ab-angle->ABCF C

Percentage Accurate: 80.0% → 80.0%

Time: 21.7s

Alternatives: 6

Speedup: 1.3×

• 36.5% 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: 80.0% 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: 80.0% accurate, 1.3× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 78.3%

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

3. Simplified 78.3%

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

6. Applied egg-rr 77.2%

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

7. Taylor expanded in angle around 0 77.7%

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

   1. associate-*l* 78.8%

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

   2. pow2 78.8%

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

   3. *-commutative 78.8%

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

9. Applied egg-rr 78.8%

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

10. Final simplification 78.8%

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

Alternative 2: 80.0% accurate, 1.3× 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 78.3%

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

3. Simplified 78.3%

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

6. Applied egg-rr 77.2%

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

7. Taylor expanded in angle around 0 77.7%

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

8. Taylor expanded in angle around inf 65.2%

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

   1. unpow2 65.2%

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

   2. *-commutative 65.2%

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

   3. associate-*r* 65.3%

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

   4. unpow2 65.3%

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

   5. swap-sqr 78.8%

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

   6. unpow2 78.8%

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

   7. associate-*r* 78.7%

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

   8. *-commutative 78.7%

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

10. Simplified 78.7%

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

11. Final simplification 78.7%

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

Alternative 3: 67.7% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq 3.1 \cdot 10^{-51}:\\ \;\;\;\;{a}^{2} + {\left(b \cdot 0\right)}^{2}\\ \mathbf{else}:\\ \;\;\;\;{\left(a \cdot \cos \left(angle \cdot \frac{\pi}{-180}\right)\right)}^{2} + \left(-0.005555555555555556 \cdot \left(\pi \cdot \left(-0.005555555555555556 \cdot \left(b \cdot angle\right)\right)\right)\right) \cdot \left(angle \cdot \left(b \cdot \pi\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (a b angle)
 :precision binary64
 (if (<= b 3.1e-51)
   (+ (pow a 2.0) (pow (* b 0.0) 2.0))
   (+
    (pow (* a (cos (* angle (/ PI -180.0)))) 2.0)
    (*
     (* -0.005555555555555556 (* PI (* -0.005555555555555556 (* b angle))))
     (* angle (* b PI))))))

C

double code(double a, double b, double angle) {
	double tmp;
	if (b <= 3.1e-51) {
		tmp = pow(a, 2.0) + pow((b * 0.0), 2.0);
	} else {
		tmp = pow((a * cos((angle * (((double) M_PI) / -180.0)))), 2.0) + ((-0.005555555555555556 * (((double) M_PI) * (-0.005555555555555556 * (b * angle)))) * (angle * (b * ((double) M_PI))));
	}
	return tmp;
}

Java

public static double code(double a, double b, double angle) {
	double tmp;
	if (b <= 3.1e-51) {
		tmp = Math.pow(a, 2.0) + Math.pow((b * 0.0), 2.0);
	} else {
		tmp = Math.pow((a * Math.cos((angle * (Math.PI / -180.0)))), 2.0) + ((-0.005555555555555556 * (Math.PI * (-0.005555555555555556 * (b * angle)))) * (angle * (b * Math.PI)));
	}
	return tmp;
}

Python

def code(a, b, angle):
	tmp = 0
	if b <= 3.1e-51:
		tmp = math.pow(a, 2.0) + math.pow((b * 0.0), 2.0)
	else:
		tmp = math.pow((a * math.cos((angle * (math.pi / -180.0)))), 2.0) + ((-0.005555555555555556 * (math.pi * (-0.005555555555555556 * (b * angle)))) * (angle * (b * math.pi)))
	return tmp

Julia

function code(a, b, angle)
	tmp = 0.0
	if (b <= 3.1e-51)
		tmp = Float64((a ^ 2.0) + (Float64(b * 0.0) ^ 2.0));
	else
		tmp = Float64((Float64(a * cos(Float64(angle * Float64(pi / -180.0)))) ^ 2.0) + Float64(Float64(-0.005555555555555556 * Float64(pi * Float64(-0.005555555555555556 * Float64(b * angle)))) * Float64(angle * Float64(b * pi))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, angle)
	tmp = 0.0;
	if (b <= 3.1e-51)
		tmp = (a ^ 2.0) + ((b * 0.0) ^ 2.0);
	else
		tmp = ((a * cos((angle * (pi / -180.0)))) ^ 2.0) + ((-0.005555555555555556 * (pi * (-0.005555555555555556 * (b * angle)))) * (angle * (b * pi)));
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, angle_] := If[LessEqual[b, 3.1e-51], N[(N[Power[a, 2.0], $MachinePrecision] + N[Power[N[(b * 0.0), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision], N[(N[Power[N[(a * N[Cos[N[(angle * N[(Pi / -180.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision] + N[(N[(-0.005555555555555556 * N[(Pi * N[(-0.005555555555555556 * N[(b * angle), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(angle * N[(b * Pi), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b < 3.0999999999999997e-51

