ab-angle->ABCF A

Percentage Accurate: 79.7% → 79.6%

Time: 36.9s

Alternatives: 9

Speedup: N/A×

• 36.7% of points produce a very large (infinite) output. You may want to add a precondition.

• could not determine a ground truth

  1. a = 7.70677570466877e+127
  2. b = 2.631484189770071e-225
  3. angle = -1.1170025481421776e+221

Specification

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 79.7% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 79.6% accurate, 1.3× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 81.1%

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

   1. associate-*l/ 81.2%

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

   2. associate-/l* 81.2%

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

   3. associate-*l/ 81.1%

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

   4. associate-/l* 81.2%

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

3. Simplified 81.2%

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

5. Taylor expanded in angle around 0 81.3%

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

   1. *-un-lft-identity 81.3%

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

   2. *-commutative 81.3%

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

   3. add-sqr-sqrt 81.3%

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

   4. pow2 81.3%

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

   5. sqrt-pow1 81.3%

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

   6. metadata-eval 81.3%

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

   7. pow1 81.3%

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

   8. clear-num 81.3%

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

   9. un-div-inv 81.4%

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

7. Applied egg-rr 81.4%

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

8. Final simplification 81.4%

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

Alternative 2: 79.5% 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 81.1%

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

   1. associate-*l/ 81.2%

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

   2. associate-/l* 81.2%

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

   3. associate-*l/ 81.1%

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

   4. associate-/l* 81.2%

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

3. Simplified 81.2%

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

5. Taylor expanded in angle around 0 81.3%

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

6. Taylor expanded in angle around inf 81.3%

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

7. Final simplification 81.3%

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

Alternative 3: 79.6% accurate, 1.3× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 81.1%

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

   1. associate-*l/ 81.2%

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

   2. associate-/l* 81.2%

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

   3. associate-*l/ 81.1%

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

   4. associate-/l* 81.2%

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

3. Simplified 81.2%

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

5. Taylor expanded in angle around 0 81.3%

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

6. Final simplification 81.3%

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

Alternative 4: 62.8% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;a \leq 2.8 \cdot 10^{-62}:\\ \;\;\;\;{b}^{2} + 0 \cdot {a}^{2}\\ \mathbf{else}:\\ \;\;\;\;{b}^{2} + {\left(angle \cdot \left(a \cdot \pi\right)\right)}^{2} \cdot 3.08641975308642 \cdot 10^{-5}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b angle)
 :precision binary64
 (if (<= a 2.8e-62)
   (+ (pow b 2.0) (* 0.0 (pow a 2.0)))
   (+ (pow b 2.0) (* (pow (* angle (* a PI)) 2.0) 3.08641975308642e-5))))

C

double code(double a, double b, double angle) {
	double tmp;
	if (a <= 2.8e-62) {
		tmp = pow(b, 2.0) + (0.0 * pow(a, 2.0));
	} else {
		tmp = pow(b, 2.0) + (pow((angle * (a * ((double) M_PI))), 2.0) * 3.08641975308642e-5);
	}
	return tmp;
}

Java

public static double code(double a, double b, double angle) {
	double tmp;
	if (a <= 2.8e-62) {
		tmp = Math.pow(b, 2.0) + (0.0 * Math.pow(a, 2.0));
	} else {
		tmp = Math.pow(b, 2.0) + (Math.pow((angle * (a * Math.PI)), 2.0) * 3.08641975308642e-5);
	}
	return tmp;
}

Python

def code(a, b, angle):
	tmp = 0
	if a <= 2.8e-62:
		tmp = math.pow(b, 2.0) + (0.0 * math.pow(a, 2.0))
	else:
		tmp = math.pow(b, 2.0) + (math.pow((angle * (a * math.pi)), 2.0) * 3.08641975308642e-5)
	return tmp

Julia

function code(a, b, angle)
	tmp = 0.0
	if (a <= 2.8e-62)
		tmp = Float64((b ^ 2.0) + Float64(0.0 * (a ^ 2.0)));
	else
		tmp = Float64((b ^ 2.0) + Float64((Float64(angle * Float64(a * pi)) ^ 2.0) * 3.08641975308642e-5));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, angle)
	tmp = 0.0;
	if (a <= 2.8e-62)
		tmp = (b ^ 2.0) + (0.0 * (a ^ 2.0));
	else
		tmp = (b ^ 2.0) + (((angle * (a * pi)) ^ 2.0) * 3.08641975308642e-5);
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, angle_] := If[LessEqual[a, 2.8e-62], N[(N[Power[b, 2.0], $MachinePrecision] + N[(0.0 * N[Power[a, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[Power[b, 2.0], $MachinePrecision] + N[(N[Power[N[(angle * N[(a * Pi), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision] * 3.08641975308642e-5), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if a < 2.80000000000000002e-62

