ab-angle->ABCF B

Percentage Accurate: 54.4% → 66.5%

Time: 45.9s

Alternatives: 13

Speedup: 5.5×

• 31.8% 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: 54.4% 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: 66.5% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \sin \left(\pi \cdot \left(angle \cdot 0.005555555555555556\right)\right)\\ t_1 := \left(b - a\right) \cdot \left(b + a\right)\\ t_2 := t_0 \cdot t_1\\ t_3 := angle \cdot \left(\pi \cdot -0.005555555555555556\right)\\ t_4 := \cos t_3\\ \mathbf{if}\;\frac{angle}{180} \leq -1 \cdot 10^{+155}:\\ \;\;\;\;2 \cdot \left(t_4 \cdot t_2\right)\\ \mathbf{elif}\;\frac{angle}{180} \leq -2 \cdot 10^{+44}:\\ \;\;\;\;2 \cdot \left(t_1 \cdot \left(t_4 \cdot \sin t_3\right)\right)\\ \mathbf{elif}\;\frac{angle}{180} \leq -5 \cdot 10^{-34}:\\ \;\;\;\;2 \cdot \left(t_2 \cdot \cos \left(\pi \cdot \left(angle \cdot -0.005555555555555556\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;2 \cdot {\left(\sqrt[3]{\left(b + a\right) \cdot \left(\left(b - a\right) \cdot t_0\right)}\right)}^{3}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b angle)
 :precision binary64
 (let* ((t_0 (sin (* PI (* angle 0.005555555555555556))))
        (t_1 (* (- b a) (+ b a)))
        (t_2 (* t_0 t_1))
        (t_3 (* angle (* PI -0.005555555555555556)))
        (t_4 (cos t_3)))
   (if (<= (/ angle 180.0) -1e+155)
     (* 2.0 (* t_4 t_2))
     (if (<= (/ angle 180.0) -2e+44)
       (* 2.0 (* t_1 (* t_4 (sin t_3))))
       (if (<= (/ angle 180.0) -5e-34)
         (* 2.0 (* t_2 (cos (* PI (* angle -0.005555555555555556)))))
         (* 2.0 (pow (cbrt (* (+ b a) (* (- b a) t_0))) 3.0)))))))

C

double code(double a, double b, double angle) {
	double t_0 = sin((((double) M_PI) * (angle * 0.005555555555555556)));
	double t_1 = (b - a) * (b + a);
	double t_2 = t_0 * t_1;
	double t_3 = angle * (((double) M_PI) * -0.005555555555555556);
	double t_4 = cos(t_3);
	double tmp;
	if ((angle / 180.0) <= -1e+155) {
		tmp = 2.0 * (t_4 * t_2);
	} else if ((angle / 180.0) <= -2e+44) {
		tmp = 2.0 * (t_1 * (t_4 * sin(t_3)));
	} else if ((angle / 180.0) <= -5e-34) {
		tmp = 2.0 * (t_2 * cos((((double) M_PI) * (angle * -0.005555555555555556))));
	} else {
		tmp = 2.0 * pow(cbrt(((b + a) * ((b - a) * t_0))), 3.0);
	}
	return tmp;
}

Java

public static double code(double a, double b, double angle) {
	double t_0 = Math.sin((Math.PI * (angle * 0.005555555555555556)));
	double t_1 = (b - a) * (b + a);
	double t_2 = t_0 * t_1;
	double t_3 = angle * (Math.PI * -0.005555555555555556);
	double t_4 = Math.cos(t_3);
	double tmp;
	if ((angle / 180.0) <= -1e+155) {
		tmp = 2.0 * (t_4 * t_2);
	} else if ((angle / 180.0) <= -2e+44) {
		tmp = 2.0 * (t_1 * (t_4 * Math.sin(t_3)));
	} else if ((angle / 180.0) <= -5e-34) {
		tmp = 2.0 * (t_2 * Math.cos((Math.PI * (angle * -0.005555555555555556))));
	} else {
		tmp = 2.0 * Math.pow(Math.cbrt(((b + a) * ((b - a) * t_0))), 3.0);
	}
	return tmp;
}

Julia

function code(a, b, angle)
	t_0 = sin(Float64(pi * Float64(angle * 0.005555555555555556)))
	t_1 = Float64(Float64(b - a) * Float64(b + a))
	t_2 = Float64(t_0 * t_1)
	t_3 = Float64(angle * Float64(pi * -0.005555555555555556))
	t_4 = cos(t_3)
	tmp = 0.0
	if (Float64(angle / 180.0) <= -1e+155)
		tmp = Float64(2.0 * Float64(t_4 * t_2));
	elseif (Float64(angle / 180.0) <= -2e+44)
		tmp = Float64(2.0 * Float64(t_1 * Float64(t_4 * sin(t_3))));
	elseif (Float64(angle / 180.0) <= -5e-34)
		tmp = Float64(2.0 * Float64(t_2 * cos(Float64(pi * Float64(angle * -0.005555555555555556)))));
	else
		tmp = Float64(2.0 * (cbrt(Float64(Float64(b + a) * Float64(Float64(b - a) * t_0))) ^ 3.0));
	end
	return tmp
end

Wolfram

code[a_, b_, angle_] := Block[{t$95$0 = N[Sin[N[(Pi * N[(angle * 0.005555555555555556), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$1 = N[(N[(b - a), $MachinePrecision] * N[(b + a), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(t$95$0 * t$95$1), $MachinePrecision]}, Block[{t$95$3 = N[(angle * N[(Pi * -0.005555555555555556), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$4 = N[Cos[t$95$3], $MachinePrecision]}, If[LessEqual[N[(angle / 180.0), $MachinePrecision], -1e+155], N[(2.0 * N[(t$95$4 * t$95$2), $MachinePrecision]), $MachinePrecision], If[LessEqual[N[(angle / 180.0), $MachinePrecision], -2e+44], N[(2.0 * N[(t$95$1 * N[(t$95$4 * N[Sin[t$95$3], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[N[(angle / 180.0), $MachinePrecision], -5e-34], N[(2.0 * N[(t$95$2 * N[Cos[N[(Pi * N[(angle * -0.005555555555555556), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(2.0 * N[Power[N[Power[N[(N[(b + a), $MachinePrecision] * N[(N[(b - a), $MachinePrecision] * t$95$0), $MachinePrecision]), $MachinePrecision], 1/3], $MachinePrecision], 3.0], $MachinePrecision]), $MachinePrecision]]]]]]]]]

Derivation

1. Split input into 4 regimes

2. if (/.f64 angle 180) < -1.00000000000000001e155

   1. Initial program 31.9%

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

      1. associate-*l* 31.9%

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

      2. associate-*l* 31.9%

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

      3. cos-neg 31.9%

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

      4. distribute-rgt-neg-out 31.9%

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

      5. distribute-frac-neg 31.9%

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

      6. neg-mul-1 31.9%

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

      7. associate-/l* 31.2%

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

      8. associate-*r/ 34.6%

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

      9. associate-/r/ 36.5%

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

      10. associate-/l* 36.5%

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

      11. metadata-eval 36.5%

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

   3. Simplified 36.5%

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

      1. unpow2 36.5%

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

      2. unpow2 36.5%

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

      3. difference-of-squares 39.9%

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

   5. Applied egg-rr 39.9%

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

   6. Taylor expanded in angle around inf 29.8%

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

   7. Taylor expanded in angle around inf 29.1%

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

      1. *-commutative 29.1%

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

      2. associate-*r* 43.1%

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

      3. +-commutative 43.1%

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

      4. *-commutative 43.1%

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

      5. +-commutative 43.1%

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

      6. *-commutative 43.1%

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

      7. *-commutative 43.1%

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

      8. *-commutative 43.1%

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

      9. associate-*r* 47.2%

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

   9. Simplified 47.2%

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

   if -1.00000000000000001e155 < (/.f64 angle 180) < -2.0000000000000002e44

   1. Initial program 22.7%

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

      1. associate-*l* 22.7%

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

      2. associate-*l* 22.7%

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

      3. cos-neg 22.7%

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

      4. distribute-rgt-neg-out 22.7%

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

      5. distribute-frac-neg 22.7%

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

      6. neg-mul-1 22.7%

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

      7. associate-/l* 31.8%

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

      8. associate-*r/ 31.8%

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

      9. associate-/r/ 30.1%

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

      10. associate-/l* 30.1%

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

      11. metadata-eval 30.1%

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

   3. Simplified 30.1%

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

      1. unpow2 30.1%

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

      2. unpow2 30.1%

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

      3. difference-of-squares 30.1%

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

   5. Applied egg-rr 30.1%

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

   6. Taylor expanded in angle around inf 22.4%

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

   8. Applied egg-rr 21.0%

      \[\leadsto 2 \cdot \color{blue}{\left(e^{\mathsf{log1p}\left(\left(b + a\right) \cdot \left(\left(b - a\right) \cdot \left(\cos \left(angle \cdot \left(\pi \cdot -0.005555555555555556\right)\right) \cdot \sin \left(angle \cdot \left(\pi \cdot -0.005555555555555556\right)\right)\right)\right)\right)} - 1\right)} \]
   9. Step-by-step derivation

      1. expm1-def 21.6%

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

      2. expm1-log1p 46.8%

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

      3. associate-*r* 46.8%

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

      4. *-commutative 46.8%

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

      5. +-commutative 46.8%

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

      6. *-commutative 46.8%

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

   10. Simplified 46.8%

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

   if -2.0000000000000002e44 < (/.f64 angle 180) < -5.0000000000000003e-34

   1. Initial program 69.6%

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

      1. associate-*l* 69.5%

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

      2. associate-*l* 69.5%

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

      3. cos-neg 69.5%

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

      4. distribute-rgt-neg-out 69.5%

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

      5. distribute-frac-neg 69.5%

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

      6. neg-mul-1 69.5%

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

      7. associate-/l* 68.6%

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

      8. associate-*r/ 63.7%

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

      9. associate-/r/ 64.7%

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

      10. associate-/l* 64.7%

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

      11. metadata-eval 64.7%

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

   3. Simplified 64.7%

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

      1. unpow2 64.7%

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

      2. unpow2 64.7%

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

      3. difference-of-squares 69.4%

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

   5. Applied egg-rr 69.4%

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

   6. Taylor expanded in angle around inf 74.0%

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

      1. *-commutative 74.0%

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

      2. *-commutative 74.0%

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

      3. associate-*r* 69.2%

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

      4. *-commutative 69.2%

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

      5. *-commutative 69.2%

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

      6. associate-*r* 75.1%

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

      7. *-commutative 75.1%

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

      8. *-commutative 75.1%

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

      9. *-commutative 75.1%

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

   8. Simplified 75.1%

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

   if -5.0000000000000003e-34 < (/.f64 angle 180)

