ab-angle->ABCF A

Percentage Accurate: 80.5% → 80.5%

Time: 19.5s

Alternatives: 12

Speedup: 1.3×

• 36.6% of points produce a very large (infinite) output. You may want to add a precondition.

Specification

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 80.5% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 80.5% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \cos \left(\frac{\pi}{\frac{180}{angle}}\right)\\ {\left(a \cdot \sin \left(angle \cdot \frac{\pi}{180}\right)\right)}^{2} + b \cdot \left(b \cdot \left({\left(\sqrt[3]{t\_0}\right)}^{2} \cdot \sqrt[3]{{t\_0}^{4}}\right)\right) \end{array} \end{array} \]

FPCore

(FPCore (a b angle)
 :precision binary64
 (let* ((t_0 (cos (/ PI (/ 180.0 angle)))))
   (+
    (pow (* a (sin (* angle (/ PI 180.0)))) 2.0)
    (* b (* b (* (pow (cbrt t_0) 2.0) (cbrt (pow t_0 4.0))))))))

C

double code(double a, double b, double angle) {
	double t_0 = cos((((double) M_PI) / (180.0 / angle)));
	return pow((a * sin((angle * (((double) M_PI) / 180.0)))), 2.0) + (b * (b * (pow(cbrt(t_0), 2.0) * cbrt(pow(t_0, 4.0)))));
}

Java

public static double code(double a, double b, double angle) {
	double t_0 = Math.cos((Math.PI / (180.0 / angle)));
	return Math.pow((a * Math.sin((angle * (Math.PI / 180.0)))), 2.0) + (b * (b * (Math.pow(Math.cbrt(t_0), 2.0) * Math.cbrt(Math.pow(t_0, 4.0)))));
}

Julia

function code(a, b, angle)
	t_0 = cos(Float64(pi / Float64(180.0 / angle)))
	return Float64((Float64(a * sin(Float64(angle * Float64(pi / 180.0)))) ^ 2.0) + Float64(b * Float64(b * Float64((cbrt(t_0) ^ 2.0) * cbrt((t_0 ^ 4.0))))))
end

Wolfram

code[a_, b_, angle_] := Block[{t$95$0 = N[Cos[N[(Pi / N[(180.0 / angle), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]}, N[(N[Power[N[(a * N[Sin[N[(angle * N[(Pi / 180.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision] + N[(b * N[(b * N[(N[Power[N[Power[t$95$0, 1/3], $MachinePrecision], 2.0], $MachinePrecision] * N[Power[N[Power[t$95$0, 4.0], $MachinePrecision], 1/3], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Initial program 79.3%

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

   1. unpow2 79.3%

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

   2. associate-*l/ 78.9%

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

   3. associate-/l* 79.4%

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

   4. unpow2 79.4%

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

3. Simplified 79.7%

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

   1. associate-*r/ 79.3%

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

   2. associate-*l/ 79.4%

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

   3. log1p-expm1-u 79.4%

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

   4. associate-*l/ 79.3%

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

   5. associate-*r/ 79.7%

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

   6. div-inv 79.7%

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

   7. metadata-eval 79.7%

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

6. Applied egg-rr 79.7%

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

   1. log1p-expm1-u 79.7%

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

   2. metadata-eval 79.7%

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

   3. div-inv 79.7%

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

   4. *-commutative 79.7%

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

   5. associate-/r/ 79.7%

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

   6. unpow-prod-down 79.7%

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

   7. unpow2 79.7%

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

   8. associate-/r/ 79.7%

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

   9. *-commutative 79.7%

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

   10. div-inv 79.7%

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

   11. metadata-eval 79.7%

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

   12. associate-*l* 79.7%

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

8. Applied egg-rr 79.7%

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

   1. add-cube-cbrt 79.7%

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

   2. cbrt-unprod 79.7%

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

10. Applied egg-rr 79.7%

    \[\leadsto {\left(a \cdot \sin \left(angle \cdot \frac{\pi}{180}\right)\right)}^{2} + b \cdot \left(b \cdot \color{blue}{\left(\sqrt[3]{{\cos \left(angle \cdot \left(\pi \cdot 0.005555555555555556\right)\right)}^{4}} \cdot {\left(\sqrt[3]{\cos \left(angle \cdot \left(\pi \cdot 0.005555555555555556\right)\right)}\right)}^{2}\right)}\right) \]
11. Step-by-step derivation

    1. *-commutative 79.7%

       \[\leadsto {\left(a \cdot \sin \left(angle \cdot \frac{\pi}{180}\right)\right)}^{2} + b \cdot \left(b \cdot \color{blue}{\left({\left(\sqrt[3]{\cos \left(angle \cdot \left(\pi \cdot 0.005555555555555556\right)\right)}\right)}^{2} \cdot \sqrt[3]{{\cos \left(angle \cdot \left(\pi \cdot 0.005555555555555556\right)\right)}^{4}}\right)}\right) \]

    2. associate-*r* 79.3%

       \[\leadsto {\left(a \cdot \sin \left(angle \cdot \frac{\pi}{180}\right)\right)}^{2} + b \cdot \left(b \cdot \left({\left(\sqrt[3]{\cos \color{blue}{\left(\left(angle \cdot \pi\right) \cdot 0.005555555555555556\right)}}\right)}^{2} \cdot \sqrt[3]{{\cos \left(angle \cdot \left(\pi \cdot 0.005555555555555556\right)\right)}^{4}}\right)\right) \]

    3. *-commutative 79.3%

       \[\leadsto {\left(a \cdot \sin \left(angle \cdot \frac{\pi}{180}\right)\right)}^{2} + b \cdot \left(b \cdot \left({\left(\sqrt[3]{\cos \color{blue}{\left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}}\right)}^{2} \cdot \sqrt[3]{{\cos \left(angle \cdot \left(\pi \cdot 0.005555555555555556\right)\right)}^{4}}\right)\right) \]

