ab-angle->ABCF B

Percentage Accurate: 54.3% → 67.2%

Time: 32.6s

Alternatives: 14

Speedup: 5.5×

• 34.3% of points produce a very large (infinite) output. You may want to add a precondition.

Specification

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 54.3% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 67.2% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \sin \left(\frac{angle \cdot \pi}{180}\right) \cdot \left(b + a\right)\\ \mathbf{if}\;\frac{angle}{180} \leq -2 \cdot 10^{+68}:\\ \;\;\;\;2 \cdot \left(t_0 \cdot \left(\left(b - a\right) \cdot \cos \left(\left(angle \cdot \pi\right) \cdot 0.005555555555555556\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \left(t_0 \cdot \left(\left(b - a\right) \cdot \cos \left(0.005555555555555556 \cdot \left(angle \cdot \sqrt[3]{e^{\log \pi \cdot 3}}\right)\right)\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (a b angle)
 :precision binary64
 (let* ((t_0 (* (sin (/ (* angle PI) 180.0)) (+ b a))))
   (if (<= (/ angle 180.0) -2e+68)
     (* 2.0 (* t_0 (* (- b a) (cos (* (* angle PI) 0.005555555555555556)))))
     (*
      2.0
      (*
       t_0
       (*
        (- b a)
        (cos
         (*
          0.005555555555555556
          (* angle (cbrt (exp (* (log PI) 3.0))))))))))))

C

double code(double a, double b, double angle) {
	double t_0 = sin(((angle * ((double) M_PI)) / 180.0)) * (b + a);
	double tmp;
	if ((angle / 180.0) <= -2e+68) {
		tmp = 2.0 * (t_0 * ((b - a) * cos(((angle * ((double) M_PI)) * 0.005555555555555556))));
	} else {
		tmp = 2.0 * (t_0 * ((b - a) * cos((0.005555555555555556 * (angle * cbrt(exp((log(((double) M_PI)) * 3.0))))))));
	}
	return tmp;
}

Java

public static double code(double a, double b, double angle) {
	double t_0 = Math.sin(((angle * Math.PI) / 180.0)) * (b + a);
	double tmp;
	if ((angle / 180.0) <= -2e+68) {
		tmp = 2.0 * (t_0 * ((b - a) * Math.cos(((angle * Math.PI) * 0.005555555555555556))));
	} else {
		tmp = 2.0 * (t_0 * ((b - a) * Math.cos((0.005555555555555556 * (angle * Math.cbrt(Math.exp((Math.log(Math.PI) * 3.0))))))));
	}
	return tmp;
}

Julia

function code(a, b, angle)
	t_0 = Float64(sin(Float64(Float64(angle * pi) / 180.0)) * Float64(b + a))
	tmp = 0.0
	if (Float64(angle / 180.0) <= -2e+68)
		tmp = Float64(2.0 * Float64(t_0 * Float64(Float64(b - a) * cos(Float64(Float64(angle * pi) * 0.005555555555555556)))));
	else
		tmp = Float64(2.0 * Float64(t_0 * Float64(Float64(b - a) * cos(Float64(0.005555555555555556 * Float64(angle * cbrt(exp(Float64(log(pi) * 3.0)))))))));
	end
	return tmp
end

Wolfram

code[a_, b_, angle_] := Block[{t$95$0 = N[(N[Sin[N[(N[(angle * Pi), $MachinePrecision] / 180.0), $MachinePrecision]], $MachinePrecision] * N[(b + a), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[(angle / 180.0), $MachinePrecision], -2e+68], N[(2.0 * N[(t$95$0 * N[(N[(b - a), $MachinePrecision] * N[Cos[N[(N[(angle * Pi), $MachinePrecision] * 0.005555555555555556), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(2.0 * N[(t$95$0 * N[(N[(b - a), $MachinePrecision] * N[Cos[N[(0.005555555555555556 * N[(angle * N[Power[N[Exp[N[(N[Log[Pi], $MachinePrecision] * 3.0), $MachinePrecision]], $MachinePrecision], 1/3], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 angle 180) < -1.99999999999999991e68

   1. Initial program 24.3%

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

      1. associate-*l* 24.3%

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

      2. unpow2 24.3%

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

      3. unpow2 24.3%

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

      4. difference-of-squares 27.9%

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

   3. Simplified 27.9%

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

   4. Taylor expanded in angle around inf 35.3%

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

      1. associate-*r* 35.3%

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

      2. *-commutative 35.3%

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

      3. +-commutative 35.3%

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

   6. Simplified 35.3%

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

      1. associate-*r* 34.8%

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

      2. *-commutative 34.8%

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

      3. metadata-eval 34.8%

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

      4. div-inv 32.6%

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

      5. *-commutative 32.6%

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

      6. associate-*r/ 38.8%

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

      7. *-commutative 38.8%

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

   8. Applied egg-rr 38.8%

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

   if -1.99999999999999991e68 < (/.f64 angle 180)

   1. Initial program 56.1%

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

      1. associate-*l* 56.1%

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

      2. unpow2 56.1%

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

      3. unpow2 56.1%

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

      4. difference-of-squares 59.8%

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

   3. Simplified 59.8%

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

   4. Taylor expanded in angle around inf 73.0%

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

      1. associate-*r* 73.0%

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

      2. *-commutative 73.0%

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

      3. +-commutative 73.0%

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

   6. Simplified 73.0%

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

      1. associate-*r* 73.3%

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

      2. *-commutative 73.3%

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

      3. metadata-eval 73.3%

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

      4. div-inv 74.9%

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

      5. *-commutative 74.9%

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

      6. associate-*r/ 73.2%

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

      7. *-commutative 73.2%

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

   8. Applied egg-rr 73.2%

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

      1. add-cbrt-cube 76.3%

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

      2. pow3 76.3%

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

   10. Applied egg-rr 76.3%

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

       1. pow-to-exp 77.1%

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

   12. Applied egg-rr 77.1%

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

4. Final simplification 68.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{angle}{180} \leq -2 \cdot 10^{+68}:\\ \;\;\;\;2 \cdot \left(\left(\sin \left(\frac{angle \cdot \pi}{180}\right) \cdot \left(b + a\right)\right) \cdot \left(\left(b - a\right) \cdot \cos \left(\left(angle \cdot \pi\right) \cdot 0.005555555555555556\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \left(\left(\sin \left(\frac{angle \cdot \pi}{180}\right) \cdot \left(b + a\right)\right) \cdot \left(\left(b - a\right) \cdot \cos \left(0.005555555555555556 \cdot \left(angle \cdot \sqrt[3]{e^{\log \pi \cdot 3}}\right)\right)\right)\right)\\ \end{array} \]

Alternative 2: 67.1% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \sin \left(\frac{angle \cdot \pi}{180}\right) \cdot \left(b + a\right)\\ t_1 := \frac{angle}{180} \cdot \pi\\ \mathbf{if}\;\left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin t_1\right) \cdot \cos t_1 \leq 2 \cdot 10^{-78}:\\ \;\;\;\;2 \cdot \left(t_0 \cdot \left(\left(b - a\right) \cdot \cos \left(\left(angle \cdot \pi\right) \cdot 0.005555555555555556\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \left(t_0 \cdot \left(\left(b - a\right) \cdot \cos \left(0.005555555555555556 \cdot \left(angle \cdot \sqrt[3]{{\pi}^{3}}\right)\right)\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (a b angle)
 :precision binary64
 (let* ((t_0 (* (sin (/ (* angle PI) 180.0)) (+ b a)))
        (t_1 (* (/ angle 180.0) PI)))
   (if (<=
        (* (* (* 2.0 (- (pow b 2.0) (pow a 2.0))) (sin t_1)) (cos t_1))
        2e-78)
     (* 2.0 (* t_0 (* (- b a) (cos (* (* angle PI) 0.005555555555555556)))))
     (*
      2.0
      (*
       t_0
       (*
        (- b a)
        (cos (* 0.005555555555555556 (* angle (cbrt (pow PI 3.0)))))))))))

C

double code(double a, double b, double angle) {
	double t_0 = sin(((angle * ((double) M_PI)) / 180.0)) * (b + a);
	double t_1 = (angle / 180.0) * ((double) M_PI);
	double tmp;
	if ((((2.0 * (pow(b, 2.0) - pow(a, 2.0))) * sin(t_1)) * cos(t_1)) <= 2e-78) {
		tmp = 2.0 * (t_0 * ((b - a) * cos(((angle * ((double) M_PI)) * 0.005555555555555556))));
	} else {
		tmp = 2.0 * (t_0 * ((b - a) * cos((0.005555555555555556 * (angle * cbrt(pow(((double) M_PI), 3.0)))))));
	}
	return tmp;
}

Java

public static double code(double a, double b, double angle) {
	double t_0 = Math.sin(((angle * Math.PI) / 180.0)) * (b + a);
	double t_1 = (angle / 180.0) * Math.PI;
	double tmp;
	if ((((2.0 * (Math.pow(b, 2.0) - Math.pow(a, 2.0))) * Math.sin(t_1)) * Math.cos(t_1)) <= 2e-78) {
		tmp = 2.0 * (t_0 * ((b - a) * Math.cos(((angle * Math.PI) * 0.005555555555555556))));
	} else {
		tmp = 2.0 * (t_0 * ((b - a) * Math.cos((0.005555555555555556 * (angle * Math.cbrt(Math.pow(Math.PI, 3.0)))))));
	}
	return tmp;
}

Julia

function code(a, b, angle)
	t_0 = Float64(sin(Float64(Float64(angle * pi) / 180.0)) * Float64(b + a))
	t_1 = Float64(Float64(angle / 180.0) * pi)
	tmp = 0.0
	if (Float64(Float64(Float64(2.0 * Float64((b ^ 2.0) - (a ^ 2.0))) * sin(t_1)) * cos(t_1)) <= 2e-78)
		tmp = Float64(2.0 * Float64(t_0 * Float64(Float64(b - a) * cos(Float64(Float64(angle * pi) * 0.005555555555555556)))));
	else
		tmp = Float64(2.0 * Float64(t_0 * Float64(Float64(b - a) * cos(Float64(0.005555555555555556 * Float64(angle * cbrt((pi ^ 3.0))))))));
	end
	return tmp
end

