ab-angle->ABCF A

Percentage Accurate: 79.8% → 79.7%

Time: 29.8s

Alternatives: 12

Speedup: 1.3×

• 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: 79.8% 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: 79.7% accurate, 0.8× speedup

Math

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

FPCore

(FPCore (a b angle)
 :precision binary64
 (+ (pow (* a (sin (* angle (/ (cbrt (pow PI 3.0)) 180.0)))) 2.0) (pow b 2.0)))

C

double code(double a, double b, double angle) {
	return pow((a * sin((angle * (cbrt(pow(((double) M_PI), 3.0)) / 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.cbrt(Math.pow(Math.PI, 3.0)) / 180.0)))), 2.0) + Math.pow(b, 2.0);
}

Julia

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

Wolfram

code[a_, b_, angle_] := N[(N[Power[N[(a * N[Sin[N[(angle * N[(N[Power[N[Power[Pi, 3.0], $MachinePrecision], 1/3], $MachinePrecision] / 180.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision] + N[Power[b, 2.0], $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 81.3%

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

   1. associate-*l/ 81.2%

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

   2. associate-/l* 81.2%

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

   3. cos-neg 81.2%

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

   4. distribute-lft-neg-out 81.2%

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

   5. distribute-frac-neg 81.2%

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

   6. distribute-frac-neg 81.2%

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

   7. distribute-lft-neg-out 81.2%

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

   8. cos-neg 81.2%

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

   9. associate-*l/ 81.2%

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

   10. associate-/l* 81.2%

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

3. Simplified 81.2%

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

5. Taylor expanded in angle around 0 81.3%

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

   1. add-cbrt-cube 81.4%

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

   2. pow3 81.4%

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

7. Applied egg-rr 81.4%

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

8. Final simplification 81.4%

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

Alternative 2: 79.8% accurate, 1.3× 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 81.3%

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

   1. associate-*l/ 81.2%

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

4. Applied egg-rr 81.2%

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

5. Taylor expanded in angle around 0 81.4%

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

6. Final simplification 81.4%

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

Alternative 3: 79.8% accurate, 1.3× speedup

Math

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

FPCore

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

C

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

Python

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

Julia

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

MATLAB

function tmp = code(a, b, angle)
	tmp = (b ^ 2.0) + ((a * sin((pi * (angle * 0.005555555555555556)))) ^ 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 * 0.005555555555555556), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 81.3%

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

   1. associate-*l/ 81.2%

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

   2. associate-/l* 81.2%

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

   3. cos-neg 81.2%

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

   4. distribute-lft-neg-out 81.2%

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

   5. distribute-frac-neg 81.2%

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

   6. distribute-frac-neg 81.2%

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

   7. distribute-lft-neg-out 81.2%

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

   8. cos-neg 81.2%

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

   9. associate-*l/ 81.2%

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

   10. associate-/l* 81.2%

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

3. Simplified 81.2%

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

5. Taylor expanded in angle around 0 81.3%

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

   1. add-cbrt-cube 81.4%

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

   2. pow3 81.4%

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

7. Applied egg-rr 81.4%

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

8. Taylor expanded in angle around inf 81.3%

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

   1. associate-*r* 81.4%

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

   2. *-commutative 81.4%

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

   3. *-commutative 81.4%

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

   4. *-commutative 81.4%

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

10. Simplified 81.4%

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

11. Final simplification 81.4%

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

Alternative 4: 79.7% accurate, 1.3× 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 81.3%