    4. associate-*r* 79.7%

       \[\leadsto {\left(a \cdot \sin \left(angle \cdot \frac{\pi}{180}\right)\right)}^{2} + b \cdot \left(b \cdot \left({\left(\sqrt[3]{\cos \color{blue}{\left(\left(0.005555555555555556 \cdot angle\right) \cdot \pi\right)}}\right)}^{2} \cdot \sqrt[3]{{\cos \left(angle \cdot \left(\pi \cdot 0.005555555555555556\right)\right)}^{4}}\right)\right) \]

    5. metadata-eval 79.7%

       \[\leadsto {\left(a \cdot \sin \left(angle \cdot \frac{\pi}{180}\right)\right)}^{2} + b \cdot \left(b \cdot \left({\left(\sqrt[3]{\cos \left(\left(\color{blue}{\frac{1}{180}} \cdot angle\right) \cdot \pi\right)}\right)}^{2} \cdot \sqrt[3]{{\cos \left(angle \cdot \left(\pi \cdot 0.005555555555555556\right)\right)}^{4}}\right)\right) \]

    6. associate-/r/ 79.7%

       \[\leadsto {\left(a \cdot \sin \left(angle \cdot \frac{\pi}{180}\right)\right)}^{2} + b \cdot \left(b \cdot \left({\left(\sqrt[3]{\cos \left(\color{blue}{\frac{1}{\frac{180}{angle}}} \cdot \pi\right)}\right)}^{2} \cdot \sqrt[3]{{\cos \left(angle \cdot \left(\pi \cdot 0.005555555555555556\right)\right)}^{4}}\right)\right) \]

    7. associate-*l/ 79.7%

       \[\leadsto {\left(a \cdot \sin \left(angle \cdot \frac{\pi}{180}\right)\right)}^{2} + b \cdot \left(b \cdot \left({\left(\sqrt[3]{\cos \color{blue}{\left(\frac{1 \cdot \pi}{\frac{180}{angle}}\right)}}\right)}^{2} \cdot \sqrt[3]{{\cos \left(angle \cdot \left(\pi \cdot 0.005555555555555556\right)\right)}^{4}}\right)\right) \]

    8. *-lft-identity 79.7%

       \[\leadsto {\left(a \cdot \sin \left(angle \cdot \frac{\pi}{180}\right)\right)}^{2} + b \cdot \left(b \cdot \left({\left(\sqrt[3]{\cos \left(\frac{\color{blue}{\pi}}{\frac{180}{angle}}\right)}\right)}^{2} \cdot \sqrt[3]{{\cos \left(angle \cdot \left(\pi \cdot 0.005555555555555556\right)\right)}^{4}}\right)\right) \]

12. Simplified 79.7%

    \[\leadsto {\left(a \cdot \sin \left(angle \cdot \frac{\pi}{180}\right)\right)}^{2} + b \cdot \left(b \cdot \color{blue}{\left({\left(\sqrt[3]{\cos \left(\frac{\pi}{\frac{180}{angle}}\right)}\right)}^{2} \cdot \sqrt[3]{{\cos \left(\frac{\pi}{\frac{180}{angle}}\right)}^{4}}\right)}\right) \]
13. Add Preprocessing

Alternative 2: 40.3% accurate, 0.7× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

code[a_, b_, angle_] := N[(N[Power[N[(a * N[Sin[N[Power[N[Sqrt[N[(angle * N[(Pi * 0.005555555555555556), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], 2.0], $MachinePrecision]], $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision] + N[Power[N[(b * N[Cos[N[(Pi / N[(180.0 / angle), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 79.3%

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

   1. *-commutative 79.3%

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

   2. clear-num 79.3%

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

   3. un-div-inv 79.6%

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

4. Applied egg-rr 79.6%

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

   1. add-sqr-sqrt 42.0%

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

   2. pow2 42.0%

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

   3. associate-*l/ 42.0%

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

   4. associate-*r/ 42.0%

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

   5. div-inv 42.0%

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

   6. metadata-eval 42.0%

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

6. Applied egg-rr 42.0%

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

Alternative 3: 80.4% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ {\left(a \cdot \sin \left(\frac{1}{\frac{\frac{180}{angle}}{\pi}}\right)\right)}^{2} + b \cdot \left(b \cdot {\cos \left(angle \cdot \left(\pi \cdot 0.005555555555555556\right)\right)}^{2}\right) \end{array} \]

FPCore

(FPCore (a b angle)
 :precision binary64
 (+
  (pow (* a (sin (/ 1.0 (/ (/ 180.0 angle) PI)))) 2.0)
  (* b (* b (pow (cos (* angle (* PI 0.005555555555555556))) 2.0)))))

C

double code(double a, double b, double angle) {
	return pow((a * sin((1.0 / ((180.0 / angle) / ((double) M_PI))))), 2.0) + (b * (b * pow(cos((angle * (((double) M_PI) * 0.005555555555555556))), 2.0)));
}

Java

public static double code(double a, double b, double angle) {
	return Math.pow((a * Math.sin((1.0 / ((180.0 / angle) / Math.PI)))), 2.0) + (b * (b * Math.pow(Math.cos((angle * (Math.PI * 0.005555555555555556))), 2.0)));
}

Python

def code(a, b, angle):
	return math.pow((a * math.sin((1.0 / ((180.0 / angle) / math.pi)))), 2.0) + (b * (b * math.pow(math.cos((angle * (math.pi * 0.005555555555555556))), 2.0)))