Wolfram

code[a_, b_, angle_] := Block[{t$95$0 = N[(N[Sin[N[(N[(angle * Pi), $MachinePrecision] / 180.0), $MachinePrecision]], $MachinePrecision] * N[(b + a), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[(angle / 180.0), $MachinePrecision] * Pi), $MachinePrecision]}, If[LessEqual[N[(N[(N[(2.0 * N[(N[Power[b, 2.0], $MachinePrecision] - N[Power[a, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[Sin[t$95$1], $MachinePrecision]), $MachinePrecision] * N[Cos[t$95$1], $MachinePrecision]), $MachinePrecision], 2e-78], N[(2.0 * N[(t$95$0 * N[(N[(b - a), $MachinePrecision] * N[Cos[N[(N[(angle * Pi), $MachinePrecision] * 0.005555555555555556), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(2.0 * N[(t$95$0 * N[(N[(b - a), $MachinePrecision] * N[Cos[N[(0.005555555555555556 * N[(angle * N[Power[N[Power[Pi, 3.0], $MachinePrecision], 1/3], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 (*.f64 (*.f64 2 (-.f64 (pow.f64 b 2) (pow.f64 a 2))) (sin.f64 (*.f64 (PI.f64) (/.f64 angle 180)))) (cos.f64 (*.f64 (PI.f64) (/.f64 angle 180)))) < 2e-78

   1. Initial program 58.4%

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

      1. associate-*l* 58.4%

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

      2. unpow2 58.4%

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

      3. unpow2 58.4%

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

      4. difference-of-squares 58.4%

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

   3. Simplified 58.4%

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

   4. Taylor expanded in angle around inf 67.5%

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

      1. associate-*r* 67.5%

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

      2. *-commutative 67.5%

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

      3. +-commutative 67.5%

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

   6. Simplified 67.5%

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

      1. associate-*r* 67.4%

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

      2. *-commutative 67.4%

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

      3. metadata-eval 67.4%

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

      4. div-inv 68.0%

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

      5. *-commutative 68.0%

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

      6. associate-*r/ 69.3%

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

      7. *-commutative 69.3%

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

   8. Applied egg-rr 69.3%

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

   if 2e-78 < (*.f64 (*.f64 (*.f64 2 (-.f64 (pow.f64 b 2) (pow.f64 a 2))) (sin.f64 (*.f64 (PI.f64) (/.f64 angle 180)))) (cos.f64 (*.f64 (PI.f64) (/.f64 angle 180))))

   1. Initial program 34.8%

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

      1. associate-*l* 34.8%

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

      2. unpow2 34.8%

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

      3. unpow2 34.8%

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

      4. difference-of-squares 44.1%

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

   3. Simplified 44.1%

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

   4. Taylor expanded in angle around inf 60.4%

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

      1. associate-*r* 60.4%

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

      2. *-commutative 60.4%

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

      3. +-commutative 60.4%

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

   6. Simplified 60.4%

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

      1. associate-*r* 60.9%

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

      2. *-commutative 60.9%

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

      3. metadata-eval 60.9%

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

      4. div-inv 61.9%

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

      5. *-commutative 61.9%

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

      6. associate-*r/ 59.8%

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

      7. *-commutative 59.8%

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

   8. Applied egg-rr 59.8%

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

      1. add-cbrt-cube 66.6%

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

      2. pow3 66.6%

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

   10. Applied egg-rr 66.6%

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

4. Final simplification 68.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right) \cdot \cos \left(\frac{angle}{180} \cdot \pi\right) \leq 2 \cdot 10^{-78}:\\ \;\;\;\;2 \cdot \left(\left(\sin \left(\frac{angle \cdot \pi}{180}\right) \cdot \left(b + a\right)\right) \cdot \left(\left(b - a\right) \cdot \cos \left(\left(angle \cdot \pi\right) \cdot 0.005555555555555556\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \left(\left(\sin \left(\frac{angle \cdot \pi}{180}\right) \cdot \left(b + a\right)\right) \cdot \left(\left(b - a\right) \cdot \cos \left(0.005555555555555556 \cdot \left(angle \cdot \sqrt[3]{{\pi}^{3}}\right)\right)\right)\right)\\ \end{array} \]

Alternative 3: 67.5% accurate, 1.5× speedup

Math

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

FPCore

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

C

double code(double a, double b, double angle) {
	double t_0 = (angle * ((double) M_PI)) * 0.005555555555555556;
	return 2.0 * (((b - a) * cos(t_0)) * ((b + a) * sin(t_0)));
}

Java

public static double code(double a, double b, double angle) {
	double t_0 = (angle * Math.PI) * 0.005555555555555556;
	return 2.0 * (((b - a) * Math.cos(t_0)) * ((b + a) * Math.sin(t_0)));
}

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 49.2%

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

   1. associate-*l* 49.2%

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

   2. unpow2 49.2%

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

   3. unpow2 49.2%

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

   4. difference-of-squares 52.8%

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

3. Simplified 52.8%

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

4. Taylor expanded in angle around inf 64.7%

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

   1. associate-*r* 64.7%

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

   2. *-commutative 64.7%

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

   3. +-commutative 64.7%

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

6. Simplified 64.7%

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

7. Final simplification 64.7%

   \[\leadsto 2 \cdot \left(\left(\left(b - a\right) \cdot \cos \left(\left(angle \cdot \pi\right) \cdot 0.005555555555555556\right)\right) \cdot \left(\left(b + a\right) \cdot \sin \left(\left(angle \cdot \pi\right) \cdot 0.005555555555555556\right)\right)\right) \]

Alternative 4: 67.4% accurate, 1.5× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 49.2%

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

   1. associate-*l* 49.2%

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

   2. unpow2 49.2%

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

   3. unpow2 49.2%

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

   4. difference-of-squares 52.8%

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

3. Simplified 52.8%

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

4. Taylor expanded in angle around inf 64.7%

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

   1. associate-*r* 64.7%

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

   2. *-commutative 64.7%

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

   3. +-commutative 64.7%

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

6. Simplified 64.7%

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

   1. associate-*r* 64.9%

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

   2. *-commutative 64.9%

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

   3. metadata-eval 64.9%

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

   4. div-inv 65.6%

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

   5. *-commutative 65.6%

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

   6. associate-*r/ 65.6%

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

   7. *-commutative 65.6%

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

8. Applied egg-rr 65.6%

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

9. Final simplification 65.6%

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

Alternative 5: 65.8% accurate, 2.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;a \leq 3.9 \cdot 10^{+215}:\\ \;\;\;\;2 \cdot \left(\left(b - a\right) \cdot \left(\left(b + a\right) \cdot \sin \left(\left(angle \cdot \pi\right) \cdot 0.005555555555555556\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;0.011111111111111112 \cdot \left(\left(\pi \cdot \left(b + a\right)\right) \cdot \left(angle \cdot \left(b - a\right)\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (a b angle)
 :precision binary64
 (if (<= a 3.9e+215)
   (* 2.0 (* (- b a) (* (+ b a) (sin (* (* angle PI) 0.005555555555555556)))))
   (* 0.011111111111111112 (* (* PI (+ b a)) (* angle (- b a))))))

C

double code(double a, double b, double angle) {
	double tmp;
	if (a <= 3.9e+215) {
		tmp = 2.0 * ((b - a) * ((b + a) * sin(((angle * ((double) M_PI)) * 0.005555555555555556))));
	} else {
		tmp = 0.011111111111111112 * ((((double) M_PI) * (b + a)) * (angle * (b - a)));
	}
	return tmp;
}

Java

public static double code(double a, double b, double angle) {
	double tmp;
	if (a <= 3.9e+215) {
		tmp = 2.0 * ((b - a) * ((b + a) * Math.sin(((angle * Math.PI) * 0.005555555555555556))));
	} else {
		tmp = 0.011111111111111112 * ((Math.PI * (b + a)) * (angle * (b - a)));
	}
	return tmp;
}

Python

def code(a, b, angle):
	tmp = 0
	if a <= 3.9e+215:
		tmp = 2.0 * ((b - a) * ((b + a) * math.sin(((angle * math.pi) * 0.005555555555555556))))
	else:
		tmp = 0.011111111111111112 * ((math.pi * (b + a)) * (angle * (b - a)))
	return tmp

Julia

function code(a, b, angle)
	tmp = 0.0
	if (a <= 3.9e+215)
		tmp = Float64(2.0 * Float64(Float64(b - a) * Float64(Float64(b + a) * sin(Float64(Float64(angle * pi) * 0.005555555555555556)))));
	else
		tmp = Float64(0.011111111111111112 * Float64(Float64(pi * Float64(b + a)) * Float64(angle * Float64(b - a))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, angle)
	tmp = 0.0;
	if (a <= 3.9e+215)
		tmp = 2.0 * ((b - a) * ((b + a) * sin(((angle * pi) * 0.005555555555555556))));
	else
		tmp = 0.011111111111111112 * ((pi * (b + a)) * (angle * (b - a)));
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, angle_] := If[LessEqual[a, 3.9e+215], N[(2.0 * N[(N[(b - a), $MachinePrecision] * N[(N[(b + a), $MachinePrecision] * N[Sin[N[(N[(angle * Pi), $MachinePrecision] * 0.005555555555555556), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(0.011111111111111112 * N[(N[(Pi * N[(b + a), $MachinePrecision]), $MachinePrecision] * N[(angle * N[(b - a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if a < 3.89999999999999965e215

   1. Initial program 48.2%

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

      1. associate-*l* 48.2%

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

      2. unpow2 48.2%

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

      3. unpow2 48.2%

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

      4. difference-of-squares 51.7%

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

   3. Simplified 51.7%

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

   4. Taylor expanded in angle around inf 63.2%

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

      1. associate-*r* 63.2%

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

      2. *-commutative 63.2%

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

      3. +-commutative 63.2%

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

   6. Simplified 63.2%

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

   7. Taylor expanded in angle around 0 62.9%

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

   if 3.89999999999999965e215 < a

   1. Initial program 63.3%

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

      1. associate-*l* 63.3%

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

      2. unpow2 63.3%

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

      3. unpow2 63.3%

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

      4. difference-of-squares 69.5%

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

   3. Simplified 69.5%

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

   4. Taylor expanded in angle around 0 69.5%

      \[\leadsto \color{blue}{0.011111111111111112 \cdot \left(angle \cdot \left(\left(b - a\right) \cdot \left(\pi \cdot \left(a + b\right)\right)\right)\right)} \]
   5. Step-by-step derivation