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

   1. associate-*l/ 81.2%

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

   2. associate-/l* 81.2%

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

   3. cos-neg 81.2%

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

   4. distribute-lft-neg-out 81.2%

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

   5. distribute-frac-neg 81.2%

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

   6. distribute-frac-neg 81.2%

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

   7. distribute-lft-neg-out 81.2%

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

   8. cos-neg 81.2%

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

   9. associate-*l/ 81.2%

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

   10. associate-/l* 81.2%

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

3. Simplified 81.2%

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

5. Taylor expanded in angle around 0 81.3%

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

6. Taylor expanded in angle around inf 81.3%

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

7. Final simplification 81.3%

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

Alternative 5: 74.2% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;a \leq 1.25 \cdot 10^{-141}:\\ \;\;\;\;{b}^{2} + {\left(\left(1 + 0.005555555555555556 \cdot \left(angle \cdot \left(a \cdot \pi\right)\right)\right) + -1\right)}^{2}\\ \mathbf{else}:\\ \;\;\;\;\left(a \cdot \left(angle \cdot \pi\right)\right) \cdot \left(0.005555555555555556 \cdot \left(angle \cdot \left(a \cdot \left(\pi \cdot 0.005555555555555556\right)\right)\right)\right) + {\left(b \cdot \cos \left(angle \cdot \frac{\pi}{180}\right)\right)}^{2}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b angle)
 :precision binary64
 (if (<= a 1.25e-141)
   (+
    (pow b 2.0)
    (pow (+ (+ 1.0 (* 0.005555555555555556 (* angle (* a PI)))) -1.0) 2.0))
   (+
    (*
     (* a (* angle PI))
     (* 0.005555555555555556 (* angle (* a (* PI 0.005555555555555556)))))
    (pow (* b (cos (* angle (/ PI 180.0)))) 2.0))))

C

double code(double a, double b, double angle) {
	double tmp;
	if (a <= 1.25e-141) {
		tmp = pow(b, 2.0) + pow(((1.0 + (0.005555555555555556 * (angle * (a * ((double) M_PI))))) + -1.0), 2.0);
	} else {
		tmp = ((a * (angle * ((double) M_PI))) * (0.005555555555555556 * (angle * (a * (((double) M_PI) * 0.005555555555555556))))) + pow((b * cos((angle * (((double) M_PI) / 180.0)))), 2.0);
	}
	return tmp;
}

Java

public static double code(double a, double b, double angle) {
	double tmp;
	if (a <= 1.25e-141) {
		tmp = Math.pow(b, 2.0) + Math.pow(((1.0 + (0.005555555555555556 * (angle * (a * Math.PI)))) + -1.0), 2.0);
	} else {
		tmp = ((a * (angle * Math.PI)) * (0.005555555555555556 * (angle * (a * (Math.PI * 0.005555555555555556))))) + Math.pow((b * Math.cos((angle * (Math.PI / 180.0)))), 2.0);
	}
	return tmp;
}

Python

def code(a, b, angle):
	tmp = 0
	if a <= 1.25e-141:
		tmp = math.pow(b, 2.0) + math.pow(((1.0 + (0.005555555555555556 * (angle * (a * math.pi)))) + -1.0), 2.0)
	else:
		tmp = ((a * (angle * math.pi)) * (0.005555555555555556 * (angle * (a * (math.pi * 0.005555555555555556))))) + math.pow((b * math.cos((angle * (math.pi / 180.0)))), 2.0)
	return tmp

Julia

function code(a, b, angle)
	tmp = 0.0
	if (a <= 1.25e-141)
		tmp = Float64((b ^ 2.0) + (Float64(Float64(1.0 + Float64(0.005555555555555556 * Float64(angle * Float64(a * pi)))) + -1.0) ^ 2.0));
	else
		tmp = Float64(Float64(Float64(a * Float64(angle * pi)) * Float64(0.005555555555555556 * Float64(angle * Float64(a * Float64(pi * 0.005555555555555556))))) + (Float64(b * cos(Float64(angle * Float64(pi / 180.0)))) ^ 2.0));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, angle)
	tmp = 0.0;
	if (a <= 1.25e-141)
		tmp = (b ^ 2.0) + (((1.0 + (0.005555555555555556 * (angle * (a * pi)))) + -1.0) ^ 2.0);
	else
		tmp = ((a * (angle * pi)) * (0.005555555555555556 * (angle * (a * (pi * 0.005555555555555556))))) + ((b * cos((angle * (pi / 180.0)))) ^ 2.0);
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, angle_] := If[LessEqual[a, 1.25e-141], N[(N[Power[b, 2.0], $MachinePrecision] + N[Power[N[(N[(1.0 + N[(0.005555555555555556 * N[(angle * N[(a * Pi), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + -1.0), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision], N[(N[(N[(a * N[(angle * Pi), $MachinePrecision]), $MachinePrecision] * N[(0.005555555555555556 * N[(angle * N[(a * N[(Pi * 0.005555555555555556), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[Power[N[(b * N[Cos[N[(angle * N[(Pi / 180.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if a < 1.25e-141

