ab-angle->ABCF B

Percentage Accurate: 54.5% → 66.5%

Time: 35.8s

Alternatives: 11

Speedup: 5.5×

• 33.0% of points produce a very large (infinite) output. You may want to add a precondition.

Specification

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 54.5% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 66.5% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;{b}^{2} \leq 2 \cdot 10^{+285}:\\ \;\;\;\;{\left(\sqrt[3]{2 \cdot \left(\left(b - a\right) \cdot \left(\left(b + a\right) \cdot \sin \left(angle \cdot \left(\pi \cdot 0.005555555555555556\right)\right)\right)\right)}\right)}^{3} \cdot \cos \left(\pi \cdot \frac{angle}{180}\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\left(b - a\right) \cdot \left(\left(b + a\right) \cdot angle\right)\right) \cdot \left(\pi \cdot 0.011111111111111112\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (a b angle)
 :precision binary64
 (if (<= (pow b 2.0) 2e+285)
   (*
    (pow
     (cbrt
      (*
       2.0
       (* (- b a) (* (+ b a) (sin (* angle (* PI 0.005555555555555556)))))))
     3.0)
    (cos (* PI (/ angle 180.0))))
   (* (* (- b a) (* (+ b a) angle)) (* PI 0.011111111111111112))))

C

double code(double a, double b, double angle) {
	double tmp;
	if (pow(b, 2.0) <= 2e+285) {
		tmp = pow(cbrt((2.0 * ((b - a) * ((b + a) * sin((angle * (((double) M_PI) * 0.005555555555555556))))))), 3.0) * cos((((double) M_PI) * (angle / 180.0)));
	} else {
		tmp = ((b - a) * ((b + a) * angle)) * (((double) M_PI) * 0.011111111111111112);
	}
	return tmp;
}

Java

public static double code(double a, double b, double angle) {
	double tmp;
	if (Math.pow(b, 2.0) <= 2e+285) {
		tmp = Math.pow(Math.cbrt((2.0 * ((b - a) * ((b + a) * Math.sin((angle * (Math.PI * 0.005555555555555556))))))), 3.0) * Math.cos((Math.PI * (angle / 180.0)));
	} else {
		tmp = ((b - a) * ((b + a) * angle)) * (Math.PI * 0.011111111111111112);
	}
	return tmp;
}

Julia

function code(a, b, angle)
	tmp = 0.0
	if ((b ^ 2.0) <= 2e+285)
		tmp = Float64((cbrt(Float64(2.0 * Float64(Float64(b - a) * Float64(Float64(b + a) * sin(Float64(angle * Float64(pi * 0.005555555555555556))))))) ^ 3.0) * cos(Float64(pi * Float64(angle / 180.0))));
	else
		tmp = Float64(Float64(Float64(b - a) * Float64(Float64(b + a) * angle)) * Float64(pi * 0.011111111111111112));
	end
	return tmp
end

Wolfram

code[a_, b_, angle_] := If[LessEqual[N[Power[b, 2.0], $MachinePrecision], 2e+285], N[(N[Power[N[Power[N[(2.0 * N[(N[(b - a), $MachinePrecision] * N[(N[(b + a), $MachinePrecision] * N[Sin[N[(angle * N[(Pi * 0.005555555555555556), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 1/3], $MachinePrecision], 3.0], $MachinePrecision] * N[Cos[N[(Pi * N[(angle / 180.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(N[(N[(b - a), $MachinePrecision] * N[(N[(b + a), $MachinePrecision] * angle), $MachinePrecision]), $MachinePrecision] * N[(Pi * 0.011111111111111112), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (pow.f64 b 2) < 2e285

   1. Initial program 58.3%

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

      1. add-cube-cbrt 57.9%

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

      2. pow3 57.9%

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

      3. *-commutative 57.9%

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

      4. div-inv 58.1%

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

      5. metadata-eval 58.1%

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

      6. unpow2 58.1%

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

      7. unpow2 58.1%

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

   3. Applied egg-rr 58.1%

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

      1. expm1-log1p-u 44.2%

         \[\leadsto {\color{blue}{\left(\mathsf{expm1}\left(\mathsf{log1p}\left(\sqrt[3]{\sin \left(\pi \cdot \left(angle \cdot 0.005555555555555556\right)\right) \cdot \left(2 \cdot \left(b \cdot b - a \cdot a\right)\right)}\right)\right)\right)}}^{3} \cdot \cos \left(\pi \cdot \frac{angle}{180}\right) \]

      2. expm1-udef 29.1%

         \[\leadsto {\color{blue}{\left(e^{\mathsf{log1p}\left(\sqrt[3]{\sin \left(\pi \cdot \left(angle \cdot 0.005555555555555556\right)\right) \cdot \left(2 \cdot \left(b \cdot b - a \cdot a\right)\right)}\right)} - 1\right)}}^{3} \cdot \cos \left(\pi \cdot \frac{angle}{180}\right) \]

   5. Applied egg-rr 29.1%

      \[\leadsto {\color{blue}{\left(e^{\mathsf{log1p}\left(\sqrt[3]{\left(\left(b \cdot b - a \cdot a\right) \cdot 2\right) \cdot \sin \left(\pi \cdot \left(angle \cdot 0.005555555555555556\right)\right)}\right)} - 1\right)}}^{3} \cdot \cos \left(\pi \cdot \frac{angle}{180}\right) \]
   6. Step-by-step derivation

      1. expm1-def 44.2%

         \[\leadsto {\color{blue}{\left(\mathsf{expm1}\left(\mathsf{log1p}\left(\sqrt[3]{\left(\left(b \cdot b - a \cdot a\right) \cdot 2\right) \cdot \sin \left(\pi \cdot \left(angle \cdot 0.005555555555555556\right)\right)}\right)\right)\right)}}^{3} \cdot \cos \left(\pi \cdot \frac{angle}{180}\right) \]

      2. expm1-log1p 58.1%

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

      3. *-commutative 58.1%

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

      4. associate-*r* 56.8%

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

      5. *-commutative 56.8%

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

      6. *-commutative 56.8%

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

      7. unpow2 56.8%

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

      8. unpow2 56.8%

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

      9. associate-*r* 56.8%

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

      10. unpow2 56.8%

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

      11. unpow2 56.8%

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

      12. difference-of-squares 56.8%

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

      13. *-commutative 56.8%

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

   7. Simplified 67.5%

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

   if 2e285 < (pow.f64 b 2)

   1. Initial program 47.1%

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

   2. Taylor expanded in angle around 0 36.4%

      \[\leadsto \color{blue}{\left(0.011111111111111112 \cdot \left(angle \cdot \left(\left({b}^{2} - {a}^{2}\right) \cdot \pi\right)\right)\right)} \cdot \cos \left(\pi \cdot \frac{angle}{180}\right) \]
   3. Step-by-step derivation

      1. associate-*r* 36.4%

         \[\leadsto \left(0.011111111111111112 \cdot \color{blue}{\left(\left(angle \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \pi\right)}\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right) \]

      2. unpow2 36.4%

         \[\leadsto \left(0.011111111111111112 \cdot \left(\left(angle \cdot \left(\color{blue}{b \cdot b} - {a}^{2}\right)\right) \cdot \pi\right)\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right) \]

      3. unpow2 36.4%

         \[\leadsto \left(0.011111111111111112 \cdot \left(\left(angle \cdot \left(b \cdot b - \color{blue}{a \cdot a}\right)\right) \cdot \pi\right)\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right) \]

   4. Simplified 36.4%

      \[\leadsto \color{blue}{\left(0.011111111111111112 \cdot \left(\left(angle \cdot \left(b \cdot b - a \cdot a\right)\right) \cdot \pi\right)\right)} \cdot \cos \left(\pi \cdot \frac{angle}{180}\right) \]
   5. Step-by-step derivation

      1. pow1 36.4%

         \[\leadsto \left(0.011111111111111112 \cdot \left(\color{blue}{{\left(angle \cdot \left(b \cdot b - a \cdot a\right)\right)}^{1}} \cdot \pi\right)\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right) \]

      2. *-commutative 36.4%

         \[\leadsto \left(0.011111111111111112 \cdot \left({\color{blue}{\left(\left(b \cdot b - a \cdot a\right) \cdot angle\right)}}^{1} \cdot \pi\right)\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right) \]

      3. difference-of-squares 45.7%

         \[\leadsto \left(0.011111111111111112 \cdot \left({\left(\color{blue}{\left(\left(b + a\right) \cdot \left(b - a\right)\right)} \cdot angle\right)}^{1} \cdot \pi\right)\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right) \]

      4. associate-*l* 62.4%

         \[\leadsto \left(0.011111111111111112 \cdot \left({\color{blue}{\left(\left(b + a\right) \cdot \left(\left(b - a\right) \cdot angle\right)\right)}}^{1} \cdot \pi\right)\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right) \]

   6. Applied egg-rr 62.4%

      \[\leadsto \left(0.011111111111111112 \cdot \left(\color{blue}{{\left(\left(b + a\right) \cdot \left(\left(b - a\right) \cdot angle\right)\right)}^{1}} \cdot \pi\right)\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right) \]

   7. Taylor expanded in angle around 0 82.1%

      \[\leadsto \left(0.011111111111111112 \cdot \left({\left(\left(b + a\right) \cdot \left(\left(b - a\right) \cdot angle\right)\right)}^{1} \cdot \pi\right)\right) \cdot \color{blue}{1} \]

   8. Taylor expanded in angle around 0 65.3%

      \[\leadsto \color{blue}{\left(0.011111111111111112 \cdot \left(angle \cdot \left(\left(b - a\right) \cdot \left(\left(a + b\right) \cdot \pi\right)\right)\right)\right)} \cdot 1 \]
   9. Step-by-step derivation

      1. *-commutative 65.3%

         \[\leadsto \color{blue}{\left(\left(angle \cdot \left(\left(b - a\right) \cdot \left(\left(a + b\right) \cdot \pi\right)\right)\right) \cdot 0.011111111111111112\right)} \cdot 1 \]

      2. associate-*r* 82.1%

         \[\leadsto \left(\color{blue}{\left(\left(angle \cdot \left(b - a\right)\right) \cdot \left(\left(a + b\right) \cdot \pi\right)\right)} \cdot 0.011111111111111112\right) \cdot 1 \]

      3. *-commutative 82.1%

         \[\leadsto \left(\left(\color{blue}{\left(\left(b - a\right) \cdot angle\right)} \cdot \left(\left(a + b\right) \cdot \pi\right)\right) \cdot 0.011111111111111112\right) \cdot 1 \]

      4. associate-*l* 82.1%

         \[\leadsto \left(\color{blue}{\left(\left(\left(\left(b - a\right) \cdot angle\right) \cdot \left(a + b\right)\right) \cdot \pi\right)} \cdot 0.011111111111111112\right) \cdot 1 \]

      5. +-commutative 82.1%

         \[\leadsto \left(\left(\left(\left(\left(b - a\right) \cdot angle\right) \cdot \color{blue}{\left(b + a\right)}\right) \cdot \pi\right) \cdot 0.011111111111111112\right) \cdot 1 \]

      6. associate-*r* 82.1%

         \[\leadsto \left(\left(\color{blue}{\left(\left(b - a\right) \cdot \left(angle \cdot \left(b + a\right)\right)\right)} \cdot \pi\right) \cdot 0.011111111111111112\right) \cdot 1 \]

      7. associate-*l* 82.1%

         \[\leadsto \color{blue}{\left(\left(\left(b - a\right) \cdot \left(angle \cdot \left(b + a\right)\right)\right) \cdot \left(\pi \cdot 0.011111111111111112\right)\right)} \cdot 1 \]

      8. +-commutative 82.1%

         \[\leadsto \left(\left(\left(b - a\right) \cdot \left(angle \cdot \color{blue}{\left(a + b\right)}\right)\right) \cdot \left(\pi \cdot 0.011111111111111112\right)\right) \cdot 1 \]

   10. Simplified 82.1%

       \[\leadsto \color{blue}{\left(\left(\left(b - a\right) \cdot \left(angle \cdot \left(a + b\right)\right)\right) \cdot \left(\pi \cdot 0.011111111111111112\right)\right)} \cdot 1 \]
3. Recombined 2 regimes into one program.