Julia

function code(a, b, angle)
	return Float64((Float64(a * sin(Float64(1.0 / Float64(Float64(180.0 / angle) / pi)))) ^ 2.0) + Float64(b * Float64(b * (cos(Float64(angle * Float64(pi * 0.005555555555555556))) ^ 2.0))))
end

MATLAB

function tmp = code(a, b, angle)
	tmp = ((a * sin((1.0 / ((180.0 / angle) / pi)))) ^ 2.0) + (b * (b * (cos((angle * (pi * 0.005555555555555556))) ^ 2.0)));
end

Wolfram

code[a_, b_, angle_] := N[(N[Power[N[(a * N[Sin[N[(1.0 / N[(N[(180.0 / angle), $MachinePrecision] / Pi), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision] + N[(b * N[(b * N[Power[N[Cos[N[(angle * N[(Pi * 0.005555555555555556), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 79.3%

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

   1. unpow2 79.3%

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

   2. associate-*l/ 78.9%

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

   3. associate-/l* 79.4%

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

   4. unpow2 79.4%

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

3. Simplified 79.7%

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

   1. associate-*r/ 79.3%

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

   2. associate-*l/ 79.4%

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

   3. log1p-expm1-u 79.4%

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

   4. associate-*l/ 79.3%

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

   5. associate-*r/ 79.7%

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

   6. div-inv 79.7%

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

   7. metadata-eval 79.7%

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

6. Applied egg-rr 79.7%

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

   1. log1p-expm1-u 79.7%

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

   2. metadata-eval 79.7%

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

   3. div-inv 79.7%

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

   4. *-commutative 79.7%

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

   5. associate-/r/ 79.7%

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

   6. unpow-prod-down 79.7%

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

   7. unpow2 79.7%

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

   8. associate-/r/ 79.7%

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

   9. *-commutative 79.7%

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

   10. div-inv 79.7%

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

   11. metadata-eval 79.7%

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

   12. associate-*l* 79.7%

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

8. Applied egg-rr 79.7%

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

   1. *-commutative 79.7%

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

   2. associate-/r/ 79.6%

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

   3. clear-num 79.7%

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

10. Applied egg-rr 79.7%

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

Alternative 4: 80.5% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 79.3%

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

   1. unpow2 79.3%

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

   2. associate-*l/ 78.9%

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

   3. associate-/l* 79.4%

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

   4. unpow2 79.4%

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

3. Simplified 79.7%

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

   1. associate-*r/ 79.3%

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

   2. associate-*l/ 79.4%

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

   3. log1p-expm1-u 79.4%

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

   4. associate-*l/ 79.3%

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

   5. associate-*r/ 79.7%

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

   6. div-inv 79.7%

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

   7. metadata-eval 79.7%

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

6. Applied egg-rr 79.7%

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

   1. log1p-expm1-u 79.7%

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

   2. metadata-eval 79.7%

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

   3. div-inv 79.7%

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

   4. *-commutative 79.7%

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

   5. associate-/r/ 79.7%

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

   6. unpow-prod-down 79.7%

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

   7. unpow2 79.7%

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

   8. associate-/r/ 79.7%

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

   9. *-commutative 79.7%

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

   10. div-inv 79.7%

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

   11. metadata-eval 79.7%

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

   12. associate-*l* 79.7%

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

8. Applied egg-rr 79.7%

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

Alternative 5: 80.5% accurate, 1.3× speedup

Math

\[\begin{array}{l} \\ {\left(a \cdot \sin \left(angle \cdot \frac{\pi}{180}\right)\right)}^{2} + b \cdot \left(b \cdot \frac{1 + \cos \left(angle \cdot \left(\pi \cdot 0.011111111111111112\right)\right)}{2}\right) \end{array} \]

FPCore

(FPCore (a b angle)
 :precision binary64
 (+
  (pow (* a (sin (* angle (/ PI 180.0)))) 2.0)
  (* b (* b (/ (+ 1.0 (cos (* angle (* PI 0.011111111111111112)))) 2.0)))))

C

double code(double a, double b, double angle) {
	return pow((a * sin((angle * (((double) M_PI) / 180.0)))), 2.0) + (b * (b * ((1.0 + cos((angle * (((double) M_PI) * 0.011111111111111112)))) / 2.0)));
}

Java

public static double code(double a, double b, double angle) {
	return Math.pow((a * Math.sin((angle * (Math.PI / 180.0)))), 2.0) + (b * (b * ((1.0 + Math.cos((angle * (Math.PI * 0.011111111111111112)))) / 2.0)));
}

Python

def code(a, b, angle):
	return math.pow((a * math.sin((angle * (math.pi / 180.0)))), 2.0) + (b * (b * ((1.0 + math.cos((angle * (math.pi * 0.011111111111111112)))) / 2.0)))

Julia

function code(a, b, angle)
	return Float64((Float64(a * sin(Float64(angle * Float64(pi / 180.0)))) ^ 2.0) + Float64(b * Float64(b * Float64(Float64(1.0 + cos(Float64(angle * Float64(pi * 0.011111111111111112)))) / 2.0))))
end

MATLAB

function tmp = code(a, b, angle)
	tmp = ((a * sin((angle * (pi / 180.0)))) ^ 2.0) + (b * (b * ((1.0 + cos((angle * (pi * 0.011111111111111112)))) / 2.0)));
end

Wolfram

code[a_, b_, angle_] := N[(N[Power[N[(a * N[Sin[N[(angle * N[(Pi / 180.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision] + N[(b * N[(b * N[(N[(1.0 + N[Cos[N[(angle * N[(Pi * 0.011111111111111112), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 79.3%