   1. Initial program 79.7%

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

      1. associate-*l/ 79.7%

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

      2. associate-/l* 79.7%

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

      3. cos-neg 79.7%

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

      4. distribute-lft-neg-out 79.7%

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

      5. distribute-frac-neg 79.7%

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

      6. distribute-frac-neg 79.7%

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

      7. distribute-lft-neg-out 79.7%

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

      8. cos-neg 79.7%

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

      9. associate-*l/ 79.7%

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

      10. associate-/l* 79.7%

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

   3. Simplified 79.7%

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

   5. Taylor expanded in angle around 0 80.3%

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

      1. expm1-log1p-u 69.2%

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

      2. expm1-undefine 64.6%

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

      3. *-commutative 64.6%

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

      4. div-inv 64.6%

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

      5. metadata-eval 64.6%

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

   7. Applied egg-rr 64.6%

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

   8. Taylor expanded in angle around 0 72.9%

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

      1. *-commutative 72.9%

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

      2. associate-*l* 72.9%

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

   10. Simplified 72.9%

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

   if 1.25e-141 < 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. Step-by-step derivation

      1. associate-*l/ 83.9%

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

      2. associate-/l* 83.9%

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

      3. cos-neg 83.9%

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

      4. distribute-lft-neg-out 83.9%

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

      5. distribute-frac-neg 83.9%

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

      6. distribute-frac-neg 83.9%

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

      7. distribute-lft-neg-out 83.9%

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

      8. cos-neg 83.9%

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

      9. associate-*l/ 83.9%

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

      10. associate-/l* 83.9%

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

   3. Simplified 83.9%

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

   5. Taylor expanded in angle around 0 80.8%

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

      1. *-commutative 80.8%

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

      2. associate-*r* 80.9%

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

      3. associate-*l* 80.8%

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

      4. *-commutative 80.8%

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

   7. Simplified 80.8%

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

      1. unpow2 80.8%

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

      2. associate-*r* 80.9%

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

      3. associate-*l* 80.8%

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

      4. *-commutative 80.8%

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

      5. associate-*l* 80.8%

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

      6. *-commutative 80.8%

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

      7. associate-*l* 80.9%

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

   9. Applied egg-rr 80.9%

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

4. Final simplification 75.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \leq 1.25 \cdot 10^{-141}:\\ \;\;\;\;{b}^{2} + {\left(\left(1 + 0.005555555555555556 \cdot \left(angle \cdot \left(a \cdot \pi\right)\right)\right) + -1\right)}^{2}\\ \mathbf{else}:\\ \;\;\;\;\left(a \cdot \left(angle \cdot \pi\right)\right) \cdot \left(0.005555555555555556 \cdot \left(angle \cdot \left(a \cdot \left(\pi \cdot 0.005555555555555556\right)\right)\right)\right) + {\left(b \cdot \cos \left(angle \cdot \frac{\pi}{180}\right)\right)}^{2}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 74.2% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := a \cdot \left(angle \cdot \pi\right)\\ \mathbf{if}\;a \leq 1.5 \cdot 10^{-141}:\\ \;\;\;\;{b}^{2} + {\left(\left(1 + 0.005555555555555556 \cdot \left(angle \cdot \left(a \cdot \pi\right)\right)\right) + -1\right)}^{2}\\ \mathbf{else}:\\ \;\;\;\;{\left(b \cdot \cos \left(angle \cdot \frac{\pi}{180}\right)\right)}^{2} + \left(t\_0 \cdot t\_0\right) \cdot 3.08641975308642 \cdot 10^{-5}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b angle)
 :precision binary64
 (let* ((t_0 (* a (* angle PI))))
   (if (<= a 1.5e-141)
     (+
      (pow b 2.0)
      (pow (+ (+ 1.0 (* 0.005555555555555556 (* angle (* a PI)))) -1.0) 2.0))
     (+
      (pow (* b (cos (* angle (/ PI 180.0)))) 2.0)
      (* (* t_0 t_0) 3.08641975308642e-5)))))