4. Final simplification 70.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;{b}^{2} \leq 2 \cdot 10^{+285}:\\ \;\;\;\;{\left(\sqrt[3]{2 \cdot \left(\left(b - a\right) \cdot \left(\left(b + a\right) \cdot \sin \left(angle \cdot \left(\pi \cdot 0.005555555555555556\right)\right)\right)\right)}\right)}^{3} \cdot \cos \left(\pi \cdot \frac{angle}{180}\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\left(b - a\right) \cdot \left(\left(b + a\right) \cdot angle\right)\right) \cdot \left(\pi \cdot 0.011111111111111112\right)\\ \end{array} \]

Alternative 2: 66.4% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := angle \cdot \left(\pi \cdot 0.005555555555555556\right)\\ t_1 := {b}^{2} - {a}^{2}\\ \mathbf{if}\;t_1 \leq -1 \cdot 10^{+297}:\\ \;\;\;\;2 \cdot \left(\left(b - a\right) \cdot \left(\left(b + a\right) \cdot \sin t_0\right)\right)\\ \mathbf{elif}\;t_1 \leq 4 \cdot 10^{+250}:\\ \;\;\;\;\left(\left(2 \cdot t_1\right) \cdot \sin \left(\pi \cdot \frac{angle}{180}\right)\right) \cdot \cos t_0\\ \mathbf{else}:\\ \;\;\;\;\left(\left(b - a\right) \cdot \left(\left(b + a\right) \cdot angle\right)\right) \cdot \left(\pi \cdot 0.011111111111111112\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (a b angle)
 :precision binary64
 (let* ((t_0 (* angle (* PI 0.005555555555555556)))
        (t_1 (- (pow b 2.0) (pow a 2.0))))
   (if (<= t_1 -1e+297)
     (* 2.0 (* (- b a) (* (+ b a) (sin t_0))))
     (if (<= t_1 4e+250)
       (* (* (* 2.0 t_1) (sin (* PI (/ angle 180.0)))) (cos t_0))
       (* (* (- b a) (* (+ b a) angle)) (* PI 0.011111111111111112))))))

C

double code(double a, double b, double angle) {
	double t_0 = angle * (((double) M_PI) * 0.005555555555555556);
	double t_1 = pow(b, 2.0) - pow(a, 2.0);
	double tmp;
	if (t_1 <= -1e+297) {
		tmp = 2.0 * ((b - a) * ((b + a) * sin(t_0)));
	} else if (t_1 <= 4e+250) {
		tmp = ((2.0 * t_1) * sin((((double) M_PI) * (angle / 180.0)))) * cos(t_0);
	} else {
		tmp = ((b - a) * ((b + a) * angle)) * (((double) M_PI) * 0.011111111111111112);
	}
	return tmp;
}

Java

public static double code(double a, double b, double angle) {
	double t_0 = angle * (Math.PI * 0.005555555555555556);
	double t_1 = Math.pow(b, 2.0) - Math.pow(a, 2.0);
	double tmp;
	if (t_1 <= -1e+297) {
		tmp = 2.0 * ((b - a) * ((b + a) * Math.sin(t_0)));
	} else if (t_1 <= 4e+250) {
		tmp = ((2.0 * t_1) * Math.sin((Math.PI * (angle / 180.0)))) * Math.cos(t_0);
	} else {
		tmp = ((b - a) * ((b + a) * angle)) * (Math.PI * 0.011111111111111112);
	}
	return tmp;
}

Python

def code(a, b, angle):
	t_0 = angle * (math.pi * 0.005555555555555556)
	t_1 = math.pow(b, 2.0) - math.pow(a, 2.0)
	tmp = 0
	if t_1 <= -1e+297:
		tmp = 2.0 * ((b - a) * ((b + a) * math.sin(t_0)))
	elif t_1 <= 4e+250:
		tmp = ((2.0 * t_1) * math.sin((math.pi * (angle / 180.0)))) * math.cos(t_0)
	else:
		tmp = ((b - a) * ((b + a) * angle)) * (math.pi * 0.011111111111111112)
	return tmp

Julia

function code(a, b, angle)
	t_0 = Float64(angle * Float64(pi * 0.005555555555555556))
	t_1 = Float64((b ^ 2.0) - (a ^ 2.0))
	tmp = 0.0
	if (t_1 <= -1e+297)
		tmp = Float64(2.0 * Float64(Float64(b - a) * Float64(Float64(b + a) * sin(t_0))));
	elseif (t_1 <= 4e+250)
		tmp = Float64(Float64(Float64(2.0 * t_1) * sin(Float64(pi * Float64(angle / 180.0)))) * cos(t_0));
	else
		tmp = Float64(Float64(Float64(b - a) * Float64(Float64(b + a) * angle)) * Float64(pi * 0.011111111111111112));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, angle)
	t_0 = angle * (pi * 0.005555555555555556);
	t_1 = (b ^ 2.0) - (a ^ 2.0);
	tmp = 0.0;
	if (t_1 <= -1e+297)
		tmp = 2.0 * ((b - a) * ((b + a) * sin(t_0)));
	elseif (t_1 <= 4e+250)
		tmp = ((2.0 * t_1) * sin((pi * (angle / 180.0)))) * cos(t_0);
	else
		tmp = ((b - a) * ((b + a) * angle)) * (pi * 0.011111111111111112);
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, angle_] := Block[{t$95$0 = N[(angle * N[(Pi * 0.005555555555555556), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[Power[b, 2.0], $MachinePrecision] - N[Power[a, 2.0], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$1, -1e+297], N[(2.0 * N[(N[(b - a), $MachinePrecision] * N[(N[(b + a), $MachinePrecision] * N[Sin[t$95$0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$1, 4e+250], N[(N[(N[(2.0 * t$95$1), $MachinePrecision] * N[Sin[N[(Pi * N[(angle / 180.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[Cos[t$95$0], $MachinePrecision]), $MachinePrecision], N[(N[(N[(b - a), $MachinePrecision] * N[(N[(b + a), $MachinePrecision] * angle), $MachinePrecision]), $MachinePrecision] * N[(Pi * 0.011111111111111112), $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

2. if (-.f64 (pow.f64 b 2) (pow.f64 a 2)) < -1e297

   1. Initial program 53.6%

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

      1. add-cube-cbrt 53.6%

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

      2. pow3 53.6%

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

      3. unpow2 53.6%

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

      4. unpow2 53.6%

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

   3. Applied egg-rr 53.6%

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

   4. Taylor expanded in angle around 0 50.7%

      \[\leadsto \left(\left(2 \cdot {\left(\sqrt[3]{b \cdot b - a \cdot a}\right)}^{3}\right) \cdot \sin \left(\pi \cdot \frac{angle}{180}\right)\right) \cdot \color{blue}{1} \]
   5. Step-by-step derivation

      1. add-sqr-sqrt 27.6%

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

      2. sqrt-unprod 35.6%

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

      3. pow2 35.6%

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

   6. Applied egg-rr 35.6%

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

   7. Taylor expanded in angle around inf 49.4%

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

      1. unpow2 49.4%

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

      2. unpow2 49.4%

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

      3. difference-of-squares 49.4%

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

      4. *-commutative 49.4%

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

      5. associate-*l* 80.1%

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

      6. +-commutative 80.1%

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

      7. *-commutative 80.1%

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

      8. associate-*l* 80.1%

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

   9. Simplified 80.1%

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

   if -1e297 < (-.f64 (pow.f64 b 2) (pow.f64 a 2)) < 3.9999999999999997e250

   1. Initial program 60.3%

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

   2. Taylor expanded in angle around inf 59.5%

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

      1. *-commutative 59.5%

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

      2. *-commutative 59.5%

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

      3. associate-*r* 60.2%

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

      4. *-commutative 60.2%

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

      5. associate-*l* 60.0%

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

   4. Simplified 60.0%

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

   if 3.9999999999999997e250 < (-.f64 (pow.f64 b 2) (pow.f64 a 2))

   1. Initial program 48.3%

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

   2. Taylor expanded in angle around 0 39.5%

      \[\leadsto \color{blue}{\left(0.011111111111111112 \cdot \left(angle \cdot \left(\left({b}^{2} - {a}^{2}\right) \cdot \pi\right)\right)\right)} \cdot \cos \left(\pi \cdot \frac{angle}{180}\right) \]
   3. Step-by-step derivation

      1. associate-*r* 39.5%

         \[\leadsto \left(0.011111111111111112 \cdot \color{blue}{\left(\left(angle \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \pi\right)}\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right) \]

      2. unpow2 39.5%

         \[\leadsto \left(0.011111111111111112 \cdot \left(\left(angle \cdot \left(\color{blue}{b \cdot b} - {a}^{2}\right)\right) \cdot \pi\right)\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right) \]

      3. unpow2 39.5%

         \[\leadsto \left(0.011111111111111112 \cdot \left(\left(angle \cdot \left(b \cdot b - \color{blue}{a \cdot a}\right)\right) \cdot \pi\right)\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right) \]

   4. Simplified 39.5%

      \[\leadsto \color{blue}{\left(0.011111111111111112 \cdot \left(\left(angle \cdot \left(b \cdot b - a \cdot a\right)\right) \cdot \pi\right)\right)} \cdot \cos \left(\pi \cdot \frac{angle}{180}\right) \]
   5. Step-by-step derivation

      1. pow1 39.5%

         \[\leadsto \left(0.011111111111111112 \cdot \left(\color{blue}{{\left(angle \cdot \left(b \cdot b - a \cdot a\right)\right)}^{1}} \cdot \pi\right)\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right) \]

      2. *-commutative 39.5%

         \[\leadsto \left(0.011111111111111112 \cdot \left({\color{blue}{\left(\left(b \cdot b - a \cdot a\right) \cdot angle\right)}}^{1} \cdot \pi\right)\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right) \]

      3. difference-of-squares 48.3%

         \[\leadsto \left(0.011111111111111112 \cdot \left({\left(\color{blue}{\left(\left(b + a\right) \cdot \left(b - a\right)\right)} \cdot angle\right)}^{1} \cdot \pi\right)\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right) \]

      4. associate-*l* 62.6%

         \[\leadsto \left(0.011111111111111112 \cdot \left({\color{blue}{\left(\left(b + a\right) \cdot \left(\left(b - a\right) \cdot angle\right)\right)}}^{1} \cdot \pi\right)\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right) \]

   6. Applied egg-rr 62.6%

      \[\leadsto \left(0.011111111111111112 \cdot \left(\color{blue}{{\left(\left(b + a\right) \cdot \left(\left(b - a\right) \cdot angle\right)\right)}^{1}} \cdot \pi\right)\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right) \]

   7. Taylor expanded in angle around 0 79.7%

      \[\leadsto \left(0.011111111111111112 \cdot \left({\left(\left(b + a\right) \cdot \left(\left(b - a\right) \cdot angle\right)\right)}^{1} \cdot \pi\right)\right) \cdot \color{blue}{1} \]

   8. Taylor expanded in angle around 0 65.4%

      \[\leadsto \color{blue}{\left(0.011111111111111112 \cdot \left(angle \cdot \left(\left(b - a\right) \cdot \left(\left(a + b\right) \cdot \pi\right)\right)\right)\right)} \cdot 1 \]
   9. Step-by-step derivation

      1. *-commutative 65.4%

         \[\leadsto \color{blue}{\left(\left(angle \cdot \left(\left(b - a\right) \cdot \left(\left(a + b\right) \cdot \pi\right)\right)\right) \cdot 0.011111111111111112\right)} \cdot 1 \]

      2. associate-*r* 79.7%

         \[\leadsto \left(\color{blue}{\left(\left(angle \cdot \left(b - a\right)\right) \cdot \left(\left(a + b\right) \cdot \pi\right)\right)} \cdot 0.011111111111111112\right) \cdot 1 \]

      3. *-commutative 79.7%

         \[\leadsto \left(\left(\color{blue}{\left(\left(b - a\right) \cdot angle\right)} \cdot \left(\left(a + b\right) \cdot \pi\right)\right) \cdot 0.011111111111111112\right) \cdot 1 \]

      4. associate-*l* 79.7%

         \[\leadsto \left(\color{blue}{\left(\left(\left(\left(b - a\right) \cdot angle\right) \cdot \left(a + b\right)\right) \cdot \pi\right)} \cdot 0.011111111111111112\right) \cdot 1 \]

      5. +-commutative 79.7%

         \[\leadsto \left(\left(\left(\left(\left(b - a\right) \cdot angle\right) \cdot \color{blue}{\left(b + a\right)}\right) \cdot \pi\right) \cdot 0.011111111111111112\right) \cdot 1 \]

      6. associate-*r* 79.7%

         \[\leadsto \left(\left(\color{blue}{\left(\left(b - a\right) \cdot \left(angle \cdot \left(b + a\right)\right)\right)} \cdot \pi\right) \cdot 0.011111111111111112\right) \cdot 1 \]

      7. associate-*l* 79.7%

         \[\leadsto \color{blue}{\left(\left(\left(b - a\right) \cdot \left(angle \cdot \left(b + a\right)\right)\right) \cdot \left(\pi \cdot 0.011111111111111112\right)\right)} \cdot 1 \]

      8. +-commutative 79.7%

         \[\leadsto \left(\left(\left(b - a\right) \cdot \left(angle \cdot \color{blue}{\left(a + b\right)}\right)\right) \cdot \left(\pi \cdot 0.011111111111111112\right)\right) \cdot 1 \]

   10. Simplified 79.7%

       \[\leadsto \color{blue}{\left(\left(\left(b - a\right) \cdot \left(angle \cdot \left(a + b\right)\right)\right) \cdot \left(\pi \cdot 0.011111111111111112\right)\right)} \cdot 1 \]
3. Recombined 3 regimes into one program.