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

   1. unpow2 79.3%

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

   2. associate-*l/ 78.9%

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

   3. associate-/l* 79.4%

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

   4. unpow2 79.4%

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

3. Simplified 79.7%

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

   1. associate-*r/ 79.3%

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

   2. associate-*l/ 79.4%

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

   3. log1p-expm1-u 79.4%

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

   4. associate-*l/ 79.3%

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

   5. associate-*r/ 79.7%

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

   6. div-inv 79.7%

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

   7. metadata-eval 79.7%

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

6. Applied egg-rr 79.7%

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

   1. log1p-expm1-u 79.7%

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

   2. metadata-eval 79.7%

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

   3. div-inv 79.7%

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

   4. *-commutative 79.7%

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

   5. associate-/r/ 79.7%

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

   6. unpow-prod-down 79.7%

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

   7. unpow2 79.7%

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

   8. associate-/r/ 79.7%

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

   9. *-commutative 79.7%

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

   10. div-inv 79.7%

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

   11. metadata-eval 79.7%

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

   12. associate-*l* 79.7%

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

8. Applied egg-rr 79.7%

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

   1. metadata-eval 79.7%

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

   2. div-inv 79.7%

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

   3. *-commutative 79.7%

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

   4. pow2 79.7%

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

   5. associate-/r/ 77.2%

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

   6. associate-/r/ 79.7%

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

   7. cos-mult 79.7%

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

10. Applied egg-rr 79.7%

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

    1. +-commutative 79.7%

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

    2. +-inverses 79.7%

       \[\leadsto {\left(a \cdot \sin \left(angle \cdot \frac{\pi}{180}\right)\right)}^{2} + b \cdot \left(b \cdot \frac{\cos \color{blue}{0} + \cos \left(angle \cdot \left(\pi \cdot 0.005555555555555556\right) + angle \cdot \left(\pi \cdot 0.005555555555555556\right)\right)}{2}\right) \]

    3. cos-0 79.7%

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

    4. distribute-lft-out 79.7%

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

    5. distribute-lft-out 79.7%

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

    6. metadata-eval 79.7%

       \[\leadsto {\left(a \cdot \sin \left(angle \cdot \frac{\pi}{180}\right)\right)}^{2} + b \cdot \left(b \cdot \frac{1 + \cos \left(angle \cdot \left(\pi \cdot \color{blue}{0.011111111111111112}\right)\right)}{2}\right) \]

12. Simplified 79.7%

    \[\leadsto {\left(a \cdot \sin \left(angle \cdot \frac{\pi}{180}\right)\right)}^{2} + b \cdot \left(b \cdot \color{blue}{\frac{1 + \cos \left(angle \cdot \left(\pi \cdot 0.011111111111111112\right)\right)}{2}}\right) \]
13. Add Preprocessing

Alternative 6: 80.4% accurate, 1.3× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 79.3%

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

   1. unpow2 79.3%

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

   2. associate-*l/ 78.9%

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

   3. associate-/l* 79.4%

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

   4. unpow2 79.4%

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

3. Simplified 79.7%

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

   1. associate-*r/ 79.3%

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

   2. associate-*l/ 79.4%

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

   3. log1p-expm1-u 79.4%

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

   4. associate-*l/ 79.3%

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

   5. associate-*r/ 79.7%

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

   6. div-inv 79.7%

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

   7. metadata-eval 79.7%

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

6. Applied egg-rr 79.7%

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

7. Taylor expanded in angle around 0 79.2%

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

Alternative 7: 59.5% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(angle \cdot \pi\right) \cdot \left(a \cdot 0.005555555555555556\right)\\ \mathbf{if}\;b \leq 5.8 \cdot 10^{-158}:\\ \;\;\;\;{\left(a \cdot \sin \left(angle \cdot \left(\pi \cdot 0.005555555555555556\right)\right)\right)}^{2}\\ \mathbf{else}:\\ \;\;\;\;{\left(b \cdot \cos \left(\frac{\pi}{\frac{180}{angle}}\right)\right)}^{2} + t\_0 \cdot t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (a b angle)
 :precision binary64
 (let* ((t_0 (* (* angle PI) (* a 0.005555555555555556))))
   (if (<= b 5.8e-158)
     (pow (* a (sin (* angle (* PI 0.005555555555555556)))) 2.0)
     (+ (pow (* b (cos (/ PI (/ 180.0 angle)))) 2.0) (* t_0 t_0)))))

C

double code(double a, double b, double angle) {
	double t_0 = (angle * ((double) M_PI)) * (a * 0.005555555555555556);
	double tmp;
	if (b <= 5.8e-158) {
		tmp = pow((a * sin((angle * (((double) M_PI) * 0.005555555555555556)))), 2.0);
	} else {
		tmp = pow((b * cos((((double) M_PI) / (180.0 / angle)))), 2.0) + (t_0 * t_0);
	}
	return tmp;
}

Java

public static double code(double a, double b, double angle) {
	double t_0 = (angle * Math.PI) * (a * 0.005555555555555556);
	double tmp;
	if (b <= 5.8e-158) {
		tmp = Math.pow((a * Math.sin((angle * (Math.PI * 0.005555555555555556)))), 2.0);
	} else {
		tmp = Math.pow((b * Math.cos((Math.PI / (180.0 / angle)))), 2.0) + (t_0 * t_0);
	}
	return tmp;
}

Python

def code(a, b, angle):
	t_0 = (angle * math.pi) * (a * 0.005555555555555556)
	tmp = 0
	if b <= 5.8e-158:
		tmp = math.pow((a * math.sin((angle * (math.pi * 0.005555555555555556)))), 2.0)
	else:
		tmp = math.pow((b * math.cos((math.pi / (180.0 / angle)))), 2.0) + (t_0 * t_0)
	return tmp

