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

      7. *-commutative 81.7%

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

      8. associate-*l* 81.8%

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

   7. Applied egg-rr 81.8%

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

   if 3.99999999999999976e-98 < angle

   1. Initial program 66.0%

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

   2. Taylor expanded in angle around 0 65.0%

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

   3. Taylor expanded in angle around 0 55.1%

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

      1. *-commutative 55.1%

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

   5. Simplified 55.1%

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

      1. unpow2 55.1%

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

      2. associate-*r* 55.1%

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

      3. *-commutative 55.1%

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

      4. associate-*l* 60.1%

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

      5. associate-*r* 60.1%

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

      6. *-commutative 60.1%

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

      7. associate-*l* 60.1%

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

   7. Applied egg-rr 60.1%

      \[\leadsto {\left(a \cdot 1\right)}^{2} + \color{blue}{\left(angle \cdot 0.005555555555555556\right) \cdot \left(\left(\pi \cdot b\right) \cdot \left(angle \cdot \left(0.005555555555555556 \cdot \left(\pi \cdot b\right)\right)\right)\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 75.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;angle \leq 4 \cdot 10^{-98}:\\ \;\;\;\;{a}^{2} + \left(b \cdot \left(\pi \cdot angle\right)\right) \cdot \left(0.005555555555555556 \cdot \left(angle \cdot \left(0.005555555555555556 \cdot \left(\pi \cdot b\right)\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;{a}^{2} + \left(angle \cdot 0.005555555555555556\right) \cdot \left(\left(\pi \cdot b\right) \cdot \left(angle \cdot \left(0.005555555555555556 \cdot \left(\pi \cdot b\right)\right)\right)\right)\\ \end{array} \]

Alternative 11: 75.4% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := angle \cdot \left(0.005555555555555556 \cdot \left(\pi \cdot b\right)\right)\\ \mathbf{if}\;angle \leq 1.6 \cdot 10^{-22}:\\ \;\;\;\;{a}^{2} + 0.005555555555555556 \cdot \left(t_0 \cdot \left(b \cdot \left(\pi \cdot angle\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;{a}^{2} + \left(angle \cdot 0.005555555555555556\right) \cdot \left(\left(\pi \cdot b\right) \cdot t_0\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (a b angle)
 :precision binary64
 (let* ((t_0 (* angle (* 0.005555555555555556 (* PI b)))))
   (if (<= angle 1.6e-22)
     (+ (pow a 2.0) (* 0.005555555555555556 (* t_0 (* b (* PI angle)))))
     (+ (pow a 2.0) (* (* angle 0.005555555555555556) (* (* PI b) t_0))))))

C

double code(double a, double b, double angle) {
	double t_0 = angle * (0.005555555555555556 * (((double) M_PI) * b));
	double tmp;
	if (angle <= 1.6e-22) {
		tmp = pow(a, 2.0) + (0.005555555555555556 * (t_0 * (b * (((double) M_PI) * angle))));
	} else {
		tmp = pow(a, 2.0) + ((angle * 0.005555555555555556) * ((((double) M_PI) * b) * t_0));
	}
	return tmp;
}

Java

public static double code(double a, double b, double angle) {
	double t_0 = angle * (0.005555555555555556 * (Math.PI * b));
	double tmp;
	if (angle <= 1.6e-22) {
		tmp = Math.pow(a, 2.0) + (0.005555555555555556 * (t_0 * (b * (Math.PI * angle))));
	} else {
		tmp = Math.pow(a, 2.0) + ((angle * 0.005555555555555556) * ((Math.PI * b) * t_0));
	}
	return tmp;
}

Python

def code(a, b, angle):
	t_0 = angle * (0.005555555555555556 * (math.pi * b))
	tmp = 0
	if angle <= 1.6e-22:
		tmp = math.pow(a, 2.0) + (0.005555555555555556 * (t_0 * (b * (math.pi * angle))))
	else:
		tmp = math.pow(a, 2.0) + ((angle * 0.005555555555555556) * ((math.pi * b) * t_0))
	return tmp

