ab-angle->ABCF B

Percentage Accurate: 53.4% → 66.9%

Time: 28.8s

Alternatives: 21

Speedup: 5.3×

• 32.3% of points produce a very large (infinite) output. You may want to add a precondition.

Specification

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 53.4% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 66.9% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \pi \cdot \frac{angle}{180}\\ t_1 := \sin t_0\\ \mathbf{if}\;\frac{angle}{180} \leq -2 \cdot 10^{+207}:\\ \;\;\;\;t_1 \cdot \left(\left(2 \cdot {\left({\left({\left(b + a\right)}^{2}\right)}^{3}\right)}^{0.3333333333333333}\right) \cdot \cos t_0\right)\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \left(\left(t_1 \cdot \left(b + a\right)\right) \cdot \left(\left(b - a\right) \cdot \cos \left(\pi \cdot \left(0.005555555555555556 \cdot angle\right)\right)\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (a b angle)
 :precision binary64
 (let* ((t_0 (* PI (/ angle 180.0))) (t_1 (sin t_0)))
   (if (<= (/ angle 180.0) -2e+207)
     (*
      t_1
      (*
       (* 2.0 (pow (pow (pow (+ b a) 2.0) 3.0) 0.3333333333333333))
       (cos t_0)))
     (*
      2.0
      (*
       (* t_1 (+ b a))
       (* (- b a) (cos (* PI (* 0.005555555555555556 angle)))))))))

C

double code(double a, double b, double angle) {
	double t_0 = ((double) M_PI) * (angle / 180.0);
	double t_1 = sin(t_0);
	double tmp;
	if ((angle / 180.0) <= -2e+207) {
		tmp = t_1 * ((2.0 * pow(pow(pow((b + a), 2.0), 3.0), 0.3333333333333333)) * cos(t_0));
	} else {
		tmp = 2.0 * ((t_1 * (b + a)) * ((b - a) * cos((((double) M_PI) * (0.005555555555555556 * angle)))));
	}
	return tmp;
}

Java

public static double code(double a, double b, double angle) {
	double t_0 = Math.PI * (angle / 180.0);
	double t_1 = Math.sin(t_0);
	double tmp;
	if ((angle / 180.0) <= -2e+207) {
		tmp = t_1 * ((2.0 * Math.pow(Math.pow(Math.pow((b + a), 2.0), 3.0), 0.3333333333333333)) * Math.cos(t_0));
	} else {
		tmp = 2.0 * ((t_1 * (b + a)) * ((b - a) * Math.cos((Math.PI * (0.005555555555555556 * angle)))));
	}
	return tmp;
}

Python

def code(a, b, angle):
	t_0 = math.pi * (angle / 180.0)
	t_1 = math.sin(t_0)
	tmp = 0
	if (angle / 180.0) <= -2e+207:
		tmp = t_1 * ((2.0 * math.pow(math.pow(math.pow((b + a), 2.0), 3.0), 0.3333333333333333)) * math.cos(t_0))
	else:
		tmp = 2.0 * ((t_1 * (b + a)) * ((b - a) * math.cos((math.pi * (0.005555555555555556 * angle)))))
	return tmp

Julia

function code(a, b, angle)
	t_0 = Float64(pi * Float64(angle / 180.0))
	t_1 = sin(t_0)
	tmp = 0.0
	if (Float64(angle / 180.0) <= -2e+207)
		tmp = Float64(t_1 * Float64(Float64(2.0 * (((Float64(b + a) ^ 2.0) ^ 3.0) ^ 0.3333333333333333)) * cos(t_0)));
	else
		tmp = Float64(2.0 * Float64(Float64(t_1 * Float64(b + a)) * Float64(Float64(b - a) * cos(Float64(pi * Float64(0.005555555555555556 * angle))))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, angle)
	t_0 = pi * (angle / 180.0);
	t_1 = sin(t_0);
	tmp = 0.0;
	if ((angle / 180.0) <= -2e+207)
		tmp = t_1 * ((2.0 * ((((b + a) ^ 2.0) ^ 3.0) ^ 0.3333333333333333)) * cos(t_0));
	else
		tmp = 2.0 * ((t_1 * (b + a)) * ((b - a) * cos((pi * (0.005555555555555556 * angle)))));
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, angle_] := Block[{t$95$0 = N[(Pi * N[(angle / 180.0), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[Sin[t$95$0], $MachinePrecision]}, If[LessEqual[N[(angle / 180.0), $MachinePrecision], -2e+207], N[(t$95$1 * N[(N[(2.0 * N[Power[N[Power[N[Power[N[(b + a), $MachinePrecision], 2.0], $MachinePrecision], 3.0], $MachinePrecision], 0.3333333333333333], $MachinePrecision]), $MachinePrecision] * N[Cos[t$95$0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(2.0 * N[(N[(t$95$1 * N[(b + a), $MachinePrecision]), $MachinePrecision] * N[(N[(b - a), $MachinePrecision] * N[Cos[N[(Pi * N[(0.005555555555555556 * angle), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 angle 180) < -2.0000000000000001e207

   1. Initial program 21.2%

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

      1. *-commutative 21.2%

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

      2. associate-*l* 21.2%

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

      3. unpow2 21.2%

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

      4. fma-neg 21.2%

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

      5. unpow2 21.2%

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

   3. Simplified 21.2%

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

      1. add-cbrt-cube 22.3%

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

      2. pow1/3 43.8%

         \[\leadsto \sin \left(\pi \cdot \frac{angle}{180}\right) \cdot \left(\left(2 \cdot \color{blue}{{\left(\left(\mathsf{fma}\left(b, b, -a \cdot a\right) \cdot \mathsf{fma}\left(b, b, -a \cdot a\right)\right) \cdot \mathsf{fma}\left(b, b, -a \cdot a\right)\right)}^{0.3333333333333333}}\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right)\right) \]

      3. pow3 43.8%

         \[\leadsto \sin \left(\pi \cdot \frac{angle}{180}\right) \cdot \left(\left(2 \cdot {\color{blue}{\left({\left(\mathsf{fma}\left(b, b, -a \cdot a\right)\right)}^{3}\right)}}^{0.3333333333333333}\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right)\right) \]

      4. fma-neg 39.8%

         \[\leadsto \sin \left(\pi \cdot \frac{angle}{180}\right) \cdot \left(\left(2 \cdot {\left({\color{blue}{\left(b \cdot b - a \cdot a\right)}}^{3}\right)}^{0.3333333333333333}\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right)\right) \]

      5. difference-of-squares 43.8%

         \[\leadsto \sin \left(\pi \cdot \frac{angle}{180}\right) \cdot \left(\left(2 \cdot {\left({\color{blue}{\left(\left(b + a\right) \cdot \left(b - a\right)\right)}}^{3}\right)}^{0.3333333333333333}\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right)\right) \]

      6. sub-neg 43.8%

         \[\leadsto \sin \left(\pi \cdot \frac{angle}{180}\right) \cdot \left(\left(2 \cdot {\left({\left(\left(b + a\right) \cdot \color{blue}{\left(b + \left(-a\right)\right)}\right)}^{3}\right)}^{0.3333333333333333}\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right)\right) \]

      7. distribute-rgt-in 39.8%

         \[\leadsto \sin \left(\pi \cdot \frac{angle}{180}\right) \cdot \left(\left(2 \cdot {\left({\color{blue}{\left(b \cdot \left(b + a\right) + \left(-a\right) \cdot \left(b + a\right)\right)}}^{3}\right)}^{0.3333333333333333}\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right)\right) \]

      8. add-sqr-sqrt 18.0%

         \[\leadsto \sin \left(\pi \cdot \frac{angle}{180}\right) \cdot \left(\left(2 \cdot {\left({\left(b \cdot \left(b + a\right) + \color{blue}{\left(\sqrt{-a} \cdot \sqrt{-a}\right)} \cdot \left(b + a\right)\right)}^{3}\right)}^{0.3333333333333333}\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right)\right) \]

      9. sqrt-unprod 44.6%

         \[\leadsto \sin \left(\pi \cdot \frac{angle}{180}\right) \cdot \left(\left(2 \cdot {\left({\left(b \cdot \left(b + a\right) + \color{blue}{\sqrt{\left(-a\right) \cdot \left(-a\right)}} \cdot \left(b + a\right)\right)}^{3}\right)}^{0.3333333333333333}\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right)\right) \]

      10. sqr-neg 44.6%

          \[\leadsto \sin \left(\pi \cdot \frac{angle}{180}\right) \cdot \left(\left(2 \cdot {\left({\left(b \cdot \left(b + a\right) + \sqrt{\color{blue}{a \cdot a}} \cdot \left(b + a\right)\right)}^{3}\right)}^{0.3333333333333333}\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right)\right) \]

      11. sqrt-prod 26.6%

          \[\leadsto \sin \left(\pi \cdot \frac{angle}{180}\right) \cdot \left(\left(2 \cdot {\left({\left(b \cdot \left(b + a\right) + \color{blue}{\left(\sqrt{a} \cdot \sqrt{a}\right)} \cdot \left(b + a\right)\right)}^{3}\right)}^{0.3333333333333333}\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right)\right) \]

      12. add-sqr-sqrt 40.7%

          \[\leadsto \sin \left(\pi \cdot \frac{angle}{180}\right) \cdot \left(\left(2 \cdot {\left({\left(b \cdot \left(b + a\right) + \color{blue}{a} \cdot \left(b + a\right)\right)}^{3}\right)}^{0.3333333333333333}\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right)\right) \]

      13. distribute-rgt-in 44.7%

          \[\leadsto \sin \left(\pi \cdot \frac{angle}{180}\right) \cdot \left(\left(2 \cdot {\left({\color{blue}{\left(\left(b + a\right) \cdot \left(b + a\right)\right)}}^{3}\right)}^{0.3333333333333333}\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right)\right) \]

      14. pow2 44.7%

          \[\leadsto \sin \left(\pi \cdot \frac{angle}{180}\right) \cdot \left(\left(2 \cdot {\left({\color{blue}{\left({\left(b + a\right)}^{2}\right)}}^{3}\right)}^{0.3333333333333333}\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right)\right) \]

   5. Applied egg-rr 44.7%

      \[\leadsto \sin \left(\pi \cdot \frac{angle}{180}\right) \cdot \left(\left(2 \cdot \color{blue}{{\left({\left({\left(b + a\right)}^{2}\right)}^{3}\right)}^{0.3333333333333333}}\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right)\right) \]

   if -2.0000000000000001e207 < (/.f64 angle 180)

   1. Initial program 56.3%

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

      1. associate-*l* 56.2%

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

      2. unpow2 56.2%

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

      3. unpow2 56.2%

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

      4. difference-of-squares 58.9%

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

   3. Simplified 58.9%

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

   4. Taylor expanded in angle around inf 71.3%

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

      1. associate-*r* 71.3%

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

      2. *-commutative 71.3%

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

      3. *-commutative 71.3%

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

      4. associate-*r* 73.1%

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

      5. *-commutative 73.1%

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

      6. *-commutative 73.1%

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

      7. *-commutative 73.1%

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

      8. associate-*r* 72.7%

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

      9. *-commutative 72.7%

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

      10. +-commutative 72.7%

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

   6. Simplified 72.7%

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

      1. *-commutative 72.7%

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

      2. metadata-eval 72.7%

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

      3. div-inv 74.4%

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

   8. Applied egg-rr 74.4%

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

4. Final simplification 71.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{angle}{180} \leq -2 \cdot 10^{+207}:\\ \;\;\;\;\sin \left(\pi \cdot \frac{angle}{180}\right) \cdot \left(\left(2 \cdot {\left({\left({\left(b + a\right)}^{2}\right)}^{3}\right)}^{0.3333333333333333}\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \left(\left(\sin \left(\pi \cdot \frac{angle}{180}\right) \cdot \left(b + a\right)\right) \cdot \left(\left(b - a\right) \cdot \cos \left(\pi \cdot \left(0.005555555555555556 \cdot angle\right)\right)\right)\right)\\ \end{array} \]

Alternative 2: 67.2% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \sqrt[3]{\cos \left(\pi \cdot \left(0.005555555555555556 \cdot angle\right)\right)}\\ 2 \cdot \left(\left(\left(b - a\right) \cdot \left(\left(t_0 \cdot t_0\right) \cdot \sqrt[3]{\cos \left(\frac{\pi \cdot angle}{180}\right)}\right)\right) \cdot \left(\sin \left(\pi \cdot \frac{angle}{180}\right) \cdot \left(b + a\right)\right)\right) \end{array} \end{array} \]

FPCore

(FPCore (a b angle)
 :precision binary64
 (let* ((t_0 (cbrt (cos (* PI (* 0.005555555555555556 angle))))))
   (*
    2.0
    (*
     (* (- b a) (* (* t_0 t_0) (cbrt (cos (/ (* PI angle) 180.0)))))
     (* (sin (* PI (/ angle 180.0))) (+ b a))))))

C

double code(double a, double b, double angle) {
	double t_0 = cbrt(cos((((double) M_PI) * (0.005555555555555556 * angle))));
	return 2.0 * (((b - a) * ((t_0 * t_0) * cbrt(cos(((((double) M_PI) * angle) / 180.0))))) * (sin((((double) M_PI) * (angle / 180.0))) * (b + a)));
}

Java

public static double code(double a, double b, double angle) {
	double t_0 = Math.cbrt(Math.cos((Math.PI * (0.005555555555555556 * angle))));
	return 2.0 * (((b - a) * ((t_0 * t_0) * Math.cbrt(Math.cos(((Math.PI * angle) / 180.0))))) * (Math.sin((Math.PI * (angle / 180.0))) * (b + a)));
}

Julia

function code(a, b, angle)
	t_0 = cbrt(cos(Float64(pi * Float64(0.005555555555555556 * angle))))
	return Float64(2.0 * Float64(Float64(Float64(b - a) * Float64(Float64(t_0 * t_0) * cbrt(cos(Float64(Float64(pi * angle) / 180.0))))) * Float64(sin(Float64(pi * Float64(angle / 180.0))) * Float64(b + a))))
end

Wolfram

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

Derivation

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 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 inf 66.8%