4. Final simplification 69.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;{b}^{2} - {a}^{2} \leq -1 \cdot 10^{+297}:\\ \;\;\;\;2 \cdot \left(\left(b - a\right) \cdot \left(\left(b + a\right) \cdot \sin \left(angle \cdot \left(\pi \cdot 0.005555555555555556\right)\right)\right)\right)\\ \mathbf{elif}\;{b}^{2} - {a}^{2} \leq 4 \cdot 10^{+250}:\\ \;\;\;\;\left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin \left(\pi \cdot \frac{angle}{180}\right)\right) \cdot \cos \left(angle \cdot \left(\pi \cdot 0.005555555555555556\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\left(b - a\right) \cdot \left(\left(b + a\right) \cdot angle\right)\right) \cdot \left(\pi \cdot 0.011111111111111112\right)\\ \end{array} \]

Alternative 3: 66.5% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := {b}^{2} - {a}^{2}\\ t_1 := \pi \cdot \frac{angle}{180}\\ \mathbf{if}\;t_0 \leq -1 \cdot 10^{+297}:\\ \;\;\;\;2 \cdot \left(\left(b - a\right) \cdot \left(\left(b + a\right) \cdot \sin \left(angle \cdot \left(\pi \cdot 0.005555555555555556\right)\right)\right)\right)\\ \mathbf{elif}\;t_0 \leq 2 \cdot 10^{+285}:\\ \;\;\;\;\cos t_1 \cdot \left(\left(2 \cdot t_0\right) \cdot \sin t_1\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\left(b - a\right) \cdot \left(\left(b + a\right) \cdot angle\right)\right) \cdot \left(\pi \cdot 0.011111111111111112\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (a b angle)
 :precision binary64
 (let* ((t_0 (- (pow b 2.0) (pow a 2.0))) (t_1 (* PI (/ angle 180.0))))
   (if (<= t_0 -1e+297)
     (*
      2.0
      (* (- b a) (* (+ b a) (sin (* angle (* PI 0.005555555555555556))))))
     (if (<= t_0 2e+285)
       (* (cos t_1) (* (* 2.0 t_0) (sin t_1)))
       (* (* (- b a) (* (+ b a) angle)) (* PI 0.011111111111111112))))))

C

double code(double a, double b, double angle) {
	double t_0 = pow(b, 2.0) - pow(a, 2.0);
	double t_1 = ((double) M_PI) * (angle / 180.0);
	double tmp;
	if (t_0 <= -1e+297) {
		tmp = 2.0 * ((b - a) * ((b + a) * sin((angle * (((double) M_PI) * 0.005555555555555556)))));
	} else if (t_0 <= 2e+285) {
		tmp = cos(t_1) * ((2.0 * t_0) * sin(t_1));
	} else {
		tmp = ((b - a) * ((b + a) * angle)) * (((double) M_PI) * 0.011111111111111112);
	}
	return tmp;
}

Java

public static double code(double a, double b, double angle) {
	double t_0 = Math.pow(b, 2.0) - Math.pow(a, 2.0);
	double t_1 = Math.PI * (angle / 180.0);
	double tmp;
	if (t_0 <= -1e+297) {
		tmp = 2.0 * ((b - a) * ((b + a) * Math.sin((angle * (Math.PI * 0.005555555555555556)))));
	} else if (t_0 <= 2e+285) {
		tmp = Math.cos(t_1) * ((2.0 * t_0) * Math.sin(t_1));
	} else {
		tmp = ((b - a) * ((b + a) * angle)) * (Math.PI * 0.011111111111111112);
	}
	return tmp;
}

Python

def code(a, b, angle):
	t_0 = math.pow(b, 2.0) - math.pow(a, 2.0)
	t_1 = math.pi * (angle / 180.0)
	tmp = 0
	if t_0 <= -1e+297:
		tmp = 2.0 * ((b - a) * ((b + a) * math.sin((angle * (math.pi * 0.005555555555555556)))))
	elif t_0 <= 2e+285:
		tmp = math.cos(t_1) * ((2.0 * t_0) * math.sin(t_1))
	else:
		tmp = ((b - a) * ((b + a) * angle)) * (math.pi * 0.011111111111111112)
	return tmp

Julia

function code(a, b, angle)
	t_0 = Float64((b ^ 2.0) - (a ^ 2.0))
	t_1 = Float64(pi * Float64(angle / 180.0))
	tmp = 0.0
	if (t_0 <= -1e+297)
		tmp = Float64(2.0 * Float64(Float64(b - a) * Float64(Float64(b + a) * sin(Float64(angle * Float64(pi * 0.005555555555555556))))));
	elseif (t_0 <= 2e+285)
		tmp = Float64(cos(t_1) * Float64(Float64(2.0 * t_0) * sin(t_1)));
	else
		tmp = Float64(Float64(Float64(b - a) * Float64(Float64(b + a) * angle)) * Float64(pi * 0.011111111111111112));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, angle)
	t_0 = (b ^ 2.0) - (a ^ 2.0);
	t_1 = pi * (angle / 180.0);
	tmp = 0.0;
	if (t_0 <= -1e+297)
		tmp = 2.0 * ((b - a) * ((b + a) * sin((angle * (pi * 0.005555555555555556)))));
	elseif (t_0 <= 2e+285)
		tmp = cos(t_1) * ((2.0 * t_0) * sin(t_1));
	else
		tmp = ((b - a) * ((b + a) * angle)) * (pi * 0.011111111111111112);
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, angle_] := Block[{t$95$0 = N[(N[Power[b, 2.0], $MachinePrecision] - N[Power[a, 2.0], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(Pi * N[(angle / 180.0), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$0, -1e+297], N[(2.0 * N[(N[(b - a), $MachinePrecision] * N[(N[(b + a), $MachinePrecision] * N[Sin[N[(angle * N[(Pi * 0.005555555555555556), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$0, 2e+285], N[(N[Cos[t$95$1], $MachinePrecision] * N[(N[(2.0 * t$95$0), $MachinePrecision] * N[Sin[t$95$1], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(b - a), $MachinePrecision] * N[(N[(b + a), $MachinePrecision] * angle), $MachinePrecision]), $MachinePrecision] * N[(Pi * 0.011111111111111112), $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

2. if (-.f64 (pow.f64 b 2) (pow.f64 a 2)) < -1e297

   1. Initial program 53.6%

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

      1. add-cube-cbrt 53.6%

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

      2. pow3 53.6%

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

      3. unpow2 53.6%

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

      4. unpow2 53.6%

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

   3. Applied egg-rr 53.6%

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

   4. Taylor expanded in angle around 0 50.7%

      \[\leadsto \left(\left(2 \cdot {\left(\sqrt[3]{b \cdot b - a \cdot a}\right)}^{3}\right) \cdot \sin \left(\pi \cdot \frac{angle}{180}\right)\right) \cdot \color{blue}{1} \]
   5. Step-by-step derivation

      1. add-sqr-sqrt 27.6%

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

      2. sqrt-unprod 35.6%

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

      3. pow2 35.6%

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

   6. Applied egg-rr 35.6%

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

   7. Taylor expanded in angle around inf 49.4%

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

      1. unpow2 49.4%

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

      2. unpow2 49.4%

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

      3. difference-of-squares 49.4%

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

      4. *-commutative 49.4%

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

      5. associate-*l* 80.1%

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

      6. +-commutative 80.1%

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

      7. *-commutative 80.1%

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

      8. associate-*l* 80.1%

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

   9. Simplified 80.1%

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

   if -1e297 < (-.f64 (pow.f64 b 2) (pow.f64 a 2)) < 2e285

   1. Initial program 60.4%

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

   if 2e285 < (-.f64 (pow.f64 b 2) (pow.f64 a 2))

   1. Initial program 46.9%

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

   2. Taylor expanded in angle around 0 37.6%

      \[\leadsto \color{blue}{\left(0.011111111111111112 \cdot \left(angle \cdot \left(\left({b}^{2} - {a}^{2}\right) \cdot \pi\right)\right)\right)} \cdot \cos \left(\pi \cdot \frac{angle}{180}\right) \]
   3. Step-by-step derivation

      1. associate-*r* 37.6%

         \[\leadsto \left(0.011111111111111112 \cdot \color{blue}{\left(\left(angle \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \pi\right)}\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right) \]

      2. unpow2 37.6%

         \[\leadsto \left(0.011111111111111112 \cdot \left(\left(angle \cdot \left(\color{blue}{b \cdot b} - {a}^{2}\right)\right) \cdot \pi\right)\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right) \]

      3. unpow2 37.6%

         \[\leadsto \left(0.011111111111111112 \cdot \left(\left(angle \cdot \left(b \cdot b - \color{blue}{a \cdot a}\right)\right) \cdot \pi\right)\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right) \]

   4. Simplified 37.6%

      \[\leadsto \color{blue}{\left(0.011111111111111112 \cdot \left(\left(angle \cdot \left(b \cdot b - a \cdot a\right)\right) \cdot \pi\right)\right)} \cdot \cos \left(\pi \cdot \frac{angle}{180}\right) \]
   5. Step-by-step derivation

      1. pow1 37.6%

         \[\leadsto \left(0.011111111111111112 \cdot \left(\color{blue}{{\left(angle \cdot \left(b \cdot b - a \cdot a\right)\right)}^{1}} \cdot \pi\right)\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right) \]

      2. *-commutative 37.6%

         \[\leadsto \left(0.011111111111111112 \cdot \left({\color{blue}{\left(\left(b \cdot b - a \cdot a\right) \cdot angle\right)}}^{1} \cdot \pi\right)\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right) \]

      3. difference-of-squares 47.2%

         \[\leadsto \left(0.011111111111111112 \cdot \left({\left(\color{blue}{\left(\left(b + a\right) \cdot \left(b - a\right)\right)} \cdot angle\right)}^{1} \cdot \pi\right)\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right) \]

      4. associate-*l* 62.9%

         \[\leadsto \left(0.011111111111111112 \cdot \left({\color{blue}{\left(\left(b + a\right) \cdot \left(\left(b - a\right) \cdot angle\right)\right)}}^{1} \cdot \pi\right)\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right) \]

   6. Applied egg-rr 62.9%

      \[\leadsto \left(0.011111111111111112 \cdot \left(\color{blue}{{\left(\left(b + a\right) \cdot \left(\left(b - a\right) \cdot angle\right)\right)}^{1}} \cdot \pi\right)\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right) \]

   7. Taylor expanded in angle around 0 81.4%

      \[\leadsto \left(0.011111111111111112 \cdot \left({\left(\left(b + a\right) \cdot \left(\left(b - a\right) \cdot angle\right)\right)}^{1} \cdot \pi\right)\right) \cdot \color{blue}{1} \]

   8. Taylor expanded in angle around 0 65.7%

      \[\leadsto \color{blue}{\left(0.011111111111111112 \cdot \left(angle \cdot \left(\left(b - a\right) \cdot \left(\left(a + b\right) \cdot \pi\right)\right)\right)\right)} \cdot 1 \]
   9. Step-by-step derivation

      1. *-commutative 65.7%

         \[\leadsto \color{blue}{\left(\left(angle \cdot \left(\left(b - a\right) \cdot \left(\left(a + b\right) \cdot \pi\right)\right)\right) \cdot 0.011111111111111112\right)} \cdot 1 \]

      2. associate-*r* 81.4%

         \[\leadsto \left(\color{blue}{\left(\left(angle \cdot \left(b - a\right)\right) \cdot \left(\left(a + b\right) \cdot \pi\right)\right)} \cdot 0.011111111111111112\right) \cdot 1 \]

      3. *-commutative 81.4%

         \[\leadsto \left(\left(\color{blue}{\left(\left(b - a\right) \cdot angle\right)} \cdot \left(\left(a + b\right) \cdot \pi\right)\right) \cdot 0.011111111111111112\right) \cdot 1 \]

      4. associate-*l* 81.4%

         \[\leadsto \left(\color{blue}{\left(\left(\left(\left(b - a\right) \cdot angle\right) \cdot \left(a + b\right)\right) \cdot \pi\right)} \cdot 0.011111111111111112\right) \cdot 1 \]

      5. +-commutative 81.4%

         \[\leadsto \left(\left(\left(\left(\left(b - a\right) \cdot angle\right) \cdot \color{blue}{\left(b + a\right)}\right) \cdot \pi\right) \cdot 0.011111111111111112\right) \cdot 1 \]

      6. associate-*r* 81.4%

         \[\leadsto \left(\left(\color{blue}{\left(\left(b - a\right) \cdot \left(angle \cdot \left(b + a\right)\right)\right)} \cdot \pi\right) \cdot 0.011111111111111112\right) \cdot 1 \]

      7. associate-*l* 81.4%

         \[\leadsto \color{blue}{\left(\left(\left(b - a\right) \cdot \left(angle \cdot \left(b + a\right)\right)\right) \cdot \left(\pi \cdot 0.011111111111111112\right)\right)} \cdot 1 \]

      8. +-commutative 81.4%

         \[\leadsto \left(\left(\left(b - a\right) \cdot \left(angle \cdot \color{blue}{\left(a + b\right)}\right)\right) \cdot \left(\pi \cdot 0.011111111111111112\right)\right) \cdot 1 \]

   10. Simplified 81.4%

       \[\leadsto \color{blue}{\left(\left(\left(b - a\right) \cdot \left(angle \cdot \left(a + b\right)\right)\right) \cdot \left(\pi \cdot 0.011111111111111112\right)\right)} \cdot 1 \]
3. Recombined 3 regimes into one program.