C

double code(double a, double b, double angle) {
	double t_0 = a * (angle * ((double) M_PI));
	double tmp;
	if (a <= 1.5e-141) {
		tmp = pow(b, 2.0) + pow(((1.0 + (0.005555555555555556 * (angle * (a * ((double) M_PI))))) + -1.0), 2.0);
	} else {
		tmp = pow((b * cos((angle * (((double) M_PI) / 180.0)))), 2.0) + ((t_0 * t_0) * 3.08641975308642e-5);
	}
	return tmp;
}

Java

public static double code(double a, double b, double angle) {
	double t_0 = a * (angle * Math.PI);
	double tmp;
	if (a <= 1.5e-141) {
		tmp = Math.pow(b, 2.0) + Math.pow(((1.0 + (0.005555555555555556 * (angle * (a * Math.PI)))) + -1.0), 2.0);
	} else {
		tmp = Math.pow((b * Math.cos((angle * (Math.PI / 180.0)))), 2.0) + ((t_0 * t_0) * 3.08641975308642e-5);
	}
	return tmp;
}

Python

def code(a, b, angle):
	t_0 = a * (angle * math.pi)
	tmp = 0
	if a <= 1.5e-141:
		tmp = math.pow(b, 2.0) + math.pow(((1.0 + (0.005555555555555556 * (angle * (a * math.pi)))) + -1.0), 2.0)
	else:
		tmp = math.pow((b * math.cos((angle * (math.pi / 180.0)))), 2.0) + ((t_0 * t_0) * 3.08641975308642e-5)
	return tmp

Julia

function code(a, b, angle)
	t_0 = Float64(a * Float64(angle * pi))
	tmp = 0.0
	if (a <= 1.5e-141)
		tmp = Float64((b ^ 2.0) + (Float64(Float64(1.0 + Float64(0.005555555555555556 * Float64(angle * Float64(a * pi)))) + -1.0) ^ 2.0));
	else
		tmp = Float64((Float64(b * cos(Float64(angle * Float64(pi / 180.0)))) ^ 2.0) + Float64(Float64(t_0 * t_0) * 3.08641975308642e-5));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, angle)
	t_0 = a * (angle * pi);
	tmp = 0.0;
	if (a <= 1.5e-141)
		tmp = (b ^ 2.0) + (((1.0 + (0.005555555555555556 * (angle * (a * pi)))) + -1.0) ^ 2.0);
	else
		tmp = ((b * cos((angle * (pi / 180.0)))) ^ 2.0) + ((t_0 * t_0) * 3.08641975308642e-5);
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, angle_] := Block[{t$95$0 = N[(a * N[(angle * Pi), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[a, 1.5e-141], N[(N[Power[b, 2.0], $MachinePrecision] + N[Power[N[(N[(1.0 + N[(0.005555555555555556 * N[(angle * N[(a * Pi), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + -1.0), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision], N[(N[Power[N[(b * N[Cos[N[(angle * N[(Pi / 180.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision] + N[(N[(t$95$0 * t$95$0), $MachinePrecision] * 3.08641975308642e-5), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if a < 1.49999999999999992e-141

   1. Initial program 79.7%

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

      1. associate-*l/ 79.7%

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

      2. associate-/l* 79.7%

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

      3. cos-neg 79.7%

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

      4. distribute-lft-neg-out 79.7%

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

      5. distribute-frac-neg 79.7%

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

      6. distribute-frac-neg 79.7%

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

      7. distribute-lft-neg-out 79.7%

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

      8. cos-neg 79.7%

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

      9. associate-*l/ 79.7%

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

      10. associate-/l* 79.7%

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

   3. Simplified 79.7%

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

   5. Taylor expanded in angle around 0 80.3%

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

      1. expm1-log1p-u 69.2%

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

      2. expm1-undefine 64.6%

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

      3. *-commutative 64.6%

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

      4. div-inv 64.6%

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

      5. metadata-eval 64.6%

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

   7. Applied egg-rr 64.6%

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

   8. Taylor expanded in angle around 0 72.9%

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

      1. *-commutative 72.9%

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

      2. associate-*l* 72.9%

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

   10. Simplified 72.9%

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

   if 1.49999999999999992e-141 < 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. Step-by-step derivation

