ab-angle->ABCF A

Percentage Accurate: 79.3% → 79.3%

Time: 28.4s

Alternatives: 5

Speedup: N/A×

• 36.3% of points produce a very large (infinite) output. You may want to add a precondition.

Specification

Math

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

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: 79.3% 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: 79.3% accurate, 1.5× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 76.5%

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

   1. unpow2 76.5%

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

   2. associate-*l/ 76.5%

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

   3. associate-*r/ 76.6%

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

   4. unpow2 76.6%

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

   5. associate-*l/ 76.6%

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

   6. associate-*r/ 76.6%

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

3. Simplified 76.6%

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

4. Taylor expanded in angle around 0 77.1%

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

5. Taylor expanded in angle around 0 77.0%

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

   1. associate-*r* 77.1%

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

7. Simplified 77.1%

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

8. Final simplification 77.1%

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

Alternative 2: 79.2% accurate, 1.5× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 76.5%

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

   1. unpow2 76.5%

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

   2. associate-*l/ 76.5%

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

   3. associate-*r/ 76.6%

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

   4. unpow2 76.6%

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

   5. associate-*l/ 76.6%

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

   6. associate-*r/ 76.6%

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

3. Simplified 76.6%

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

4. Taylor expanded in angle around 0 77.1%

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

5. Taylor expanded in angle around 0 77.0%

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

6. Final simplification 77.0%

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

Alternative 3: 79.3% accurate, 1.5× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 76.5%

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

   1. unpow2 76.5%

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

   2. associate-*l/ 76.5%

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

   3. associate-*r/ 76.6%

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

   4. unpow2 76.6%

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

   5. associate-*l/ 76.6%

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

   6. associate-*r/ 76.6%

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

3. Simplified 76.6%

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

4. Taylor expanded in angle around 0 77.1%

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

5. Final simplification 77.1%

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

Alternative 4: 74.1% accurate, 2.0× speedup

Math

\[\begin{array}{l} \\ {b}^{2} + \frac{{\left(a \cdot \left(angle \cdot \pi\right)\right)}^{2}}{32400} \end{array} \]

FPCore

(FPCore (a b angle)
 :precision binary64
 (+ (pow b 2.0) (/ (pow (* a (* angle PI)) 2.0) 32400.0)))

C

double code(double a, double b, double angle) {
	return pow(b, 2.0) + (pow((a * (angle * ((double) M_PI))), 2.0) / 32400.0);
}

Java

public static double code(double a, double b, double angle) {
	return Math.pow(b, 2.0) + (Math.pow((a * (angle * Math.PI)), 2.0) / 32400.0);
}

Python

def code(a, b, angle):
	return math.pow(b, 2.0) + (math.pow((a * (angle * math.pi)), 2.0) / 32400.0)

Julia

function code(a, b, angle)
	return Float64((b ^ 2.0) + Float64((Float64(a * Float64(angle * pi)) ^ 2.0) / 32400.0))
end

MATLAB

function tmp = code(a, b, angle)
	tmp = (b ^ 2.0) + (((a * (angle * pi)) ^ 2.0) / 32400.0);
end

Wolfram

code[a_, b_, angle_] := N[(N[Power[b, 2.0], $MachinePrecision] + N[(N[Power[N[(a * N[(angle * Pi), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision] / 32400.0), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 76.5%