   1. Initial program 65.3%

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

      1. associate-*l* 65.3%

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

      2. associate-*l* 65.3%

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

      3. cos-neg 65.3%

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

      4. distribute-rgt-neg-out 65.3%

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

      5. distribute-frac-neg 65.3%

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

      6. neg-mul-1 65.3%

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

      7. associate-/l* 66.4%

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

      8. associate-*r/ 65.8%

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

      9. associate-/r/ 65.3%

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

      10. associate-/l* 65.3%

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

      11. metadata-eval 65.3%

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

   3. Simplified 65.3%

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

      1. unpow2 65.3%

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

      2. unpow2 65.3%

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

      3. difference-of-squares 68.1%

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

   5. Applied egg-rr 68.1%

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

      1. add-cube-cbrt 67.8%

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

      2. pow3 67.7%

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

   7. Applied egg-rr 78.9%

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

   8. Taylor expanded in angle around 0 79.7%

      \[\leadsto 2 \cdot {\left(\sqrt[3]{\left(b + a\right) \cdot \left(\left(b - a\right) \cdot \left(\sin \left(\pi \cdot \left(angle \cdot 0.005555555555555556\right)\right) \cdot \color{blue}{1}\right)\right)}\right)}^{3} \]
3. Recombined 4 regimes into one program.

4. Final simplification 72.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{angle}{180} \leq -1 \cdot 10^{+155}:\\ \;\;\;\;2 \cdot \left(\cos \left(angle \cdot \left(\pi \cdot -0.005555555555555556\right)\right) \cdot \left(\sin \left(\pi \cdot \left(angle \cdot 0.005555555555555556\right)\right) \cdot \left(\left(b - a\right) \cdot \left(b + a\right)\right)\right)\right)\\ \mathbf{elif}\;\frac{angle}{180} \leq -2 \cdot 10^{+44}:\\ \;\;\;\;2 \cdot \left(\left(\left(b - a\right) \cdot \left(b + a\right)\right) \cdot \left(\cos \left(angle \cdot \left(\pi \cdot -0.005555555555555556\right)\right) \cdot \sin \left(angle \cdot \left(\pi \cdot -0.005555555555555556\right)\right)\right)\right)\\ \mathbf{elif}\;\frac{angle}{180} \leq -5 \cdot 10^{-34}:\\ \;\;\;\;2 \cdot \left(\left(\sin \left(\pi \cdot \left(angle \cdot 0.005555555555555556\right)\right) \cdot \left(\left(b - a\right) \cdot \left(b + a\right)\right)\right) \cdot \cos \left(\pi \cdot \left(angle \cdot -0.005555555555555556\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;2 \cdot {\left(\sqrt[3]{\left(b + a\right) \cdot \left(\left(b - a\right) \cdot \sin \left(\pi \cdot \left(angle \cdot 0.005555555555555556\right)\right)\right)}\right)}^{3}\\ \end{array} \]

Alternative 2: 67.2% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \sqrt[3]{{\pi}^{3}}\\ \mathbf{if}\;{b}^{2} \leq 2 \cdot 10^{+109}:\\ \;\;\;\;2 \cdot \left(\left(\left(b - a\right) \cdot \left(\sin \left(\pi \cdot \left(angle \cdot 0.005555555555555556\right)\right) \cdot \cos \left(-0.005555555555555556 \cdot \left(\pi \cdot angle\right)\right)\right)\right) \cdot \left(b + a\right)\right)\\ \mathbf{else}:\\ \;\;\;\;2 \cdot {\left(\sqrt[3]{\left(b + a\right) \cdot \left(\left(b - a\right) \cdot \left(\sin \left(\left(angle \cdot 0.005555555555555556\right) \cdot t_0\right) \cdot \cos \left(angle \cdot \left(-0.005555555555555556 \cdot t_0\right)\right)\right)\right)}\right)}^{3}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b angle)
 :precision binary64
 (let* ((t_0 (cbrt (pow PI 3.0))))
   (if (<= (pow b 2.0) 2e+109)
     (*
      2.0
      (*
       (*
        (- b a)
        (*
         (sin (* PI (* angle 0.005555555555555556)))
         (cos (* -0.005555555555555556 (* PI angle)))))
       (+ b a)))
     (*
      2.0
      (pow
       (cbrt
        (*
         (+ b a)
         (*
          (- b a)
          (*
           (sin (* (* angle 0.005555555555555556) t_0))
           (cos (* angle (* -0.005555555555555556 t_0)))))))
       3.0)))))

C

double code(double a, double b, double angle) {
	double t_0 = cbrt(pow(((double) M_PI), 3.0));
	double tmp;
	if (pow(b, 2.0) <= 2e+109) {
		tmp = 2.0 * (((b - a) * (sin((((double) M_PI) * (angle * 0.005555555555555556))) * cos((-0.005555555555555556 * (((double) M_PI) * angle))))) * (b + a));
	} else {
		tmp = 2.0 * pow(cbrt(((b + a) * ((b - a) * (sin(((angle * 0.005555555555555556) * t_0)) * cos((angle * (-0.005555555555555556 * t_0))))))), 3.0);
	}
	return tmp;
}

Java

public static double code(double a, double b, double angle) {
	double t_0 = Math.cbrt(Math.pow(Math.PI, 3.0));
	double tmp;
	if (Math.pow(b, 2.0) <= 2e+109) {
		tmp = 2.0 * (((b - a) * (Math.sin((Math.PI * (angle * 0.005555555555555556))) * Math.cos((-0.005555555555555556 * (Math.PI * angle))))) * (b + a));
	} else {
		tmp = 2.0 * Math.pow(Math.cbrt(((b + a) * ((b - a) * (Math.sin(((angle * 0.005555555555555556) * t_0)) * Math.cos((angle * (-0.005555555555555556 * t_0))))))), 3.0);
	}
	return tmp;
}

Julia

function code(a, b, angle)
	t_0 = cbrt((pi ^ 3.0))
	tmp = 0.0
	if ((b ^ 2.0) <= 2e+109)
		tmp = Float64(2.0 * Float64(Float64(Float64(b - a) * Float64(sin(Float64(pi * Float64(angle * 0.005555555555555556))) * cos(Float64(-0.005555555555555556 * Float64(pi * angle))))) * Float64(b + a)));
	else
		tmp = Float64(2.0 * (cbrt(Float64(Float64(b + a) * Float64(Float64(b - a) * Float64(sin(Float64(Float64(angle * 0.005555555555555556) * t_0)) * cos(Float64(angle * Float64(-0.005555555555555556 * t_0))))))) ^ 3.0));
	end
	return tmp
end

Wolfram

code[a_, b_, angle_] := Block[{t$95$0 = N[Power[N[Power[Pi, 3.0], $MachinePrecision], 1/3], $MachinePrecision]}, If[LessEqual[N[Power[b, 2.0], $MachinePrecision], 2e+109], N[(2.0 * N[(N[(N[(b - a), $MachinePrecision] * N[(N[Sin[N[(Pi * N[(angle * 0.005555555555555556), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * N[Cos[N[(-0.005555555555555556 * N[(Pi * angle), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(b + a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(2.0 * N[Power[N[Power[N[(N[(b + a), $MachinePrecision] * N[(N[(b - a), $MachinePrecision] * N[(N[Sin[N[(N[(angle * 0.005555555555555556), $MachinePrecision] * t$95$0), $MachinePrecision]], $MachinePrecision] * N[Cos[N[(angle * N[(-0.005555555555555556 * t$95$0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 1/3], $MachinePrecision], 3.0], $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (pow.f64 b 2) < 1.99999999999999996e109

   1. Initial program 64.0%

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

      1. associate-*l* 64.0%

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

      2. associate-*l* 64.0%

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

      3. cos-neg 64.0%

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

      4. distribute-rgt-neg-out 64.0%

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

      5. distribute-frac-neg 64.0%

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

      6. neg-mul-1 64.0%

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

      7. associate-/l* 63.9%

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

      8. associate-*r/ 63.1%

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

      9. associate-/r/ 63.2%

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

      10. associate-/l* 63.2%

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

      11. metadata-eval 63.2%

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

   3. Simplified 63.2%

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

      1. unpow2 63.2%

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

      2. unpow2 63.2%

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

      3. difference-of-squares 63.2%

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

   5. Applied egg-rr 63.2%

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

      1. expm1-log1p-u 51.6%

         \[\leadsto 2 \cdot \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\left(\left(b + a\right) \cdot \left(b - a\right)\right) \cdot \left(\sin \left(\pi \cdot \frac{angle}{180}\right) \cdot \cos \left(\frac{\pi}{-180} \cdot angle\right)\right)\right)\right)} \]

      2. expm1-udef 32.7%

         \[\leadsto 2 \cdot \color{blue}{\left(e^{\mathsf{log1p}\left(\left(\left(b + a\right) \cdot \left(b - a\right)\right) \cdot \left(\sin \left(\pi \cdot \frac{angle}{180}\right) \cdot \cos \left(\frac{\pi}{-180} \cdot angle\right)\right)\right)} - 1\right)} \]

   7. Applied egg-rr 36.5%

      \[\leadsto 2 \cdot \color{blue}{\left(e^{\mathsf{log1p}\left(\left(b + a\right) \cdot \left(\left(b - a\right) \cdot \left(\sin \left(\pi \cdot \left(angle \cdot 0.005555555555555556\right)\right) \cdot \cos \left(angle \cdot \left(\pi \cdot -0.005555555555555556\right)\right)\right)\right)\right)} - 1\right)} \]
   8. Step-by-step derivation