   1. Initial program 81.0%

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

      1. associate-*l/ 81.2%

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

      2. associate-/l* 81.1%

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

      3. associate-*l/ 80.9%

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

      4. associate-/l* 81.0%

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

   3. Simplified 81.0%

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

   5. Taylor expanded in angle around 0 81.1%

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

      1. *-commutative 81.1%

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

      2. unpow-prod-down 75.7%

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

      3. clear-num 75.7%

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

      4. un-div-inv 75.8%

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

   7. Applied egg-rr 75.8%

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

      1. unpow2 75.8%

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

      2. div-inv 74.9%

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

      3. clear-num 74.9%

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

      4. div-inv 74.9%

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

      5. metadata-eval 74.9%

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

      6. div-inv 75.7%

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

      7. clear-num 75.7%

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

      8. div-inv 75.7%

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

      9. metadata-eval 75.7%

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

      10. sqr-sin-a 72.4%

          \[\leadsto \color{blue}{\left(0.5 - 0.5 \cdot \cos \left(2 \cdot \left(angle \cdot \left(\pi \cdot 0.005555555555555556\right)\right)\right)\right)} \cdot {a}^{2} + {\left(b \cdot 1\right)}^{2} \]

      11. add-sqr-sqrt 33.6%

          \[\leadsto \left(0.5 - 0.5 \cdot \cos \left(2 \cdot \color{blue}{\left(\sqrt{angle \cdot \left(\pi \cdot 0.005555555555555556\right)} \cdot \sqrt{angle \cdot \left(\pi \cdot 0.005555555555555556\right)}\right)}\right)\right) \cdot {a}^{2} + {\left(b \cdot 1\right)}^{2} \]

      12. sqrt-unprod 56.5%

          \[\leadsto \left(0.5 - 0.5 \cdot \cos \left(2 \cdot \color{blue}{\sqrt{\left(angle \cdot \left(\pi \cdot 0.005555555555555556\right)\right) \cdot \left(angle \cdot \left(\pi \cdot 0.005555555555555556\right)\right)}}\right)\right) \cdot {a}^{2} + {\left(b \cdot 1\right)}^{2} \]

      13. associate-*r* 56.5%

          \[\leadsto \left(0.5 - 0.5 \cdot \cos \left(2 \cdot \sqrt{\color{blue}{\left(\left(angle \cdot \pi\right) \cdot 0.005555555555555556\right)} \cdot \left(angle \cdot \left(\pi \cdot 0.005555555555555556\right)\right)}\right)\right) \cdot {a}^{2} + {\left(b \cdot 1\right)}^{2} \]

      14. *-commutative 56.5%

          \[\leadsto \left(0.5 - 0.5 \cdot \cos \left(2 \cdot \sqrt{\color{blue}{\left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)} \cdot \left(angle \cdot \left(\pi \cdot 0.005555555555555556\right)\right)}\right)\right) \cdot {a}^{2} + {\left(b \cdot 1\right)}^{2} \]

      15. associate-*r* 56.5%

          \[\leadsto \left(0.5 - 0.5 \cdot \cos \left(2 \cdot \sqrt{\left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right) \cdot \color{blue}{\left(\left(angle \cdot \pi\right) \cdot 0.005555555555555556\right)}}\right)\right) \cdot {a}^{2} + {\left(b \cdot 1\right)}^{2} \]

      16. *-commutative 56.5%

          \[\leadsto \left(0.5 - 0.5 \cdot \cos \left(2 \cdot \sqrt{\left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right) \cdot \color{blue}{\left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}}\right)\right) \cdot {a}^{2} + {\left(b \cdot 1\right)}^{2} \]

   9. Applied egg-rr 72.4%

      \[\leadsto \color{blue}{\left(0.5 - 0.5 \cdot \cos \left(2 \cdot \left(\pi \cdot \left(angle \cdot -0.005555555555555556\right)\right)\right)\right)} \cdot {a}^{2} + {\left(b \cdot 1\right)}^{2} \]