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

   1. associate-*r* 66.8%

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

   2. *-commutative 66.8%

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

   3. *-commutative 66.8%

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

   4. associate-*r* 68.3%

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

   5. *-commutative 68.3%

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

   6. *-commutative 68.3%

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

   7. *-commutative 68.3%

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

   8. associate-*r* 67.7%

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

   9. *-commutative 67.7%

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

   10. +-commutative 67.7%

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

6. Simplified 67.7%

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

   1. *-commutative 67.7%

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

   2. metadata-eval 67.7%

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

   3. div-inv 69.2%

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

8. Applied egg-rr 69.2%

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

   1. add-cube-cbrt 69.2%

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

10. Applied egg-rr 69.2%

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

    1. *-commutative 69.2%

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

    2. metadata-eval 69.2%

       \[\leadsto 2 \cdot \left(\left(\left(b - a\right) \cdot \left(\left(\sqrt[3]{\cos \left(\pi \cdot \left(0.005555555555555556 \cdot angle\right)\right)} \cdot \sqrt[3]{\cos \left(\pi \cdot \left(0.005555555555555556 \cdot angle\right)\right)}\right) \cdot \sqrt[3]{\cos \left(\pi \cdot \left(angle \cdot \color{blue}{\frac{1}{180}}\right)\right)}\right)\right) \cdot \left(\sin \left(\pi \cdot \frac{angle}{180}\right) \cdot \left(b + a\right)\right)\right) \]

    3. div-inv 67.1%

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

    4. associate-*r/ 69.4%

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

12. Applied egg-rr 69.4%

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

13. Final simplification 69.4%

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

Alternative 3: 67.1% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \pi \cdot \frac{angle}{180}\\ t_1 := \sin t_0\\ \mathbf{if}\;\frac{angle}{180} \leq -2 \cdot 10^{+207}:\\ \;\;\;\;t_1 \cdot \left(\cos t_0 \cdot \left(2 \cdot \left({\left(b + a\right)}^{2} + a \cdot \left(a + a\right)\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \left(\left(t_1 \cdot \left(b + a\right)\right) \cdot \left(\left(b - a\right) \cdot \cos \left(\pi \cdot \left(0.005555555555555556 \cdot angle\right)\right)\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (a b angle)
 :precision binary64
 (let* ((t_0 (* PI (/ angle 180.0))) (t_1 (sin t_0)))
   (if (<= (/ angle 180.0) -2e+207)
     (* t_1 (* (cos t_0) (* 2.0 (+ (pow (+ b a) 2.0) (* a (+ a a))))))
     (*
      2.0
      (*
       (* t_1 (+ b a))
       (* (- b a) (cos (* PI (* 0.005555555555555556 angle)))))))))

C

double code(double a, double b, double angle) {
	double t_0 = ((double) M_PI) * (angle / 180.0);
	double t_1 = sin(t_0);
	double tmp;
	if ((angle / 180.0) <= -2e+207) {
		tmp = t_1 * (cos(t_0) * (2.0 * (pow((b + a), 2.0) + (a * (a + a)))));
	} else {
		tmp = 2.0 * ((t_1 * (b + a)) * ((b - a) * cos((((double) M_PI) * (0.005555555555555556 * angle)))));
	}
	return tmp;
}

Java

public static double code(double a, double b, double angle) {
	double t_0 = Math.PI * (angle / 180.0);
	double t_1 = Math.sin(t_0);
	double tmp;
	if ((angle / 180.0) <= -2e+207) {
		tmp = t_1 * (Math.cos(t_0) * (2.0 * (Math.pow((b + a), 2.0) + (a * (a + a)))));
	} else {
		tmp = 2.0 * ((t_1 * (b + a)) * ((b - a) * Math.cos((Math.PI * (0.005555555555555556 * angle)))));
	}
	return tmp;
}

Python

def code(a, b, angle):
	t_0 = math.pi * (angle / 180.0)
	t_1 = math.sin(t_0)
	tmp = 0
	if (angle / 180.0) <= -2e+207:
		tmp = t_1 * (math.cos(t_0) * (2.0 * (math.pow((b + a), 2.0) + (a * (a + a)))))
	else:
		tmp = 2.0 * ((t_1 * (b + a)) * ((b - a) * math.cos((math.pi * (0.005555555555555556 * angle)))))
	return tmp

Julia

function code(a, b, angle)
	t_0 = Float64(pi * Float64(angle / 180.0))
	t_1 = sin(t_0)
	tmp = 0.0
	if (Float64(angle / 180.0) <= -2e+207)
		tmp = Float64(t_1 * Float64(cos(t_0) * Float64(2.0 * Float64((Float64(b + a) ^ 2.0) + Float64(a * Float64(a + a))))));
	else
		tmp = Float64(2.0 * Float64(Float64(t_1 * Float64(b + a)) * Float64(Float64(b - a) * cos(Float64(pi * Float64(0.005555555555555556 * angle))))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, angle)
	t_0 = pi * (angle / 180.0);
	t_1 = sin(t_0);
	tmp = 0.0;
	if ((angle / 180.0) <= -2e+207)
		tmp = t_1 * (cos(t_0) * (2.0 * (((b + a) ^ 2.0) + (a * (a + a)))));
	else
		tmp = 2.0 * ((t_1 * (b + a)) * ((b - a) * cos((pi * (0.005555555555555556 * angle)))));
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, angle_] := Block[{t$95$0 = N[(Pi * N[(angle / 180.0), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[Sin[t$95$0], $MachinePrecision]}, If[LessEqual[N[(angle / 180.0), $MachinePrecision], -2e+207], N[(t$95$1 * N[(N[Cos[t$95$0], $MachinePrecision] * N[(2.0 * N[(N[Power[N[(b + a), $MachinePrecision], 2.0], $MachinePrecision] + N[(a * N[(a + a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(2.0 * N[(N[(t$95$1 * N[(b + a), $MachinePrecision]), $MachinePrecision] * N[(N[(b - a), $MachinePrecision] * N[Cos[N[(Pi * N[(0.005555555555555556 * angle), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 angle 180) < -2.0000000000000001e207

   1. Initial program 21.2%

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

      1. *-commutative 21.2%

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

      2. associate-*l* 21.2%

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

      3. unpow2 21.2%

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

      4. fma-neg 21.2%

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

      5. unpow2 21.2%

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

   3. Simplified 21.2%

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

      1. fma-neg 21.2%

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

      2. prod-diff 21.2%

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

      3. fma-neg 21.2%

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

      4. difference-of-squares 25.2%

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

      5. sub-neg 25.2%

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

      6. distribute-rgt-in 21.2%

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

      7. add-sqr-sqrt 12.0%

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

      8. sqrt-unprod 35.3%

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

      9. sqr-neg 35.3%

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

      10. sqrt-prod 23.3%

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

      11. add-sqr-sqrt 38.6%

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

      12. distribute-rgt-in 42.6%

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

      13. pow2 42.6%

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

      14. add-sqr-sqrt 19.3%

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

      15. sqrt-unprod 30.5%

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

      16. sqr-neg 30.5%

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

      17. sqrt-prod 23.3%

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

      18. add-sqr-sqrt 42.6%

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

   5. Applied egg-rr 42.6%

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

      1. fma-udef 42.6%

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

      2. distribute-lft-out 42.6%

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

   7. Simplified 42.6%

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

   if -2.0000000000000001e207 < (/.f64 angle 180)

   1. Initial program 56.3%

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

      1. associate-*l* 56.2%

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

      2. unpow2 56.2%

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

      3. unpow2 56.2%

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

      4. difference-of-squares 58.9%

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

   3. Simplified 58.9%

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

   4. Taylor expanded in angle around inf 71.3%

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

      1. associate-*r* 71.3%

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

      2. *-commutative 71.3%

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

      3. *-commutative 71.3%

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

      4. associate-*r* 73.1%

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

      5. *-commutative 73.1%

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

      6. *-commutative 73.1%

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

      7. *-commutative 73.1%

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

      8. associate-*r* 72.7%

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

      9. *-commutative 72.7%

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

      10. +-commutative 72.7%

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

   6. Simplified 72.7%

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

      1. *-commutative 72.7%

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

      2. metadata-eval 72.7%

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

      3. div-inv 74.4%

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

   8. Applied egg-rr 74.4%

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

4. Final simplification 71.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{angle}{180} \leq -2 \cdot 10^{+207}:\\ \;\;\;\;\sin \left(\pi \cdot \frac{angle}{180}\right) \cdot \left(\cos \left(\pi \cdot \frac{angle}{180}\right) \cdot \left(2 \cdot \left({\left(b + a\right)}^{2} + a \cdot \left(a + a\right)\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \left(\left(\sin \left(\pi \cdot \frac{angle}{180}\right) \cdot \left(b + a\right)\right) \cdot \left(\left(b - a\right) \cdot \cos \left(\pi \cdot \left(0.005555555555555556 \cdot angle\right)\right)\right)\right)\\ \end{array} \]

Alternative 4: 66.5% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := 0.005555555555555556 \cdot \left(\pi \cdot angle\right)\\ t_1 := \left(b + a\right) \cdot \sin t_0\\ \mathbf{if}\;{a}^{2} \leq 10^{+156}:\\ \;\;\;\;2 \cdot \left(\left(\left(b - a\right) \cdot \cos t_0\right) \cdot t_1\right)\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \left(\left(b - a\right) \cdot t_1\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (a b angle)
 :precision binary64
 (let* ((t_0 (* 0.005555555555555556 (* PI angle)))
        (t_1 (* (+ b a) (sin t_0))))
   (if (<= (pow a 2.0) 1e+156)
     (* 2.0 (* (* (- b a) (cos t_0)) t_1))
     (* 2.0 (* (- b a) t_1)))))

C

double code(double a, double b, double angle) {
	double t_0 = 0.005555555555555556 * (((double) M_PI) * angle);
	double t_1 = (b + a) * sin(t_0);
	double tmp;
	if (pow(a, 2.0) <= 1e+156) {
		tmp = 2.0 * (((b - a) * cos(t_0)) * t_1);
	} else {
		tmp = 2.0 * ((b - a) * t_1);
	}
	return tmp;
}

Java

public static double code(double a, double b, double angle) {
	double t_0 = 0.005555555555555556 * (Math.PI * angle);
	double t_1 = (b + a) * Math.sin(t_0);
	double tmp;
	if (Math.pow(a, 2.0) <= 1e+156) {
		tmp = 2.0 * (((b - a) * Math.cos(t_0)) * t_1);
	} else {
		tmp = 2.0 * ((b - a) * t_1);
	}
	return tmp;
}

Python

def code(a, b, angle):
	t_0 = 0.005555555555555556 * (math.pi * angle)
	t_1 = (b + a) * math.sin(t_0)
	tmp = 0
	if math.pow(a, 2.0) <= 1e+156:
		tmp = 2.0 * (((b - a) * math.cos(t_0)) * t_1)
	else:
		tmp = 2.0 * ((b - a) * t_1)
	return tmp

Julia

function code(a, b, angle)
	t_0 = Float64(0.005555555555555556 * Float64(pi * angle))
	t_1 = Float64(Float64(b + a) * sin(t_0))
	tmp = 0.0
	if ((a ^ 2.0) <= 1e+156)
		tmp = Float64(2.0 * Float64(Float64(Float64(b - a) * cos(t_0)) * t_1));
	else
		tmp = Float64(2.0 * Float64(Float64(b - a) * t_1));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, angle)
	t_0 = 0.005555555555555556 * (pi * angle);
	t_1 = (b + a) * sin(t_0);
	tmp = 0.0;
	if ((a ^ 2.0) <= 1e+156)
		tmp = 2.0 * (((b - a) * cos(t_0)) * t_1);
	else
		tmp = 2.0 * ((b - a) * t_1);
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, angle_] := Block[{t$95$0 = N[(0.005555555555555556 * N[(Pi * angle), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[(b + a), $MachinePrecision] * N[Sin[t$95$0], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[Power[a, 2.0], $MachinePrecision], 1e+156], N[(2.0 * N[(N[(N[(b - a), $MachinePrecision] * N[Cos[t$95$0], $MachinePrecision]), $MachinePrecision] * t$95$1), $MachinePrecision]), $MachinePrecision], N[(2.0 * N[(N[(b - a), $MachinePrecision] * t$95$1), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 2 regimes

2. if (pow.f64 a 2) < 9.9999999999999998e155

   1. Initial program 57.0%

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

      1. associate-*l* 57.0%

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

      2. unpow2 57.0%

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

      3. unpow2 57.0%

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

      4. difference-of-squares 57.0%

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

   3. Simplified 57.0%

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

   4. Taylor expanded in angle around inf 64.7%

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

      1. associate-*r* 64.7%

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

      2. *-commutative 64.7%

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

      3. +-commutative 64.7%

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

   6. Simplified 64.7%

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

   if 9.9999999999999998e155 < (pow.f64 a 2)

   1. Initial program 45.5%

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

      1. associate-*l* 45.5%

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

      2. unpow2 45.5%

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

      3. unpow2 45.5%

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

      4. difference-of-squares 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. Step-by-step derivation

      1. add-cube-cbrt 56.7%

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

      2. pow3 56.7%

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

      3. div-inv 56.7%

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

      4. metadata-eval 56.7%

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

   5. Applied egg-rr 56.7%

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

   6. Taylor expanded in angle around inf 75.7%

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

4. Final simplification 68.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;{a}^{2} \leq 10^{+156}:\\ \;\;\;\;2 \cdot \left(\left(\left(b - a\right) \cdot \cos \left(0.005555555555555556 \cdot \left(\pi \cdot angle\right)\right)\right) \cdot \left(\left(b + a\right) \cdot \sin \left(0.005555555555555556 \cdot \left(\pi \cdot angle\right)\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \left(\left(b - a\right) \cdot \left(\left(b + a\right) \cdot \sin \left(0.005555555555555556 \cdot \left(\pi \cdot angle\right)\right)\right)\right)\\ \end{array} \]

Alternative 5: 67.1% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{angle}{180} \leq -2 \cdot 10^{+207}:\\ \;\;\;\;2 \cdot \left({\left(b + a\right)}^{2} \cdot \left(\sin \left(\pi \cdot \left(2 \cdot \left(0.005555555555555556 \cdot angle\right)\right)\right) \cdot 0.5\right)\right)\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \left(\left(\sin \left(\pi \cdot \frac{angle}{180}\right) \cdot \left(b + a\right)\right) \cdot \left(\left(b - a\right) \cdot \cos \left(\pi \cdot \left(0.005555555555555556 \cdot angle\right)\right)\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (a b angle)
 :precision binary64
 (if (<= (/ angle 180.0) -2e+207)
   (*
    2.0
    (*
     (pow (+ b a) 2.0)
     (* (sin (* PI (* 2.0 (* 0.005555555555555556 angle)))) 0.5)))
   (*
    2.0
    (*
     (* (sin (* PI (/ angle 180.0))) (+ b a))
     (* (- b a) (cos (* PI (* 0.005555555555555556 angle))))))))

C

double code(double a, double b, double angle) {
	double tmp;
	if ((angle / 180.0) <= -2e+207) {
		tmp = 2.0 * (pow((b + a), 2.0) * (sin((((double) M_PI) * (2.0 * (0.005555555555555556 * angle)))) * 0.5));
	} else {
		tmp = 2.0 * ((sin((((double) M_PI) * (angle / 180.0))) * (b + a)) * ((b - a) * cos((((double) M_PI) * (0.005555555555555556 * angle)))));
	}
	return tmp;
}

