ab-angle->ABCF B

Percentage Accurate: 53.5% → 67.8%

Time: 43.6s

Alternatives: 20

Speedup: 5.4×

• 32.4% of points produce a very large (infinite) output. You may want to add a precondition.

Specification

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 53.5% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 67.8% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := 0.005555555555555556 \cdot \left(angle \cdot \pi\right)\\ t_1 := \pi \cdot \left(angle \cdot 0.005555555555555556\right)\\ \mathbf{if}\;\frac{angle}{180} \leq 50000000000000:\\ \;\;\;\;2 \cdot \left(\left(b - a\right) \cdot \left(\cos t_0 \cdot \left(\sin t_0 \cdot \left(b + a\right)\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\sin t_1 \cdot \left(2 \cdot {\left(b + a\right)}^{2}\right)\right) \cdot \cos \left({\left(\sqrt[3]{t_1}\right)}^{3}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (a b angle)
 :precision binary64
 (let* ((t_0 (* 0.005555555555555556 (* angle PI)))
        (t_1 (* PI (* angle 0.005555555555555556))))
   (if (<= (/ angle 180.0) 50000000000000.0)
     (* 2.0 (* (- b a) (* (cos t_0) (* (sin t_0) (+ b a)))))
     (* (* (sin t_1) (* 2.0 (pow (+ b a) 2.0))) (cos (pow (cbrt t_1) 3.0))))))

C

double code(double a, double b, double angle) {
	double t_0 = 0.005555555555555556 * (angle * ((double) M_PI));
	double t_1 = ((double) M_PI) * (angle * 0.005555555555555556);
	double tmp;
	if ((angle / 180.0) <= 50000000000000.0) {
		tmp = 2.0 * ((b - a) * (cos(t_0) * (sin(t_0) * (b + a))));
	} else {
		tmp = (sin(t_1) * (2.0 * pow((b + a), 2.0))) * cos(pow(cbrt(t_1), 3.0));
	}
	return tmp;
}

Java

public static double code(double a, double b, double angle) {
	double t_0 = 0.005555555555555556 * (angle * Math.PI);
	double t_1 = Math.PI * (angle * 0.005555555555555556);
	double tmp;
	if ((angle / 180.0) <= 50000000000000.0) {
		tmp = 2.0 * ((b - a) * (Math.cos(t_0) * (Math.sin(t_0) * (b + a))));
	} else {
		tmp = (Math.sin(t_1) * (2.0 * Math.pow((b + a), 2.0))) * Math.cos(Math.pow(Math.cbrt(t_1), 3.0));
	}
	return tmp;
}

Julia

function code(a, b, angle)
	t_0 = Float64(0.005555555555555556 * Float64(angle * pi))
	t_1 = Float64(pi * Float64(angle * 0.005555555555555556))
	tmp = 0.0
	if (Float64(angle / 180.0) <= 50000000000000.0)
		tmp = Float64(2.0 * Float64(Float64(b - a) * Float64(cos(t_0) * Float64(sin(t_0) * Float64(b + a)))));
	else
		tmp = Float64(Float64(sin(t_1) * Float64(2.0 * (Float64(b + a) ^ 2.0))) * cos((cbrt(t_1) ^ 3.0)));
	end
	return tmp
end

Wolfram

code[a_, b_, angle_] := Block[{t$95$0 = N[(0.005555555555555556 * N[(angle * Pi), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(Pi * N[(angle * 0.005555555555555556), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[(angle / 180.0), $MachinePrecision], 50000000000000.0], N[(2.0 * N[(N[(b - a), $MachinePrecision] * N[(N[Cos[t$95$0], $MachinePrecision] * N[(N[Sin[t$95$0], $MachinePrecision] * N[(b + a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[Sin[t$95$1], $MachinePrecision] * N[(2.0 * N[Power[N[(b + a), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[Cos[N[Power[N[Power[t$95$1, 1/3], $MachinePrecision], 3.0], $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 angle 180) < 5e13

   1. Initial program 64.8%

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

      1. associate-*l* 64.8%

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

      2. associate-*l* 64.8%

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

      3. unpow2 64.8%

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

      4. unpow2 64.8%

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

      5. difference-of-squares 69.4%

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

   3. Simplified 69.4%

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

   4. Taylor expanded in angle around inf 85.1%

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

   if 5e13 < (/.f64 angle 180)

   1. Initial program 28.6%

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

      1. *-commutative 28.6%

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

      2. associate-*l* 28.6%

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

      3. unpow2 28.6%

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

      4. fma-neg 32.0%

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

      5. unpow2 32.0%

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

      6. distribute-rgt-neg-in 32.0%

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

   3. Simplified 32.0%

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

      1. expm1-log1p-u 24.0%

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

      2. expm1-udef 22.7%

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

   5. Applied egg-rr 18.2%

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

      1. expm1-def 19.6%

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

      2. expm1-log1p 30.8%

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

      3. rem-exp-log 30.8%

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

      4. rem-exp-log 30.8%

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

      5. +-commutative 30.8%

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

   7. Simplified 30.8%

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

      1. add-cube-cbrt 36.6%

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

      2. pow3 42.0%

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

      3. div-inv 45.4%

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

      4. metadata-eval 45.4%

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

   9. Applied egg-rr 45.4%

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

4. Final simplification 76.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{angle}{180} \leq 50000000000000:\\ \;\;\;\;2 \cdot \left(\left(b - a\right) \cdot \left(\cos \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right) \cdot \left(\sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right) \cdot \left(b + a\right)\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\sin \left(\pi \cdot \left(angle \cdot 0.005555555555555556\right)\right) \cdot \left(2 \cdot {\left(b + a\right)}^{2}\right)\right) \cdot \cos \left({\left(\sqrt[3]{\pi \cdot \left(angle \cdot 0.005555555555555556\right)}\right)}^{3}\right)\\ \end{array} \]

Alternative 2: 67.6% accurate, 1.5× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 56.5%

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

   1. associate-*l* 56.5%

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

   2. associate-*l* 56.5%

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

   3. unpow2 56.5%

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

   4. unpow2 56.5%

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

   5. difference-of-squares 61.2%

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

3. Simplified 61.2%

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

4. Taylor expanded in angle around inf 73.0%

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

   1. associate-*r* 73.0%

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

   2. *-commutative 73.0%

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

   3. associate-*r* 73.2%

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

   4. *-commutative 73.2%

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

   5. *-commutative 73.2%

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

   6. *-commutative 73.2%

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

   7. associate-*r* 72.8%

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

   8. *-commutative 72.8%

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

   9. *-commutative 72.8%

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

6. Simplified 72.8%

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

7. Final simplification 72.8%

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

Alternative 3: 67.6% accurate, 1.5× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 56.5%

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

   1. associate-*l* 56.5%

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

   2. associate-*l* 56.5%

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

   3. unpow2 56.5%

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

   4. unpow2 56.5%

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

   5. difference-of-squares 61.2%

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

3. Simplified 61.2%

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

4. Taylor expanded in angle around inf 73.0%

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

5. Final simplification 73.0%

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

Alternative 4: 67.6% accurate, 1.5× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 56.5%

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

   1. associate-*l* 56.5%

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

   2. associate-*l* 56.5%

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

   3. unpow2 56.5%

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

   4. unpow2 56.5%

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

   5. difference-of-squares 61.2%

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

3. Simplified 61.2%

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

4. Taylor expanded in angle around inf 73.0%

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

   1. associate-*r* 73.2%

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

   2. *-commutative 73.2%

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

   3. metadata-eval 73.2%

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

   4. div-inv 73.2%

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

   5. *-commutative 73.2%

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

   6. clear-num 72.5%

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

   7. div-inv 72.9%

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

   8. clear-num 73.0%

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

6. Applied egg-rr 73.0%

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

   1. clear-num 72.9%

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

   2. associate-/r/ 74.1%

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

8. Applied egg-rr 74.1%

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

9. Final simplification 74.1%

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

Alternative 5: 65.7% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq 3.05 \cdot 10^{+164}:\\ \;\;\;\;2 \cdot \left(\left(b - a\right) \cdot \left(\left(b + a\right) \cdot \sin \left(\frac{1}{\frac{\frac{180}{angle}}{\pi}}\right)\right)\right)\\ \mathbf{elif}\;b \leq 5.8 \cdot 10^{+266}:\\ \;\;\;\;2 \cdot \left(\left(b - a\right) \cdot \left(\cos \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right) \cdot \left(\left(b + a\right) \cdot \left(angle \cdot \frac{\pi}{180}\right)\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;0.011111111111111112 \cdot \left(angle \cdot \left(\pi \cdot \left(b \cdot b\right)\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (a b angle)
 :precision binary64
 (if (<= b 3.05e+164)
   (* 2.0 (* (- b a) (* (+ b a) (sin (/ 1.0 (/ (/ 180.0 angle) PI))))))
   (if (<= b 5.8e+266)
     (*
      2.0
      (*
       (- b a)
       (*
        (cos (* 0.005555555555555556 (* angle PI)))
        (* (+ b a) (* angle (/ PI 180.0))))))
     (* 0.011111111111111112 (* angle (* PI (* b b)))))))

C

double code(double a, double b, double angle) {
	double tmp;
	if (b <= 3.05e+164) {
		tmp = 2.0 * ((b - a) * ((b + a) * sin((1.0 / ((180.0 / angle) / ((double) M_PI))))));
	} else if (b <= 5.8e+266) {
		tmp = 2.0 * ((b - a) * (cos((0.005555555555555556 * (angle * ((double) M_PI)))) * ((b + a) * (angle * (((double) M_PI) / 180.0)))));
	} else {
		tmp = 0.011111111111111112 * (angle * (((double) M_PI) * (b * b)));
	}
	return tmp;
}

Java

public static double code(double a, double b, double angle) {
	double tmp;
	if (b <= 3.05e+164) {
		tmp = 2.0 * ((b - a) * ((b + a) * Math.sin((1.0 / ((180.0 / angle) / Math.PI)))));
	} else if (b <= 5.8e+266) {
		tmp = 2.0 * ((b - a) * (Math.cos((0.005555555555555556 * (angle * Math.PI))) * ((b + a) * (angle * (Math.PI / 180.0)))));
	} else {
		tmp = 0.011111111111111112 * (angle * (Math.PI * (b * b)));
	}
	return tmp;
}

Python

def code(a, b, angle):
	tmp = 0
	if b <= 3.05e+164:
		tmp = 2.0 * ((b - a) * ((b + a) * math.sin((1.0 / ((180.0 / angle) / math.pi)))))
	elif b <= 5.8e+266:
		tmp = 2.0 * ((b - a) * (math.cos((0.005555555555555556 * (angle * math.pi))) * ((b + a) * (angle * (math.pi / 180.0)))))
	else:
		tmp = 0.011111111111111112 * (angle * (math.pi * (b * b)))
	return tmp

Julia

function code(a, b, angle)
	tmp = 0.0
	if (b <= 3.05e+164)
		tmp = Float64(2.0 * Float64(Float64(b - a) * Float64(Float64(b + a) * sin(Float64(1.0 / Float64(Float64(180.0 / angle) / pi))))));
	elseif (b <= 5.8e+266)
		tmp = Float64(2.0 * Float64(Float64(b - a) * Float64(cos(Float64(0.005555555555555556 * Float64(angle * pi))) * Float64(Float64(b + a) * Float64(angle * Float64(pi / 180.0))))));
	else
		tmp = Float64(0.011111111111111112 * Float64(angle * Float64(pi * Float64(b * b))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, angle)
	tmp = 0.0;
	if (b <= 3.05e+164)
		tmp = 2.0 * ((b - a) * ((b + a) * sin((1.0 / ((180.0 / angle) / pi)))));
	elseif (b <= 5.8e+266)
		tmp = 2.0 * ((b - a) * (cos((0.005555555555555556 * (angle * pi))) * ((b + a) * (angle * (pi / 180.0)))));
	else
		tmp = 0.011111111111111112 * (angle * (pi * (b * b)));
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, angle_] := If[LessEqual[b, 3.05e+164], N[(2.0 * N[(N[(b - a), $MachinePrecision] * N[(N[(b + a), $MachinePrecision] * N[Sin[N[(1.0 / N[(N[(180.0 / angle), $MachinePrecision] / Pi), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[b, 5.8e+266], N[(2.0 * N[(N[(b - a), $MachinePrecision] * N[(N[Cos[N[(0.005555555555555556 * N[(angle * Pi), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * N[(N[(b + a), $MachinePrecision] * N[(angle * N[(Pi / 180.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(0.011111111111111112 * N[(angle * N[(Pi * N[(b * b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if b < 3.0499999999999998e164

