ab-angle->ABCF B

Percentage Accurate: 54.5% → 67.2%

Time: 24.6s

Alternatives: 14

Speedup: 5.5×

• 32.6% of points produce a very large (infinite) output. You may want to add a precondition.

Specification

Math

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

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: 54.5% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 67.2% accurate, 0.9× speedup

Math

\[\begin{array}{l} a = |a|\\ b = |b|\\ \\ \begin{array}{l} t_0 := \pi \cdot \left(angle \cdot 0.005555555555555556\right)\\ t_1 := 0.005555555555555556 \cdot \left(angle \cdot \pi\right)\\ t_2 := \left(b - a\right) \cdot \cos t_0\\ \mathbf{if}\;\frac{angle}{180} \leq -2 \cdot 10^{+22}:\\ \;\;\;\;2 \cdot \left(t_2 \cdot \left(\sin \left({\left({\left(\sqrt[3]{t_0}\right)}^{2}\right)}^{1.5}\right) \cdot \left(b + a\right)\right)\right)\\ \mathbf{elif}\;\frac{angle}{180} \leq 10^{+175}:\\ \;\;\;\;2 \cdot \left(t_2 \cdot \left(\left(b + a\right) \cdot \sin t_1\right)\right)\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \left(\left(b - a\right) \cdot \left(\cos t_1 \cdot \left(\left(b + a\right) \cdot \sin \left(0.005555555555555556 \cdot \mathsf{expm1}\left(\mathsf{log1p}\left(angle \cdot \pi\right)\right)\right)\right)\right)\right)\\ \end{array} \end{array} \]

FPCore

NOTE: a should be positive before calling this function
NOTE: b should be positive before calling this function
(FPCore (a b angle)
 :precision binary64
 (let* ((t_0 (* PI (* angle 0.005555555555555556)))
        (t_1 (* 0.005555555555555556 (* angle PI)))
        (t_2 (* (- b a) (cos t_0))))
   (if (<= (/ angle 180.0) -2e+22)
     (* 2.0 (* t_2 (* (sin (pow (pow (cbrt t_0) 2.0) 1.5)) (+ b a))))
     (if (<= (/ angle 180.0) 1e+175)
       (* 2.0 (* t_2 (* (+ b a) (sin t_1))))
       (*
        2.0
        (*
         (- b a)
         (*
          (cos t_1)
          (*
           (+ b a)
           (sin (* 0.005555555555555556 (expm1 (log1p (* angle PI)))))))))))))

C

a = abs(a);
b = abs(b);
double code(double a, double b, double angle) {
	double t_0 = ((double) M_PI) * (angle * 0.005555555555555556);
	double t_1 = 0.005555555555555556 * (angle * ((double) M_PI));
	double t_2 = (b - a) * cos(t_0);
	double tmp;
	if ((angle / 180.0) <= -2e+22) {
		tmp = 2.0 * (t_2 * (sin(pow(pow(cbrt(t_0), 2.0), 1.5)) * (b + a)));
	} else if ((angle / 180.0) <= 1e+175) {
		tmp = 2.0 * (t_2 * ((b + a) * sin(t_1)));
	} else {
		tmp = 2.0 * ((b - a) * (cos(t_1) * ((b + a) * sin((0.005555555555555556 * expm1(log1p((angle * ((double) M_PI)))))))));
	}
	return tmp;
}

Java

a = Math.abs(a);
b = Math.abs(b);
public static double code(double a, double b, double angle) {
	double t_0 = Math.PI * (angle * 0.005555555555555556);
	double t_1 = 0.005555555555555556 * (angle * Math.PI);
	double t_2 = (b - a) * Math.cos(t_0);
	double tmp;
	if ((angle / 180.0) <= -2e+22) {
		tmp = 2.0 * (t_2 * (Math.sin(Math.pow(Math.pow(Math.cbrt(t_0), 2.0), 1.5)) * (b + a)));
	} else if ((angle / 180.0) <= 1e+175) {
		tmp = 2.0 * (t_2 * ((b + a) * Math.sin(t_1)));
	} else {
		tmp = 2.0 * ((b - a) * (Math.cos(t_1) * ((b + a) * Math.sin((0.005555555555555556 * Math.expm1(Math.log1p((angle * Math.PI))))))));
	}
	return tmp;
}

Julia

a = abs(a)
b = abs(b)
function code(a, b, angle)
	t_0 = Float64(pi * Float64(angle * 0.005555555555555556))
	t_1 = Float64(0.005555555555555556 * Float64(angle * pi))
	t_2 = Float64(Float64(b - a) * cos(t_0))
	tmp = 0.0
	if (Float64(angle / 180.0) <= -2e+22)
		tmp = Float64(2.0 * Float64(t_2 * Float64(sin(((cbrt(t_0) ^ 2.0) ^ 1.5)) * Float64(b + a))));
	elseif (Float64(angle / 180.0) <= 1e+175)
		tmp = Float64(2.0 * Float64(t_2 * Float64(Float64(b + a) * sin(t_1))));
	else
		tmp = Float64(2.0 * Float64(Float64(b - a) * Float64(cos(t_1) * Float64(Float64(b + a) * sin(Float64(0.005555555555555556 * expm1(log1p(Float64(angle * pi)))))))));
	end
	return tmp
end

Wolfram

NOTE: a should be positive before calling this function
NOTE: b should be positive before calling this function
code[a_, b_, angle_] := Block[{t$95$0 = N[(Pi * N[(angle * 0.005555555555555556), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(0.005555555555555556 * N[(angle * Pi), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(N[(b - a), $MachinePrecision] * N[Cos[t$95$0], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[(angle / 180.0), $MachinePrecision], -2e+22], N[(2.0 * N[(t$95$2 * N[(N[Sin[N[Power[N[Power[N[Power[t$95$0, 1/3], $MachinePrecision], 2.0], $MachinePrecision], 1.5], $MachinePrecision]], $MachinePrecision] * N[(b + a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[N[(angle / 180.0), $MachinePrecision], 1e+175], N[(2.0 * N[(t$95$2 * N[(N[(b + a), $MachinePrecision] * N[Sin[t$95$1], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(2.0 * N[(N[(b - a), $MachinePrecision] * N[(N[Cos[t$95$1], $MachinePrecision] * N[(N[(b + a), $MachinePrecision] * N[Sin[N[(0.005555555555555556 * N[(Exp[N[Log[1 + N[(angle * Pi), $MachinePrecision]], $MachinePrecision]] - 1), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]]

Derivation

1. Split input into 3 regimes

2. if (/.f64 angle 180) < -2e22

   1. Initial program 25.3%

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

      1. associate-*l* 25.3%

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

      2. unpow2 25.3%

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

      3. unpow2 25.3%

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

      4. difference-of-squares 25.3%

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

   3. Simplified 25.3%

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

   4. Taylor expanded in angle around inf 26.5%

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

      1. associate-*r* 26.5%

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

      2. *-commutative 26.5%

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

      3. *-commutative 26.5%

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

      4. associate-*r* 23.2%

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

      5. *-commutative 23.2%

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

      6. *-commutative 23.2%

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

      7. *-commutative 23.2%

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

      8. associate-*r* 26.0%

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

      9. *-commutative 26.0%

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

      10. +-commutative 26.0%

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

   6. Simplified 26.0%

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

   7. Taylor expanded in angle around inf 23.2%

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

      1. associate-*r* 26.0%

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

      2. *-commutative 26.0%

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

      3. rem-cube-cbrt 21.6%

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

      4. sqr-pow 0.0%

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

      5. pow-prod-down 36.6%

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

      6. pow2 36.6%

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

      7. *-commutative 36.6%

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

      8. associate-*r* 33.9%

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

      9. metadata-eval 33.9%

         \[\leadsto 2 \cdot \left(\left(\left(b - a\right) \cdot \cos \left(\pi \cdot \left(0.005555555555555556 \cdot angle\right)\right)\right) \cdot \left(\sin \left({\left({\left(\sqrt[3]{0.005555555555555556 \cdot \left(angle \cdot \pi\right)}\right)}^{2}\right)}^{\color{blue}{1.5}}\right) \cdot \left(b + a\right)\right)\right) \]

   9. Applied egg-rr 33.9%

      \[\leadsto 2 \cdot \left(\left(\left(b - a\right) \cdot \cos \left(\pi \cdot \left(0.005555555555555556 \cdot angle\right)\right)\right) \cdot \left(\sin \color{blue}{\left({\left({\left(\sqrt[3]{0.005555555555555556 \cdot \left(angle \cdot \pi\right)}\right)}^{2}\right)}^{1.5}\right)} \cdot \left(b + a\right)\right)\right) \]
   10. Step-by-step derivation

       1. associate-*r* 36.6%

          \[\leadsto 2 \cdot \left(\left(\left(b - a\right) \cdot \cos \left(\pi \cdot \left(0.005555555555555556 \cdot angle\right)\right)\right) \cdot \left(\sin \left({\left({\left(\sqrt[3]{\color{blue}{\left(0.005555555555555556 \cdot angle\right) \cdot \pi}}\right)}^{2}\right)}^{1.5}\right) \cdot \left(b + a\right)\right)\right) \]