Java

public static double code(double a, double b, double angle) {
	double tmp;
	if ((angle / 180.0) <= -2e+207) {
		tmp = 2.0 * (Math.pow((b + a), 2.0) * (Math.sin((Math.PI * (2.0 * (0.005555555555555556 * angle)))) * 0.5));
	} else {
		tmp = 2.0 * ((Math.sin((Math.PI * (angle / 180.0))) * (b + a)) * ((b - a) * Math.cos((Math.PI * (0.005555555555555556 * angle)))));
	}
	return tmp;
}

Python

def code(a, b, angle):
	tmp = 0
	if (angle / 180.0) <= -2e+207:
		tmp = 2.0 * (math.pow((b + a), 2.0) * (math.sin((math.pi * (2.0 * (0.005555555555555556 * angle)))) * 0.5))
	else:
		tmp = 2.0 * ((math.sin((math.pi * (angle / 180.0))) * (b + a)) * ((b - a) * math.cos((math.pi * (0.005555555555555556 * angle)))))
	return tmp

Julia

function code(a, b, angle)
	tmp = 0.0
	if (Float64(angle / 180.0) <= -2e+207)
		tmp = Float64(2.0 * Float64((Float64(b + a) ^ 2.0) * Float64(sin(Float64(pi * Float64(2.0 * Float64(0.005555555555555556 * angle)))) * 0.5)));
	else
		tmp = Float64(2.0 * Float64(Float64(sin(Float64(pi * Float64(angle / 180.0))) * Float64(b + a)) * Float64(Float64(b - a) * cos(Float64(pi * Float64(0.005555555555555556 * angle))))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, angle)
	tmp = 0.0;
	if ((angle / 180.0) <= -2e+207)
		tmp = 2.0 * (((b + a) ^ 2.0) * (sin((pi * (2.0 * (0.005555555555555556 * angle)))) * 0.5));
	else
		tmp = 2.0 * ((sin((pi * (angle / 180.0))) * (b + a)) * ((b - a) * cos((pi * (0.005555555555555556 * angle)))));
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, angle_] := If[LessEqual[N[(angle / 180.0), $MachinePrecision], -2e+207], N[(2.0 * N[(N[Power[N[(b + a), $MachinePrecision], 2.0], $MachinePrecision] * N[(N[Sin[N[(Pi * N[(2.0 * N[(0.005555555555555556 * angle), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * 0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(2.0 * N[(N[(N[Sin[N[(Pi * N[(angle / 180.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * N[(b + a), $MachinePrecision]), $MachinePrecision] * N[(N[(b - a), $MachinePrecision] * N[Cos[N[(Pi * N[(0.005555555555555556 * angle), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 angle 180) < -2.0000000000000001e207

   1. Initial program 21.2%

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

      1. *-commutative 21.2%

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

      2. associate-*l* 21.2%

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

      3. unpow2 21.2%

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

      4. fma-neg 21.2%

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

      5. unpow2 21.2%

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

   3. Simplified 21.2%

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

   5. Applied egg-rr 24.3%

      \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(\left(2 \cdot {\left(b + a\right)}^{2}\right) \cdot \left(\left(\sin 0 + \sin \left(\left(\pi \cdot \left(angle \cdot 0.005555555555555556\right)\right) \cdot 2\right)\right) \cdot 0.5\right)\right)} - 1} \]
   6. Step-by-step derivation

      1. expm1-def 21.5%

         \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\left(2 \cdot {\left(b + a\right)}^{2}\right) \cdot \left(\left(\sin 0 + \sin \left(\left(\pi \cdot \left(angle \cdot 0.005555555555555556\right)\right) \cdot 2\right)\right) \cdot 0.5\right)\right)\right)} \]

      2. expm1-log1p 38.5%

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

      3. associate-*l* 38.5%

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

      4. sin-0 38.5%

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

      5. +-lft-identity 38.5%

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

      6. associate-*l* 38.5%

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

      7. *-commutative 38.5%

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

      8. *-commutative 38.5%

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

   7. Simplified 38.5%

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

   if -2.0000000000000001e207 < (/.f64 angle 180)

   1. Initial program 56.3%

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

      1. associate-*l* 56.2%

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

      2. unpow2 56.2%

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

      3. unpow2 56.2%

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

      4. difference-of-squares 58.9%

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

   3. Simplified 58.9%

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

   4. Taylor expanded in angle around inf 71.3%

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

      1. associate-*r* 71.3%

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

      2. *-commutative 71.3%

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

      3. *-commutative 71.3%

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

      4. associate-*r* 73.1%

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

      5. *-commutative 73.1%

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

      6. *-commutative 73.1%

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

      7. *-commutative 73.1%

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

      8. associate-*r* 72.7%

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

      9. *-commutative 72.7%

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

      10. +-commutative 72.7%

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

   6. Simplified 72.7%

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

      1. *-commutative 72.7%

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

      2. metadata-eval 72.7%

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

      3. div-inv 74.4%

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

   8. Applied egg-rr 74.4%

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

4. Final simplification 70.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{angle}{180} \leq -2 \cdot 10^{+207}:\\ \;\;\;\;2 \cdot \left({\left(b + a\right)}^{2} \cdot \left(\sin \left(\pi \cdot \left(2 \cdot \left(0.005555555555555556 \cdot angle\right)\right)\right) \cdot 0.5\right)\right)\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \left(\left(\sin \left(\pi \cdot \frac{angle}{180}\right) \cdot \left(b + a\right)\right) \cdot \left(\left(b - a\right) \cdot \cos \left(\pi \cdot \left(0.005555555555555556 \cdot angle\right)\right)\right)\right)\\ \end{array} \]

Alternative 6: 67.1% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \pi \cdot \frac{angle}{180}\\ t_1 := \sin t_0\\ \mathbf{if}\;\frac{angle}{180} \leq -2 \cdot 10^{+207}:\\ \;\;\;\;t_1 \cdot \left(\cos t_0 \cdot \left(2 \cdot \left(b \cdot b + a \cdot a\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \left(\left(t_1 \cdot \left(b + a\right)\right) \cdot \left(\left(b - a\right) \cdot \cos \left(\pi \cdot \left(0.005555555555555556 \cdot angle\right)\right)\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (a b angle)
 :precision binary64
 (let* ((t_0 (* PI (/ angle 180.0))) (t_1 (sin t_0)))
   (if (<= (/ angle 180.0) -2e+207)
     (* t_1 (* (cos t_0) (* 2.0 (+ (* b b) (* a a)))))
     (*
      2.0
      (*
       (* t_1 (+ b a))
       (* (- b a) (cos (* PI (* 0.005555555555555556 angle)))))))))

C

double code(double a, double b, double angle) {
	double t_0 = ((double) M_PI) * (angle / 180.0);
	double t_1 = sin(t_0);
	double tmp;
	if ((angle / 180.0) <= -2e+207) {
		tmp = t_1 * (cos(t_0) * (2.0 * ((b * b) + (a * a))));
	} else {
		tmp = 2.0 * ((t_1 * (b + a)) * ((b - a) * cos((((double) M_PI) * (0.005555555555555556 * angle)))));
	}
	return tmp;
}

Java

public static double code(double a, double b, double angle) {
	double t_0 = Math.PI * (angle / 180.0);
	double t_1 = Math.sin(t_0);
	double tmp;
	if ((angle / 180.0) <= -2e+207) {
		tmp = t_1 * (Math.cos(t_0) * (2.0 * ((b * b) + (a * a))));
	} else {
		tmp = 2.0 * ((t_1 * (b + a)) * ((b - a) * Math.cos((Math.PI * (0.005555555555555556 * angle)))));
	}
	return tmp;
}

Python

def code(a, b, angle):
	t_0 = math.pi * (angle / 180.0)
	t_1 = math.sin(t_0)
	tmp = 0
	if (angle / 180.0) <= -2e+207:
		tmp = t_1 * (math.cos(t_0) * (2.0 * ((b * b) + (a * a))))
	else:
		tmp = 2.0 * ((t_1 * (b + a)) * ((b - a) * math.cos((math.pi * (0.005555555555555556 * angle)))))
	return tmp

Julia

function code(a, b, angle)
	t_0 = Float64(pi * Float64(angle / 180.0))
	t_1 = sin(t_0)
	tmp = 0.0
	if (Float64(angle / 180.0) <= -2e+207)
		tmp = Float64(t_1 * Float64(cos(t_0) * Float64(2.0 * Float64(Float64(b * b) + Float64(a * a)))));
	else
		tmp = Float64(2.0 * Float64(Float64(t_1 * Float64(b + a)) * Float64(Float64(b - a) * cos(Float64(pi * Float64(0.005555555555555556 * angle))))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, angle)
	t_0 = pi * (angle / 180.0);
	t_1 = sin(t_0);
	tmp = 0.0;
	if ((angle / 180.0) <= -2e+207)
		tmp = t_1 * (cos(t_0) * (2.0 * ((b * b) + (a * a))));
	else
		tmp = 2.0 * ((t_1 * (b + a)) * ((b - a) * cos((pi * (0.005555555555555556 * angle)))));
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, angle_] := Block[{t$95$0 = N[(Pi * N[(angle / 180.0), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[Sin[t$95$0], $MachinePrecision]}, If[LessEqual[N[(angle / 180.0), $MachinePrecision], -2e+207], N[(t$95$1 * N[(N[Cos[t$95$0], $MachinePrecision] * N[(2.0 * N[(N[(b * b), $MachinePrecision] + N[(a * a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(2.0 * N[(N[(t$95$1 * N[(b + a), $MachinePrecision]), $MachinePrecision] * N[(N[(b - a), $MachinePrecision] * N[Cos[N[(Pi * N[(0.005555555555555556 * angle), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 angle 180) < -2.0000000000000001e207

   1. Initial program 21.2%

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

      1. *-commutative 21.2%

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

      2. associate-*l* 21.2%

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

      3. unpow2 21.2%

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

      4. fma-neg 21.2%

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

      5. unpow2 21.2%

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

   3. Simplified 21.2%

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

      1. fma-udef 21.2%

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

      2. add-sqr-sqrt 6.0%

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

      3. sqrt-unprod 46.1%

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

      4. sqr-neg 46.1%

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

      5. sqrt-unprod 42.6%

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

      6. add-sqr-sqrt 42.6%

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

   5. Applied egg-rr 42.6%

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

   if -2.0000000000000001e207 < (/.f64 angle 180)

   1. Initial program 56.3%

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

      1. associate-*l* 56.2%

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

      2. unpow2 56.2%

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

      3. unpow2 56.2%

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

      4. difference-of-squares 58.9%

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

   3. Simplified 58.9%

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

   4. Taylor expanded in angle around inf 71.3%

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

      1. associate-*r* 71.3%

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

      2. *-commutative 71.3%

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

      3. *-commutative 71.3%

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

      4. associate-*r* 73.1%

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

      5. *-commutative 73.1%

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

      6. *-commutative 73.1%

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

      7. *-commutative 73.1%

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

      8. associate-*r* 72.7%

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

      9. *-commutative 72.7%

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

      10. +-commutative 72.7%

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

   6. Simplified 72.7%

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

      1. *-commutative 72.7%

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

      2. metadata-eval 72.7%

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

      3. div-inv 74.4%

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

   8. Applied egg-rr 74.4%

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

4. Final simplification 71.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{angle}{180} \leq -2 \cdot 10^{+207}:\\ \;\;\;\;\sin \left(\pi \cdot \frac{angle}{180}\right) \cdot \left(\cos \left(\pi \cdot \frac{angle}{180}\right) \cdot \left(2 \cdot \left(b \cdot b + a \cdot a\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \left(\left(\sin \left(\pi \cdot \frac{angle}{180}\right) \cdot \left(b + a\right)\right) \cdot \left(\left(b - a\right) \cdot \cos \left(\pi \cdot \left(0.005555555555555556 \cdot angle\right)\right)\right)\right)\\ \end{array} \]

Alternative 7: 67.0% accurate, 1.5× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

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 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 inf 66.8%

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

   1. associate-*r* 66.8%

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

   2. *-commutative 66.8%

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

   3. *-commutative 66.8%

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

   4. associate-*r* 68.3%

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

   5. *-commutative 68.3%

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

   6. *-commutative 68.3%

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

   7. *-commutative 68.3%

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

   8. associate-*r* 67.7%

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

   9. *-commutative 67.7%

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

   10. +-commutative 67.7%

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

6. Simplified 67.7%

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

7. Taylor expanded in angle around inf 68.5%

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

8. Final simplification 68.5%

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

Alternative 8: 66.0% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{angle}{180} \leq -5 \cdot 10^{+20}:\\ \;\;\;\;2 \cdot \left({\left(b + a\right)}^{2} \cdot \left(\sin \left(\pi \cdot \left(2 \cdot \left(0.005555555555555556 \cdot angle\right)\right)\right) \cdot 0.5\right)\right)\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \left(\left(b - a\right) \cdot \left(\left(b + a\right) \cdot \sin \left(\pi \cdot \left(0.005555555555555556 \cdot angle\right)\right)\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (a b angle)
 :precision binary64
 (if (<= (/ angle 180.0) -5e+20)
   (*
    2.0
    (*
     (pow (+ b a) 2.0)
     (* (sin (* PI (* 2.0 (* 0.005555555555555556 angle)))) 0.5)))
   (*
    2.0
    (* (- b a) (* (+ b a) (sin (* PI (* 0.005555555555555556 angle))))))))

C

double code(double a, double b, double angle) {
	double tmp;
	if ((angle / 180.0) <= -5e+20) {
		tmp = 2.0 * (pow((b + a), 2.0) * (sin((((double) M_PI) * (2.0 * (0.005555555555555556 * angle)))) * 0.5));
	} else {
		tmp = 2.0 * ((b - a) * ((b + a) * sin((((double) M_PI) * (0.005555555555555556 * angle)))));
	}
	return tmp;
}

Java

public static double code(double a, double b, double angle) {
	double tmp;
	if ((angle / 180.0) <= -5e+20) {
		tmp = 2.0 * (Math.pow((b + a), 2.0) * (Math.sin((Math.PI * (2.0 * (0.005555555555555556 * angle)))) * 0.5));
	} else {
		tmp = 2.0 * ((b - a) * ((b + a) * Math.sin((Math.PI * (0.005555555555555556 * angle)))));
	}
	return tmp;
}