      1. associate-*l/ 83.9%

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

      2. associate-/l* 83.9%

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

      3. cos-neg 83.9%

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

      4. distribute-lft-neg-out 83.9%

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

      5. distribute-frac-neg 83.9%

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

      6. distribute-frac-neg 83.9%

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

      7. distribute-lft-neg-out 83.9%

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

      8. cos-neg 83.9%

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

      9. associate-*l/ 83.9%

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

      10. associate-/l* 83.9%

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

   3. Simplified 83.9%

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

   5. Taylor expanded in angle around 0 80.8%

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

      1. *-commutative 80.8%

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

      2. associate-*r* 80.9%

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

      3. associate-*l* 80.8%

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

      4. *-commutative 80.8%

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

   7. Simplified 80.8%

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

      1. unpow2 80.8%

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

      2. associate-*r* 80.9%

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

      3. associate-*r* 80.9%

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

      4. swap-sqr 80.8%

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

      5. *-commutative 80.8%

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

      6. associate-*l* 80.8%

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

      7. *-commutative 80.8%

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

      8. *-commutative 80.8%

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

      9. associate-*l* 80.8%

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

      10. *-commutative 80.8%

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

      11. metadata-eval 80.8%

          \[\leadsto \left(\left(a \cdot \left(\pi \cdot angle\right)\right) \cdot \left(a \cdot \left(\pi \cdot angle\right)\right)\right) \cdot \color{blue}{3.08641975308642 \cdot 10^{-5}} + {\left(b \cdot \cos \left(angle \cdot \frac{\pi}{180}\right)\right)}^{2} \]

   9. Applied egg-rr 80.8%

      \[\leadsto \color{blue}{\left(\left(a \cdot \left(\pi \cdot angle\right)\right) \cdot \left(a \cdot \left(\pi \cdot angle\right)\right)\right) \cdot 3.08641975308642 \cdot 10^{-5}} + {\left(b \cdot \cos \left(angle \cdot \frac{\pi}{180}\right)\right)}^{2} \]
3. Recombined 2 regimes into one program.

4. Final simplification 75.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \leq 1.5 \cdot 10^{-141}:\\ \;\;\;\;{b}^{2} + {\left(\left(1 + 0.005555555555555556 \cdot \left(angle \cdot \left(a \cdot \pi\right)\right)\right) + -1\right)}^{2}\\ \mathbf{else}:\\ \;\;\;\;{\left(b \cdot \cos \left(angle \cdot \frac{\pi}{180}\right)\right)}^{2} + \left(\left(a \cdot \left(angle \cdot \pi\right)\right) \cdot \left(a \cdot \left(angle \cdot \pi\right)\right)\right) \cdot 3.08641975308642 \cdot 10^{-5}\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 74.2% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := angle \cdot \left(a \cdot \pi\right)\\ \mathbf{if}\;a \leq 2.5 \cdot 10^{-101}:\\ \;\;\;\;{b}^{2} + {\left(\left(1 + 0.005555555555555556 \cdot t\_0\right) + -1\right)}^{2}\\ \mathbf{else}:\\ \;\;\;\;{b}^{2} + t\_0 \cdot \left(0.005555555555555556 \cdot \left(\left(angle \cdot \pi\right) \cdot \left(a \cdot 0.005555555555555556\right)\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (a b angle)
 :precision binary64
 (let* ((t_0 (* angle (* a PI))))
   (if (<= a 2.5e-101)
     (+ (pow b 2.0) (pow (+ (+ 1.0 (* 0.005555555555555556 t_0)) -1.0) 2.0))
     (+
      (pow b 2.0)
      (*
       t_0
       (*
        0.005555555555555556
        (* (* angle PI) (* a 0.005555555555555556))))))))