      1. expm1-def 55.6%

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

      2. expm1-log1p 70.4%

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

      3. +-commutative 70.4%

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

      4. associate-*r* 70.4%

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

      5. *-commutative 70.4%

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

   9. Simplified 70.4%

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

   if 1.99999999999999996e109 < (pow.f64 b 2)

   1. Initial program 49.4%

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

      1. associate-*l* 49.4%

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

      2. associate-*l* 49.4%

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

      3. cos-neg 49.4%

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

      4. distribute-rgt-neg-out 49.4%

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

      5. distribute-frac-neg 49.4%

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

      6. neg-mul-1 49.4%

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

      7. associate-/l* 53.0%

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

      8. associate-*r/ 53.1%

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

      9. associate-/r/ 52.4%

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

      10. associate-/l* 52.4%

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

      11. metadata-eval 52.4%

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

   3. Simplified 52.4%

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

      1. unpow2 52.4%

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

      2. unpow2 52.4%

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

      3. difference-of-squares 59.5%

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

   5. Applied egg-rr 59.5%

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

      1. add-cube-cbrt 59.3%

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

      2. pow3 59.3%

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

   7. Applied egg-rr 70.3%

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

      1. add-cbrt-cube 73.2%

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

      2. pow3 73.2%

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

   9. Applied egg-rr 73.2%

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

       1. add-cbrt-cube 73.2%

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

       2. pow3 73.2%

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

   11. Applied egg-rr 75.9%

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

4. Final simplification 72.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;{b}^{2} \leq 2 \cdot 10^{+109}:\\ \;\;\;\;2 \cdot \left(\left(\left(b - a\right) \cdot \left(\sin \left(\pi \cdot \left(angle \cdot 0.005555555555555556\right)\right) \cdot \cos \left(-0.005555555555555556 \cdot \left(\pi \cdot angle\right)\right)\right)\right) \cdot \left(b + a\right)\right)\\ \mathbf{else}:\\ \;\;\;\;2 \cdot {\left(\sqrt[3]{\left(b + a\right) \cdot \left(\left(b - a\right) \cdot \left(\sin \left(\left(angle \cdot 0.005555555555555556\right) \cdot \sqrt[3]{{\pi}^{3}}\right) \cdot \cos \left(angle \cdot \left(-0.005555555555555556 \cdot \sqrt[3]{{\pi}^{3}}\right)\right)\right)\right)}\right)}^{3}\\ \end{array} \]

Alternative 3: 66.3% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \sqrt[3]{{\pi}^{3}}\\ \mathbf{if}\;\frac{angle}{180} \leq -1 \cdot 10^{+18}:\\ \;\;\;\;\left(\left(b - a\right) \cdot \left(b + a\right)\right) \cdot \left(\left(2 \cdot \sin \left(\pi \cdot \frac{angle}{180}\right)\right) \cdot \cos \left(t_0 \cdot \frac{angle}{180}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;2 \cdot {\left(\sqrt[3]{\left(b + a\right) \cdot \left(\left(b - a\right) \cdot \left(\sin \left(\left(angle \cdot 0.005555555555555556\right) \cdot t_0\right) \cdot \cos \left(angle \cdot \left(\pi \cdot -0.005555555555555556\right)\right)\right)\right)}\right)}^{3}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b angle)
 :precision binary64
 (let* ((t_0 (cbrt (pow PI 3.0))))
   (if (<= (/ angle 180.0) -1e+18)
     (*
      (* (- b a) (+ b a))
      (* (* 2.0 (sin (* PI (/ angle 180.0)))) (cos (* t_0 (/ angle 180.0)))))
     (*
      2.0
      (pow
       (cbrt
        (*
         (+ b a)
         (*
          (- b a)
          (*
           (sin (* (* angle 0.005555555555555556) t_0))
           (cos (* angle (* PI -0.005555555555555556)))))))
       3.0)))))

C

double code(double a, double b, double angle) {
	double t_0 = cbrt(pow(((double) M_PI), 3.0));
	double tmp;
	if ((angle / 180.0) <= -1e+18) {
		tmp = ((b - a) * (b + a)) * ((2.0 * sin((((double) M_PI) * (angle / 180.0)))) * cos((t_0 * (angle / 180.0))));
	} else {
		tmp = 2.0 * pow(cbrt(((b + a) * ((b - a) * (sin(((angle * 0.005555555555555556) * t_0)) * cos((angle * (((double) M_PI) * -0.005555555555555556))))))), 3.0);
	}
	return tmp;
}

Java

public static double code(double a, double b, double angle) {
	double t_0 = Math.cbrt(Math.pow(Math.PI, 3.0));
	double tmp;
	if ((angle / 180.0) <= -1e+18) {
		tmp = ((b - a) * (b + a)) * ((2.0 * Math.sin((Math.PI * (angle / 180.0)))) * Math.cos((t_0 * (angle / 180.0))));
	} else {
		tmp = 2.0 * Math.pow(Math.cbrt(((b + a) * ((b - a) * (Math.sin(((angle * 0.005555555555555556) * t_0)) * Math.cos((angle * (Math.PI * -0.005555555555555556))))))), 3.0);
	}
	return tmp;
}

Julia

function code(a, b, angle)
	t_0 = cbrt((pi ^ 3.0))
	tmp = 0.0
	if (Float64(angle / 180.0) <= -1e+18)
		tmp = Float64(Float64(Float64(b - a) * Float64(b + a)) * Float64(Float64(2.0 * sin(Float64(pi * Float64(angle / 180.0)))) * cos(Float64(t_0 * Float64(angle / 180.0)))));
	else
		tmp = Float64(2.0 * (cbrt(Float64(Float64(b + a) * Float64(Float64(b - a) * Float64(sin(Float64(Float64(angle * 0.005555555555555556) * t_0)) * cos(Float64(angle * Float64(pi * -0.005555555555555556))))))) ^ 3.0));
	end
	return tmp
end

Wolfram

code[a_, b_, angle_] := Block[{t$95$0 = N[Power[N[Power[Pi, 3.0], $MachinePrecision], 1/3], $MachinePrecision]}, If[LessEqual[N[(angle / 180.0), $MachinePrecision], -1e+18], N[(N[(N[(b - a), $MachinePrecision] * N[(b + a), $MachinePrecision]), $MachinePrecision] * N[(N[(2.0 * N[Sin[N[(Pi * N[(angle / 180.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[Cos[N[(t$95$0 * N[(angle / 180.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(2.0 * N[Power[N[Power[N[(N[(b + a), $MachinePrecision] * N[(N[(b - a), $MachinePrecision] * N[(N[Sin[N[(N[(angle * 0.005555555555555556), $MachinePrecision] * t$95$0), $MachinePrecision]], $MachinePrecision] * N[Cos[N[(angle * N[(Pi * -0.005555555555555556), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 1/3], $MachinePrecision], 3.0], $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 angle 180) < -1e18

   1. Initial program 29.0%

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

      1. *-commutative 29.0%

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

      2. associate-*l* 29.0%

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

      3. associate-*l* 29.0%

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

   3. Simplified 29.0%

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

      1. unpow2 32.4%

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

      2. unpow2 32.4%

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

      3. difference-of-squares 34.1%

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

   5. Applied egg-rr 30.7%

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

      1. add-cbrt-cube 29.5%

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

      2. pow3 29.5%

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

   7. Applied egg-rr 41.1%

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

   if -1e18 < (/.f64 angle 180)

   1. Initial program 66.8%

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

      1. associate-*l* 66.8%

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

      2. associate-*l* 66.8%

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

      3. cos-neg 66.8%

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

      4. distribute-rgt-neg-out 66.8%

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

      5. distribute-frac-neg 66.8%

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

      6. neg-mul-1 66.8%

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

      7. associate-/l* 67.7%

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

      8. associate-*r/ 67.2%

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

      9. associate-/r/ 66.7%

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

      10. associate-/l* 66.7%

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

      11. metadata-eval 66.7%

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

   3. Simplified 66.7%

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

      1. unpow2 66.7%

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

      2. unpow2 66.7%

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

      3. difference-of-squares 69.8%

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

   5. Applied egg-rr 69.8%

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

      1. add-cube-cbrt 69.5%

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

      2. pow3 69.5%

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

   7. Applied egg-rr 79.8%

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

      1. add-cbrt-cube 81.8%

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

      2. pow3 81.8%

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

   9. Applied egg-rr 81.8%

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

4. Final simplification 72.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{angle}{180} \leq -1 \cdot 10^{+18}:\\ \;\;\;\;\left(\left(b - a\right) \cdot \left(b + a\right)\right) \cdot \left(\left(2 \cdot \sin \left(\pi \cdot \frac{angle}{180}\right)\right) \cdot \cos \left(\sqrt[3]{{\pi}^{3}} \cdot \frac{angle}{180}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;2 \cdot {\left(\sqrt[3]{\left(b + a\right) \cdot \left(\left(b - a\right) \cdot \left(\sin \left(\left(angle \cdot 0.005555555555555556\right) \cdot \sqrt[3]{{\pi}^{3}}\right) \cdot \cos \left(angle \cdot \left(\pi \cdot -0.005555555555555556\right)\right)\right)\right)}\right)}^{3}\\ \end{array} \]

