ab-angle->ABCF C

Percentage Accurate: 80.4% → 80.4%

Time: 23.8s

Alternatives: 9

Speedup: 1.5×

• 36.4% of points produce a very large (infinite) output. You may want to add a precondition.

Specification

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 80.4% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 80.4% accurate, 0.7× speedup

Math

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

FPCore

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

C

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

Java

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

Julia

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

Wolfram

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

Derivation

1. Initial program 81.8%

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

   1. add-cube-cbrt 81.9%

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

   2. pow3 81.9%

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

   3. div-inv 81.9%

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

   4. metadata-eval 81.9%

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

3. Applied egg-rr 81.9%

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

   1. metadata-eval 81.9%

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

   2. div-inv 81.9%

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

   3. *-commutative 81.9%

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

   4. cbrt-prod 82.0%

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

5. Applied egg-rr 82.0%

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

6. Final simplification 82.0%

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

Alternative 2: 80.4% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ {\left(b \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} + {\left(a \cdot \cos \left({\left(\sqrt[3]{\pi \cdot \left(angle \cdot 0.005555555555555556\right)}\right)}^{3}\right)\right)}^{2} \end{array} \]

FPCore

(FPCore (a b angle)
 :precision binary64
 (+
  (pow (* b (sin (* (/ angle 180.0) PI))) 2.0)
  (pow
   (* a (cos (pow (cbrt (* PI (* angle 0.005555555555555556))) 3.0)))
   2.0)))

C

double code(double a, double b, double angle) {
	return pow((b * sin(((angle / 180.0) * ((double) M_PI)))), 2.0) + pow((a * cos(pow(cbrt((((double) M_PI) * (angle * 0.005555555555555556))), 3.0))), 2.0);
}

Java

public static double code(double a, double b, double angle) {
	return Math.pow((b * Math.sin(((angle / 180.0) * Math.PI))), 2.0) + Math.pow((a * Math.cos(Math.pow(Math.cbrt((Math.PI * (angle * 0.005555555555555556))), 3.0))), 2.0);
}

Julia

function code(a, b, angle)
	return Float64((Float64(b * sin(Float64(Float64(angle / 180.0) * pi))) ^ 2.0) + (Float64(a * cos((cbrt(Float64(pi * Float64(angle * 0.005555555555555556))) ^ 3.0))) ^ 2.0))
end

Wolfram

code[a_, b_, angle_] := N[(N[Power[N[(b * N[Sin[N[(N[(angle / 180.0), $MachinePrecision] * Pi), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision] + N[Power[N[(a * N[Cos[N[Power[N[Power[N[(Pi * N[(angle * 0.005555555555555556), $MachinePrecision]), $MachinePrecision], 1/3], $MachinePrecision], 3.0], $MachinePrecision]], $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 81.8%

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

   1. add-cube-cbrt 81.9%

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

   2. pow3 81.9%

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

   3. div-inv 81.9%

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

   4. metadata-eval 81.9%

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

3. Applied egg-rr 81.9%

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

4. Final simplification 81.9%

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

Alternative 3: 80.4% accurate, 1.5× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 81.8%

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

2. Taylor expanded in angle around 0 81.9%

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

3. Taylor expanded in b around 0 81.5%

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

4. Final simplification 81.5%

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

Alternative 4: 80.4% accurate, 1.5× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 81.8%

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

2. Taylor expanded in angle around 0 81.9%

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

3. Final simplification 81.9%

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

Alternative 5: 75.7% accurate, 1.9× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 81.8%

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

2. Taylor expanded in angle around 0 81.9%

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

3. Taylor expanded in angle around 0 77.5%

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

   1. unpow2 77.5%

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

   2. associate-*l* 77.2%

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

   3. *-commutative 77.2%

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

   4. associate-*r* 77.2%

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

   5. *-commutative 77.2%

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

   6. metadata-eval 77.2%

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

   7. div-inv 77.2%

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

   8. *-commutative 77.2%

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

   9. associate-*r* 77.2%

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

   10. div-inv 77.2%

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

   11. metadata-eval 77.2%

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

5. Applied egg-rr 77.2%

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

   1. associate-*r* 77.5%

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

   2. *-commutative 77.5%

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

   3. associate-*r* 77.5%

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

   4. *-commutative 77.5%

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

   5. *-commutative 77.5%

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

   6. *-commutative 77.5%

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

   7. metadata-eval 77.5%

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

   8. associate-/r/ 77.5%

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

   9. associate-*r/ 77.5%

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

   10. *-rgt-identity 77.5%

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

   11. associate-/r/ 77.5%

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

   12. associate-/l* 77.5%

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

   13. associate-/r/ 77.5%

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

   14. *-commutative 77.5%

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

   15. *-commutative 77.5%

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

   16. associate-/l* 77.5%

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

7. Simplified 77.5%

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

8. Final simplification 77.5%

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

Alternative 6: 75.6% accurate, 2.0× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 81.8%

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

2. Taylor expanded in angle around 0 81.9%

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

3. Taylor expanded in angle around 0 77.5%

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

   1. *-commutative 77.5%

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

   2. unpow-prod-down 77.2%

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

   3. *-commutative 77.2%

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

   4. metadata-eval 77.2%

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

5. Applied egg-rr 77.2%

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

6. Final simplification 77.2%

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

Alternative 7: 75.6% accurate, 2.0× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 81.8%

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

2. Taylor expanded in angle around 0 81.9%

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

3. Taylor expanded in angle around 0 77.5%

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

   1. *-commutative 77.5%

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

   2. unpow-prod-down 77.2%

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

   3. *-commutative 77.2%

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

   4. metadata-eval 77.2%

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

5. Applied egg-rr 77.2%

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

6. Taylor expanded in angle around 0 77.2%

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

   1. associate-*r* 77.2%

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

8. Simplified 77.2%

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

9. Final simplification 77.2%

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

Alternative 8: 75.7% accurate, 2.0× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 81.8%

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

2. Taylor expanded in angle around 0 81.9%

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

3. Taylor expanded in angle around 0 77.5%

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

4. Final simplification 77.5%

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

Alternative 9: 75.7% accurate, 2.0× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 81.8%

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

2. Taylor expanded in angle around 0 81.9%

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

3. Taylor expanded in angle around 0 77.5%

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

4. Taylor expanded in angle around 0 67.9%

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

   1. associate-*r* 67.8%

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

   2. *-commutative 67.8%

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

   3. unpow2 67.8%

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

   4. metadata-eval 67.8%

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

   5. swap-sqr 67.8%

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

   6. *-commutative 67.8%

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

   7. unpow2 67.8%

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

   8. unpow2 67.8%

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

   9. swap-sqr 67.8%

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

   10. swap-sqr 77.5%

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

   11. associate-*l* 77.5%

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

   12. associate-*l* 77.5%

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

   13. unpow2 77.5%

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

6. Simplified 77.5%

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

7. Final simplification 77.5%

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

Reproduce

herbie shell --seed 2023224 
(FPCore (a b angle)
  :name "ab-angle->ABCF C"
  :precision binary64
  (+ (pow (* a (cos (* PI (/ angle 180.0)))) 2.0) (pow (* b (sin (* PI (/ angle 180.0)))) 2.0)))