4. Final simplification 69.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;{b}^{2} - {a}^{2} \leq -1 \cdot 10^{+297}:\\ \;\;\;\;2 \cdot \left(\left(b - a\right) \cdot \left(\left(b + a\right) \cdot \sin \left(angle \cdot \left(\pi \cdot 0.005555555555555556\right)\right)\right)\right)\\ \mathbf{elif}\;{b}^{2} - {a}^{2} \leq 2 \cdot 10^{+285}:\\ \;\;\;\;\cos \left(\pi \cdot \frac{angle}{180}\right) \cdot \left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin \left(\pi \cdot \frac{angle}{180}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\left(b - a\right) \cdot \left(\left(b + a\right) \cdot angle\right)\right) \cdot \left(\pi \cdot 0.011111111111111112\right)\\ \end{array} \]

Alternative 4: 64.0% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{angle}{180} \leq 10^{-10}:\\ \;\;\;\;\left(\left(b - a\right) \cdot \left(\left(b + a\right) \cdot angle\right)\right) \cdot \left(\pi \cdot 0.011111111111111112\right)\\ \mathbf{else}:\\ \;\;\;\;\cos \left(\pi \cdot \frac{angle}{180}\right) \cdot \left(\left(2 \cdot {\left(\sqrt[3]{b \cdot b - a \cdot a}\right)}^{3}\right) \cdot \sin \left(\frac{\pi}{\frac{180}{angle}}\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (a b angle)
 :precision binary64
 (if (<= (/ angle 180.0) 1e-10)
   (* (* (- b a) (* (+ b a) angle)) (* PI 0.011111111111111112))
   (*
    (cos (* PI (/ angle 180.0)))
    (*
     (* 2.0 (pow (cbrt (- (* b b) (* a a))) 3.0))
     (sin (/ PI (/ 180.0 angle)))))))

C

double code(double a, double b, double angle) {
	double tmp;
	if ((angle / 180.0) <= 1e-10) {
		tmp = ((b - a) * ((b + a) * angle)) * (((double) M_PI) * 0.011111111111111112);
	} else {
		tmp = cos((((double) M_PI) * (angle / 180.0))) * ((2.0 * pow(cbrt(((b * b) - (a * a))), 3.0)) * sin((((double) M_PI) / (180.0 / angle))));
	}
	return tmp;
}

Java

public static double code(double a, double b, double angle) {
	double tmp;
	if ((angle / 180.0) <= 1e-10) {
		tmp = ((b - a) * ((b + a) * angle)) * (Math.PI * 0.011111111111111112);
	} else {
		tmp = Math.cos((Math.PI * (angle / 180.0))) * ((2.0 * Math.pow(Math.cbrt(((b * b) - (a * a))), 3.0)) * Math.sin((Math.PI / (180.0 / angle))));
	}
	return tmp;
}

Julia

function code(a, b, angle)
	tmp = 0.0
	if (Float64(angle / 180.0) <= 1e-10)
		tmp = Float64(Float64(Float64(b - a) * Float64(Float64(b + a) * angle)) * Float64(pi * 0.011111111111111112));
	else
		tmp = Float64(cos(Float64(pi * Float64(angle / 180.0))) * Float64(Float64(2.0 * (cbrt(Float64(Float64(b * b) - Float64(a * a))) ^ 3.0)) * sin(Float64(pi / Float64(180.0 / angle)))));
	end
	return tmp
end

Wolfram

code[a_, b_, angle_] := If[LessEqual[N[(angle / 180.0), $MachinePrecision], 1e-10], N[(N[(N[(b - a), $MachinePrecision] * N[(N[(b + a), $MachinePrecision] * angle), $MachinePrecision]), $MachinePrecision] * N[(Pi * 0.011111111111111112), $MachinePrecision]), $MachinePrecision], N[(N[Cos[N[(Pi * N[(angle / 180.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * N[(N[(2.0 * N[Power[N[Power[N[(N[(b * b), $MachinePrecision] - N[(a * a), $MachinePrecision]), $MachinePrecision], 1/3], $MachinePrecision], 3.0], $MachinePrecision]), $MachinePrecision] * N[Sin[N[(Pi / N[(180.0 / angle), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 angle 180) < 1.00000000000000004e-10

   1. Initial program 61.9%

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

   2. Taylor expanded in angle around 0 57.5%

      \[\leadsto \color{blue}{\left(0.011111111111111112 \cdot \left(angle \cdot \left(\left({b}^{2} - {a}^{2}\right) \cdot \pi\right)\right)\right)} \cdot \cos \left(\pi \cdot \frac{angle}{180}\right) \]
   3. Step-by-step derivation

      1. associate-*r* 57.6%

         \[\leadsto \left(0.011111111111111112 \cdot \color{blue}{\left(\left(angle \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \pi\right)}\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right) \]

      2. unpow2 57.6%

         \[\leadsto \left(0.011111111111111112 \cdot \left(\left(angle \cdot \left(\color{blue}{b \cdot b} - {a}^{2}\right)\right) \cdot \pi\right)\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right) \]

      3. unpow2 57.6%

         \[\leadsto \left(0.011111111111111112 \cdot \left(\left(angle \cdot \left(b \cdot b - \color{blue}{a \cdot a}\right)\right) \cdot \pi\right)\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right) \]

   4. Simplified 57.6%

      \[\leadsto \color{blue}{\left(0.011111111111111112 \cdot \left(\left(angle \cdot \left(b \cdot b - a \cdot a\right)\right) \cdot \pi\right)\right)} \cdot \cos \left(\pi \cdot \frac{angle}{180}\right) \]
   5. Step-by-step derivation

      1. pow1 57.6%

         \[\leadsto \left(0.011111111111111112 \cdot \left(\color{blue}{{\left(angle \cdot \left(b \cdot b - a \cdot a\right)\right)}^{1}} \cdot \pi\right)\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right) \]

      2. *-commutative 57.6%

         \[\leadsto \left(0.011111111111111112 \cdot \left({\color{blue}{\left(\left(b \cdot b - a \cdot a\right) \cdot angle\right)}}^{1} \cdot \pi\right)\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right) \]

      3. difference-of-squares 60.0%

         \[\leadsto \left(0.011111111111111112 \cdot \left({\left(\color{blue}{\left(\left(b + a\right) \cdot \left(b - a\right)\right)} \cdot angle\right)}^{1} \cdot \pi\right)\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right) \]

      4. associate-*l* 74.9%

         \[\leadsto \left(0.011111111111111112 \cdot \left({\color{blue}{\left(\left(b + a\right) \cdot \left(\left(b - a\right) \cdot angle\right)\right)}}^{1} \cdot \pi\right)\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right) \]

   6. Applied egg-rr 74.9%

      \[\leadsto \left(0.011111111111111112 \cdot \left(\color{blue}{{\left(\left(b + a\right) \cdot \left(\left(b - a\right) \cdot angle\right)\right)}^{1}} \cdot \pi\right)\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right) \]

   7. Taylor expanded in angle around 0 80.5%

      \[\leadsto \left(0.011111111111111112 \cdot \left({\left(\left(b + a\right) \cdot \left(\left(b - a\right) \cdot angle\right)\right)}^{1} \cdot \pi\right)\right) \cdot \color{blue}{1} \]

   8. Taylor expanded in angle around 0 65.6%

      \[\leadsto \color{blue}{\left(0.011111111111111112 \cdot \left(angle \cdot \left(\left(b - a\right) \cdot \left(\left(a + b\right) \cdot \pi\right)\right)\right)\right)} \cdot 1 \]
   9. Step-by-step derivation

      1. *-commutative 65.6%

         \[\leadsto \color{blue}{\left(\left(angle \cdot \left(\left(b - a\right) \cdot \left(\left(a + b\right) \cdot \pi\right)\right)\right) \cdot 0.011111111111111112\right)} \cdot 1 \]

      2. associate-*r* 80.5%

         \[\leadsto \left(\color{blue}{\left(\left(angle \cdot \left(b - a\right)\right) \cdot \left(\left(a + b\right) \cdot \pi\right)\right)} \cdot 0.011111111111111112\right) \cdot 1 \]

      3. *-commutative 80.5%

         \[\leadsto \left(\left(\color{blue}{\left(\left(b - a\right) \cdot angle\right)} \cdot \left(\left(a + b\right) \cdot \pi\right)\right) \cdot 0.011111111111111112\right) \cdot 1 \]

      4. associate-*l* 80.5%

         \[\leadsto \left(\color{blue}{\left(\left(\left(\left(b - a\right) \cdot angle\right) \cdot \left(a + b\right)\right) \cdot \pi\right)} \cdot 0.011111111111111112\right) \cdot 1 \]

      5. +-commutative 80.5%

         \[\leadsto \left(\left(\left(\left(\left(b - a\right) \cdot angle\right) \cdot \color{blue}{\left(b + a\right)}\right) \cdot \pi\right) \cdot 0.011111111111111112\right) \cdot 1 \]

      6. associate-*r* 80.5%

         \[\leadsto \left(\left(\color{blue}{\left(\left(b - a\right) \cdot \left(angle \cdot \left(b + a\right)\right)\right)} \cdot \pi\right) \cdot 0.011111111111111112\right) \cdot 1 \]

      7. associate-*l* 80.6%

         \[\leadsto \color{blue}{\left(\left(\left(b - a\right) \cdot \left(angle \cdot \left(b + a\right)\right)\right) \cdot \left(\pi \cdot 0.011111111111111112\right)\right)} \cdot 1 \]

      8. +-commutative 80.6%

         \[\leadsto \left(\left(\left(b - a\right) \cdot \left(angle \cdot \color{blue}{\left(a + b\right)}\right)\right) \cdot \left(\pi \cdot 0.011111111111111112\right)\right) \cdot 1 \]

   10. Simplified 80.6%

       \[\leadsto \color{blue}{\left(\left(\left(b - a\right) \cdot \left(angle \cdot \left(a + b\right)\right)\right) \cdot \left(\pi \cdot 0.011111111111111112\right)\right)} \cdot 1 \]

   if 1.00000000000000004e-10 < (/.f64 angle 180)

   1. Initial program 42.5%

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

      1. add-cube-cbrt 42.4%

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

      2. pow3 42.5%

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

      3. unpow2 42.5%

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

      4. unpow2 42.5%

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

   3. Applied egg-rr 42.5%

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

      1. clear-num 41.6%

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

      2. un-div-inv 45.6%

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

   5. Applied egg-rr 45.6%

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

4. Final simplification 69.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{angle}{180} \leq 10^{-10}:\\ \;\;\;\;\left(\left(b - a\right) \cdot \left(\left(b + a\right) \cdot angle\right)\right) \cdot \left(\pi \cdot 0.011111111111111112\right)\\ \mathbf{else}:\\ \;\;\;\;\cos \left(\pi \cdot \frac{angle}{180}\right) \cdot \left(\left(2 \cdot {\left(\sqrt[3]{b \cdot b - a \cdot a}\right)}^{3}\right) \cdot \sin \left(\frac{\pi}{\frac{180}{angle}}\right)\right)\\ \end{array} \]

Alternative 5: 65.5% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;{b}^{2} \leq 2 \cdot 10^{+285}:\\ \;\;\;\;2 \cdot \left(\left(b - a\right) \cdot \left(\left(b + a\right) \cdot \sin \left(angle \cdot \left(\pi \cdot 0.005555555555555556\right)\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\left(b - a\right) \cdot \left(\left(b + a\right) \cdot angle\right)\right) \cdot \left(\pi \cdot 0.011111111111111112\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (a b angle)
 :precision binary64
 (if (<= (pow b 2.0) 2e+285)
   (* 2.0 (* (- b a) (* (+ b a) (sin (* angle (* PI 0.005555555555555556))))))
   (* (* (- b a) (* (+ b a) angle)) (* PI 0.011111111111111112))))

C

double code(double a, double b, double angle) {
	double tmp;
	if (pow(b, 2.0) <= 2e+285) {
		tmp = 2.0 * ((b - a) * ((b + a) * sin((angle * (((double) M_PI) * 0.005555555555555556)))));
	} else {
		tmp = ((b - a) * ((b + a) * angle)) * (((double) M_PI) * 0.011111111111111112);
	}
	return tmp;
}

Java

public static double code(double a, double b, double angle) {
	double tmp;
	if (Math.pow(b, 2.0) <= 2e+285) {
		tmp = 2.0 * ((b - a) * ((b + a) * Math.sin((angle * (Math.PI * 0.005555555555555556)))));
	} else {
		tmp = ((b - a) * ((b + a) * angle)) * (Math.PI * 0.011111111111111112);
	}
	return tmp;
}

Python

def code(a, b, angle):
	tmp = 0
	if math.pow(b, 2.0) <= 2e+285:
		tmp = 2.0 * ((b - a) * ((b + a) * math.sin((angle * (math.pi * 0.005555555555555556)))))
	else:
		tmp = ((b - a) * ((b + a) * angle)) * (math.pi * 0.011111111111111112)
	return tmp

Julia

function code(a, b, angle)
	tmp = 0.0
	if ((b ^ 2.0) <= 2e+285)
		tmp = Float64(2.0 * Float64(Float64(b - a) * Float64(Float64(b + a) * sin(Float64(angle * Float64(pi * 0.005555555555555556))))));
	else
		tmp = Float64(Float64(Float64(b - a) * Float64(Float64(b + a) * angle)) * Float64(pi * 0.011111111111111112));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, angle)
	tmp = 0.0;
	if ((b ^ 2.0) <= 2e+285)
		tmp = 2.0 * ((b - a) * ((b + a) * sin((angle * (pi * 0.005555555555555556)))));
	else
		tmp = ((b - a) * ((b + a) * angle)) * (pi * 0.011111111111111112);
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, angle_] := If[LessEqual[N[Power[b, 2.0], $MachinePrecision], 2e+285], N[(2.0 * N[(N[(b - a), $MachinePrecision] * N[(N[(b + a), $MachinePrecision] * N[Sin[N[(angle * N[(Pi * 0.005555555555555556), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(b - a), $MachinePrecision] * N[(N[(b + a), $MachinePrecision] * angle), $MachinePrecision]), $MachinePrecision] * N[(Pi * 0.011111111111111112), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (pow.f64 b 2) < 2e285

   1. Initial program 58.3%

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

      1. add-cube-cbrt 58.0%

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

      2. pow3 58.0%

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

      3. unpow2 58.0%

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

      4. unpow2 58.0%

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

   3. Applied egg-rr 58.0%

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

   4. Taylor expanded in angle around 0 54.6%

      \[\leadsto \left(\left(2 \cdot {\left(\sqrt[3]{b \cdot b - a \cdot a}\right)}^{3}\right) \cdot \sin \left(\pi \cdot \frac{angle}{180}\right)\right) \cdot \color{blue}{1} \]
   5. Step-by-step derivation