   1. Initial program 59.1%

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

      1. associate-*l* 59.1%

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

      2. associate-*l* 59.1%

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

      3. unpow2 59.1%

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

      4. unpow2 59.1%

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

      5. difference-of-squares 61.7%

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

   3. Simplified 61.7%

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

   4. Taylor expanded in angle around inf 72.5%

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

      1. associate-*r* 73.1%

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

      2. *-commutative 73.1%

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

      3. metadata-eval 73.1%

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

      4. div-inv 72.7%

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

      5. *-commutative 72.7%

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

      6. clear-num 72.4%

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

      7. div-inv 72.8%

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

      8. clear-num 72.9%

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

   6. Applied egg-rr 72.9%

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

   7. Taylor expanded in angle around 0 68.2%

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

   if 3.0499999999999998e164 < b < 5.80000000000000035e266

   1. Initial program 27.4%

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

      1. associate-*l* 27.4%

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

      2. associate-*l* 27.4%

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

      3. unpow2 27.4%

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

      4. unpow2 27.4%

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

      5. difference-of-squares 49.0%

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

   3. Simplified 49.0%

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

   4. Taylor expanded in angle around inf 83.9%

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

      1. associate-*r* 84.0%

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

      2. *-commutative 84.0%

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

      3. metadata-eval 84.0%

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

      4. div-inv 78.8%

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

      5. *-commutative 78.8%

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

      6. clear-num 84.0%

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

      7. div-inv 78.8%

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

      8. clear-num 78.8%

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

   6. Applied egg-rr 78.8%

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

   7. Taylor expanded in angle around 0 94.4%

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

      1. associate-*r* 94.6%

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

      2. metadata-eval 94.6%

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

      3. associate-/r/ 94.6%

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

      4. associate-*l/ 94.6%

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

      5. *-lft-identity 94.6%

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

      6. associate-/r/ 94.6%

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

      7. *-commutative 94.6%

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

   9. Simplified 94.6%

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

   if 5.80000000000000035e266 < b

   1. Initial program 50.0%

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

      1. *-commutative 50.0%

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

      2. associate-*l* 50.0%

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

      3. unpow2 50.0%

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

      4. fma-neg 62.5%

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

      5. unpow2 62.5%

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

   3. Simplified 62.5%

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

   4. Taylor expanded in angle around 0 62.5%

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

      1. *-commutative 62.5%

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

      2. unpow2 62.5%

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

      3. unpow2 62.5%

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

   6. Simplified 62.5%

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

   7. Taylor expanded in b around inf 100.0%

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

      1. *-commutative 100.0%

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

      2. unpow2 100.0%

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

   9. Simplified 100.0%

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

4. Final simplification 71.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq 3.05 \cdot 10^{+164}:\\ \;\;\;\;2 \cdot \left(\left(b - a\right) \cdot \left(\left(b + a\right) \cdot \sin \left(\frac{1}{\frac{\frac{180}{angle}}{\pi}}\right)\right)\right)\\ \mathbf{elif}\;b \leq 5.8 \cdot 10^{+266}:\\ \;\;\;\;2 \cdot \left(\left(b - a\right) \cdot \left(\cos \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right) \cdot \left(\left(b + a\right) \cdot \left(angle \cdot \frac{\pi}{180}\right)\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;0.011111111111111112 \cdot \left(angle \cdot \left(\pi \cdot \left(b \cdot b\right)\right)\right)\\ \end{array} \]

Alternative 6: 65.9% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{angle}{180} \leq -4 \cdot 10^{+179}:\\ \;\;\;\;2 \cdot \left(\sin \left(2 \cdot \left(\pi \cdot \left(angle \cdot 0.005555555555555556\right)\right)\right) \cdot \left({\left(b + a\right)}^{2} \cdot 0.5\right)\right)\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \left(\left(b - a\right) \cdot \left(\left(b + a\right) \cdot \sin \left(\frac{1}{\frac{\frac{180}{angle}}{\pi}}\right)\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (a b angle)
 :precision binary64
 (if (<= (/ angle 180.0) -4e+179)
   (*
    2.0
    (*
     (sin (* 2.0 (* PI (* angle 0.005555555555555556))))
     (* (pow (+ b a) 2.0) 0.5)))
   (* 2.0 (* (- b a) (* (+ b a) (sin (/ 1.0 (/ (/ 180.0 angle) PI))))))))

C

double code(double a, double b, double angle) {
	double tmp;
	if ((angle / 180.0) <= -4e+179) {
		tmp = 2.0 * (sin((2.0 * (((double) M_PI) * (angle * 0.005555555555555556)))) * (pow((b + a), 2.0) * 0.5));
	} else {
		tmp = 2.0 * ((b - a) * ((b + a) * sin((1.0 / ((180.0 / angle) / ((double) M_PI))))));
	}
	return tmp;
}

Java

public static double code(double a, double b, double angle) {
	double tmp;
	if ((angle / 180.0) <= -4e+179) {
		tmp = 2.0 * (Math.sin((2.0 * (Math.PI * (angle * 0.005555555555555556)))) * (Math.pow((b + a), 2.0) * 0.5));
	} else {
		tmp = 2.0 * ((b - a) * ((b + a) * Math.sin((1.0 / ((180.0 / angle) / Math.PI)))));
	}
	return tmp;
}

Python

def code(a, b, angle):
	tmp = 0
	if (angle / 180.0) <= -4e+179:
		tmp = 2.0 * (math.sin((2.0 * (math.pi * (angle * 0.005555555555555556)))) * (math.pow((b + a), 2.0) * 0.5))
	else:
		tmp = 2.0 * ((b - a) * ((b + a) * math.sin((1.0 / ((180.0 / angle) / math.pi)))))
	return tmp

Julia

function code(a, b, angle)
	tmp = 0.0
	if (Float64(angle / 180.0) <= -4e+179)
		tmp = Float64(2.0 * Float64(sin(Float64(2.0 * Float64(pi * Float64(angle * 0.005555555555555556)))) * Float64((Float64(b + a) ^ 2.0) * 0.5)));
	else
		tmp = Float64(2.0 * Float64(Float64(b - a) * Float64(Float64(b + a) * sin(Float64(1.0 / Float64(Float64(180.0 / angle) / pi))))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, angle)
	tmp = 0.0;
	if ((angle / 180.0) <= -4e+179)
		tmp = 2.0 * (sin((2.0 * (pi * (angle * 0.005555555555555556)))) * (((b + a) ^ 2.0) * 0.5));
	else
		tmp = 2.0 * ((b - a) * ((b + a) * sin((1.0 / ((180.0 / angle) / pi)))));
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, angle_] := If[LessEqual[N[(angle / 180.0), $MachinePrecision], -4e+179], N[(2.0 * N[(N[Sin[N[(2.0 * N[(Pi * N[(angle * 0.005555555555555556), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * N[(N[Power[N[(b + a), $MachinePrecision], 2.0], $MachinePrecision] * 0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(2.0 * N[(N[(b - a), $MachinePrecision] * N[(N[(b + a), $MachinePrecision] * N[Sin[N[(1.0 / N[(N[(180.0 / angle), $MachinePrecision] / Pi), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 angle 180) < -3.99999999999999992e179

   1. Initial program 28.9%

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

      1. *-commutative 28.9%

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

      2. associate-*l* 28.9%

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

      3. unpow2 28.9%

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

      4. fma-neg 40.0%

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

      5. unpow2 40.0%

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

   3. Simplified 40.0%

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

   5. Applied egg-rr 14.6%

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

      1. unpow2 14.6%

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

      2. associate-*r* 14.6%

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

      3. *-commutative 14.6%

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

      4. associate-*r* 14.6%

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

      5. *-commutative 14.6%

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

      6. swap-sqr 14.6%

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

   7. Simplified 45.0%

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

   if -3.99999999999999992e179 < (/.f64 angle 180)

   1. Initial program 58.6%

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

      1. associate-*l* 58.6%

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

      2. associate-*l* 58.5%

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

      3. unpow2 58.5%

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

      4. unpow2 58.6%

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

      5. difference-of-squares 62.8%

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

   3. Simplified 62.8%

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

   4. Taylor expanded in angle around inf 75.1%

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

      1. associate-*r* 75.7%

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

      2. *-commutative 75.7%

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

      3. metadata-eval 75.7%

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

      4. div-inv 75.7%

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

      5. *-commutative 75.7%

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

      6. clear-num 75.4%

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

      7. div-inv 75.0%

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

      8. clear-num 75.1%

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

   6. Applied egg-rr 75.1%

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

   7. Taylor expanded in angle around 0 73.3%

      \[\leadsto 2 \cdot \left(\left(b - a\right) \cdot \left(\color{blue}{1} \cdot \left(\sin \left(\frac{1}{\frac{\frac{180}{angle}}{\pi}}\right) \cdot \left(a + b\right)\right)\right)\right) \]
3. Recombined 2 regimes into one program.

4. Final simplification 71.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{angle}{180} \leq -4 \cdot 10^{+179}:\\ \;\;\;\;2 \cdot \left(\sin \left(2 \cdot \left(\pi \cdot \left(angle \cdot 0.005555555555555556\right)\right)\right) \cdot \left({\left(b + a\right)}^{2} \cdot 0.5\right)\right)\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \left(\left(b - a\right) \cdot \left(\left(b + a\right) \cdot \sin \left(\frac{1}{\frac{\frac{180}{angle}}{\pi}}\right)\right)\right)\\ \end{array} \]

Alternative 7: 65.8% accurate, 2.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{angle}{180} \leq -4 \cdot 10^{+179}:\\ \;\;\;\;0.011111111111111112 \cdot \left(angle \cdot \left(\pi \cdot \mathsf{fma}\left(1, b \cdot b, a \cdot a\right)\right)\right)\\ \mathbf{elif}\;\frac{angle}{180} \leq 50000000000000:\\ \;\;\;\;2 \cdot \left(\left(b - a\right) \cdot \left(\sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right) \cdot \left(b + a\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \left(\left(\left(b - a\right) \cdot \left(b + a\right)\right) \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (a b angle)
 :precision binary64
 (if (<= (/ angle 180.0) -4e+179)
   (* 0.011111111111111112 (* angle (* PI (fma 1.0 (* b b) (* a a)))))
   (if (<= (/ angle 180.0) 50000000000000.0)
     (*
      2.0
      (* (- b a) (* (sin (* 0.005555555555555556 (* angle PI))) (+ b a))))
     (* 2.0 (* (* (- b a) (+ b a)) (sin (* (/ angle 180.0) PI)))))))

C

double code(double a, double b, double angle) {
	double tmp;
	if ((angle / 180.0) <= -4e+179) {
		tmp = 0.011111111111111112 * (angle * (((double) M_PI) * fma(1.0, (b * b), (a * a))));
	} else if ((angle / 180.0) <= 50000000000000.0) {
		tmp = 2.0 * ((b - a) * (sin((0.005555555555555556 * (angle * ((double) M_PI)))) * (b + a)));
	} else {
		tmp = 2.0 * (((b - a) * (b + a)) * sin(((angle / 180.0) * ((double) M_PI))));
	}
	return tmp;
}

Julia

function code(a, b, angle)
	tmp = 0.0
	if (Float64(angle / 180.0) <= -4e+179)
		tmp = Float64(0.011111111111111112 * Float64(angle * Float64(pi * fma(1.0, Float64(b * b), Float64(a * a)))));
	elseif (Float64(angle / 180.0) <= 50000000000000.0)
		tmp = Float64(2.0 * Float64(Float64(b - a) * Float64(sin(Float64(0.005555555555555556 * Float64(angle * pi))) * Float64(b + a))));
	else
		tmp = Float64(2.0 * Float64(Float64(Float64(b - a) * Float64(b + a)) * sin(Float64(Float64(angle / 180.0) * pi))));
	end
	return tmp
end