C

double code(double a, double b, double angle) {
	double t_0 = angle * (a * ((double) M_PI));
	double tmp;
	if (a <= 2.5e-101) {
		tmp = pow(b, 2.0) + pow(((1.0 + (0.005555555555555556 * t_0)) + -1.0), 2.0);
	} else {
		tmp = pow(b, 2.0) + (t_0 * (0.005555555555555556 * ((angle * ((double) M_PI)) * (a * 0.005555555555555556))));
	}
	return tmp;
}

Java

public static double code(double a, double b, double angle) {
	double t_0 = angle * (a * Math.PI);
	double tmp;
	if (a <= 2.5e-101) {
		tmp = Math.pow(b, 2.0) + Math.pow(((1.0 + (0.005555555555555556 * t_0)) + -1.0), 2.0);
	} else {
		tmp = Math.pow(b, 2.0) + (t_0 * (0.005555555555555556 * ((angle * Math.PI) * (a * 0.005555555555555556))));
	}
	return tmp;
}

Python

def code(a, b, angle):
	t_0 = angle * (a * math.pi)
	tmp = 0
	if a <= 2.5e-101:
		tmp = math.pow(b, 2.0) + math.pow(((1.0 + (0.005555555555555556 * t_0)) + -1.0), 2.0)
	else:
		tmp = math.pow(b, 2.0) + (t_0 * (0.005555555555555556 * ((angle * math.pi) * (a * 0.005555555555555556))))
	return tmp

Julia

function code(a, b, angle)
	t_0 = Float64(angle * Float64(a * pi))
	tmp = 0.0
	if (a <= 2.5e-101)
		tmp = Float64((b ^ 2.0) + (Float64(Float64(1.0 + Float64(0.005555555555555556 * t_0)) + -1.0) ^ 2.0));
	else
		tmp = Float64((b ^ 2.0) + Float64(t_0 * Float64(0.005555555555555556 * Float64(Float64(angle * pi) * Float64(a * 0.005555555555555556)))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, angle)
	t_0 = angle * (a * pi);
	tmp = 0.0;
	if (a <= 2.5e-101)
		tmp = (b ^ 2.0) + (((1.0 + (0.005555555555555556 * t_0)) + -1.0) ^ 2.0);
	else
		tmp = (b ^ 2.0) + (t_0 * (0.005555555555555556 * ((angle * pi) * (a * 0.005555555555555556))));
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, angle_] := Block[{t$95$0 = N[(angle * N[(a * Pi), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[a, 2.5e-101], N[(N[Power[b, 2.0], $MachinePrecision] + N[Power[N[(N[(1.0 + N[(0.005555555555555556 * t$95$0), $MachinePrecision]), $MachinePrecision] + -1.0), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision], N[(N[Power[b, 2.0], $MachinePrecision] + N[(t$95$0 * N[(0.005555555555555556 * N[(N[(angle * Pi), $MachinePrecision] * N[(a * 0.005555555555555556), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if a < 2.5e-101