Python

def code(a, b, angle):
	tmp = 0
	if (angle / 180.0) <= -5e+20:
		tmp = 2.0 * (math.pow((b + a), 2.0) * (math.sin((math.pi * (2.0 * (0.005555555555555556 * angle)))) * 0.5))
	else:
		tmp = 2.0 * ((b - a) * ((b + a) * math.sin((math.pi * (0.005555555555555556 * angle)))))
	return tmp

Julia

function code(a, b, angle)
	tmp = 0.0
	if (Float64(angle / 180.0) <= -5e+20)
		tmp = Float64(2.0 * Float64((Float64(b + a) ^ 2.0) * Float64(sin(Float64(pi * Float64(2.0 * Float64(0.005555555555555556 * angle)))) * 0.5)));
	else
		tmp = Float64(2.0 * Float64(Float64(b - a) * Float64(Float64(b + a) * sin(Float64(pi * Float64(0.005555555555555556 * angle))))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, angle)
	tmp = 0.0;
	if ((angle / 180.0) <= -5e+20)
		tmp = 2.0 * (((b + a) ^ 2.0) * (sin((pi * (2.0 * (0.005555555555555556 * angle)))) * 0.5));
	else
		tmp = 2.0 * ((b - a) * ((b + a) * sin((pi * (0.005555555555555556 * angle)))));
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, angle_] := If[LessEqual[N[(angle / 180.0), $MachinePrecision], -5e+20], N[(2.0 * N[(N[Power[N[(b + a), $MachinePrecision], 2.0], $MachinePrecision] * N[(N[Sin[N[(Pi * N[(2.0 * N[(0.005555555555555556 * angle), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * 0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(2.0 * N[(N[(b - a), $MachinePrecision] * N[(N[(b + a), $MachinePrecision] * N[Sin[N[(Pi * N[(0.005555555555555556 * angle), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 angle 180) < -5e20

   1. Initial program 24.8%

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

      1. *-commutative 24.8%

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

      2. associate-*l* 24.8%

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

      3. unpow2 24.8%

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

      4. fma-neg 26.7%

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

      5. unpow2 26.7%

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

   3. Simplified 26.7%

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

   5. Applied egg-rr 25.4%

      \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(\left(2 \cdot {\left(b + a\right)}^{2}\right) \cdot \left(\left(\sin 0 + \sin \left(\left(\pi \cdot \left(angle \cdot 0.005555555555555556\right)\right) \cdot 2\right)\right) \cdot 0.5\right)\right)} - 1} \]
   6. Step-by-step derivation

      1. expm1-def 24.0%

         \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\left(2 \cdot {\left(b + a\right)}^{2}\right) \cdot \left(\left(\sin 0 + \sin \left(\left(\pi \cdot \left(angle \cdot 0.005555555555555556\right)\right) \cdot 2\right)\right) \cdot 0.5\right)\right)\right)} \]

      2. expm1-log1p 36.2%

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

      3. associate-*l* 36.2%

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

      4. sin-0 36.2%

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

      5. +-lft-identity 36.2%

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

      6. associate-*l* 36.2%

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

      7. *-commutative 36.2%

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

      8. *-commutative 36.2%

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

   7. Simplified 36.2%

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

   if -5e20 < (/.f64 angle 180)

   1. Initial program 60.5%

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

      1. associate-*l* 60.5%

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

      2. unpow2 60.5%

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

      3. unpow2 60.5%

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

      4. difference-of-squares 63.5%

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

   3. Simplified 63.5%

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

   4. Taylor expanded in angle around inf 77.7%

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

      1. associate-*r* 77.7%

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

      2. *-commutative 77.7%

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

      3. *-commutative 77.7%

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

      4. associate-*r* 79.3%

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

      5. *-commutative 79.3%

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

      6. *-commutative 79.3%

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

      7. *-commutative 79.3%

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

      8. associate-*r* 78.8%

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

      9. *-commutative 78.8%

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

      10. +-commutative 78.8%

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

   6. Simplified 78.8%

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

   7. Taylor expanded in angle around 0 78.8%

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

4. Final simplification 69.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{angle}{180} \leq -5 \cdot 10^{+20}:\\ \;\;\;\;2 \cdot \left({\left(b + a\right)}^{2} \cdot \left(\sin \left(\pi \cdot \left(2 \cdot \left(0.005555555555555556 \cdot angle\right)\right)\right) \cdot 0.5\right)\right)\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \left(\left(b - a\right) \cdot \left(\left(b + a\right) \cdot \sin \left(\pi \cdot \left(0.005555555555555556 \cdot angle\right)\right)\right)\right)\\ \end{array} \]

Alternative 9: 63.4% accurate, 2.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;angle \leq -5.6 \cdot 10^{+102}:\\ \;\;\;\;2 \cdot \left(a \cdot \left(\left(a \cdot 0.5\right) \cdot \sin \left(angle \cdot \left(\pi \cdot 0.011111111111111112\right)\right)\right)\right)\\ \mathbf{elif}\;angle \leq 4.8 \cdot 10^{+78}:\\ \;\;\;\;0.011111111111111112 \cdot \left(\left(\pi \cdot \left(b + a\right)\right) \cdot \left(\left(b - a\right) \cdot angle\right)\right)\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \left(\sin \left(\pi \cdot \left(0.005555555555555556 \cdot angle\right)\right) \cdot \left(\left(b - a\right) \cdot \left(b + a\right)\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (a b angle)
 :precision binary64
 (if (<= angle -5.6e+102)
   (* 2.0 (* a (* (* a 0.5) (sin (* angle (* PI 0.011111111111111112))))))
   (if (<= angle 4.8e+78)
     (* 0.011111111111111112 (* (* PI (+ b a)) (* (- b a) angle)))
     (*
      2.0
      (* (sin (* PI (* 0.005555555555555556 angle))) (* (- b a) (+ b a)))))))

C

double code(double a, double b, double angle) {
	double tmp;
	if (angle <= -5.6e+102) {
		tmp = 2.0 * (a * ((a * 0.5) * sin((angle * (((double) M_PI) * 0.011111111111111112)))));
	} else if (angle <= 4.8e+78) {
		tmp = 0.011111111111111112 * ((((double) M_PI) * (b + a)) * ((b - a) * angle));
	} else {
		tmp = 2.0 * (sin((((double) M_PI) * (0.005555555555555556 * angle))) * ((b - a) * (b + a)));
	}
	return tmp;
}

Java

public static double code(double a, double b, double angle) {
	double tmp;
	if (angle <= -5.6e+102) {
		tmp = 2.0 * (a * ((a * 0.5) * Math.sin((angle * (Math.PI * 0.011111111111111112)))));
	} else if (angle <= 4.8e+78) {
		tmp = 0.011111111111111112 * ((Math.PI * (b + a)) * ((b - a) * angle));
	} else {
		tmp = 2.0 * (Math.sin((Math.PI * (0.005555555555555556 * angle))) * ((b - a) * (b + a)));
	}
	return tmp;
}

Python

def code(a, b, angle):
	tmp = 0
	if angle <= -5.6e+102:
		tmp = 2.0 * (a * ((a * 0.5) * math.sin((angle * (math.pi * 0.011111111111111112)))))
	elif angle <= 4.8e+78:
		tmp = 0.011111111111111112 * ((math.pi * (b + a)) * ((b - a) * angle))
	else:
		tmp = 2.0 * (math.sin((math.pi * (0.005555555555555556 * angle))) * ((b - a) * (b + a)))
	return tmp

Julia

function code(a, b, angle)
	tmp = 0.0
	if (angle <= -5.6e+102)
		tmp = Float64(2.0 * Float64(a * Float64(Float64(a * 0.5) * sin(Float64(angle * Float64(pi * 0.011111111111111112))))));
	elseif (angle <= 4.8e+78)
		tmp = Float64(0.011111111111111112 * Float64(Float64(pi * Float64(b + a)) * Float64(Float64(b - a) * angle)));
	else
		tmp = Float64(2.0 * Float64(sin(Float64(pi * Float64(0.005555555555555556 * angle))) * Float64(Float64(b - a) * Float64(b + a))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, angle)
	tmp = 0.0;
	if (angle <= -5.6e+102)
		tmp = 2.0 * (a * ((a * 0.5) * sin((angle * (pi * 0.011111111111111112)))));
	elseif (angle <= 4.8e+78)
		tmp = 0.011111111111111112 * ((pi * (b + a)) * ((b - a) * angle));
	else
		tmp = 2.0 * (sin((pi * (0.005555555555555556 * angle))) * ((b - a) * (b + a)));
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, angle_] := If[LessEqual[angle, -5.6e+102], N[(2.0 * N[(a * N[(N[(a * 0.5), $MachinePrecision] * N[Sin[N[(angle * N[(Pi * 0.011111111111111112), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[angle, 4.8e+78], N[(0.011111111111111112 * N[(N[(Pi * N[(b + a), $MachinePrecision]), $MachinePrecision] * N[(N[(b - a), $MachinePrecision] * angle), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(2.0 * N[(N[Sin[N[(Pi * N[(0.005555555555555556 * angle), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * N[(N[(b - a), $MachinePrecision] * N[(b + a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if angle < -5.60000000000000037e102

   1. Initial program 24.4%

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

      1. *-commutative 24.4%

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

      2. associate-*l* 24.4%

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

      3. unpow2 24.4%

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

      4. fma-neg 26.9%

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

      5. unpow2 26.9%

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

   3. Simplified 26.9%

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

   5. Applied egg-rr 38.4%

      \[\leadsto \color{blue}{\log \left({\left(\sqrt{e^{\sin 0 + \sin \left(\left(\pi \cdot \left(angle \cdot 0.005555555555555556\right)\right) \cdot 2\right)}}\right)}^{\left(2 \cdot {\left(b + a\right)}^{2}\right)}\right)} \]
   6. Step-by-step derivation

      1. log-pow 38.4%

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

      2. sin-0 38.4%

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

      3. +-lft-identity 38.4%

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

      4. associate-*l* 38.4%

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

      5. *-commutative 38.4%

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

      6. *-commutative 38.4%

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

   7. Simplified 38.4%

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

   8. Taylor expanded in b around 0 30.0%

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

      1. expm1-log1p-u 20.0%

         \[\leadsto 2 \cdot \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\log \left(\sqrt{e^{\sin \left(0.011111111111111112 \cdot \left(angle \cdot \pi\right)\right)}}\right) \cdot {a}^{2}\right)\right)} \]

      2. expm1-udef 22.3%

         \[\leadsto 2 \cdot \color{blue}{\left(e^{\mathsf{log1p}\left(\log \left(\sqrt{e^{\sin \left(0.011111111111111112 \cdot \left(angle \cdot \pi\right)\right)}}\right) \cdot {a}^{2}\right)} - 1\right)} \]

      3. pow2 22.3%

         \[\leadsto 2 \cdot \left(e^{\mathsf{log1p}\left(\log \left(\sqrt{e^{\sin \left(0.011111111111111112 \cdot \left(angle \cdot \pi\right)\right)}}\right) \cdot \color{blue}{\left(a \cdot a\right)}\right)} - 1\right) \]

      4. *-commutative 22.3%

         \[\leadsto 2 \cdot \left(e^{\mathsf{log1p}\left(\color{blue}{\left(a \cdot a\right) \cdot \log \left(\sqrt{e^{\sin \left(0.011111111111111112 \cdot \left(angle \cdot \pi\right)\right)}}\right)}\right)} - 1\right) \]

      5. pow1/2 22.3%

         \[\leadsto 2 \cdot \left(e^{\mathsf{log1p}\left(\left(a \cdot a\right) \cdot \log \color{blue}{\left({\left(e^{\sin \left(0.011111111111111112 \cdot \left(angle \cdot \pi\right)\right)}\right)}^{0.5}\right)}\right)} - 1\right) \]

      6. log-pow 22.3%

         \[\leadsto 2 \cdot \left(e^{\mathsf{log1p}\left(\left(a \cdot a\right) \cdot \color{blue}{\left(0.5 \cdot \log \left(e^{\sin \left(0.011111111111111112 \cdot \left(angle \cdot \pi\right)\right)}\right)\right)}\right)} - 1\right) \]

      7. add-log-exp 22.3%

         \[\leadsto 2 \cdot \left(e^{\mathsf{log1p}\left(\left(a \cdot a\right) \cdot \left(0.5 \cdot \color{blue}{\sin \left(0.011111111111111112 \cdot \left(angle \cdot \pi\right)\right)}\right)\right)} - 1\right) \]

   10. Applied egg-rr 22.3%

       \[\leadsto 2 \cdot \color{blue}{\left(e^{\mathsf{log1p}\left(\left(a \cdot a\right) \cdot \left(0.5 \cdot \sin \left(0.011111111111111112 \cdot \left(angle \cdot \pi\right)\right)\right)\right)} - 1\right)} \]
   11. Step-by-step derivation

       1. expm1-def 20.0%

          \[\leadsto 2 \cdot \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\left(a \cdot a\right) \cdot \left(0.5 \cdot \sin \left(0.011111111111111112 \cdot \left(angle \cdot \pi\right)\right)\right)\right)\right)} \]

       2. expm1-log1p 30.0%

          \[\leadsto 2 \cdot \color{blue}{\left(\left(a \cdot a\right) \cdot \left(0.5 \cdot \sin \left(0.011111111111111112 \cdot \left(angle \cdot \pi\right)\right)\right)\right)} \]

       3. associate-*l* 30.0%

          \[\leadsto 2 \cdot \color{blue}{\left(a \cdot \left(a \cdot \left(0.5 \cdot \sin \left(0.011111111111111112 \cdot \left(angle \cdot \pi\right)\right)\right)\right)\right)} \]

       4. associate-*r* 30.0%

          \[\leadsto 2 \cdot \left(a \cdot \color{blue}{\left(\left(a \cdot 0.5\right) \cdot \sin \left(0.011111111111111112 \cdot \left(angle \cdot \pi\right)\right)\right)}\right) \]

       5. *-commutative 30.0%

          \[\leadsto 2 \cdot \left(a \cdot \left(\left(a \cdot 0.5\right) \cdot \sin \color{blue}{\left(\left(angle \cdot \pi\right) \cdot 0.011111111111111112\right)}\right)\right) \]

       6. associate-*l* 32.2%

          \[\leadsto 2 \cdot \left(a \cdot \left(\left(a \cdot 0.5\right) \cdot \sin \color{blue}{\left(angle \cdot \left(\pi \cdot 0.011111111111111112\right)\right)}\right)\right) \]

   12. Simplified 32.2%

       \[\leadsto 2 \cdot \color{blue}{\left(a \cdot \left(\left(a \cdot 0.5\right) \cdot \sin \left(angle \cdot \left(\pi \cdot 0.011111111111111112\right)\right)\right)\right)} \]

   if -5.60000000000000037e102 < angle < 4.7999999999999997e78

   1. Initial program 65.0%

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

      1. associate-*l* 65.0%

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

      2. unpow2 65.0%

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

      3. unpow2 65.0%

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

      4. difference-of-squares 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 0 70.5%

