ab-angle->ABCF A

Percentage Accurate: 80.2% → 80.2%

Time: 42.6s

Alternatives: 7

Speedup: 1.3×

• 36.2% of points produce a very large (infinite) output. You may want to add a precondition.

Specification

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 80.2% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 80.2% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 78.8%

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

   1. unpow2 78.8%

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

   2. associate-*l/ 78.9%

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

   3. associate-/l* 78.9%

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

   4. unpow2 78.9%

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

3. Simplified 79.0%

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

5. Taylor expanded in angle around inf 79.0%

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

6. Final simplification 79.0%

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

Alternative 2: 80.1% accurate, 1.0× speedup

Math

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

FPCore

(FPCore (a b angle)
 :precision binary64
 (+
  (pow (* a (sin (* 0.005555555555555556 (* angle PI)))) 2.0)
  (pow (* b (cos (* angle (/ PI 180.0)))) 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 * cos((angle * (((double) M_PI) / 180.0)))), 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 * Math.cos((angle * (Math.PI / 180.0)))), 2.0);
}

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 78.8%

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

   1. unpow2 78.8%

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

   2. associate-*l/ 78.9%

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

   3. associate-/l* 78.9%

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

   4. unpow2 78.9%

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

3. Simplified 79.0%

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

5. Taylor expanded in angle around inf 79.0%

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

6. Final simplification 79.0%

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

Alternative 3: 80.1% accurate, 1.3× speedup

Math

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

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(0.005555555555555556 * Float64(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[(0.005555555555555556 * N[(angle * Pi), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision] + N[Power[b, 2.0], $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 78.8%

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

   1. unpow2 78.8%

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

   2. associate-*l/ 78.9%

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

   3. associate-/l* 78.9%

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

   4. unpow2 78.9%

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

3. Simplified 79.0%

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

5. Taylor expanded in angle around 0 78.4%

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

6. Taylor expanded in angle around 0 78.4%

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

7. Final simplification 78.4%

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

Alternative 4: 80.2% accurate, 1.3× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 78.8%

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

   1. unpow2 78.8%

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

   2. associate-*l/ 78.9%

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

   3. associate-/l* 78.9%

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

   4. unpow2 78.9%

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

3. Simplified 79.0%

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

5. Taylor expanded in angle around 0 78.4%

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

6. Final simplification 78.4%

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

Alternative 5: 74.9% accurate, 2.0× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 78.8%

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

   1. unpow2 78.8%

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

   2. associate-*l/ 78.9%

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

   3. associate-/l* 78.9%

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

   4. unpow2 78.9%

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

3. Simplified 79.0%

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

5. Taylor expanded in angle around 0 78.4%

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

6. Taylor expanded in angle around 0 72.5%

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

   1. unpow2 72.5%

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

   2. rem-cube-cbrt 72.3%

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

   3. associate-*r* 72.3%

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

   4. associate-*l* 71.2%

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

   5. rem-cube-cbrt 71.4%

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

   6. rem-cube-cbrt 71.2%

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

   7. *-commutative 71.2%

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

   8. associate-*r* 71.2%

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

   9. rem-cube-cbrt 71.5%

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

8. Applied egg-rr 71.5%

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

9. Taylor expanded in a around 0 59.5%

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

    1. *-commutative 59.5%

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

    2. associate-*r* 59.5%

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

    3. associate-*l* 59.5%

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

    4. unpow2 59.5%

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

    5. metadata-eval 59.5%

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

    6. swap-sqr 59.5%

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

    7. *-commutative 59.5%

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

    8. unpow2 59.5%

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

    9. unpow2 59.5%

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

    10. swap-sqr 72.5%

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

    11. swap-sqr 72.5%

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

    12. *-commutative 72.5%

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

    13. *-commutative 72.5%

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

    14. unpow2 72.5%

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

    15. *-commutative 72.5%

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

    16. associate-*l* 72.6%

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

11. Simplified 72.6%

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

12. Final simplification 72.6%

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

Alternative 6: 74.1% accurate, 3.5× speedup

Math

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

FPCore

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

C

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

Python

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

Julia

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

MATLAB

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

Derivation

1. Initial program 78.8%

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

   1. unpow2 78.8%

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

   2. associate-*l/ 78.9%

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

   3. associate-/l* 78.9%

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

   4. unpow2 78.9%

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

3. Simplified 79.0%

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

5. Taylor expanded in angle around 0 78.4%

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

6. Taylor expanded in angle around 0 72.5%

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

   1. unpow2 72.5%

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

   2. rem-cube-cbrt 72.3%

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

   3. associate-*r* 72.3%

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

   4. associate-*l* 71.2%

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

   5. rem-cube-cbrt 71.4%

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

   6. rem-cube-cbrt 71.2%

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

   7. *-commutative 71.2%

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

   8. associate-*r* 71.2%

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

   9. rem-cube-cbrt 71.5%

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

8. Applied egg-rr 71.5%

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

9. Final simplification 71.5%

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

Alternative 7: 74.9% accurate, 3.5× speedup

Math

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

FPCore

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

C

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 78.8%

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

   1. unpow2 78.8%

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

   2. associate-*l/ 78.9%

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

   3. associate-/l* 78.9%

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

   4. unpow2 78.9%

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

3. Simplified 79.0%

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

5. Taylor expanded in angle around 0 78.4%

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

6. Taylor expanded in angle around 0 72.5%

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

   1. unpow2 72.5%

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

   2. rem-cube-cbrt 72.3%

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

   3. *-commutative 72.3%

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

   4. associate-*r* 72.2%

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

   5. rem-cube-cbrt 72.5%

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

   6. rem-cube-cbrt 72.2%

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

   7. *-commutative 72.2%

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

   8. associate-*r* 72.2%

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

   9. rem-cube-cbrt 72.5%

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

8. Applied egg-rr 72.5%

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

9. Final simplification 72.5%

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

Reproduce

herbie shell --seed 2024076 
(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)))