      1. add-sqr-sqrt 35.0%

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

      2. sqrt-unprod 35.5%

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

      3. pow2 35.5%

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

   6. Applied egg-rr 35.7%

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

   7. Taylor expanded in angle around inf 54.8%

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

      1. unpow2 54.8%

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

      2. unpow2 54.8%

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

      3. difference-of-squares 54.8%

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

      4. *-commutative 54.8%

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

      5. associate-*l* 64.1%

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

      6. +-commutative 64.1%

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

      7. *-commutative 64.1%

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

      8. associate-*l* 64.0%

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

   9. Simplified 64.0%

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

   if 2e285 < (pow.f64 b 2)

   1. Initial program 47.1%

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

   2. Taylor expanded in angle around 0 36.4%

      \[\leadsto \color{blue}{\left(0.011111111111111112 \cdot \left(angle \cdot \left(\left({b}^{2} - {a}^{2}\right) \cdot \pi\right)\right)\right)} \cdot \cos \left(\pi \cdot \frac{angle}{180}\right) \]
   3. Step-by-step derivation

      1. associate-*r* 36.4%

         \[\leadsto \left(0.011111111111111112 \cdot \color{blue}{\left(\left(angle \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \pi\right)}\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right) \]

      2. unpow2 36.4%

         \[\leadsto \left(0.011111111111111112 \cdot \left(\left(angle \cdot \left(\color{blue}{b \cdot b} - {a}^{2}\right)\right) \cdot \pi\right)\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right) \]

      3. unpow2 36.4%

         \[\leadsto \left(0.011111111111111112 \cdot \left(\left(angle \cdot \left(b \cdot b - \color{blue}{a \cdot a}\right)\right) \cdot \pi\right)\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right) \]

   4. Simplified 36.4%

      \[\leadsto \color{blue}{\left(0.011111111111111112 \cdot \left(\left(angle \cdot \left(b \cdot b - a \cdot a\right)\right) \cdot \pi\right)\right)} \cdot \cos \left(\pi \cdot \frac{angle}{180}\right) \]
   5. Step-by-step derivation

      1. pow1 36.4%

         \[\leadsto \left(0.011111111111111112 \cdot \left(\color{blue}{{\left(angle \cdot \left(b \cdot b - a \cdot a\right)\right)}^{1}} \cdot \pi\right)\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right) \]

      2. *-commutative 36.4%

         \[\leadsto \left(0.011111111111111112 \cdot \left({\color{blue}{\left(\left(b \cdot b - a \cdot a\right) \cdot angle\right)}}^{1} \cdot \pi\right)\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right) \]

      3. difference-of-squares 45.7%

         \[\leadsto \left(0.011111111111111112 \cdot \left({\left(\color{blue}{\left(\left(b + a\right) \cdot \left(b - a\right)\right)} \cdot angle\right)}^{1} \cdot \pi\right)\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right) \]

      4. associate-*l* 62.4%

         \[\leadsto \left(0.011111111111111112 \cdot \left({\color{blue}{\left(\left(b + a\right) \cdot \left(\left(b - a\right) \cdot angle\right)\right)}}^{1} \cdot \pi\right)\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right) \]

   6. Applied egg-rr 62.4%

      \[\leadsto \left(0.011111111111111112 \cdot \left(\color{blue}{{\left(\left(b + a\right) \cdot \left(\left(b - a\right) \cdot angle\right)\right)}^{1}} \cdot \pi\right)\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right) \]

   7. Taylor expanded in angle around 0 82.1%

      \[\leadsto \left(0.011111111111111112 \cdot \left({\left(\left(b + a\right) \cdot \left(\left(b - a\right) \cdot angle\right)\right)}^{1} \cdot \pi\right)\right) \cdot \color{blue}{1} \]

   8. Taylor expanded in angle around 0 65.3%

      \[\leadsto \color{blue}{\left(0.011111111111111112 \cdot \left(angle \cdot \left(\left(b - a\right) \cdot \left(\left(a + b\right) \cdot \pi\right)\right)\right)\right)} \cdot 1 \]
   9. Step-by-step derivation

      1. *-commutative 65.3%

         \[\leadsto \color{blue}{\left(\left(angle \cdot \left(\left(b - a\right) \cdot \left(\left(a + b\right) \cdot \pi\right)\right)\right) \cdot 0.011111111111111112\right)} \cdot 1 \]

      2. associate-*r* 82.1%

         \[\leadsto \left(\color{blue}{\left(\left(angle \cdot \left(b - a\right)\right) \cdot \left(\left(a + b\right) \cdot \pi\right)\right)} \cdot 0.011111111111111112\right) \cdot 1 \]

      3. *-commutative 82.1%

         \[\leadsto \left(\left(\color{blue}{\left(\left(b - a\right) \cdot angle\right)} \cdot \left(\left(a + b\right) \cdot \pi\right)\right) \cdot 0.011111111111111112\right) \cdot 1 \]

      4. associate-*l* 82.1%

         \[\leadsto \left(\color{blue}{\left(\left(\left(\left(b - a\right) \cdot angle\right) \cdot \left(a + b\right)\right) \cdot \pi\right)} \cdot 0.011111111111111112\right) \cdot 1 \]

      5. +-commutative 82.1%

         \[\leadsto \left(\left(\left(\left(\left(b - a\right) \cdot angle\right) \cdot \color{blue}{\left(b + a\right)}\right) \cdot \pi\right) \cdot 0.011111111111111112\right) \cdot 1 \]

      6. associate-*r* 82.1%

         \[\leadsto \left(\left(\color{blue}{\left(\left(b - a\right) \cdot \left(angle \cdot \left(b + a\right)\right)\right)} \cdot \pi\right) \cdot 0.011111111111111112\right) \cdot 1 \]

      7. associate-*l* 82.1%

         \[\leadsto \color{blue}{\left(\left(\left(b - a\right) \cdot \left(angle \cdot \left(b + a\right)\right)\right) \cdot \left(\pi \cdot 0.011111111111111112\right)\right)} \cdot 1 \]

      8. +-commutative 82.1%

         \[\leadsto \left(\left(\left(b - a\right) \cdot \left(angle \cdot \color{blue}{\left(a + b\right)}\right)\right) \cdot \left(\pi \cdot 0.011111111111111112\right)\right) \cdot 1 \]

   10. Simplified 82.1%

       \[\leadsto \color{blue}{\left(\left(\left(b - a\right) \cdot \left(angle \cdot \left(a + b\right)\right)\right) \cdot \left(\pi \cdot 0.011111111111111112\right)\right)} \cdot 1 \]
3. Recombined 2 regimes into one program.

4. Final simplification 68.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;{b}^{2} \leq 2 \cdot 10^{+285}:\\ \;\;\;\;2 \cdot \left(\left(b - a\right) \cdot \left(\left(b + a\right) \cdot \sin \left(angle \cdot \left(\pi \cdot 0.005555555555555556\right)\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\left(b - a\right) \cdot \left(\left(b + a\right) \cdot angle\right)\right) \cdot \left(\pi \cdot 0.011111111111111112\right)\\ \end{array} \]

Alternative 6: 62.0% accurate, 2.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;angle \leq 2.1 \cdot 10^{+99}:\\ \;\;\;\;\left(\left(b - a\right) \cdot \left(\left(b + a\right) \cdot angle\right)\right) \cdot \left(\pi \cdot 0.011111111111111112\right)\\ \mathbf{else}:\\ \;\;\;\;\sin \left(\pi \cdot \left(angle \cdot 0.005555555555555556\right)\right) \cdot \left(\left(a \cdot a\right) \cdot -2\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (a b angle)
 :precision binary64
 (if (<= angle 2.1e+99)
   (* (* (- b a) (* (+ b a) angle)) (* PI 0.011111111111111112))
   (* (sin (* PI (* angle 0.005555555555555556))) (* (* a a) -2.0))))

C

double code(double a, double b, double angle) {
	double tmp;
	if (angle <= 2.1e+99) {
		tmp = ((b - a) * ((b + a) * angle)) * (((double) M_PI) * 0.011111111111111112);
	} else {
		tmp = sin((((double) M_PI) * (angle * 0.005555555555555556))) * ((a * a) * -2.0);
	}
	return tmp;
}

Java

public static double code(double a, double b, double angle) {
	double tmp;
	if (angle <= 2.1e+99) {
		tmp = ((b - a) * ((b + a) * angle)) * (Math.PI * 0.011111111111111112);
	} else {
		tmp = Math.sin((Math.PI * (angle * 0.005555555555555556))) * ((a * a) * -2.0);
	}
	return tmp;
}

Python

def code(a, b, angle):
	tmp = 0
	if angle <= 2.1e+99:
		tmp = ((b - a) * ((b + a) * angle)) * (math.pi * 0.011111111111111112)
	else:
		tmp = math.sin((math.pi * (angle * 0.005555555555555556))) * ((a * a) * -2.0)
	return tmp

Julia

function code(a, b, angle)
	tmp = 0.0
	if (angle <= 2.1e+99)
		tmp = Float64(Float64(Float64(b - a) * Float64(Float64(b + a) * angle)) * Float64(pi * 0.011111111111111112));
	else
		tmp = Float64(sin(Float64(pi * Float64(angle * 0.005555555555555556))) * Float64(Float64(a * a) * -2.0));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, angle)
	tmp = 0.0;
	if (angle <= 2.1e+99)
		tmp = ((b - a) * ((b + a) * angle)) * (pi * 0.011111111111111112);
	else
		tmp = sin((pi * (angle * 0.005555555555555556))) * ((a * a) * -2.0);
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, angle_] := If[LessEqual[angle, 2.1e+99], N[(N[(N[(b - a), $MachinePrecision] * N[(N[(b + a), $MachinePrecision] * angle), $MachinePrecision]), $MachinePrecision] * N[(Pi * 0.011111111111111112), $MachinePrecision]), $MachinePrecision], N[(N[Sin[N[(Pi * N[(angle * 0.005555555555555556), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * N[(N[(a * a), $MachinePrecision] * -2.0), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if angle < 2.1000000000000001e99

   1. Initial program 58.7%

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

   2. Taylor expanded in angle around 0 51.1%

      \[\leadsto \color{blue}{\left(0.011111111111111112 \cdot \left(angle \cdot \left(\left({b}^{2} - {a}^{2}\right) \cdot \pi\right)\right)\right)} \cdot \cos \left(\pi \cdot \frac{angle}{180}\right) \]
   3. Step-by-step derivation

      1. associate-*r* 51.2%

         \[\leadsto \left(0.011111111111111112 \cdot \color{blue}{\left(\left(angle \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \pi\right)}\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right) \]

      2. unpow2 51.2%

         \[\leadsto \left(0.011111111111111112 \cdot \left(\left(angle \cdot \left(\color{blue}{b \cdot b} - {a}^{2}\right)\right) \cdot \pi\right)\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right) \]

      3. unpow2 51.2%

         \[\leadsto \left(0.011111111111111112 \cdot \left(\left(angle \cdot \left(b \cdot b - \color{blue}{a \cdot a}\right)\right) \cdot \pi\right)\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right) \]

   4. Simplified 51.2%

      \[\leadsto \color{blue}{\left(0.011111111111111112 \cdot \left(\left(angle \cdot \left(b \cdot b - a \cdot a\right)\right) \cdot \pi\right)\right)} \cdot \cos \left(\pi \cdot \frac{angle}{180}\right) \]
   5. Step-by-step derivation

      1. pow1 51.2%

         \[\leadsto \left(0.011111111111111112 \cdot \left(\color{blue}{{\left(angle \cdot \left(b \cdot b - a \cdot a\right)\right)}^{1}} \cdot \pi\right)\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right) \]

      2. *-commutative 51.2%

         \[\leadsto \left(0.011111111111111112 \cdot \left({\color{blue}{\left(\left(b \cdot b - a \cdot a\right) \cdot angle\right)}}^{1} \cdot \pi\right)\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right) \]

      3. difference-of-squares 53.2%

         \[\leadsto \left(0.011111111111111112 \cdot \left({\left(\color{blue}{\left(\left(b + a\right) \cdot \left(b - a\right)\right)} \cdot angle\right)}^{1} \cdot \pi\right)\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right) \]

      4. associate-*l* 65.6%

         \[\leadsto \left(0.011111111111111112 \cdot \left({\color{blue}{\left(\left(b + a\right) \cdot \left(\left(b - a\right) \cdot angle\right)\right)}}^{1} \cdot \pi\right)\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right) \]

   6. Applied egg-rr 65.6%

      \[\leadsto \left(0.011111111111111112 \cdot \left(\color{blue}{{\left(\left(b + a\right) \cdot \left(\left(b - a\right) \cdot angle\right)\right)}^{1}} \cdot \pi\right)\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right) \]

   7. Taylor expanded in angle around 0 72.9%

      \[\leadsto \left(0.011111111111111112 \cdot \left({\left(\left(b + a\right) \cdot \left(\left(b - a\right) \cdot angle\right)\right)}^{1} \cdot \pi\right)\right) \cdot \color{blue}{1} \]

   8. Taylor expanded in angle around 0 60.4%

      \[\leadsto \color{blue}{\left(0.011111111111111112 \cdot \left(angle \cdot \left(\left(b - a\right) \cdot \left(\left(a + b\right) \cdot \pi\right)\right)\right)\right)} \cdot 1 \]
   9. Step-by-step derivation

      1. *-commutative 60.4%

         \[\leadsto \color{blue}{\left(\left(angle \cdot \left(\left(b - a\right) \cdot \left(\left(a + b\right) \cdot \pi\right)\right)\right) \cdot 0.011111111111111112\right)} \cdot 1 \]

      2. associate-*r* 72.9%

         \[\leadsto \left(\color{blue}{\left(\left(angle \cdot \left(b - a\right)\right) \cdot \left(\left(a + b\right) \cdot \pi\right)\right)} \cdot 0.011111111111111112\right) \cdot 1 \]