   11. Simplified 61.7%

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

   if 2.80000000000000002e-62 < a

   1. Initial program 81.4%

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

      1. associate-*l/ 81.4%

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

      2. associate-/l* 81.6%

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

      3. associate-*l/ 81.6%

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

      4. associate-/l* 81.6%

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

   3. Simplified 81.6%

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

   5. Taylor expanded in angle around 0 81.9%

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

   6. Taylor expanded in angle around 0 76.5%

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

      1. *-commutative 76.5%

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

      2. *-commutative 76.5%

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

      3. associate-*l* 76.5%

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

   8. Simplified 76.5%

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

      1. *-commutative 76.5%

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

      2. unpow-prod-down 76.6%

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

      3. *-commutative 76.6%

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

      4. associate-*l* 76.6%

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

      5. metadata-eval 76.6%

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

   10. Applied egg-rr 76.6%

       \[\leadsto \color{blue}{{\left(angle \cdot \left(a \cdot \pi\right)\right)}^{2} \cdot 3.08641975308642 \cdot 10^{-5}} + {\left(b \cdot 1\right)}^{2} \]
3. Recombined 2 regimes into one program.

4. Final simplification 66.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \leq 2.8 \cdot 10^{-62}:\\ \;\;\;\;{b}^{2} + 0 \cdot {a}^{2}\\ \mathbf{else}:\\ \;\;\;\;{b}^{2} + {\left(angle \cdot \left(a \cdot \pi\right)\right)}^{2} \cdot 3.08641975308642 \cdot 10^{-5}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 62.9% accurate, 2.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;a \leq 1.26 \cdot 10^{-61}:\\ \;\;\;\;{b}^{2} + 0 \cdot {a}^{2}\\ \mathbf{else}:\\ \;\;\;\;{b}^{2} + \left(\left(\pi \cdot 0.005555555555555556\right) \cdot \left(a \cdot angle\right)\right) \cdot \left(angle \cdot \left(a \cdot \left(\pi \cdot 0.005555555555555556\right)\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (a b angle)
 :precision binary64
 (if (<= a 1.26e-61)
   (+ (pow b 2.0) (* 0.0 (pow a 2.0)))
   (+
    (pow b 2.0)
    (*
     (* (* PI 0.005555555555555556) (* a angle))
     (* angle (* a (* PI 0.005555555555555556)))))))

C

double code(double a, double b, double angle) {
	double tmp;
	if (a <= 1.26e-61) {
		tmp = pow(b, 2.0) + (0.0 * pow(a, 2.0));
	} else {
		tmp = pow(b, 2.0) + (((((double) M_PI) * 0.005555555555555556) * (a * angle)) * (angle * (a * (((double) M_PI) * 0.005555555555555556))));
	}
	return tmp;
}

Java

public static double code(double a, double b, double angle) {
	double tmp;
	if (a <= 1.26e-61) {
		tmp = Math.pow(b, 2.0) + (0.0 * Math.pow(a, 2.0));
	} else {
		tmp = Math.pow(b, 2.0) + (((Math.PI * 0.005555555555555556) * (a * angle)) * (angle * (a * (Math.PI * 0.005555555555555556))));
	}
	return tmp;
}

Python

def code(a, b, angle):
	tmp = 0
	if a <= 1.26e-61:
		tmp = math.pow(b, 2.0) + (0.0 * math.pow(a, 2.0))
	else:
		tmp = math.pow(b, 2.0) + (((math.pi * 0.005555555555555556) * (a * angle)) * (angle * (a * (math.pi * 0.005555555555555556))))
	return tmp

Julia

function code(a, b, angle)
	tmp = 0.0
	if (a <= 1.26e-61)
		tmp = Float64((b ^ 2.0) + Float64(0.0 * (a ^ 2.0)));
	else
		tmp = Float64((b ^ 2.0) + Float64(Float64(Float64(pi * 0.005555555555555556) * Float64(a * angle)) * Float64(angle * Float64(a * Float64(pi * 0.005555555555555556)))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, angle)
	tmp = 0.0;
	if (a <= 1.26e-61)
		tmp = (b ^ 2.0) + (0.0 * (a ^ 2.0));
	else
		tmp = (b ^ 2.0) + (((pi * 0.005555555555555556) * (a * angle)) * (angle * (a * (pi * 0.005555555555555556))));
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, angle_] := If[LessEqual[a, 1.26e-61], N[(N[Power[b, 2.0], $MachinePrecision] + N[(0.0 * N[Power[a, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[Power[b, 2.0], $MachinePrecision] + N[(N[(N[(Pi * 0.005555555555555556), $MachinePrecision] * N[(a * angle), $MachinePrecision]), $MachinePrecision] * N[(angle * N[(a * N[(Pi * 0.005555555555555556), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if a < 1.2599999999999999e-61