Wolfram

code[a_, b_, angle_] := If[LessEqual[N[(angle / 180.0), $MachinePrecision], -4e+179], N[(0.011111111111111112 * N[(angle * N[(Pi * N[(1.0 * N[(b * b), $MachinePrecision] + N[(a * a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[N[(angle / 180.0), $MachinePrecision], 50000000000000.0], N[(2.0 * N[(N[(b - a), $MachinePrecision] * N[(N[Sin[N[(0.005555555555555556 * N[(angle * Pi), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * N[(b + a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(2.0 * N[(N[(N[(b - a), $MachinePrecision] * N[(b + a), $MachinePrecision]), $MachinePrecision] * N[Sin[N[(N[(angle / 180.0), $MachinePrecision] * Pi), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if (/.f64 angle 180) < -3.99999999999999992e179

   1. Initial program 28.9%

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

      1. *-commutative 28.9%

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

      2. associate-*l* 28.9%

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

      3. unpow2 28.9%

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

      4. fma-neg 40.0%

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

      5. unpow2 40.0%

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

   3. Simplified 40.0%

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

   4. Taylor expanded in angle around 0 18.2%

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

      1. *-commutative 18.2%

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

      2. unpow2 18.2%

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

      3. unpow2 18.2%

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

   6. Simplified 18.2%

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

      1. *-un-lft-identity 18.2%

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

      2. fma-neg 18.2%

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

      3. add-sqr-sqrt 11.5%

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

      4. sqrt-unprod 40.5%

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

      5. sqr-neg 40.5%

         \[\leadsto 0.011111111111111112 \cdot \left(angle \cdot \left(\pi \cdot \mathsf{fma}\left(1, b \cdot b, \sqrt{\color{blue}{\left(a \cdot a\right) \cdot \left(a \cdot a\right)}}\right)\right)\right) \]

      6. sqrt-unprod 40.4%

         \[\leadsto 0.011111111111111112 \cdot \left(angle \cdot \left(\pi \cdot \mathsf{fma}\left(1, b \cdot b, \color{blue}{\sqrt{a \cdot a} \cdot \sqrt{a \cdot a}}\right)\right)\right) \]

      7. add-sqr-sqrt 40.4%

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

   8. Applied egg-rr 40.4%

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

   if -3.99999999999999992e179 < (/.f64 angle 180) < 5e13

   1. Initial program 68.4%

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

      1. associate-*l* 68.4%

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

      2. associate-*l* 68.4%

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

      3. unpow2 68.4%

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

      4. unpow2 68.4%

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

      5. difference-of-squares 72.4%

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

   3. Simplified 72.4%

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

   4. Taylor expanded in angle around inf 89.1%

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

   5. Taylor expanded in angle around 0 83.9%

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

   if 5e13 < (/.f64 angle 180)

   1. Initial program 28.6%

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

      1. associate-*l* 28.6%

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

      2. associate-*l* 28.6%

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

      3. unpow2 28.6%

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

      4. unpow2 28.6%

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

      5. difference-of-squares 33.7%

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

   3. Simplified 33.7%

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

   4. Taylor expanded in angle around 0 42.8%

      \[\leadsto 2 \cdot \left(\left(\left(b + a\right) \cdot \left(b - a\right)\right) \cdot \left(\sin \left(\pi \cdot \frac{angle}{180}\right) \cdot \color{blue}{1}\right)\right) \]
3. Recombined 3 regimes into one program.

4. Final simplification 71.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{angle}{180} \leq -4 \cdot 10^{+179}:\\ \;\;\;\;0.011111111111111112 \cdot \left(angle \cdot \left(\pi \cdot \mathsf{fma}\left(1, b \cdot b, a \cdot a\right)\right)\right)\\ \mathbf{elif}\;\frac{angle}{180} \leq 50000000000000:\\ \;\;\;\;2 \cdot \left(\left(b - a\right) \cdot \left(\sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right) \cdot \left(b + a\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \left(\left(\left(b - a\right) \cdot \left(b + a\right)\right) \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right)\\ \end{array} \]

Alternative 8: 65.6% accurate, 2.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{angle}{180} \leq -4 \cdot 10^{+179}:\\ \;\;\;\;0.011111111111111112 \cdot \left(angle \cdot \left(\pi \cdot \mathsf{fma}\left(1, b \cdot b, a \cdot a\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \left(\left(b - a\right) \cdot \left(\left(b + a\right) \cdot \sin \left(\frac{1}{\frac{\frac{180}{angle}}{\pi}}\right)\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (a b angle)
 :precision binary64
 (if (<= (/ angle 180.0) -4e+179)
   (* 0.011111111111111112 (* angle (* PI (fma 1.0 (* b b) (* a a)))))
   (* 2.0 (* (- b a) (* (+ b a) (sin (/ 1.0 (/ (/ 180.0 angle) PI))))))))

C

double code(double a, double b, double angle) {
	double tmp;
	if ((angle / 180.0) <= -4e+179) {
		tmp = 0.011111111111111112 * (angle * (((double) M_PI) * fma(1.0, (b * b), (a * a))));
	} else {
		tmp = 2.0 * ((b - a) * ((b + a) * sin((1.0 / ((180.0 / angle) / ((double) M_PI))))));
	}
	return tmp;
}

Julia

function code(a, b, angle)
	tmp = 0.0
	if (Float64(angle / 180.0) <= -4e+179)
		tmp = Float64(0.011111111111111112 * Float64(angle * Float64(pi * fma(1.0, Float64(b * b), Float64(a * a)))));
	else
		tmp = Float64(2.0 * Float64(Float64(b - a) * Float64(Float64(b + a) * sin(Float64(1.0 / Float64(Float64(180.0 / angle) / pi))))));
	end
	return tmp
end

Wolfram

code[a_, b_, angle_] := If[LessEqual[N[(angle / 180.0), $MachinePrecision], -4e+179], N[(0.011111111111111112 * N[(angle * N[(Pi * N[(1.0 * N[(b * b), $MachinePrecision] + N[(a * a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(2.0 * N[(N[(b - a), $MachinePrecision] * N[(N[(b + a), $MachinePrecision] * N[Sin[N[(1.0 / N[(N[(180.0 / angle), $MachinePrecision] / Pi), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 angle 180) < -3.99999999999999992e179

   1. Initial program 28.9%

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

      1. *-commutative 28.9%

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

      2. associate-*l* 28.9%

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

      3. unpow2 28.9%

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

      4. fma-neg 40.0%

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

      5. unpow2 40.0%

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

   3. Simplified 40.0%

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

   4. Taylor expanded in angle around 0 18.2%

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

      1. *-commutative 18.2%

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

      2. unpow2 18.2%

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

      3. unpow2 18.2%

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

   6. Simplified 18.2%

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

      1. *-un-lft-identity 18.2%

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

      2. fma-neg 18.2%

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

      3. add-sqr-sqrt 11.5%

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

      4. sqrt-unprod 40.5%

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

      5. sqr-neg 40.5%

         \[\leadsto 0.011111111111111112 \cdot \left(angle \cdot \left(\pi \cdot \mathsf{fma}\left(1, b \cdot b, \sqrt{\color{blue}{\left(a \cdot a\right) \cdot \left(a \cdot a\right)}}\right)\right)\right) \]

      6. sqrt-unprod 40.4%

         \[\leadsto 0.011111111111111112 \cdot \left(angle \cdot \left(\pi \cdot \mathsf{fma}\left(1, b \cdot b, \color{blue}{\sqrt{a \cdot a} \cdot \sqrt{a \cdot a}}\right)\right)\right) \]

      7. add-sqr-sqrt 40.4%

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

   8. Applied egg-rr 40.4%

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

   if -3.99999999999999992e179 < (/.f64 angle 180)

   1. Initial program 58.6%

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

      1. associate-*l* 58.6%

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

      2. associate-*l* 58.5%

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

      3. unpow2 58.5%

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

      4. unpow2 58.6%

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

      5. difference-of-squares 62.8%

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

   3. Simplified 62.8%

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

   4. Taylor expanded in angle around inf 75.1%

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

      1. associate-*r* 75.7%

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

      2. *-commutative 75.7%

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

      3. metadata-eval 75.7%

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

      4. div-inv 75.7%

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

      5. *-commutative 75.7%

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

      6. clear-num 75.4%

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

      7. div-inv 75.0%

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

      8. clear-num 75.1%

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

   6. Applied egg-rr 75.1%

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

   7. Taylor expanded in angle around 0 73.3%

      \[\leadsto 2 \cdot \left(\left(b - a\right) \cdot \left(\color{blue}{1} \cdot \left(\sin \left(\frac{1}{\frac{\frac{180}{angle}}{\pi}}\right) \cdot \left(a + b\right)\right)\right)\right) \]
3. Recombined 2 regimes into one program.

4. Final simplification 71.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{angle}{180} \leq -4 \cdot 10^{+179}:\\ \;\;\;\;0.011111111111111112 \cdot \left(angle \cdot \left(\pi \cdot \mathsf{fma}\left(1, b \cdot b, a \cdot a\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \left(\left(b - a\right) \cdot \left(\left(b + a\right) \cdot \sin \left(\frac{1}{\frac{\frac{180}{angle}}{\pi}}\right)\right)\right)\\ \end{array} \]

Alternative 9: 53.7% accurate, 2.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;angle \leq 0.38:\\ \;\;\;\;2 \cdot \left(\left(angle \cdot 0.005555555555555556\right) \cdot \left(\left(b + a\right) \cdot \left(\left(b - a\right) \cdot \pi\right)\right)\right)\\ \mathbf{elif}\;angle \leq 1.65 \cdot 10^{+53}:\\ \;\;\;\;2 \cdot \left(0.5 \cdot \left(\left(b \cdot b\right) \cdot \sin \left(\pi \cdot \left(angle \cdot 0.011111111111111112\right)\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;0.011111111111111112 \cdot \left(angle \cdot \left(\pi \cdot \mathsf{fma}\left(1, b \cdot b, a \cdot a\right)\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (a b angle)
 :precision binary64
 (if (<= angle 0.38)
   (* 2.0 (* (* angle 0.005555555555555556) (* (+ b a) (* (- b a) PI))))
   (if (<= angle 1.65e+53)
     (* 2.0 (* 0.5 (* (* b b) (sin (* PI (* angle 0.011111111111111112))))))
     (* 0.011111111111111112 (* angle (* PI (fma 1.0 (* b b) (* a a))))))))

C

double code(double a, double b, double angle) {
	double tmp;
	if (angle <= 0.38) {
		tmp = 2.0 * ((angle * 0.005555555555555556) * ((b + a) * ((b - a) * ((double) M_PI))));
	} else if (angle <= 1.65e+53) {
		tmp = 2.0 * (0.5 * ((b * b) * sin((((double) M_PI) * (angle * 0.011111111111111112)))));
	} else {
		tmp = 0.011111111111111112 * (angle * (((double) M_PI) * fma(1.0, (b * b), (a * a))));
	}
	return tmp;
}

Julia

function code(a, b, angle)
	tmp = 0.0
	if (angle <= 0.38)
		tmp = Float64(2.0 * Float64(Float64(angle * 0.005555555555555556) * Float64(Float64(b + a) * Float64(Float64(b - a) * pi))));
	elseif (angle <= 1.65e+53)
		tmp = Float64(2.0 * Float64(0.5 * Float64(Float64(b * b) * sin(Float64(pi * Float64(angle * 0.011111111111111112))))));
	else
		tmp = Float64(0.011111111111111112 * Float64(angle * Float64(pi * fma(1.0, Float64(b * b), Float64(a * a)))));
	end
	return tmp
end

Wolfram

code[a_, b_, angle_] := If[LessEqual[angle, 0.38], N[(2.0 * N[(N[(angle * 0.005555555555555556), $MachinePrecision] * N[(N[(b + a), $MachinePrecision] * N[(N[(b - a), $MachinePrecision] * Pi), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[angle, 1.65e+53], N[(2.0 * N[(0.5 * N[(N[(b * b), $MachinePrecision] * N[Sin[N[(Pi * N[(angle * 0.011111111111111112), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(0.011111111111111112 * N[(angle * N[(Pi * N[(1.0 * N[(b * b), $MachinePrecision] + N[(a * a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if angle < 0.38