       2. *-commutative 36.6%

          \[\leadsto 2 \cdot \left(\left(\left(b - a\right) \cdot \cos \left(\pi \cdot \left(0.005555555555555556 \cdot angle\right)\right)\right) \cdot \left(\sin \left({\left({\left(\sqrt[3]{\color{blue}{\pi \cdot \left(0.005555555555555556 \cdot angle\right)}}\right)}^{2}\right)}^{1.5}\right) \cdot \left(b + a\right)\right)\right) \]

   11. Simplified 36.6%

       \[\leadsto 2 \cdot \left(\left(\left(b - a\right) \cdot \cos \left(\pi \cdot \left(0.005555555555555556 \cdot angle\right)\right)\right) \cdot \left(\sin \color{blue}{\left({\left({\left(\sqrt[3]{\pi \cdot \left(0.005555555555555556 \cdot angle\right)}\right)}^{2}\right)}^{1.5}\right)} \cdot \left(b + a\right)\right)\right) \]

   if -2e22 < (/.f64 angle 180) < 9.9999999999999994e174

   1. Initial program 65.1%

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

      1. associate-*l* 65.1%

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

      2. unpow2 65.1%

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

      3. unpow2 65.1%

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

      4. difference-of-squares 68.1%

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

   3. Simplified 68.1%

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

   4. Taylor expanded in angle around inf 89.1%

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

      1. associate-*r* 89.1%

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

      2. *-commutative 89.1%

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

      3. *-commutative 89.1%

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

      4. associate-*r* 91.2%

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

      5. *-commutative 91.2%

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

      6. *-commutative 91.2%

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

      7. *-commutative 91.2%

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

      8. associate-*r* 89.5%

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

      9. *-commutative 89.5%

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

      10. +-commutative 89.5%

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

   6. Simplified 89.5%

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

   7. Taylor expanded in angle around inf 91.2%

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

   if 9.9999999999999994e174 < (/.f64 angle 180)

   1. Initial program 27.8%

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

      1. associate-*l* 27.8%

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

      2. unpow2 27.8%

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

      3. unpow2 27.8%

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

      4. difference-of-squares 31.7%

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

   3. Simplified 31.7%

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

   4. Taylor expanded in angle around inf 25.7%

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

      1. expm1-log1p-u 52.1%

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

      2. *-commutative 52.1%

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

   6. Applied egg-rr 52.1%

      \[\leadsto 2 \cdot \left(\left(b - a\right) \cdot \left(\cos \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right) \cdot \left(\sin \left(0.005555555555555556 \cdot \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\pi \cdot angle\right)\right)}\right) \cdot \left(a + b\right)\right)\right)\right) \]
3. Recombined 3 regimes into one program.

4. Final simplification 73.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{angle}{180} \leq -2 \cdot 10^{+22}:\\ \;\;\;\;2 \cdot \left(\left(\left(b - a\right) \cdot \cos \left(\pi \cdot \left(angle \cdot 0.005555555555555556\right)\right)\right) \cdot \left(\sin \left({\left({\left(\sqrt[3]{\pi \cdot \left(angle \cdot 0.005555555555555556\right)}\right)}^{2}\right)}^{1.5}\right) \cdot \left(b + a\right)\right)\right)\\ \mathbf{elif}\;\frac{angle}{180} \leq 10^{+175}:\\ \;\;\;\;2 \cdot \left(\left(\left(b - a\right) \cdot \cos \left(\pi \cdot \left(angle \cdot 0.005555555555555556\right)\right)\right) \cdot \left(\left(b + a\right) \cdot \sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \left(\left(b - a\right) \cdot \left(\cos \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right) \cdot \left(\left(b + a\right) \cdot \sin \left(0.005555555555555556 \cdot \mathsf{expm1}\left(\mathsf{log1p}\left(angle \cdot \pi\right)\right)\right)\right)\right)\right)\\ \end{array} \]

Alternative 2: 67.5% accurate, 1.0× speedup

Math

\[\begin{array}{l} a = |a|\\ b = |b|\\ \\ \begin{array}{l} t_0 := 0.005555555555555556 \cdot \left(angle \cdot \pi\right)\\ t_1 := \pi \cdot \left(angle \cdot 0.005555555555555556\right)\\ \mathbf{if}\;\frac{angle}{180} \leq -2 \cdot 10^{+22}:\\ \;\;\;\;2 \cdot \left(\left(b - a\right) \cdot \left(\left(b + a\right) \cdot \sin \left({\left(\sqrt[3]{t_1}\right)}^{3}\right)\right)\right)\\ \mathbf{elif}\;\frac{angle}{180} \leq 10^{+175}:\\ \;\;\;\;2 \cdot \left(\left(\left(b - a\right) \cdot \cos t_1\right) \cdot \left(\left(b + a\right) \cdot \sin t_0\right)\right)\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \left(\left(b - a\right) \cdot \left(\cos t_0 \cdot \left(\left(b + a\right) \cdot \sin \left(0.005555555555555556 \cdot \mathsf{expm1}\left(\mathsf{log1p}\left(angle \cdot \pi\right)\right)\right)\right)\right)\right)\\ \end{array} \end{array} \]

FPCore

NOTE: a should be positive before calling this function
NOTE: b should be positive before calling this function
(FPCore (a b angle)
 :precision binary64
 (let* ((t_0 (* 0.005555555555555556 (* angle PI)))
        (t_1 (* PI (* angle 0.005555555555555556))))
   (if (<= (/ angle 180.0) -2e+22)
     (* 2.0 (* (- b a) (* (+ b a) (sin (pow (cbrt t_1) 3.0)))))
     (if (<= (/ angle 180.0) 1e+175)
       (* 2.0 (* (* (- b a) (cos t_1)) (* (+ b a) (sin t_0))))
       (*
        2.0
        (*
         (- b a)
         (*
          (cos t_0)
          (*
           (+ b a)
           (sin (* 0.005555555555555556 (expm1 (log1p (* angle PI)))))))))))))

C

a = abs(a);
b = abs(b);
double code(double a, double b, double angle) {
	double t_0 = 0.005555555555555556 * (angle * ((double) M_PI));
	double t_1 = ((double) M_PI) * (angle * 0.005555555555555556);
	double tmp;
	if ((angle / 180.0) <= -2e+22) {
		tmp = 2.0 * ((b - a) * ((b + a) * sin(pow(cbrt(t_1), 3.0))));
	} else if ((angle / 180.0) <= 1e+175) {
		tmp = 2.0 * (((b - a) * cos(t_1)) * ((b + a) * sin(t_0)));
	} else {
		tmp = 2.0 * ((b - a) * (cos(t_0) * ((b + a) * sin((0.005555555555555556 * expm1(log1p((angle * ((double) M_PI)))))))));
	}
	return tmp;
}

Java

a = Math.abs(a);
b = Math.abs(b);
public static double code(double a, double b, double angle) {
	double t_0 = 0.005555555555555556 * (angle * Math.PI);
	double t_1 = Math.PI * (angle * 0.005555555555555556);
	double tmp;
	if ((angle / 180.0) <= -2e+22) {
		tmp = 2.0 * ((b - a) * ((b + a) * Math.sin(Math.pow(Math.cbrt(t_1), 3.0))));
	} else if ((angle / 180.0) <= 1e+175) {
		tmp = 2.0 * (((b - a) * Math.cos(t_1)) * ((b + a) * Math.sin(t_0)));
	} else {
		tmp = 2.0 * ((b - a) * (Math.cos(t_0) * ((b + a) * Math.sin((0.005555555555555556 * Math.expm1(Math.log1p((angle * Math.PI))))))));
	}
	return tmp;
}

Julia

a = abs(a)
b = abs(b)
function code(a, b, angle)
	t_0 = Float64(0.005555555555555556 * Float64(angle * pi))
	t_1 = Float64(pi * Float64(angle * 0.005555555555555556))
	tmp = 0.0
	if (Float64(angle / 180.0) <= -2e+22)
		tmp = Float64(2.0 * Float64(Float64(b - a) * Float64(Float64(b + a) * sin((cbrt(t_1) ^ 3.0)))));
	elseif (Float64(angle / 180.0) <= 1e+175)
		tmp = Float64(2.0 * Float64(Float64(Float64(b - a) * cos(t_1)) * Float64(Float64(b + a) * sin(t_0))));
	else
		tmp = Float64(2.0 * Float64(Float64(b - a) * Float64(cos(t_0) * Float64(Float64(b + a) * sin(Float64(0.005555555555555556 * expm1(log1p(Float64(angle * pi)))))))));
	end
	return tmp
end

Wolfram

NOTE: a should be positive before calling this function
NOTE: b should be positive before calling this function
code[a_, b_, angle_] := Block[{t$95$0 = N[(0.005555555555555556 * N[(angle * Pi), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(Pi * N[(angle * 0.005555555555555556), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[(angle / 180.0), $MachinePrecision], -2e+22], N[(2.0 * N[(N[(b - a), $MachinePrecision] * N[(N[(b + a), $MachinePrecision] * N[Sin[N[Power[N[Power[t$95$1, 1/3], $MachinePrecision], 3.0], $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[N[(angle / 180.0), $MachinePrecision], 1e+175], N[(2.0 * N[(N[(N[(b - a), $MachinePrecision] * N[Cos[t$95$1], $MachinePrecision]), $MachinePrecision] * N[(N[(b + a), $MachinePrecision] * N[Sin[t$95$0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(2.0 * N[(N[(b - a), $MachinePrecision] * N[(N[Cos[t$95$0], $MachinePrecision] * N[(N[(b + a), $MachinePrecision] * N[Sin[N[(0.005555555555555556 * N[(Exp[N[Log[1 + N[(angle * Pi), $MachinePrecision]], $MachinePrecision]] - 1), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