Alternative 4: 67.1% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{angle}{180} \leq 2 \cdot 10^{+40}:\\ \;\;\;\;2 \cdot \left(\left(\left(b - a\right) \cdot \left(\sin \left(\pi \cdot \left(angle \cdot 0.005555555555555556\right)\right) \cdot \cos \left(-0.005555555555555556 \cdot \left(\pi \cdot angle\right)\right)\right)\right) \cdot \left(b + a\right)\right)\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \left(\left(\left(b - a\right) \cdot \left(b + a\right)\right) \cdot \left(\sin \left(\pi \cdot \frac{angle}{180}\right) \cdot \cos \left(angle \cdot \frac{{\left(\sqrt{\pi}\right)}^{2}}{-180}\right)\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (a b angle)
 :precision binary64
 (if (<= (/ angle 180.0) 2e+40)
   (*
    2.0
    (*
     (*
      (- b a)
      (*
       (sin (* PI (* angle 0.005555555555555556)))
       (cos (* -0.005555555555555556 (* PI angle)))))
     (+ b a)))
   (*
    2.0
    (*
     (* (- b a) (+ b a))
     (*
      (sin (* PI (/ angle 180.0)))
      (cos (* angle (/ (pow (sqrt PI) 2.0) -180.0))))))))

C

double code(double a, double b, double angle) {
	double tmp;
	if ((angle / 180.0) <= 2e+40) {
		tmp = 2.0 * (((b - a) * (sin((((double) M_PI) * (angle * 0.005555555555555556))) * cos((-0.005555555555555556 * (((double) M_PI) * angle))))) * (b + a));
	} else {
		tmp = 2.0 * (((b - a) * (b + a)) * (sin((((double) M_PI) * (angle / 180.0))) * cos((angle * (pow(sqrt(((double) M_PI)), 2.0) / -180.0)))));
	}
	return tmp;
}

Java

public static double code(double a, double b, double angle) {
	double tmp;
	if ((angle / 180.0) <= 2e+40) {
		tmp = 2.0 * (((b - a) * (Math.sin((Math.PI * (angle * 0.005555555555555556))) * Math.cos((-0.005555555555555556 * (Math.PI * angle))))) * (b + a));
	} else {
		tmp = 2.0 * (((b - a) * (b + a)) * (Math.sin((Math.PI * (angle / 180.0))) * Math.cos((angle * (Math.pow(Math.sqrt(Math.PI), 2.0) / -180.0)))));
	}
	return tmp;
}

Python

def code(a, b, angle):
	tmp = 0
	if (angle / 180.0) <= 2e+40:
		tmp = 2.0 * (((b - a) * (math.sin((math.pi * (angle * 0.005555555555555556))) * math.cos((-0.005555555555555556 * (math.pi * angle))))) * (b + a))
	else:
		tmp = 2.0 * (((b - a) * (b + a)) * (math.sin((math.pi * (angle / 180.0))) * math.cos((angle * (math.pow(math.sqrt(math.pi), 2.0) / -180.0)))))
	return tmp

Julia

function code(a, b, angle)
	tmp = 0.0
	if (Float64(angle / 180.0) <= 2e+40)
		tmp = Float64(2.0 * Float64(Float64(Float64(b - a) * Float64(sin(Float64(pi * Float64(angle * 0.005555555555555556))) * cos(Float64(-0.005555555555555556 * Float64(pi * angle))))) * Float64(b + a)));
	else
		tmp = Float64(2.0 * Float64(Float64(Float64(b - a) * Float64(b + a)) * Float64(sin(Float64(pi * Float64(angle / 180.0))) * cos(Float64(angle * Float64((sqrt(pi) ^ 2.0) / -180.0))))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, angle)
	tmp = 0.0;
	if ((angle / 180.0) <= 2e+40)
		tmp = 2.0 * (((b - a) * (sin((pi * (angle * 0.005555555555555556))) * cos((-0.005555555555555556 * (pi * angle))))) * (b + a));
	else
		tmp = 2.0 * (((b - a) * (b + a)) * (sin((pi * (angle / 180.0))) * cos((angle * ((sqrt(pi) ^ 2.0) / -180.0)))));
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, angle_] := If[LessEqual[N[(angle / 180.0), $MachinePrecision], 2e+40], N[(2.0 * N[(N[(N[(b - a), $MachinePrecision] * N[(N[Sin[N[(Pi * N[(angle * 0.005555555555555556), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * N[Cos[N[(-0.005555555555555556 * N[(Pi * angle), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(b + a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(2.0 * N[(N[(N[(b - a), $MachinePrecision] * N[(b + a), $MachinePrecision]), $MachinePrecision] * N[(N[Sin[N[(Pi * N[(angle / 180.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * N[Cos[N[(angle * N[(N[Power[N[Sqrt[Pi], $MachinePrecision], 2.0], $MachinePrecision] / -180.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 angle 180) < 2.00000000000000006e40

   1. Initial program 63.9%

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

      1. associate-*l* 63.9%

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

      2. associate-*l* 63.9%

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

      3. cos-neg 63.9%

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

      4. distribute-rgt-neg-out 63.9%

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

      5. distribute-frac-neg 63.9%

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

      6. neg-mul-1 63.9%

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

      7. associate-/l* 64.6%

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

      8. associate-*r/ 64.5%

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

      9. associate-/r/ 64.7%

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

      10. associate-/l* 64.7%

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

      11. metadata-eval 64.7%

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

   3. Simplified 64.7%

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

      1. unpow2 64.7%

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

      2. unpow2 64.7%

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

      3. difference-of-squares 68.2%

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

   5. Applied egg-rr 68.2%

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

      1. expm1-log1p-u 49.9%

         \[\leadsto 2 \cdot \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\left(\left(b + a\right) \cdot \left(b - a\right)\right) \cdot \left(\sin \left(\pi \cdot \frac{angle}{180}\right) \cdot \cos \left(\frac{\pi}{-180} \cdot angle\right)\right)\right)\right)} \]

      2. expm1-udef 33.8%

         \[\leadsto 2 \cdot \color{blue}{\left(e^{\mathsf{log1p}\left(\left(\left(b + a\right) \cdot \left(b - a\right)\right) \cdot \left(\sin \left(\pi \cdot \frac{angle}{180}\right) \cdot \cos \left(\frac{\pi}{-180} \cdot angle\right)\right)\right)} - 1\right)} \]

   7. Applied egg-rr 39.5%

      \[\leadsto 2 \cdot \color{blue}{\left(e^{\mathsf{log1p}\left(\left(b + a\right) \cdot \left(\left(b - a\right) \cdot \left(\sin \left(\pi \cdot \left(angle \cdot 0.005555555555555556\right)\right) \cdot \cos \left(angle \cdot \left(\pi \cdot -0.005555555555555556\right)\right)\right)\right)\right)} - 1\right)} \]
   8. Step-by-step derivation

      1. expm1-def 55.8%

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

      2. expm1-log1p 79.8%

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

      3. +-commutative 79.8%

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

      4. associate-*r* 79.2%

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

      5. *-commutative 79.2%

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

   9. Simplified 79.2%

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

   if 2.00000000000000006e40 < (/.f64 angle 180)

   1. Initial program 34.3%

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

      1. associate-*l* 34.3%

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

      2. associate-*l* 34.3%

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

      3. cos-neg 34.3%

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

      4. distribute-rgt-neg-out 34.3%

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

      5. distribute-frac-neg 34.3%

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

      6. neg-mul-1 34.3%

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

      7. associate-/l* 38.6%

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

      8. associate-*r/ 36.4%

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

      9. associate-/r/ 34.5%

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

      10. associate-/l* 34.5%

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

      11. metadata-eval 34.5%

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

   3. Simplified 34.5%

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

      1. unpow2 34.5%

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

      2. unpow2 34.5%

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

      3. difference-of-squares 34.5%

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

   5. Applied egg-rr 34.5%

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

      1. add-sqr-sqrt 41.2%

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

      2. pow2 41.2%

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

   7. Applied egg-rr 41.2%

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

4. Final simplification 71.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{angle}{180} \leq 2 \cdot 10^{+40}:\\ \;\;\;\;2 \cdot \left(\left(\left(b - a\right) \cdot \left(\sin \left(\pi \cdot \left(angle \cdot 0.005555555555555556\right)\right) \cdot \cos \left(-0.005555555555555556 \cdot \left(\pi \cdot angle\right)\right)\right)\right) \cdot \left(b + a\right)\right)\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \left(\left(\left(b - a\right) \cdot \left(b + a\right)\right) \cdot \left(\sin \left(\pi \cdot \frac{angle}{180}\right) \cdot \cos \left(angle \cdot \frac{{\left(\sqrt{\pi}\right)}^{2}}{-180}\right)\right)\right)\\ \end{array} \]

Alternative 5: 66.6% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{angle}{180} \leq 5 \cdot 10^{-45}:\\ \;\;\;\;2 \cdot {\left(\sqrt[3]{\left(b + a\right) \cdot \left(\left(b - a\right) \cdot \left(\sin \left(\pi \cdot \left(angle \cdot 0.005555555555555556\right)\right) \cdot \cos \left(angle \cdot \left(\pi \cdot -0.005555555555555556\right)\right)\right)\right)}\right)}^{3}\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \left(\left(\left(b - a\right) \cdot \left(b + a\right)\right) \cdot \left(\sin \left(\pi \cdot \frac{angle}{180}\right) \cdot \cos \left(angle \cdot \frac{{\left(\sqrt{\pi}\right)}^{2}}{-180}\right)\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (a b angle)
 :precision binary64
 (if (<= (/ angle 180.0) 5e-45)
   (*
    2.0
    (pow
     (cbrt
      (*
       (+ b a)
       (*
        (- b a)
        (*
         (sin (* PI (* angle 0.005555555555555556)))
         (cos (* angle (* PI -0.005555555555555556)))))))
     3.0))
   (*
    2.0
    (*
     (* (- b a) (+ b a))
     (*
      (sin (* PI (/ angle 180.0)))
      (cos (* angle (/ (pow (sqrt PI) 2.0) -180.0))))))))

C

double code(double a, double b, double angle) {
	double tmp;
	if ((angle / 180.0) <= 5e-45) {
		tmp = 2.0 * pow(cbrt(((b + a) * ((b - a) * (sin((((double) M_PI) * (angle * 0.005555555555555556))) * cos((angle * (((double) M_PI) * -0.005555555555555556))))))), 3.0);
	} else {
		tmp = 2.0 * (((b - a) * (b + a)) * (sin((((double) M_PI) * (angle / 180.0))) * cos((angle * (pow(sqrt(((double) M_PI)), 2.0) / -180.0)))));
	}
	return tmp;
}