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

      1. associate-*r* 87.2%

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

      2. *-commutative 87.2%

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

      3. +-commutative 87.2%

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

   6. Simplified 87.2%

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

   if 4.7999999999999997e78 < angle

   1. Initial program 33.7%

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

      1. associate-*l* 33.7%

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

      2. unpow2 33.7%

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

      3. unpow2 33.7%

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

      4. difference-of-squares 35.8%

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

   3. Simplified 35.8%

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

      1. add-cube-cbrt 34.6%

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

      2. pow3 38.5%

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

      3. div-inv 36.4%

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

      4. metadata-eval 36.4%

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

   5. Applied egg-rr 36.4%

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

   6. Taylor expanded in angle around inf 38.1%

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

      1. *-commutative 38.1%

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

      2. associate-*l* 38.1%

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

      3. associate-*r* 41.9%

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

      4. *-commutative 41.9%

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

      5. *-commutative 41.9%

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

      6. +-commutative 41.9%

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

   8. Simplified 41.9%

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

4. Final simplification 70.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;angle \leq -5.6 \cdot 10^{+102}:\\ \;\;\;\;2 \cdot \left(a \cdot \left(\left(a \cdot 0.5\right) \cdot \sin \left(angle \cdot \left(\pi \cdot 0.011111111111111112\right)\right)\right)\right)\\ \mathbf{elif}\;angle \leq 4.8 \cdot 10^{+78}:\\ \;\;\;\;0.011111111111111112 \cdot \left(\left(\pi \cdot \left(b + a\right)\right) \cdot \left(\left(b - a\right) \cdot angle\right)\right)\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \left(\sin \left(\pi \cdot \left(0.005555555555555556 \cdot angle\right)\right) \cdot \left(\left(b - a\right) \cdot \left(b + a\right)\right)\right)\\ \end{array} \]

Alternative 10: 64.9% accurate, 2.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \pi \cdot \left(0.005555555555555556 \cdot angle\right)\\ \mathbf{if}\;\frac{angle}{180} \leq -1 \cdot 10^{+16}:\\ \;\;\;\;t_0 \cdot \left(2 \cdot \mathsf{fma}\left(b, b, a \cdot \left(-a\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \left(\left(b - a\right) \cdot \left(\left(b + a\right) \cdot \sin t_0\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (a b angle)
 :precision binary64
 (let* ((t_0 (* PI (* 0.005555555555555556 angle))))
   (if (<= (/ angle 180.0) -1e+16)
     (* t_0 (* 2.0 (fma b b (* a (- a)))))
     (* 2.0 (* (- b a) (* (+ b a) (sin t_0)))))))

C

double code(double a, double b, double angle) {
	double t_0 = ((double) M_PI) * (0.005555555555555556 * angle);
	double tmp;
	if ((angle / 180.0) <= -1e+16) {
		tmp = t_0 * (2.0 * fma(b, b, (a * -a)));
	} else {
		tmp = 2.0 * ((b - a) * ((b + a) * sin(t_0)));
	}
	return tmp;
}

Julia

function code(a, b, angle)
	t_0 = Float64(pi * Float64(0.005555555555555556 * angle))
	tmp = 0.0
	if (Float64(angle / 180.0) <= -1e+16)
		tmp = Float64(t_0 * Float64(2.0 * fma(b, b, Float64(a * Float64(-a)))));
	else
		tmp = Float64(2.0 * Float64(Float64(b - a) * Float64(Float64(b + a) * sin(t_0))));
	end
	return tmp
end

Wolfram

code[a_, b_, angle_] := Block[{t$95$0 = N[(Pi * N[(0.005555555555555556 * angle), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[(angle / 180.0), $MachinePrecision], -1e+16], N[(t$95$0 * N[(2.0 * N[(b * b + N[(a * (-a)), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(2.0 * N[(N[(b - a), $MachinePrecision] * N[(N[(b + a), $MachinePrecision] * N[Sin[t$95$0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 angle 180) < -1e16

   1. Initial program 24.4%

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

      1. *-commutative 24.4%

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

      2. associate-*l* 24.4%

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

      3. unpow2 24.4%

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

      4. fma-neg 26.2%

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

      5. unpow2 26.2%

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

   3. Simplified 26.2%

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

   4. Taylor expanded in angle around 0 25.3%

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

      1. *-commutative 25.3%

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

      2. *-commutative 25.3%

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

      3. associate-*r* 25.3%

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

      4. *-commutative 25.3%

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

   6. Simplified 25.3%

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

   7. Taylor expanded in angle around 0 32.2%

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

   if -1e16 < (/.f64 angle 180)

   1. Initial program 60.8%

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

      1. associate-*l* 60.8%

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

      2. unpow2 60.8%

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

      3. unpow2 60.8%

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

      4. difference-of-squares 63.9%

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

   3. Simplified 63.9%

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

   4. Taylor expanded in angle around inf 78.1%

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

      1. associate-*r* 78.1%

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

      2. *-commutative 78.1%

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

      3. *-commutative 78.1%

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

      4. associate-*r* 79.7%

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

      5. *-commutative 79.7%

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

      6. *-commutative 79.7%

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

      7. *-commutative 79.7%

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

      8. associate-*r* 79.2%

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

      9. *-commutative 79.2%

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

      10. +-commutative 79.2%

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

   6. Simplified 79.2%

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

   7. Taylor expanded in angle around 0 79.2%

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

4. Final simplification 68.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{angle}{180} \leq -1 \cdot 10^{+16}:\\ \;\;\;\;\left(\pi \cdot \left(0.005555555555555556 \cdot angle\right)\right) \cdot \left(2 \cdot \mathsf{fma}\left(b, b, a \cdot \left(-a\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \left(\left(b - a\right) \cdot \left(\left(b + a\right) \cdot \sin \left(\pi \cdot \left(0.005555555555555556 \cdot angle\right)\right)\right)\right)\\ \end{array} \]

Alternative 11: 62.5% accurate, 2.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;angle \leq -5.6 \cdot 10^{+102}:\\ \;\;\;\;2 \cdot \left(a \cdot \left(\left(a \cdot 0.5\right) \cdot \sin \left(angle \cdot \left(\pi \cdot 0.011111111111111112\right)\right)\right)\right)\\ \mathbf{elif}\;angle \leq 6.8 \cdot 10^{+68}:\\ \;\;\;\;0.011111111111111112 \cdot \left(\left(\pi \cdot \left(b + a\right)\right) \cdot \left(\left(b - a\right) \cdot angle\right)\right)\\ \mathbf{else}:\\ \;\;\;\;0.011111111111111112 \cdot \left({\left(b + a\right)}^{2} \cdot \left(\pi \cdot angle\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (a b angle)
 :precision binary64
 (if (<= angle -5.6e+102)
   (* 2.0 (* a (* (* a 0.5) (sin (* angle (* PI 0.011111111111111112))))))
   (if (<= angle 6.8e+68)
     (* 0.011111111111111112 (* (* PI (+ b a)) (* (- b a) angle)))
     (* 0.011111111111111112 (* (pow (+ b a) 2.0) (* PI angle))))))

C

double code(double a, double b, double angle) {
	double tmp;
	if (angle <= -5.6e+102) {
		tmp = 2.0 * (a * ((a * 0.5) * sin((angle * (((double) M_PI) * 0.011111111111111112)))));
	} else if (angle <= 6.8e+68) {
		tmp = 0.011111111111111112 * ((((double) M_PI) * (b + a)) * ((b - a) * angle));
	} else {
		tmp = 0.011111111111111112 * (pow((b + a), 2.0) * (((double) M_PI) * angle));
	}
	return tmp;
}

Java

public static double code(double a, double b, double angle) {
	double tmp;
	if (angle <= -5.6e+102) {
		tmp = 2.0 * (a * ((a * 0.5) * Math.sin((angle * (Math.PI * 0.011111111111111112)))));
	} else if (angle <= 6.8e+68) {
		tmp = 0.011111111111111112 * ((Math.PI * (b + a)) * ((b - a) * angle));
	} else {
		tmp = 0.011111111111111112 * (Math.pow((b + a), 2.0) * (Math.PI * angle));
	}
	return tmp;
}

Python

def code(a, b, angle):
	tmp = 0
	if angle <= -5.6e+102:
		tmp = 2.0 * (a * ((a * 0.5) * math.sin((angle * (math.pi * 0.011111111111111112)))))
	elif angle <= 6.8e+68:
		tmp = 0.011111111111111112 * ((math.pi * (b + a)) * ((b - a) * angle))
	else:
		tmp = 0.011111111111111112 * (math.pow((b + a), 2.0) * (math.pi * angle))
	return tmp

Julia

function code(a, b, angle)
	tmp = 0.0
	if (angle <= -5.6e+102)
		tmp = Float64(2.0 * Float64(a * Float64(Float64(a * 0.5) * sin(Float64(angle * Float64(pi * 0.011111111111111112))))));
	elseif (angle <= 6.8e+68)
		tmp = Float64(0.011111111111111112 * Float64(Float64(pi * Float64(b + a)) * Float64(Float64(b - a) * angle)));
	else
		tmp = Float64(0.011111111111111112 * Float64((Float64(b + a) ^ 2.0) * Float64(pi * angle)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, angle)
	tmp = 0.0;
	if (angle <= -5.6e+102)
		tmp = 2.0 * (a * ((a * 0.5) * sin((angle * (pi * 0.011111111111111112)))));
	elseif (angle <= 6.8e+68)
		tmp = 0.011111111111111112 * ((pi * (b + a)) * ((b - a) * angle));
	else
		tmp = 0.011111111111111112 * (((b + a) ^ 2.0) * (pi * angle));
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, angle_] := If[LessEqual[angle, -5.6e+102], N[(2.0 * N[(a * N[(N[(a * 0.5), $MachinePrecision] * N[Sin[N[(angle * N[(Pi * 0.011111111111111112), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[angle, 6.8e+68], N[(0.011111111111111112 * N[(N[(Pi * N[(b + a), $MachinePrecision]), $MachinePrecision] * N[(N[(b - a), $MachinePrecision] * angle), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(0.011111111111111112 * N[(N[Power[N[(b + a), $MachinePrecision], 2.0], $MachinePrecision] * N[(Pi * angle), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if angle < -5.60000000000000037e102

   1. Initial program 24.4%

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

      1. *-commutative 24.4%

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

      2. associate-*l* 24.4%

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

      3. unpow2 24.4%

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

      4. fma-neg 26.9%

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

      5. unpow2 26.9%

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

   3. Simplified 26.9%

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

   5. Applied egg-rr 38.4%

      \[\leadsto \color{blue}{\log \left({\left(\sqrt{e^{\sin 0 + \sin \left(\left(\pi \cdot \left(angle \cdot 0.005555555555555556\right)\right) \cdot 2\right)}}\right)}^{\left(2 \cdot {\left(b + a\right)}^{2}\right)}\right)} \]
   6. Step-by-step derivation

      1. log-pow 38.4%

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

      2. sin-0 38.4%

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

      3. +-lft-identity 38.4%

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

      4. associate-*l* 38.4%

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

      5. *-commutative 38.4%

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

      6. *-commutative 38.4%

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

   7. Simplified 38.4%

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

   8. Taylor expanded in b around 0 30.0%

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

      1. expm1-log1p-u 20.0%

         \[\leadsto 2 \cdot \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\log \left(\sqrt{e^{\sin \left(0.011111111111111112 \cdot \left(angle \cdot \pi\right)\right)}}\right) \cdot {a}^{2}\right)\right)} \]

      2. expm1-udef 22.3%

         \[\leadsto 2 \cdot \color{blue}{\left(e^{\mathsf{log1p}\left(\log \left(\sqrt{e^{\sin \left(0.011111111111111112 \cdot \left(angle \cdot \pi\right)\right)}}\right) \cdot {a}^{2}\right)} - 1\right)} \]

      3. pow2 22.3%

         \[\leadsto 2 \cdot \left(e^{\mathsf{log1p}\left(\log \left(\sqrt{e^{\sin \left(0.011111111111111112 \cdot \left(angle \cdot \pi\right)\right)}}\right) \cdot \color{blue}{\left(a \cdot a\right)}\right)} - 1\right) \]

      4. *-commutative 22.3%

         \[\leadsto 2 \cdot \left(e^{\mathsf{log1p}\left(\color{blue}{\left(a \cdot a\right) \cdot \log \left(\sqrt{e^{\sin \left(0.011111111111111112 \cdot \left(angle \cdot \pi\right)\right)}}\right)}\right)} - 1\right) \]

      5. pow1/2 22.3%

         \[\leadsto 2 \cdot \left(e^{\mathsf{log1p}\left(\left(a \cdot a\right) \cdot \log \color{blue}{\left({\left(e^{\sin \left(0.011111111111111112 \cdot \left(angle \cdot \pi\right)\right)}\right)}^{0.5}\right)}\right)} - 1\right) \]

      6. log-pow 22.3%

         \[\leadsto 2 \cdot \left(e^{\mathsf{log1p}\left(\left(a \cdot a\right) \cdot \color{blue}{\left(0.5 \cdot \log \left(e^{\sin \left(0.011111111111111112 \cdot \left(angle \cdot \pi\right)\right)}\right)\right)}\right)} - 1\right) \]

      7. add-log-exp 22.3%

         \[\leadsto 2 \cdot \left(e^{\mathsf{log1p}\left(\left(a \cdot a\right) \cdot \left(0.5 \cdot \color{blue}{\sin \left(0.011111111111111112 \cdot \left(angle \cdot \pi\right)\right)}\right)\right)} - 1\right) \]

   10. Applied egg-rr 22.3%

       \[\leadsto 2 \cdot \color{blue}{\left(e^{\mathsf{log1p}\left(\left(a \cdot a\right) \cdot \left(0.5 \cdot \sin \left(0.011111111111111112 \cdot \left(angle \cdot \pi\right)\right)\right)\right)} - 1\right)} \]
   11. Step-by-step derivation

       1. expm1-def 20.0%

          \[\leadsto 2 \cdot \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\left(a \cdot a\right) \cdot \left(0.5 \cdot \sin \left(0.011111111111111112 \cdot \left(angle \cdot \pi\right)\right)\right)\right)\right)} \]

       2. expm1-log1p 30.0%

          \[\leadsto 2 \cdot \color{blue}{\left(\left(a \cdot a\right) \cdot \left(0.5 \cdot \sin \left(0.011111111111111112 \cdot \left(angle \cdot \pi\right)\right)\right)\right)} \]