2. if (/.f64 angle 180) < -2e22

   1. Initial program 25.3%

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

      1. associate-*l* 25.3%

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

      2. unpow2 25.3%

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

      3. unpow2 25.3%

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

      4. difference-of-squares 25.3%

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

   3. Simplified 25.3%

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

   4. Taylor expanded in angle around inf 26.5%

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

      1. associate-*r* 26.5%

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

      2. *-commutative 26.5%

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

      3. *-commutative 26.5%

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

      4. associate-*r* 23.2%

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

      5. *-commutative 23.2%

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

      6. *-commutative 23.2%

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

      7. *-commutative 23.2%

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

      8. associate-*r* 26.0%

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

      9. *-commutative 26.0%

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

      10. +-commutative 26.0%

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

   6. Simplified 26.0%

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

      1. *-commutative 26.0%

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

      2. add-cube-cbrt 20.1%

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

      3. unpow3 21.6%

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

      4. associate-*r* 24.5%

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

      5. *-commutative 24.5%

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

      6. *-commutative 24.5%

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

      7. associate-*r* 21.6%

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

      8. *-commutative 21.6%

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

   8. Applied egg-rr 21.6%

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

   9. Taylor expanded in angle around 0 36.2%

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

   if -2e22 < (/.f64 angle 180) < 9.9999999999999994e174

   1. Initial program 65.1%

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

      1. associate-*l* 65.1%

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

      2. unpow2 65.1%

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

      3. unpow2 65.1%

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

      4. difference-of-squares 68.1%

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

   3. Simplified 68.1%

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

   4. Taylor expanded in angle around inf 89.1%

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

      1. associate-*r* 89.1%

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

      2. *-commutative 89.1%

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

      3. *-commutative 89.1%

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

      4. associate-*r* 91.2%

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

      5. *-commutative 91.2%

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

      6. *-commutative 91.2%

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

      7. *-commutative 91.2%

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

      8. associate-*r* 89.5%

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

      9. *-commutative 89.5%

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

      10. +-commutative 89.5%

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

   6. Simplified 89.5%

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

   7. Taylor expanded in angle around inf 91.2%

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

   if 9.9999999999999994e174 < (/.f64 angle 180)

   1. Initial program 27.8%

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

      1. associate-*l* 27.8%

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

      2. unpow2 27.8%

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

      3. unpow2 27.8%

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

      4. difference-of-squares 31.7%

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

   3. Simplified 31.7%

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

   4. Taylor expanded in angle around inf 25.7%

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

      1. expm1-log1p-u 52.1%

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

      2. *-commutative 52.1%

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

   6. Applied egg-rr 52.1%

      \[\leadsto 2 \cdot \left(\left(b - a\right) \cdot \left(\cos \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right) \cdot \left(\sin \left(0.005555555555555556 \cdot \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\pi \cdot angle\right)\right)}\right) \cdot \left(a + b\right)\right)\right)\right) \]
3. Recombined 3 regimes into one program.

4. Final simplification 73.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{angle}{180} \leq -2 \cdot 10^{+22}:\\ \;\;\;\;2 \cdot \left(\left(b - a\right) \cdot \left(\left(b + a\right) \cdot \sin \left({\left(\sqrt[3]{\pi \cdot \left(angle \cdot 0.005555555555555556\right)}\right)}^{3}\right)\right)\right)\\ \mathbf{elif}\;\frac{angle}{180} \leq 10^{+175}:\\ \;\;\;\;2 \cdot \left(\left(\left(b - a\right) \cdot \cos \left(\pi \cdot \left(angle \cdot 0.005555555555555556\right)\right)\right) \cdot \left(\left(b + a\right) \cdot \sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \left(\left(b - a\right) \cdot \left(\cos \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right) \cdot \left(\left(b + a\right) \cdot \sin \left(0.005555555555555556 \cdot \mathsf{expm1}\left(\mathsf{log1p}\left(angle \cdot \pi\right)\right)\right)\right)\right)\right)\\ \end{array} \]

Alternative 3: 67.4% accurate, 1.5× speedup

Math

\[\begin{array}{l} a = |a|\\ b = |b|\\ \\ \begin{array}{l} t_0 := \pi \cdot \left(angle \cdot 0.005555555555555556\right)\\ \mathbf{if}\;\frac{angle}{180} \leq -2 \cdot 10^{+22}:\\ \;\;\;\;2 \cdot \left(\left(b - a\right) \cdot \left(\left(b + a\right) \cdot \sin \left({\left(\sqrt[3]{t_0}\right)}^{3}\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \left(\left(\left(b - a\right) \cdot \cos t_0\right) \cdot \left(\left(b + a\right) \cdot \sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)\right)\right)\\ \end{array} \end{array} \]

FPCore

NOTE: a should be positive before calling this function
NOTE: b should be positive before calling this function
(FPCore (a b angle)
 :precision binary64
 (let* ((t_0 (* PI (* angle 0.005555555555555556))))
   (if (<= (/ angle 180.0) -2e+22)
     (* 2.0 (* (- b a) (* (+ b a) (sin (pow (cbrt t_0) 3.0)))))
     (*
      2.0
      (*
       (* (- b a) (cos t_0))
       (* (+ b a) (sin (* 0.005555555555555556 (* angle PI)))))))))

C

a = abs(a);
b = abs(b);
double code(double a, double b, double angle) {
	double t_0 = ((double) M_PI) * (angle * 0.005555555555555556);
	double tmp;
	if ((angle / 180.0) <= -2e+22) {
		tmp = 2.0 * ((b - a) * ((b + a) * sin(pow(cbrt(t_0), 3.0))));
	} else {
		tmp = 2.0 * (((b - a) * cos(t_0)) * ((b + a) * sin((0.005555555555555556 * (angle * ((double) M_PI))))));
	}
	return tmp;
}

Java

a = Math.abs(a);
b = Math.abs(b);
public static double code(double a, double b, double angle) {
	double t_0 = Math.PI * (angle * 0.005555555555555556);
	double tmp;
	if ((angle / 180.0) <= -2e+22) {
		tmp = 2.0 * ((b - a) * ((b + a) * Math.sin(Math.pow(Math.cbrt(t_0), 3.0))));
	} else {
		tmp = 2.0 * (((b - a) * Math.cos(t_0)) * ((b + a) * Math.sin((0.005555555555555556 * (angle * Math.PI)))));
	}
	return tmp;
}

Julia

a = abs(a)
b = abs(b)
function code(a, b, angle)
	t_0 = Float64(pi * Float64(angle * 0.005555555555555556))
	tmp = 0.0
	if (Float64(angle / 180.0) <= -2e+22)
		tmp = Float64(2.0 * Float64(Float64(b - a) * Float64(Float64(b + a) * sin((cbrt(t_0) ^ 3.0)))));
	else
		tmp = Float64(2.0 * Float64(Float64(Float64(b - a) * cos(t_0)) * Float64(Float64(b + a) * sin(Float64(0.005555555555555556 * Float64(angle * pi))))));
	end
	return tmp
end

Wolfram

NOTE: a should be positive before calling this function
NOTE: b should be positive before calling this function
code[a_, b_, angle_] := Block[{t$95$0 = N[(Pi * N[(angle * 0.005555555555555556), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[(angle / 180.0), $MachinePrecision], -2e+22], N[(2.0 * N[(N[(b - a), $MachinePrecision] * N[(N[(b + a), $MachinePrecision] * N[Sin[N[Power[N[Power[t$95$0, 1/3], $MachinePrecision], 3.0], $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(2.0 * N[(N[(N[(b - a), $MachinePrecision] * N[Cos[t$95$0], $MachinePrecision]), $MachinePrecision] * N[(N[(b + a), $MachinePrecision] * N[Sin[N[(0.005555555555555556 * N[(angle * Pi), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 angle 180) < -2e22

   1. Initial program 25.3%

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

      1. associate-*l* 25.3%

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

      2. unpow2 25.3%

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

      3. unpow2 25.3%

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

      4. difference-of-squares 25.3%

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

   3. Simplified 25.3%

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

   4. Taylor expanded in angle around inf 26.5%

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

      1. associate-*r* 26.5%

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

      2. *-commutative 26.5%

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

      3. *-commutative 26.5%

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

      4. associate-*r* 23.2%

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

      5. *-commutative 23.2%

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

      6. *-commutative 23.2%

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

      7. *-commutative 23.2%

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

      8. associate-*r* 26.0%

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

      9. *-commutative 26.0%

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

      10. +-commutative 26.0%

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

   6. Simplified 26.0%

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

      1. *-commutative 26.0%

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

      2. add-cube-cbrt 20.1%

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

      3. unpow3 21.6%

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

      4. associate-*r* 24.5%

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

      5. *-commutative 24.5%

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

      6. *-commutative 24.5%

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

      7. associate-*r* 21.6%

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

      8. *-commutative 21.6%

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

   8. Applied egg-rr 21.6%

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

   9. Taylor expanded in angle around 0 36.2%

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

   if -2e22 < (/.f64 angle 180)