Julia

function code(a, b, angle)
	t_0 = Float64(Float64(angle * pi) * Float64(a * 0.005555555555555556))
	tmp = 0.0
	if (b <= 5.8e-158)
		tmp = Float64(a * sin(Float64(angle * Float64(pi * 0.005555555555555556)))) ^ 2.0;
	else
		tmp = Float64((Float64(b * cos(Float64(pi / Float64(180.0 / angle)))) ^ 2.0) + Float64(t_0 * t_0));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, angle)
	t_0 = (angle * pi) * (a * 0.005555555555555556);
	tmp = 0.0;
	if (b <= 5.8e-158)
		tmp = (a * sin((angle * (pi * 0.005555555555555556)))) ^ 2.0;
	else
		tmp = ((b * cos((pi / (180.0 / angle)))) ^ 2.0) + (t_0 * t_0);
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, angle_] := Block[{t$95$0 = N[(N[(angle * Pi), $MachinePrecision] * N[(a * 0.005555555555555556), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[b, 5.8e-158], N[Power[N[(a * N[Sin[N[(angle * N[(Pi * 0.005555555555555556), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision], N[(N[Power[N[(b * N[Cos[N[(Pi / N[(180.0 / angle), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision] + N[(t$95$0 * t$95$0), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if b < 5.79999999999999961e-158

   1. Initial program 78.9%

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

      1. unpow2 78.9%

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

      2. associate-*l/ 78.4%

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

      3. associate-/l* 79.1%

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

      4. unpow2 79.1%

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

   3. Simplified 79.5%

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

   6. Applied egg-rr 63.5%

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

      1. expm1-define 78.4%

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

      2. associate-*r* 77.7%

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

      3. *-commutative 77.7%

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

      4. associate-*r* 78.2%

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

      5. associate-*r* 77.5%

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

      6. *-commutative 77.5%

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

      7. associate-*r* 77.7%

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

   8. Simplified 77.7%

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

   9. Taylor expanded in b around 0 35.4%

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

       1. log1p-define 39.2%

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

       2. unpow2 39.2%

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

       3. *-commutative 39.2%

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

       4. associate-*r* 39.9%

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

       5. unpow2 39.9%

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

       6. swap-sqr 46.1%

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

       7. unpow2 46.1%

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

       8. associate-*r* 45.4%

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

       9. *-commutative 45.4%

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

       10. associate-*r* 45.9%

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

       11. associate-*r* 45.4%

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

       12. *-commutative 45.4%

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

       13. *-commutative 45.4%

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

   11. Simplified 45.4%

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

       1. expm1-log1p-u 45.7%

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

       2. *-commutative 45.7%

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

       3. associate-*r* 46.4%

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

       4. add-cube-cbrt 46.1%

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

       5. unpow2 46.1%

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

       6. add-sqr-sqrt 46.1%

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

       7. pow2 46.1%

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

   13. Applied egg-rr 46.4%

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

   if 5.79999999999999961e-158 < b

   1. Initial program 79.9%

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

      1. *-commutative 79.9%

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

      2. clear-num 80.0%

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

      3. un-div-inv 80.0%

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

   4. Applied egg-rr 80.0%

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

   5. Taylor expanded in angle around 0 77.6%

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

      1. associate-*r* 77.6%

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

   7. Simplified 77.6%

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

      1. unpow2 77.6%

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

      2. *-commutative 77.6%

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

      3. *-commutative 77.6%

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

   9. Applied egg-rr 77.6%

      \[\leadsto \color{blue}{\left(\left(a \cdot 0.005555555555555556\right) \cdot \left(angle \cdot \pi\right)\right) \cdot \left(\left(a \cdot 0.005555555555555556\right) \cdot \left(angle \cdot \pi\right)\right)} + {\left(b \cdot \cos \left(\frac{\pi}{\frac{180}{angle}}\right)\right)}^{2} \]
3. Recombined 2 regimes into one program.

4. Final simplification 58.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq 5.8 \cdot 10^{-158}:\\ \;\;\;\;{\left(a \cdot \sin \left(angle \cdot \left(\pi \cdot 0.005555555555555556\right)\right)\right)}^{2}\\ \mathbf{else}:\\ \;\;\;\;{\left(b \cdot \cos \left(\frac{\pi}{\frac{180}{angle}}\right)\right)}^{2} + \left(\left(angle \cdot \pi\right) \cdot \left(a \cdot 0.005555555555555556\right)\right) \cdot \left(\left(angle \cdot \pi\right) \cdot \left(a \cdot 0.005555555555555556\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 67.5% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;a \leq 3 \cdot 10^{-22}:\\ \;\;\;\;{\left(b \cdot \cos \left(\frac{\pi}{\frac{180}{angle}}\right)\right)}^{2}\\ \mathbf{else}:\\ \;\;\;\;{b}^{2} + {\left(\left(angle \cdot \pi\right) \cdot \left(a \cdot 0.005555555555555556\right)\right)}^{2}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b angle)
 :precision binary64
 (if (<= a 3e-22)
   (pow (* b (cos (/ PI (/ 180.0 angle)))) 2.0)
   (+ (pow b 2.0) (pow (* (* angle PI) (* a 0.005555555555555556)) 2.0))))

C

double code(double a, double b, double angle) {
	double tmp;
	if (a <= 3e-22) {
		tmp = pow((b * cos((((double) M_PI) / (180.0 / angle)))), 2.0);
	} else {
		tmp = pow(b, 2.0) + pow(((angle * ((double) M_PI)) * (a * 0.005555555555555556)), 2.0);
	}
	return tmp;
}

