ab-angle->ABCF B

Percentage Accurate: 53.9% → 67.2%

Time: 35.7s

Alternatives: 17

Speedup: 5.3×

• 32.8% of points produce a very large (infinite) output. You may want to add a precondition.

Specification

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 53.9% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 67.2% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{angle}{180} \cdot \pi\\ t_1 := \sqrt[3]{\sqrt{t_0}}\\ t_2 := \sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right) \cdot \left(b + a\right)\\ t_3 := \sqrt[3]{\sqrt[3]{\pi \cdot \left(angle \cdot 0.005555555555555556\right)}}\\ \mathbf{if}\;\frac{angle}{180} \leq -1 \cdot 10^{-66}:\\ \;\;\;\;\left(2 \cdot \left(\left(b - a\right) \cdot \left(b + a\right)\right)\right) \cdot \left(\sin t_0 \cdot \cos \left(\frac{\pi}{\frac{180}{angle}}\right)\right)\\ \mathbf{elif}\;\frac{angle}{180} \leq 4 \cdot 10^{+121}:\\ \;\;\;\;2 \cdot \left(t_2 \cdot \left(\left(b - a\right) \cdot \cos \left(\mathsf{expm1}\left(\mathsf{log1p}\left(t_0\right)\right)\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \left(t_2 \cdot \left(\left(b - a\right) \cdot \cos \left({\left(t_3 \cdot t_3\right)}^{3} \cdot \left(t_1 \cdot t_1\right)\right)\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (a b angle)
 :precision binary64
 (let* ((t_0 (* (/ angle 180.0) PI))
        (t_1 (cbrt (sqrt t_0)))
        (t_2 (* (sin (* 0.005555555555555556 (* angle PI))) (+ b a)))
        (t_3 (cbrt (cbrt (* PI (* angle 0.005555555555555556))))))
   (if (<= (/ angle 180.0) -1e-66)
     (* (* 2.0 (* (- b a) (+ b a))) (* (sin t_0) (cos (/ PI (/ 180.0 angle)))))
     (if (<= (/ angle 180.0) 4e+121)
       (* 2.0 (* t_2 (* (- b a) (cos (expm1 (log1p t_0))))))
       (*
        2.0
        (* t_2 (* (- b a) (cos (* (pow (* t_3 t_3) 3.0) (* t_1 t_1))))))))))

C

double code(double a, double b, double angle) {
	double t_0 = (angle / 180.0) * ((double) M_PI);
	double t_1 = cbrt(sqrt(t_0));
	double t_2 = sin((0.005555555555555556 * (angle * ((double) M_PI)))) * (b + a);
	double t_3 = cbrt(cbrt((((double) M_PI) * (angle * 0.005555555555555556))));
	double tmp;
	if ((angle / 180.0) <= -1e-66) {
		tmp = (2.0 * ((b - a) * (b + a))) * (sin(t_0) * cos((((double) M_PI) / (180.0 / angle))));
	} else if ((angle / 180.0) <= 4e+121) {
		tmp = 2.0 * (t_2 * ((b - a) * cos(expm1(log1p(t_0)))));
	} else {
		tmp = 2.0 * (t_2 * ((b - a) * cos((pow((t_3 * t_3), 3.0) * (t_1 * t_1)))));
	}
	return tmp;
}

Java

public static double code(double a, double b, double angle) {
	double t_0 = (angle / 180.0) * Math.PI;
	double t_1 = Math.cbrt(Math.sqrt(t_0));
	double t_2 = Math.sin((0.005555555555555556 * (angle * Math.PI))) * (b + a);
	double t_3 = Math.cbrt(Math.cbrt((Math.PI * (angle * 0.005555555555555556))));
	double tmp;
	if ((angle / 180.0) <= -1e-66) {
		tmp = (2.0 * ((b - a) * (b + a))) * (Math.sin(t_0) * Math.cos((Math.PI / (180.0 / angle))));
	} else if ((angle / 180.0) <= 4e+121) {
		tmp = 2.0 * (t_2 * ((b - a) * Math.cos(Math.expm1(Math.log1p(t_0)))));
	} else {
		tmp = 2.0 * (t_2 * ((b - a) * Math.cos((Math.pow((t_3 * t_3), 3.0) * (t_1 * t_1)))));
	}
	return tmp;
}