   1. Initial program 65.1%

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

      1. associate-*l* 65.2%

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

      2. associate-*l* 65.1%

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

      3. unpow2 65.1%

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

      4. unpow2 65.1%

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

      5. difference-of-squares 69.5%

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

   3. Simplified 69.5%

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

   4. Taylor expanded in angle around 0 65.5%

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

      1. associate-*r* 65.5%

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

      2. *-commutative 65.5%

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

      3. *-commutative 65.5%

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

      4. associate-*r* 65.5%

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

      5. +-commutative 65.5%

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

      6. *-commutative 65.5%

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

      7. +-commutative 65.5%

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

      8. *-commutative 65.5%

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

   6. Simplified 65.5%

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

   if 0.38 < angle < 1.6500000000000001e53

   1. Initial program 51.9%

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

      1. associate-*l* 51.9%

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

      2. associate-*l* 51.9%

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

      3. unpow2 51.9%

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

      4. unpow2 51.9%

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

      5. difference-of-squares 58.1%

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

   3. Simplified 58.1%

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

   5. Applied egg-rr 15.6%

      \[\leadsto 2 \cdot \color{blue}{\frac{\left({b}^{4} + {a}^{4}\right) \cdot \left(\sin 0 + \sin \left(\left(\pi \cdot \left(angle \cdot 0.005555555555555556\right)\right) \cdot 2\right)\right)}{2 \cdot {\left(b + a\right)}^{2}}} \]

   6. Taylor expanded in b around inf 49.7%

      \[\leadsto 2 \cdot \color{blue}{\left(0.5 \cdot \left(\sin \left(0.011111111111111112 \cdot \left(angle \cdot \pi\right)\right) \cdot {b}^{2}\right)\right)} \]
   7. Step-by-step derivation

      1. *-commutative 49.7%

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

      2. *-commutative 49.7%

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

      3. unpow2 49.7%

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

      4. associate-*r* 49.9%

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

   8. Simplified 49.9%

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

   if 1.6500000000000001e53 < angle

   1. Initial program 27.2%

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

      1. *-commutative 27.2%

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

      2. associate-*l* 27.2%

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

      3. unpow2 27.2%

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

      4. fma-neg 31.0%

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

      5. unpow2 31.0%

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

   3. Simplified 31.0%

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

   4. Taylor expanded in angle around 0 25.8%

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

      1. *-commutative 25.8%

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

      2. unpow2 25.8%

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

      3. unpow2 25.8%

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

   6. Simplified 25.8%

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

      1. *-un-lft-identity 25.8%

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

      2. fma-neg 25.8%

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

      3. add-sqr-sqrt 11.6%

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

      4. sqrt-unprod 41.5%

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

      5. sqr-neg 41.5%

         \[\leadsto 0.011111111111111112 \cdot \left(angle \cdot \left(\pi \cdot \mathsf{fma}\left(1, b \cdot b, \sqrt{\color{blue}{\left(a \cdot a\right) \cdot \left(a \cdot a\right)}}\right)\right)\right) \]

      6. sqrt-unprod 41.5%

         \[\leadsto 0.011111111111111112 \cdot \left(angle \cdot \left(\pi \cdot \mathsf{fma}\left(1, b \cdot b, \color{blue}{\sqrt{a \cdot a} \cdot \sqrt{a \cdot a}}\right)\right)\right) \]

      7. add-sqr-sqrt 41.5%

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

   8. Applied egg-rr 41.5%

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

4. Final simplification 59.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;angle \leq 0.38:\\ \;\;\;\;2 \cdot \left(\left(angle \cdot 0.005555555555555556\right) \cdot \left(\left(b + a\right) \cdot \left(\left(b - a\right) \cdot \pi\right)\right)\right)\\ \mathbf{elif}\;angle \leq 1.65 \cdot 10^{+53}:\\ \;\;\;\;2 \cdot \left(0.5 \cdot \left(\left(b \cdot b\right) \cdot \sin \left(\pi \cdot \left(angle \cdot 0.011111111111111112\right)\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;0.011111111111111112 \cdot \left(angle \cdot \left(\pi \cdot \mathsf{fma}\left(1, b \cdot b, a \cdot a\right)\right)\right)\\ \end{array} \]

Alternative 10: 55.7% accurate, 2.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq 2.5 \cdot 10^{+199}:\\ \;\;\;\;2 \cdot \left(\left(\left(b - a\right) \cdot \left(b + a\right)\right) \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right)\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \left(0.005555555555555556 \cdot \left(angle \cdot \left(\left(b - a\right) \cdot \left(\pi \cdot \left(b + a\right)\right)\right)\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (a b angle)
 :precision binary64
 (if (<= b 2.5e+199)
   (* 2.0 (* (* (- b a) (+ b a)) (sin (* (/ angle 180.0) PI))))
   (* 2.0 (* 0.005555555555555556 (* angle (* (- b a) (* PI (+ b a))))))))

C

double code(double a, double b, double angle) {
	double tmp;
	if (b <= 2.5e+199) {
		tmp = 2.0 * (((b - a) * (b + a)) * sin(((angle / 180.0) * ((double) M_PI))));
	} else {
		tmp = 2.0 * (0.005555555555555556 * (angle * ((b - a) * (((double) M_PI) * (b + a)))));
	}
	return tmp;
}

Java

public static double code(double a, double b, double angle) {
	double tmp;
	if (b <= 2.5e+199) {
		tmp = 2.0 * (((b - a) * (b + a)) * Math.sin(((angle / 180.0) * Math.PI)));
	} else {
		tmp = 2.0 * (0.005555555555555556 * (angle * ((b - a) * (Math.PI * (b + a)))));
	}
	return tmp;
}

Python

def code(a, b, angle):
	tmp = 0
	if b <= 2.5e+199:
		tmp = 2.0 * (((b - a) * (b + a)) * math.sin(((angle / 180.0) * math.pi)))
	else:
		tmp = 2.0 * (0.005555555555555556 * (angle * ((b - a) * (math.pi * (b + a)))))
	return tmp

Julia

function code(a, b, angle)
	tmp = 0.0
	if (b <= 2.5e+199)
		tmp = Float64(2.0 * Float64(Float64(Float64(b - a) * Float64(b + a)) * sin(Float64(Float64(angle / 180.0) * pi))));
	else
		tmp = Float64(2.0 * Float64(0.005555555555555556 * Float64(angle * Float64(Float64(b - a) * Float64(pi * Float64(b + a))))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, angle)
	tmp = 0.0;
	if (b <= 2.5e+199)
		tmp = 2.0 * (((b - a) * (b + a)) * sin(((angle / 180.0) * pi)));
	else
		tmp = 2.0 * (0.005555555555555556 * (angle * ((b - a) * (pi * (b + a)))));
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, angle_] := If[LessEqual[b, 2.5e+199], N[(2.0 * N[(N[(N[(b - a), $MachinePrecision] * N[(b + a), $MachinePrecision]), $MachinePrecision] * N[Sin[N[(N[(angle / 180.0), $MachinePrecision] * Pi), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(2.0 * N[(0.005555555555555556 * N[(angle * N[(N[(b - a), $MachinePrecision] * N[(Pi * N[(b + a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b < 2.4999999999999999e199

   1. Initial program 58.1%

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

      1. associate-*l* 58.1%

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

      2. associate-*l* 58.1%

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

      3. unpow2 58.1%

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

      4. unpow2 58.1%

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

      5. difference-of-squares 61.5%

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

   3. Simplified 61.5%

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

   4. Taylor expanded in angle around 0 58.9%

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

   if 2.4999999999999999e199 < b

   1. Initial program 39.9%

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

      1. associate-*l* 39.9%

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

      2. associate-*l* 39.9%

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

      3. unpow2 39.9%

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

      4. unpow2 39.9%

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

      5. difference-of-squares 57.5%

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

   3. Simplified 57.5%

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

   4. Taylor expanded in angle around 0 70.6%

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

4. Final simplification 59.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq 2.5 \cdot 10^{+199}:\\ \;\;\;\;2 \cdot \left(\left(\left(b - a\right) \cdot \left(b + a\right)\right) \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right)\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \left(0.005555555555555556 \cdot \left(angle \cdot \left(\left(b - a\right) \cdot \left(\pi \cdot \left(b + a\right)\right)\right)\right)\right)\\ \end{array} \]

Alternative 11: 40.1% accurate, 5.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq 2.9 \cdot 10^{-40} \lor \neg \left(b \leq 6.8 \cdot 10^{-17}\right) \land b \leq 1.46 \cdot 10^{+48}:\\ \;\;\;\;-0.011111111111111112 \cdot \left(\pi \cdot \left(angle \cdot \left(a \cdot a\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;0.011111111111111112 \cdot \left(angle \cdot \left(\pi \cdot \left(b \cdot b\right)\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (a b angle)
 :precision binary64
 (if (or (<= b 2.9e-40) (and (not (<= b 6.8e-17)) (<= b 1.46e+48)))
   (* -0.011111111111111112 (* PI (* angle (* a a))))
   (* 0.011111111111111112 (* angle (* PI (* b b))))))

C

double code(double a, double b, double angle) {
	double tmp;
	if ((b <= 2.9e-40) || (!(b <= 6.8e-17) && (b <= 1.46e+48))) {
		tmp = -0.011111111111111112 * (((double) M_PI) * (angle * (a * a)));
	} else {
		tmp = 0.011111111111111112 * (angle * (((double) M_PI) * (b * b)));
	}
	return tmp;
}

Java

public static double code(double a, double b, double angle) {
	double tmp;
	if ((b <= 2.9e-40) || (!(b <= 6.8e-17) && (b <= 1.46e+48))) {
		tmp = -0.011111111111111112 * (Math.PI * (angle * (a * a)));
	} else {
		tmp = 0.011111111111111112 * (angle * (Math.PI * (b * b)));
	}
	return tmp;
}

Python

def code(a, b, angle):
	tmp = 0
	if (b <= 2.9e-40) or (not (b <= 6.8e-17) and (b <= 1.46e+48)):
		tmp = -0.011111111111111112 * (math.pi * (angle * (a * a)))
	else:
		tmp = 0.011111111111111112 * (angle * (math.pi * (b * b)))
	return tmp

Julia

function code(a, b, angle)
	tmp = 0.0
	if ((b <= 2.9e-40) || (!(b <= 6.8e-17) && (b <= 1.46e+48)))
		tmp = Float64(-0.011111111111111112 * Float64(pi * Float64(angle * Float64(a * a))));
	else
		tmp = Float64(0.011111111111111112 * Float64(angle * Float64(pi * Float64(b * b))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, angle)
	tmp = 0.0;
	if ((b <= 2.9e-40) || (~((b <= 6.8e-17)) && (b <= 1.46e+48)))
		tmp = -0.011111111111111112 * (pi * (angle * (a * a)));
	else
		tmp = 0.011111111111111112 * (angle * (pi * (b * b)));
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, angle_] := If[Or[LessEqual[b, 2.9e-40], And[N[Not[LessEqual[b, 6.8e-17]], $MachinePrecision], LessEqual[b, 1.46e+48]]], N[(-0.011111111111111112 * N[(Pi * N[(angle * N[(a * a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(0.011111111111111112 * N[(angle * N[(Pi * N[(b * b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b < 2.8999999999999999e-40 or 6.7999999999999996e-17 < b < 1.46e48

   1. Initial program 57.3%

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

      1. associate-*l* 57.3%

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

      2. unpow2 57.3%

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

      3. unpow2 57.3%

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

   3. Simplified 57.3%

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

      1. clear-num 56.2%

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

      2. un-div-inv 58.4%

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

   5. Applied egg-rr 58.4%

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

   6. Taylor expanded in b around 0 43.0%

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

      1. associate-*r* 43.0%

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

      2. *-commutative 43.0%

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

      3. associate-*r* 44.4%

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

      4. metadata-eval 44.4%

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

      5. associate-/r/ 43.9%

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

      6. associate-*l/ 44.3%

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

      7. *-lft-identity 44.3%

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

      8. associate-*r* 45.5%

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

      9. metadata-eval 45.5%

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

      10. associate-/r/ 44.9%

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

      11. associate-*l/ 43.9%

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

      12. *-lft-identity 43.9%

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

      13. *-commutative 43.9%

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

      14. unpow2 43.9%

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

   8. Simplified 43.9%

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

   9. Taylor expanded in angle around 0 40.4%

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

       1. associate-*r* 40.4%

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

       2. unpow2 40.4%

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

   11. Simplified 40.4%

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

   if 2.8999999999999999e-40 < b < 6.7999999999999996e-17 or 1.46e48 < b

   1. Initial program 54.0%

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

      1. *-commutative 54.0%

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

      2. associate-*l* 54.0%

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

      3. unpow2 54.0%

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

      4. fma-neg 60.5%

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

      5. unpow2 60.5%

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

   3. Simplified 60.5%

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

   4. Taylor expanded in angle around 0 50.4%

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

      1. *-commutative 50.4%

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

      2. unpow2 50.4%

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

      3. unpow2 50.4%

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

   6. Simplified 50.4%

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

   7. Taylor expanded in b around inf 52.2%

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

      1. *-commutative 52.2%

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

      2. unpow2 52.2%

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

   9. Simplified 52.2%

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

4. Final simplification 43.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq 2.9 \cdot 10^{-40} \lor \neg \left(b \leq 6.8 \cdot 10^{-17}\right) \land b \leq 1.46 \cdot 10^{+48}:\\ \;\;\;\;-0.011111111111111112 \cdot \left(\pi \cdot \left(angle \cdot \left(a \cdot a\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;0.011111111111111112 \cdot \left(angle \cdot \left(\pi \cdot \left(b \cdot b\right)\right)\right)\\ \end{array} \]