Java

public static double code(double a, double b, double angle) {
	double tmp;
	if (a <= 3e-22) {
		tmp = Math.pow((b * Math.cos((Math.PI / (180.0 / angle)))), 2.0);
	} else {
		tmp = Math.pow(b, 2.0) + Math.pow(((angle * Math.PI) * (a * 0.005555555555555556)), 2.0);
	}
	return tmp;
}

Python

def code(a, b, angle):
	tmp = 0
	if a <= 3e-22:
		tmp = math.pow((b * math.cos((math.pi / (180.0 / angle)))), 2.0)
	else:
		tmp = math.pow(b, 2.0) + math.pow(((angle * math.pi) * (a * 0.005555555555555556)), 2.0)
	return tmp

Julia

function code(a, b, angle)
	tmp = 0.0
	if (a <= 3e-22)
		tmp = Float64(b * cos(Float64(pi / Float64(180.0 / angle)))) ^ 2.0;
	else
		tmp = Float64((b ^ 2.0) + (Float64(Float64(angle * pi) * Float64(a * 0.005555555555555556)) ^ 2.0));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, angle)
	tmp = 0.0;
	if (a <= 3e-22)
		tmp = (b * cos((pi / (180.0 / angle)))) ^ 2.0;
	else
		tmp = (b ^ 2.0) + (((angle * pi) * (a * 0.005555555555555556)) ^ 2.0);
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, angle_] := If[LessEqual[a, 3e-22], N[Power[N[(b * N[Cos[N[(Pi / N[(180.0 / angle), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision], N[(N[Power[b, 2.0], $MachinePrecision] + N[Power[N[(N[(angle * Pi), $MachinePrecision] * N[(a * 0.005555555555555556), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if a < 2.9999999999999999e-22

   1. Initial program 77.9%

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

      1. *-commutative 77.9%

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

      2. clear-num 77.9%

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

      3. un-div-inv 78.3%

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

   4. Applied egg-rr 78.3%

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

   5. Taylor expanded in angle around 0 69.5%

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

      1. associate-*r* 69.5%

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

   7. Simplified 69.5%

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

      1. expm1-log1p-u 69.5%

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

      2. expm1-undefine 69.5%

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

      3. associate-/r/ 69.5%

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

   9. Applied egg-rr 69.5%

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

   10. Taylor expanded in a around 0 67.6%

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

       1. unpow2 67.6%

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

       2. *-commutative 67.6%

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

       3. associate-*r* 68.1%

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

       4. unpow2 68.1%

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

       5. swap-sqr 68.0%

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

       6. unpow2 68.0%

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

       7. associate-*r* 67.6%

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

       8. *-commutative 67.6%

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

       9. associate-*r* 67.7%

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

       10. metadata-eval 67.7%

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

       11. associate-/r/ 67.7%

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

       12. associate-*l/ 68.1%

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

       13. *-lft-identity 68.1%

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

   12. Simplified 68.1%

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

   if 2.9999999999999999e-22 < a

   1. Initial program 84.0%

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

      1. *-commutative 84.0%

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

      2. clear-num 83.9%

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

      3. un-div-inv 84.0%

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

   4. Applied egg-rr 84.0%

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

   5. Taylor expanded in angle around 0 81.9%

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

      1. associate-*r* 81.9%

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

   7. Simplified 81.9%

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

   8. Taylor expanded in angle around 0 81.9%

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

4. Final simplification 71.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \leq 3 \cdot 10^{-22}:\\ \;\;\;\;{\left(b \cdot \cos \left(\frac{\pi}{\frac{180}{angle}}\right)\right)}^{2}\\ \mathbf{else}:\\ \;\;\;\;{b}^{2} + {\left(\left(angle \cdot \pi\right) \cdot \left(a \cdot 0.005555555555555556\right)\right)}^{2}\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 63.0% accurate, 2.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;a \leq 5.2 \cdot 10^{+80}:\\ \;\;\;\;{\left(b \cdot \cos \left(\frac{\pi}{\frac{180}{angle}}\right)\right)}^{2}\\ \mathbf{else}:\\ \;\;\;\;3.08641975308642 \cdot 10^{-5} \cdot {\left(a \cdot \left(angle \cdot \pi\right)\right)}^{2}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b angle)
 :precision binary64
 (if (<= a 5.2e+80)
   (pow (* b (cos (/ PI (/ 180.0 angle)))) 2.0)
   (* 3.08641975308642e-5 (pow (* a (* angle PI)) 2.0))))

C

double code(double a, double b, double angle) {
	double tmp;
	if (a <= 5.2e+80) {
		tmp = pow((b * cos((((double) M_PI) / (180.0 / angle)))), 2.0);
	} else {
		tmp = 3.08641975308642e-5 * pow((a * (angle * ((double) M_PI))), 2.0);
	}
	return tmp;
}

Java

public static double code(double a, double b, double angle) {
	double tmp;
	if (a <= 5.2e+80) {
		tmp = Math.pow((b * Math.cos((Math.PI / (180.0 / angle)))), 2.0);
	} else {
		tmp = 3.08641975308642e-5 * Math.pow((a * (angle * Math.PI)), 2.0);
	}
	return tmp;
}

Python

def code(a, b, angle):
	tmp = 0
	if a <= 5.2e+80:
		tmp = math.pow((b * math.cos((math.pi / (180.0 / angle)))), 2.0)
	else:
		tmp = 3.08641975308642e-5 * math.pow((a * (angle * math.pi)), 2.0)
	return tmp