Julia

function code(a, b, angle)
	t_0 = Float64(Float64(angle / 180.0) * pi)
	t_1 = cbrt(sqrt(t_0))
	t_2 = Float64(sin(Float64(0.005555555555555556 * Float64(angle * pi))) * Float64(b + a))
	t_3 = cbrt(cbrt(Float64(pi * Float64(angle * 0.005555555555555556))))
	tmp = 0.0
	if (Float64(angle / 180.0) <= -1e-66)
		tmp = Float64(Float64(2.0 * Float64(Float64(b - a) * Float64(b + a))) * Float64(sin(t_0) * cos(Float64(pi / Float64(180.0 / angle)))));
	elseif (Float64(angle / 180.0) <= 4e+121)
		tmp = Float64(2.0 * Float64(t_2 * Float64(Float64(b - a) * cos(expm1(log1p(t_0))))));
	else
		tmp = Float64(2.0 * Float64(t_2 * Float64(Float64(b - a) * cos(Float64((Float64(t_3 * t_3) ^ 3.0) * Float64(t_1 * t_1))))));
	end
	return tmp
end

Wolfram

code[a_, b_, angle_] := Block[{t$95$0 = N[(N[(angle / 180.0), $MachinePrecision] * Pi), $MachinePrecision]}, Block[{t$95$1 = N[Power[N[Sqrt[t$95$0], $MachinePrecision], 1/3], $MachinePrecision]}, Block[{t$95$2 = N[(N[Sin[N[(0.005555555555555556 * N[(angle * Pi), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * N[(b + a), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$3 = N[Power[N[Power[N[(Pi * N[(angle * 0.005555555555555556), $MachinePrecision]), $MachinePrecision], 1/3], $MachinePrecision], 1/3], $MachinePrecision]}, If[LessEqual[N[(angle / 180.0), $MachinePrecision], -1e-66], N[(N[(2.0 * N[(N[(b - a), $MachinePrecision] * N[(b + a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(N[Sin[t$95$0], $MachinePrecision] * N[Cos[N[(Pi / N[(180.0 / angle), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[N[(angle / 180.0), $MachinePrecision], 4e+121], N[(2.0 * N[(t$95$2 * N[(N[(b - a), $MachinePrecision] * N[Cos[N[(Exp[N[Log[1 + t$95$0], $MachinePrecision]] - 1), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(2.0 * N[(t$95$2 * N[(N[(b - a), $MachinePrecision] * N[Cos[N[(N[Power[N[(t$95$3 * t$95$3), $MachinePrecision], 3.0], $MachinePrecision] * N[(t$95$1 * t$95$1), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]]]

Derivation

1. Split input into 3 regimes

2. if (/.f64 angle 180) < -9.9999999999999998e-67

   1. Initial program 47.0%

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

      1. associate-*l* 47.0%

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

      2. unpow2 47.0%

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

      3. unpow2 47.0%

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

      4. difference-of-squares 52.5%

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

   3. Simplified 52.5%

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

      1. clear-num 55.2%

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

      2. un-div-inv 56.7%

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

   5. Applied egg-rr 56.7%

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

   if -9.9999999999999998e-67 < (/.f64 angle 180) < 4.00000000000000015e121

   1. Initial program 67.1%

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

      1. associate-*l* 67.1%

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

      2. associate-*l* 67.1%

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

      3. unpow2 67.1%

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

      4. unpow2 67.1%

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

      5. difference-of-squares 71.6%

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

   3. Simplified 71.6%

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

   4. Taylor expanded in angle around inf 90.1%

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

      1. associate-*r* 90.1%

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

      2. *-commutative 90.1%

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

      3. *-commutative 90.1%

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

   6. Simplified 90.1%

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

      1. associate-*r* 90.0%

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

      2. *-commutative 90.0%

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

      3. metadata-eval 90.0%

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

      4. div-inv 90.5%

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

      5. *-commutative 90.5%

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

      6. expm1-log1p-u 93.4%

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

   8. Applied egg-rr 93.4%

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

   if 4.00000000000000015e121 < (/.f64 angle 180)