Java

public static double code(double a, double b, double angle) {
	double tmp;
	if ((angle / 180.0) <= 5e-45) {
		tmp = 2.0 * Math.pow(Math.cbrt(((b + a) * ((b - a) * (Math.sin((Math.PI * (angle * 0.005555555555555556))) * Math.cos((angle * (Math.PI * -0.005555555555555556))))))), 3.0);
	} else {
		tmp = 2.0 * (((b - a) * (b + a)) * (Math.sin((Math.PI * (angle / 180.0))) * Math.cos((angle * (Math.pow(Math.sqrt(Math.PI), 2.0) / -180.0)))));
	}
	return tmp;
}

Julia

function code(a, b, angle)
	tmp = 0.0
	if (Float64(angle / 180.0) <= 5e-45)
		tmp = Float64(2.0 * (cbrt(Float64(Float64(b + a) * Float64(Float64(b - a) * Float64(sin(Float64(pi * Float64(angle * 0.005555555555555556))) * cos(Float64(angle * Float64(pi * -0.005555555555555556))))))) ^ 3.0));
	else
		tmp = Float64(2.0 * Float64(Float64(Float64(b - a) * Float64(b + a)) * Float64(sin(Float64(pi * Float64(angle / 180.0))) * cos(Float64(angle * Float64((sqrt(pi) ^ 2.0) / -180.0))))));
	end
	return tmp
end

Wolfram

code[a_, b_, angle_] := If[LessEqual[N[(angle / 180.0), $MachinePrecision], 5e-45], N[(2.0 * N[Power[N[Power[N[(N[(b + a), $MachinePrecision] * N[(N[(b - a), $MachinePrecision] * N[(N[Sin[N[(Pi * N[(angle * 0.005555555555555556), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * N[Cos[N[(angle * N[(Pi * -0.005555555555555556), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 1/3], $MachinePrecision], 3.0], $MachinePrecision]), $MachinePrecision], N[(2.0 * N[(N[(N[(b - a), $MachinePrecision] * N[(b + a), $MachinePrecision]), $MachinePrecision] * N[(N[Sin[N[(Pi * N[(angle / 180.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * N[Cos[N[(angle * N[(N[Power[N[Sqrt[Pi], $MachinePrecision], 2.0], $MachinePrecision] / -180.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 angle 180) < 4.99999999999999976e-45

   1. Initial program 62.0%

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

      1. associate-*l* 62.0%

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

      2. associate-*l* 62.0%

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

      3. cos-neg 62.0%

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

      4. distribute-rgt-neg-out 62.0%

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

      5. distribute-frac-neg 62.0%

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

      6. neg-mul-1 62.0%

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

      7. associate-/l* 62.9%

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

      8. associate-*r/ 62.9%

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

      9. associate-/r/ 63.1%

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

      10. associate-/l* 63.1%

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

      11. metadata-eval 63.1%

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

   3. Simplified 63.1%

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

      1. unpow2 63.1%

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

      2. unpow2 63.1%

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

      3. difference-of-squares 66.4%

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

   5. Applied egg-rr 66.4%

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

      1. add-cube-cbrt 66.1%

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

      2. pow3 66.0%

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

   7. Applied egg-rr 78.8%

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

   if 4.99999999999999976e-45 < (/.f64 angle 180)

   1. Initial program 48.4%

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

      1. associate-*l* 48.4%

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

      2. associate-*l* 48.3%

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

      3. cos-neg 48.3%

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

      4. distribute-rgt-neg-out 48.3%

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

      5. distribute-frac-neg 48.3%

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

      6. neg-mul-1 48.3%

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

      7. associate-/l* 51.0%

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

      8. associate-*r/ 49.5%

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

      9. associate-/r/ 48.2%

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

      10. associate-/l* 48.2%

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

      11. metadata-eval 48.2%

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

   3. Simplified 48.2%

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

      1. unpow2 48.2%

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

      2. unpow2 48.2%

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

      3. difference-of-squares 49.6%

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

   5. Applied egg-rr 49.6%

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

      1. add-sqr-sqrt 54.4%

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

      2. pow2 54.4%

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

   7. Applied egg-rr 54.4%

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

4. Final simplification 72.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{angle}{180} \leq 5 \cdot 10^{-45}:\\ \;\;\;\;2 \cdot {\left(\sqrt[3]{\left(b + a\right) \cdot \left(\left(b - a\right) \cdot \left(\sin \left(\pi \cdot \left(angle \cdot 0.005555555555555556\right)\right) \cdot \cos \left(angle \cdot \left(\pi \cdot -0.005555555555555556\right)\right)\right)\right)}\right)}^{3}\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \left(\left(\left(b - a\right) \cdot \left(b + a\right)\right) \cdot \left(\sin \left(\pi \cdot \frac{angle}{180}\right) \cdot \cos \left(angle \cdot \frac{{\left(\sqrt{\pi}\right)}^{2}}{-180}\right)\right)\right)\\ \end{array} \]

Alternative 6: 66.8% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{angle}{180} \leq -2 \cdot 10^{+44}:\\ \;\;\;\;\left(\left(b - a\right) \cdot \left(b + a\right)\right) \cdot \left(\left(2 \cdot \sin \left(\pi \cdot \frac{angle}{180}\right)\right) \cdot \cos \left(\sqrt[3]{{\pi}^{3}} \cdot \frac{angle}{180}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;2 \cdot {\left(\sqrt[3]{\left(b + a\right) \cdot \left(\left(b - a\right) \cdot \left(\cos \left(angle \cdot \left(\pi \cdot -0.005555555555555556\right)\right) \cdot \sin \left(0.005555555555555556 \cdot \left(\pi \cdot angle\right)\right)\right)\right)}\right)}^{3}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b angle)
 :precision binary64
 (if (<= (/ angle 180.0) -2e+44)
   (*
    (* (- b a) (+ b a))
    (*
     (* 2.0 (sin (* PI (/ angle 180.0))))
     (cos (* (cbrt (pow PI 3.0)) (/ angle 180.0)))))
   (*
    2.0
    (pow
     (cbrt
      (*
       (+ b a)
       (*
        (- b a)
        (*
         (cos (* angle (* PI -0.005555555555555556)))
         (sin (* 0.005555555555555556 (* PI angle)))))))
     3.0))))

C

double code(double a, double b, double angle) {
	double tmp;
	if ((angle / 180.0) <= -2e+44) {
		tmp = ((b - a) * (b + a)) * ((2.0 * sin((((double) M_PI) * (angle / 180.0)))) * cos((cbrt(pow(((double) M_PI), 3.0)) * (angle / 180.0))));
	} else {
		tmp = 2.0 * pow(cbrt(((b + a) * ((b - a) * (cos((angle * (((double) M_PI) * -0.005555555555555556))) * sin((0.005555555555555556 * (((double) M_PI) * angle))))))), 3.0);
	}
	return tmp;
}

Java

public static double code(double a, double b, double angle) {
	double tmp;
	if ((angle / 180.0) <= -2e+44) {
		tmp = ((b - a) * (b + a)) * ((2.0 * Math.sin((Math.PI * (angle / 180.0)))) * Math.cos((Math.cbrt(Math.pow(Math.PI, 3.0)) * (angle / 180.0))));
	} else {
		tmp = 2.0 * Math.pow(Math.cbrt(((b + a) * ((b - a) * (Math.cos((angle * (Math.PI * -0.005555555555555556))) * Math.sin((0.005555555555555556 * (Math.PI * angle))))))), 3.0);
	}
	return tmp;
}

Julia

function code(a, b, angle)
	tmp = 0.0
	if (Float64(angle / 180.0) <= -2e+44)
		tmp = Float64(Float64(Float64(b - a) * Float64(b + a)) * Float64(Float64(2.0 * sin(Float64(pi * Float64(angle / 180.0)))) * cos(Float64(cbrt((pi ^ 3.0)) * Float64(angle / 180.0)))));
	else
		tmp = Float64(2.0 * (cbrt(Float64(Float64(b + a) * Float64(Float64(b - a) * Float64(cos(Float64(angle * Float64(pi * -0.005555555555555556))) * sin(Float64(0.005555555555555556 * Float64(pi * angle))))))) ^ 3.0));
	end
	return tmp
end

Wolfram

code[a_, b_, angle_] := If[LessEqual[N[(angle / 180.0), $MachinePrecision], -2e+44], N[(N[(N[(b - a), $MachinePrecision] * N[(b + a), $MachinePrecision]), $MachinePrecision] * N[(N[(2.0 * N[Sin[N[(Pi * N[(angle / 180.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[Cos[N[(N[Power[N[Power[Pi, 3.0], $MachinePrecision], 1/3], $MachinePrecision] * N[(angle / 180.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(2.0 * N[Power[N[Power[N[(N[(b + a), $MachinePrecision] * N[(N[(b - a), $MachinePrecision] * N[(N[Cos[N[(angle * N[(Pi * -0.005555555555555556), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * N[Sin[N[(0.005555555555555556 * N[(Pi * angle), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 1/3], $MachinePrecision], 3.0], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 angle 180) < -2.0000000000000002e44

   1. Initial program 27.9%

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

      1. *-commutative 27.9%

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

      2. associate-*l* 27.9%

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

      3. associate-*l* 27.9%

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

   3. Simplified 27.9%

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

      1. unpow2 33.7%

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

      2. unpow2 33.7%

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

      3. difference-of-squares 35.7%

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

   5. Applied egg-rr 29.9%

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

      1. add-cbrt-cube 30.9%

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

      2. pow3 30.9%

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

   7. Applied egg-rr 43.7%

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

   if -2.0000000000000002e44 < (/.f64 angle 180)

