ab-angle->ABCF A

Percentage Accurate: 80.2% → 80.1%

Time: 32.8s

Alternatives: 5

Speedup: N/A×

• 37.6% 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: 80.1% accurate, 1.3× speedup

Math

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

FPCore

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

C

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 78.4%

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

   1. associate-*l/ 78.4%

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

   2. associate-/l* 78.5%

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

   3. *-rgt-identity 78.5%

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

   4. metadata-eval 78.5%

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

   5. associate-*l/ 78.5%

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

   6. associate-/l* 78.5%

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

   7. metadata-eval 78.5%

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

   8. metadata-eval 78.5%

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

3. Simplified 78.4%

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

5. Taylor expanded in angle around 0 78.8%

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

6. Final simplification 78.8%

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

Alternative 2: 80.1% accurate, 1.3× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 78.4%

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

   1. associate-*l/ 78.4%

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

   2. associate-/l* 78.5%

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

   3. *-rgt-identity 78.5%

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

   4. metadata-eval 78.5%

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

   5. associate-*l/ 78.5%

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

   6. associate-/l* 78.5%

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

   7. metadata-eval 78.5%

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

   8. metadata-eval 78.5%

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

3. Simplified 78.4%

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

5. Taylor expanded in angle around 0 78.8%

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

6. Taylor expanded in a around 0 71.1%

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

   1. unpow2 71.1%

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

   2. *-commutative 71.1%

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

   3. associate-*r* 71.1%

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

   4. unpow2 71.1%

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

   5. swap-sqr 78.8%

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

   6. unpow2 78.8%

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

   7. associate-*r* 78.7%

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

   8. *-commutative 78.7%

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

8. Simplified 78.7%

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

9. Final simplification 78.7%

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

Alternative 3: 75.1% accurate, 3.6× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 78.4%

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

   1. associate-*l/ 78.4%

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

   2. associate-/l* 78.5%

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

   3. *-rgt-identity 78.5%

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

   4. metadata-eval 78.5%

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

   5. associate-*l/ 78.5%

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

   6. associate-/l* 78.5%

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

   7. metadata-eval 78.5%

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

   8. metadata-eval 78.5%

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

3. Simplified 78.4%

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

5. Taylor expanded in angle around 0 78.8%

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

   1. expm1-log1p-u 78.8%

      \[\leadsto {\left(a \cdot \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\sin \left(angle \cdot \left(\pi \cdot 0.005555555555555556\right)\right)\right)\right)}\right)}^{2} + {\left(b \cdot 1\right)}^{2} \]

7. Applied egg-rr 78.8%

   \[\leadsto {\left(a \cdot \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\sin \left(angle \cdot \left(\pi \cdot 0.005555555555555556\right)\right)\right)\right)}\right)}^{2} + {\left(b \cdot 1\right)}^{2} \]

8. Taylor expanded in angle around 0 63.1%

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

   1. unpow2 63.1%

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

   2. unpow2 63.1%

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

   3. unpow2 63.1%

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

   4. swap-sqr 63.1%

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

   5. swap-sqr 72.7%

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

   6. unpow2 72.7%

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

10. Simplified 72.7%

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

    1. unpow2 72.7%

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

    2. associate-*r* 72.7%

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

    3. associate-*r* 72.7%

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

    4. *-commutative 72.7%

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

    5. associate-*l* 72.7%

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

12. Applied egg-rr 72.7%

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

13. Final simplification 72.7%

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

Alternative 4: 75.1% accurate, 3.6× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 78.4%

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

   1. associate-*l/ 78.4%

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

   2. associate-/l* 78.5%

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

   3. *-rgt-identity 78.5%

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

   4. metadata-eval 78.5%

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

   5. associate-*l/ 78.5%

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

   6. associate-/l* 78.5%

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

   7. metadata-eval 78.5%

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

   8. metadata-eval 78.5%

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

3. Simplified 78.4%

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

5. Taylor expanded in angle around 0 78.8%

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

   1. expm1-log1p-u 78.8%

      \[\leadsto {\left(a \cdot \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\sin \left(angle \cdot \left(\pi \cdot 0.005555555555555556\right)\right)\right)\right)}\right)}^{2} + {\left(b \cdot 1\right)}^{2} \]