Julia

function code(a, b, angle)
	t_0 = Float64(angle * Float64(0.005555555555555556 * Float64(pi * b)))
	tmp = 0.0
	if (angle <= 1.6e-22)
		tmp = Float64((a ^ 2.0) + Float64(0.005555555555555556 * Float64(t_0 * Float64(b * Float64(pi * angle)))));
	else
		tmp = Float64((a ^ 2.0) + Float64(Float64(angle * 0.005555555555555556) * Float64(Float64(pi * b) * t_0)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, angle)
	t_0 = angle * (0.005555555555555556 * (pi * b));
	tmp = 0.0;
	if (angle <= 1.6e-22)
		tmp = (a ^ 2.0) + (0.005555555555555556 * (t_0 * (b * (pi * angle))));
	else
		tmp = (a ^ 2.0) + ((angle * 0.005555555555555556) * ((pi * b) * t_0));
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, angle_] := Block[{t$95$0 = N[(angle * N[(0.005555555555555556 * N[(Pi * b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[angle, 1.6e-22], N[(N[Power[a, 2.0], $MachinePrecision] + N[(0.005555555555555556 * N[(t$95$0 * N[(b * N[(Pi * angle), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[Power[a, 2.0], $MachinePrecision] + N[(N[(angle * 0.005555555555555556), $MachinePrecision] * N[(N[(Pi * b), $MachinePrecision] * t$95$0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if angle < 1.59999999999999994e-22

   1. Initial program 86.2%

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

   2. Taylor expanded in angle around 0 86.3%

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

   3. Taylor expanded in angle around 0 83.0%

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

      1. *-commutative 83.0%

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

   5. Simplified 83.0%

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

      1. unpow2 83.0%

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

      2. *-commutative 83.0%

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

      3. associate-*r* 83.0%

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

      4. associate-*r* 83.0%

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

      5. *-commutative 83.0%

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

      6. associate-*l* 83.0%

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

      7. *-commutative 83.0%

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

      8. *-commutative 83.0%

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

      9. associate-*l* 83.1%

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

   7. Applied egg-rr 83.1%

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

   if 1.59999999999999994e-22 < angle

   1. Initial program 58.2%

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

   2. Taylor expanded in angle around 0 57.1%

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

   3. Taylor expanded in angle around 0 44.8%

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

      1. *-commutative 44.8%

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

   5. Simplified 44.8%

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

      1. unpow2 44.8%

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

      2. associate-*r* 44.8%

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

      3. *-commutative 44.8%

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

      4. associate-*l* 50.9%

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

      5. associate-*r* 50.9%

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

      6. *-commutative 50.9%

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

      7. associate-*l* 50.9%

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

   7. Applied egg-rr 50.9%

      \[\leadsto {\left(a \cdot 1\right)}^{2} + \color{blue}{\left(angle \cdot 0.005555555555555556\right) \cdot \left(\left(\pi \cdot b\right) \cdot \left(angle \cdot \left(0.005555555555555556 \cdot \left(\pi \cdot b\right)\right)\right)\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 75.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;angle \leq 1.6 \cdot 10^{-22}:\\ \;\;\;\;{a}^{2} + 0.005555555555555556 \cdot \left(\left(angle \cdot \left(0.005555555555555556 \cdot \left(\pi \cdot b\right)\right)\right) \cdot \left(b \cdot \left(\pi \cdot angle\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;{a}^{2} + \left(angle \cdot 0.005555555555555556\right) \cdot \left(\left(\pi \cdot b\right) \cdot \left(angle \cdot \left(0.005555555555555556 \cdot \left(\pi \cdot b\right)\right)\right)\right)\\ \end{array} \]