   1. Initial program 65.8%

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

      1. associate-*l* 65.8%

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

      2. associate-*l* 65.8%

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

      3. cos-neg 65.8%

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

      4. distribute-rgt-neg-out 65.8%

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

      5. distribute-frac-neg 65.8%

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

      6. neg-mul-1 65.8%

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

      7. associate-/l* 66.6%

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

      8. associate-*r/ 65.6%

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

      9. associate-/r/ 65.2%

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

      10. associate-/l* 65.2%

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

      11. metadata-eval 65.2%

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

   3. Simplified 65.2%

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

      1. unpow2 65.2%

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

      2. unpow2 65.2%

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

      3. difference-of-squares 68.2%

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

   5. Applied egg-rr 68.2%

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

      1. add-cube-cbrt 67.9%

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

      2. pow3 67.9%

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

   7. Applied egg-rr 77.9%

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

   8. Taylor expanded in angle around inf 79.3%

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

4. Final simplification 72.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{angle}{180} \leq -2 \cdot 10^{+44}:\\ \;\;\;\;\left(\left(b - a\right) \cdot \left(b + a\right)\right) \cdot \left(\left(2 \cdot \sin \left(\pi \cdot \frac{angle}{180}\right)\right) \cdot \cos \left(\sqrt[3]{{\pi}^{3}} \cdot \frac{angle}{180}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;2 \cdot {\left(\sqrt[3]{\left(b + a\right) \cdot \left(\left(b - a\right) \cdot \left(\cos \left(angle \cdot \left(\pi \cdot -0.005555555555555556\right)\right) \cdot \sin \left(0.005555555555555556 \cdot \left(\pi \cdot angle\right)\right)\right)\right)}\right)}^{3}\\ \end{array} \]

Alternative 7: 66.2% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \sin \left(\pi \cdot \left(angle \cdot 0.005555555555555556\right)\right)\\ \mathbf{if}\;\frac{angle}{180} \leq -5 \cdot 10^{-34}:\\ \;\;\;\;2 \cdot \left(\cos \left(angle \cdot \left(\pi \cdot -0.005555555555555556\right)\right) \cdot \left(t_0 \cdot \left(\left(b - a\right) \cdot \left(b + a\right)\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;2 \cdot {\left(\sqrt[3]{\left(b + a\right) \cdot \left(\left(b - a\right) \cdot t_0\right)}\right)}^{3}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b angle)
 :precision binary64
 (let* ((t_0 (sin (* PI (* angle 0.005555555555555556)))))
   (if (<= (/ angle 180.0) -5e-34)
     (*
      2.0
      (*
       (cos (* angle (* PI -0.005555555555555556)))
       (* t_0 (* (- b a) (+ b a)))))
     (* 2.0 (pow (cbrt (* (+ b a) (* (- b a) t_0))) 3.0)))))

C

double code(double a, double b, double angle) {
	double t_0 = sin((((double) M_PI) * (angle * 0.005555555555555556)));
	double tmp;
	if ((angle / 180.0) <= -5e-34) {
		tmp = 2.0 * (cos((angle * (((double) M_PI) * -0.005555555555555556))) * (t_0 * ((b - a) * (b + a))));
	} else {
		tmp = 2.0 * pow(cbrt(((b + a) * ((b - a) * t_0))), 3.0);
	}
	return tmp;
}

Java

public static double code(double a, double b, double angle) {
	double t_0 = Math.sin((Math.PI * (angle * 0.005555555555555556)));
	double tmp;
	if ((angle / 180.0) <= -5e-34) {
		tmp = 2.0 * (Math.cos((angle * (Math.PI * -0.005555555555555556))) * (t_0 * ((b - a) * (b + a))));
	} else {
		tmp = 2.0 * Math.pow(Math.cbrt(((b + a) * ((b - a) * t_0))), 3.0);
	}
	return tmp;
}

Julia

function code(a, b, angle)
	t_0 = sin(Float64(pi * Float64(angle * 0.005555555555555556)))
	tmp = 0.0
	if (Float64(angle / 180.0) <= -5e-34)
		tmp = Float64(2.0 * Float64(cos(Float64(angle * Float64(pi * -0.005555555555555556))) * Float64(t_0 * Float64(Float64(b - a) * Float64(b + a)))));
	else
		tmp = Float64(2.0 * (cbrt(Float64(Float64(b + a) * Float64(Float64(b - a) * t_0))) ^ 3.0));
	end
	return tmp
end

Wolfram

code[a_, b_, angle_] := Block[{t$95$0 = N[Sin[N[(Pi * N[(angle * 0.005555555555555556), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[N[(angle / 180.0), $MachinePrecision], -5e-34], N[(2.0 * N[(N[Cos[N[(angle * N[(Pi * -0.005555555555555556), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * N[(t$95$0 * N[(N[(b - a), $MachinePrecision] * N[(b + a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(2.0 * N[Power[N[Power[N[(N[(b + a), $MachinePrecision] * N[(N[(b - a), $MachinePrecision] * t$95$0), $MachinePrecision]), $MachinePrecision], 1/3], $MachinePrecision], 3.0], $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 angle 180) < -5.0000000000000003e-34

   1. Initial program 40.1%

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

      1. associate-*l* 40.1%

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

      2. associate-*l* 40.1%

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

      3. cos-neg 40.1%

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

      4. distribute-rgt-neg-out 40.1%

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

      5. distribute-frac-neg 40.1%

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

      6. neg-mul-1 40.1%

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

      7. associate-/l* 42.3%

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

      8. associate-*r/ 42.2%

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

      9. associate-/r/ 42.8%

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

      10. associate-/l* 42.8%

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

      11. metadata-eval 42.8%

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

   3. Simplified 42.8%

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

      1. unpow2 42.8%

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

      2. unpow2 42.8%

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

      3. difference-of-squares 45.5%

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

   5. Applied egg-rr 45.5%

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

   6. Taylor expanded in angle around inf 40.5%

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

   7. Taylor expanded in angle around inf 37.4%

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

      1. *-commutative 37.4%

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

      2. associate-*r* 46.0%

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

      3. +-commutative 46.0%

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

      4. *-commutative 46.0%

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

      5. +-commutative 46.0%

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

      6. *-commutative 46.0%

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

      7. *-commutative 46.0%

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

      8. *-commutative 46.0%

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

      9. associate-*r* 47.8%

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

   9. Simplified 47.8%

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

   if -5.0000000000000003e-34 < (/.f64 angle 180)

   1. Initial program 65.3%

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

      1. associate-*l* 65.3%

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

      2. associate-*l* 65.3%

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

      3. cos-neg 65.3%

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

      4. distribute-rgt-neg-out 65.3%

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

      5. distribute-frac-neg 65.3%

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

      6. neg-mul-1 65.3%

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

      7. associate-/l* 66.4%

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

      8. associate-*r/ 65.8%

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

      9. associate-/r/ 65.3%

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

      10. associate-/l* 65.3%

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

      11. metadata-eval 65.3%

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

   3. Simplified 65.3%

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

      1. unpow2 65.3%

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

      2. unpow2 65.3%

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

      3. difference-of-squares 68.1%

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

   5. Applied egg-rr 68.1%

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

      1. add-cube-cbrt 67.8%

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

      2. pow3 67.7%

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

   7. Applied egg-rr 78.9%

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

   8. Taylor expanded in angle around 0 79.7%

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

4. Final simplification 70.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{angle}{180} \leq -5 \cdot 10^{-34}:\\ \;\;\;\;2 \cdot \left(\cos \left(angle \cdot \left(\pi \cdot -0.005555555555555556\right)\right) \cdot \left(\sin \left(\pi \cdot \left(angle \cdot 0.005555555555555556\right)\right) \cdot \left(\left(b - a\right) \cdot \left(b + a\right)\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;2 \cdot {\left(\sqrt[3]{\left(b + a\right) \cdot \left(\left(b - a\right) \cdot \sin \left(\pi \cdot \left(angle \cdot 0.005555555555555556\right)\right)\right)}\right)}^{3}\\ \end{array} \]

Alternative 8: 67.3% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{angle}{180} \leq 40000000000000:\\ \;\;\;\;2 \cdot \left(\left(\left(b - a\right) \cdot \left(\sin \left(\pi \cdot \left(angle \cdot 0.005555555555555556\right)\right) \cdot \cos \left(-0.005555555555555556 \cdot \left(\pi \cdot angle\right)\right)\right)\right) \cdot \left(b + a\right)\right)\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \left(\left(\left(b - a\right) \cdot \left(b + a\right)\right) \cdot \sin \left(0.005555555555555556 \cdot \left(\pi \cdot angle\right)\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (a b angle)
 :precision binary64
 (if (<= (/ angle 180.0) 40000000000000.0)
   (*
    2.0
    (*
     (*
      (- b a)
      (*
       (sin (* PI (* angle 0.005555555555555556)))
       (cos (* -0.005555555555555556 (* PI angle)))))
     (+ b a)))
   (*
    2.0
    (* (* (- b a) (+ b a)) (sin (* 0.005555555555555556 (* PI angle)))))))

C

double code(double a, double b, double angle) {
	double tmp;
	if ((angle / 180.0) <= 40000000000000.0) {
		tmp = 2.0 * (((b - a) * (sin((((double) M_PI) * (angle * 0.005555555555555556))) * cos((-0.005555555555555556 * (((double) M_PI) * angle))))) * (b + a));
	} else {
		tmp = 2.0 * (((b - a) * (b + a)) * sin((0.005555555555555556 * (((double) M_PI) * angle))));
	}
	return tmp;
}

Java

public static double code(double a, double b, double angle) {
	double tmp;
	if ((angle / 180.0) <= 40000000000000.0) {
		tmp = 2.0 * (((b - a) * (Math.sin((Math.PI * (angle * 0.005555555555555556))) * Math.cos((-0.005555555555555556 * (Math.PI * angle))))) * (b + a));
	} else {
		tmp = 2.0 * (((b - a) * (b + a)) * Math.sin((0.005555555555555556 * (Math.PI * angle))));
	}
	return tmp;
}

Python

def code(a, b, angle):
	tmp = 0
	if (angle / 180.0) <= 40000000000000.0:
		tmp = 2.0 * (((b - a) * (math.sin((math.pi * (angle * 0.005555555555555556))) * math.cos((-0.005555555555555556 * (math.pi * angle))))) * (b + a))
	else:
		tmp = 2.0 * (((b - a) * (b + a)) * math.sin((0.005555555555555556 * (math.pi * angle))))
	return tmp