7. Applied egg-rr 78.8%

   \[\leadsto {\left(a \cdot \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\sin \left(angle \cdot \left(\pi \cdot 0.005555555555555556\right)\right)\right)\right)}\right)}^{2} + {\left(b \cdot 1\right)}^{2} \]

8. Taylor expanded in angle around 0 63.1%

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

   1. unpow2 63.1%

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

   2. unpow2 63.1%

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

   3. unpow2 63.1%

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

   4. swap-sqr 63.1%

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

   5. swap-sqr 72.7%

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

   6. unpow2 72.7%

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

10. Simplified 72.7%

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

    1. unpow2 72.7%

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

    2. associate-*r* 72.7%

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

    3. associate-*r* 72.7%

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

    4. swap-sqr 72.7%

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

12. Applied egg-rr 72.7%

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

13. Final simplification 72.7%

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

Alternative 5: 75.1% accurate, 3.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := angle \cdot \left(a \cdot \pi\right)\\ {b}^{2} + 3.08641975308642 \cdot 10^{-5} \cdot \left(t\_0 \cdot t\_0\right) \end{array} \end{array} \]

FPCore

(FPCore (a b angle)
 :precision binary64
 (let* ((t_0 (* angle (* a PI))))
   (+ (pow b 2.0) (* 3.08641975308642e-5 (* t_0 t_0)))))

C

double code(double a, double b, double angle) {
	double t_0 = angle * (a * ((double) M_PI));
	return pow(b, 2.0) + (3.08641975308642e-5 * (t_0 * t_0));
}

Java

public static double code(double a, double b, double angle) {
	double t_0 = angle * (a * Math.PI);
	return Math.pow(b, 2.0) + (3.08641975308642e-5 * (t_0 * t_0));
}

Python

def code(a, b, angle):
	t_0 = angle * (a * math.pi)
	return math.pow(b, 2.0) + (3.08641975308642e-5 * (t_0 * t_0))

Julia

function code(a, b, angle)
	t_0 = Float64(angle * Float64(a * pi))
	return Float64((b ^ 2.0) + Float64(3.08641975308642e-5 * Float64(t_0 * t_0)))
end

MATLAB

function tmp = code(a, b, angle)
	t_0 = angle * (a * pi);
	tmp = (b ^ 2.0) + (3.08641975308642e-5 * (t_0 * t_0));
end

Wolfram

code[a_, b_, angle_] := Block[{t$95$0 = N[(angle * N[(a * Pi), $MachinePrecision]), $MachinePrecision]}, N[(N[Power[b, 2.0], $MachinePrecision] + N[(3.08641975308642e-5 * N[(t$95$0 * t$95$0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Initial program 78.4%

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

   1. associate-*l/ 78.4%

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

   2. associate-/l* 78.5%

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

   3. *-rgt-identity 78.5%

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

   4. metadata-eval 78.5%

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

   5. associate-*l/ 78.5%

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

   6. associate-/l* 78.5%

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

   7. metadata-eval 78.5%

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

   8. metadata-eval 78.5%

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

3. Simplified 78.4%

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

5. Taylor expanded in angle around 0 78.8%

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

   1. expm1-log1p-u 78.8%

      \[\leadsto {\left(a \cdot \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\sin \left(angle \cdot \left(\pi \cdot 0.005555555555555556\right)\right)\right)\right)}\right)}^{2} + {\left(b \cdot 1\right)}^{2} \]

7. Applied egg-rr 78.8%

   \[\leadsto {\left(a \cdot \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\sin \left(angle \cdot \left(\pi \cdot 0.005555555555555556\right)\right)\right)\right)}\right)}^{2} + {\left(b \cdot 1\right)}^{2} \]

8. Taylor expanded in angle around 0 63.1%

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

   1. unpow2 63.1%

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

   2. unpow2 63.1%

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

   3. unpow2 63.1%

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

   4. swap-sqr 63.1%

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

   5. swap-sqr 72.7%

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

   6. unpow2 72.7%

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

10. Simplified 72.7%

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

    1. unpow2 72.7%

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

    2. *-commutative 72.7%

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

    3. associate-*l* 72.7%

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

    4. *-commutative 72.7%

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

    5. associate-*l* 72.7%

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

12. Applied egg-rr 72.7%

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

13. Final simplification 72.7%

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

Reproduce

herbie shell --seed 2024096 
(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)))