   1. Initial program 60.0%

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

      1. associate-*l* 60.0%

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

      2. unpow2 60.0%

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

      3. unpow2 60.0%

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

      4. difference-of-squares 63.2%

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

   3. Simplified 63.2%

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

   4. Taylor expanded in angle around inf 80.5%

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

      1. associate-*r* 80.5%

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

      2. *-commutative 80.5%

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

      3. *-commutative 80.5%

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

      4. associate-*r* 83.8%

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

      5. *-commutative 83.8%

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

      6. *-commutative 83.8%

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

      7. *-commutative 83.8%

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

      8. associate-*r* 82.3%

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

      9. *-commutative 82.3%

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

      10. +-commutative 82.3%

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

   6. Simplified 82.3%

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

   7. Taylor expanded in angle around inf 83.8%

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

4. Final simplification 71.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{angle}{180} \leq -2 \cdot 10^{+22}:\\ \;\;\;\;2 \cdot \left(\left(b - a\right) \cdot \left(\left(b + a\right) \cdot \sin \left({\left(\sqrt[3]{\pi \cdot \left(angle \cdot 0.005555555555555556\right)}\right)}^{3}\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \left(\left(\left(b - a\right) \cdot \cos \left(\pi \cdot \left(angle \cdot 0.005555555555555556\right)\right)\right) \cdot \left(\left(b + a\right) \cdot \sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)\right)\right)\\ \end{array} \]

Alternative 4: 67.1% accurate, 1.5× speedup

Math

\[\begin{array}{l} a = |a|\\ b = |b|\\ \\ \begin{array}{l} t_0 := 0.005555555555555556 \cdot \left(angle \cdot \pi\right)\\ \mathbf{if}\;b \leq 6.2 \cdot 10^{+173}:\\ \;\;\;\;2 \cdot \left(\left(\left(b + a\right) \cdot \sin t_0\right) \cdot \left(\left(b - a\right) \cdot \cos t_0\right)\right)\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \left(\left(b - a\right) \cdot \left(\left(b + a\right) \cdot \sin \left({\left(\sqrt[3]{\pi \cdot \left(angle \cdot 0.005555555555555556\right)}\right)}^{3}\right)\right)\right)\\ \end{array} \end{array} \]

FPCore

NOTE: a should be positive before calling this function
NOTE: b should be positive before calling this function
(FPCore (a b angle)
 :precision binary64
 (let* ((t_0 (* 0.005555555555555556 (* angle PI))))
   (if (<= b 6.2e+173)
     (* 2.0 (* (* (+ b a) (sin t_0)) (* (- b a) (cos t_0))))
     (*
      2.0
      (*
       (- b a)
       (*
        (+ b a)
        (sin (pow (cbrt (* PI (* angle 0.005555555555555556))) 3.0))))))))

C

a = abs(a);
b = abs(b);
double code(double a, double b, double angle) {
	double t_0 = 0.005555555555555556 * (angle * ((double) M_PI));
	double tmp;
	if (b <= 6.2e+173) {
		tmp = 2.0 * (((b + a) * sin(t_0)) * ((b - a) * cos(t_0)));
	} else {
		tmp = 2.0 * ((b - a) * ((b + a) * sin(pow(cbrt((((double) M_PI) * (angle * 0.005555555555555556))), 3.0))));
	}
	return tmp;
}

Java

a = Math.abs(a);
b = Math.abs(b);
public static double code(double a, double b, double angle) {
	double t_0 = 0.005555555555555556 * (angle * Math.PI);
	double tmp;
	if (b <= 6.2e+173) {
		tmp = 2.0 * (((b + a) * Math.sin(t_0)) * ((b - a) * Math.cos(t_0)));
	} else {
		tmp = 2.0 * ((b - a) * ((b + a) * Math.sin(Math.pow(Math.cbrt((Math.PI * (angle * 0.005555555555555556))), 3.0))));
	}
	return tmp;
}

Julia

a = abs(a)
b = abs(b)
function code(a, b, angle)
	t_0 = Float64(0.005555555555555556 * Float64(angle * pi))
	tmp = 0.0
	if (b <= 6.2e+173)
		tmp = Float64(2.0 * Float64(Float64(Float64(b + a) * sin(t_0)) * Float64(Float64(b - a) * cos(t_0))));
	else
		tmp = Float64(2.0 * Float64(Float64(b - a) * Float64(Float64(b + a) * sin((cbrt(Float64(pi * Float64(angle * 0.005555555555555556))) ^ 3.0)))));
	end
	return tmp
end

Wolfram

NOTE: a should be positive before calling this function
NOTE: b should be positive before calling this function
code[a_, b_, angle_] := Block[{t$95$0 = N[(0.005555555555555556 * N[(angle * Pi), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[b, 6.2e+173], N[(2.0 * N[(N[(N[(b + a), $MachinePrecision] * N[Sin[t$95$0], $MachinePrecision]), $MachinePrecision] * N[(N[(b - a), $MachinePrecision] * N[Cos[t$95$0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(2.0 * N[(N[(b - a), $MachinePrecision] * N[(N[(b + a), $MachinePrecision] * N[Sin[N[Power[N[Power[N[(Pi * N[(angle * 0.005555555555555556), $MachinePrecision]), $MachinePrecision], 1/3], $MachinePrecision], 3.0], $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if b < 6.2e173

   1. Initial program 50.9%

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

      1. associate-*l* 50.9%

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

      2. unpow2 50.9%

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

      3. unpow2 50.9%

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

      4. difference-of-squares 52.7%

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

   3. Simplified 52.7%

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

   4. Taylor expanded in angle around inf 66.1%

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

      1. associate-*r* 66.1%

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

      2. *-commutative 66.1%

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

      3. +-commutative 66.1%

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

   6. Simplified 66.1%

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

   if 6.2e173 < b

   1. Initial program 56.1%

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

      1. associate-*l* 56.1%

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

      2. unpow2 56.1%

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

      3. unpow2 56.1%

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

      4. difference-of-squares 65.2%

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

   3. Simplified 65.2%

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

   4. Taylor expanded in angle around inf 77.1%

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

      1. associate-*r* 77.1%

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

      2. *-commutative 77.1%

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

      3. *-commutative 77.1%

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

      4. associate-*r* 77.1%

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

      5. *-commutative 77.1%

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

      6. *-commutative 77.1%

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

      7. *-commutative 77.1%

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

      8. associate-*r* 77.2%

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

      9. *-commutative 77.2%

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

      10. +-commutative 77.2%

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

   6. Simplified 77.2%

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

      1. *-commutative 77.2%

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

      2. add-cube-cbrt 67.6%

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

      3. unpow3 72.1%

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

      4. associate-*r* 76.6%

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

      5. *-commutative 76.6%

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

      6. *-commutative 76.6%

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

      7. associate-*r* 72.1%

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

      8. *-commutative 72.1%

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

   8. Applied egg-rr 72.1%

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

   9. Taylor expanded in angle around 0 90.3%

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

4. Final simplification 68.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq 6.2 \cdot 10^{+173}:\\ \;\;\;\;2 \cdot \left(\left(\left(b + a\right) \cdot \sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)\right) \cdot \left(\left(b - a\right) \cdot \cos \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \left(\left(b - a\right) \cdot \left(\left(b + a\right) \cdot \sin \left({\left(\sqrt[3]{\pi \cdot \left(angle \cdot 0.005555555555555556\right)}\right)}^{3}\right)\right)\right)\\ \end{array} \]

Alternative 5: 65.3% accurate, 1.5× speedup

Math

\[\begin{array}{l} a = |a|\\ b = |b|\\ \\ 2 \cdot \left(\left(b - a\right) \cdot \left(\left(b + a\right) \cdot \sin \left({\left(\sqrt[3]{\pi \cdot \left(angle \cdot 0.005555555555555556\right)}\right)}^{3}\right)\right)\right) \end{array} \]

FPCore

NOTE: a should be positive before calling this function
NOTE: b should be positive before calling this function
(FPCore (a b angle)
 :precision binary64
 (*
  2.0
  (*
   (- b a)
   (* (+ b a) (sin (pow (cbrt (* PI (* angle 0.005555555555555556))) 3.0))))))

C

a = abs(a);
b = abs(b);
double code(double a, double b, double angle) {
	return 2.0 * ((b - a) * ((b + a) * sin(pow(cbrt((((double) M_PI) * (angle * 0.005555555555555556))), 3.0))));
}

Java

a = Math.abs(a);
b = Math.abs(b);
public static double code(double a, double b, double angle) {
	return 2.0 * ((b - a) * ((b + a) * Math.sin(Math.pow(Math.cbrt((Math.PI * (angle * 0.005555555555555556))), 3.0))));
}

Julia

a = abs(a)
b = abs(b)
function code(a, b, angle)
	return Float64(2.0 * Float64(Float64(b - a) * Float64(Float64(b + a) * sin((cbrt(Float64(pi * Float64(angle * 0.005555555555555556))) ^ 3.0)))))
end

Wolfram

NOTE: a should be positive before calling this function
NOTE: b should be positive before calling this function
code[a_, b_, angle_] := N[(2.0 * N[(N[(b - a), $MachinePrecision] * N[(N[(b + a), $MachinePrecision] * N[Sin[N[Power[N[Power[N[(Pi * N[(angle * 0.005555555555555556), $MachinePrecision]), $MachinePrecision], 1/3], $MachinePrecision], 3.0], $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 51.4%