      1. associate-*r* 75.1%

         \[\leadsto 0.011111111111111112 \cdot \color{blue}{\left(\left(angle \cdot \left(b - a\right)\right) \cdot \left(\pi \cdot \left(a + b\right)\right)\right)} \]

      2. *-commutative 75.1%

         \[\leadsto 0.011111111111111112 \cdot \color{blue}{\left(\left(\pi \cdot \left(a + b\right)\right) \cdot \left(angle \cdot \left(b - a\right)\right)\right)} \]

      3. +-commutative 75.1%

         \[\leadsto 0.011111111111111112 \cdot \left(\left(\pi \cdot \color{blue}{\left(b + a\right)}\right) \cdot \left(angle \cdot \left(b - a\right)\right)\right) \]

   6. Simplified 75.1%

      \[\leadsto \color{blue}{0.011111111111111112 \cdot \left(\left(\pi \cdot \left(b + a\right)\right) \cdot \left(angle \cdot \left(b - a\right)\right)\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 63.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \leq 3.9 \cdot 10^{+215}:\\ \;\;\;\;2 \cdot \left(\left(b - a\right) \cdot \left(\left(b + a\right) \cdot \sin \left(\left(angle \cdot \pi\right) \cdot 0.005555555555555556\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;0.011111111111111112 \cdot \left(\left(\pi \cdot \left(b + a\right)\right) \cdot \left(angle \cdot \left(b - a\right)\right)\right)\\ \end{array} \]

Alternative 6: 53.4% accurate, 2.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;a \leq 5.4 \cdot 10^{-94}:\\ \;\;\;\;b \cdot \left(b \cdot \sin \left(\pi \cdot \left(angle \cdot 0.011111111111111112\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;0.011111111111111112 \cdot \left(\left(\pi \cdot \left(b + a\right)\right) \cdot \left(angle \cdot \left(b - a\right)\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (a b angle)
 :precision binary64
 (if (<= a 5.4e-94)
   (* b (* b (sin (* PI (* angle 0.011111111111111112)))))
   (* 0.011111111111111112 (* (* PI (+ b a)) (* angle (- b a))))))

C

double code(double a, double b, double angle) {
	double tmp;
	if (a <= 5.4e-94) {
		tmp = b * (b * sin((((double) M_PI) * (angle * 0.011111111111111112))));
	} else {
		tmp = 0.011111111111111112 * ((((double) M_PI) * (b + a)) * (angle * (b - a)));
	}
	return tmp;
}

Java

public static double code(double a, double b, double angle) {
	double tmp;
	if (a <= 5.4e-94) {
		tmp = b * (b * Math.sin((Math.PI * (angle * 0.011111111111111112))));
	} else {
		tmp = 0.011111111111111112 * ((Math.PI * (b + a)) * (angle * (b - a)));
	}
	return tmp;
}

Python

def code(a, b, angle):
	tmp = 0
	if a <= 5.4e-94:
		tmp = b * (b * math.sin((math.pi * (angle * 0.011111111111111112))))
	else:
		tmp = 0.011111111111111112 * ((math.pi * (b + a)) * (angle * (b - a)))
	return tmp

Julia

function code(a, b, angle)
	tmp = 0.0
	if (a <= 5.4e-94)
		tmp = Float64(b * Float64(b * sin(Float64(pi * Float64(angle * 0.011111111111111112)))));
	else
		tmp = Float64(0.011111111111111112 * Float64(Float64(pi * Float64(b + a)) * Float64(angle * Float64(b - a))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, angle)
	tmp = 0.0;
	if (a <= 5.4e-94)
		tmp = b * (b * sin((pi * (angle * 0.011111111111111112))));
	else
		tmp = 0.011111111111111112 * ((pi * (b + a)) * (angle * (b - a)));
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, angle_] := If[LessEqual[a, 5.4e-94], N[(b * N[(b * N[Sin[N[(Pi * N[(angle * 0.011111111111111112), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(0.011111111111111112 * N[(N[(Pi * N[(b + a), $MachinePrecision]), $MachinePrecision] * N[(angle * N[(b - a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if a < 5.4000000000000002e-94

   1. Initial program 51.0%

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

      1. *-commutative 51.0%

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

      2. associate-*l* 51.0%

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

      3. unpow2 51.0%

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

      4. fma-neg 51.6%

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

      5. unpow2 51.6%

         \[\leadsto \sin \left(\pi \cdot \frac{angle}{180}\right) \cdot \left(\left(2 \cdot \mathsf{fma}\left(b, b, -\color{blue}{a \cdot a}\right)\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right)\right) \]

   3. Simplified 51.6%

      \[\leadsto \color{blue}{\sin \left(\pi \cdot \frac{angle}{180}\right) \cdot \left(\left(2 \cdot \mathsf{fma}\left(b, b, -a \cdot a\right)\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right)\right)} \]

   5. Applied egg-rr 32.3%

      \[\leadsto \color{blue}{\sqrt{{\left(\left(2 \cdot {\left(b + a\right)}^{2}\right) \cdot \left(\left(\sin 0 + \sin \left(\left(\pi \cdot \left(angle \cdot 0.005555555555555556\right)\right) \cdot 2\right)\right) \cdot 0.5\right)\right)}^{2}}} \]

   6. Taylor expanded in b around inf 26.9%

      \[\leadsto \sqrt{{\color{blue}{\left(\sin \left(0.011111111111111112 \cdot \left(angle \cdot \pi\right)\right) \cdot {b}^{2}\right)}}^{2}} \]
   7. Step-by-step derivation

      1. *-commutative 26.9%

         \[\leadsto \sqrt{{\color{blue}{\left({b}^{2} \cdot \sin \left(0.011111111111111112 \cdot \left(angle \cdot \pi\right)\right)\right)}}^{2}} \]

      2. unpow2 26.9%

         \[\leadsto \sqrt{{\left(\color{blue}{\left(b \cdot b\right)} \cdot \sin \left(0.011111111111111112 \cdot \left(angle \cdot \pi\right)\right)\right)}^{2}} \]

      3. associate-*r* 26.8%

         \[\leadsto \sqrt{{\left(\left(b \cdot b\right) \cdot \sin \color{blue}{\left(\left(0.011111111111111112 \cdot angle\right) \cdot \pi\right)}\right)}^{2}} \]

      4. *-commutative 26.8%

         \[\leadsto \sqrt{{\left(\left(b \cdot b\right) \cdot \sin \color{blue}{\left(\pi \cdot \left(0.011111111111111112 \cdot angle\right)\right)}\right)}^{2}} \]

      5. *-commutative 26.8%

         \[\leadsto \sqrt{{\left(\left(b \cdot b\right) \cdot \sin \left(\pi \cdot \color{blue}{\left(angle \cdot 0.011111111111111112\right)}\right)\right)}^{2}} \]

   8. Simplified 26.8%

      \[\leadsto \sqrt{{\color{blue}{\left(\left(b \cdot b\right) \cdot \sin \left(\pi \cdot \left(angle \cdot 0.011111111111111112\right)\right)\right)}}^{2}} \]

   9. Taylor expanded in b around 0 42.5%

      \[\leadsto \color{blue}{\sin \left(0.011111111111111112 \cdot \left(angle \cdot \pi\right)\right) \cdot {b}^{2}} \]
   10. Step-by-step derivation

       1. *-commutative 42.5%

          \[\leadsto \color{blue}{{b}^{2} \cdot \sin \left(0.011111111111111112 \cdot \left(angle \cdot \pi\right)\right)} \]

       2. unpow2 42.5%

          \[\leadsto \color{blue}{\left(b \cdot b\right)} \cdot \sin \left(0.011111111111111112 \cdot \left(angle \cdot \pi\right)\right) \]

       3. *-commutative 42.5%

          \[\leadsto \left(b \cdot b\right) \cdot \sin \color{blue}{\left(\left(angle \cdot \pi\right) \cdot 0.011111111111111112\right)} \]

       4. *-commutative 42.5%

          \[\leadsto \left(b \cdot b\right) \cdot \sin \left(\color{blue}{\left(\pi \cdot angle\right)} \cdot 0.011111111111111112\right) \]

       5. associate-*r* 42.5%

          \[\leadsto \left(b \cdot b\right) \cdot \sin \color{blue}{\left(\pi \cdot \left(angle \cdot 0.011111111111111112\right)\right)} \]

       6. associate-*r* 47.4%

          \[\leadsto \color{blue}{b \cdot \left(b \cdot \sin \left(\pi \cdot \left(angle \cdot 0.011111111111111112\right)\right)\right)} \]

   11. Simplified 47.4%

       \[\leadsto \color{blue}{b \cdot \left(b \cdot \sin \left(\pi \cdot \left(angle \cdot 0.011111111111111112\right)\right)\right)} \]

   if 5.4000000000000002e-94 < a

   1. Initial program 45.5%

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

      1. associate-*l* 45.5%

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

      2. unpow2 45.5%

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

      3. unpow2 45.5%

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

      4. difference-of-squares 51.6%

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

   3. Simplified 51.6%

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

   4. Taylor expanded in angle around 0 51.5%

      \[\leadsto \color{blue}{0.011111111111111112 \cdot \left(angle \cdot \left(\left(b - a\right) \cdot \left(\pi \cdot \left(a + b\right)\right)\right)\right)} \]
   5. Step-by-step derivation

      1. associate-*r* 64.8%

         \[\leadsto 0.011111111111111112 \cdot \color{blue}{\left(\left(angle \cdot \left(b - a\right)\right) \cdot \left(\pi \cdot \left(a + b\right)\right)\right)} \]