Alternative 12: 40.1% accurate, 5.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := -0.011111111111111112 \cdot \left(\pi \cdot \left(angle \cdot \left(a \cdot a\right)\right)\right)\\ \mathbf{if}\;b \leq 2.1 \cdot 10^{-40}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;b \leq 6.2 \cdot 10^{-17}:\\ \;\;\;\;0.011111111111111112 \cdot \left(\pi \cdot \left(angle \cdot \left(b \cdot b\right)\right)\right)\\ \mathbf{elif}\;b \leq 1.7 \cdot 10^{+47}:\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;0.011111111111111112 \cdot \left(angle \cdot \left(\pi \cdot \left(b \cdot b\right)\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (a b angle)
 :precision binary64
 (let* ((t_0 (* -0.011111111111111112 (* PI (* angle (* a a))))))
   (if (<= b 2.1e-40)
     t_0
     (if (<= b 6.2e-17)
       (* 0.011111111111111112 (* PI (* angle (* b b))))
       (if (<= b 1.7e+47)
         t_0
         (* 0.011111111111111112 (* angle (* PI (* b b)))))))))

C

double code(double a, double b, double angle) {
	double t_0 = -0.011111111111111112 * (((double) M_PI) * (angle * (a * a)));
	double tmp;
	if (b <= 2.1e-40) {
		tmp = t_0;
	} else if (b <= 6.2e-17) {
		tmp = 0.011111111111111112 * (((double) M_PI) * (angle * (b * b)));
	} else if (b <= 1.7e+47) {
		tmp = t_0;
	} else {
		tmp = 0.011111111111111112 * (angle * (((double) M_PI) * (b * b)));
	}
	return tmp;
}

Java

public static double code(double a, double b, double angle) {
	double t_0 = -0.011111111111111112 * (Math.PI * (angle * (a * a)));
	double tmp;
	if (b <= 2.1e-40) {
		tmp = t_0;
	} else if (b <= 6.2e-17) {
		tmp = 0.011111111111111112 * (Math.PI * (angle * (b * b)));
	} else if (b <= 1.7e+47) {
		tmp = t_0;
	} else {
		tmp = 0.011111111111111112 * (angle * (Math.PI * (b * b)));
	}
	return tmp;
}

Python

def code(a, b, angle):
	t_0 = -0.011111111111111112 * (math.pi * (angle * (a * a)))
	tmp = 0
	if b <= 2.1e-40:
		tmp = t_0
	elif b <= 6.2e-17:
		tmp = 0.011111111111111112 * (math.pi * (angle * (b * b)))
	elif b <= 1.7e+47:
		tmp = t_0
	else:
		tmp = 0.011111111111111112 * (angle * (math.pi * (b * b)))
	return tmp

Julia

function code(a, b, angle)
	t_0 = Float64(-0.011111111111111112 * Float64(pi * Float64(angle * Float64(a * a))))
	tmp = 0.0
	if (b <= 2.1e-40)
		tmp = t_0;
	elseif (b <= 6.2e-17)
		tmp = Float64(0.011111111111111112 * Float64(pi * Float64(angle * Float64(b * b))));
	elseif (b <= 1.7e+47)
		tmp = t_0;
	else
		tmp = Float64(0.011111111111111112 * Float64(angle * Float64(pi * Float64(b * b))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, angle)
	t_0 = -0.011111111111111112 * (pi * (angle * (a * a)));
	tmp = 0.0;
	if (b <= 2.1e-40)
		tmp = t_0;
	elseif (b <= 6.2e-17)
		tmp = 0.011111111111111112 * (pi * (angle * (b * b)));
	elseif (b <= 1.7e+47)
		tmp = t_0;
	else
		tmp = 0.011111111111111112 * (angle * (pi * (b * b)));
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, angle_] := Block[{t$95$0 = N[(-0.011111111111111112 * N[(Pi * N[(angle * N[(a * a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[b, 2.1e-40], t$95$0, If[LessEqual[b, 6.2e-17], N[(0.011111111111111112 * N[(Pi * N[(angle * N[(b * b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[b, 1.7e+47], t$95$0, N[(0.011111111111111112 * N[(angle * N[(Pi * N[(b * b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

2. if b < 2.10000000000000018e-40 or 6.1999999999999997e-17 < b < 1.6999999999999999e47

   1. Initial program 57.3%

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

      1. associate-*l* 57.3%

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

      2. unpow2 57.3%

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

      3. unpow2 57.3%

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

   3. Simplified 57.3%

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

      1. clear-num 56.2%

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

      2. un-div-inv 58.4%

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

   5. Applied egg-rr 58.4%

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

   6. Taylor expanded in b around 0 43.0%

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

      1. associate-*r* 43.0%

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

      2. *-commutative 43.0%

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

      3. associate-*r* 44.4%

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

      4. metadata-eval 44.4%

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

      5. associate-/r/ 43.9%

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

      6. associate-*l/ 44.3%

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

      7. *-lft-identity 44.3%

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

      8. associate-*r* 45.5%

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

      9. metadata-eval 45.5%

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

      10. associate-/r/ 44.9%

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

      11. associate-*l/ 43.9%

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

      12. *-lft-identity 43.9%

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

      13. *-commutative 43.9%

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

      14. unpow2 43.9%

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

   8. Simplified 43.9%

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

   9. Taylor expanded in angle around 0 40.4%

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

       1. associate-*r* 40.4%

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

       2. unpow2 40.4%

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

   11. Simplified 40.4%

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

   if 2.10000000000000018e-40 < b < 6.1999999999999997e-17

   1. Initial program 84.5%

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

      1. *-commutative 84.5%

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

      2. associate-*l* 84.5%

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

      3. unpow2 84.5%

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

      4. fma-neg 84.5%

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

      5. unpow2 84.5%

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

   3. Simplified 84.5%

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

   4. Taylor expanded in angle around 0 66.9%

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

      1. *-commutative 66.9%

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

      2. unpow2 66.9%

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

      3. unpow2 66.9%

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

   6. Simplified 66.9%

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

   7. Taylor expanded in b around inf 51.2%

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

      1. *-commutative 51.2%

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

      2. unpow2 51.2%

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

   9. Simplified 51.2%

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

   10. Taylor expanded in angle around 0 51.2%

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

       1. associate-*r* 51.3%

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

       2. *-commutative 51.3%

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

       3. unpow2 51.3%

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

   12. Simplified 51.3%

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

   if 1.6999999999999999e47 < b

   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. Step-by-step derivation

      1. *-commutative 46.9%

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

      2. associate-*l* 46.9%

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

      3. unpow2 46.9%

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

      4. fma-neg 54.8%

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

      5. unpow2 54.8%

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

   3. Simplified 54.8%

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

   4. Taylor expanded in angle around 0 46.5%

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

      1. *-commutative 46.5%

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

      2. unpow2 46.5%

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

      3. unpow2 46.5%

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

   6. Simplified 46.5%

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

   7. Taylor expanded in b around inf 52.5%

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

      1. *-commutative 52.5%

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

      2. unpow2 52.5%

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

   9. Simplified 52.5%

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

4. Final simplification 43.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq 2.1 \cdot 10^{-40}:\\ \;\;\;\;-0.011111111111111112 \cdot \left(\pi \cdot \left(angle \cdot \left(a \cdot a\right)\right)\right)\\ \mathbf{elif}\;b \leq 6.2 \cdot 10^{-17}:\\ \;\;\;\;0.011111111111111112 \cdot \left(\pi \cdot \left(angle \cdot \left(b \cdot b\right)\right)\right)\\ \mathbf{elif}\;b \leq 1.7 \cdot 10^{+47}:\\ \;\;\;\;-0.011111111111111112 \cdot \left(\pi \cdot \left(angle \cdot \left(a \cdot a\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;0.011111111111111112 \cdot \left(angle \cdot \left(\pi \cdot \left(b \cdot b\right)\right)\right)\\ \end{array} \]

Alternative 13: 40.0% accurate, 5.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \pi \cdot \left(b \cdot b\right)\\ t_1 := -0.011111111111111112 \cdot \left(\pi \cdot \left(angle \cdot \left(a \cdot a\right)\right)\right)\\ \mathbf{if}\;b \leq 7.5 \cdot 10^{-41}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;b \leq 3.4 \cdot 10^{-16}:\\ \;\;\;\;angle \cdot \left(0.011111111111111112 \cdot t_0\right)\\ \mathbf{elif}\;b \leq 8.4 \cdot 10^{+46}:\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;0.011111111111111112 \cdot \left(angle \cdot t_0\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (a b angle)
 :precision binary64
 (let* ((t_0 (* PI (* b b)))
        (t_1 (* -0.011111111111111112 (* PI (* angle (* a a))))))
   (if (<= b 7.5e-41)
     t_1
     (if (<= b 3.4e-16)
       (* angle (* 0.011111111111111112 t_0))
       (if (<= b 8.4e+46) t_1 (* 0.011111111111111112 (* angle t_0)))))))

C

double code(double a, double b, double angle) {
	double t_0 = ((double) M_PI) * (b * b);
	double t_1 = -0.011111111111111112 * (((double) M_PI) * (angle * (a * a)));
	double tmp;
	if (b <= 7.5e-41) {
		tmp = t_1;
	} else if (b <= 3.4e-16) {
		tmp = angle * (0.011111111111111112 * t_0);
	} else if (b <= 8.4e+46) {
		tmp = t_1;
	} else {
		tmp = 0.011111111111111112 * (angle * t_0);
	}
	return tmp;
}

Java

public static double code(double a, double b, double angle) {
	double t_0 = Math.PI * (b * b);
	double t_1 = -0.011111111111111112 * (Math.PI * (angle * (a * a)));
	double tmp;
	if (b <= 7.5e-41) {
		tmp = t_1;
	} else if (b <= 3.4e-16) {
		tmp = angle * (0.011111111111111112 * t_0);
	} else if (b <= 8.4e+46) {
		tmp = t_1;
	} else {
		tmp = 0.011111111111111112 * (angle * t_0);
	}
	return tmp;
}

Python

def code(a, b, angle):
	t_0 = math.pi * (b * b)
	t_1 = -0.011111111111111112 * (math.pi * (angle * (a * a)))
	tmp = 0
	if b <= 7.5e-41:
		tmp = t_1
	elif b <= 3.4e-16:
		tmp = angle * (0.011111111111111112 * t_0)
	elif b <= 8.4e+46:
		tmp = t_1
	else:
		tmp = 0.011111111111111112 * (angle * t_0)
	return tmp

Julia

function code(a, b, angle)
	t_0 = Float64(pi * Float64(b * b))
	t_1 = Float64(-0.011111111111111112 * Float64(pi * Float64(angle * Float64(a * a))))
	tmp = 0.0
	if (b <= 7.5e-41)
		tmp = t_1;
	elseif (b <= 3.4e-16)
		tmp = Float64(angle * Float64(0.011111111111111112 * t_0));
	elseif (b <= 8.4e+46)
		tmp = t_1;
	else
		tmp = Float64(0.011111111111111112 * Float64(angle * t_0));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, angle)
	t_0 = pi * (b * b);
	t_1 = -0.011111111111111112 * (pi * (angle * (a * a)));
	tmp = 0.0;
	if (b <= 7.5e-41)
		tmp = t_1;
	elseif (b <= 3.4e-16)
		tmp = angle * (0.011111111111111112 * t_0);
	elseif (b <= 8.4e+46)
		tmp = t_1;
	else
		tmp = 0.011111111111111112 * (angle * t_0);
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, angle_] := Block[{t$95$0 = N[(Pi * N[(b * b), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(-0.011111111111111112 * N[(Pi * N[(angle * N[(a * a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[b, 7.5e-41], t$95$1, If[LessEqual[b, 3.4e-16], N[(angle * N[(0.011111111111111112 * t$95$0), $MachinePrecision]), $MachinePrecision], If[LessEqual[b, 8.4e+46], t$95$1, N[(0.011111111111111112 * N[(angle * t$95$0), $MachinePrecision]), $MachinePrecision]]]]]]