   1. Initial program 81.0%

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

      1. associate-*l/ 81.2%

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

      2. associate-/l* 81.1%

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

      3. associate-*l/ 80.9%

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

      4. associate-/l* 81.0%

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

   3. Simplified 81.0%

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

   5. Taylor expanded in angle around 0 81.1%

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

      1. *-commutative 81.1%

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

      2. unpow-prod-down 75.7%

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

      3. clear-num 75.7%

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

      4. un-div-inv 75.8%

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

   7. Applied egg-rr 75.8%

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

      1. unpow2 75.8%

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

      2. div-inv 74.9%

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

      3. clear-num 74.9%

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

      4. div-inv 74.9%

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

      5. metadata-eval 74.9%

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

      6. div-inv 75.7%

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

      7. clear-num 75.7%

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

      8. div-inv 75.7%

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

      9. metadata-eval 75.7%

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

      10. sqr-sin-a 72.4%

          \[\leadsto \color{blue}{\left(0.5 - 0.5 \cdot \cos \left(2 \cdot \left(angle \cdot \left(\pi \cdot 0.005555555555555556\right)\right)\right)\right)} \cdot {a}^{2} + {\left(b \cdot 1\right)}^{2} \]

      11. add-sqr-sqrt 33.6%

          \[\leadsto \left(0.5 - 0.5 \cdot \cos \left(2 \cdot \color{blue}{\left(\sqrt{angle \cdot \left(\pi \cdot 0.005555555555555556\right)} \cdot \sqrt{angle \cdot \left(\pi \cdot 0.005555555555555556\right)}\right)}\right)\right) \cdot {a}^{2} + {\left(b \cdot 1\right)}^{2} \]

      12. sqrt-unprod 56.5%

          \[\leadsto \left(0.5 - 0.5 \cdot \cos \left(2 \cdot \color{blue}{\sqrt{\left(angle \cdot \left(\pi \cdot 0.005555555555555556\right)\right) \cdot \left(angle \cdot \left(\pi \cdot 0.005555555555555556\right)\right)}}\right)\right) \cdot {a}^{2} + {\left(b \cdot 1\right)}^{2} \]

      13. associate-*r* 56.5%

          \[\leadsto \left(0.5 - 0.5 \cdot \cos \left(2 \cdot \sqrt{\color{blue}{\left(\left(angle \cdot \pi\right) \cdot 0.005555555555555556\right)} \cdot \left(angle \cdot \left(\pi \cdot 0.005555555555555556\right)\right)}\right)\right) \cdot {a}^{2} + {\left(b \cdot 1\right)}^{2} \]

      14. *-commutative 56.5%

          \[\leadsto \left(0.5 - 0.5 \cdot \cos \left(2 \cdot \sqrt{\color{blue}{\left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)} \cdot \left(angle \cdot \left(\pi \cdot 0.005555555555555556\right)\right)}\right)\right) \cdot {a}^{2} + {\left(b \cdot 1\right)}^{2} \]

      15. associate-*r* 56.5%

          \[\leadsto \left(0.5 - 0.5 \cdot \cos \left(2 \cdot \sqrt{\left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right) \cdot \color{blue}{\left(\left(angle \cdot \pi\right) \cdot 0.005555555555555556\right)}}\right)\right) \cdot {a}^{2} + {\left(b \cdot 1\right)}^{2} \]

      16. *-commutative 56.5%

          \[\leadsto \left(0.5 - 0.5 \cdot \cos \left(2 \cdot \sqrt{\left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right) \cdot \color{blue}{\left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}}\right)\right) \cdot {a}^{2} + {\left(b \cdot 1\right)}^{2} \]

   9. Applied egg-rr 72.4%

      \[\leadsto \color{blue}{\left(0.5 - 0.5 \cdot \cos \left(2 \cdot \left(\pi \cdot \left(angle \cdot -0.005555555555555556\right)\right)\right)\right)} \cdot {a}^{2} + {\left(b \cdot 1\right)}^{2} \]