   1. Initial program 77.4%

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

   3. Simplified 77.4%

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

   5. Taylor expanded in angle around 0 78.0%

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

      1. div-inv 78.0%

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

      2. metadata-eval 78.0%

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

      3. rem-cube-cbrt 77.7%

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

   7. Applied egg-rr 77.7%

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

   8. Taylor expanded in angle around 0 60.7%

      \[\leadsto {\left(a \cdot 1\right)}^{2} + {\left(b \cdot \color{blue}{0}\right)}^{2} \]

   if 3.0999999999999997e-51 < b

   1. Initial program 80.5%

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

   3. Simplified 80.5%

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

   5. Taylor expanded in angle around 0 77.5%

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

      1. associate-*r* 77.5%

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

      2. *-commutative 77.5%

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

   7. Simplified 77.5%

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

      1. unpow2 77.5%

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

      2. associate-*r* 77.6%

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

      3. *-commutative 77.6%

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

      4. associate-*l* 77.6%

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

      5. *-commutative 77.6%

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

      6. associate-*l* 77.6%

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

   9. Applied egg-rr 77.6%

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

4. Final simplification 65.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq 3.1 \cdot 10^{-51}:\\ \;\;\;\;{a}^{2} + {\left(b \cdot 0\right)}^{2}\\ \mathbf{else}:\\ \;\;\;\;{\left(a \cdot \cos \left(angle \cdot \frac{\pi}{-180}\right)\right)}^{2} + \left(-0.005555555555555556 \cdot \left(\pi \cdot \left(-0.005555555555555556 \cdot \left(b \cdot angle\right)\right)\right)\right) \cdot \left(angle \cdot \left(b \cdot \pi\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 67.7% accurate, 2.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq 3.9 \cdot 10^{-51}:\\ \;\;\;\;{a}^{2} + {\left(b \cdot 0\right)}^{2}\\ \mathbf{else}:\\ \;\;\;\;\left(-0.005555555555555556 \cdot \left(\pi \cdot \left(-0.005555555555555556 \cdot \left(b \cdot angle\right)\right)\right)\right) \cdot \left(angle \cdot \left(b \cdot \pi\right)\right) + a \cdot a\\ \end{array} \end{array} \]

FPCore

(FPCore (a b angle)
 :precision binary64
 (if (<= b 3.9e-51)
   (+ (pow a 2.0) (pow (* b 0.0) 2.0))
   (+
    (*
     (* -0.005555555555555556 (* PI (* -0.005555555555555556 (* b angle))))
     (* angle (* b PI)))
    (* a a))))

C

double code(double a, double b, double angle) {
	double tmp;
	if (b <= 3.9e-51) {
		tmp = pow(a, 2.0) + pow((b * 0.0), 2.0);
	} else {
		tmp = ((-0.005555555555555556 * (((double) M_PI) * (-0.005555555555555556 * (b * angle)))) * (angle * (b * ((double) M_PI)))) + (a * a);
	}
	return tmp;
}

Java

public static double code(double a, double b, double angle) {
	double tmp;
	if (b <= 3.9e-51) {
		tmp = Math.pow(a, 2.0) + Math.pow((b * 0.0), 2.0);
	} else {
		tmp = ((-0.005555555555555556 * (Math.PI * (-0.005555555555555556 * (b * angle)))) * (angle * (b * Math.PI))) + (a * a);
	}
	return tmp;
}

Python

def code(a, b, angle):
	tmp = 0
	if b <= 3.9e-51:
		tmp = math.pow(a, 2.0) + math.pow((b * 0.0), 2.0)
	else:
		tmp = ((-0.005555555555555556 * (math.pi * (-0.005555555555555556 * (b * angle)))) * (angle * (b * math.pi))) + (a * a)
	return tmp