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

   1. associate-*l* 51.4%

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

   2. unpow2 51.4%

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

   3. unpow2 51.4%

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

   4. difference-of-squares 53.7%

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

3. Simplified 53.7%

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

4. Taylor expanded in angle around inf 67.0%

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

   1. associate-*r* 67.0%

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

   2. *-commutative 67.0%

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

   3. *-commutative 67.0%

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

   4. associate-*r* 68.6%

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

   5. *-commutative 68.6%

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

   6. *-commutative 68.6%

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

   7. *-commutative 68.6%

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

   8. associate-*r* 68.2%

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

   9. *-commutative 68.2%

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

   10. +-commutative 68.2%

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

6. Simplified 68.2%

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

   1. *-commutative 68.2%

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

   2. add-cube-cbrt 65.0%

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

   3. unpow3 65.9%

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

   4. associate-*r* 66.3%

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

   5. *-commutative 66.3%

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

   6. *-commutative 66.3%

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

   7. associate-*r* 65.9%

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

   8. *-commutative 65.9%

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

8. Applied egg-rr 65.9%

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

9. Taylor expanded in angle around 0 68.0%

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

10. Final simplification 68.0%

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

Alternative 6: 64.6% accurate, 2.8× speedup

Math

\[\begin{array}{l} a = |a|\\ b = |b|\\ \\ \begin{array}{l} \mathbf{if}\;angle \leq -2.95 \cdot 10^{+144} \lor \neg \left(angle \leq 4 \cdot 10^{+197}\right):\\ \;\;\;\;\sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right) \cdot \left(2 \cdot \left(\left(b - a\right) \cdot \left(b + a\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;0.011111111111111112 \cdot \left(\left(\pi \cdot \left(b + a\right)\right) \cdot \left(angle \cdot \left(b - a\right)\right)\right)\\ \end{array} \end{array} \]

FPCore

NOTE: a should be positive before calling this function
NOTE: b should be positive before calling this function
(FPCore (a b angle)
 :precision binary64
 (if (or (<= angle -2.95e+144) (not (<= angle 4e+197)))
   (* (sin (* 0.005555555555555556 (* angle PI))) (* 2.0 (* (- b a) (+ b a))))
   (* 0.011111111111111112 (* (* PI (+ b a)) (* angle (- b a))))))

C

a = abs(a);
b = abs(b);
double code(double a, double b, double angle) {
	double tmp;
	if ((angle <= -2.95e+144) || !(angle <= 4e+197)) {
		tmp = sin((0.005555555555555556 * (angle * ((double) M_PI)))) * (2.0 * ((b - a) * (b + a)));
	} else {
		tmp = 0.011111111111111112 * ((((double) M_PI) * (b + a)) * (angle * (b - a)));
	}
	return tmp;
}

Java

a = Math.abs(a);
b = Math.abs(b);
public static double code(double a, double b, double angle) {
	double tmp;
	if ((angle <= -2.95e+144) || !(angle <= 4e+197)) {
		tmp = Math.sin((0.005555555555555556 * (angle * Math.PI))) * (2.0 * ((b - a) * (b + a)));
	} else {
		tmp = 0.011111111111111112 * ((Math.PI * (b + a)) * (angle * (b - a)));
	}
	return tmp;
}

Python

a = abs(a)
b = abs(b)
def code(a, b, angle):
	tmp = 0
	if (angle <= -2.95e+144) or not (angle <= 4e+197):
		tmp = math.sin((0.005555555555555556 * (angle * math.pi))) * (2.0 * ((b - a) * (b + a)))
	else:
		tmp = 0.011111111111111112 * ((math.pi * (b + a)) * (angle * (b - a)))
	return tmp

Julia

a = abs(a)
b = abs(b)
function code(a, b, angle)
	tmp = 0.0
	if ((angle <= -2.95e+144) || !(angle <= 4e+197))
		tmp = Float64(sin(Float64(0.005555555555555556 * Float64(angle * pi))) * Float64(2.0 * Float64(Float64(b - a) * Float64(b + a))));
	else
		tmp = Float64(0.011111111111111112 * Float64(Float64(pi * Float64(b + a)) * Float64(angle * Float64(b - a))));
	end
	return tmp
end

MATLAB

a = abs(a)
b = abs(b)
function tmp_2 = code(a, b, angle)
	tmp = 0.0;
	if ((angle <= -2.95e+144) || ~((angle <= 4e+197)))
		tmp = sin((0.005555555555555556 * (angle * pi))) * (2.0 * ((b - a) * (b + a)));
	else
		tmp = 0.011111111111111112 * ((pi * (b + a)) * (angle * (b - a)));
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: a should be positive before calling this function
NOTE: b should be positive before calling this function
code[a_, b_, angle_] := If[Or[LessEqual[angle, -2.95e+144], N[Not[LessEqual[angle, 4e+197]], $MachinePrecision]], N[(N[Sin[N[(0.005555555555555556 * N[(angle * Pi), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * N[(2.0 * N[(N[(b - a), $MachinePrecision] * N[(b + a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(0.011111111111111112 * N[(N[(Pi * N[(b + a), $MachinePrecision]), $MachinePrecision] * N[(angle * N[(b - a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if angle < -2.94999999999999994e144 or 3.9999999999999998e197 < angle

   1. Initial program 29.1%

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

      1. associate-*l* 29.1%

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

      2. unpow2 29.1%

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

      3. unpow2 29.1%

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

      4. difference-of-squares 30.9%

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

   3. Simplified 30.9%

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

      1. add-cube-cbrt 33.3%

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

      2. pow3 31.7%

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

      3. div-inv 32.0%

         \[\leadsto \left(2 \cdot \left(\left(b + a\right) \cdot \left(b - a\right)\right)\right) \cdot \left(\sin \left(\pi \cdot \frac{angle}{180}\right) \cdot \cos \left({\left(\sqrt[3]{\pi \cdot \color{blue}{\left(angle \cdot \frac{1}{180}\right)}}\right)}^{3}\right)\right) \]

      4. metadata-eval 32.0%

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

   5. Applied egg-rr 32.0%

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

   6. Taylor expanded in angle around inf 37.6%

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

      1. associate-*r* 37.6%

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

      2. +-commutative 37.6%

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

      3. *-commutative 37.6%

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

      4. associate-*l* 37.6%

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

      5. associate-*r* 37.6%

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

      6. *-commutative 37.6%

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

   8. Simplified 37.6%

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

   if -2.94999999999999994e144 < angle < 3.9999999999999998e197

   1. Initial program 57.5%

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

      1. associate-*l* 57.5%

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

      2. unpow2 57.5%

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

      3. unpow2 57.5%

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

      4. difference-of-squares 60.0%

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

   3. Simplified 60.0%

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

   4. Taylor expanded in angle around 0 57.8%

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

      1. associate-*r* 74.4%

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

      2. *-commutative 74.4%

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

      3. +-commutative 74.4%

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

   6. Simplified 74.4%

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

4. Final simplification 66.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;angle \leq -2.95 \cdot 10^{+144} \lor \neg \left(angle \leq 4 \cdot 10^{+197}\right):\\ \;\;\;\;\sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right) \cdot \left(2 \cdot \left(\left(b - a\right) \cdot \left(b + a\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;0.011111111111111112 \cdot \left(\left(\pi \cdot \left(b + a\right)\right) \cdot \left(angle \cdot \left(b - a\right)\right)\right)\\ \end{array} \]

Alternative 7: 65.5% accurate, 2.9× speedup

Math

\[\begin{array}{l} a = |a|\\ b = |b|\\ \\ \begin{array}{l} \mathbf{if}\;b \leq 3.5 \cdot 10^{+140}:\\ \;\;\;\;\left(\left(b + a\right) \cdot \sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)\right) \cdot \left(2 \cdot \left(b - a\right)\right)\\ \mathbf{else}:\\ \;\;\;\;0.011111111111111112 \cdot \left(\left(\pi \cdot \left(b + a\right)\right) \cdot \left(angle \cdot \left(b - a\right)\right)\right)\\ \end{array} \end{array} \]

FPCore

NOTE: a should be positive before calling this function
NOTE: b should be positive before calling this function
(FPCore (a b angle)
 :precision binary64
 (if (<= b 3.5e+140)
   (* (* (+ b a) (sin (* 0.005555555555555556 (* angle PI)))) (* 2.0 (- b a)))
   (* 0.011111111111111112 (* (* PI (+ b a)) (* angle (- b a))))))

C

a = abs(a);
b = abs(b);
double code(double a, double b, double angle) {
	double tmp;
	if (b <= 3.5e+140) {
		tmp = ((b + a) * sin((0.005555555555555556 * (angle * ((double) M_PI))))) * (2.0 * (b - a));
	} else {
		tmp = 0.011111111111111112 * ((((double) M_PI) * (b + a)) * (angle * (b - a)));
	}
	return tmp;
}