      2. *-commutative 64.8%

         \[\leadsto 0.011111111111111112 \cdot \color{blue}{\left(\left(\pi \cdot \left(a + b\right)\right) \cdot \left(angle \cdot \left(b - a\right)\right)\right)} \]

      3. +-commutative 64.8%

         \[\leadsto 0.011111111111111112 \cdot \left(\left(\pi \cdot \color{blue}{\left(b + a\right)}\right) \cdot \left(angle \cdot \left(b - a\right)\right)\right) \]

   6. Simplified 64.8%

      \[\leadsto \color{blue}{0.011111111111111112 \cdot \left(\left(\pi \cdot \left(b + a\right)\right) \cdot \left(angle \cdot \left(b - a\right)\right)\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 53.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \leq 5.4 \cdot 10^{-94}:\\ \;\;\;\;b \cdot \left(b \cdot \sin \left(\pi \cdot \left(angle \cdot 0.011111111111111112\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;0.011111111111111112 \cdot \left(\left(\pi \cdot \left(b + a\right)\right) \cdot \left(angle \cdot \left(b - a\right)\right)\right)\\ \end{array} \]

Alternative 7: 42.9% accurate, 5.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;a \leq 3.5 \cdot 10^{-39}:\\ \;\;\;\;angle \cdot \left(0.011111111111111112 \cdot \left(\pi \cdot \left(b \cdot b\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;0.011111111111111112 \cdot \left(angle \cdot \left(\left(b - a\right) \cdot \left(\pi \cdot a\right)\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (a b angle)
 :precision binary64
 (if (<= a 3.5e-39)
   (* angle (* 0.011111111111111112 (* PI (* b b))))
   (* 0.011111111111111112 (* angle (* (- b a) (* PI a))))))

C

double code(double a, double b, double angle) {
	double tmp;
	if (a <= 3.5e-39) {
		tmp = angle * (0.011111111111111112 * (((double) M_PI) * (b * b)));
	} else {
		tmp = 0.011111111111111112 * (angle * ((b - a) * (((double) M_PI) * a)));
	}
	return tmp;
}

Java

public static double code(double a, double b, double angle) {
	double tmp;
	if (a <= 3.5e-39) {
		tmp = angle * (0.011111111111111112 * (Math.PI * (b * b)));
	} else {
		tmp = 0.011111111111111112 * (angle * ((b - a) * (Math.PI * a)));
	}
	return tmp;
}

Python

def code(a, b, angle):
	tmp = 0
	if a <= 3.5e-39:
		tmp = angle * (0.011111111111111112 * (math.pi * (b * b)))
	else:
		tmp = 0.011111111111111112 * (angle * ((b - a) * (math.pi * a)))
	return tmp

Julia

function code(a, b, angle)
	tmp = 0.0
	if (a <= 3.5e-39)
		tmp = Float64(angle * Float64(0.011111111111111112 * Float64(pi * Float64(b * b))));
	else
		tmp = Float64(0.011111111111111112 * Float64(angle * Float64(Float64(b - a) * Float64(pi * a))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, angle)
	tmp = 0.0;
	if (a <= 3.5e-39)
		tmp = angle * (0.011111111111111112 * (pi * (b * b)));
	else
		tmp = 0.011111111111111112 * (angle * ((b - a) * (pi * a)));
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, angle_] := If[LessEqual[a, 3.5e-39], N[(angle * N[(0.011111111111111112 * N[(Pi * N[(b * b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(0.011111111111111112 * N[(angle * N[(N[(b - a), $MachinePrecision] * N[(Pi * a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if a < 3.5e-39

   1. Initial program 51.1%

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

      1. *-commutative 51.1%

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

      2. associate-*l* 51.1%

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

      3. unpow2 51.1%

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

      4. fma-neg 51.7%

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

      5. unpow2 51.7%

         \[\leadsto \sin \left(\pi \cdot \frac{angle}{180}\right) \cdot \left(\left(2 \cdot \mathsf{fma}\left(b, b, -\color{blue}{a \cdot a}\right)\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right)\right) \]

   3. Simplified 51.7%

      \[\leadsto \color{blue}{\sin \left(\pi \cdot \frac{angle}{180}\right) \cdot \left(\left(2 \cdot \mathsf{fma}\left(b, b, -a \cdot a\right)\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right)\right)} \]

   5. Applied egg-rr 31.3%

      \[\leadsto \color{blue}{\sqrt{{\left(\left(2 \cdot {\left(b + a\right)}^{2}\right) \cdot \left(\left(\sin 0 + \sin \left(\left(\pi \cdot \left(angle \cdot 0.005555555555555556\right)\right) \cdot 2\right)\right) \cdot 0.5\right)\right)}^{2}}} \]

   6. Taylor expanded in b around inf 26.1%

      \[\leadsto \sqrt{{\color{blue}{\left(\sin \left(0.011111111111111112 \cdot \left(angle \cdot \pi\right)\right) \cdot {b}^{2}\right)}}^{2}} \]
   7. Step-by-step derivation

      1. *-commutative 26.1%

         \[\leadsto \sqrt{{\color{blue}{\left({b}^{2} \cdot \sin \left(0.011111111111111112 \cdot \left(angle \cdot \pi\right)\right)\right)}}^{2}} \]

      2. unpow2 26.1%

         \[\leadsto \sqrt{{\left(\color{blue}{\left(b \cdot b\right)} \cdot \sin \left(0.011111111111111112 \cdot \left(angle \cdot \pi\right)\right)\right)}^{2}} \]

      3. associate-*r* 26.0%

         \[\leadsto \sqrt{{\left(\left(b \cdot b\right) \cdot \sin \color{blue}{\left(\left(0.011111111111111112 \cdot angle\right) \cdot \pi\right)}\right)}^{2}} \]

      4. *-commutative 26.0%

         \[\leadsto \sqrt{{\left(\left(b \cdot b\right) \cdot \sin \color{blue}{\left(\pi \cdot \left(0.011111111111111112 \cdot angle\right)\right)}\right)}^{2}} \]

      5. *-commutative 26.0%

         \[\leadsto \sqrt{{\left(\left(b \cdot b\right) \cdot \sin \left(\pi \cdot \color{blue}{\left(angle \cdot 0.011111111111111112\right)}\right)\right)}^{2}} \]

   8. Simplified 26.0%

      \[\leadsto \sqrt{{\color{blue}{\left(\left(b \cdot b\right) \cdot \sin \left(\pi \cdot \left(angle \cdot 0.011111111111111112\right)\right)\right)}}^{2}} \]

   9. Taylor expanded in angle around 0 37.5%

      \[\leadsto \color{blue}{0.011111111111111112 \cdot \left(angle \cdot \left({b}^{2} \cdot \pi\right)\right)} \]
   10. Step-by-step derivation

       1. *-commutative 37.5%

          \[\leadsto \color{blue}{\left(angle \cdot \left({b}^{2} \cdot \pi\right)\right) \cdot 0.011111111111111112} \]

       2. associate-*l* 38.0%

          \[\leadsto \color{blue}{angle \cdot \left(\left({b}^{2} \cdot \pi\right) \cdot 0.011111111111111112\right)} \]

       3. *-commutative 38.0%

          \[\leadsto angle \cdot \left(\color{blue}{\left(\pi \cdot {b}^{2}\right)} \cdot 0.011111111111111112\right) \]

       4. unpow2 38.0%

          \[\leadsto angle \cdot \left(\left(\pi \cdot \color{blue}{\left(b \cdot b\right)}\right) \cdot 0.011111111111111112\right) \]

   11. Simplified 38.0%

       \[\leadsto \color{blue}{angle \cdot \left(\left(\pi \cdot \left(b \cdot b\right)\right) \cdot 0.011111111111111112\right)} \]

   if 3.5e-39 < a

   1. Initial program 44.3%

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

      1. associate-*l* 44.3%

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

      2. unpow2 44.3%

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

      3. unpow2 44.3%

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

      4. difference-of-squares 51.4%

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

   3. Simplified 51.4%

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

   4. Taylor expanded in angle around 0 51.6%

      \[\leadsto \color{blue}{0.011111111111111112 \cdot \left(angle \cdot \left(\left(b - a\right) \cdot \left(\pi \cdot \left(a + b\right)\right)\right)\right)} \]

   5. Taylor expanded in a around inf 44.7%

      \[\leadsto 0.011111111111111112 \cdot \left(angle \cdot \left(\left(b - a\right) \cdot \color{blue}{\left(a \cdot \pi\right)}\right)\right) \]
   6. Step-by-step derivation

      1. *-commutative 44.7%

         \[\leadsto 0.011111111111111112 \cdot \left(angle \cdot \left(\left(b - a\right) \cdot \color{blue}{\left(\pi \cdot a\right)}\right)\right) \]

   7. Simplified 44.7%

      \[\leadsto 0.011111111111111112 \cdot \left(angle \cdot \left(\left(b - a\right) \cdot \color{blue}{\left(\pi \cdot a\right)}\right)\right) \]
3. Recombined 2 regimes into one program.