Derivation

1. Split input into 3 regimes

2. if b < 7.50000000000000049e-41 or 3.4e-16 < b < 8.4e46

   1. Initial program 57.3%

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

      1. associate-*l* 57.3%

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

      2. unpow2 57.3%

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

      3. unpow2 57.3%

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

   3. Simplified 57.3%

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

      1. clear-num 56.2%

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

      2. un-div-inv 58.4%

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

   5. Applied egg-rr 58.4%

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

   6. Taylor expanded in b around 0 43.0%

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

      1. associate-*r* 43.0%

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

      2. *-commutative 43.0%

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

      3. associate-*r* 44.4%

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

      4. metadata-eval 44.4%

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

      5. associate-/r/ 43.9%

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

      6. associate-*l/ 44.3%

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

      7. *-lft-identity 44.3%

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

      8. associate-*r* 45.5%

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

      9. metadata-eval 45.5%

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

      10. associate-/r/ 44.9%

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

      11. associate-*l/ 43.9%

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

      12. *-lft-identity 43.9%

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

      13. *-commutative 43.9%

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

      14. unpow2 43.9%

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

   8. Simplified 43.9%

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

   9. Taylor expanded in angle around 0 40.4%

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

       1. associate-*r* 40.4%

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

       2. unpow2 40.4%

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

   11. Simplified 40.4%

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

   if 7.50000000000000049e-41 < b < 3.4e-16

   1. Initial program 84.5%

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

      1. *-commutative 84.5%

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

      2. associate-*l* 84.5%

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

      3. unpow2 84.5%

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

      4. fma-neg 84.5%

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

      5. unpow2 84.5%

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

   3. Simplified 84.5%

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

   4. Taylor expanded in angle around 0 66.9%

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

      1. *-commutative 66.9%

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

      2. unpow2 66.9%

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

      3. unpow2 66.9%

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

   6. Simplified 66.9%

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

   7. Taylor expanded in b around inf 51.2%

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

      1. *-commutative 51.2%

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

      2. unpow2 51.2%

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

   9. Simplified 51.2%

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

   10. Taylor expanded in angle around 0 51.2%

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

       1. *-commutative 51.2%

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

       2. *-commutative 51.2%

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

       3. unpow2 51.2%

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

       4. associate-*l* 51.3%

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

   12. Simplified 51.3%

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

   if 8.4e46 < b

   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. Step-by-step derivation

      1. *-commutative 46.9%

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

      2. associate-*l* 46.9%

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

      3. unpow2 46.9%

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

      4. fma-neg 54.8%

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

      5. unpow2 54.8%

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

   3. Simplified 54.8%

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

   4. Taylor expanded in angle around 0 46.5%

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

      1. *-commutative 46.5%

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

      2. unpow2 46.5%

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

      3. unpow2 46.5%

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

   6. Simplified 46.5%

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

   7. Taylor expanded in b around inf 52.5%

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

      1. *-commutative 52.5%

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

      2. unpow2 52.5%

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

   9. Simplified 52.5%

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

4. Final simplification 43.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq 7.5 \cdot 10^{-41}:\\ \;\;\;\;-0.011111111111111112 \cdot \left(\pi \cdot \left(angle \cdot \left(a \cdot a\right)\right)\right)\\ \mathbf{elif}\;b \leq 3.4 \cdot 10^{-16}:\\ \;\;\;\;angle \cdot \left(0.011111111111111112 \cdot \left(\pi \cdot \left(b \cdot b\right)\right)\right)\\ \mathbf{elif}\;b \leq 8.4 \cdot 10^{+46}:\\ \;\;\;\;-0.011111111111111112 \cdot \left(\pi \cdot \left(angle \cdot \left(a \cdot a\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;0.011111111111111112 \cdot \left(angle \cdot \left(\pi \cdot \left(b \cdot b\right)\right)\right)\\ \end{array} \]

Alternative 14: 40.1% accurate, 5.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \pi \cdot \left(b \cdot b\right)\\ \mathbf{if}\;b \leq 9 \cdot 10^{-42}:\\ \;\;\;\;-0.011111111111111112 \cdot \left(\pi \cdot \left(angle \cdot \left(a \cdot a\right)\right)\right)\\ \mathbf{elif}\;b \leq 8.5 \cdot 10^{-17}:\\ \;\;\;\;angle \cdot \left(0.011111111111111112 \cdot t_0\right)\\ \mathbf{elif}\;b \leq 4.9 \cdot 10^{+45}:\\ \;\;\;\;-0.011111111111111112 \cdot \left(angle \cdot \left(\pi \cdot \left(a \cdot a\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;0.011111111111111112 \cdot \left(angle \cdot t_0\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (a b angle)
 :precision binary64
 (let* ((t_0 (* PI (* b b))))
   (if (<= b 9e-42)
     (* -0.011111111111111112 (* PI (* angle (* a a))))
     (if (<= b 8.5e-17)
       (* angle (* 0.011111111111111112 t_0))
       (if (<= b 4.9e+45)
         (* -0.011111111111111112 (* angle (* PI (* a a))))
         (* 0.011111111111111112 (* angle t_0)))))))

C

double code(double a, double b, double angle) {
	double t_0 = ((double) M_PI) * (b * b);
	double tmp;
	if (b <= 9e-42) {
		tmp = -0.011111111111111112 * (((double) M_PI) * (angle * (a * a)));
	} else if (b <= 8.5e-17) {
		tmp = angle * (0.011111111111111112 * t_0);
	} else if (b <= 4.9e+45) {
		tmp = -0.011111111111111112 * (angle * (((double) M_PI) * (a * a)));
	} else {
		tmp = 0.011111111111111112 * (angle * t_0);
	}
	return tmp;
}

Java

public static double code(double a, double b, double angle) {
	double t_0 = Math.PI * (b * b);
	double tmp;
	if (b <= 9e-42) {
		tmp = -0.011111111111111112 * (Math.PI * (angle * (a * a)));
	} else if (b <= 8.5e-17) {
		tmp = angle * (0.011111111111111112 * t_0);
	} else if (b <= 4.9e+45) {
		tmp = -0.011111111111111112 * (angle * (Math.PI * (a * a)));
	} else {
		tmp = 0.011111111111111112 * (angle * t_0);
	}
	return tmp;
}

Python

def code(a, b, angle):
	t_0 = math.pi * (b * b)
	tmp = 0
	if b <= 9e-42:
		tmp = -0.011111111111111112 * (math.pi * (angle * (a * a)))
	elif b <= 8.5e-17:
		tmp = angle * (0.011111111111111112 * t_0)
	elif b <= 4.9e+45:
		tmp = -0.011111111111111112 * (angle * (math.pi * (a * a)))
	else:
		tmp = 0.011111111111111112 * (angle * t_0)
	return tmp

Julia

function code(a, b, angle)
	t_0 = Float64(pi * Float64(b * b))
	tmp = 0.0
	if (b <= 9e-42)
		tmp = Float64(-0.011111111111111112 * Float64(pi * Float64(angle * Float64(a * a))));
	elseif (b <= 8.5e-17)
		tmp = Float64(angle * Float64(0.011111111111111112 * t_0));
	elseif (b <= 4.9e+45)
		tmp = Float64(-0.011111111111111112 * Float64(angle * Float64(pi * Float64(a * a))));
	else
		tmp = Float64(0.011111111111111112 * Float64(angle * t_0));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, angle)
	t_0 = pi * (b * b);
	tmp = 0.0;
	if (b <= 9e-42)
		tmp = -0.011111111111111112 * (pi * (angle * (a * a)));
	elseif (b <= 8.5e-17)
		tmp = angle * (0.011111111111111112 * t_0);
	elseif (b <= 4.9e+45)
		tmp = -0.011111111111111112 * (angle * (pi * (a * a)));
	else
		tmp = 0.011111111111111112 * (angle * t_0);
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, angle_] := Block[{t$95$0 = N[(Pi * N[(b * b), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[b, 9e-42], N[(-0.011111111111111112 * N[(Pi * N[(angle * N[(a * a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[b, 8.5e-17], N[(angle * N[(0.011111111111111112 * t$95$0), $MachinePrecision]), $MachinePrecision], If[LessEqual[b, 4.9e+45], N[(-0.011111111111111112 * N[(angle * N[(Pi * N[(a * a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(0.011111111111111112 * N[(angle * t$95$0), $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 4 regimes

2. if b < 9e-42

   1. Initial program 57.9%

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

      1. associate-*l* 57.9%

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

      2. unpow2 57.9%

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

      3. unpow2 57.9%

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

   3. Simplified 57.9%

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

      1. clear-num 56.7%

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

      2. un-div-inv 59.1%

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

   5. Applied egg-rr 59.1%

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

   6. Taylor expanded in b around 0 43.1%

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

      1. associate-*r* 43.1%

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

      2. *-commutative 43.1%

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

      3. associate-*r* 44.1%

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

      4. metadata-eval 44.1%

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

      5. associate-/r/ 43.5%

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

      6. associate-*l/ 44.0%

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

      7. *-lft-identity 44.0%

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

      8. associate-*r* 45.7%

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

      9. metadata-eval 45.7%

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

      10. associate-/r/ 45.2%

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

      11. associate-*l/ 44.1%

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

      12. *-lft-identity 44.1%

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

      13. *-commutative 44.1%

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

      14. unpow2 44.1%

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

   8. Simplified 44.1%

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

   9. Taylor expanded in angle around 0 39.9%

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

       1. associate-*r* 39.9%

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

       2. unpow2 39.9%

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

   11. Simplified 39.9%

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

   if 9e-42 < b < 8.5e-17

   1. Initial program 84.5%

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

      1. *-commutative 84.5%

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

      2. associate-*l* 84.5%

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

      3. unpow2 84.5%

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

      4. fma-neg 84.5%

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

      5. unpow2 84.5%

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

   3. Simplified 84.5%

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

   4. Taylor expanded in angle around 0 66.9%

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

      1. *-commutative 66.9%

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

      2. unpow2 66.9%

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

      3. unpow2 66.9%

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

   6. Simplified 66.9%

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

   7. Taylor expanded in b around inf 51.2%

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

      1. *-commutative 51.2%

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

      2. unpow2 51.2%

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

   9. Simplified 51.2%

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

   10. Taylor expanded in angle around 0 51.2%

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

       1. *-commutative 51.2%

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

       2. *-commutative 51.2%

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

       3. unpow2 51.2%

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

       4. associate-*l* 51.3%

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

   12. Simplified 51.3%

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

   if 8.5e-17 < b < 4.9000000000000002e45

   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. Step-by-step derivation

      1. *-commutative 48.3%

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

      2. associate-*l* 48.3%

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

      3. unpow2 48.3%

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

      4. fma-neg 48.3%

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

      5. unpow2 48.3%

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

   3. Simplified 48.3%

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

   4. Taylor expanded in angle around 0 56.1%

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

      1. *-commutative 56.1%

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

      2. unpow2 56.1%

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

      3. unpow2 56.1%

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

   6. Simplified 56.1%

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

   7. Taylor expanded in b around 0 48.5%

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

      1. *-commutative 48.5%

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

      2. *-commutative 48.5%

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

      3. unpow2 48.5%

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

   9. Simplified 48.5%

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

   if 4.9000000000000002e45 < b

   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. Step-by-step derivation

      1. *-commutative 46.9%

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

      2. associate-*l* 46.9%

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

      3. unpow2 46.9%

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

      4. fma-neg 54.8%

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

      5. unpow2 54.8%

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

   3. Simplified 54.8%

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

   4. Taylor expanded in angle around 0 46.5%

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

      1. *-commutative 46.5%

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

      2. unpow2 46.5%

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

      3. unpow2 46.5%

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

   6. Simplified 46.5%

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

   7. Taylor expanded in b around inf 52.5%

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

      1. *-commutative 52.5%

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

      2. unpow2 52.5%

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

   9. Simplified 52.5%

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

4. Final simplification 43.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq 9 \cdot 10^{-42}:\\ \;\;\;\;-0.011111111111111112 \cdot \left(\pi \cdot \left(angle \cdot \left(a \cdot a\right)\right)\right)\\ \mathbf{elif}\;b \leq 8.5 \cdot 10^{-17}:\\ \;\;\;\;angle \cdot \left(0.011111111111111112 \cdot \left(\pi \cdot \left(b \cdot b\right)\right)\right)\\ \mathbf{elif}\;b \leq 4.9 \cdot 10^{+45}:\\ \;\;\;\;-0.011111111111111112 \cdot \left(angle \cdot \left(\pi \cdot \left(a \cdot a\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;0.011111111111111112 \cdot \left(angle \cdot \left(\pi \cdot \left(b \cdot b\right)\right)\right)\\ \end{array} \]