Java

a = Math.abs(a);
b = Math.abs(b);
public static double code(double a, double b, double angle) {
	double tmp;
	if (b <= 3.5e+140) {
		tmp = ((b + a) * Math.sin((0.005555555555555556 * (angle * Math.PI)))) * (2.0 * (b - a));
	} else {
		tmp = 0.011111111111111112 * ((Math.PI * (b + a)) * (angle * (b - a)));
	}
	return tmp;
}

Python

a = abs(a)
b = abs(b)
def code(a, b, angle):
	tmp = 0
	if b <= 3.5e+140:
		tmp = ((b + a) * math.sin((0.005555555555555556 * (angle * math.pi)))) * (2.0 * (b - a))
	else:
		tmp = 0.011111111111111112 * ((math.pi * (b + a)) * (angle * (b - a)))
	return tmp

Julia

a = abs(a)
b = abs(b)
function code(a, b, angle)
	tmp = 0.0
	if (b <= 3.5e+140)
		tmp = Float64(Float64(Float64(b + a) * sin(Float64(0.005555555555555556 * Float64(angle * pi)))) * Float64(2.0 * Float64(b - a)));
	else
		tmp = Float64(0.011111111111111112 * Float64(Float64(pi * Float64(b + a)) * Float64(angle * Float64(b - a))));
	end
	return tmp
end

MATLAB

a = abs(a)
b = abs(b)
function tmp_2 = code(a, b, angle)
	tmp = 0.0;
	if (b <= 3.5e+140)
		tmp = ((b + a) * sin((0.005555555555555556 * (angle * pi)))) * (2.0 * (b - a));
	else
		tmp = 0.011111111111111112 * ((pi * (b + a)) * (angle * (b - a)));
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: a should be positive before calling this function
NOTE: b should be positive before calling this function
code[a_, b_, angle_] := If[LessEqual[b, 3.5e+140], N[(N[(N[(b + a), $MachinePrecision] * N[Sin[N[(0.005555555555555556 * N[(angle * Pi), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[(2.0 * N[(b - a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(0.011111111111111112 * N[(N[(Pi * N[(b + a), $MachinePrecision]), $MachinePrecision] * N[(angle * N[(b - a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b < 3.49999999999999989e140

   1. Initial program 51.2%

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

      1. associate-*l* 51.2%

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

      2. unpow2 51.2%

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

      3. unpow2 51.2%

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

      4. difference-of-squares 53.0%

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

   3. Simplified 53.0%

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

      1. add-cube-cbrt 57.0%

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

      2. pow3 55.1%

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

      3. div-inv 56.0%

         \[\leadsto \left(2 \cdot \left(\left(b + a\right) \cdot \left(b - a\right)\right)\right) \cdot \left(\sin \left(\pi \cdot \frac{angle}{180}\right) \cdot \cos \left({\left(\sqrt[3]{\pi \cdot \color{blue}{\left(angle \cdot \frac{1}{180}\right)}}\right)}^{3}\right)\right) \]

      4. metadata-eval 56.0%

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

   5. Applied egg-rr 56.0%

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

   6. Taylor expanded in angle around inf 64.3%

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

      1. associate-*r* 64.3%

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

      2. +-commutative 64.3%

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

   8. Simplified 64.3%

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

   if 3.49999999999999989e140 < b

   1. Initial program 52.7%

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

      1. associate-*l* 52.7%

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

      2. unpow2 52.7%

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

      3. unpow2 52.7%

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

      4. difference-of-squares 59.1%

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

   3. Simplified 59.1%

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

   4. Taylor expanded in angle around 0 50.8%

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

      1. associate-*r* 78.1%

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

      2. *-commutative 78.1%

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

      3. +-commutative 78.1%

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

   6. Simplified 78.1%

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

4. Final simplification 66.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq 3.5 \cdot 10^{+140}:\\ \;\;\;\;\left(\left(b + a\right) \cdot \sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)\right) \cdot \left(2 \cdot \left(b - a\right)\right)\\ \mathbf{else}:\\ \;\;\;\;0.011111111111111112 \cdot \left(\left(\pi \cdot \left(b + a\right)\right) \cdot \left(angle \cdot \left(b - a\right)\right)\right)\\ \end{array} \]

Alternative 8: 65.9% accurate, 2.9× speedup

Math

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

FPCore

NOTE: a should be positive before calling this function
NOTE: b should be positive before calling this function
(FPCore (a b angle)
 :precision binary64
 (* 2.0 (* (- b a) (* (+ b a) (sin (* PI (* angle 0.005555555555555556)))))))

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

NOTE: a should be positive before calling this function
NOTE: b should be positive before calling this function
code[a_, b_, angle_] := N[(2.0 * N[(N[(b - a), $MachinePrecision] * N[(N[(b + a), $MachinePrecision] * N[Sin[N[(Pi * N[(angle * 0.005555555555555556), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 51.4%

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

   1. associate-*l* 51.4%

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

   2. unpow2 51.4%

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

   3. unpow2 51.4%

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

   4. difference-of-squares 53.7%

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

3. Simplified 53.7%

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

4. Taylor expanded in angle around inf 67.0%

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

   1. associate-*r* 67.0%

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

   2. *-commutative 67.0%

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

   3. *-commutative 67.0%

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

   4. associate-*r* 68.6%

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

   5. *-commutative 68.6%

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

   6. *-commutative 68.6%

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

   7. *-commutative 68.6%

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

   8. associate-*r* 68.2%

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

   9. *-commutative 68.2%

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

   10. +-commutative 68.2%

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

6. Simplified 68.2%

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

7. Taylor expanded in angle around 0 66.6%

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

8. Final simplification 66.6%

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

Alternative 9: 58.6% accurate, 5.4× speedup

Math

\[\begin{array}{l} a = |a|\\ b = |b|\\ \\ \begin{array}{l} \mathbf{if}\;b \leq 5.2 \cdot 10^{+153}:\\ \;\;\;\;0.011111111111111112 \cdot \left(angle \cdot \left(\pi \cdot \left(\left(b - a\right) \cdot \left(b + a\right)\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;0.011111111111111112 \cdot \left(\left(b \cdot \pi\right) \cdot \left(angle \cdot b\right)\right)\\ \end{array} \end{array} \]

FPCore

NOTE: a should be positive before calling this function
NOTE: b should be positive before calling this function
(FPCore (a b angle)
 :precision binary64
 (if (<= b 5.2e+153)
   (* 0.011111111111111112 (* angle (* PI (* (- b a) (+ b a)))))
   (* 0.011111111111111112 (* (* b PI) (* angle b)))))

C

a = abs(a);
b = abs(b);
double code(double a, double b, double angle) {
	double tmp;
	if (b <= 5.2e+153) {
		tmp = 0.011111111111111112 * (angle * (((double) M_PI) * ((b - a) * (b + a))));
	} else {
		tmp = 0.011111111111111112 * ((b * ((double) M_PI)) * (angle * b));
	}
	return tmp;
}

Java

a = Math.abs(a);
b = Math.abs(b);
public static double code(double a, double b, double angle) {
	double tmp;
	if (b <= 5.2e+153) {
		tmp = 0.011111111111111112 * (angle * (Math.PI * ((b - a) * (b + a))));
	} else {
		tmp = 0.011111111111111112 * ((b * Math.PI) * (angle * b));
	}
	return tmp;
}

Python

a = abs(a)
b = abs(b)
def code(a, b, angle):
	tmp = 0
	if b <= 5.2e+153:
		tmp = 0.011111111111111112 * (angle * (math.pi * ((b - a) * (b + a))))
	else:
		tmp = 0.011111111111111112 * ((b * math.pi) * (angle * b))
	return tmp

Julia

a = abs(a)
b = abs(b)
function code(a, b, angle)
	tmp = 0.0
	if (b <= 5.2e+153)
		tmp = Float64(0.011111111111111112 * Float64(angle * Float64(pi * Float64(Float64(b - a) * Float64(b + a)))));
	else
		tmp = Float64(0.011111111111111112 * Float64(Float64(b * pi) * Float64(angle * b)));
	end
	return tmp
end

MATLAB

a = abs(a)
b = abs(b)
function tmp_2 = code(a, b, angle)
	tmp = 0.0;
	if (b <= 5.2e+153)
		tmp = 0.011111111111111112 * (angle * (pi * ((b - a) * (b + a))));
	else
		tmp = 0.011111111111111112 * ((b * pi) * (angle * b));
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: a should be positive before calling this function
NOTE: b should be positive before calling this function
code[a_, b_, angle_] := If[LessEqual[b, 5.2e+153], N[(0.011111111111111112 * N[(angle * N[(Pi * N[(N[(b - a), $MachinePrecision] * N[(b + a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(0.011111111111111112 * N[(N[(b * Pi), $MachinePrecision] * N[(angle * b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b < 5.1999999999999998e153