4. Final simplification 39.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \leq 3.5 \cdot 10^{-39}:\\ \;\;\;\;angle \cdot \left(0.011111111111111112 \cdot \left(\pi \cdot \left(b \cdot b\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;0.011111111111111112 \cdot \left(angle \cdot \left(\left(b - a\right) \cdot \left(\pi \cdot a\right)\right)\right)\\ \end{array} \]

Alternative 8: 42.8% accurate, 5.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq 550000000:\\ \;\;\;\;angle \cdot \left(\pi \cdot \left(a \cdot \left(a \cdot -0.011111111111111112\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;0.011111111111111112 \cdot \left(angle \cdot \left(\left(b - a\right) \cdot \left(\pi \cdot b\right)\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (a b angle)
 :precision binary64
 (if (<= b 550000000.0)
   (* angle (* PI (* a (* a -0.011111111111111112))))
   (* 0.011111111111111112 (* angle (* (- b a) (* PI b))))))

C

double code(double a, double b, double angle) {
	double tmp;
	if (b <= 550000000.0) {
		tmp = angle * (((double) M_PI) * (a * (a * -0.011111111111111112)));
	} else {
		tmp = 0.011111111111111112 * (angle * ((b - a) * (((double) M_PI) * b)));
	}
	return tmp;
}

Java

public static double code(double a, double b, double angle) {
	double tmp;
	if (b <= 550000000.0) {
		tmp = angle * (Math.PI * (a * (a * -0.011111111111111112)));
	} else {
		tmp = 0.011111111111111112 * (angle * ((b - a) * (Math.PI * b)));
	}
	return tmp;
}

Python

def code(a, b, angle):
	tmp = 0
	if b <= 550000000.0:
		tmp = angle * (math.pi * (a * (a * -0.011111111111111112)))
	else:
		tmp = 0.011111111111111112 * (angle * ((b - a) * (math.pi * b)))
	return tmp

Julia

function code(a, b, angle)
	tmp = 0.0
	if (b <= 550000000.0)
		tmp = Float64(angle * Float64(pi * Float64(a * Float64(a * -0.011111111111111112))));
	else
		tmp = Float64(0.011111111111111112 * Float64(angle * Float64(Float64(b - a) * Float64(pi * b))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, angle)
	tmp = 0.0;
	if (b <= 550000000.0)
		tmp = angle * (pi * (a * (a * -0.011111111111111112)));
	else
		tmp = 0.011111111111111112 * (angle * ((b - a) * (pi * b)));
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, angle_] := If[LessEqual[b, 550000000.0], N[(angle * N[(Pi * N[(a * N[(a * -0.011111111111111112), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(0.011111111111111112 * N[(angle * N[(N[(b - a), $MachinePrecision] * N[(Pi * b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b < 5.5e8

   1. Initial program 51.0%

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

      1. associate-*l* 51.0%

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

      2. unpow2 51.0%

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

      3. unpow2 51.0%

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

      4. difference-of-squares 54.2%

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

   3. Simplified 54.2%

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

   4. Taylor expanded in angle around 0 50.2%

      \[\leadsto \color{blue}{0.011111111111111112 \cdot \left(angle \cdot \left(\left(b - a\right) \cdot \left(\pi \cdot \left(a + b\right)\right)\right)\right)} \]

   5. Taylor expanded in b around 0 36.3%

      \[\leadsto \color{blue}{-0.011111111111111112 \cdot \left(angle \cdot \left({a}^{2} \cdot \pi\right)\right)} \]
   6. Step-by-step derivation

      1. *-commutative 36.3%

         \[\leadsto \color{blue}{\left(angle \cdot \left({a}^{2} \cdot \pi\right)\right) \cdot -0.011111111111111112} \]

      2. *-commutative 36.3%

         \[\leadsto \left(angle \cdot \color{blue}{\left(\pi \cdot {a}^{2}\right)}\right) \cdot -0.011111111111111112 \]

      3. unpow2 36.3%

         \[\leadsto \left(angle \cdot \left(\pi \cdot \color{blue}{\left(a \cdot a\right)}\right)\right) \cdot -0.011111111111111112 \]

   7. Simplified 36.3%

      \[\leadsto \color{blue}{\left(angle \cdot \left(\pi \cdot \left(a \cdot a\right)\right)\right) \cdot -0.011111111111111112} \]

   8. Taylor expanded in angle around 0 36.3%

      \[\leadsto \color{blue}{-0.011111111111111112 \cdot \left(angle \cdot \left({a}^{2} \cdot \pi\right)\right)} \]
   9. Step-by-step derivation

      1. *-commutative 36.3%

         \[\leadsto \color{blue}{\left(angle \cdot \left({a}^{2} \cdot \pi\right)\right) \cdot -0.011111111111111112} \]

      2. *-commutative 36.3%

         \[\leadsto \left(angle \cdot \color{blue}{\left(\pi \cdot {a}^{2}\right)}\right) \cdot -0.011111111111111112 \]

      3. associate-*r* 36.3%

         \[\leadsto \color{blue}{\left(\left(angle \cdot \pi\right) \cdot {a}^{2}\right)} \cdot -0.011111111111111112 \]

      4. associate-*l* 36.3%

         \[\leadsto \color{blue}{\left(angle \cdot \pi\right) \cdot \left({a}^{2} \cdot -0.011111111111111112\right)} \]

      5. associate-*r* 36.3%

         \[\leadsto \color{blue}{angle \cdot \left(\pi \cdot \left({a}^{2} \cdot -0.011111111111111112\right)\right)} \]

      6. unpow2 36.3%

         \[\leadsto angle \cdot \left(\pi \cdot \left(\color{blue}{\left(a \cdot a\right)} \cdot -0.011111111111111112\right)\right) \]

      7. associate-*l* 36.3%

         \[\leadsto angle \cdot \left(\pi \cdot \color{blue}{\left(a \cdot \left(a \cdot -0.011111111111111112\right)\right)}\right) \]

   10. Simplified 36.3%

       \[\leadsto \color{blue}{angle \cdot \left(\pi \cdot \left(a \cdot \left(a \cdot -0.011111111111111112\right)\right)\right)} \]

   if 5.5e8 < b

   1. Initial program 43.2%

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

      1. associate-*l* 43.2%

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

      2. unpow2 43.2%

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

      3. unpow2 43.2%

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

      4. difference-of-squares 48.3%

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

   3. Simplified 48.3%

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

   4. Taylor expanded in angle around 0 49.2%

      \[\leadsto \color{blue}{0.011111111111111112 \cdot \left(angle \cdot \left(\left(b - a\right) \cdot \left(\pi \cdot \left(a + b\right)\right)\right)\right)} \]

   5. Taylor expanded in a around 0 50.7%

      \[\leadsto 0.011111111111111112 \cdot \left(angle \cdot \left(\left(b - a\right) \cdot \color{blue}{\left(b \cdot \pi\right)}\right)\right) \]
   6. Step-by-step derivation

      1. *-commutative 50.7%

         \[\leadsto 0.011111111111111112 \cdot \left(angle \cdot \left(\left(b - a\right) \cdot \color{blue}{\left(\pi \cdot b\right)}\right)\right) \]

   7. Simplified 50.7%

      \[\leadsto 0.011111111111111112 \cdot \left(angle \cdot \left(\left(b - a\right) \cdot \color{blue}{\left(\pi \cdot b\right)}\right)\right) \]
3. Recombined 2 regimes into one program.

4. Final simplification 39.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq 550000000:\\ \;\;\;\;angle \cdot \left(\pi \cdot \left(a \cdot \left(a \cdot -0.011111111111111112\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;0.011111111111111112 \cdot \left(angle \cdot \left(\left(b - a\right) \cdot \left(\pi \cdot b\right)\right)\right)\\ \end{array} \]

Alternative 9: 55.1% accurate, 5.5× speedup

Math

\[\begin{array}{l} \\ 0.011111111111111112 \cdot \left(angle \cdot \left(\left(b - a\right) \cdot \left(\pi \cdot \left(b + a\right)\right)\right)\right) \end{array} \]

FPCore

(FPCore (a b angle)
 :precision binary64
 (* 0.011111111111111112 (* angle (* (- b a) (* PI (+ b a))))))

C

double code(double a, double b, double angle) {
	return 0.011111111111111112 * (angle * ((b - a) * (((double) M_PI) * (b + a))));
}

Java

public static double code(double a, double b, double angle) {
	return 0.011111111111111112 * (angle * ((b - a) * (Math.PI * (b + a))));
}

Python

def code(a, b, angle):
	return 0.011111111111111112 * (angle * ((b - a) * (math.pi * (b + a))))

Julia

function code(a, b, angle)
	return Float64(0.011111111111111112 * Float64(angle * Float64(Float64(b - a) * Float64(pi * Float64(b + a)))))
end

MATLAB

function tmp = code(a, b, angle)
	tmp = 0.011111111111111112 * (angle * ((b - a) * (pi * (b + a))));
end

Wolfram

code[a_, b_, angle_] := N[(0.011111111111111112 * N[(angle * N[(N[(b - a), $MachinePrecision] * N[(Pi * N[(b + a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 49.2%

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

   1. associate-*l* 49.2%

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

   2. unpow2 49.2%

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

   3. unpow2 49.2%

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

   4. difference-of-squares 52.8%

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

3. Simplified 52.8%

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

4. Taylor expanded in angle around 0 50.0%

   \[\leadsto \color{blue}{0.011111111111111112 \cdot \left(angle \cdot \left(\left(b - a\right) \cdot \left(\pi \cdot \left(a + b\right)\right)\right)\right)} \]

5. Final simplification 50.0%

   \[\leadsto 0.011111111111111112 \cdot \left(angle \cdot \left(\left(b - a\right) \cdot \left(\pi \cdot \left(b + a\right)\right)\right)\right) \]

Alternative 10: 62.5% accurate, 5.5× speedup

Math

\[\begin{array}{l} \\ 0.011111111111111112 \cdot \left(\left(\pi \cdot \left(b + a\right)\right) \cdot \left(angle \cdot \left(b - a\right)\right)\right) \end{array} \]

FPCore

(FPCore (a b angle)
 :precision binary64
 (* 0.011111111111111112 (* (* PI (+ b a)) (* angle (- b a)))))

C

double code(double a, double b, double angle) {
	return 0.011111111111111112 * ((((double) M_PI) * (b + a)) * (angle * (b - a)));
}

Java

public static double code(double a, double b, double angle) {
	return 0.011111111111111112 * ((Math.PI * (b + a)) * (angle * (b - a)));
}

Python

def code(a, b, angle):
	return 0.011111111111111112 * ((math.pi * (b + a)) * (angle * (b - a)))

Julia

function code(a, b, angle)
	return Float64(0.011111111111111112 * Float64(Float64(pi * Float64(b + a)) * Float64(angle * Float64(b - a))))
end

MATLAB

function tmp = code(a, b, angle)
	tmp = 0.011111111111111112 * ((pi * (b + a)) * (angle * (b - a)));
end

Wolfram

code[a_, b_, angle_] := N[(0.011111111111111112 * N[(N[(Pi * N[(b + a), $MachinePrecision]), $MachinePrecision] * N[(angle * N[(b - a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 49.2%