       3. associate-*l* 30.0%

          \[\leadsto 2 \cdot \color{blue}{\left(a \cdot \left(a \cdot \left(0.5 \cdot \sin \left(0.011111111111111112 \cdot \left(angle \cdot \pi\right)\right)\right)\right)\right)} \]

       4. associate-*r* 30.0%

          \[\leadsto 2 \cdot \left(a \cdot \color{blue}{\left(\left(a \cdot 0.5\right) \cdot \sin \left(0.011111111111111112 \cdot \left(angle \cdot \pi\right)\right)\right)}\right) \]

       5. *-commutative 30.0%

          \[\leadsto 2 \cdot \left(a \cdot \left(\left(a \cdot 0.5\right) \cdot \sin \color{blue}{\left(\left(angle \cdot \pi\right) \cdot 0.011111111111111112\right)}\right)\right) \]

       6. associate-*l* 32.2%

          \[\leadsto 2 \cdot \left(a \cdot \left(\left(a \cdot 0.5\right) \cdot \sin \color{blue}{\left(angle \cdot \left(\pi \cdot 0.011111111111111112\right)\right)}\right)\right) \]

   12. Simplified 32.2%

       \[\leadsto 2 \cdot \color{blue}{\left(a \cdot \left(\left(a \cdot 0.5\right) \cdot \sin \left(angle \cdot \left(\pi \cdot 0.011111111111111112\right)\right)\right)\right)} \]

   if -5.60000000000000037e102 < angle < 6.8000000000000003e68

   1. Initial program 65.6%

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

      1. associate-*l* 65.6%

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

      2. unpow2 65.6%

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

      3. unpow2 65.6%

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

      4. difference-of-squares 68.7%

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

   3. Simplified 68.7%

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

   4. Taylor expanded in angle around 0 71.2%

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

      1. associate-*r* 88.2%

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

      2. *-commutative 88.2%

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

      3. +-commutative 88.2%

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

   6. Simplified 88.2%

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

   if 6.8000000000000003e68 < angle

   1. Initial program 33.7%

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

      1. *-commutative 33.7%

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

      2. associate-*l* 33.7%

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

      3. unpow2 33.7%

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

      4. fma-neg 35.7%

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

      5. unpow2 35.7%

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

   3. Simplified 35.7%

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

   5. Applied egg-rr 25.7%

      \[\leadsto \color{blue}{\log \left({\left(\sqrt{e^{\sin 0 + \sin \left(\left(\pi \cdot \left(angle \cdot 0.005555555555555556\right)\right) \cdot 2\right)}}\right)}^{\left(2 \cdot {\left(b + a\right)}^{2}\right)}\right)} \]
   6. Step-by-step derivation

      1. log-pow 29.7%

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

      2. sin-0 29.7%

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

      3. +-lft-identity 29.7%

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

      4. associate-*l* 29.7%

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

      5. *-commutative 29.7%

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

      6. *-commutative 29.7%

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

   7. Simplified 29.7%

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

   8. Taylor expanded in angle around 0 31.0%

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

      1. associate-*r* 31.0%

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

      2. +-commutative 31.0%

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

   10. Simplified 31.0%

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

4. Final simplification 68.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;angle \leq -5.6 \cdot 10^{+102}:\\ \;\;\;\;2 \cdot \left(a \cdot \left(\left(a \cdot 0.5\right) \cdot \sin \left(angle \cdot \left(\pi \cdot 0.011111111111111112\right)\right)\right)\right)\\ \mathbf{elif}\;angle \leq 6.8 \cdot 10^{+68}:\\ \;\;\;\;0.011111111111111112 \cdot \left(\left(\pi \cdot \left(b + a\right)\right) \cdot \left(\left(b - a\right) \cdot angle\right)\right)\\ \mathbf{else}:\\ \;\;\;\;0.011111111111111112 \cdot \left({\left(b + a\right)}^{2} \cdot \left(\pi \cdot angle\right)\right)\\ \end{array} \]

Alternative 12: 62.5% accurate, 2.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;angle \leq -3.1 \cdot 10^{+229}:\\ \;\;\;\;0.011111111111111112 \cdot \left(angle \cdot \left(b \cdot \left(b \cdot \pi\right)\right)\right)\\ \mathbf{elif}\;angle \leq 6.8 \cdot 10^{+68}:\\ \;\;\;\;0.011111111111111112 \cdot \left(\left(\pi \cdot \left(b + a\right)\right) \cdot \left(\left(b - a\right) \cdot angle\right)\right)\\ \mathbf{else}:\\ \;\;\;\;0.011111111111111112 \cdot \left({\left(b + a\right)}^{2} \cdot \left(\pi \cdot angle\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (a b angle)
 :precision binary64
 (if (<= angle -3.1e+229)
   (* 0.011111111111111112 (* angle (* b (* b PI))))
   (if (<= angle 6.8e+68)
     (* 0.011111111111111112 (* (* PI (+ b a)) (* (- b a) angle)))
     (* 0.011111111111111112 (* (pow (+ b a) 2.0) (* PI angle))))))

C

double code(double a, double b, double angle) {
	double tmp;
	if (angle <= -3.1e+229) {
		tmp = 0.011111111111111112 * (angle * (b * (b * ((double) M_PI))));
	} else if (angle <= 6.8e+68) {
		tmp = 0.011111111111111112 * ((((double) M_PI) * (b + a)) * ((b - a) * angle));
	} else {
		tmp = 0.011111111111111112 * (pow((b + a), 2.0) * (((double) M_PI) * angle));
	}
	return tmp;
}

Java

public static double code(double a, double b, double angle) {
	double tmp;
	if (angle <= -3.1e+229) {
		tmp = 0.011111111111111112 * (angle * (b * (b * Math.PI)));
	} else if (angle <= 6.8e+68) {
		tmp = 0.011111111111111112 * ((Math.PI * (b + a)) * ((b - a) * angle));
	} else {
		tmp = 0.011111111111111112 * (Math.pow((b + a), 2.0) * (Math.PI * angle));
	}
	return tmp;
}

Python

def code(a, b, angle):
	tmp = 0
	if angle <= -3.1e+229:
		tmp = 0.011111111111111112 * (angle * (b * (b * math.pi)))
	elif angle <= 6.8e+68:
		tmp = 0.011111111111111112 * ((math.pi * (b + a)) * ((b - a) * angle))
	else:
		tmp = 0.011111111111111112 * (math.pow((b + a), 2.0) * (math.pi * angle))
	return tmp

Julia

function code(a, b, angle)
	tmp = 0.0
	if (angle <= -3.1e+229)
		tmp = Float64(0.011111111111111112 * Float64(angle * Float64(b * Float64(b * pi))));
	elseif (angle <= 6.8e+68)
		tmp = Float64(0.011111111111111112 * Float64(Float64(pi * Float64(b + a)) * Float64(Float64(b - a) * angle)));
	else
		tmp = Float64(0.011111111111111112 * Float64((Float64(b + a) ^ 2.0) * Float64(pi * angle)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, angle)
	tmp = 0.0;
	if (angle <= -3.1e+229)
		tmp = 0.011111111111111112 * (angle * (b * (b * pi)));
	elseif (angle <= 6.8e+68)
		tmp = 0.011111111111111112 * ((pi * (b + a)) * ((b - a) * angle));
	else
		tmp = 0.011111111111111112 * (((b + a) ^ 2.0) * (pi * angle));
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, angle_] := If[LessEqual[angle, -3.1e+229], N[(0.011111111111111112 * N[(angle * N[(b * N[(b * Pi), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[angle, 6.8e+68], N[(0.011111111111111112 * N[(N[(Pi * N[(b + a), $MachinePrecision]), $MachinePrecision] * N[(N[(b - a), $MachinePrecision] * angle), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(0.011111111111111112 * N[(N[Power[N[(b + a), $MachinePrecision], 2.0], $MachinePrecision] * N[(Pi * angle), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if angle < -3.10000000000000014e229

   1. Initial program 24.8%

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

      1. associate-*l* 24.8%

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

      2. unpow2 24.8%

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

      3. unpow2 24.8%

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

      4. difference-of-squares 30.1%

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

   3. Simplified 30.1%

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

   4. Taylor expanded in angle around 0 17.8%

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

   5. Taylor expanded in b around inf 35.3%

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

      1. *-commutative 35.3%

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

      2. unpow2 35.3%

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

   7. Simplified 35.3%

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

   8. Taylor expanded in angle around 0 35.3%

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

      1. unpow2 35.3%

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

      2. associate-*l* 35.3%

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

   10. Simplified 35.3%

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

   if -3.10000000000000014e229 < angle < 6.8000000000000003e68

   1. Initial program 60.9%

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

      1. associate-*l* 60.9%

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

      2. unpow2 60.9%

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

      3. unpow2 60.9%

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

      4. difference-of-squares 63.7%

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

   3. Simplified 63.7%

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

   4. Taylor expanded in angle around 0 66.0%

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

      1. associate-*r* 81.1%

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

      2. *-commutative 81.1%

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

      3. +-commutative 81.1%

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

   6. Simplified 81.1%

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

   if 6.8000000000000003e68 < angle

   1. Initial program 33.7%

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

      1. *-commutative 33.7%

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

      2. associate-*l* 33.7%

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

      3. unpow2 33.7%

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

      4. fma-neg 35.7%

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

      5. unpow2 35.7%

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

   3. Simplified 35.7%

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

   5. Applied egg-rr 25.7%

      \[\leadsto \color{blue}{\log \left({\left(\sqrt{e^{\sin 0 + \sin \left(\left(\pi \cdot \left(angle \cdot 0.005555555555555556\right)\right) \cdot 2\right)}}\right)}^{\left(2 \cdot {\left(b + a\right)}^{2}\right)}\right)} \]
   6. Step-by-step derivation

      1. log-pow 29.7%

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

      2. sin-0 29.7%

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

      3. +-lft-identity 29.7%

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

      4. associate-*l* 29.7%

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

      5. *-commutative 29.7%

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

      6. *-commutative 29.7%

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

   7. Simplified 29.7%

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

   8. Taylor expanded in angle around 0 31.0%

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

      1. associate-*r* 31.0%

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

      2. +-commutative 31.0%

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

   10. Simplified 31.0%

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

4. Final simplification 67.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;angle \leq -3.1 \cdot 10^{+229}:\\ \;\;\;\;0.011111111111111112 \cdot \left(angle \cdot \left(b \cdot \left(b \cdot \pi\right)\right)\right)\\ \mathbf{elif}\;angle \leq 6.8 \cdot 10^{+68}:\\ \;\;\;\;0.011111111111111112 \cdot \left(\left(\pi \cdot \left(b + a\right)\right) \cdot \left(\left(b - a\right) \cdot angle\right)\right)\\ \mathbf{else}:\\ \;\;\;\;0.011111111111111112 \cdot \left({\left(b + a\right)}^{2} \cdot \left(\pi \cdot angle\right)\right)\\ \end{array} \]

Alternative 13: 41.4% accurate, 5.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;a \leq 0.7:\\ \;\;\;\;0.011111111111111112 \cdot \left(\pi \cdot \left(angle \cdot \left(b \cdot b\right)\right)\right)\\ \mathbf{elif}\;a \leq 1.58 \cdot 10^{+184}:\\ \;\;\;\;0.011111111111111112 \cdot \left(angle \cdot \left(\left(b - a\right) \cdot \left(a \cdot \pi\right)\right)\right)\\ \mathbf{elif}\;a \leq 5 \cdot 10^{+217}:\\ \;\;\;\;2 \cdot \left(\pi \cdot \left(0.005555555555555556 \cdot \left(angle \cdot \left(a \cdot a\right)\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(angle \cdot -0.011111111111111112\right) \cdot \left(\pi \cdot \left(a \cdot a\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (a b angle)
 :precision binary64
 (if (<= a 0.7)
   (* 0.011111111111111112 (* PI (* angle (* b b))))
   (if (<= a 1.58e+184)
     (* 0.011111111111111112 (* angle (* (- b a) (* a PI))))
     (if (<= a 5e+217)
       (* 2.0 (* PI (* 0.005555555555555556 (* angle (* a a)))))
       (* (* angle -0.011111111111111112) (* PI (* a a)))))))

C

double code(double a, double b, double angle) {
	double tmp;
	if (a <= 0.7) {
		tmp = 0.011111111111111112 * (((double) M_PI) * (angle * (b * b)));
	} else if (a <= 1.58e+184) {
		tmp = 0.011111111111111112 * (angle * ((b - a) * (a * ((double) M_PI))));
	} else if (a <= 5e+217) {
		tmp = 2.0 * (((double) M_PI) * (0.005555555555555556 * (angle * (a * a))));
	} else {
		tmp = (angle * -0.011111111111111112) * (((double) M_PI) * (a * a));
	}
	return tmp;
}

Java

public static double code(double a, double b, double angle) {
	double tmp;
	if (a <= 0.7) {
		tmp = 0.011111111111111112 * (Math.PI * (angle * (b * b)));
	} else if (a <= 1.58e+184) {
		tmp = 0.011111111111111112 * (angle * ((b - a) * (a * Math.PI)));
	} else if (a <= 5e+217) {
		tmp = 2.0 * (Math.PI * (0.005555555555555556 * (angle * (a * a))));
	} else {
		tmp = (angle * -0.011111111111111112) * (Math.PI * (a * a));
	}
	return tmp;
}

Python

def code(a, b, angle):
	tmp = 0
	if a <= 0.7:
		tmp = 0.011111111111111112 * (math.pi * (angle * (b * b)))
	elif a <= 1.58e+184:
		tmp = 0.011111111111111112 * (angle * ((b - a) * (a * math.pi)))
	elif a <= 5e+217:
		tmp = 2.0 * (math.pi * (0.005555555555555556 * (angle * (a * a))))
	else:
		tmp = (angle * -0.011111111111111112) * (math.pi * (a * a))
	return tmp

Julia

function code(a, b, angle)
	tmp = 0.0
	if (a <= 0.7)
		tmp = Float64(0.011111111111111112 * Float64(pi * Float64(angle * Float64(b * b))));
	elseif (a <= 1.58e+184)
		tmp = Float64(0.011111111111111112 * Float64(angle * Float64(Float64(b - a) * Float64(a * pi))));
	elseif (a <= 5e+217)
		tmp = Float64(2.0 * Float64(pi * Float64(0.005555555555555556 * Float64(angle * Float64(a * a)))));
	else
		tmp = Float64(Float64(angle * -0.011111111111111112) * Float64(pi * Float64(a * a)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, angle)
	tmp = 0.0;
	if (a <= 0.7)
		tmp = 0.011111111111111112 * (pi * (angle * (b * b)));
	elseif (a <= 1.58e+184)
		tmp = 0.011111111111111112 * (angle * ((b - a) * (a * pi)));
	elseif (a <= 5e+217)
		tmp = 2.0 * (pi * (0.005555555555555556 * (angle * (a * a))));
	else
		tmp = (angle * -0.011111111111111112) * (pi * (a * a));
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, angle_] := If[LessEqual[a, 0.7], N[(0.011111111111111112 * N[(Pi * N[(angle * N[(b * b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[a, 1.58e+184], N[(0.011111111111111112 * N[(angle * N[(N[(b - a), $MachinePrecision] * N[(a * Pi), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[a, 5e+217], N[(2.0 * N[(Pi * N[(0.005555555555555556 * N[(angle * N[(a * a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(angle * -0.011111111111111112), $MachinePrecision] * N[(Pi * N[(a * a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 4 regimes