   11. Simplified 61.7%

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

   if 1.2599999999999999e-61 < a

   1. Initial program 81.4%

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

      1. associate-*l/ 81.4%

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

      2. associate-/l* 81.6%

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

      3. associate-*l/ 81.6%

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

      4. associate-/l* 81.6%

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

   3. Simplified 81.6%

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

   5. Taylor expanded in angle around 0 81.9%

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

   6. Taylor expanded in angle around 0 76.5%

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

      1. *-commutative 76.5%

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

      2. *-commutative 76.5%

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

      3. associate-*l* 76.5%

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

   8. Simplified 76.5%

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

      1. unpow2 76.5%

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

      2. associate-*r* 76.6%

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

      3. associate-*l* 76.6%

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

      4. *-commutative 76.6%

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

      5. associate-*r* 76.6%

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

      6. associate-*r* 76.6%

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

      7. metadata-eval 76.6%

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

      8. associate-/r/ 76.6%

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

      9. associate-*l/ 76.6%

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

      10. *-un-lft-identity 76.6%

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

      11. div-inv 76.6%

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

      12. clear-num 76.6%

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

      13. div-inv 76.6%

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

      14. metadata-eval 76.6%

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

   10. Applied egg-rr 76.6%

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

       1. associate-*r* 76.6%

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

       2. *-commutative 76.6%

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

       3. associate-*l* 76.6%

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

       4. *-commutative 76.6%

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

   12. Simplified 76.6%

       \[\leadsto \color{blue}{\left(\left(0.005555555555555556 \cdot \pi\right) \cdot \left(angle \cdot a\right)\right) \cdot \left(angle \cdot \left(\left(0.005555555555555556 \cdot \pi\right) \cdot a\right)\right)} + {\left(b \cdot 1\right)}^{2} \]
3. Recombined 2 regimes into one program.

4. Final simplification 66.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \leq 1.26 \cdot 10^{-61}:\\ \;\;\;\;{b}^{2} + 0 \cdot {a}^{2}\\ \mathbf{else}:\\ \;\;\;\;{b}^{2} + \left(\left(\pi \cdot 0.005555555555555556\right) \cdot \left(a \cdot angle\right)\right) \cdot \left(angle \cdot \left(a \cdot \left(\pi \cdot 0.005555555555555556\right)\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 73.5% accurate, 3.5× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 81.1%

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

   1. associate-*l/ 81.2%

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

   2. associate-/l* 81.2%

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

   3. associate-*l/ 81.1%

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

   4. associate-/l* 81.2%

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

3. Simplified 81.2%

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

5. Taylor expanded in angle around 0 81.3%

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

6. Taylor expanded in angle around 0 76.1%

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

   1. *-commutative 76.1%

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

   2. *-commutative 76.1%

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

   3. associate-*l* 76.1%

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

8. Simplified 76.1%

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

   1. unpow2 76.1%

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

   2. *-commutative 76.1%

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

   3. associate-*l* 76.1%

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

   4. *-commutative 76.1%

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

   5. associate-*l* 76.1%

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

   6. associate-*r* 76.1%

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

   7. associate-*r* 76.1%

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

   8. metadata-eval 76.1%

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

   9. associate-/r/ 76.1%

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

   10. associate-*l/ 76.1%

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

   11. *-un-lft-identity 76.1%

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

   12. div-inv 76.1%

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

   13. clear-num 76.1%

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

   14. div-inv 76.1%

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

   15. metadata-eval 76.1%

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

10. Applied egg-rr 76.1%

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

    1. associate-*r* 76.1%

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

    2. associate-*r* 76.1%

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

    3. *-commutative 76.1%

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

    4. associate-*r* 76.1%

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

    5. *-commutative 76.1%

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

    6. associate-*r* 76.1%

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

    7. *-commutative 76.1%

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

    8. associate-*r* 76.1%

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

    9. associate-*r* 76.1%

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

    10. associate-*r* 75.4%

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

    11. *-commutative 75.4%

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

    12. *-commutative 75.4%

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

    13. associate-*r* 75.4%

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

    14. *-commutative 75.4%

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

    15. associate-*r* 75.4%

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

    16. *-commutative 75.4%

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

12. Simplified 75.4%

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

13. Final simplification 75.4%

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

Alternative 7: 73.5% accurate, 3.5× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 81.1%

