ab-angle->ABCF A

Percentage Accurate: 79.9% → 79.9%

Time: 23.9s

Alternatives: 7

Speedup: N/A×

• 36.8% 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: 79.9% 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: 79.9% accurate, 1.5× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 79.1%

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

   1. *-commutative 79.1%

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

   2. associate-*r/ 79.0%

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

   3. associate-*l/ 79.1%

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

   4. *-commutative 79.1%

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

   5. *-commutative 79.1%

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

   6. associate-*r/ 79.1%

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

   7. associate-*l/ 79.0%

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

   8. *-commutative 79.0%

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

3. Simplified 79.0%

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

4. Taylor expanded in angle around 0 79.2%

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

   1. associate-*r/ 79.2%

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

   2. associate-*l/ 79.3%

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

   3. add-sqr-sqrt 46.7%

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

   4. sqrt-prod 79.3%

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

   5. unpow2 79.3%

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

   6. expm1-log1p-u 78.8%

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

   7. sqrt-pow1 64.5%

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

   8. metadata-eval 64.5%

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

   9. pow1 64.5%

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

   10. associate-/r/ 63.9%

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

   11. div-inv 63.4%

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

   12. clear-num 63.4%

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

   13. div-inv 63.4%

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

   14. metadata-eval 63.4%

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

6. Applied egg-rr 63.4%

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

   1. expm1-log1p-u 79.2%

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

   2. *-commutative 79.2%

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

   3. *-commutative 79.2%

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

   4. associate-*r* 79.3%

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

8. Applied egg-rr 79.3%

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

9. Final simplification 79.3%

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

Alternative 2: 79.8% accurate, 1.5× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 79.1%

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

   1. *-commutative 79.1%

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

   2. associate-*r/ 79.0%

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

   3. associate-*l/ 79.1%

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

   4. *-commutative 79.1%

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

   5. *-commutative 79.1%

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

   6. associate-*r/ 79.1%

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

   7. associate-*l/ 79.0%

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

   8. *-commutative 79.0%

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

3. Simplified 79.0%

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

4. Taylor expanded in angle around 0 79.2%

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

5. Taylor expanded in angle around 0 79.2%

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

6. Final simplification 79.2%

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

Alternative 3: 79.9% accurate, 1.5× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 79.1%

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

   1. *-commutative 79.1%

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

   2. associate-*r/ 79.0%

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

   3. associate-*l/ 79.1%

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

   4. *-commutative 79.1%

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

   5. *-commutative 79.1%

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

   6. associate-*r/ 79.1%

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

   7. associate-*l/ 79.0%

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

   8. *-commutative 79.0%

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

3. Simplified 79.0%

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

4. Taylor expanded in angle around 0 79.2%

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

5. Final simplification 79.2%

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

Alternative 4: 79.9% accurate, 1.5× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 79.1%

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

   1. associate-/r/ 79.1%

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

   2. add-sqr-sqrt 39.0%

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

   3. associate-/l* 38.9%

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

3. Applied egg-rr 38.9%

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

4. Taylor expanded in angle around 0 79.3%

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

5. Final simplification 79.3%

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

Alternative 5: 74.7% accurate, 2.0× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 79.1%

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

   1. associate-/r/ 79.1%

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

   2. add-sqr-sqrt 39.0%

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

   3. associate-/l* 38.9%

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

3. Applied egg-rr 38.9%

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

4. Taylor expanded in angle around 0 79.3%

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

5. Taylor expanded in angle around 0 62.4%

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

   1. *-commutative 62.4%

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

   2. unpow2 62.4%

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

   3. unpow2 62.4%

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

   4. unswap-sqr 62.4%

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

   5. unpow2 62.4%

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

   6. swap-sqr 74.8%

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

   7. associate-*r* 74.8%

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

   8. associate-*r* 74.8%

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

   9. unpow2 74.8%

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

   10. associate-*r* 74.8%

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

   11. *-commutative 74.8%

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

   12. associate-*r* 74.8%

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

   13. *-commutative 74.8%

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

   14. *-commutative 74.8%

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

7. Simplified 74.8%

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

   1. unpow2 74.8%

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

   2. associate-*r* 74.9%

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

   3. *-commutative 74.9%

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

   4. *-commutative 74.9%

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

   5. associate-*l* 74.9%

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

9. Applied egg-rr 74.9%

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

10. Final simplification 74.9%

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

Alternative 6: 74.7% accurate, 2.0× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 79.1%

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

   1. associate-/r/ 79.1%

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

   2. add-sqr-sqrt 39.0%

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

   3. associate-/l* 38.9%

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

3. Applied egg-rr 38.9%

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

4. Taylor expanded in angle around 0 79.3%

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

5. Taylor expanded in angle around 0 62.4%

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

   1. *-commutative 62.4%

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

   2. unpow2 62.4%

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

   3. unpow2 62.4%

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

   4. unswap-sqr 62.4%

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

   5. unpow2 62.4%

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

   6. swap-sqr 74.8%

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

   7. associate-*r* 74.8%

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

   8. associate-*r* 74.8%

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

   9. unpow2 74.8%

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

   10. associate-*r* 74.8%

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

   11. *-commutative 74.8%

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

   12. associate-*r* 74.8%

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

   13. *-commutative 74.8%

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

   14. *-commutative 74.8%

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

7. Simplified 74.8%

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

8. Final simplification 74.8%

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

Alternative 7: 74.7% accurate, 2.0× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 79.1%

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

   1. *-commutative 79.1%

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

   2. associate-*r/ 79.0%

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

   3. associate-*l/ 79.1%

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

   4. *-commutative 79.1%

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

   5. *-commutative 79.1%

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

   6. associate-*r/ 79.1%

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

   7. associate-*l/ 79.0%

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

   8. *-commutative 79.0%

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

3. Simplified 79.0%

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

4. Taylor expanded in angle around 0 79.2%

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

5. Taylor expanded in angle around 0 74.9%

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

   1. *-commutative 74.9%

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

   2. associate-*l* 74.8%

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

7. Simplified 74.8%

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

8. Final simplification 74.8%

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

Reproduce

herbie shell --seed 2023318 
(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)))