Alternative 12: 75.2% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;angle \leq 9.8 \cdot 10^{-98}:\\ \;\;\;\;{a}^{2} + {\left(b \cdot \left(0.005555555555555556 \cdot \left(\pi \cdot angle\right)\right)\right)}^{2}\\ \mathbf{else}:\\ \;\;\;\;{a}^{2} + \left(\left(\pi \cdot angle\right) \cdot \left(b \cdot \left(b \cdot \left(\pi \cdot angle\right)\right)\right)\right) \cdot 3.08641975308642 \cdot 10^{-5}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b angle)
 :precision binary64
 (if (<= angle 9.8e-98)
   (+ (pow a 2.0) (pow (* b (* 0.005555555555555556 (* PI angle))) 2.0))
   (+
    (pow a 2.0)
    (* (* (* PI angle) (* b (* b (* PI angle)))) 3.08641975308642e-5))))

C

double code(double a, double b, double angle) {
	double tmp;
	if (angle <= 9.8e-98) {
		tmp = pow(a, 2.0) + pow((b * (0.005555555555555556 * (((double) M_PI) * angle))), 2.0);
	} else {
		tmp = pow(a, 2.0) + (((((double) M_PI) * angle) * (b * (b * (((double) M_PI) * angle)))) * 3.08641975308642e-5);
	}
	return tmp;
}

Java

public static double code(double a, double b, double angle) {
	double tmp;
	if (angle <= 9.8e-98) {
		tmp = Math.pow(a, 2.0) + Math.pow((b * (0.005555555555555556 * (Math.PI * angle))), 2.0);
	} else {
		tmp = Math.pow(a, 2.0) + (((Math.PI * angle) * (b * (b * (Math.PI * angle)))) * 3.08641975308642e-5);
	}
	return tmp;
}

Python

def code(a, b, angle):
	tmp = 0
	if angle <= 9.8e-98:
		tmp = math.pow(a, 2.0) + math.pow((b * (0.005555555555555556 * (math.pi * angle))), 2.0)
	else:
		tmp = math.pow(a, 2.0) + (((math.pi * angle) * (b * (b * (math.pi * angle)))) * 3.08641975308642e-5)
	return tmp

Julia

function code(a, b, angle)
	tmp = 0.0
	if (angle <= 9.8e-98)
		tmp = Float64((a ^ 2.0) + (Float64(b * Float64(0.005555555555555556 * Float64(pi * angle))) ^ 2.0));
	else
		tmp = Float64((a ^ 2.0) + Float64(Float64(Float64(pi * angle) * Float64(b * Float64(b * Float64(pi * angle)))) * 3.08641975308642e-5));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, angle)
	tmp = 0.0;
	if (angle <= 9.8e-98)
		tmp = (a ^ 2.0) + ((b * (0.005555555555555556 * (pi * angle))) ^ 2.0);
	else
		tmp = (a ^ 2.0) + (((pi * angle) * (b * (b * (pi * angle)))) * 3.08641975308642e-5);
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, angle_] := If[LessEqual[angle, 9.8e-98], N[(N[Power[a, 2.0], $MachinePrecision] + N[Power[N[(b * N[(0.005555555555555556 * N[(Pi * angle), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision], N[(N[Power[a, 2.0], $MachinePrecision] + N[(N[(N[(Pi * angle), $MachinePrecision] * N[(b * N[(b * N[(Pi * angle), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * 3.08641975308642e-5), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if angle < 9.80000000000000028e-98