2. if a < 0.69999999999999996

   1. Initial program 55.8%

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

      1. associate-*l* 55.8%

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

      2. unpow2 55.8%

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

      3. unpow2 55.8%

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

      4. difference-of-squares 58.0%

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

   3. Simplified 58.0%

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

   4. Taylor expanded in angle around 0 56.3%

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

   5. Taylor expanded in b around inf 41.1%

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

      1. *-commutative 41.1%

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

      2. unpow2 41.1%

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

   7. Simplified 41.1%

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

   8. Taylor expanded in angle around 0 41.1%

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

      1. unpow2 41.1%

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

      2. associate-*r* 41.2%

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

   10. Simplified 41.2%

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

   if 0.69999999999999996 < a < 1.58e184

   1. Initial program 54.7%

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

      1. associate-*l* 54.7%

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

      2. unpow2 54.7%

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

      3. unpow2 54.7%

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

      4. difference-of-squares 54.8%

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

   3. Simplified 54.8%

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

   4. Taylor expanded in angle around 0 48.9%

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

   5. Taylor expanded in a around inf 48.4%

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

      1. *-commutative 48.4%

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

   7. Simplified 48.4%

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

   if 1.58e184 < a < 5.00000000000000041e217

   1. Initial program 20.6%

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

      1. *-commutative 20.6%

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

      2. associate-*l* 20.6%

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

      3. unpow2 20.6%

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

      4. fma-neg 21.0%

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

      5. unpow2 21.0%

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

   3. Simplified 21.0%

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

      1. fma-neg 20.6%

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

      2. difference-of-squares 21.0%

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

      3. sub-neg 21.0%

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

      4. distribute-rgt-in 21.0%

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

      5. add-sqr-sqrt 0.0%

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

      6. sqrt-unprod 50.2%

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

      7. sqr-neg 50.2%

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

      8. sqrt-prod 50.2%

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

      9. add-sqr-sqrt 50.2%

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

      10. distribute-rgt-in 50.2%

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

      11. add-sqr-sqrt 50.2%

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

      12. associate-*r* 50.2%

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

   5. Applied egg-rr 50.2%

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

   6. Taylor expanded in b around 0 50.2%

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

      1. associate-*r* 50.2%

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

      2. unpow2 50.2%

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

      3. associate-*r* 40.2%

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

      4. *-commutative 40.2%

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

      5. associate-*r* 40.2%

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

      6. *-commutative 40.2%

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

   8. Simplified 40.2%

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

   9. Taylor expanded in angle around 0 60.2%

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

       1. associate-*r* 60.2%

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

       2. associate-*r* 60.2%

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

       3. unpow2 60.2%

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

   11. Simplified 60.2%

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

   if 5.00000000000000041e217 < a

   1. Initial program 35.4%

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

      1. associate-*l* 35.4%

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

      2. unpow2 35.4%

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

      3. unpow2 35.4%

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

      4. difference-of-squares 52.1%

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

   3. Simplified 52.1%

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

   4. Taylor expanded in angle around 0 57.6%

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

   5. Taylor expanded in b around 0 57.6%

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

      1. associate-*r* 57.6%

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

      2. *-commutative 57.6%

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

      3. unpow2 57.6%

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

   7. Simplified 57.6%

      \[\leadsto \color{blue}{\left(-0.011111111111111112 \cdot angle\right) \cdot \left(\pi \cdot \left(a \cdot a\right)\right)} \]
3. Recombined 4 regimes into one program.

4. Final simplification 44.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \leq 0.7:\\ \;\;\;\;0.011111111111111112 \cdot \left(\pi \cdot \left(angle \cdot \left(b \cdot b\right)\right)\right)\\ \mathbf{elif}\;a \leq 1.58 \cdot 10^{+184}:\\ \;\;\;\;0.011111111111111112 \cdot \left(angle \cdot \left(\left(b - a\right) \cdot \left(a \cdot \pi\right)\right)\right)\\ \mathbf{elif}\;a \leq 5 \cdot 10^{+217}:\\ \;\;\;\;2 \cdot \left(\pi \cdot \left(0.005555555555555556 \cdot \left(angle \cdot \left(a \cdot a\right)\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(angle \cdot -0.011111111111111112\right) \cdot \left(\pi \cdot \left(a \cdot a\right)\right)\\ \end{array} \]

Alternative 14: 61.5% accurate, 5.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;angle \leq -4.2 \cdot 10^{+229}:\\ \;\;\;\;0.011111111111111112 \cdot \left(angle \cdot \left(b \cdot \left(b \cdot \pi\right)\right)\right)\\ \mathbf{elif}\;angle \leq 2.4 \cdot 10^{+87}:\\ \;\;\;\;0.011111111111111112 \cdot \left(\left(\pi \cdot \left(b + a\right)\right) \cdot \left(\left(b - a\right) \cdot angle\right)\right)\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \left(\pi \cdot \left(0.005555555555555556 \cdot \left(angle \cdot \left(a \cdot a\right)\right)\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (a b angle)
 :precision binary64
 (if (<= angle -4.2e+229)
   (* 0.011111111111111112 (* angle (* b (* b PI))))
   (if (<= angle 2.4e+87)
     (* 0.011111111111111112 (* (* PI (+ b a)) (* (- b a) angle)))
     (* 2.0 (* PI (* 0.005555555555555556 (* angle (* a a))))))))

C

double code(double a, double b, double angle) {
	double tmp;
	if (angle <= -4.2e+229) {
		tmp = 0.011111111111111112 * (angle * (b * (b * ((double) M_PI))));
	} else if (angle <= 2.4e+87) {
		tmp = 0.011111111111111112 * ((((double) M_PI) * (b + a)) * ((b - a) * angle));
	} else {
		tmp = 2.0 * (((double) M_PI) * (0.005555555555555556 * (angle * (a * a))));
	}
	return tmp;
}

Java

public static double code(double a, double b, double angle) {
	double tmp;
	if (angle <= -4.2e+229) {
		tmp = 0.011111111111111112 * (angle * (b * (b * Math.PI)));
	} else if (angle <= 2.4e+87) {
		tmp = 0.011111111111111112 * ((Math.PI * (b + a)) * ((b - a) * angle));
	} else {
		tmp = 2.0 * (Math.PI * (0.005555555555555556 * (angle * (a * a))));
	}
	return tmp;
}

Python

def code(a, b, angle):
	tmp = 0
	if angle <= -4.2e+229:
		tmp = 0.011111111111111112 * (angle * (b * (b * math.pi)))
	elif angle <= 2.4e+87:
		tmp = 0.011111111111111112 * ((math.pi * (b + a)) * ((b - a) * angle))
	else:
		tmp = 2.0 * (math.pi * (0.005555555555555556 * (angle * (a * a))))
	return tmp

Julia

function code(a, b, angle)
	tmp = 0.0
	if (angle <= -4.2e+229)
		tmp = Float64(0.011111111111111112 * Float64(angle * Float64(b * Float64(b * pi))));
	elseif (angle <= 2.4e+87)
		tmp = Float64(0.011111111111111112 * Float64(Float64(pi * Float64(b + a)) * Float64(Float64(b - a) * angle)));
	else
		tmp = Float64(2.0 * Float64(pi * Float64(0.005555555555555556 * Float64(angle * Float64(a * a)))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, angle)
	tmp = 0.0;
	if (angle <= -4.2e+229)
		tmp = 0.011111111111111112 * (angle * (b * (b * pi)));
	elseif (angle <= 2.4e+87)
		tmp = 0.011111111111111112 * ((pi * (b + a)) * ((b - a) * angle));
	else
		tmp = 2.0 * (pi * (0.005555555555555556 * (angle * (a * a))));
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, angle_] := If[LessEqual[angle, -4.2e+229], N[(0.011111111111111112 * N[(angle * N[(b * N[(b * Pi), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[angle, 2.4e+87], N[(0.011111111111111112 * N[(N[(Pi * N[(b + a), $MachinePrecision]), $MachinePrecision] * N[(N[(b - a), $MachinePrecision] * angle), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(2.0 * N[(Pi * N[(0.005555555555555556 * N[(angle * N[(a * a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if angle < -4.19999999999999975e229

   1. Initial program 24.8%

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

      1. associate-*l* 24.8%

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

      2. unpow2 24.8%

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

      3. unpow2 24.8%

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

      4. difference-of-squares 30.1%

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

   3. Simplified 30.1%

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

   4. Taylor expanded in angle around 0 17.8%

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

   5. Taylor expanded in b around inf 35.3%

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

      1. *-commutative 35.3%

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

      2. unpow2 35.3%

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

   7. Simplified 35.3%

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

   8. Taylor expanded in angle around 0 35.3%

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

      1. unpow2 35.3%

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

      2. associate-*l* 35.3%

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

   10. Simplified 35.3%

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

   if -4.19999999999999975e229 < angle < 2.39999999999999981e87

   1. Initial program 60.4%

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

      1. associate-*l* 60.4%

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

      2. unpow2 60.4%

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

      3. unpow2 60.4%

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

      4. difference-of-squares 63.1%

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

   3. Simplified 63.1%

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

   4. Taylor expanded in angle around 0 65.3%

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

      1. associate-*r* 80.0%

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

      2. *-commutative 80.0%

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

      3. +-commutative 80.0%

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

   6. Simplified 80.0%

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

   if 2.39999999999999981e87 < angle

   1. Initial program 33.0%

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

      1. *-commutative 33.0%

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

      2. associate-*l* 33.0%

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

      3. unpow2 33.0%

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

      4. fma-neg 35.2%

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

      5. unpow2 35.2%

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

   3. Simplified 35.2%

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

      1. fma-neg 33.0%

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

      2. difference-of-squares 35.2%

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

      3. sub-neg 35.2%

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

      4. distribute-rgt-in 28.7%

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

      5. add-sqr-sqrt 8.8%

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

      6. sqrt-unprod 27.2%

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

      7. sqr-neg 27.2%

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

      8. sqrt-prod 18.3%

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

      9. add-sqr-sqrt 26.0%

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

      10. distribute-rgt-in 26.0%

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

      11. add-sqr-sqrt 16.2%

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

      12. associate-*r* 16.2%

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

   5. Applied egg-rr 16.2%

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

   6. Taylor expanded in b around 0 18.9%

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

      1. associate-*r* 18.9%

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

      2. unpow2 18.9%

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

      3. associate-*r* 13.4%

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

      4. *-commutative 13.4%

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

      5. associate-*r* 17.1%

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

      6. *-commutative 17.1%

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

   8. Simplified 17.1%

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

   9. Taylor expanded in angle around 0 27.7%

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

       1. associate-*r* 27.7%

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

       2. associate-*r* 27.7%

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

       3. unpow2 27.7%

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

   11. Simplified 27.7%

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

4. Final simplification 67.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;angle \leq -4.2 \cdot 10^{+229}:\\ \;\;\;\;0.011111111111111112 \cdot \left(angle \cdot \left(b \cdot \left(b \cdot \pi\right)\right)\right)\\ \mathbf{elif}\;angle \leq 2.4 \cdot 10^{+87}:\\ \;\;\;\;0.011111111111111112 \cdot \left(\left(\pi \cdot \left(b + a\right)\right) \cdot \left(\left(b - a\right) \cdot angle\right)\right)\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \left(\pi \cdot \left(0.005555555555555556 \cdot \left(angle \cdot \left(a \cdot a\right)\right)\right)\right)\\ \end{array} \]

Alternative 15: 52.8% accurate, 5.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;angle \leq 2.4 \cdot 10^{+87}:\\ \;\;\;\;0.011111111111111112 \cdot \left(angle \cdot \left(\left(b - a\right) \cdot \left(\pi \cdot \left(b + a\right)\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \left(\pi \cdot \left(0.005555555555555556 \cdot \left(angle \cdot \left(a \cdot a\right)\right)\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (a b angle)
 :precision binary64
 (if (<= angle 2.4e+87)
   (* 0.011111111111111112 (* angle (* (- b a) (* PI (+ b a)))))
   (* 2.0 (* PI (* 0.005555555555555556 (* angle (* a a)))))))

C

double code(double a, double b, double angle) {
	double tmp;
	if (angle <= 2.4e+87) {
		tmp = 0.011111111111111112 * (angle * ((b - a) * (((double) M_PI) * (b + a))));
	} else {
		tmp = 2.0 * (((double) M_PI) * (0.005555555555555556 * (angle * (a * a))));
	}
	return tmp;
}

Java

public static double code(double a, double b, double angle) {
	double tmp;
	if (angle <= 2.4e+87) {
		tmp = 0.011111111111111112 * (angle * ((b - a) * (Math.PI * (b + a))));
	} else {
		tmp = 2.0 * (Math.PI * (0.005555555555555556 * (angle * (a * a))));
	}
	return tmp;
}

Python

def code(a, b, angle):
	tmp = 0
	if angle <= 2.4e+87:
		tmp = 0.011111111111111112 * (angle * ((b - a) * (math.pi * (b + a))))
	else:
		tmp = 2.0 * (math.pi * (0.005555555555555556 * (angle * (a * a))))
	return tmp

Julia

function code(a, b, angle)
	tmp = 0.0
	if (angle <= 2.4e+87)
		tmp = Float64(0.011111111111111112 * Float64(angle * Float64(Float64(b - a) * Float64(pi * Float64(b + a)))));
	else
		tmp = Float64(2.0 * Float64(pi * Float64(0.005555555555555556 * Float64(angle * Float64(a * a)))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, angle)
	tmp = 0.0;
	if (angle <= 2.4e+87)
		tmp = 0.011111111111111112 * (angle * ((b - a) * (pi * (b + a))));
	else
		tmp = 2.0 * (pi * (0.005555555555555556 * (angle * (a * a))));
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, angle_] := If[LessEqual[angle, 2.4e+87], N[(0.011111111111111112 * N[(angle * N[(N[(b - a), $MachinePrecision] * N[(Pi * N[(b + a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(2.0 * N[(Pi * N[(0.005555555555555556 * N[(angle * N[(a * a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if angle < 2.39999999999999981e87