      3. *-commutative 72.9%

         \[\leadsto \left(\left(\color{blue}{\left(\left(b - a\right) \cdot angle\right)} \cdot \left(\left(a + b\right) \cdot \pi\right)\right) \cdot 0.011111111111111112\right) \cdot 1 \]

      4. associate-*l* 72.9%

         \[\leadsto \left(\color{blue}{\left(\left(\left(\left(b - a\right) \cdot angle\right) \cdot \left(a + b\right)\right) \cdot \pi\right)} \cdot 0.011111111111111112\right) \cdot 1 \]

      5. +-commutative 72.9%

         \[\leadsto \left(\left(\left(\left(\left(b - a\right) \cdot angle\right) \cdot \color{blue}{\left(b + a\right)}\right) \cdot \pi\right) \cdot 0.011111111111111112\right) \cdot 1 \]

      6. associate-*r* 72.9%

         \[\leadsto \left(\left(\color{blue}{\left(\left(b - a\right) \cdot \left(angle \cdot \left(b + a\right)\right)\right)} \cdot \pi\right) \cdot 0.011111111111111112\right) \cdot 1 \]

      7. associate-*l* 73.0%

         \[\leadsto \color{blue}{\left(\left(\left(b - a\right) \cdot \left(angle \cdot \left(b + a\right)\right)\right) \cdot \left(\pi \cdot 0.011111111111111112\right)\right)} \cdot 1 \]

      8. +-commutative 73.0%

         \[\leadsto \left(\left(\left(b - a\right) \cdot \left(angle \cdot \color{blue}{\left(a + b\right)}\right)\right) \cdot \left(\pi \cdot 0.011111111111111112\right)\right) \cdot 1 \]

   10. Simplified 73.0%

       \[\leadsto \color{blue}{\left(\left(\left(b - a\right) \cdot \left(angle \cdot \left(a + b\right)\right)\right) \cdot \left(\pi \cdot 0.011111111111111112\right)\right)} \cdot 1 \]

   if 2.1000000000000001e99 < angle

   1. Initial program 42.9%

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

      1. add-cube-cbrt 42.9%

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

      2. pow3 42.9%

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

      3. unpow2 42.9%

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

      4. unpow2 42.9%

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

   3. Applied egg-rr 42.9%

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

   4. Taylor expanded in angle around 0 43.1%

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

   5. Taylor expanded in b around 0 31.3%

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

      1. unpow2 31.3%

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

      2. associate-*r* 31.3%

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

      3. associate-*r* 32.0%

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

      4. *-commutative 32.0%

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

      5. associate-*r* 29.5%

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

      6. *-commutative 29.5%

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

      7. associate-*r* 32.0%

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

      8. *-commutative 32.0%

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

      9. associate-*r* 31.3%

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

      10. *-commutative 31.3%

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

      11. *-commutative 31.3%

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

      12. associate-*l* 32.0%

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

   7. Simplified 32.0%

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

4. Final simplification 65.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;angle \leq 2.1 \cdot 10^{+99}:\\ \;\;\;\;\left(\left(b - a\right) \cdot \left(\left(b + a\right) \cdot angle\right)\right) \cdot \left(\pi \cdot 0.011111111111111112\right)\\ \mathbf{else}:\\ \;\;\;\;\sin \left(\pi \cdot \left(angle \cdot 0.005555555555555556\right)\right) \cdot \left(\left(a \cdot a\right) \cdot -2\right)\\ \end{array} \]

Alternative 7: 62.9% accurate, 5.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;angle \leq 5 \cdot 10^{+87}:\\ \;\;\;\;\left(\left(b - a\right) \cdot \left(\left(b + a\right) \cdot angle\right)\right) \cdot \left(\pi \cdot 0.011111111111111112\right)\\ \mathbf{else}:\\ \;\;\;\;0.011111111111111112 \cdot \left(angle \cdot \left(\left(b - a\right) \cdot \left(\left(b + a\right) \cdot \pi\right)\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (a b angle)
 :precision binary64
 (if (<= angle 5e+87)
   (* (* (- b a) (* (+ b a) angle)) (* PI 0.011111111111111112))
   (* 0.011111111111111112 (* angle (* (- b a) (* (+ b a) PI))))))

C

double code(double a, double b, double angle) {
	double tmp;
	if (angle <= 5e+87) {
		tmp = ((b - a) * ((b + a) * angle)) * (((double) M_PI) * 0.011111111111111112);
	} else {
		tmp = 0.011111111111111112 * (angle * ((b - a) * ((b + a) * ((double) M_PI))));
	}
	return tmp;
}

Java

public static double code(double a, double b, double angle) {
	double tmp;
	if (angle <= 5e+87) {
		tmp = ((b - a) * ((b + a) * angle)) * (Math.PI * 0.011111111111111112);
	} else {
		tmp = 0.011111111111111112 * (angle * ((b - a) * ((b + a) * Math.PI)));
	}
	return tmp;
}

Python

def code(a, b, angle):
	tmp = 0
	if angle <= 5e+87:
		tmp = ((b - a) * ((b + a) * angle)) * (math.pi * 0.011111111111111112)
	else:
		tmp = 0.011111111111111112 * (angle * ((b - a) * ((b + a) * math.pi)))
	return tmp

Julia

function code(a, b, angle)
	tmp = 0.0
	if (angle <= 5e+87)
		tmp = Float64(Float64(Float64(b - a) * Float64(Float64(b + a) * angle)) * Float64(pi * 0.011111111111111112));
	else
		tmp = Float64(0.011111111111111112 * Float64(angle * Float64(Float64(b - a) * Float64(Float64(b + a) * pi))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, angle)
	tmp = 0.0;
	if (angle <= 5e+87)
		tmp = ((b - a) * ((b + a) * angle)) * (pi * 0.011111111111111112);
	else
		tmp = 0.011111111111111112 * (angle * ((b - a) * ((b + a) * pi)));
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, angle_] := If[LessEqual[angle, 5e+87], N[(N[(N[(b - a), $MachinePrecision] * N[(N[(b + a), $MachinePrecision] * angle), $MachinePrecision]), $MachinePrecision] * N[(Pi * 0.011111111111111112), $MachinePrecision]), $MachinePrecision], N[(0.011111111111111112 * N[(angle * N[(N[(b - a), $MachinePrecision] * N[(N[(b + a), $MachinePrecision] * Pi), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if angle < 4.9999999999999998e87

   1. Initial program 59.2%

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

   2. Taylor expanded in angle around 0 51.6%

      \[\leadsto \color{blue}{\left(0.011111111111111112 \cdot \left(angle \cdot \left(\left({b}^{2} - {a}^{2}\right) \cdot \pi\right)\right)\right)} \cdot \cos \left(\pi \cdot \frac{angle}{180}\right) \]
   3. Step-by-step derivation

      1. associate-*r* 51.6%

         \[\leadsto \left(0.011111111111111112 \cdot \color{blue}{\left(\left(angle \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \pi\right)}\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right) \]

      2. unpow2 51.6%

         \[\leadsto \left(0.011111111111111112 \cdot \left(\left(angle \cdot \left(\color{blue}{b \cdot b} - {a}^{2}\right)\right) \cdot \pi\right)\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right) \]

      3. unpow2 51.6%

         \[\leadsto \left(0.011111111111111112 \cdot \left(\left(angle \cdot \left(b \cdot b - \color{blue}{a \cdot a}\right)\right) \cdot \pi\right)\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right) \]

   4. Simplified 51.6%

      \[\leadsto \color{blue}{\left(0.011111111111111112 \cdot \left(\left(angle \cdot \left(b \cdot b - a \cdot a\right)\right) \cdot \pi\right)\right)} \cdot \cos \left(\pi \cdot \frac{angle}{180}\right) \]
   5. Step-by-step derivation

      1. pow1 51.6%

         \[\leadsto \left(0.011111111111111112 \cdot \left(\color{blue}{{\left(angle \cdot \left(b \cdot b - a \cdot a\right)\right)}^{1}} \cdot \pi\right)\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right) \]

      2. *-commutative 51.6%

         \[\leadsto \left(0.011111111111111112 \cdot \left({\color{blue}{\left(\left(b \cdot b - a \cdot a\right) \cdot angle\right)}}^{1} \cdot \pi\right)\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right) \]

      3. difference-of-squares 53.7%

         \[\leadsto \left(0.011111111111111112 \cdot \left({\left(\color{blue}{\left(\left(b + a\right) \cdot \left(b - a\right)\right)} \cdot angle\right)}^{1} \cdot \pi\right)\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right) \]

      4. associate-*l* 66.4%

         \[\leadsto \left(0.011111111111111112 \cdot \left({\color{blue}{\left(\left(b + a\right) \cdot \left(\left(b - a\right) \cdot angle\right)\right)}}^{1} \cdot \pi\right)\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right) \]

   6. Applied egg-rr 66.4%

      \[\leadsto \left(0.011111111111111112 \cdot \left(\color{blue}{{\left(\left(b + a\right) \cdot \left(\left(b - a\right) \cdot angle\right)\right)}^{1}} \cdot \pi\right)\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right) \]

   7. Taylor expanded in angle around 0 73.3%

      \[\leadsto \left(0.011111111111111112 \cdot \left({\left(\left(b + a\right) \cdot \left(\left(b - a\right) \cdot angle\right)\right)}^{1} \cdot \pi\right)\right) \cdot \color{blue}{1} \]

   8. Taylor expanded in angle around 0 60.6%

      \[\leadsto \color{blue}{\left(0.011111111111111112 \cdot \left(angle \cdot \left(\left(b - a\right) \cdot \left(\left(a + b\right) \cdot \pi\right)\right)\right)\right)} \cdot 1 \]
   9. Step-by-step derivation

      1. *-commutative 60.6%

         \[\leadsto \color{blue}{\left(\left(angle \cdot \left(\left(b - a\right) \cdot \left(\left(a + b\right) \cdot \pi\right)\right)\right) \cdot 0.011111111111111112\right)} \cdot 1 \]

      2. associate-*r* 73.3%

         \[\leadsto \left(\color{blue}{\left(\left(angle \cdot \left(b - a\right)\right) \cdot \left(\left(a + b\right) \cdot \pi\right)\right)} \cdot 0.011111111111111112\right) \cdot 1 \]

      3. *-commutative 73.3%

         \[\leadsto \left(\left(\color{blue}{\left(\left(b - a\right) \cdot angle\right)} \cdot \left(\left(a + b\right) \cdot \pi\right)\right) \cdot 0.011111111111111112\right) \cdot 1 \]

      4. associate-*l* 73.3%

         \[\leadsto \left(\color{blue}{\left(\left(\left(\left(b - a\right) \cdot angle\right) \cdot \left(a + b\right)\right) \cdot \pi\right)} \cdot 0.011111111111111112\right) \cdot 1 \]

      5. +-commutative 73.3%

         \[\leadsto \left(\left(\left(\left(\left(b - a\right) \cdot angle\right) \cdot \color{blue}{\left(b + a\right)}\right) \cdot \pi\right) \cdot 0.011111111111111112\right) \cdot 1 \]

      6. associate-*r* 73.3%

         \[\leadsto \left(\left(\color{blue}{\left(\left(b - a\right) \cdot \left(angle \cdot \left(b + a\right)\right)\right)} \cdot \pi\right) \cdot 0.011111111111111112\right) \cdot 1 \]

      7. associate-*l* 73.4%

         \[\leadsto \color{blue}{\left(\left(\left(b - a\right) \cdot \left(angle \cdot \left(b + a\right)\right)\right) \cdot \left(\pi \cdot 0.011111111111111112\right)\right)} \cdot 1 \]

      8. +-commutative 73.4%

         \[\leadsto \left(\left(\left(b - a\right) \cdot \left(angle \cdot \color{blue}{\left(a + b\right)}\right)\right) \cdot \left(\pi \cdot 0.011111111111111112\right)\right) \cdot 1 \]

   10. Simplified 73.4%

       \[\leadsto \color{blue}{\left(\left(\left(b - a\right) \cdot \left(angle \cdot \left(a + b\right)\right)\right) \cdot \left(\pi \cdot 0.011111111111111112\right)\right)} \cdot 1 \]

   if 4.9999999999999998e87 < angle

   1. Initial program 42.0%

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

   2. Taylor expanded in angle around 0 29.3%

      \[\leadsto \color{blue}{\left(0.011111111111111112 \cdot \left(angle \cdot \left(\left({b}^{2} - {a}^{2}\right) \cdot \pi\right)\right)\right)} \cdot \cos \left(\pi \cdot \frac{angle}{180}\right) \]
   3. Step-by-step derivation

      1. associate-*r* 29.3%

         \[\leadsto \left(0.011111111111111112 \cdot \color{blue}{\left(\left(angle \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \pi\right)}\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right) \]

      2. unpow2 29.3%

         \[\leadsto \left(0.011111111111111112 \cdot \left(\left(angle \cdot \left(\color{blue}{b \cdot b} - {a}^{2}\right)\right) \cdot \pi\right)\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right) \]

      3. unpow2 29.3%

         \[\leadsto \left(0.011111111111111112 \cdot \left(\left(angle \cdot \left(b \cdot b - \color{blue}{a \cdot a}\right)\right) \cdot \pi\right)\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right) \]

   4. Simplified 29.3%

      \[\leadsto \color{blue}{\left(0.011111111111111112 \cdot \left(\left(angle \cdot \left(b \cdot b - a \cdot a\right)\right) \cdot \pi\right)\right)} \cdot \cos \left(\pi \cdot \frac{angle}{180}\right) \]
   5. Step-by-step derivation

      1. pow1 29.3%

         \[\leadsto \left(0.011111111111111112 \cdot \left(\color{blue}{{\left(angle \cdot \left(b \cdot b - a \cdot a\right)\right)}^{1}} \cdot \pi\right)\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right) \]

      2. *-commutative 29.3%

         \[\leadsto \left(0.011111111111111112 \cdot \left({\color{blue}{\left(\left(b \cdot b - a \cdot a\right) \cdot angle\right)}}^{1} \cdot \pi\right)\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right) \]

      3. difference-of-squares 31.3%

         \[\leadsto \left(0.011111111111111112 \cdot \left({\left(\color{blue}{\left(\left(b + a\right) \cdot \left(b - a\right)\right)} \cdot angle\right)}^{1} \cdot \pi\right)\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right) \]

      4. associate-*l* 21.8%

         \[\leadsto \left(0.011111111111111112 \cdot \left({\color{blue}{\left(\left(b + a\right) \cdot \left(\left(b - a\right) \cdot angle\right)\right)}}^{1} \cdot \pi\right)\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right) \]

   6. Applied egg-rr 21.8%

      \[\leadsto \left(0.011111111111111112 \cdot \left(\color{blue}{{\left(\left(b + a\right) \cdot \left(\left(b - a\right) \cdot angle\right)\right)}^{1}} \cdot \pi\right)\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right) \]

   7. Taylor expanded in angle around 0 21.9%

      \[\leadsto \left(0.011111111111111112 \cdot \left({\left(\left(b + a\right) \cdot \left(\left(b - a\right) \cdot angle\right)\right)}^{1} \cdot \pi\right)\right) \cdot \color{blue}{1} \]

   8. Taylor expanded in angle around 0 31.4%

      \[\leadsto \color{blue}{\left(0.011111111111111112 \cdot \left(angle \cdot \left(\left(b - a\right) \cdot \left(\left(a + b\right) \cdot \pi\right)\right)\right)\right)} \cdot 1 \]
3. Recombined 2 regimes into one program.