Julia

function code(a, b, angle)
	tmp = 0.0
	if (Float64(angle / 180.0) <= 40000000000000.0)
		tmp = Float64(2.0 * Float64(Float64(Float64(b - a) * Float64(sin(Float64(pi * Float64(angle * 0.005555555555555556))) * cos(Float64(-0.005555555555555556 * Float64(pi * angle))))) * Float64(b + a)));
	else
		tmp = Float64(2.0 * Float64(Float64(Float64(b - a) * Float64(b + a)) * sin(Float64(0.005555555555555556 * Float64(pi * angle)))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, angle)
	tmp = 0.0;
	if ((angle / 180.0) <= 40000000000000.0)
		tmp = 2.0 * (((b - a) * (sin((pi * (angle * 0.005555555555555556))) * cos((-0.005555555555555556 * (pi * angle))))) * (b + a));
	else
		tmp = 2.0 * (((b - a) * (b + a)) * sin((0.005555555555555556 * (pi * angle))));
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, angle_] := If[LessEqual[N[(angle / 180.0), $MachinePrecision], 40000000000000.0], N[(2.0 * N[(N[(N[(b - a), $MachinePrecision] * N[(N[Sin[N[(Pi * N[(angle * 0.005555555555555556), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * N[Cos[N[(-0.005555555555555556 * N[(Pi * angle), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(b + a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(2.0 * N[(N[(N[(b - a), $MachinePrecision] * N[(b + a), $MachinePrecision]), $MachinePrecision] * N[Sin[N[(0.005555555555555556 * N[(Pi * angle), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 angle 180) < 4e13

   1. Initial program 64.4%

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

      1. associate-*l* 64.4%

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

      2. associate-*l* 64.4%

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

      3. cos-neg 64.4%

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

      4. distribute-rgt-neg-out 64.4%

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

      5. distribute-frac-neg 64.4%

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

      6. neg-mul-1 64.4%

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

      7. associate-/l* 65.1%

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

      8. associate-*r/ 65.1%

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

      9. associate-/r/ 65.3%

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

      10. associate-/l* 65.3%

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

      11. metadata-eval 65.3%

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

   3. Simplified 65.3%

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

      1. unpow2 65.3%

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

      2. unpow2 65.3%

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

      3. difference-of-squares 68.9%

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

   5. Applied egg-rr 68.9%

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

      1. expm1-log1p-u 50.1%

         \[\leadsto 2 \cdot \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\left(\left(b + a\right) \cdot \left(b - a\right)\right) \cdot \left(\sin \left(\pi \cdot \frac{angle}{180}\right) \cdot \cos \left(\frac{\pi}{-180} \cdot angle\right)\right)\right)\right)} \]

      2. expm1-udef 33.6%

         \[\leadsto 2 \cdot \color{blue}{\left(e^{\mathsf{log1p}\left(\left(\left(b + a\right) \cdot \left(b - a\right)\right) \cdot \left(\sin \left(\pi \cdot \frac{angle}{180}\right) \cdot \cos \left(\frac{\pi}{-180} \cdot angle\right)\right)\right)} - 1\right)} \]

   7. Applied egg-rr 39.4%

      \[\leadsto 2 \cdot \color{blue}{\left(e^{\mathsf{log1p}\left(\left(b + a\right) \cdot \left(\left(b - a\right) \cdot \left(\sin \left(\pi \cdot \left(angle \cdot 0.005555555555555556\right)\right) \cdot \cos \left(angle \cdot \left(\pi \cdot -0.005555555555555556\right)\right)\right)\right)\right)} - 1\right)} \]
   8. Step-by-step derivation

      1. expm1-def 56.0%

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

      2. expm1-log1p 80.7%

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

      3. +-commutative 80.7%

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

      4. associate-*r* 80.0%

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

      5. *-commutative 80.0%

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

   9. Simplified 80.0%

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

   if 4e13 < (/.f64 angle 180)

   1. Initial program 35.3%

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

      1. associate-*l* 35.3%

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

      2. associate-*l* 35.3%

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

      3. cos-neg 35.3%

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

      4. distribute-rgt-neg-out 35.3%

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

      5. distribute-frac-neg 35.3%

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

      6. neg-mul-1 35.3%

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

      7. associate-/l* 38.8%

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

      8. associate-*r/ 36.9%

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

      9. associate-/r/ 35.2%

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

      10. associate-/l* 35.2%

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

      11. metadata-eval 35.2%

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

   3. Simplified 35.2%

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

      1. unpow2 35.2%

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

      2. unpow2 35.2%

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

      3. difference-of-squares 35.2%

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

   5. Applied egg-rr 35.2%

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

   6. Taylor expanded in angle around inf 35.9%

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

   7. Taylor expanded in angle around 0 39.5%

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

4. Final simplification 71.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{angle}{180} \leq 40000000000000:\\ \;\;\;\;2 \cdot \left(\left(\left(b - a\right) \cdot \left(\sin \left(\pi \cdot \left(angle \cdot 0.005555555555555556\right)\right) \cdot \cos \left(-0.005555555555555556 \cdot \left(\pi \cdot angle\right)\right)\right)\right) \cdot \left(b + a\right)\right)\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \left(\left(\left(b - a\right) \cdot \left(b + a\right)\right) \cdot \sin \left(0.005555555555555556 \cdot \left(\pi \cdot angle\right)\right)\right)\\ \end{array} \]

Alternative 9: 66.2% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{angle}{180} \leq -5 \cdot 10^{-34}:\\ \;\;\;\;\left(\left(b - a\right) \cdot \left(b + a\right)\right) \cdot \sin \left(\pi \cdot \left(2 \cdot \left(angle \cdot 0.005555555555555556\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;2 \cdot {\left(\sqrt[3]{\left(b + a\right) \cdot \left(\left(b - a\right) \cdot \sin \left(\pi \cdot \left(angle \cdot 0.005555555555555556\right)\right)\right)}\right)}^{3}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b angle)
 :precision binary64
 (if (<= (/ angle 180.0) -5e-34)
   (* (* (- b a) (+ b a)) (sin (* PI (* 2.0 (* angle 0.005555555555555556)))))
   (*
    2.0
    (pow
     (cbrt (* (+ b a) (* (- b a) (sin (* PI (* angle 0.005555555555555556))))))
     3.0))))

C

double code(double a, double b, double angle) {
	double tmp;
	if ((angle / 180.0) <= -5e-34) {
		tmp = ((b - a) * (b + a)) * sin((((double) M_PI) * (2.0 * (angle * 0.005555555555555556))));
	} else {
		tmp = 2.0 * pow(cbrt(((b + a) * ((b - a) * sin((((double) M_PI) * (angle * 0.005555555555555556)))))), 3.0);
	}
	return tmp;
}

Java

public static double code(double a, double b, double angle) {
	double tmp;
	if ((angle / 180.0) <= -5e-34) {
		tmp = ((b - a) * (b + a)) * Math.sin((Math.PI * (2.0 * (angle * 0.005555555555555556))));
	} else {
		tmp = 2.0 * Math.pow(Math.cbrt(((b + a) * ((b - a) * Math.sin((Math.PI * (angle * 0.005555555555555556)))))), 3.0);
	}
	return tmp;
}

Julia

function code(a, b, angle)
	tmp = 0.0
	if (Float64(angle / 180.0) <= -5e-34)
		tmp = Float64(Float64(Float64(b - a) * Float64(b + a)) * sin(Float64(pi * Float64(2.0 * Float64(angle * 0.005555555555555556)))));
	else
		tmp = Float64(2.0 * (cbrt(Float64(Float64(b + a) * Float64(Float64(b - a) * sin(Float64(pi * Float64(angle * 0.005555555555555556)))))) ^ 3.0));
	end
	return tmp
end

Wolfram

code[a_, b_, angle_] := If[LessEqual[N[(angle / 180.0), $MachinePrecision], -5e-34], N[(N[(N[(b - a), $MachinePrecision] * N[(b + a), $MachinePrecision]), $MachinePrecision] * N[Sin[N[(Pi * N[(2.0 * N[(angle * 0.005555555555555556), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(2.0 * N[Power[N[Power[N[(N[(b + a), $MachinePrecision] * N[(N[(b - a), $MachinePrecision] * N[Sin[N[(Pi * N[(angle * 0.005555555555555556), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 1/3], $MachinePrecision], 3.0], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 angle 180) < -5.0000000000000003e-34

   1. Initial program 40.1%

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

      1. *-commutative 40.1%

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

      2. associate-*l* 40.1%

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

      3. associate-*l* 40.1%

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

   3. Simplified 40.1%

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

      1. unpow2 42.8%

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

      2. unpow2 42.8%

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

      3. difference-of-squares 45.5%

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

   5. Applied egg-rr 42.8%

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

      1. expm1-log1p-u 22.5%

         \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\left(\left(b + a\right) \cdot \left(b - a\right)\right) \cdot \left(\left(2 \cdot \sin \left(\pi \cdot \frac{angle}{180}\right)\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right)\right)\right)\right)} \]

      2. expm1-udef 18.5%

         \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(\left(\left(b + a\right) \cdot \left(b - a\right)\right) \cdot \left(\left(2 \cdot \sin \left(\pi \cdot \frac{angle}{180}\right)\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right)\right)\right)} - 1} \]

   7. Applied egg-rr 19.8%

      \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(\left(\left(b + a\right) \cdot \left(b - a\right)\right) \cdot \sin \left(\left(\pi \cdot \left(angle \cdot 0.005555555555555556\right)\right) \cdot 2\right)\right)} - 1} \]
   8. Step-by-step derivation

      1. expm1-def 24.0%

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

      2. expm1-log1p 46.6%

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

      3. *-commutative 46.6%

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

      4. +-commutative 46.6%

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

      5. associate-*l* 46.6%

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

   9. Simplified 46.6%

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

   if -5.0000000000000003e-34 < (/.f64 angle 180)