Alternative 15: 40.0% accurate, 5.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \pi \cdot \left(b \cdot b\right)\\ \mathbf{if}\;b \leq 2.8 \cdot 10^{-41}:\\ \;\;\;\;-0.011111111111111112 \cdot \left(\pi \cdot \left(angle \cdot \left(a \cdot a\right)\right)\right)\\ \mathbf{elif}\;b \leq 3.6 \cdot 10^{-16}:\\ \;\;\;\;t_0 \cdot \left(angle \cdot 0.011111111111111112\right)\\ \mathbf{elif}\;b \leq 8.5 \cdot 10^{+47}:\\ \;\;\;\;-0.011111111111111112 \cdot \left(angle \cdot \left(\pi \cdot \left(a \cdot a\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;0.011111111111111112 \cdot \left(angle \cdot t_0\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (a b angle)
 :precision binary64
 (let* ((t_0 (* PI (* b b))))
   (if (<= b 2.8e-41)
     (* -0.011111111111111112 (* PI (* angle (* a a))))
     (if (<= b 3.6e-16)
       (* t_0 (* angle 0.011111111111111112))
       (if (<= b 8.5e+47)
         (* -0.011111111111111112 (* angle (* PI (* a a))))
         (* 0.011111111111111112 (* angle t_0)))))))

C

double code(double a, double b, double angle) {
	double t_0 = ((double) M_PI) * (b * b);
	double tmp;
	if (b <= 2.8e-41) {
		tmp = -0.011111111111111112 * (((double) M_PI) * (angle * (a * a)));
	} else if (b <= 3.6e-16) {
		tmp = t_0 * (angle * 0.011111111111111112);
	} else if (b <= 8.5e+47) {
		tmp = -0.011111111111111112 * (angle * (((double) M_PI) * (a * a)));
	} else {
		tmp = 0.011111111111111112 * (angle * t_0);
	}
	return tmp;
}

Java

public static double code(double a, double b, double angle) {
	double t_0 = Math.PI * (b * b);
	double tmp;
	if (b <= 2.8e-41) {
		tmp = -0.011111111111111112 * (Math.PI * (angle * (a * a)));
	} else if (b <= 3.6e-16) {
		tmp = t_0 * (angle * 0.011111111111111112);
	} else if (b <= 8.5e+47) {
		tmp = -0.011111111111111112 * (angle * (Math.PI * (a * a)));
	} else {
		tmp = 0.011111111111111112 * (angle * t_0);
	}
	return tmp;
}

Python

def code(a, b, angle):
	t_0 = math.pi * (b * b)
	tmp = 0
	if b <= 2.8e-41:
		tmp = -0.011111111111111112 * (math.pi * (angle * (a * a)))
	elif b <= 3.6e-16:
		tmp = t_0 * (angle * 0.011111111111111112)
	elif b <= 8.5e+47:
		tmp = -0.011111111111111112 * (angle * (math.pi * (a * a)))
	else:
		tmp = 0.011111111111111112 * (angle * t_0)
	return tmp

Julia

function code(a, b, angle)
	t_0 = Float64(pi * Float64(b * b))
	tmp = 0.0
	if (b <= 2.8e-41)
		tmp = Float64(-0.011111111111111112 * Float64(pi * Float64(angle * Float64(a * a))));
	elseif (b <= 3.6e-16)
		tmp = Float64(t_0 * Float64(angle * 0.011111111111111112));
	elseif (b <= 8.5e+47)
		tmp = Float64(-0.011111111111111112 * Float64(angle * Float64(pi * Float64(a * a))));
	else
		tmp = Float64(0.011111111111111112 * Float64(angle * t_0));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, angle)
	t_0 = pi * (b * b);
	tmp = 0.0;
	if (b <= 2.8e-41)
		tmp = -0.011111111111111112 * (pi * (angle * (a * a)));
	elseif (b <= 3.6e-16)
		tmp = t_0 * (angle * 0.011111111111111112);
	elseif (b <= 8.5e+47)
		tmp = -0.011111111111111112 * (angle * (pi * (a * a)));
	else
		tmp = 0.011111111111111112 * (angle * t_0);
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, angle_] := Block[{t$95$0 = N[(Pi * N[(b * b), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[b, 2.8e-41], N[(-0.011111111111111112 * N[(Pi * N[(angle * N[(a * a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[b, 3.6e-16], N[(t$95$0 * N[(angle * 0.011111111111111112), $MachinePrecision]), $MachinePrecision], If[LessEqual[b, 8.5e+47], N[(-0.011111111111111112 * N[(angle * N[(Pi * N[(a * a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(0.011111111111111112 * N[(angle * t$95$0), $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 4 regimes

2. if b < 2.8000000000000002e-41

   1. Initial program 57.9%

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

      1. associate-*l* 57.9%

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

      2. unpow2 57.9%

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

      3. unpow2 57.9%

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

   3. Simplified 57.9%

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

      1. clear-num 56.7%

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

      2. un-div-inv 59.1%

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

   5. Applied egg-rr 59.1%

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

   6. Taylor expanded in b around 0 43.1%

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

      1. associate-*r* 43.1%

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

      2. *-commutative 43.1%

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

      3. associate-*r* 44.1%

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

      4. metadata-eval 44.1%

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

      5. associate-/r/ 43.5%

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

      6. associate-*l/ 44.0%

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

      7. *-lft-identity 44.0%

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

      8. associate-*r* 45.7%

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

      9. metadata-eval 45.7%

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

      10. associate-/r/ 45.2%

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

      11. associate-*l/ 44.1%

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

      12. *-lft-identity 44.1%

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

      13. *-commutative 44.1%

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

      14. unpow2 44.1%

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

   8. Simplified 44.1%

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

   9. Taylor expanded in angle around 0 39.9%

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

       1. associate-*r* 39.9%

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

       2. unpow2 39.9%

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

   11. Simplified 39.9%

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

   if 2.8000000000000002e-41 < b < 3.59999999999999983e-16

   1. Initial program 84.5%

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

      1. *-commutative 84.5%

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

      2. associate-*l* 84.5%

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

      3. unpow2 84.5%

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

      4. fma-neg 84.5%

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

      5. unpow2 84.5%

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

   3. Simplified 84.5%

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

   4. Taylor expanded in angle around 0 66.9%

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

      1. *-commutative 66.9%

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

      2. unpow2 66.9%

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

      3. unpow2 66.9%

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

   6. Simplified 66.9%

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

   7. Taylor expanded in b around inf 51.2%

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

      1. associate-*r* 51.5%

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

      2. *-commutative 51.5%

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

      3. *-commutative 51.5%

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

      4. unpow2 51.5%

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

      5. *-commutative 51.5%

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

   9. Simplified 51.5%

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

   if 3.59999999999999983e-16 < b < 8.5000000000000008e47

   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. Step-by-step derivation

      1. *-commutative 48.3%

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

      2. associate-*l* 48.3%

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

      3. unpow2 48.3%

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

      4. fma-neg 48.3%

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

      5. unpow2 48.3%

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

   3. Simplified 48.3%

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

   4. Taylor expanded in angle around 0 56.1%

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

      1. *-commutative 56.1%

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

      2. unpow2 56.1%

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

      3. unpow2 56.1%

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

   6. Simplified 56.1%

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

   7. Taylor expanded in b around 0 48.5%

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

      1. *-commutative 48.5%

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

      2. *-commutative 48.5%

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

      3. unpow2 48.5%

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

   9. Simplified 48.5%

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

   if 8.5000000000000008e47 < b

   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. Step-by-step derivation

      1. *-commutative 46.9%

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

      2. associate-*l* 46.9%

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

      3. unpow2 46.9%

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

      4. fma-neg 54.8%

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

      5. unpow2 54.8%

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

   3. Simplified 54.8%

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

   4. Taylor expanded in angle around 0 46.5%

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

      1. *-commutative 46.5%

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

      2. unpow2 46.5%

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

      3. unpow2 46.5%

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

   6. Simplified 46.5%

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

   7. Taylor expanded in b around inf 52.5%

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

      1. *-commutative 52.5%

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

      2. unpow2 52.5%

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

   9. Simplified 52.5%

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

4. Final simplification 43.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq 2.8 \cdot 10^{-41}:\\ \;\;\;\;-0.011111111111111112 \cdot \left(\pi \cdot \left(angle \cdot \left(a \cdot a\right)\right)\right)\\ \mathbf{elif}\;b \leq 3.6 \cdot 10^{-16}:\\ \;\;\;\;\left(\pi \cdot \left(b \cdot b\right)\right) \cdot \left(angle \cdot 0.011111111111111112\right)\\ \mathbf{elif}\;b \leq 8.5 \cdot 10^{+47}:\\ \;\;\;\;-0.011111111111111112 \cdot \left(angle \cdot \left(\pi \cdot \left(a \cdot a\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;0.011111111111111112 \cdot \left(angle \cdot \left(\pi \cdot \left(b \cdot b\right)\right)\right)\\ \end{array} \]

Alternative 16: 51.4% accurate, 5.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq 3 \cdot 10^{+139}:\\ \;\;\;\;0.011111111111111112 \cdot \left(angle \cdot \left(\pi \cdot \left(b \cdot b - a \cdot a\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;0.011111111111111112 \cdot \left(angle \cdot \left(\pi \cdot \left(b \cdot b\right)\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (a b angle)
 :precision binary64
 (if (<= b 3e+139)
   (* 0.011111111111111112 (* angle (* PI (- (* b b) (* a a)))))
   (* 0.011111111111111112 (* angle (* PI (* b b))))))

C

double code(double a, double b, double angle) {
	double tmp;
	if (b <= 3e+139) {
		tmp = 0.011111111111111112 * (angle * (((double) M_PI) * ((b * b) - (a * a))));
	} else {
		tmp = 0.011111111111111112 * (angle * (((double) M_PI) * (b * b)));
	}
	return tmp;
}

Java

public static double code(double a, double b, double angle) {
	double tmp;
	if (b <= 3e+139) {
		tmp = 0.011111111111111112 * (angle * (Math.PI * ((b * b) - (a * a))));
	} else {
		tmp = 0.011111111111111112 * (angle * (Math.PI * (b * b)));
	}
	return tmp;
}

Python

def code(a, b, angle):
	tmp = 0
	if b <= 3e+139:
		tmp = 0.011111111111111112 * (angle * (math.pi * ((b * b) - (a * a))))
	else:
		tmp = 0.011111111111111112 * (angle * (math.pi * (b * b)))
	return tmp

Julia

function code(a, b, angle)
	tmp = 0.0
	if (b <= 3e+139)
		tmp = Float64(0.011111111111111112 * Float64(angle * Float64(pi * Float64(Float64(b * b) - Float64(a * a)))));
	else
		tmp = Float64(0.011111111111111112 * Float64(angle * Float64(pi * Float64(b * b))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, angle)
	tmp = 0.0;
	if (b <= 3e+139)
		tmp = 0.011111111111111112 * (angle * (pi * ((b * b) - (a * a))));
	else
		tmp = 0.011111111111111112 * (angle * (pi * (b * b)));
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, angle_] := If[LessEqual[b, 3e+139], N[(0.011111111111111112 * N[(angle * N[(Pi * N[(N[(b * b), $MachinePrecision] - N[(a * a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(0.011111111111111112 * N[(angle * N[(Pi * N[(b * b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b < 3e139