4. Final simplification 65.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;angle \leq 5 \cdot 10^{+87}:\\ \;\;\;\;\left(\left(b - a\right) \cdot \left(\left(b + a\right) \cdot angle\right)\right) \cdot \left(\pi \cdot 0.011111111111111112\right)\\ \mathbf{else}:\\ \;\;\;\;0.011111111111111112 \cdot \left(angle \cdot \left(\left(b - a\right) \cdot \left(\left(b + a\right) \cdot \pi\right)\right)\right)\\ \end{array} \]

Alternative 8: 46.4% accurate, 5.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq -9 \cdot 10^{-17} \lor \neg \left(b \leq 1.25 \cdot 10^{+100}\right):\\ \;\;\;\;0.011111111111111112 \cdot \left(\pi \cdot \left(angle \cdot \left(b \cdot b\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;-0.011111111111111112 \cdot \left(\pi \cdot \left(angle \cdot \left(a \cdot a\right)\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (a b angle)
 :precision binary64
 (if (or (<= b -9e-17) (not (<= b 1.25e+100)))
   (* 0.011111111111111112 (* PI (* angle (* b b))))
   (* -0.011111111111111112 (* PI (* angle (* a a))))))

C

double code(double a, double b, double angle) {
	double tmp;
	if ((b <= -9e-17) || !(b <= 1.25e+100)) {
		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 ((b <= -9e-17) || !(b <= 1.25e+100)) {
		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 (b <= -9e-17) or not (b <= 1.25e+100):
		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 ((b <= -9e-17) || !(b <= 1.25e+100))
		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 ((b <= -9e-17) || ~((b <= 1.25e+100)))
		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[Or[LessEqual[b, -9e-17], N[Not[LessEqual[b, 1.25e+100]], $MachinePrecision]], 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 b < -8.99999999999999957e-17 or 1.25e100 < b

   1. Initial program 50.3%

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

   2. Taylor expanded in angle around 0 39.6%

      \[\leadsto \color{blue}{\left(0.011111111111111112 \cdot \left(angle \cdot \left(\left({b}^{2} - {a}^{2}\right) \cdot \pi\right)\right)\right)} \cdot \cos \left(\pi \cdot \frac{angle}{180}\right) \]
   3. Step-by-step derivation

      1. associate-*r* 39.6%

         \[\leadsto \left(0.011111111111111112 \cdot \color{blue}{\left(\left(angle \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \pi\right)}\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right) \]

      2. unpow2 39.6%

         \[\leadsto \left(0.011111111111111112 \cdot \left(\left(angle \cdot \left(\color{blue}{b \cdot b} - {a}^{2}\right)\right) \cdot \pi\right)\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right) \]

      3. unpow2 39.6%

         \[\leadsto \left(0.011111111111111112 \cdot \left(\left(angle \cdot \left(b \cdot b - \color{blue}{a \cdot a}\right)\right) \cdot \pi\right)\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right) \]

   4. Simplified 39.6%

      \[\leadsto \color{blue}{\left(0.011111111111111112 \cdot \left(\left(angle \cdot \left(b \cdot b - a \cdot a\right)\right) \cdot \pi\right)\right)} \cdot \cos \left(\pi \cdot \frac{angle}{180}\right) \]

   5. Taylor expanded in angle around 0 48.7%

      \[\leadsto \left(0.011111111111111112 \cdot \left(\left(angle \cdot \left(b \cdot b - a \cdot a\right)\right) \cdot \pi\right)\right) \cdot \color{blue}{1} \]

   6. Taylor expanded in b around inf 45.0%

      \[\leadsto \left(0.011111111111111112 \cdot \left(\color{blue}{\left(angle \cdot {b}^{2}\right)} \cdot \pi\right)\right) \cdot 1 \]
   7. Step-by-step derivation

      1. unpow2 45.0%

         \[\leadsto \left(0.011111111111111112 \cdot \left(\left(angle \cdot \color{blue}{\left(b \cdot b\right)}\right) \cdot \pi\right)\right) \cdot 1 \]

   8. Simplified 45.0%

      \[\leadsto \left(0.011111111111111112 \cdot \left(\color{blue}{\left(angle \cdot \left(b \cdot b\right)\right)} \cdot \pi\right)\right) \cdot 1 \]

   if -8.99999999999999957e-17 < b < 1.25e100

   1. Initial program 60.1%

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

   2. Taylor expanded in angle around 0 53.1%

      \[\leadsto \color{blue}{\left(0.011111111111111112 \cdot \left(angle \cdot \left(\left({b}^{2} - {a}^{2}\right) \cdot \pi\right)\right)\right)} \cdot \cos \left(\pi \cdot \frac{angle}{180}\right) \]
   3. Step-by-step derivation

      1. associate-*r* 53.2%

         \[\leadsto \left(0.011111111111111112 \cdot \color{blue}{\left(\left(angle \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \pi\right)}\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right) \]

      2. unpow2 53.2%

         \[\leadsto \left(0.011111111111111112 \cdot \left(\left(angle \cdot \left(\color{blue}{b \cdot b} - {a}^{2}\right)\right) \cdot \pi\right)\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right) \]

      3. unpow2 53.2%

         \[\leadsto \left(0.011111111111111112 \cdot \left(\left(angle \cdot \left(b \cdot b - \color{blue}{a \cdot a}\right)\right) \cdot \pi\right)\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right) \]

   4. Simplified 53.2%

      \[\leadsto \color{blue}{\left(0.011111111111111112 \cdot \left(\left(angle \cdot \left(b \cdot b - a \cdot a\right)\right) \cdot \pi\right)\right)} \cdot \cos \left(\pi \cdot \frac{angle}{180}\right) \]
   5. Step-by-step derivation

      1. pow1 53.2%

         \[\leadsto \left(0.011111111111111112 \cdot \left(\color{blue}{{\left(angle \cdot \left(b \cdot b - a \cdot a\right)\right)}^{1}} \cdot \pi\right)\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right) \]

      2. *-commutative 53.2%

         \[\leadsto \left(0.011111111111111112 \cdot \left({\color{blue}{\left(\left(b \cdot b - a \cdot a\right) \cdot angle\right)}}^{1} \cdot \pi\right)\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right) \]

      3. difference-of-squares 53.2%

         \[\leadsto \left(0.011111111111111112 \cdot \left({\left(\color{blue}{\left(\left(b + a\right) \cdot \left(b - a\right)\right)} \cdot angle\right)}^{1} \cdot \pi\right)\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right) \]

      4. associate-*l* 58.9%

         \[\leadsto \left(0.011111111111111112 \cdot \left({\color{blue}{\left(\left(b + a\right) \cdot \left(\left(b - a\right) \cdot angle\right)\right)}}^{1} \cdot \pi\right)\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right) \]

   6. Applied egg-rr 58.9%

      \[\leadsto \left(0.011111111111111112 \cdot \left(\color{blue}{{\left(\left(b + a\right) \cdot \left(\left(b - a\right) \cdot angle\right)\right)}^{1}} \cdot \pi\right)\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right) \]

   7. Taylor expanded in angle around 0 60.4%

      \[\leadsto \left(0.011111111111111112 \cdot \left({\left(\left(b + a\right) \cdot \left(\left(b - a\right) \cdot angle\right)\right)}^{1} \cdot \pi\right)\right) \cdot \color{blue}{1} \]

   8. Taylor expanded in b around 0 51.4%

      \[\leadsto \color{blue}{\left(-0.011111111111111112 \cdot \left(angle \cdot \left({a}^{2} \cdot \pi\right)\right)\right)} \cdot 1 \]
   9. Step-by-step derivation

      1. associate-*r* 51.4%

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

      2. unpow2 51.4%

         \[\leadsto \left(-0.011111111111111112 \cdot \left(\left(angle \cdot \color{blue}{\left(a \cdot a\right)}\right) \cdot \pi\right)\right) \cdot 1 \]

   10. Simplified 51.4%

       \[\leadsto \color{blue}{\left(-0.011111111111111112 \cdot \left(\left(angle \cdot \left(a \cdot a\right)\right) \cdot \pi\right)\right)} \cdot 1 \]
3. Recombined 2 regimes into one program.

4. Final simplification 48.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -9 \cdot 10^{-17} \lor \neg \left(b \leq 1.25 \cdot 10^{+100}\right):\\ \;\;\;\;0.011111111111111112 \cdot \left(\pi \cdot \left(angle \cdot \left(b \cdot b\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;-0.011111111111111112 \cdot \left(\pi \cdot \left(angle \cdot \left(a \cdot a\right)\right)\right)\\ \end{array} \]

Alternative 9: 50.9% accurate, 5.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq -1.7 \cdot 10^{-16} \lor \neg \left(b \leq 1.15 \cdot 10^{+100}\right):\\ \;\;\;\;0.011111111111111112 \cdot \left(\pi \cdot \left(b \cdot \left(b \cdot angle\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;-0.011111111111111112 \cdot \left(\pi \cdot \left(angle \cdot \left(a \cdot a\right)\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (a b angle)
 :precision binary64
 (if (or (<= b -1.7e-16) (not (<= b 1.15e+100)))
   (* 0.011111111111111112 (* PI (* b (* b angle))))
   (* -0.011111111111111112 (* PI (* angle (* a a))))))

C

double code(double a, double b, double angle) {
	double tmp;
	if ((b <= -1.7e-16) || !(b <= 1.15e+100)) {
		tmp = 0.011111111111111112 * (((double) M_PI) * (b * (b * angle)));
	} 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 ((b <= -1.7e-16) || !(b <= 1.15e+100)) {
		tmp = 0.011111111111111112 * (Math.PI * (b * (b * angle)));
	} else {
		tmp = -0.011111111111111112 * (Math.PI * (angle * (a * a)));
	}
	return tmp;
}

Python

def code(a, b, angle):
	tmp = 0
	if (b <= -1.7e-16) or not (b <= 1.15e+100):
		tmp = 0.011111111111111112 * (math.pi * (b * (b * angle)))
	else:
		tmp = -0.011111111111111112 * (math.pi * (angle * (a * a)))
	return tmp

Julia

function code(a, b, angle)
	tmp = 0.0
	if ((b <= -1.7e-16) || !(b <= 1.15e+100))
		tmp = Float64(0.011111111111111112 * Float64(pi * Float64(b * Float64(b * angle))));
	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 ((b <= -1.7e-16) || ~((b <= 1.15e+100)))
		tmp = 0.011111111111111112 * (pi * (b * (b * angle)));
	else
		tmp = -0.011111111111111112 * (pi * (angle * (a * a)));
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, angle_] := If[Or[LessEqual[b, -1.7e-16], N[Not[LessEqual[b, 1.15e+100]], $MachinePrecision]], N[(0.011111111111111112 * N[(Pi * N[(b * N[(b * angle), $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 b < -1.7e-16 or 1.14999999999999995e100 < b

   1. Initial program 50.3%

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

   2. Taylor expanded in angle around 0 39.6%

      \[\leadsto \color{blue}{\left(0.011111111111111112 \cdot \left(angle \cdot \left(\left({b}^{2} - {a}^{2}\right) \cdot \pi\right)\right)\right)} \cdot \cos \left(\pi \cdot \frac{angle}{180}\right) \]
   3. Step-by-step derivation

      1. associate-*r* 39.6%

         \[\leadsto \left(0.011111111111111112 \cdot \color{blue}{\left(\left(angle \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \pi\right)}\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right) \]

      2. unpow2 39.6%

         \[\leadsto \left(0.011111111111111112 \cdot \left(\left(angle \cdot \left(\color{blue}{b \cdot b} - {a}^{2}\right)\right) \cdot \pi\right)\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right) \]