   1. Initial program 79.7%

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

      1. associate-*l/ 79.7%

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

      2. associate-/l* 79.7%

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

      3. cos-neg 79.7%

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

      4. distribute-lft-neg-out 79.7%

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

      5. distribute-frac-neg 79.7%

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

      6. distribute-frac-neg 79.7%

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

      7. distribute-lft-neg-out 79.7%

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

      8. cos-neg 79.7%

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

      9. associate-*l/ 79.7%

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

      10. associate-/l* 79.7%

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

   3. Simplified 79.7%

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

   5. Taylor expanded in angle around 0 80.3%

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

      1. expm1-log1p-u 69.4%

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

      2. expm1-undefine 64.9%

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

      3. *-commutative 64.9%

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

      4. div-inv 64.9%

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

      5. metadata-eval 64.9%

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

   7. Applied egg-rr 64.9%

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

   8. Taylor expanded in angle around 0 72.9%

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

      1. *-commutative 72.9%

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

      2. associate-*l* 72.9%

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

   10. Simplified 72.9%

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

   if 2.5e-101 < a

   1. Initial program 84.3%

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

      1. associate-*l/ 84.2%

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

      2. associate-/l* 84.2%

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

      3. cos-neg 84.2%

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

      4. distribute-lft-neg-out 84.2%

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

      5. distribute-frac-neg 84.2%

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

      6. distribute-frac-neg 84.2%

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

      7. distribute-lft-neg-out 84.2%

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

      8. cos-neg 84.2%

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

      9. associate-*l/ 84.1%

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

      10. associate-/l* 84.1%

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

   3. Simplified 84.1%

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

   5. Taylor expanded in angle around 0 83.3%

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

   6. Taylor expanded in angle around 0 80.1%

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

      1. unpow2 80.1%

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

      2. associate-*r* 80.1%

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

      3. associate-*r* 80.1%

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

      4. *-commutative 80.1%

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

      5. *-commutative 80.1%

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

      6. associate-*l* 80.2%

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

   8. Applied egg-rr 80.2%

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

4. Final simplification 75.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \leq 2.5 \cdot 10^{-101}:\\ \;\;\;\;{b}^{2} + {\left(\left(1 + 0.005555555555555556 \cdot \left(angle \cdot \left(a \cdot \pi\right)\right)\right) + -1\right)}^{2}\\ \mathbf{else}:\\ \;\;\;\;{b}^{2} + \left(angle \cdot \left(a \cdot \pi\right)\right) \cdot \left(0.005555555555555556 \cdot \left(\left(angle \cdot \pi\right) \cdot \left(a \cdot 0.005555555555555556\right)\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 73.2% accurate, 3.5× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 81.3%

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

   1. associate-*l/ 81.2%

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

   2. associate-/l* 81.2%

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

   3. cos-neg 81.2%

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

   4. distribute-lft-neg-out 81.2%

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

   5. distribute-frac-neg 81.2%

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

   6. distribute-frac-neg 81.2%

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

   7. distribute-lft-neg-out 81.2%

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

   8. cos-neg 81.2%

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

   9. associate-*l/ 81.2%

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

   10. associate-/l* 81.2%

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

3. Simplified 81.2%

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

5. Taylor expanded in angle around 0 81.3%

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

6. Taylor expanded in angle around 0 76.3%

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

   1. unpow2 76.3%

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

   2. associate-*r* 76.3%

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

   3. associate-*r* 76.7%

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

   4. associate-*r* 76.6%

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

   5. *-commutative 76.6%

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

8. Applied egg-rr 76.6%

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

9. Final simplification 76.6%

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

Alternative 9: 74.9% accurate, 3.5× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 81.3%

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

   1. associate-*l/ 81.2%

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

   2. associate-/l* 81.2%

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

   3. cos-neg 81.2%

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

   4. distribute-lft-neg-out 81.2%

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

   5. distribute-frac-neg 81.2%

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

   6. distribute-frac-neg 81.2%

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

   7. distribute-lft-neg-out 81.2%

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

   8. cos-neg 81.2%

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

   9. associate-*l/ 81.2%

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

   10. associate-/l* 81.2%

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

3. Simplified 81.2%

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

5. Taylor expanded in angle around 0 81.3%

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

6. Taylor expanded in angle around 0 76.3%

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

   1. unpow2 76.3%

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

   2. associate-*r* 76.3%

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

   3. associate-*r* 76.3%

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

   4. *-commutative 76.3%

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

   5. *-commutative 76.3%

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

   6. associate-*l* 76.3%

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

8. Applied egg-rr 76.3%

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

9. Final simplification 76.3%

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

Alternative 10: 74.9% accurate, 3.5× speedup

Math

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

FPCore

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 81.3%

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

   1. associate-*l/ 81.2%

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

   2. associate-/l* 81.2%

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

   3. cos-neg 81.2%

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

   4. distribute-lft-neg-out 81.2%

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

   5. distribute-frac-neg 81.2%

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

   6. distribute-frac-neg 81.2%

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

   7. distribute-lft-neg-out 81.2%

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

   8. cos-neg 81.2%

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

   9. associate-*l/ 81.2%

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

   10. associate-/l* 81.2%

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

3. Simplified 81.2%

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

5. Taylor expanded in angle around 0 81.3%

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

6. Taylor expanded in angle around 0 76.3%

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

   1. unpow2 76.3%

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

   2. associate-*r* 76.3%

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

   3. associate-*r* 76.3%

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

   4. *-commutative 76.3%

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

   5. *-commutative 76.3%

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

8. Applied egg-rr 76.3%

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

9. Final simplification 76.3%

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

Alternative 11: 74.0% accurate, 3.5× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 81.3%