Julia

function code(a, b, angle)
	tmp = 0.0
	if (a <= 5.2e+80)
		tmp = Float64(b * cos(Float64(pi / Float64(180.0 / angle)))) ^ 2.0;
	else
		tmp = Float64(3.08641975308642e-5 * (Float64(a * Float64(angle * pi)) ^ 2.0));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, angle)
	tmp = 0.0;
	if (a <= 5.2e+80)
		tmp = (b * cos((pi / (180.0 / angle)))) ^ 2.0;
	else
		tmp = 3.08641975308642e-5 * ((a * (angle * pi)) ^ 2.0);
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, angle_] := If[LessEqual[a, 5.2e+80], N[Power[N[(b * N[Cos[N[(Pi / N[(180.0 / angle), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision], N[(3.08641975308642e-5 * N[Power[N[(a * N[(angle * Pi), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if a < 5.19999999999999963e80

   1. Initial program 76.9%

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

      1. *-commutative 76.9%

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

      2. clear-num 76.9%

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

      3. un-div-inv 77.3%

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

   4. Applied egg-rr 77.3%

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

   5. Taylor expanded in angle around 0 68.8%

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

      1. associate-*r* 68.8%

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

   7. Simplified 68.8%

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

      1. expm1-log1p-u 68.8%

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

      2. expm1-undefine 68.8%

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

      3. associate-/r/ 68.8%

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

   9. Applied egg-rr 68.8%

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

   10. Taylor expanded in a around 0 66.6%

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

       1. unpow2 66.6%

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

       2. *-commutative 66.6%

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

       3. associate-*r* 67.1%

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

       4. unpow2 67.1%

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

       5. swap-sqr 67.1%

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

       6. unpow2 67.1%

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

       7. associate-*r* 66.6%

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

       8. *-commutative 66.6%

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

       9. associate-*r* 66.7%

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

       10. metadata-eval 66.7%

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

       11. associate-/r/ 66.7%

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

       12. associate-*l/ 67.1%

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

       13. *-lft-identity 67.1%

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

   12. Simplified 67.1%

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

   if 5.19999999999999963e80 < a

   1. Initial program 91.6%

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

      1. *-commutative 91.6%

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

      2. clear-num 91.6%

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

      3. un-div-inv 91.6%

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

   4. Applied egg-rr 91.6%

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

   5. Taylor expanded in angle around 0 90.7%

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

      1. associate-*r* 90.7%

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

   7. Simplified 90.7%

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

   8. Taylor expanded in a around inf 70.3%

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

      1. unpow2 70.3%

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

      2. unpow2 70.3%

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

      3. unpow2 70.3%

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

      4. swap-sqr 70.3%

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

      5. swap-sqr 86.1%

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

      6. unpow2 86.1%

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

      7. *-commutative 86.1%

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

   10. Simplified 86.1%

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

4. Final simplification 70.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \leq 5.2 \cdot 10^{+80}:\\ \;\;\;\;{\left(b \cdot \cos \left(\frac{\pi}{\frac{180}{angle}}\right)\right)}^{2}\\ \mathbf{else}:\\ \;\;\;\;3.08641975308642 \cdot 10^{-5} \cdot {\left(a \cdot \left(angle \cdot \pi\right)\right)}^{2}\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 63.0% accurate, 2.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;a \leq 9 \cdot 10^{+80}:\\ \;\;\;\;{\left(b \cdot \cos \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)\right)}^{2}\\ \mathbf{else}:\\ \;\;\;\;3.08641975308642 \cdot 10^{-5} \cdot {\left(a \cdot \left(angle \cdot \pi\right)\right)}^{2}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b angle)
 :precision binary64
 (if (<= a 9e+80)
   (pow (* b (cos (* 0.005555555555555556 (* angle PI)))) 2.0)
   (* 3.08641975308642e-5 (pow (* a (* angle PI)) 2.0))))

C

double code(double a, double b, double angle) {
	double tmp;
	if (a <= 9e+80) {
		tmp = pow((b * cos((0.005555555555555556 * (angle * ((double) M_PI))))), 2.0);
	} else {
		tmp = 3.08641975308642e-5 * pow((a * (angle * ((double) M_PI))), 2.0);
	}
	return tmp;
}

Java

public static double code(double a, double b, double angle) {
	double tmp;
	if (a <= 9e+80) {
		tmp = Math.pow((b * Math.cos((0.005555555555555556 * (angle * Math.PI)))), 2.0);
	} else {
		tmp = 3.08641975308642e-5 * Math.pow((a * (angle * Math.PI)), 2.0);
	}
	return tmp;
}

Python

def code(a, b, angle):
	tmp = 0
	if a <= 9e+80:
		tmp = math.pow((b * math.cos((0.005555555555555556 * (angle * math.pi)))), 2.0)
	else:
		tmp = 3.08641975308642e-5 * math.pow((a * (angle * math.pi)), 2.0)
	return tmp

Julia

function code(a, b, angle)
	tmp = 0.0
	if (a <= 9e+80)
		tmp = Float64(b * cos(Float64(0.005555555555555556 * Float64(angle * pi)))) ^ 2.0;
	else
		tmp = Float64(3.08641975308642e-5 * (Float64(a * Float64(angle * pi)) ^ 2.0));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, angle)
	tmp = 0.0;
	if (a <= 9e+80)
		tmp = (b * cos((0.005555555555555556 * (angle * pi)))) ^ 2.0;
	else
		tmp = 3.08641975308642e-5 * ((a * (angle * pi)) ^ 2.0);
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, angle_] := If[LessEqual[a, 9e+80], N[Power[N[(b * N[Cos[N[(0.005555555555555556 * N[(angle * Pi), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision], N[(3.08641975308642e-5 * N[Power[N[(a * N[(angle * Pi), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if a < 9.00000000000000013e80