   1. Initial program 59.6%

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

      1. *-commutative 59.6%

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

      2. associate-*l* 59.6%

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

      3. unpow2 59.6%

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

      4. fma-neg 60.9%

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

      5. unpow2 60.9%

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

   3. Simplified 60.9%

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

   4. Taylor expanded in angle around 0 53.1%

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

      1. *-commutative 53.1%

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

      2. unpow2 53.1%

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

      3. unpow2 53.1%

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

   6. Simplified 53.1%

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

   if 3e139 < b

   1. Initial program 31.9%

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

      1. *-commutative 31.9%

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

      2. associate-*l* 31.9%

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

      3. unpow2 31.9%

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

      4. fma-neg 45.9%

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

      5. unpow2 45.9%

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

   3. Simplified 45.9%

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

   4. Taylor expanded in angle around 0 35.4%

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

      1. *-commutative 35.4%

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

      2. unpow2 35.4%

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

      3. unpow2 35.4%

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

   6. Simplified 35.4%

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

   7. Taylor expanded in b around inf 63.1%

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

      1. *-commutative 63.1%

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

      2. unpow2 63.1%

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

   9. Simplified 63.1%

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

4. Final simplification 54.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq 3 \cdot 10^{+139}:\\ \;\;\;\;0.011111111111111112 \cdot \left(angle \cdot \left(\pi \cdot \left(b \cdot b - a \cdot a\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;0.011111111111111112 \cdot \left(angle \cdot \left(\pi \cdot \left(b \cdot b\right)\right)\right)\\ \end{array} \]

Alternative 17: 51.4% accurate, 5.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq 1.4 \cdot 10^{+139}:\\ \;\;\;\;angle \cdot \left(0.011111111111111112 \cdot \left(\pi \cdot \left(b \cdot b - a \cdot a\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;0.011111111111111112 \cdot \left(angle \cdot \left(\pi \cdot \left(b \cdot b\right)\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (a b angle)
 :precision binary64
 (if (<= b 1.4e+139)
   (* angle (* 0.011111111111111112 (* PI (- (* b b) (* a a)))))
   (* 0.011111111111111112 (* angle (* PI (* b b))))))

C

double code(double a, double b, double angle) {
	double tmp;
	if (b <= 1.4e+139) {
		tmp = angle * (0.011111111111111112 * (((double) M_PI) * ((b * b) - (a * a))));
	} else {
		tmp = 0.011111111111111112 * (angle * (((double) M_PI) * (b * b)));
	}
	return tmp;
}

Java

public static double code(double a, double b, double angle) {
	double tmp;
	if (b <= 1.4e+139) {
		tmp = angle * (0.011111111111111112 * (Math.PI * ((b * b) - (a * a))));
	} else {
		tmp = 0.011111111111111112 * (angle * (Math.PI * (b * b)));
	}
	return tmp;
}

Python

def code(a, b, angle):
	tmp = 0
	if b <= 1.4e+139:
		tmp = angle * (0.011111111111111112 * (math.pi * ((b * b) - (a * a))))
	else:
		tmp = 0.011111111111111112 * (angle * (math.pi * (b * b)))
	return tmp

Julia

function code(a, b, angle)
	tmp = 0.0
	if (b <= 1.4e+139)
		tmp = Float64(angle * Float64(0.011111111111111112 * Float64(pi * Float64(Float64(b * b) - Float64(a * a)))));
	else
		tmp = Float64(0.011111111111111112 * Float64(angle * Float64(pi * Float64(b * b))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, angle)
	tmp = 0.0;
	if (b <= 1.4e+139)
		tmp = angle * (0.011111111111111112 * (pi * ((b * b) - (a * a))));
	else
		tmp = 0.011111111111111112 * (angle * (pi * (b * b)));
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, angle_] := If[LessEqual[b, 1.4e+139], N[(angle * N[(0.011111111111111112 * N[(Pi * N[(N[(b * b), $MachinePrecision] - N[(a * a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(0.011111111111111112 * N[(angle * N[(Pi * N[(b * b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b < 1.3999999999999999e139

   1. Initial program 59.6%

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

      1. *-commutative 59.6%

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

      2. associate-*l* 59.6%

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

      3. unpow2 59.6%

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

      4. fma-neg 60.9%

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

      5. unpow2 60.9%

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

   3. Simplified 60.9%

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

   4. Taylor expanded in angle around 0 53.1%

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

      1. *-commutative 53.1%

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

      2. unpow2 53.1%

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

      3. unpow2 53.1%

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

   6. Simplified 53.1%

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

   7. Taylor expanded in angle around 0 53.1%

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

      1. *-commutative 53.1%

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

      2. unpow2 53.1%

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

      3. unpow2 53.1%

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

      4. *-commutative 53.1%

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

      5. associate-*l* 53.1%

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

   9. Simplified 53.1%

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

   if 1.3999999999999999e139 < b

   1. Initial program 31.9%

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

      1. *-commutative 31.9%

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

      2. associate-*l* 31.9%

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

      3. unpow2 31.9%

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

      4. fma-neg 45.9%

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

      5. unpow2 45.9%

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

   3. Simplified 45.9%

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

   4. Taylor expanded in angle around 0 35.4%

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

      1. *-commutative 35.4%

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

      2. unpow2 35.4%

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

      3. unpow2 35.4%

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

   6. Simplified 35.4%

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

   7. Taylor expanded in b around inf 63.1%

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

      1. *-commutative 63.1%

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

      2. unpow2 63.1%

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

   9. Simplified 63.1%

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

4. Final simplification 54.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq 1.4 \cdot 10^{+139}:\\ \;\;\;\;angle \cdot \left(0.011111111111111112 \cdot \left(\pi \cdot \left(b \cdot b - a \cdot a\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;0.011111111111111112 \cdot \left(angle \cdot \left(\pi \cdot \left(b \cdot b\right)\right)\right)\\ \end{array} \]

Alternative 18: 53.6% accurate, 5.4× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 56.5%

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

   1. associate-*l* 56.5%

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

   2. associate-*l* 56.5%

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

   3. unpow2 56.5%

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

   4. unpow2 56.5%

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

   5. difference-of-squares 61.2%

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

3. Simplified 61.2%

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

4. Taylor expanded in angle around 0 56.2%

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

5. Final simplification 56.2%

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

Alternative 19: 53.6% accurate, 5.4× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 56.5%

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

   1. associate-*l* 56.5%

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

   2. associate-*l* 56.5%

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

   3. unpow2 56.5%

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

   4. unpow2 56.5%

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

   5. difference-of-squares 61.2%

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

3. Simplified 61.2%

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

4. Taylor expanded in angle around 0 56.2%

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

   1. associate-*r* 56.2%

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

   2. *-commutative 56.2%

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

   3. *-commutative 56.2%

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

   4. associate-*r* 56.2%

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

   5. +-commutative 56.2%

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

   6. *-commutative 56.2%

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

   7. +-commutative 56.2%

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

   8. *-commutative 56.2%

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

6. Simplified 56.2%

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

7. Final simplification 56.2%

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

Alternative 20: 34.0% accurate, 5.7× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 56.5%

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

   1. associate-*l* 56.5%

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

   2. unpow2 56.5%

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

   3. unpow2 56.5%

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

3. Simplified 56.5%

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

   1. clear-num 55.5%

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

   2. un-div-inv 57.3%

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

5. Applied egg-rr 57.3%

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

6. Taylor expanded in b around 0 38.2%

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

   1. associate-*r* 38.2%

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

   2. *-commutative 38.2%

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

   3. associate-*r* 40.0%

      \[\leadsto \left(\cos \color{blue}{\left(\left(0.005555555555555556 \cdot angle\right) \cdot \pi\right)} \cdot \sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)\right) \cdot \left(-2 \cdot {a}^{2}\right) \]

   4. metadata-eval 40.0%

      \[\leadsto \left(\cos \left(\left(\color{blue}{\frac{1}{180}} \cdot angle\right) \cdot \pi\right) \cdot \sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)\right) \cdot \left(-2 \cdot {a}^{2}\right) \]

   5. associate-/r/ 39.2%

      \[\leadsto \left(\cos \left(\color{blue}{\frac{1}{\frac{180}{angle}}} \cdot \pi\right) \cdot \sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)\right) \cdot \left(-2 \cdot {a}^{2}\right) \]

   6. associate-*l/ 40.0%

      \[\leadsto \left(\cos \color{blue}{\left(\frac{1 \cdot \pi}{\frac{180}{angle}}\right)} \cdot \sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)\right) \cdot \left(-2 \cdot {a}^{2}\right) \]

   7. *-lft-identity 40.0%

      \[\leadsto \left(\cos \left(\frac{\color{blue}{\pi}}{\frac{180}{angle}}\right) \cdot \sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)\right) \cdot \left(-2 \cdot {a}^{2}\right) \]

   8. associate-*r* 40.5%

      \[\leadsto \left(\cos \left(\frac{\pi}{\frac{180}{angle}}\right) \cdot \sin \color{blue}{\left(\left(0.005555555555555556 \cdot angle\right) \cdot \pi\right)}\right) \cdot \left(-2 \cdot {a}^{2}\right) \]

   9. metadata-eval 40.5%

      \[\leadsto \left(\cos \left(\frac{\pi}{\frac{180}{angle}}\right) \cdot \sin \left(\left(\color{blue}{\frac{1}{180}} \cdot angle\right) \cdot \pi\right)\right) \cdot \left(-2 \cdot {a}^{2}\right) \]

   10. associate-/r/ 40.1%

       \[\leadsto \left(\cos \left(\frac{\pi}{\frac{180}{angle}}\right) \cdot \sin \left(\color{blue}{\frac{1}{\frac{180}{angle}}} \cdot \pi\right)\right) \cdot \left(-2 \cdot {a}^{2}\right) \]

   11. associate-*l/ 38.5%

       \[\leadsto \left(\cos \left(\frac{\pi}{\frac{180}{angle}}\right) \cdot \sin \color{blue}{\left(\frac{1 \cdot \pi}{\frac{180}{angle}}\right)}\right) \cdot \left(-2 \cdot {a}^{2}\right) \]

   12. *-lft-identity 38.5%

       \[\leadsto \left(\cos \left(\frac{\pi}{\frac{180}{angle}}\right) \cdot \sin \left(\frac{\color{blue}{\pi}}{\frac{180}{angle}}\right)\right) \cdot \left(-2 \cdot {a}^{2}\right) \]

   13. *-commutative 38.5%

       \[\leadsto \left(\cos \left(\frac{\pi}{\frac{180}{angle}}\right) \cdot \sin \left(\frac{\pi}{\frac{180}{angle}}\right)\right) \cdot \color{blue}{\left({a}^{2} \cdot -2\right)} \]

   14. unpow2 38.6%

       \[\leadsto \left(\cos \left(\frac{\pi}{\frac{180}{angle}}\right) \cdot \sin \left(\frac{\pi}{\frac{180}{angle}}\right)\right) \cdot \left(\color{blue}{\left(a \cdot a\right)} \cdot -2\right) \]

8. Simplified 38.6%

   \[\leadsto \color{blue}{\left(\cos \left(\frac{\pi}{\frac{180}{angle}}\right) \cdot \sin \left(\frac{\pi}{\frac{180}{angle}}\right)\right) \cdot \left(\left(a \cdot a\right) \cdot -2\right)} \]

9. Taylor expanded in angle around 0 35.1%

   \[\leadsto \color{blue}{-0.011111111111111112 \cdot \left(angle \cdot \left({a}^{2} \cdot \pi\right)\right)} \]
10. Step-by-step derivation

    1. associate-*r* 35.1%

       \[\leadsto -0.011111111111111112 \cdot \color{blue}{\left(\left(angle \cdot {a}^{2}\right) \cdot \pi\right)} \]

    2. unpow2 35.1%

       \[\leadsto -0.011111111111111112 \cdot \left(\left(angle \cdot \color{blue}{\left(a \cdot a\right)}\right) \cdot \pi\right) \]

11. Simplified 35.1%

    \[\leadsto \color{blue}{-0.011111111111111112 \cdot \left(\left(angle \cdot \left(a \cdot a\right)\right) \cdot \pi\right)} \]

12. Final simplification 35.1%

    \[\leadsto -0.011111111111111112 \cdot \left(\pi \cdot \left(angle \cdot \left(a \cdot a\right)\right)\right) \]

Reproduce

herbie shell --seed 2023238 
(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)))))