   1. Initial program 22.2%

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

      1. associate-*l* 22.2%

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

      2. associate-*l* 22.2%

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

      3. unpow2 22.2%

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

      4. unpow2 22.2%

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

      5. difference-of-squares 22.2%

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

   3. Simplified 22.2%

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

   4. Taylor expanded in angle around inf 28.5%

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

      1. associate-*r* 28.5%

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

      2. *-commutative 28.5%

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

      3. *-commutative 28.5%

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

   6. Simplified 28.5%

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

      1. *-commutative 28.5%

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

      2. *-commutative 28.5%

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

      3. associate-*r* 24.4%

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

      4. add-cube-cbrt 36.5%

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

      5. unpow3 34.9%

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

      6. associate-*r* 34.4%

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

      7. *-commutative 34.4%

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

      8. *-commutative 34.4%

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

      9. associate-*r* 34.9%

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

      10. *-commutative 34.9%

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

      11. metadata-eval 34.9%

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

      12. div-inv 34.0%

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

      13. *-commutative 34.0%

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

   8. Applied egg-rr 34.0%

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

      1. add-cube-cbrt 37.3%

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

      2. unpow-prod-down 31.0%

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

      3. div-inv 30.6%

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

      4. metadata-eval 30.6%

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

      5. div-inv 31.1%

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

      6. metadata-eval 31.1%

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

   10. Applied egg-rr 31.0%

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

       1. rem-cube-cbrt 28.1%

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

       2. pow1/3 26.1%

          \[\leadsto 2 \cdot \left(\left(\sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right) \cdot \left(a + b\right)\right) \cdot \left(\cos \left({\left(\sqrt[3]{\sqrt[3]{\pi \cdot \left(angle \cdot 0.005555555555555556\right)}} \cdot \sqrt[3]{\sqrt[3]{\pi \cdot \left(angle \cdot 0.005555555555555556\right)}}\right)}^{3} \cdot \color{blue}{{\left(\pi \cdot \left(angle \cdot 0.005555555555555556\right)\right)}^{0.3333333333333333}}\right) \cdot \left(b - a\right)\right)\right) \]

       3. metadata-eval 26.1%

          \[\leadsto 2 \cdot \left(\left(\sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right) \cdot \left(a + b\right)\right) \cdot \left(\cos \left({\left(\sqrt[3]{\sqrt[3]{\pi \cdot \left(angle \cdot 0.005555555555555556\right)}} \cdot \sqrt[3]{\sqrt[3]{\pi \cdot \left(angle \cdot 0.005555555555555556\right)}}\right)}^{3} \cdot {\left(\pi \cdot \left(angle \cdot \color{blue}{\frac{1}{180}}\right)\right)}^{0.3333333333333333}\right) \cdot \left(b - a\right)\right)\right) \]

       4. div-inv 26.1%

          \[\leadsto 2 \cdot \left(\left(\sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right) \cdot \left(a + b\right)\right) \cdot \left(\cos \left({\left(\sqrt[3]{\sqrt[3]{\pi \cdot \left(angle \cdot 0.005555555555555556\right)}} \cdot \sqrt[3]{\sqrt[3]{\pi \cdot \left(angle \cdot 0.005555555555555556\right)}}\right)}^{3} \cdot {\left(\pi \cdot \color{blue}{\frac{angle}{180}}\right)}^{0.3333333333333333}\right) \cdot \left(b - a\right)\right)\right) \]

       5. add-sqr-sqrt 25.7%

          \[\leadsto 2 \cdot \left(\left(\sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right) \cdot \left(a + b\right)\right) \cdot \left(\cos \left({\left(\sqrt[3]{\sqrt[3]{\pi \cdot \left(angle \cdot 0.005555555555555556\right)}} \cdot \sqrt[3]{\sqrt[3]{\pi \cdot \left(angle \cdot 0.005555555555555556\right)}}\right)}^{3} \cdot {\color{blue}{\left(\sqrt{\pi \cdot \frac{angle}{180}} \cdot \sqrt{\pi \cdot \frac{angle}{180}}\right)}}^{0.3333333333333333}\right) \cdot \left(b - a\right)\right)\right) \]

       6. unpow-prod-down 26.2%

          \[\leadsto 2 \cdot \left(\left(\sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right) \cdot \left(a + b\right)\right) \cdot \left(\cos \left({\left(\sqrt[3]{\sqrt[3]{\pi \cdot \left(angle \cdot 0.005555555555555556\right)}} \cdot \sqrt[3]{\sqrt[3]{\pi \cdot \left(angle \cdot 0.005555555555555556\right)}}\right)}^{3} \cdot \color{blue}{\left({\left(\sqrt{\pi \cdot \frac{angle}{180}}\right)}^{0.3333333333333333} \cdot {\left(\sqrt{\pi \cdot \frac{angle}{180}}\right)}^{0.3333333333333333}\right)}\right) \cdot \left(b - a\right)\right)\right) \]