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

   1. associate-*l* 49.2%

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

   2. unpow2 49.2%

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

   3. unpow2 49.2%

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

   4. difference-of-squares 52.8%

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

3. Simplified 52.8%

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

4. Taylor expanded in angle around 0 50.0%

   \[\leadsto \color{blue}{0.011111111111111112 \cdot \left(angle \cdot \left(\left(b - a\right) \cdot \left(\pi \cdot \left(a + b\right)\right)\right)\right)} \]
5. Step-by-step derivation

   1. associate-*r* 58.7%

      \[\leadsto 0.011111111111111112 \cdot \color{blue}{\left(\left(angle \cdot \left(b - a\right)\right) \cdot \left(\pi \cdot \left(a + b\right)\right)\right)} \]

   2. *-commutative 58.7%

      \[\leadsto 0.011111111111111112 \cdot \color{blue}{\left(\left(\pi \cdot \left(a + b\right)\right) \cdot \left(angle \cdot \left(b - a\right)\right)\right)} \]

   3. +-commutative 58.7%

      \[\leadsto 0.011111111111111112 \cdot \left(\left(\pi \cdot \color{blue}{\left(b + a\right)}\right) \cdot \left(angle \cdot \left(b - a\right)\right)\right) \]

6. Simplified 58.7%

   \[\leadsto \color{blue}{0.011111111111111112 \cdot \left(\left(\pi \cdot \left(b + a\right)\right) \cdot \left(angle \cdot \left(b - a\right)\right)\right)} \]

7. Final simplification 58.7%

   \[\leadsto 0.011111111111111112 \cdot \left(\left(\pi \cdot \left(b + a\right)\right) \cdot \left(angle \cdot \left(b - a\right)\right)\right) \]

Alternative 11: 42.1% accurate, 5.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq 600000000:\\ \;\;\;\;angle \cdot \left(\pi \cdot \left(a \cdot \left(a \cdot -0.011111111111111112\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;angle \cdot \left(0.011111111111111112 \cdot \left(\pi \cdot \left(b \cdot b\right)\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (a b angle)
 :precision binary64
 (if (<= b 600000000.0)
   (* angle (* PI (* a (* a -0.011111111111111112))))
   (* angle (* 0.011111111111111112 (* PI (* b b))))))

C

double code(double a, double b, double angle) {
	double tmp;
	if (b <= 600000000.0) {
		tmp = angle * (((double) M_PI) * (a * (a * -0.011111111111111112)));
	} else {
		tmp = angle * (0.011111111111111112 * (((double) M_PI) * (b * b)));
	}
	return tmp;
}

Java

public static double code(double a, double b, double angle) {
	double tmp;
	if (b <= 600000000.0) {
		tmp = angle * (Math.PI * (a * (a * -0.011111111111111112)));
	} else {
		tmp = angle * (0.011111111111111112 * (Math.PI * (b * b)));
	}
	return tmp;
}

Python

def code(a, b, angle):
	tmp = 0
	if b <= 600000000.0:
		tmp = angle * (math.pi * (a * (a * -0.011111111111111112)))
	else:
		tmp = angle * (0.011111111111111112 * (math.pi * (b * b)))
	return tmp

Julia

function code(a, b, angle)
	tmp = 0.0
	if (b <= 600000000.0)
		tmp = Float64(angle * Float64(pi * Float64(a * Float64(a * -0.011111111111111112))));
	else
		tmp = Float64(angle * Float64(0.011111111111111112 * Float64(pi * Float64(b * b))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, angle)
	tmp = 0.0;
	if (b <= 600000000.0)
		tmp = angle * (pi * (a * (a * -0.011111111111111112)));
	else
		tmp = angle * (0.011111111111111112 * (pi * (b * b)));
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, angle_] := If[LessEqual[b, 600000000.0], N[(angle * N[(Pi * N[(a * N[(a * -0.011111111111111112), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(angle * N[(0.011111111111111112 * N[(Pi * N[(b * b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b < 6e8

   1. Initial program 51.0%

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

      1. associate-*l* 51.0%

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

      2. unpow2 51.0%

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

      3. unpow2 51.0%

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

      4. difference-of-squares 54.2%

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

   3. Simplified 54.2%

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

   4. Taylor expanded in angle around 0 50.2%

      \[\leadsto \color{blue}{0.011111111111111112 \cdot \left(angle \cdot \left(\left(b - a\right) \cdot \left(\pi \cdot \left(a + b\right)\right)\right)\right)} \]

   5. Taylor expanded in b around 0 36.3%

      \[\leadsto \color{blue}{-0.011111111111111112 \cdot \left(angle \cdot \left({a}^{2} \cdot \pi\right)\right)} \]
   6. Step-by-step derivation

      1. *-commutative 36.3%

         \[\leadsto \color{blue}{\left(angle \cdot \left({a}^{2} \cdot \pi\right)\right) \cdot -0.011111111111111112} \]

      2. *-commutative 36.3%

         \[\leadsto \left(angle \cdot \color{blue}{\left(\pi \cdot {a}^{2}\right)}\right) \cdot -0.011111111111111112 \]

      3. unpow2 36.3%

         \[\leadsto \left(angle \cdot \left(\pi \cdot \color{blue}{\left(a \cdot a\right)}\right)\right) \cdot -0.011111111111111112 \]

   7. Simplified 36.3%

      \[\leadsto \color{blue}{\left(angle \cdot \left(\pi \cdot \left(a \cdot a\right)\right)\right) \cdot -0.011111111111111112} \]

   8. Taylor expanded in angle around 0 36.3%

      \[\leadsto \color{blue}{-0.011111111111111112 \cdot \left(angle \cdot \left({a}^{2} \cdot \pi\right)\right)} \]
   9. Step-by-step derivation

      1. *-commutative 36.3%

         \[\leadsto \color{blue}{\left(angle \cdot \left({a}^{2} \cdot \pi\right)\right) \cdot -0.011111111111111112} \]

      2. *-commutative 36.3%

         \[\leadsto \left(angle \cdot \color{blue}{\left(\pi \cdot {a}^{2}\right)}\right) \cdot -0.011111111111111112 \]

      3. associate-*r* 36.3%

         \[\leadsto \color{blue}{\left(\left(angle \cdot \pi\right) \cdot {a}^{2}\right)} \cdot -0.011111111111111112 \]

      4. associate-*l* 36.3%

         \[\leadsto \color{blue}{\left(angle \cdot \pi\right) \cdot \left({a}^{2} \cdot -0.011111111111111112\right)} \]

      5. associate-*r* 36.3%

         \[\leadsto \color{blue}{angle \cdot \left(\pi \cdot \left({a}^{2} \cdot -0.011111111111111112\right)\right)} \]

      6. unpow2 36.3%

         \[\leadsto angle \cdot \left(\pi \cdot \left(\color{blue}{\left(a \cdot a\right)} \cdot -0.011111111111111112\right)\right) \]

      7. associate-*l* 36.3%

         \[\leadsto angle \cdot \left(\pi \cdot \color{blue}{\left(a \cdot \left(a \cdot -0.011111111111111112\right)\right)}\right) \]

   10. Simplified 36.3%

       \[\leadsto \color{blue}{angle \cdot \left(\pi \cdot \left(a \cdot \left(a \cdot -0.011111111111111112\right)\right)\right)} \]

   if 6e8 < b

   1. Initial program 43.2%

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

      1. *-commutative 43.2%

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

      2. associate-*l* 43.2%

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

      3. unpow2 43.2%

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

      4. fma-neg 46.7%

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

      5. unpow2 46.7%

         \[\leadsto \sin \left(\pi \cdot \frac{angle}{180}\right) \cdot \left(\left(2 \cdot \mathsf{fma}\left(b, b, -\color{blue}{a \cdot a}\right)\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right)\right) \]

   3. Simplified 46.7%

      \[\leadsto \color{blue}{\sin \left(\pi \cdot \frac{angle}{180}\right) \cdot \left(\left(2 \cdot \mathsf{fma}\left(b, b, -a \cdot a\right)\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right)\right)} \]

   5. Applied egg-rr 34.2%

      \[\leadsto \color{blue}{\sqrt{{\left(\left(2 \cdot {\left(b + a\right)}^{2}\right) \cdot \left(\left(\sin 0 + \sin \left(\left(\pi \cdot \left(angle \cdot 0.005555555555555556\right)\right) \cdot 2\right)\right) \cdot 0.5\right)\right)}^{2}}} \]

   6. Taylor expanded in b around inf 32.7%

      \[\leadsto \sqrt{{\color{blue}{\left(\sin \left(0.011111111111111112 \cdot \left(angle \cdot \pi\right)\right) \cdot {b}^{2}\right)}}^{2}} \]
   7. Step-by-step derivation

      1. *-commutative 32.7%

         \[\leadsto \sqrt{{\color{blue}{\left({b}^{2} \cdot \sin \left(0.011111111111111112 \cdot \left(angle \cdot \pi\right)\right)\right)}}^{2}} \]

      2. unpow2 32.7%

         \[\leadsto \sqrt{{\left(\color{blue}{\left(b \cdot b\right)} \cdot \sin \left(0.011111111111111112 \cdot \left(angle \cdot \pi\right)\right)\right)}^{2}} \]

      3. associate-*r* 32.6%

         \[\leadsto \sqrt{{\left(\left(b \cdot b\right) \cdot \sin \color{blue}{\left(\left(0.011111111111111112 \cdot angle\right) \cdot \pi\right)}\right)}^{2}} \]

      4. *-commutative 32.6%

         \[\leadsto \sqrt{{\left(\left(b \cdot b\right) \cdot \sin \color{blue}{\left(\pi \cdot \left(0.011111111111111112 \cdot angle\right)\right)}\right)}^{2}} \]

      5. *-commutative 32.6%

         \[\leadsto \sqrt{{\left(\left(b \cdot b\right) \cdot \sin \left(\pi \cdot \color{blue}{\left(angle \cdot 0.011111111111111112\right)}\right)\right)}^{2}} \]