Julia

function code(a, b, angle)
	tmp = 0.0
	if (b <= 3.9e-51)
		tmp = Float64((a ^ 2.0) + (Float64(b * 0.0) ^ 2.0));
	else
		tmp = Float64(Float64(Float64(-0.005555555555555556 * Float64(pi * Float64(-0.005555555555555556 * Float64(b * angle)))) * Float64(angle * Float64(b * pi))) + Float64(a * a));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, angle)
	tmp = 0.0;
	if (b <= 3.9e-51)
		tmp = (a ^ 2.0) + ((b * 0.0) ^ 2.0);
	else
		tmp = ((-0.005555555555555556 * (pi * (-0.005555555555555556 * (b * angle)))) * (angle * (b * pi))) + (a * a);
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, angle_] := If[LessEqual[b, 3.9e-51], N[(N[Power[a, 2.0], $MachinePrecision] + N[Power[N[(b * 0.0), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision], N[(N[(N[(-0.005555555555555556 * N[(Pi * N[(-0.005555555555555556 * N[(b * angle), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(angle * N[(b * Pi), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(a * a), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b < 3.8999999999999997e-51

   1. Initial program 77.4%

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

   3. Simplified 77.4%

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

   5. Taylor expanded in angle around 0 78.0%

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

      1. div-inv 78.0%

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

      2. metadata-eval 78.0%

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

      3. rem-cube-cbrt 77.7%

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

   7. Applied egg-rr 77.7%

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

   8. Taylor expanded in angle around 0 60.7%

      \[\leadsto {\left(a \cdot 1\right)}^{2} + {\left(b \cdot \color{blue}{0}\right)}^{2} \]

   if 3.8999999999999997e-51 < b

   1. Initial program 80.5%

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

   3. Simplified 80.5%

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

   5. Taylor expanded in angle around 0 77.5%

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

      1. associate-*r* 77.5%

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

      2. *-commutative 77.5%

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

   7. Simplified 77.5%

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

   8. Taylor expanded in angle around 0 77.5%

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

      1. unpow2 77.5%

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

   10. Applied egg-rr 77.5%

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

       1. unpow2 77.5%

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

       2. associate-*r* 77.6%

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

       3. *-commutative 77.6%

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

       4. associate-*l* 77.6%

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

       5. *-commutative 77.6%

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

       6. associate-*l* 77.6%

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

   12. Applied egg-rr 77.6%

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

4. Final simplification 65.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq 3.9 \cdot 10^{-51}:\\ \;\;\;\;{a}^{2} + {\left(b \cdot 0\right)}^{2}\\ \mathbf{else}:\\ \;\;\;\;\left(-0.005555555555555556 \cdot \left(\pi \cdot \left(-0.005555555555555556 \cdot \left(b \cdot angle\right)\right)\right)\right) \cdot \left(angle \cdot \left(b \cdot \pi\right)\right) + a \cdot a\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 75.0% accurate, 21.9× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 78.3%

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

3. Simplified 78.3%

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

5. Taylor expanded in angle around 0 72.6%

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

   1. associate-*r* 72.6%

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

   2. *-commutative 72.6%

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

7. Simplified 72.6%

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

8. Taylor expanded in angle around 0 72.9%

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

   1. unpow2 72.9%

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

10. Applied egg-rr 72.9%

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

    1. unpow2 72.9%

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

    2. associate-*r* 72.9%

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

    3. *-commutative 72.9%

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

    4. metadata-eval 72.9%

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

    5. div-inv 72.9%

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

    6. associate-*l* 73.0%

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

    7. div-inv 73.0%

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

    8. metadata-eval 73.0%

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

    9. *-commutative 73.0%

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

    10. associate-*l* 72.9%

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

12. Applied egg-rr 72.9%

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

13. Final simplification 72.9%

    \[\leadsto a \cdot a + \left(\pi \cdot -0.005555555555555556\right) \cdot \left(\left(b \cdot angle\right) \cdot \left(\pi \cdot \left(-0.005555555555555556 \cdot \left(b \cdot angle\right)\right)\right)\right) \]
14. Add Preprocessing

Alternative 6: 75.0% accurate, 21.9× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 78.3%

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

3. Simplified 78.3%

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

5. Taylor expanded in angle around 0 72.6%

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

   1. associate-*r* 72.6%

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

   2. *-commutative 72.6%

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

7. Simplified 72.6%

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

8. Taylor expanded in angle around 0 72.9%

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

   1. unpow2 72.9%

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

10. Applied egg-rr 72.9%

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

    1. unpow2 72.9%

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

    2. associate-*r* 73.0%

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

    3. *-commutative 73.0%

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

    4. associate-*l* 73.0%

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

    5. *-commutative 73.0%

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

    6. associate-*l* 73.0%

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

12. Applied egg-rr 73.0%

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

13. Final simplification 73.0%

    \[\leadsto \left(-0.005555555555555556 \cdot \left(\pi \cdot \left(-0.005555555555555556 \cdot \left(b \cdot angle\right)\right)\right)\right) \cdot \left(angle \cdot \left(b \cdot \pi\right)\right) + a \cdot a \]
14. Add Preprocessing

Reproduce

herbie shell --seed 2024043 
(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)))