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

   1. associate-*l/ 81.2%

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

   2. associate-/l* 81.2%

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

   3. associate-*l/ 81.1%

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

   4. associate-/l* 81.2%

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

3. Simplified 81.2%

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

5. Taylor expanded in angle around 0 81.3%

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

6. Taylor expanded in angle around 0 76.1%

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

   1. *-commutative 76.1%

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

   2. *-commutative 76.1%

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

   3. associate-*l* 76.1%

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

8. Simplified 76.1%

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

   1. unpow2 76.1%

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

   2. *-commutative 76.1%

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

   3. associate-*l* 76.1%

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

   4. *-commutative 76.1%

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

   5. associate-*l* 76.1%

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

   6. associate-*r* 76.1%

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

   7. associate-*r* 76.1%

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

   8. metadata-eval 76.1%

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

   9. associate-/r/ 76.1%

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

   10. associate-*l/ 76.1%

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

   11. *-un-lft-identity 76.1%

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

   12. div-inv 76.1%

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

   13. clear-num 76.1%

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

   14. div-inv 76.1%

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

   15. metadata-eval 76.1%

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

10. Applied egg-rr 76.1%

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

    1. associate-*r* 76.1%

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

    2. associate-*r* 76.1%

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

    3. *-commutative 76.1%

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

    4. associate-*r* 76.1%

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

    5. *-commutative 76.1%

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

    6. associate-*r* 76.1%

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

    7. *-commutative 76.1%

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

    8. associate-*r* 76.1%

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

    9. associate-*r* 76.1%

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

    10. associate-*r* 75.4%

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

    11. *-commutative 75.4%

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

    12. *-commutative 75.4%

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

    13. associate-*r* 75.4%

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

    14. *-commutative 75.4%

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

    15. associate-*r* 75.4%

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

    16. *-commutative 75.4%

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

12. Simplified 75.4%

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

    1. *-commutative 75.4%

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

    2. metadata-eval 75.4%

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

    3. div-inv 75.4%

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

14. Applied egg-rr 75.4%

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

15. Final simplification 75.4%

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

Alternative 8: 74.4% accurate, 3.5× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 81.1%

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

   1. associate-*l/ 81.2%

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

   2. associate-/l* 81.2%

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

   3. associate-*l/ 81.1%

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

   4. associate-/l* 81.2%

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

3. Simplified 81.2%

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

5. Taylor expanded in angle around 0 81.3%

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

6. Taylor expanded in angle around 0 76.1%

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

   1. *-commutative 76.1%

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

   2. *-commutative 76.1%

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

   3. associate-*l* 76.1%

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

8. Simplified 76.1%

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

   1. unpow2 76.1%

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

   2. *-commutative 76.1%

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

   3. associate-*l* 76.1%

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

   4. *-commutative 76.1%

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

   5. associate-*l* 76.1%

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

   6. associate-*r* 76.1%

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

   7. associate-*r* 76.1%

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

   8. metadata-eval 76.1%

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

   9. associate-/r/ 76.1%

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

   10. associate-*l/ 76.1%

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

   11. *-un-lft-identity 76.1%

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

   12. div-inv 76.1%

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

   13. clear-num 76.1%

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

   14. div-inv 76.1%

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

   15. metadata-eval 76.1%

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

10. Applied egg-rr 76.1%

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

    1. *-commutative 76.1%

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

    2. *-commutative 76.1%

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

    3. associate-*l* 76.1%

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

    4. associate-*r* 76.1%

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

    5. *-commutative 76.1%

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

12. Simplified 76.1%

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

13. Final simplification 76.1%

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

Alternative 9: 74.4% accurate, 3.5× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 81.1%

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

   1. associate-*l/ 81.2%

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

   2. associate-/l* 81.2%

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

   3. associate-*l/ 81.1%

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

   4. associate-/l* 81.2%

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

3. Simplified 81.2%

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

5. Taylor expanded in angle around 0 81.3%

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

6. Taylor expanded in angle around 0 76.1%

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

   1. *-commutative 76.1%

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

   2. *-commutative 76.1%

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

   3. associate-*l* 76.1%

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

8. Simplified 76.1%

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

   1. unpow2 76.1%

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

   2. associate-*r* 76.1%

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

   3. associate-*l* 76.1%

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

   4. *-commutative 76.1%

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

   5. associate-*r* 76.2%

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

   6. associate-*r* 76.2%

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

   7. metadata-eval 76.2%

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

   8. associate-/r/ 76.2%

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

   9. associate-*l/ 76.2%

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

   10. *-un-lft-identity 76.2%

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

   11. div-inv 76.2%

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

   12. clear-num 76.2%

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

   13. div-inv 76.2%

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

   14. metadata-eval 76.2%

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

10. Applied egg-rr 76.2%

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

    1. associate-*r* 76.2%

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

    2. *-commutative 76.2%

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

    3. associate-*l* 76.2%

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

    4. *-commutative 76.2%

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

12. Simplified 76.2%

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

13. Final simplification 76.2%

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

Reproduce

herbie shell --seed 2024046 
(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)))