   1. Initial program 57.2%

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

      1. associate-*l* 57.2%

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

      2. unpow2 57.2%

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

      3. unpow2 57.2%

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

      4. difference-of-squares 60.1%

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

   3. Simplified 60.1%

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

   4. Taylor expanded in angle around 0 61.0%

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

   if 2.39999999999999981e87 < angle

   1. Initial program 33.0%

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

      1. *-commutative 33.0%

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

      2. associate-*l* 33.0%

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

      3. unpow2 33.0%

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

      4. fma-neg 35.2%

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

      5. unpow2 35.2%

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

   3. Simplified 35.2%

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

      1. fma-neg 33.0%

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

      2. difference-of-squares 35.2%

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

      3. sub-neg 35.2%

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

      4. distribute-rgt-in 28.7%

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

      5. add-sqr-sqrt 8.8%

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

      6. sqrt-unprod 27.2%

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

      7. sqr-neg 27.2%

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

      8. sqrt-prod 18.3%

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

      9. add-sqr-sqrt 26.0%

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

      10. distribute-rgt-in 26.0%

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

      11. add-sqr-sqrt 16.2%

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

      12. associate-*r* 16.2%

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

   5. Applied egg-rr 16.2%

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

   6. Taylor expanded in b around 0 18.9%

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

      1. associate-*r* 18.9%

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

      2. unpow2 18.9%

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

      3. associate-*r* 13.4%

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

      4. *-commutative 13.4%

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

      5. associate-*r* 17.1%

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

      6. *-commutative 17.1%

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

   8. Simplified 17.1%

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

   9. Taylor expanded in angle around 0 27.7%

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

       1. associate-*r* 27.7%

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

       2. associate-*r* 27.7%

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

       3. unpow2 27.7%

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

   11. Simplified 27.7%

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

4. Final simplification 55.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;angle \leq 2.4 \cdot 10^{+87}:\\ \;\;\;\;0.011111111111111112 \cdot \left(angle \cdot \left(\left(b - a\right) \cdot \left(\pi \cdot \left(b + a\right)\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \left(\pi \cdot \left(0.005555555555555556 \cdot \left(angle \cdot \left(a \cdot a\right)\right)\right)\right)\\ \end{array} \]

Alternative 16: 41.9% accurate, 5.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;a \leq 0.028:\\ \;\;\;\;0.011111111111111112 \cdot \left(\pi \cdot \left(angle \cdot \left(b \cdot b\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;0.011111111111111112 \cdot \left(angle \cdot \left(\left(b - a\right) \cdot \left(a \cdot \pi\right)\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (a b angle)
 :precision binary64
 (if (<= a 0.028)
   (* 0.011111111111111112 (* PI (* angle (* b b))))
   (* 0.011111111111111112 (* angle (* (- b a) (* a PI))))))

C

double code(double a, double b, double angle) {
	double tmp;
	if (a <= 0.028) {
		tmp = 0.011111111111111112 * (((double) M_PI) * (angle * (b * b)));
	} else {
		tmp = 0.011111111111111112 * (angle * ((b - a) * (a * ((double) M_PI))));
	}
	return tmp;
}

Java

public static double code(double a, double b, double angle) {
	double tmp;
	if (a <= 0.028) {
		tmp = 0.011111111111111112 * (Math.PI * (angle * (b * b)));
	} else {
		tmp = 0.011111111111111112 * (angle * ((b - a) * (a * Math.PI)));
	}
	return tmp;
}

Python

def code(a, b, angle):
	tmp = 0
	if a <= 0.028:
		tmp = 0.011111111111111112 * (math.pi * (angle * (b * b)))
	else:
		tmp = 0.011111111111111112 * (angle * ((b - a) * (a * math.pi)))
	return tmp

Julia

function code(a, b, angle)
	tmp = 0.0
	if (a <= 0.028)
		tmp = Float64(0.011111111111111112 * Float64(pi * Float64(angle * Float64(b * b))));
	else
		tmp = Float64(0.011111111111111112 * Float64(angle * Float64(Float64(b - a) * Float64(a * pi))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, angle)
	tmp = 0.0;
	if (a <= 0.028)
		tmp = 0.011111111111111112 * (pi * (angle * (b * b)));
	else
		tmp = 0.011111111111111112 * (angle * ((b - a) * (a * pi)));
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, angle_] := If[LessEqual[a, 0.028], N[(0.011111111111111112 * N[(Pi * N[(angle * N[(b * b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(0.011111111111111112 * N[(angle * N[(N[(b - a), $MachinePrecision] * N[(a * Pi), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if a < 0.0280000000000000006

   1. Initial program 55.8%

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

      1. associate-*l* 55.8%

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

      2. unpow2 55.8%

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

      3. unpow2 55.8%

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

      4. difference-of-squares 58.0%

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

   3. Simplified 58.0%

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

   4. Taylor expanded in angle around 0 56.3%

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

   5. Taylor expanded in b around inf 41.1%

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

      1. *-commutative 41.1%

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

      2. unpow2 41.1%

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

   7. Simplified 41.1%

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

   8. Taylor expanded in angle around 0 41.1%

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

      1. unpow2 41.1%

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

      2. associate-*r* 41.2%

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

   10. Simplified 41.2%

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

   if 0.0280000000000000006 < a

   1. Initial program 44.9%

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

      1. associate-*l* 44.9%

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

      2. unpow2 44.9%

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

      3. unpow2 44.9%

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

      4. difference-of-squares 49.3%

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

   3. Simplified 49.3%

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

   4. Taylor expanded in angle around 0 45.7%

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

   5. Taylor expanded in a around inf 45.4%

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

      1. *-commutative 45.4%

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

   7. Simplified 45.4%

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

4. Final simplification 42.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \leq 0.028:\\ \;\;\;\;0.011111111111111112 \cdot \left(\pi \cdot \left(angle \cdot \left(b \cdot b\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;0.011111111111111112 \cdot \left(angle \cdot \left(\left(b - a\right) \cdot \left(a \cdot \pi\right)\right)\right)\\ \end{array} \]

Alternative 17: 41.0% accurate, 5.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;a \leq 130000000000:\\ \;\;\;\;0.011111111111111112 \cdot \left(\pi \cdot \left(angle \cdot \left(b \cdot b\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(angle \cdot -0.011111111111111112\right) \cdot \left(\pi \cdot \left(a \cdot a\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (a b angle)
 :precision binary64
 (if (<= a 130000000000.0)
   (* 0.011111111111111112 (* PI (* angle (* b b))))
   (* (* angle -0.011111111111111112) (* PI (* a a)))))

C

double code(double a, double b, double angle) {
	double tmp;
	if (a <= 130000000000.0) {
		tmp = 0.011111111111111112 * (((double) M_PI) * (angle * (b * b)));
	} else {
		tmp = (angle * -0.011111111111111112) * (((double) M_PI) * (a * a));
	}
	return tmp;
}

Java

public static double code(double a, double b, double angle) {
	double tmp;
	if (a <= 130000000000.0) {
		tmp = 0.011111111111111112 * (Math.PI * (angle * (b * b)));
	} else {
		tmp = (angle * -0.011111111111111112) * (Math.PI * (a * a));
	}
	return tmp;
}

Python

def code(a, b, angle):
	tmp = 0
	if a <= 130000000000.0:
		tmp = 0.011111111111111112 * (math.pi * (angle * (b * b)))
	else:
		tmp = (angle * -0.011111111111111112) * (math.pi * (a * a))
	return tmp

Julia

function code(a, b, angle)
	tmp = 0.0
	if (a <= 130000000000.0)
		tmp = Float64(0.011111111111111112 * Float64(pi * Float64(angle * Float64(b * b))));
	else
		tmp = Float64(Float64(angle * -0.011111111111111112) * Float64(pi * Float64(a * a)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, angle)
	tmp = 0.0;
	if (a <= 130000000000.0)
		tmp = 0.011111111111111112 * (pi * (angle * (b * b)));
	else
		tmp = (angle * -0.011111111111111112) * (pi * (a * a));
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, angle_] := If[LessEqual[a, 130000000000.0], N[(0.011111111111111112 * N[(Pi * N[(angle * N[(b * b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(angle * -0.011111111111111112), $MachinePrecision] * N[(Pi * N[(a * a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if a < 1.3e11

   1. Initial program 55.8%

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

      1. associate-*l* 55.8%

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

      2. unpow2 55.8%

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

      3. unpow2 55.8%

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

      4. difference-of-squares 57.9%

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

   3. Simplified 57.9%

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

   4. Taylor expanded in angle around 0 56.3%

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

   5. Taylor expanded in b around inf 40.9%

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

      1. *-commutative 40.9%

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

      2. unpow2 40.9%

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

   7. Simplified 40.9%

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

   8. Taylor expanded in angle around 0 40.9%

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

      1. unpow2 40.9%

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

      2. associate-*r* 40.9%

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

   10. Simplified 40.9%

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

   if 1.3e11 < a

   1. Initial program 44.3%

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

      1. associate-*l* 44.3%

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

      2. unpow2 44.3%

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

      3. unpow2 44.3%

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

      4. difference-of-squares 49.0%

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

   3. Simplified 49.0%

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

   4. Taylor expanded in angle around 0 45.3%

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

   5. Taylor expanded in b around 0 45.3%

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

      1. associate-*r* 45.3%

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

      2. *-commutative 45.3%

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

      3. unpow2 45.3%

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

   7. Simplified 45.3%

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

4. Final simplification 42.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \leq 130000000000:\\ \;\;\;\;0.011111111111111112 \cdot \left(\pi \cdot \left(angle \cdot \left(b \cdot b\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(angle \cdot -0.011111111111111112\right) \cdot \left(\pi \cdot \left(a \cdot a\right)\right)\\ \end{array} \]

Alternative 18: 41.0% accurate, 5.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;a \leq 136000000000:\\ \;\;\;\;0.011111111111111112 \cdot \left(\pi \cdot \left(angle \cdot \left(b \cdot b\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;-0.011111111111111112 \cdot \left(\pi \cdot \left(angle \cdot \left(a \cdot a\right)\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (a b angle)
 :precision binary64
 (if (<= a 136000000000.0)
   (* 0.011111111111111112 (* PI (* angle (* b b))))
   (* -0.011111111111111112 (* PI (* angle (* a a))))))

C

double code(double a, double b, double angle) {
	double tmp;
	if (a <= 136000000000.0) {
		tmp = 0.011111111111111112 * (((double) M_PI) * (angle * (b * b)));
	} else {
		tmp = -0.011111111111111112 * (((double) M_PI) * (angle * (a * a)));
	}
	return tmp;
}

Java

public static double code(double a, double b, double angle) {
	double tmp;
	if (a <= 136000000000.0) {
		tmp = 0.011111111111111112 * (Math.PI * (angle * (b * b)));
	} else {
		tmp = -0.011111111111111112 * (Math.PI * (angle * (a * a)));
	}
	return tmp;
}

Python

def code(a, b, angle):
	tmp = 0
	if a <= 136000000000.0:
		tmp = 0.011111111111111112 * (math.pi * (angle * (b * b)))
	else:
		tmp = -0.011111111111111112 * (math.pi * (angle * (a * a)))
	return tmp

Julia

function code(a, b, angle)
	tmp = 0.0
	if (a <= 136000000000.0)
		tmp = Float64(0.011111111111111112 * Float64(pi * Float64(angle * Float64(b * b))));
	else
		tmp = Float64(-0.011111111111111112 * Float64(pi * Float64(angle * Float64(a * a))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, angle)
	tmp = 0.0;
	if (a <= 136000000000.0)
		tmp = 0.011111111111111112 * (pi * (angle * (b * b)));
	else
		tmp = -0.011111111111111112 * (pi * (angle * (a * a)));
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, angle_] := If[LessEqual[a, 136000000000.0], N[(0.011111111111111112 * N[(Pi * N[(angle * N[(b * b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(-0.011111111111111112 * N[(Pi * N[(angle * N[(a * a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if a < 1.36e11

   1. Initial program 55.8%

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

      1. associate-*l* 55.8%

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

      2. unpow2 55.8%

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

      3. unpow2 55.8%

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

      4. difference-of-squares 57.9%

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

   3. Simplified 57.9%

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

   4. Taylor expanded in angle around 0 56.3%

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

   5. Taylor expanded in b around inf 40.9%

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

      1. *-commutative 40.9%

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

      2. unpow2 40.9%

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

   7. Simplified 40.9%

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

   8. Taylor expanded in angle around 0 40.9%

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

      1. unpow2 40.9%

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

      2. associate-*r* 40.9%

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

   10. Simplified 40.9%

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

   if 1.36e11 < a

   1. Initial program 44.3%

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

      1. associate-*l* 44.3%

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

      2. unpow2 44.3%

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

      3. unpow2 44.3%

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

      4. difference-of-squares 49.0%

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

   3. Simplified 49.0%

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

   4. Taylor expanded in angle around 0 45.3%

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

   5. Taylor expanded in b around 0 45.3%

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

      1. *-commutative 45.3%

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

      2. associate-*r* 45.4%

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

      3. *-commutative 45.4%

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

      4. unpow2 45.4%

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

   7. Simplified 45.4%

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

4. Final simplification 42.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \leq 136000000000:\\ \;\;\;\;0.011111111111111112 \cdot \left(\pi \cdot \left(angle \cdot \left(b \cdot b\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;-0.011111111111111112 \cdot \left(\pi \cdot \left(angle \cdot \left(a \cdot a\right)\right)\right)\\ \end{array} \]

Alternative 19: 34.6% accurate, 5.7× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

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 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 53.4%

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

5. Taylor expanded in b around inf 33.2%

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

   1. *-commutative 33.2%

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

   2. unpow2 33.2%

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

7. Simplified 33.2%

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

8. Taylor expanded in angle around 0 33.2%

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

   1. unpow2 33.2%

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

   2. associate-*l* 33.1%

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

10. Simplified 33.1%

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

11. Final simplification 33.1%

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

Alternative 20: 34.6% accurate, 5.7× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

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 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 53.4%

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

5. Taylor expanded in b around inf 33.2%

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

   1. *-commutative 33.2%

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

   2. unpow2 33.2%

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

7. Simplified 33.2%

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

8. Final simplification 33.2%

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

Alternative 21: 34.6% accurate, 5.7× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

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 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 53.4%

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

5. Taylor expanded in b around inf 33.2%

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

   1. *-commutative 33.2%

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

   2. unpow2 33.2%

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

7. Simplified 33.2%

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

8. Taylor expanded in angle around 0 33.2%

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

   1. unpow2 33.2%

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

   2. associate-*r* 33.2%

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

10. Simplified 33.2%

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

11. Final simplification 33.2%

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

Reproduce

herbie shell --seed 2023214 
(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)))))