   1. Initial program 51.1%

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

      1. associate-*l* 51.1%

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

      2. unpow2 51.1%

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

      3. unpow2 51.1%

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

      4. difference-of-squares 52.9%

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

   3. Simplified 52.9%

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

   4. Taylor expanded in angle around 0 50.0%

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

   5. Taylor expanded in angle around 0 50.0%

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

      1. associate-*r* 50.1%

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

      2. *-commutative 50.1%

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

      3. +-commutative 50.1%

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

   7. Simplified 50.1%

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

   if 5.1999999999999998e153 < b

   1. Initial program 53.6%

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

      1. associate-*l* 53.6%

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

      2. unpow2 53.6%

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

      3. unpow2 53.6%

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

      4. difference-of-squares 61.0%

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

   3. Simplified 61.0%

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

   4. Taylor expanded in angle around 0 53.6%

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

   5. Taylor expanded in b around inf 57.3%

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

      1. *-commutative 57.3%

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

      2. unpow2 57.3%

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

   7. Simplified 57.3%

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

   8. Taylor expanded in angle around 0 57.3%

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

      1. *-commutative 57.3%

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

      2. unpow2 57.3%

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

      3. *-commutative 57.3%

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

      4. associate-*l* 57.3%

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

   10. Simplified 57.3%

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

   11. Taylor expanded in angle around 0 57.3%

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

       1. *-commutative 57.3%

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

       2. *-commutative 57.3%

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

       3. unpow2 57.3%

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

       4. associate-*r* 57.3%

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

       5. associate-*l* 85.2%

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

       6. *-commutative 85.2%

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

   13. Simplified 85.2%

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

4. Final simplification 53.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq 5.2 \cdot 10^{+153}:\\ \;\;\;\;0.011111111111111112 \cdot \left(angle \cdot \left(\pi \cdot \left(\left(b - a\right) \cdot \left(b + a\right)\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;0.011111111111111112 \cdot \left(\left(b \cdot \pi\right) \cdot \left(angle \cdot b\right)\right)\\ \end{array} \]

Alternative 10: 62.5% accurate, 5.5× speedup

Math

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

FPCore

NOTE: a should be positive before calling this function
NOTE: b should be positive before calling this function
(FPCore (a b angle)
 :precision binary64
 (* 0.011111111111111112 (* (* PI (+ b a)) (* angle (- b a)))))

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

NOTE: a should be positive before calling this function
NOTE: b should be positive before calling this function
code[a_, b_, angle_] := N[(0.011111111111111112 * N[(N[(Pi * N[(b + a), $MachinePrecision]), $MachinePrecision] * N[(angle * N[(b - a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 51.4%

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

   1. associate-*l* 51.4%

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

   2. unpow2 51.4%

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

   3. unpow2 51.4%

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

   4. difference-of-squares 53.7%

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

3. Simplified 53.7%

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

4. Taylor expanded in angle around 0 50.4%

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

   1. associate-*r* 61.6%

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

   2. *-commutative 61.6%

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

   3. +-commutative 61.6%

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

6. Simplified 61.6%

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

7. Final simplification 61.6%

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

Alternative 11: 39.5% accurate, 5.6× speedup

Math

\[\begin{array}{l} a = |a|\\ b = |b|\\ \\ \begin{array}{l} \mathbf{if}\;b \leq 1.2 \cdot 10^{+55}:\\ \;\;\;\;0.011111111111111112 \cdot \left(angle \cdot \left(\pi \cdot \left(b \cdot b\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;0.011111111111111112 \cdot \left(\left(b \cdot \pi\right) \cdot \left(angle \cdot b\right)\right)\\ \end{array} \end{array} \]

FPCore

NOTE: a should be positive before calling this function
NOTE: b should be positive before calling this function
(FPCore (a b angle)
 :precision binary64
 (if (<= b 1.2e+55)
   (* 0.011111111111111112 (* angle (* PI (* b b))))
   (* 0.011111111111111112 (* (* b PI) (* angle b)))))

C

a = abs(a);
b = abs(b);
double code(double a, double b, double angle) {
	double tmp;
	if (b <= 1.2e+55) {
		tmp = 0.011111111111111112 * (angle * (((double) M_PI) * (b * b)));
	} else {
		tmp = 0.011111111111111112 * ((b * ((double) M_PI)) * (angle * b));
	}
	return tmp;
}

Java

a = Math.abs(a);
b = Math.abs(b);
public static double code(double a, double b, double angle) {
	double tmp;
	if (b <= 1.2e+55) {
		tmp = 0.011111111111111112 * (angle * (Math.PI * (b * b)));
	} else {
		tmp = 0.011111111111111112 * ((b * Math.PI) * (angle * b));
	}
	return tmp;
}

Python

a = abs(a)
b = abs(b)
def code(a, b, angle):
	tmp = 0
	if b <= 1.2e+55:
		tmp = 0.011111111111111112 * (angle * (math.pi * (b * b)))
	else:
		tmp = 0.011111111111111112 * ((b * math.pi) * (angle * b))
	return tmp

Julia

a = abs(a)
b = abs(b)
function code(a, b, angle)
	tmp = 0.0
	if (b <= 1.2e+55)
		tmp = Float64(0.011111111111111112 * Float64(angle * Float64(pi * Float64(b * b))));
	else
		tmp = Float64(0.011111111111111112 * Float64(Float64(b * pi) * Float64(angle * b)));
	end
	return tmp
end

MATLAB

a = abs(a)
b = abs(b)
function tmp_2 = code(a, b, angle)
	tmp = 0.0;
	if (b <= 1.2e+55)
		tmp = 0.011111111111111112 * (angle * (pi * (b * b)));
	else
		tmp = 0.011111111111111112 * ((b * pi) * (angle * b));
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: a should be positive before calling this function
NOTE: b should be positive before calling this function
code[a_, b_, angle_] := If[LessEqual[b, 1.2e+55], N[(0.011111111111111112 * N[(angle * N[(Pi * N[(b * b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(0.011111111111111112 * N[(N[(b * Pi), $MachinePrecision] * N[(angle * b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b < 1.2e55

   1. Initial program 51.7%

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

      1. associate-*l* 51.7%

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

      2. unpow2 51.7%

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

      3. unpow2 51.7%

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

      4. difference-of-squares 53.7%

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

   3. Simplified 53.7%

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

   4. Taylor expanded in angle around 0 51.4%

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

   5. Taylor expanded in b around inf 29.6%

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

      1. *-commutative 29.6%

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

      2. unpow2 29.6%

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

   7. Simplified 29.6%

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

   if 1.2e55 < b

   1. Initial program 49.9%

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

      1. associate-*l* 49.9%

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

      2. unpow2 49.9%

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

      3. unpow2 49.9%

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

      4. difference-of-squares 53.9%

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

   3. Simplified 53.9%

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

   4. Taylor expanded in angle around 0 46.5%

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

   5. Taylor expanded in b around inf 46.2%

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

      1. *-commutative 46.2%

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

      2. unpow2 46.2%

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

   7. Simplified 46.2%

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

   8. Taylor expanded in angle around 0 46.2%

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

      1. *-commutative 46.2%

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

      2. unpow2 46.2%

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

      3. *-commutative 46.2%

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

      4. associate-*l* 46.2%

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

   10. Simplified 46.2%

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

   11. Taylor expanded in angle around 0 46.2%

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

       1. *-commutative 46.2%

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

       2. *-commutative 46.2%

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

       3. unpow2 46.2%

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

       4. associate-*r* 46.2%

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

       5. associate-*l* 61.3%

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

       6. *-commutative 61.3%

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

   13. Simplified 61.3%

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

4. Final simplification 35.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq 1.2 \cdot 10^{+55}:\\ \;\;\;\;0.011111111111111112 \cdot \left(angle \cdot \left(\pi \cdot \left(b \cdot b\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;0.011111111111111112 \cdot \left(\left(b \cdot \pi\right) \cdot \left(angle \cdot b\right)\right)\\ \end{array} \]

Alternative 12: 39.5% accurate, 5.6× speedup

Math

\[\begin{array}{l} a = |a|\\ b = |b|\\ \\ \begin{array}{l} \mathbf{if}\;b \leq 3.8 \cdot 10^{+42}:\\ \;\;\;\;angle \cdot \left(0.011111111111111112 \cdot \left(\pi \cdot \left(b \cdot b\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;0.011111111111111112 \cdot \left(\left(b \cdot \pi\right) \cdot \left(angle \cdot b\right)\right)\\ \end{array} \end{array} \]

FPCore

NOTE: a should be positive before calling this function
NOTE: b should be positive before calling this function
(FPCore (a b angle)
 :precision binary64
 (if (<= b 3.8e+42)
   (* angle (* 0.011111111111111112 (* PI (* b b))))
   (* 0.011111111111111112 (* (* b PI) (* angle b)))))

C

a = abs(a);
b = abs(b);
double code(double a, double b, double angle) {
	double tmp;
	if (b <= 3.8e+42) {
		tmp = angle * (0.011111111111111112 * (((double) M_PI) * (b * b)));
	} else {
		tmp = 0.011111111111111112 * ((b * ((double) M_PI)) * (angle * b));
	}
	return tmp;
}

Java

a = Math.abs(a);
b = Math.abs(b);
public static double code(double a, double b, double angle) {
	double tmp;
	if (b <= 3.8e+42) {
		tmp = angle * (0.011111111111111112 * (Math.PI * (b * b)));
	} else {
		tmp = 0.011111111111111112 * ((b * Math.PI) * (angle * b));
	}
	return tmp;
}

Python

a = abs(a)
b = abs(b)
def code(a, b, angle):
	tmp = 0
	if b <= 3.8e+42:
		tmp = angle * (0.011111111111111112 * (math.pi * (b * b)))
	else:
		tmp = 0.011111111111111112 * ((b * math.pi) * (angle * b))
	return tmp