   8. Simplified 32.6%

      \[\leadsto \sqrt{{\color{blue}{\left(\left(b \cdot b\right) \cdot \sin \left(\pi \cdot \left(angle \cdot 0.011111111111111112\right)\right)\right)}}^{2}} \]

   9. Taylor expanded in angle around 0 45.5%

      \[\leadsto \color{blue}{0.011111111111111112 \cdot \left(angle \cdot \left({b}^{2} \cdot \pi\right)\right)} \]
   10. Step-by-step derivation

       1. *-commutative 45.5%

          \[\leadsto \color{blue}{\left(angle \cdot \left({b}^{2} \cdot \pi\right)\right) \cdot 0.011111111111111112} \]

       2. associate-*l* 45.6%

          \[\leadsto \color{blue}{angle \cdot \left(\left({b}^{2} \cdot \pi\right) \cdot 0.011111111111111112\right)} \]

       3. *-commutative 45.6%

          \[\leadsto angle \cdot \left(\color{blue}{\left(\pi \cdot {b}^{2}\right)} \cdot 0.011111111111111112\right) \]

       4. unpow2 45.6%

          \[\leadsto angle \cdot \left(\left(\pi \cdot \color{blue}{\left(b \cdot b\right)}\right) \cdot 0.011111111111111112\right) \]

   11. Simplified 45.6%

       \[\leadsto \color{blue}{angle \cdot \left(\left(\pi \cdot \left(b \cdot b\right)\right) \cdot 0.011111111111111112\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 38.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq 600000000:\\ \;\;\;\;angle \cdot \left(\pi \cdot \left(a \cdot \left(a \cdot -0.011111111111111112\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;angle \cdot \left(0.011111111111111112 \cdot \left(\pi \cdot \left(b \cdot b\right)\right)\right)\\ \end{array} \]

Alternative 12: 42.1% accurate, 5.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq 520000000:\\ \;\;\;\;angle \cdot \left(\pi \cdot \left(a \cdot \left(a \cdot -0.011111111111111112\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(angle \cdot 0.011111111111111112\right) \cdot \left(\pi \cdot \left(b \cdot b\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (a b angle)
 :precision binary64
 (if (<= b 520000000.0)
   (* angle (* PI (* a (* a -0.011111111111111112))))
   (* (* angle 0.011111111111111112) (* PI (* b b)))))

C

double code(double a, double b, double angle) {
	double tmp;
	if (b <= 520000000.0) {
		tmp = angle * (((double) M_PI) * (a * (a * -0.011111111111111112)));
	} else {
		tmp = (angle * 0.011111111111111112) * (((double) M_PI) * (b * b));
	}
	return tmp;
}

Java

public static double code(double a, double b, double angle) {
	double tmp;
	if (b <= 520000000.0) {
		tmp = angle * (Math.PI * (a * (a * -0.011111111111111112)));
	} else {
		tmp = (angle * 0.011111111111111112) * (Math.PI * (b * b));
	}
	return tmp;
}

Python

def code(a, b, angle):
	tmp = 0
	if b <= 520000000.0:
		tmp = angle * (math.pi * (a * (a * -0.011111111111111112)))
	else:
		tmp = (angle * 0.011111111111111112) * (math.pi * (b * b))
	return tmp

Julia

function code(a, b, angle)
	tmp = 0.0
	if (b <= 520000000.0)
		tmp = Float64(angle * Float64(pi * Float64(a * Float64(a * -0.011111111111111112))));
	else
		tmp = Float64(Float64(angle * 0.011111111111111112) * Float64(pi * Float64(b * b)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, angle)
	tmp = 0.0;
	if (b <= 520000000.0)
		tmp = angle * (pi * (a * (a * -0.011111111111111112)));
	else
		tmp = (angle * 0.011111111111111112) * (pi * (b * b));
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, angle_] := If[LessEqual[b, 520000000.0], N[(angle * N[(Pi * N[(a * N[(a * -0.011111111111111112), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(angle * 0.011111111111111112), $MachinePrecision] * N[(Pi * N[(b * b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b < 5.2e8

   1. Initial program 51.0%

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

      1. associate-*l* 51.0%

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

      2. unpow2 51.0%

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

      3. unpow2 51.0%

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

      4. difference-of-squares 54.2%

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

   3. Simplified 54.2%

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

   4. Taylor expanded in angle around 0 50.2%

      \[\leadsto \color{blue}{0.011111111111111112 \cdot \left(angle \cdot \left(\left(b - a\right) \cdot \left(\pi \cdot \left(a + b\right)\right)\right)\right)} \]

   5. Taylor expanded in b around 0 36.3%

      \[\leadsto \color{blue}{-0.011111111111111112 \cdot \left(angle \cdot \left({a}^{2} \cdot \pi\right)\right)} \]
   6. Step-by-step derivation

      1. *-commutative 36.3%

         \[\leadsto \color{blue}{\left(angle \cdot \left({a}^{2} \cdot \pi\right)\right) \cdot -0.011111111111111112} \]

      2. *-commutative 36.3%

         \[\leadsto \left(angle \cdot \color{blue}{\left(\pi \cdot {a}^{2}\right)}\right) \cdot -0.011111111111111112 \]

      3. unpow2 36.3%

         \[\leadsto \left(angle \cdot \left(\pi \cdot \color{blue}{\left(a \cdot a\right)}\right)\right) \cdot -0.011111111111111112 \]

   7. Simplified 36.3%

      \[\leadsto \color{blue}{\left(angle \cdot \left(\pi \cdot \left(a \cdot a\right)\right)\right) \cdot -0.011111111111111112} \]

   8. Taylor expanded in angle around 0 36.3%

      \[\leadsto \color{blue}{-0.011111111111111112 \cdot \left(angle \cdot \left({a}^{2} \cdot \pi\right)\right)} \]
   9. Step-by-step derivation

      1. *-commutative 36.3%

         \[\leadsto \color{blue}{\left(angle \cdot \left({a}^{2} \cdot \pi\right)\right) \cdot -0.011111111111111112} \]

      2. *-commutative 36.3%

         \[\leadsto \left(angle \cdot \color{blue}{\left(\pi \cdot {a}^{2}\right)}\right) \cdot -0.011111111111111112 \]

      3. associate-*r* 36.3%

         \[\leadsto \color{blue}{\left(\left(angle \cdot \pi\right) \cdot {a}^{2}\right)} \cdot -0.011111111111111112 \]

      4. associate-*l* 36.3%

         \[\leadsto \color{blue}{\left(angle \cdot \pi\right) \cdot \left({a}^{2} \cdot -0.011111111111111112\right)} \]

      5. associate-*r* 36.3%

         \[\leadsto \color{blue}{angle \cdot \left(\pi \cdot \left({a}^{2} \cdot -0.011111111111111112\right)\right)} \]

      6. unpow2 36.3%

         \[\leadsto angle \cdot \left(\pi \cdot \left(\color{blue}{\left(a \cdot a\right)} \cdot -0.011111111111111112\right)\right) \]

      7. associate-*l* 36.3%

         \[\leadsto angle \cdot \left(\pi \cdot \color{blue}{\left(a \cdot \left(a \cdot -0.011111111111111112\right)\right)}\right) \]

   10. Simplified 36.3%

       \[\leadsto \color{blue}{angle \cdot \left(\pi \cdot \left(a \cdot \left(a \cdot -0.011111111111111112\right)\right)\right)} \]

   if 5.2e8 < b

   1. Initial program 43.2%

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

      1. associate-*l* 43.2%

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

      2. unpow2 43.2%

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

      3. unpow2 43.2%

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

      4. difference-of-squares 48.3%

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

   3. Simplified 48.3%

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

   4. Taylor expanded in angle around 0 49.2%

      \[\leadsto \color{blue}{0.011111111111111112 \cdot \left(angle \cdot \left(\left(b - a\right) \cdot \left(\pi \cdot \left(a + b\right)\right)\right)\right)} \]

   5. Taylor expanded in b around inf 45.5%

      \[\leadsto \color{blue}{0.011111111111111112 \cdot \left(angle \cdot \left({b}^{2} \cdot \pi\right)\right)} \]
   6. Step-by-step derivation

      1. associate-*r* 45.6%

         \[\leadsto \color{blue}{\left(0.011111111111111112 \cdot angle\right) \cdot \left({b}^{2} \cdot \pi\right)} \]

      2. *-commutative 45.6%

         \[\leadsto \color{blue}{\left({b}^{2} \cdot \pi\right) \cdot \left(0.011111111111111112 \cdot angle\right)} \]

      3. *-commutative 45.6%

         \[\leadsto \color{blue}{\left(\pi \cdot {b}^{2}\right)} \cdot \left(0.011111111111111112 \cdot angle\right) \]

      4. unpow2 45.6%

         \[\leadsto \left(\pi \cdot \color{blue}{\left(b \cdot b\right)}\right) \cdot \left(0.011111111111111112 \cdot angle\right) \]

      5. *-commutative 45.6%

         \[\leadsto \left(\pi \cdot \left(b \cdot b\right)\right) \cdot \color{blue}{\left(angle \cdot 0.011111111111111112\right)} \]

   7. Simplified 45.6%

      \[\leadsto \color{blue}{\left(\pi \cdot \left(b \cdot b\right)\right) \cdot \left(angle \cdot 0.011111111111111112\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 38.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq 520000000:\\ \;\;\;\;angle \cdot \left(\pi \cdot \left(a \cdot \left(a \cdot -0.011111111111111112\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(angle \cdot 0.011111111111111112\right) \cdot \left(\pi \cdot \left(b \cdot b\right)\right)\\ \end{array} \]

Alternative 13: 35.0% accurate, 5.7× speedup

Math

\[\begin{array}{l} \\ -0.011111111111111112 \cdot \left(\pi \cdot \left(angle \cdot \left(a \cdot a\right)\right)\right) \end{array} \]

FPCore

(FPCore (a b angle)
 :precision binary64
 (* -0.011111111111111112 (* PI (* angle (* a a)))))

C

double code(double a, double b, double angle) {
	return -0.011111111111111112 * (((double) M_PI) * (angle * (a * a)));
}