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

   1. associate-*l/ 81.2%

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

   2. associate-/l* 81.2%

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

   3. cos-neg 81.2%

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

   4. distribute-lft-neg-out 81.2%

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

   5. distribute-frac-neg 81.2%

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

   6. distribute-frac-neg 81.2%

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

   7. distribute-lft-neg-out 81.2%

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

   8. cos-neg 81.2%

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

   9. associate-*l/ 81.2%

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

   10. associate-/l* 81.2%

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

3. Simplified 81.2%

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

5. Taylor expanded in angle around 0 81.3%

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

6. Taylor expanded in angle around 0 76.3%

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

   1. unpow2 76.3%

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

   2. associate-*r* 76.3%

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

   3. associate-*l* 75.5%

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

   4. associate-*r* 75.5%

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

   5. *-commutative 75.5%

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

8. Applied egg-rr 75.5%

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

   1. pow1 75.5%

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

   2. associate-*r* 75.5%

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

   3. *-commutative 75.5%

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

   4. associate-*r* 75.6%

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

   5. metadata-eval 75.6%

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

   6. div-inv 75.6%

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

   7. associate-*l* 75.5%

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

   8. div-inv 75.6%

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

   9. metadata-eval 75.6%

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

10. Applied egg-rr 75.6%

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

    1. unpow1 75.6%

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

    2. *-commutative 75.6%

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

12. Simplified 75.6%

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

13. Final simplification 75.6%

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

Alternative 12: 74.0% accurate, 3.5× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 81.3%

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

   1. associate-*l/ 81.2%

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

   2. associate-/l* 81.2%

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

   3. cos-neg 81.2%

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

   4. distribute-lft-neg-out 81.2%

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

   5. distribute-frac-neg 81.2%

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

   6. distribute-frac-neg 81.2%

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

   7. distribute-lft-neg-out 81.2%

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

   8. cos-neg 81.2%

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

   9. associate-*l/ 81.2%

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

   10. associate-/l* 81.2%

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

3. Simplified 81.2%

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

5. Taylor expanded in angle around 0 81.3%

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

6. Taylor expanded in angle around 0 76.3%

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

   1. unpow2 76.3%

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

   2. associate-*r* 76.3%

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

   3. associate-*l* 75.5%

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

   4. associate-*r* 75.5%

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

   5. *-commutative 75.5%

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

8. Applied egg-rr 75.5%

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

   1. associate-*r* 75.5%

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

   2. add-cube-cbrt 75.5%

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

   3. unpow2 75.5%

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

   4. metadata-eval 75.5%

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

   5. div-inv 75.5%

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

   6. clear-num 75.5%

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

   7. associate-*l/ 75.5%

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

   8. *-un-lft-identity 75.5%

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

   9. unpow2 75.5%

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

   10. add-cube-cbrt 75.5%

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

10. Applied egg-rr 75.5%

    \[\leadsto \left(0.005555555555555556 \cdot a\right) \cdot \left(\left(angle \cdot \pi\right) \cdot \color{blue}{\frac{a}{\frac{180}{angle \cdot \pi}}}\right) + {\left(b \cdot 1\right)}^{2} \]
11. Step-by-step derivation

    1. associate-/r/ 75.5%

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

    2. associate-*l/ 75.5%

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

    3. *-commutative 75.5%

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

    4. associate-*l* 75.5%

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

    5. *-commutative 75.5%

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

    6. associate-/l* 75.6%

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

    7. *-commutative 75.6%

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

12. Simplified 75.6%

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

13. Final simplification 75.6%

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

Reproduce

herbie shell --seed 2024111 
(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)))