Julia

a = abs(a)
b = abs(b)
function code(a, b, angle)
	tmp = 0.0
	if (b <= 3.8e+42)
		tmp = Float64(angle * Float64(0.011111111111111112 * Float64(pi * Float64(b * b))));
	else
		tmp = Float64(0.011111111111111112 * Float64(Float64(b * pi) * Float64(angle * b)));
	end
	return tmp
end

MATLAB

a = abs(a)
b = abs(b)
function tmp_2 = code(a, b, angle)
	tmp = 0.0;
	if (b <= 3.8e+42)
		tmp = angle * (0.011111111111111112 * (pi * (b * b)));
	else
		tmp = 0.011111111111111112 * ((b * pi) * (angle * b));
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: a should be positive before calling this function
NOTE: b should be positive before calling this function
code[a_, b_, angle_] := If[LessEqual[b, 3.8e+42], N[(angle * N[(0.011111111111111112 * N[(Pi * N[(b * b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(0.011111111111111112 * N[(N[(b * Pi), $MachinePrecision] * N[(angle * b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b < 3.7999999999999998e42

   1. Initial program 51.2%

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

      1. associate-*l* 51.3%

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

      2. unpow2 51.3%

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

      3. unpow2 51.3%

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

      4. difference-of-squares 53.2%

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

   3. Simplified 53.2%

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

   4. Taylor expanded in angle around 0 50.9%

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

   5. Taylor expanded in b around inf 29.4%

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

      1. *-commutative 29.4%

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

      2. unpow2 29.4%

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

   7. Simplified 29.4%

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

   8. Taylor expanded in angle around 0 29.4%

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

      1. *-commutative 29.4%

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

      2. unpow2 29.4%

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

      3. *-commutative 29.4%

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

      4. associate-*l* 29.4%

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

   10. Simplified 29.4%

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

   if 3.7999999999999998e42 < b

   1. Initial program 51.8%

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

      1. associate-*l* 51.8%

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

      2. unpow2 51.8%

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

      3. unpow2 51.8%

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

      4. difference-of-squares 55.6%

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

   3. Simplified 55.6%

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

   4. Taylor expanded in angle around 0 48.6%

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

   5. Taylor expanded in b around inf 46.4%

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

      1. *-commutative 46.4%

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

      2. unpow2 46.4%

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

   7. Simplified 46.4%

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

   8. Taylor expanded in angle around 0 46.4%

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

      1. *-commutative 46.4%

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

      2. unpow2 46.4%

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

      3. *-commutative 46.4%

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

      4. associate-*l* 46.4%

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

   10. Simplified 46.4%

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

   11. Taylor expanded in angle around 0 46.4%

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

       1. *-commutative 46.4%

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

       2. *-commutative 46.4%

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

       3. unpow2 46.4%

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

       4. associate-*r* 46.4%

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

       5. associate-*l* 60.9%

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

       6. *-commutative 60.9%

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

   13. Simplified 60.9%

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

4. Final simplification 35.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq 3.8 \cdot 10^{+42}:\\ \;\;\;\;angle \cdot \left(0.011111111111111112 \cdot \left(\pi \cdot \left(b \cdot b\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;0.011111111111111112 \cdot \left(\left(b \cdot \pi\right) \cdot \left(angle \cdot b\right)\right)\\ \end{array} \]

Alternative 13: 51.9% accurate, 5.6× speedup

Math

\[\begin{array}{l} a = |a|\\ b = |b|\\ \\ \begin{array}{l} \mathbf{if}\;b \leq 5 \cdot 10^{+60}:\\ \;\;\;\;\left(angle \cdot -0.011111111111111112\right) \cdot \left(\pi \cdot \left(a \cdot a\right)\right)\\ \mathbf{else}:\\ \;\;\;\;0.011111111111111112 \cdot \left(\left(b \cdot \pi\right) \cdot \left(angle \cdot b\right)\right)\\ \end{array} \end{array} \]

FPCore

NOTE: a should be positive before calling this function
NOTE: b should be positive before calling this function
(FPCore (a b angle)
 :precision binary64
 (if (<= b 5e+60)
   (* (* angle -0.011111111111111112) (* PI (* a a)))
   (* 0.011111111111111112 (* (* b PI) (* angle b)))))

C

a = abs(a);
b = abs(b);
double code(double a, double b, double angle) {
	double tmp;
	if (b <= 5e+60) {
		tmp = (angle * -0.011111111111111112) * (((double) M_PI) * (a * a));
	} else {
		tmp = 0.011111111111111112 * ((b * ((double) M_PI)) * (angle * b));
	}
	return tmp;
}

Java

a = Math.abs(a);
b = Math.abs(b);
public static double code(double a, double b, double angle) {
	double tmp;
	if (b <= 5e+60) {
		tmp = (angle * -0.011111111111111112) * (Math.PI * (a * a));
	} else {
		tmp = 0.011111111111111112 * ((b * Math.PI) * (angle * b));
	}
	return tmp;
}

Python

a = abs(a)
b = abs(b)
def code(a, b, angle):
	tmp = 0
	if b <= 5e+60:
		tmp = (angle * -0.011111111111111112) * (math.pi * (a * a))
	else:
		tmp = 0.011111111111111112 * ((b * math.pi) * (angle * b))
	return tmp

Julia

a = abs(a)
b = abs(b)
function code(a, b, angle)
	tmp = 0.0
	if (b <= 5e+60)
		tmp = Float64(Float64(angle * -0.011111111111111112) * Float64(pi * Float64(a * a)));
	else
		tmp = Float64(0.011111111111111112 * Float64(Float64(b * pi) * Float64(angle * b)));
	end
	return tmp
end

MATLAB

a = abs(a)
b = abs(b)
function tmp_2 = code(a, b, angle)
	tmp = 0.0;
	if (b <= 5e+60)
		tmp = (angle * -0.011111111111111112) * (pi * (a * a));
	else
		tmp = 0.011111111111111112 * ((b * pi) * (angle * b));
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: a should be positive before calling this function
NOTE: b should be positive before calling this function
code[a_, b_, angle_] := If[LessEqual[b, 5e+60], N[(N[(angle * -0.011111111111111112), $MachinePrecision] * N[(Pi * N[(a * a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(0.011111111111111112 * N[(N[(b * Pi), $MachinePrecision] * N[(angle * b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b < 4.99999999999999975e60

   1. Initial program 51.0%

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

      1. associate-*l* 51.0%

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

      2. unpow2 51.0%

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

      3. unpow2 51.0%

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

      4. difference-of-squares 53.0%

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

   3. Simplified 53.0%

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

   4. Taylor expanded in angle around 0 50.7%

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

   5. Taylor expanded in b around 0 39.6%

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

      1. associate-*r* 39.2%

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

      2. *-commutative 39.2%

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

      3. unpow2 39.2%

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

   7. Simplified 39.2%

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

   if 4.99999999999999975e60 < b

   1. Initial program 52.8%

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

      1. associate-*l* 52.8%

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

      2. unpow2 52.8%

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

      3. unpow2 52.8%

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

      4. difference-of-squares 57.1%

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

   3. Simplified 57.1%

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

   4. Taylor expanded in angle around 0 49.1%

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

   5. Taylor expanded in b around inf 48.9%

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

      1. *-commutative 48.9%

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

      2. unpow2 48.9%

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

   7. Simplified 48.9%

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

   8. Taylor expanded in angle around 0 48.9%

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

      1. *-commutative 48.9%

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

      2. unpow2 48.9%

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

      3. *-commutative 48.9%

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

      4. associate-*l* 49.0%

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

   10. Simplified 49.0%

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

   11. Taylor expanded in angle around 0 48.9%

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

       1. *-commutative 48.9%

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

       2. *-commutative 48.9%

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

       3. unpow2 48.9%

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

       4. associate-*r* 48.9%

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

       5. associate-*l* 65.0%

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

       6. *-commutative 65.0%

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

   13. Simplified 65.0%

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

4. Final simplification 43.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq 5 \cdot 10^{+60}:\\ \;\;\;\;\left(angle \cdot -0.011111111111111112\right) \cdot \left(\pi \cdot \left(a \cdot a\right)\right)\\ \mathbf{else}:\\ \;\;\;\;0.011111111111111112 \cdot \left(\left(b \cdot \pi\right) \cdot \left(angle \cdot b\right)\right)\\ \end{array} \]

Alternative 14: 34.3% accurate, 5.7× speedup

Math

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

FPCore

NOTE: a should be positive before calling this function
NOTE: b should be positive before calling this function
(FPCore (a b angle)
 :precision binary64
 (* 0.011111111111111112 (* angle (* PI (* b b)))))

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

NOTE: a should be positive before calling this function
NOTE: b should be positive before calling this function
code[a_, b_, angle_] := N[(0.011111111111111112 * N[(angle * N[(Pi * N[(b * b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 51.4%

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

   1. associate-*l* 51.4%

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

   2. unpow2 51.4%

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

   3. unpow2 51.4%

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

   4. difference-of-squares 53.7%

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

3. Simplified 53.7%

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

4. Taylor expanded in angle around 0 50.4%

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

5. Taylor expanded in b around inf 32.8%

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

   1. *-commutative 32.8%

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

   2. unpow2 32.8%

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

7. Simplified 32.8%

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

8. Final simplification 32.8%

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

Reproduce

herbie shell --seed 2023257 
(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)))))