   1. Initial program 65.3%

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

      1. associate-*l* 65.3%

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

      2. associate-*l* 65.3%

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

      3. cos-neg 65.3%

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

      4. distribute-rgt-neg-out 65.3%

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

      5. distribute-frac-neg 65.3%

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

      6. neg-mul-1 65.3%

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

      7. associate-/l* 66.4%

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

      8. associate-*r/ 65.8%

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

      9. associate-/r/ 65.3%

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

      10. associate-/l* 65.3%

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

      11. metadata-eval 65.3%

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

   3. Simplified 65.3%

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

      1. unpow2 65.3%

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

      2. unpow2 65.3%

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

      3. difference-of-squares 68.1%

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

   5. Applied egg-rr 68.1%

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

      1. add-cube-cbrt 67.8%

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

      2. pow3 67.7%

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

   7. Applied egg-rr 78.9%

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

   8. Taylor expanded in angle around 0 79.7%

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

4. Final simplification 70.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{angle}{180} \leq -5 \cdot 10^{-34}:\\ \;\;\;\;\left(\left(b - a\right) \cdot \left(b + a\right)\right) \cdot \sin \left(\pi \cdot \left(2 \cdot \left(angle \cdot 0.005555555555555556\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;2 \cdot {\left(\sqrt[3]{\left(b + a\right) \cdot \left(\left(b - a\right) \cdot \sin \left(\pi \cdot \left(angle \cdot 0.005555555555555556\right)\right)\right)}\right)}^{3}\\ \end{array} \]

Alternative 10: 58.5% accurate, 2.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(b - a\right) \cdot \left(b + a\right)\\ \mathbf{if}\;a \leq 1.15 \cdot 10^{+232}:\\ \;\;\;\;t_0 \cdot \sin \left(\pi \cdot \left(2 \cdot \left(angle \cdot 0.005555555555555556\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \left(t_0 \cdot \sin \left(0.005555555555555556 \cdot \left(\pi \cdot angle\right)\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (a b angle)
 :precision binary64
 (let* ((t_0 (* (- b a) (+ b a))))
   (if (<= a 1.15e+232)
     (* t_0 (sin (* PI (* 2.0 (* angle 0.005555555555555556)))))
     (* 2.0 (* t_0 (sin (* 0.005555555555555556 (* PI angle))))))))

C

double code(double a, double b, double angle) {
	double t_0 = (b - a) * (b + a);
	double tmp;
	if (a <= 1.15e+232) {
		tmp = t_0 * sin((((double) M_PI) * (2.0 * (angle * 0.005555555555555556))));
	} else {
		tmp = 2.0 * (t_0 * sin((0.005555555555555556 * (((double) M_PI) * angle))));
	}
	return tmp;
}

Java

public static double code(double a, double b, double angle) {
	double t_0 = (b - a) * (b + a);
	double tmp;
	if (a <= 1.15e+232) {
		tmp = t_0 * Math.sin((Math.PI * (2.0 * (angle * 0.005555555555555556))));
	} else {
		tmp = 2.0 * (t_0 * Math.sin((0.005555555555555556 * (Math.PI * angle))));
	}
	return tmp;
}

Python

def code(a, b, angle):
	t_0 = (b - a) * (b + a)
	tmp = 0
	if a <= 1.15e+232:
		tmp = t_0 * math.sin((math.pi * (2.0 * (angle * 0.005555555555555556))))
	else:
		tmp = 2.0 * (t_0 * math.sin((0.005555555555555556 * (math.pi * angle))))
	return tmp

Julia

function code(a, b, angle)
	t_0 = Float64(Float64(b - a) * Float64(b + a))
	tmp = 0.0
	if (a <= 1.15e+232)
		tmp = Float64(t_0 * sin(Float64(pi * Float64(2.0 * Float64(angle * 0.005555555555555556)))));
	else
		tmp = Float64(2.0 * Float64(t_0 * sin(Float64(0.005555555555555556 * Float64(pi * angle)))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, angle)
	t_0 = (b - a) * (b + a);
	tmp = 0.0;
	if (a <= 1.15e+232)
		tmp = t_0 * sin((pi * (2.0 * (angle * 0.005555555555555556))));
	else
		tmp = 2.0 * (t_0 * sin((0.005555555555555556 * (pi * angle))));
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, angle_] := Block[{t$95$0 = N[(N[(b - a), $MachinePrecision] * N[(b + a), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[a, 1.15e+232], N[(t$95$0 * N[Sin[N[(Pi * N[(2.0 * N[(angle * 0.005555555555555556), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(2.0 * N[(t$95$0 * N[Sin[N[(0.005555555555555556 * N[(Pi * angle), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if a < 1.15000000000000003e232

   1. Initial program 58.0%

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

      1. *-commutative 58.0%

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

      2. associate-*l* 58.0%

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

      3. associate-*l* 58.0%

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

   3. Simplified 58.0%

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

      1. unpow2 58.7%

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

      2. unpow2 58.7%

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

      3. difference-of-squares 61.7%

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

   5. Applied egg-rr 60.9%

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

      1. expm1-log1p-u 43.8%

         \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\left(\left(b + a\right) \cdot \left(b - a\right)\right) \cdot \left(\left(2 \cdot \sin \left(\pi \cdot \frac{angle}{180}\right)\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right)\right)\right)\right)} \]

      2. expm1-udef 30.0%

         \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(\left(\left(b + a\right) \cdot \left(b - a\right)\right) \cdot \left(\left(2 \cdot \sin \left(\pi \cdot \frac{angle}{180}\right)\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right)\right)\right)} - 1} \]

   7. Applied egg-rr 30.8%

      \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(\left(\left(b + a\right) \cdot \left(b - a\right)\right) \cdot \sin \left(\left(\pi \cdot \left(angle \cdot 0.005555555555555556\right)\right) \cdot 2\right)\right)} - 1} \]
   8. Step-by-step derivation

      1. expm1-def 44.7%

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

      2. expm1-log1p 61.8%

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

      3. *-commutative 61.8%

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

      4. +-commutative 61.8%

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

      5. associate-*l* 61.8%

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

   9. Simplified 61.8%

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

   if 1.15000000000000003e232 < a

   1. Initial program 63.6%

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

      1. associate-*l* 63.6%

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

      2. associate-*l* 63.6%

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

      3. cos-neg 63.6%

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

      4. distribute-rgt-neg-out 63.6%

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

      5. distribute-frac-neg 63.6%

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

      6. neg-mul-1 63.6%

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

      7. associate-/l* 72.7%

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

      8. associate-*r/ 63.6%

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

      9. associate-/r/ 63.6%

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

      10. associate-/l* 63.6%

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

      11. metadata-eval 63.6%

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

   3. Simplified 63.6%

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

      1. unpow2 63.6%

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

      2. unpow2 63.6%

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

      3. difference-of-squares 63.6%

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

   5. Applied egg-rr 63.6%

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

   6. Taylor expanded in angle around inf 36.4%

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

   7. Taylor expanded in angle around 0 72.7%

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

4. Final simplification 62.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \leq 1.15 \cdot 10^{+232}:\\ \;\;\;\;\left(\left(b - a\right) \cdot \left(b + a\right)\right) \cdot \sin \left(\pi \cdot \left(2 \cdot \left(angle \cdot 0.005555555555555556\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \left(\left(\left(b - a\right) \cdot \left(b + a\right)\right) \cdot \sin \left(0.005555555555555556 \cdot \left(\pi \cdot angle\right)\right)\right)\\ \end{array} \]

Alternative 11: 58.3% accurate, 2.9× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 58.2%

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

   1. *-commutative 58.2%

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

   2. associate-*l* 58.2%

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

   3. associate-*l* 58.2%

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

3. Simplified 58.2%

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

   1. unpow2 58.9%

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

   2. unpow2 58.9%

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

   3. difference-of-squares 61.7%

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

5. Applied egg-rr 61.0%

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

   1. expm1-log1p-u 43.0%

      \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\left(\left(b + a\right) \cdot \left(b - a\right)\right) \cdot \left(\left(2 \cdot \sin \left(\pi \cdot \frac{angle}{180}\right)\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right)\right)\right)\right)} \]

   2. expm1-udef 29.8%

      \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(\left(\left(b + a\right) \cdot \left(b - a\right)\right) \cdot \left(\left(2 \cdot \sin \left(\pi \cdot \frac{angle}{180}\right)\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right)\right)\right)} - 1} \]

7. Applied egg-rr 30.2%

   \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(\left(\left(b + a\right) \cdot \left(b - a\right)\right) \cdot \sin \left(\left(\pi \cdot \left(angle \cdot 0.005555555555555556\right)\right) \cdot 2\right)\right)} - 1} \]
8. Step-by-step derivation

   1. expm1-def 43.6%

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

   2. expm1-log1p 61.1%

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

   3. *-commutative 61.1%

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

   4. +-commutative 61.1%

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

   5. associate-*l* 61.1%

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

9. Simplified 61.1%

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

10. Final simplification 61.1%

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

Alternative 12: 55.0% accurate, 5.5× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 58.2%

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

3. Simplified 57.1%

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

4. Taylor expanded in angle around 0 53.3%

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

   1. unpow2 53.3%

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

   2. unpow2 53.3%

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

   3. difference-of-squares 55.7%

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

6. Applied egg-rr 55.7%

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

7. Final simplification 55.7%

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

Alternative 13: 55.0% accurate, 5.5× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 58.2%

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

   1. *-commutative 58.2%

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

   2. associate-*l* 58.2%

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

   3. associate-*l* 58.2%

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

3. Simplified 58.2%

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

   1. unpow2 58.9%

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

   2. unpow2 58.9%

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

   3. difference-of-squares 61.7%

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

5. Applied egg-rr 61.0%

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

6. Taylor expanded in angle around 0 55.7%

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

   1. associate-*r* 55.8%

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

   2. +-commutative 55.8%

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

   3. *-commutative 55.8%

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

   4. +-commutative 55.8%

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

8. Simplified 55.8%

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

9. Final simplification 55.8%

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

Reproduce

herbie shell --seed 2023309 
(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)))))