   12. Applied egg-rr 26.2%

       \[\leadsto 2 \cdot \left(\left(\sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right) \cdot \left(a + b\right)\right) \cdot \left(\cos \left({\left(\sqrt[3]{\sqrt[3]{\pi \cdot \left(angle \cdot 0.005555555555555556\right)}} \cdot \sqrt[3]{\sqrt[3]{\pi \cdot \left(angle \cdot 0.005555555555555556\right)}}\right)}^{3} \cdot \color{blue}{\left({\left(\sqrt{\pi \cdot \frac{angle}{180}}\right)}^{0.3333333333333333} \cdot {\left(\sqrt{\pi \cdot \frac{angle}{180}}\right)}^{0.3333333333333333}\right)}\right) \cdot \left(b - a\right)\right)\right) \]
   13. Step-by-step derivation

       1. unpow1/3 21.9%

          \[\leadsto 2 \cdot \left(\left(\sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right) \cdot \left(a + b\right)\right) \cdot \left(\cos \left({\left(\sqrt[3]{\sqrt[3]{\pi \cdot \left(angle \cdot 0.005555555555555556\right)}} \cdot \sqrt[3]{\sqrt[3]{\pi \cdot \left(angle \cdot 0.005555555555555556\right)}}\right)}^{3} \cdot \left(\color{blue}{\sqrt[3]{\sqrt{\pi \cdot \frac{angle}{180}}}} \cdot {\left(\sqrt{\pi \cdot \frac{angle}{180}}\right)}^{0.3333333333333333}\right)\right) \cdot \left(b - a\right)\right)\right) \]

       2. unpow1/3 43.7%

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

   14. Simplified 43.7%

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

4. Final simplification 75.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{angle}{180} \leq -1 \cdot 10^{-66}:\\ \;\;\;\;\left(2 \cdot \left(\left(b - a\right) \cdot \left(b + a\right)\right)\right) \cdot \left(\sin \left(\frac{angle}{180} \cdot \pi\right) \cdot \cos \left(\frac{\pi}{\frac{180}{angle}}\right)\right)\\ \mathbf{elif}\;\frac{angle}{180} \leq 4 \cdot 10^{+121}:\\ \;\;\;\;2 \cdot \left(\left(\sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right) \cdot \left(b + a\right)\right) \cdot \left(\left(b - a\right) \cdot \cos \left(\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{angle}{180} \cdot \pi\right)\right)\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \left(\left(\sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right) \cdot \left(b + a\right)\right) \cdot \left(\left(b - a\right) \cdot \cos \left({\left(\sqrt[3]{\sqrt[3]{\pi \cdot \left(angle \cdot 0.005555555555555556\right)}} \cdot \sqrt[3]{\sqrt[3]{\pi \cdot \left(angle \cdot 0.005555555555555556\right)}}\right)}^{3} \cdot \left(\sqrt[3]{\sqrt{\frac{angle}{180} \cdot \pi}} \cdot \sqrt[3]{\sqrt{\frac{angle}{180} \cdot \pi}}\right)\right)\right)\right)\\ \end{array} \]

Alternative 2: 67.6% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \sqrt[3]{\sqrt[3]{\pi \cdot \left(angle \cdot 0.005555555555555556\right)}}\\ t_1 := \sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right) \cdot \left(b + a\right)\\ \mathbf{if}\;{a}^{2} \leq 5 \cdot 10^{+176}:\\ \;\;\;\;2 \cdot \left(t_1 \cdot \left(\left(b - a\right) \cdot \log \left(e^{\cos \left(\frac{angle}{180} \cdot \pi\right)}\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \left(t_1 \cdot \left(\left(b - a\right) \cdot \cos \left({t_0}^{3} \cdot {\left(t_0 \cdot \sqrt[3]{\sqrt[3]{\left(angle \cdot 0.005555555555555556\right) \cdot \sqrt[3]{{\pi}^{3}}}}\right)}^{3}\right)\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (a b angle)
 :precision binary64
 (let* ((t_0 (cbrt (cbrt (* PI (* angle 0.005555555555555556)))))
        (t_1 (* (sin (* 0.005555555555555556 (* angle PI))) (+ b a))))
   (if (<= (pow a 2.0) 5e+176)
     (* 2.0 (* t_1 (* (- b a) (log (exp (cos (* (/ angle 180.0) PI)))))))
     (*
      2.0
      (*
       t_1
       (*
        (- b a)
        (cos
         (*
          (pow t_0 3.0)
          (pow
           (*
            t_0
            (cbrt
             (cbrt (* (* angle 0.005555555555555556) (cbrt (pow PI 3.0))))))
           3.0)))))))))