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

   1. unpow2 76.5%

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

   2. associate-*l/ 76.5%

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

   3. associate-*r/ 76.6%

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

   4. unpow2 76.6%

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

   5. associate-*l/ 76.6%

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

   6. associate-*r/ 76.6%

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

3. Simplified 76.6%

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

4. Taylor expanded in angle around 0 77.1%

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

5. Taylor expanded in angle around 0 73.2%

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

   1. unpow2 73.2%

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

   2. swap-sqr 73.3%

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

   3. metadata-eval 73.3%

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

   4. unpow2 73.3%

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

   5. pow-prod-down 61.4%

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

   6. pow-prod-down 61.3%

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

   7. *-commutative 61.3%

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

   8. associate-*r* 61.4%

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

   9. associate-*l* 61.4%

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

   10. unpow2 61.4%

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

   11. metadata-eval 61.4%

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

   12. swap-sqr 61.4%

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

   13. metadata-eval 61.4%

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

   14. div-inv 61.4%

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

   15. metadata-eval 61.4%

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

   16. div-inv 61.4%

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

   17. frac-times 61.4%

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

   18. unpow2 61.4%

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

7. Applied egg-rr 73.3%

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

8. Taylor expanded in angle around 0 61.4%

   \[\leadsto \frac{\color{blue}{{a}^{2} \cdot \left({angle}^{2} \cdot {\pi}^{2}\right)}}{32400} + {\left(b \cdot 1\right)}^{2} \]
9. Step-by-step derivation

   1. unpow2 61.4%

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

   2. unpow2 61.4%

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

   3. swap-sqr 61.4%

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

   4. unpow2 61.4%

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

   5. swap-sqr 73.3%

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

   6. unpow2 73.3%

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

10. Simplified 73.3%

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

11. Final simplification 73.3%

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

Alternative 5: 74.1% accurate, 2.0× speedup

Math

\[\begin{array}{l} \\ {b}^{2} + \frac{{\left(angle \cdot \left(a \cdot \pi\right)\right)}^{2}}{32400} \end{array} \]

FPCore

(FPCore (a b angle)
 :precision binary64
 (+ (pow b 2.0) (/ (pow (* angle (* a PI)) 2.0) 32400.0)))

C

double code(double a, double b, double angle) {
	return pow(b, 2.0) + (pow((angle * (a * ((double) M_PI))), 2.0) / 32400.0);
}

Java

public static double code(double a, double b, double angle) {
	return Math.pow(b, 2.0) + (Math.pow((angle * (a * Math.PI)), 2.0) / 32400.0);
}

Python

def code(a, b, angle):
	return math.pow(b, 2.0) + (math.pow((angle * (a * math.pi)), 2.0) / 32400.0)

Julia

function code(a, b, angle)
	return Float64((b ^ 2.0) + Float64((Float64(angle * Float64(a * pi)) ^ 2.0) / 32400.0))
end

MATLAB

function tmp = code(a, b, angle)
	tmp = (b ^ 2.0) + (((angle * (a * pi)) ^ 2.0) / 32400.0);
end

Wolfram

code[a_, b_, angle_] := N[(N[Power[b, 2.0], $MachinePrecision] + N[(N[Power[N[(angle * N[(a * Pi), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision] / 32400.0), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 76.5%

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

   1. unpow2 76.5%

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

   2. associate-*l/ 76.5%

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

   3. associate-*r/ 76.6%

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

   4. unpow2 76.6%

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

   5. associate-*l/ 76.6%

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

   6. associate-*r/ 76.6%

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

3. Simplified 76.6%

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

4. Taylor expanded in angle around 0 77.1%

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

5. Taylor expanded in angle around 0 73.2%

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

   1. unpow2 73.2%

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

   2. swap-sqr 73.3%

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

   3. metadata-eval 73.3%

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

   4. unpow2 73.3%

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

   5. pow-prod-down 61.4%

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

   6. pow-prod-down 61.3%

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

   7. *-commutative 61.3%

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

   8. associate-*r* 61.4%

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

   9. associate-*l* 61.4%

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

   10. unpow2 61.4%

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

   11. metadata-eval 61.4%

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

   12. swap-sqr 61.4%

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

   13. metadata-eval 61.4%

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

   14. div-inv 61.4%

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

   15. metadata-eval 61.4%

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

   16. div-inv 61.4%

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

   17. frac-times 61.4%

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

   18. unpow2 61.4%

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

7. Applied egg-rr 73.3%

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

8. Final simplification 73.3%

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

Reproduce

herbie shell --seed 2023301 
(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)))