Java

public static double code(double a, double b, double angle) {
	return -0.011111111111111112 * (Math.PI * (angle * (a * a)));
}

Python

def code(a, b, angle):
	return -0.011111111111111112 * (math.pi * (angle * (a * a)))

Julia

function code(a, b, angle)
	return Float64(-0.011111111111111112 * Float64(pi * Float64(angle * Float64(a * a))))
end

MATLAB

function tmp = code(a, b, angle)
	tmp = -0.011111111111111112 * (pi * (angle * (a * a)));
end

Wolfram

code[a_, b_, angle_] := N[(-0.011111111111111112 * N[(Pi * N[(angle * N[(a * a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 49.2%

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

   1. associate-*l* 49.2%

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

   2. unpow2 49.2%

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

   3. unpow2 49.2%

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

   4. difference-of-squares 52.8%

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

3. Simplified 52.8%

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

4. Taylor expanded in angle around 0 50.0%

   \[\leadsto \color{blue}{0.011111111111111112 \cdot \left(angle \cdot \left(\left(b - a\right) \cdot \left(\pi \cdot \left(a + b\right)\right)\right)\right)} \]

5. Taylor expanded in b around 0 31.0%

   \[\leadsto \color{blue}{-0.011111111111111112 \cdot \left(angle \cdot \left({a}^{2} \cdot \pi\right)\right)} \]
6. Step-by-step derivation

   1. *-commutative 31.0%

      \[\leadsto \color{blue}{\left(angle \cdot \left({a}^{2} \cdot \pi\right)\right) \cdot -0.011111111111111112} \]

   2. *-commutative 31.0%

      \[\leadsto \left(angle \cdot \color{blue}{\left(\pi \cdot {a}^{2}\right)}\right) \cdot -0.011111111111111112 \]

   3. unpow2 31.0%

      \[\leadsto \left(angle \cdot \left(\pi \cdot \color{blue}{\left(a \cdot a\right)}\right)\right) \cdot -0.011111111111111112 \]

7. Simplified 31.0%

   \[\leadsto \color{blue}{\left(angle \cdot \left(\pi \cdot \left(a \cdot a\right)\right)\right) \cdot -0.011111111111111112} \]

8. Taylor expanded in angle around 0 31.0%

   \[\leadsto \color{blue}{-0.011111111111111112 \cdot \left(angle \cdot \left({a}^{2} \cdot \pi\right)\right)} \]
9. Step-by-step derivation

   1. *-commutative 31.0%

      \[\leadsto \color{blue}{\left(angle \cdot \left({a}^{2} \cdot \pi\right)\right) \cdot -0.011111111111111112} \]

   2. *-commutative 31.0%

      \[\leadsto \left(angle \cdot \color{blue}{\left(\pi \cdot {a}^{2}\right)}\right) \cdot -0.011111111111111112 \]

   3. associate-*r* 31.0%

      \[\leadsto \color{blue}{\left(\left(angle \cdot \pi\right) \cdot {a}^{2}\right)} \cdot -0.011111111111111112 \]

   4. associate-*l* 31.0%

      \[\leadsto \color{blue}{\left(angle \cdot \pi\right) \cdot \left({a}^{2} \cdot -0.011111111111111112\right)} \]

   5. associate-*r* 31.0%

      \[\leadsto \color{blue}{angle \cdot \left(\pi \cdot \left({a}^{2} \cdot -0.011111111111111112\right)\right)} \]

   6. unpow2 31.0%

      \[\leadsto angle \cdot \left(\pi \cdot \left(\color{blue}{\left(a \cdot a\right)} \cdot -0.011111111111111112\right)\right) \]

   7. associate-*l* 31.0%

      \[\leadsto angle \cdot \left(\pi \cdot \color{blue}{\left(a \cdot \left(a \cdot -0.011111111111111112\right)\right)}\right) \]

10. Simplified 31.0%

    \[\leadsto \color{blue}{angle \cdot \left(\pi \cdot \left(a \cdot \left(a \cdot -0.011111111111111112\right)\right)\right)} \]

11. Taylor expanded in angle around 0 31.0%

    \[\leadsto \color{blue}{-0.011111111111111112 \cdot \left(angle \cdot \left({a}^{2} \cdot \pi\right)\right)} \]
12. Step-by-step derivation

    1. associate-*r* 31.0%

       \[\leadsto -0.011111111111111112 \cdot \color{blue}{\left(\left(angle \cdot {a}^{2}\right) \cdot \pi\right)} \]

    2. *-commutative 31.0%

       \[\leadsto -0.011111111111111112 \cdot \color{blue}{\left(\pi \cdot \left(angle \cdot {a}^{2}\right)\right)} \]

    3. unpow2 31.0%

       \[\leadsto -0.011111111111111112 \cdot \left(\pi \cdot \left(angle \cdot \color{blue}{\left(a \cdot a\right)}\right)\right) \]

13. Simplified 31.0%

    \[\leadsto \color{blue}{-0.011111111111111112 \cdot \left(\pi \cdot \left(angle \cdot \left(a \cdot a\right)\right)\right)} \]

14. Final simplification 31.0%

    \[\leadsto -0.011111111111111112 \cdot \left(\pi \cdot \left(angle \cdot \left(a \cdot a\right)\right)\right) \]

Alternative 14: 35.0% accurate, 5.7× speedup

Math

\[\begin{array}{l} \\ angle \cdot \left(\pi \cdot \left(a \cdot \left(a \cdot -0.011111111111111112\right)\right)\right) \end{array} \]

FPCore

(FPCore (a b angle)
 :precision binary64
 (* angle (* PI (* a (* a -0.011111111111111112)))))

C

double code(double a, double b, double angle) {
	return angle * (((double) M_PI) * (a * (a * -0.011111111111111112)));
}

Java

public static double code(double a, double b, double angle) {
	return angle * (Math.PI * (a * (a * -0.011111111111111112)));
}

Python

def code(a, b, angle):
	return angle * (math.pi * (a * (a * -0.011111111111111112)))

Julia

function code(a, b, angle)
	return Float64(angle * Float64(pi * Float64(a * Float64(a * -0.011111111111111112))))
end

MATLAB

function tmp = code(a, b, angle)
	tmp = angle * (pi * (a * (a * -0.011111111111111112)));
end

Wolfram

code[a_, b_, angle_] := N[(angle * N[(Pi * N[(a * N[(a * -0.011111111111111112), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 49.2%

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

   1. associate-*l* 49.2%

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

   2. unpow2 49.2%

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

   3. unpow2 49.2%

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

   4. difference-of-squares 52.8%

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

3. Simplified 52.8%

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

4. Taylor expanded in angle around 0 50.0%

   \[\leadsto \color{blue}{0.011111111111111112 \cdot \left(angle \cdot \left(\left(b - a\right) \cdot \left(\pi \cdot \left(a + b\right)\right)\right)\right)} \]

5. Taylor expanded in b around 0 31.0%

   \[\leadsto \color{blue}{-0.011111111111111112 \cdot \left(angle \cdot \left({a}^{2} \cdot \pi\right)\right)} \]
6. Step-by-step derivation

   1. *-commutative 31.0%

      \[\leadsto \color{blue}{\left(angle \cdot \left({a}^{2} \cdot \pi\right)\right) \cdot -0.011111111111111112} \]

   2. *-commutative 31.0%

      \[\leadsto \left(angle \cdot \color{blue}{\left(\pi \cdot {a}^{2}\right)}\right) \cdot -0.011111111111111112 \]

   3. unpow2 31.0%

      \[\leadsto \left(angle \cdot \left(\pi \cdot \color{blue}{\left(a \cdot a\right)}\right)\right) \cdot -0.011111111111111112 \]

7. Simplified 31.0%

   \[\leadsto \color{blue}{\left(angle \cdot \left(\pi \cdot \left(a \cdot a\right)\right)\right) \cdot -0.011111111111111112} \]

8. Taylor expanded in angle around 0 31.0%

   \[\leadsto \color{blue}{-0.011111111111111112 \cdot \left(angle \cdot \left({a}^{2} \cdot \pi\right)\right)} \]
9. Step-by-step derivation

   1. *-commutative 31.0%

      \[\leadsto \color{blue}{\left(angle \cdot \left({a}^{2} \cdot \pi\right)\right) \cdot -0.011111111111111112} \]

   2. *-commutative 31.0%

      \[\leadsto \left(angle \cdot \color{blue}{\left(\pi \cdot {a}^{2}\right)}\right) \cdot -0.011111111111111112 \]

   3. associate-*r* 31.0%

      \[\leadsto \color{blue}{\left(\left(angle \cdot \pi\right) \cdot {a}^{2}\right)} \cdot -0.011111111111111112 \]

   4. associate-*l* 31.0%

      \[\leadsto \color{blue}{\left(angle \cdot \pi\right) \cdot \left({a}^{2} \cdot -0.011111111111111112\right)} \]

   5. associate-*r* 31.0%

      \[\leadsto \color{blue}{angle \cdot \left(\pi \cdot \left({a}^{2} \cdot -0.011111111111111112\right)\right)} \]

   6. unpow2 31.0%

      \[\leadsto angle \cdot \left(\pi \cdot \left(\color{blue}{\left(a \cdot a\right)} \cdot -0.011111111111111112\right)\right) \]

   7. associate-*l* 31.0%

      \[\leadsto angle \cdot \left(\pi \cdot \color{blue}{\left(a \cdot \left(a \cdot -0.011111111111111112\right)\right)}\right) \]

10. Simplified 31.0%

    \[\leadsto \color{blue}{angle \cdot \left(\pi \cdot \left(a \cdot \left(a \cdot -0.011111111111111112\right)\right)\right)} \]

11. Final simplification 31.0%

    \[\leadsto angle \cdot \left(\pi \cdot \left(a \cdot \left(a \cdot -0.011111111111111112\right)\right)\right) \]

Reproduce

herbie shell --seed 2023199 
(FPCore (a b angle)
  :name "ab-angle->ABCF B"
  :precision binary64
  (* (* (* 2.0 (- (pow b 2.0) (pow a 2.0))) (sin (* PI (/ angle 180.0)))) (cos (* PI (/ angle 180.0)))))