   1. Initial program 85.2%

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

   2. Taylor expanded in angle around 0 85.3%

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

   3. Taylor expanded in angle around 0 81.8%

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

   if 9.80000000000000028e-98 < angle

   1. Initial program 66.0%

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

   2. Taylor expanded in angle around 0 65.0%

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

   3. Taylor expanded in angle around 0 55.1%

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

      1. *-commutative 55.1%

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

   5. Simplified 55.1%

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

      1. *-commutative 55.1%

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

      2. unpow-prod-down 55.1%

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

      3. *-commutative 55.1%

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

      4. *-commutative 55.1%

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

      5. associate-*l* 55.1%

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

      6. metadata-eval 55.1%

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

   7. Applied egg-rr 55.1%

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

      1. unpow2 55.1%

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

      2. *-commutative 55.1%

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

      3. associate-*r* 60.1%

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

      4. *-commutative 60.1%

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

   9. Applied egg-rr 60.1%

      \[\leadsto {\left(a \cdot 1\right)}^{2} + \color{blue}{\left(\left(\left(b \cdot \left(\pi \cdot angle\right)\right) \cdot b\right) \cdot \left(\pi \cdot angle\right)\right)} \cdot 3.08641975308642 \cdot 10^{-5} \]
3. Recombined 2 regimes into one program.

4. Final simplification 75.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;angle \leq 9.8 \cdot 10^{-98}:\\ \;\;\;\;{a}^{2} + {\left(b \cdot \left(0.005555555555555556 \cdot \left(\pi \cdot angle\right)\right)\right)}^{2}\\ \mathbf{else}:\\ \;\;\;\;{a}^{2} + \left(\left(\pi \cdot angle\right) \cdot \left(b \cdot \left(b \cdot \left(\pi \cdot angle\right)\right)\right)\right) \cdot 3.08641975308642 \cdot 10^{-5}\\ \end{array} \]

Alternative 13: 75.1% accurate, 2.0× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 79.6%

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

2. Taylor expanded in angle around 0 79.4%

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

3. Taylor expanded in angle around 0 73.9%

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

   1. *-commutative 73.9%

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

5. Simplified 73.9%

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

   1. *-commutative 73.9%

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

   2. unpow-prod-down 73.9%

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

   3. *-commutative 73.9%

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

   4. *-commutative 73.9%

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

   5. associate-*l* 73.9%

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

   6. metadata-eval 73.9%

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

7. Applied egg-rr 73.9%

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

8. Taylor expanded in b around 0 73.9%

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

9. Final simplification 73.9%

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

Alternative 14: 75.1% accurate, 2.0× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 79.6%

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

2. Taylor expanded in angle around 0 79.4%

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

3. Taylor expanded in angle around 0 73.9%

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

   1. *-commutative 73.9%

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

5. Simplified 73.9%

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

   1. *-commutative 73.9%

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

   2. unpow-prod-down 73.9%

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

   3. *-commutative 73.9%

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

   4. *-commutative 73.9%

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

   5. associate-*l* 73.9%

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

   6. metadata-eval 73.9%

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

7. Applied egg-rr 73.9%

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

8. Taylor expanded in b around 0 73.9%

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

   1. associate-*r* 73.9%

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

10. Simplified 73.9%

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

11. Final simplification 73.9%

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

Alternative 15: 75.1% accurate, 2.0× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 79.6%

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

2. Taylor expanded in angle around 0 79.4%

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

3. Taylor expanded in angle around 0 73.9%

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

   1. *-commutative 73.9%

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

5. Simplified 73.9%

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

6. Final simplification 73.9%

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

Alternative 16: 75.1% accurate, 2.0× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 79.6%

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

2. Taylor expanded in angle around 0 79.4%

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

3. Taylor expanded in angle around 0 73.9%

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

   1. *-commutative 73.9%

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

5. Simplified 73.9%

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

6. Taylor expanded in angle around 0 73.9%

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

   1. associate-*r* 73.9%

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

   2. *-commutative 73.9%

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

   3. *-commutative 73.9%

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

   4. associate-*r* 73.9%

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

   5. associate-*r* 73.9%

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

8. Simplified 73.9%

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

9. Final simplification 73.9%

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

Alternative 17: 75.1% accurate, 2.0× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 79.6%

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

2. Taylor expanded in angle around 0 79.4%

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

3. Taylor expanded in angle around 0 73.9%

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

4. Final simplification 73.9%

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

Reproduce

herbie shell --seed 2023338 
(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)))