   1. Initial program 76.9%

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

      1. unpow2 76.9%

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

      2. associate-*l/ 76.5%

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

      3. associate-/l* 77.0%

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

      4. unpow2 77.0%

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

   3. Simplified 77.4%

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

   5. Taylor expanded in a around 0 66.6%

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

      1. *-commutative 66.6%

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

      2. associate-*r* 67.1%

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

      3. unpow2 67.1%

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

      4. unpow2 67.1%

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

      5. swap-sqr 67.1%

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

      6. unpow2 67.1%

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

      7. associate-*r* 66.6%

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

      8. *-commutative 66.6%

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

   7. Simplified 66.6%

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

   if 9.00000000000000013e80 < a

   1. Initial program 91.6%

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

      1. *-commutative 91.6%

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

      2. clear-num 91.6%

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

      3. un-div-inv 91.6%

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

   4. Applied egg-rr 91.6%

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

   5. Taylor expanded in angle around 0 90.7%

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

      1. associate-*r* 90.7%

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

   7. Simplified 90.7%

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

   8. Taylor expanded in a around inf 70.3%

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

      1. unpow2 70.3%

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

      2. unpow2 70.3%

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

      3. unpow2 70.3%

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

      4. swap-sqr 70.3%

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

      5. swap-sqr 86.1%

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

      6. unpow2 86.1%

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

      7. *-commutative 86.1%

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

   10. Simplified 86.1%

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

4. Final simplification 69.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \leq 9 \cdot 10^{+80}:\\ \;\;\;\;{\left(b \cdot \cos \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)\right)}^{2}\\ \mathbf{else}:\\ \;\;\;\;3.08641975308642 \cdot 10^{-5} \cdot {\left(a \cdot \left(angle \cdot \pi\right)\right)}^{2}\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 63.1% accurate, 3.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;a \leq 9.5 \cdot 10^{+80}:\\ \;\;\;\;b \cdot b\\ \mathbf{else}:\\ \;\;\;\;3.08641975308642 \cdot 10^{-5} \cdot {\left(a \cdot \left(angle \cdot \pi\right)\right)}^{2}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b angle)
 :precision binary64
 (if (<= a 9.5e+80)
   (* b b)
   (* 3.08641975308642e-5 (pow (* a (* angle PI)) 2.0))))

C

double code(double a, double b, double angle) {
	double tmp;
	if (a <= 9.5e+80) {
		tmp = b * b;
	} else {
		tmp = 3.08641975308642e-5 * pow((a * (angle * ((double) M_PI))), 2.0);
	}
	return tmp;
}

Java

public static double code(double a, double b, double angle) {
	double tmp;
	if (a <= 9.5e+80) {
		tmp = b * b;
	} else {
		tmp = 3.08641975308642e-5 * Math.pow((a * (angle * Math.PI)), 2.0);
	}
	return tmp;
}

Python

def code(a, b, angle):
	tmp = 0
	if a <= 9.5e+80:
		tmp = b * b
	else:
		tmp = 3.08641975308642e-5 * math.pow((a * (angle * math.pi)), 2.0)
	return tmp

Julia

function code(a, b, angle)
	tmp = 0.0
	if (a <= 9.5e+80)
		tmp = Float64(b * b);
	else
		tmp = Float64(3.08641975308642e-5 * (Float64(a * Float64(angle * pi)) ^ 2.0));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, angle)
	tmp = 0.0;
	if (a <= 9.5e+80)
		tmp = b * b;
	else
		tmp = 3.08641975308642e-5 * ((a * (angle * pi)) ^ 2.0);
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, angle_] := If[LessEqual[a, 9.5e+80], N[(b * b), $MachinePrecision], N[(3.08641975308642e-5 * N[Power[N[(a * N[(angle * Pi), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if a < 9.499999999999999e80

   1. Initial program 76.9%

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

      1. unpow2 76.9%

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

      2. associate-*l/ 76.5%

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

      3. associate-/l* 77.0%

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

      4. unpow2 77.0%

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

   3. Simplified 77.4%

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

   5. Taylor expanded in angle around 0 66.6%

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

      1. unpow2 66.6%

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

   7. Applied egg-rr 66.6%

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

   if 9.499999999999999e80 < a

   1. Initial program 91.6%

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

      1. *-commutative 91.6%

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

      2. clear-num 91.6%

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

      3. un-div-inv 91.6%

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

   4. Applied egg-rr 91.6%

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

   5. Taylor expanded in angle around 0 90.7%

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

      1. associate-*r* 90.7%

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

   7. Simplified 90.7%

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

   8. Taylor expanded in a around inf 70.3%

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

      1. unpow2 70.3%

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

      2. unpow2 70.3%

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

      3. unpow2 70.3%

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

      4. swap-sqr 70.3%

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

      5. swap-sqr 86.1%

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

      6. unpow2 86.1%

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

      7. *-commutative 86.1%

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

   10. Simplified 86.1%

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

4. Final simplification 69.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \leq 9.5 \cdot 10^{+80}:\\ \;\;\;\;b \cdot b\\ \mathbf{else}:\\ \;\;\;\;3.08641975308642 \cdot 10^{-5} \cdot {\left(a \cdot \left(angle \cdot \pi\right)\right)}^{2}\\ \end{array} \]
5. Add Preprocessing

Alternative 12: 56.9% accurate, 139.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 79.3%

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

   1. unpow2 79.3%

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

   2. associate-*l/ 78.9%

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

   3. associate-/l* 79.4%

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

   4. unpow2 79.4%

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

3. Simplified 79.7%

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

5. Taylor expanded in angle around 0 61.4%

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

   1. unpow2 61.4%

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

7. Applied egg-rr 61.4%

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

Reproduce

herbie shell --seed 2024170 
(FPCore (a b angle)
  :name "ab-angle->ABCF A"
  :precision binary64
  (+ (pow (* a (sin (* (/ angle 180.0) PI))) 2.0) (pow (* b (cos (* (/ angle 180.0) PI))) 2.0)))