      3. unpow2 39.6%

         \[\leadsto \left(0.011111111111111112 \cdot \left(\left(angle \cdot \left(b \cdot b - \color{blue}{a \cdot a}\right)\right) \cdot \pi\right)\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right) \]

   4. Simplified 39.6%

      \[\leadsto \color{blue}{\left(0.011111111111111112 \cdot \left(\left(angle \cdot \left(b \cdot b - a \cdot a\right)\right) \cdot \pi\right)\right)} \cdot \cos \left(\pi \cdot \frac{angle}{180}\right) \]

   5. Taylor expanded in angle around 0 48.7%

      \[\leadsto \left(0.011111111111111112 \cdot \left(\left(angle \cdot \left(b \cdot b - a \cdot a\right)\right) \cdot \pi\right)\right) \cdot \color{blue}{1} \]

   6. Taylor expanded in b around inf 45.0%

      \[\leadsto \left(0.011111111111111112 \cdot \left(\color{blue}{\left(angle \cdot {b}^{2}\right)} \cdot \pi\right)\right) \cdot 1 \]
   7. Step-by-step derivation

      1. unpow2 45.0%

         \[\leadsto \left(0.011111111111111112 \cdot \left(\left(angle \cdot \color{blue}{\left(b \cdot b\right)}\right) \cdot \pi\right)\right) \cdot 1 \]

      2. *-commutative 45.0%

         \[\leadsto \left(0.011111111111111112 \cdot \left(\color{blue}{\left(\left(b \cdot b\right) \cdot angle\right)} \cdot \pi\right)\right) \cdot 1 \]

      3. associate-*l* 50.9%

         \[\leadsto \left(0.011111111111111112 \cdot \left(\color{blue}{\left(b \cdot \left(b \cdot angle\right)\right)} \cdot \pi\right)\right) \cdot 1 \]

   8. Simplified 50.9%

      \[\leadsto \left(0.011111111111111112 \cdot \left(\color{blue}{\left(b \cdot \left(b \cdot angle\right)\right)} \cdot \pi\right)\right) \cdot 1 \]

   if -1.7e-16 < b < 1.14999999999999995e100

   1. Initial program 60.1%

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

   2. Taylor expanded in angle around 0 53.1%

      \[\leadsto \color{blue}{\left(0.011111111111111112 \cdot \left(angle \cdot \left(\left({b}^{2} - {a}^{2}\right) \cdot \pi\right)\right)\right)} \cdot \cos \left(\pi \cdot \frac{angle}{180}\right) \]
   3. Step-by-step derivation

      1. associate-*r* 53.2%

         \[\leadsto \left(0.011111111111111112 \cdot \color{blue}{\left(\left(angle \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \pi\right)}\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right) \]

      2. unpow2 53.2%

         \[\leadsto \left(0.011111111111111112 \cdot \left(\left(angle \cdot \left(\color{blue}{b \cdot b} - {a}^{2}\right)\right) \cdot \pi\right)\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right) \]

      3. unpow2 53.2%

         \[\leadsto \left(0.011111111111111112 \cdot \left(\left(angle \cdot \left(b \cdot b - \color{blue}{a \cdot a}\right)\right) \cdot \pi\right)\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right) \]

   4. Simplified 53.2%

      \[\leadsto \color{blue}{\left(0.011111111111111112 \cdot \left(\left(angle \cdot \left(b \cdot b - a \cdot a\right)\right) \cdot \pi\right)\right)} \cdot \cos \left(\pi \cdot \frac{angle}{180}\right) \]
   5. Step-by-step derivation

      1. pow1 53.2%

         \[\leadsto \left(0.011111111111111112 \cdot \left(\color{blue}{{\left(angle \cdot \left(b \cdot b - a \cdot a\right)\right)}^{1}} \cdot \pi\right)\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right) \]

      2. *-commutative 53.2%

         \[\leadsto \left(0.011111111111111112 \cdot \left({\color{blue}{\left(\left(b \cdot b - a \cdot a\right) \cdot angle\right)}}^{1} \cdot \pi\right)\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right) \]

      3. difference-of-squares 53.2%

         \[\leadsto \left(0.011111111111111112 \cdot \left({\left(\color{blue}{\left(\left(b + a\right) \cdot \left(b - a\right)\right)} \cdot angle\right)}^{1} \cdot \pi\right)\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right) \]

      4. associate-*l* 58.9%

         \[\leadsto \left(0.011111111111111112 \cdot \left({\color{blue}{\left(\left(b + a\right) \cdot \left(\left(b - a\right) \cdot angle\right)\right)}}^{1} \cdot \pi\right)\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right) \]

   6. Applied egg-rr 58.9%

      \[\leadsto \left(0.011111111111111112 \cdot \left(\color{blue}{{\left(\left(b + a\right) \cdot \left(\left(b - a\right) \cdot angle\right)\right)}^{1}} \cdot \pi\right)\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right) \]

   7. Taylor expanded in angle around 0 60.4%

      \[\leadsto \left(0.011111111111111112 \cdot \left({\left(\left(b + a\right) \cdot \left(\left(b - a\right) \cdot angle\right)\right)}^{1} \cdot \pi\right)\right) \cdot \color{blue}{1} \]

   8. Taylor expanded in b around 0 51.4%

      \[\leadsto \color{blue}{\left(-0.011111111111111112 \cdot \left(angle \cdot \left({a}^{2} \cdot \pi\right)\right)\right)} \cdot 1 \]
   9. Step-by-step derivation

      1. associate-*r* 51.4%

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

      2. unpow2 51.4%

         \[\leadsto \left(-0.011111111111111112 \cdot \left(\left(angle \cdot \color{blue}{\left(a \cdot a\right)}\right) \cdot \pi\right)\right) \cdot 1 \]

   10. Simplified 51.4%

       \[\leadsto \color{blue}{\left(-0.011111111111111112 \cdot \left(\left(angle \cdot \left(a \cdot a\right)\right) \cdot \pi\right)\right)} \cdot 1 \]
3. Recombined 2 regimes into one program.

4. Final simplification 51.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -1.7 \cdot 10^{-16} \lor \neg \left(b \leq 1.15 \cdot 10^{+100}\right):\\ \;\;\;\;0.011111111111111112 \cdot \left(\pi \cdot \left(b \cdot \left(b \cdot angle\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;-0.011111111111111112 \cdot \left(\pi \cdot \left(angle \cdot \left(a \cdot a\right)\right)\right)\\ \end{array} \]

Alternative 10: 54.7% accurate, 5.5× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

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. Taylor expanded in angle around 0 47.2%

   \[\leadsto \color{blue}{\left(0.011111111111111112 \cdot \left(angle \cdot \left(\left({b}^{2} - {a}^{2}\right) \cdot \pi\right)\right)\right)} \cdot \cos \left(\pi \cdot \frac{angle}{180}\right) \]
3. Step-by-step derivation

   1. associate-*r* 47.3%

      \[\leadsto \left(0.011111111111111112 \cdot \color{blue}{\left(\left(angle \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \pi\right)}\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right) \]

   2. unpow2 47.3%

      \[\leadsto \left(0.011111111111111112 \cdot \left(\left(angle \cdot \left(\color{blue}{b \cdot b} - {a}^{2}\right)\right) \cdot \pi\right)\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right) \]

   3. unpow2 47.3%

      \[\leadsto \left(0.011111111111111112 \cdot \left(\left(angle \cdot \left(b \cdot b - \color{blue}{a \cdot a}\right)\right) \cdot \pi\right)\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right) \]

4. Simplified 47.3%

   \[\leadsto \color{blue}{\left(0.011111111111111112 \cdot \left(\left(angle \cdot \left(b \cdot b - a \cdot a\right)\right) \cdot \pi\right)\right)} \cdot \cos \left(\pi \cdot \frac{angle}{180}\right) \]
5. Step-by-step derivation

   1. pow1 47.3%

      \[\leadsto \left(0.011111111111111112 \cdot \left(\color{blue}{{\left(angle \cdot \left(b \cdot b - a \cdot a\right)\right)}^{1}} \cdot \pi\right)\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right) \]

   2. *-commutative 47.3%

      \[\leadsto \left(0.011111111111111112 \cdot \left({\color{blue}{\left(\left(b \cdot b - a \cdot a\right) \cdot angle\right)}}^{1} \cdot \pi\right)\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right) \]

   3. difference-of-squares 49.3%

      \[\leadsto \left(0.011111111111111112 \cdot \left({\left(\color{blue}{\left(\left(b + a\right) \cdot \left(b - a\right)\right)} \cdot angle\right)}^{1} \cdot \pi\right)\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right) \]

   4. associate-*l* 57.7%

      \[\leadsto \left(0.011111111111111112 \cdot \left({\color{blue}{\left(\left(b + a\right) \cdot \left(\left(b - a\right) \cdot angle\right)\right)}}^{1} \cdot \pi\right)\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right) \]

6. Applied egg-rr 57.7%

   \[\leadsto \left(0.011111111111111112 \cdot \left(\color{blue}{{\left(\left(b + a\right) \cdot \left(\left(b - a\right) \cdot angle\right)\right)}^{1}} \cdot \pi\right)\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right) \]

7. Taylor expanded in angle around 0 63.3%

   \[\leadsto \left(0.011111111111111112 \cdot \left({\left(\left(b + a\right) \cdot \left(\left(b - a\right) \cdot angle\right)\right)}^{1} \cdot \pi\right)\right) \cdot \color{blue}{1} \]

8. Taylor expanded in angle around 0 54.9%

   \[\leadsto \color{blue}{\left(0.011111111111111112 \cdot \left(angle \cdot \left(\left(b - a\right) \cdot \left(\left(a + b\right) \cdot \pi\right)\right)\right)\right)} \cdot 1 \]

9. Final simplification 54.9%

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

Alternative 11: 35.4% accurate, 5.7× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 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. Taylor expanded in angle around 0 47.2%

   \[\leadsto \color{blue}{\left(0.011111111111111112 \cdot \left(angle \cdot \left(\left({b}^{2} - {a}^{2}\right) \cdot \pi\right)\right)\right)} \cdot \cos \left(\pi \cdot \frac{angle}{180}\right) \]
3. Step-by-step derivation

   1. associate-*r* 47.3%

      \[\leadsto \left(0.011111111111111112 \cdot \color{blue}{\left(\left(angle \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \pi\right)}\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right) \]

   2. unpow2 47.3%

      \[\leadsto \left(0.011111111111111112 \cdot \left(\left(angle \cdot \left(\color{blue}{b \cdot b} - {a}^{2}\right)\right) \cdot \pi\right)\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right) \]

   3. unpow2 47.3%

      \[\leadsto \left(0.011111111111111112 \cdot \left(\left(angle \cdot \left(b \cdot b - \color{blue}{a \cdot a}\right)\right) \cdot \pi\right)\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right) \]

4. Simplified 47.3%

   \[\leadsto \color{blue}{\left(0.011111111111111112 \cdot \left(\left(angle \cdot \left(b \cdot b - a \cdot a\right)\right) \cdot \pi\right)\right)} \cdot \cos \left(\pi \cdot \frac{angle}{180}\right) \]
5. Step-by-step derivation

   1. pow1 47.3%

      \[\leadsto \left(0.011111111111111112 \cdot \left(\color{blue}{{\left(angle \cdot \left(b \cdot b - a \cdot a\right)\right)}^{1}} \cdot \pi\right)\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right) \]

   2. *-commutative 47.3%

      \[\leadsto \left(0.011111111111111112 \cdot \left({\color{blue}{\left(\left(b \cdot b - a \cdot a\right) \cdot angle\right)}}^{1} \cdot \pi\right)\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right) \]

   3. difference-of-squares 49.3%

      \[\leadsto \left(0.011111111111111112 \cdot \left({\left(\color{blue}{\left(\left(b + a\right) \cdot \left(b - a\right)\right)} \cdot angle\right)}^{1} \cdot \pi\right)\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right) \]

   4. associate-*l* 57.7%

      \[\leadsto \left(0.011111111111111112 \cdot \left({\color{blue}{\left(\left(b + a\right) \cdot \left(\left(b - a\right) \cdot angle\right)\right)}}^{1} \cdot \pi\right)\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right) \]

6. Applied egg-rr 57.7%

   \[\leadsto \left(0.011111111111111112 \cdot \left(\color{blue}{{\left(\left(b + a\right) \cdot \left(\left(b - a\right) \cdot angle\right)\right)}^{1}} \cdot \pi\right)\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right) \]

7. Taylor expanded in angle around 0 63.3%

   \[\leadsto \left(0.011111111111111112 \cdot \left({\left(\left(b + a\right) \cdot \left(\left(b - a\right) \cdot angle\right)\right)}^{1} \cdot \pi\right)\right) \cdot \color{blue}{1} \]

8. Taylor expanded in b around 0 34.7%

   \[\leadsto \color{blue}{\left(-0.011111111111111112 \cdot \left(angle \cdot \left({a}^{2} \cdot \pi\right)\right)\right)} \cdot 1 \]
9. Step-by-step derivation

   1. associate-*r* 34.7%

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

   2. unpow2 34.7%

      \[\leadsto \left(-0.011111111111111112 \cdot \left(\left(angle \cdot \color{blue}{\left(a \cdot a\right)}\right) \cdot \pi\right)\right) \cdot 1 \]

10. Simplified 34.7%

    \[\leadsto \color{blue}{\left(-0.011111111111111112 \cdot \left(\left(angle \cdot \left(a \cdot a\right)\right) \cdot \pi\right)\right)} \cdot 1 \]

11. Final simplification 34.7%

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

Reproduce

herbie shell --seed 2023187 
(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)))))