C

double code(double a, double b, double angle) {
	double t_0 = cbrt(cbrt((((double) M_PI) * (angle * 0.005555555555555556))));
	double t_1 = sin((0.005555555555555556 * (angle * ((double) M_PI)))) * (b + a);
	double tmp;
	if (pow(a, 2.0) <= 5e+176) {
		tmp = 2.0 * (t_1 * ((b - a) * log(exp(cos(((angle / 180.0) * ((double) M_PI)))))));
	} else {
		tmp = 2.0 * (t_1 * ((b - a) * cos((pow(t_0, 3.0) * pow((t_0 * cbrt(cbrt(((angle * 0.005555555555555556) * cbrt(pow(((double) M_PI), 3.0)))))), 3.0)))));
	}
	return tmp;
}

Java

public static double code(double a, double b, double angle) {
	double t_0 = Math.cbrt(Math.cbrt((Math.PI * (angle * 0.005555555555555556))));
	double t_1 = Math.sin((0.005555555555555556 * (angle * Math.PI))) * (b + a);
	double tmp;
	if (Math.pow(a, 2.0) <= 5e+176) {
		tmp = 2.0 * (t_1 * ((b - a) * Math.log(Math.exp(Math.cos(((angle / 180.0) * Math.PI))))));
	} else {
		tmp = 2.0 * (t_1 * ((b - a) * Math.cos((Math.pow(t_0, 3.0) * Math.pow((t_0 * Math.cbrt(Math.cbrt(((angle * 0.005555555555555556) * Math.cbrt(Math.pow(Math.PI, 3.0)))))), 3.0)))));
	}
	return tmp;
}

Julia

function code(a, b, angle)
	t_0 = cbrt(cbrt(Float64(pi * Float64(angle * 0.005555555555555556))))
	t_1 = Float64(sin(Float64(0.005555555555555556 * Float64(angle * pi))) * Float64(b + a))
	tmp = 0.0
	if ((a ^ 2.0) <= 5e+176)
		tmp = Float64(2.0 * Float64(t_1 * Float64(Float64(b - a) * log(exp(cos(Float64(Float64(angle / 180.0) * pi)))))));
	else
		tmp = Float64(2.0 * Float64(t_1 * Float64(Float64(b - a) * cos(Float64((t_0 ^ 3.0) * (Float64(t_0 * cbrt(cbrt(Float64(Float64(angle * 0.005555555555555556) * cbrt((pi ^ 3.0)))))) ^ 3.0))))));
	end
	return tmp
end

Wolfram

code[a_, b_, angle_] := Block[{t$95$0 = N[Power[N[Power[N[(Pi * N[(angle * 0.005555555555555556), $MachinePrecision]), $MachinePrecision], 1/3], $MachinePrecision], 1/3], $MachinePrecision]}, Block[{t$95$1 = N[(N[Sin[N[(0.005555555555555556 * N[(angle * Pi), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * N[(b + a), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[Power[a, 2.0], $MachinePrecision], 5e+176], N[(2.0 * N[(t$95$1 * N[(N[(b - a), $MachinePrecision] * N[Log[N[Exp[N[Cos[N[(N[(angle / 180.0), $MachinePrecision] * Pi), $MachinePrecision]], $MachinePrecision]], $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(2.0 * N[(t$95$1 * N[(N[(b - a), $MachinePrecision] * N[Cos[N[(N[Power[t$95$0, 3.0], $MachinePrecision] * N[Power[N[(t$95$0 * N[Power[N[Power[N[(N[(angle * 0.005555555555555556), $MachinePrecision] * N[Power[N[Power[Pi, 3.0], $MachinePrecision], 1/3], $MachinePrecision]), $MachinePrecision], 1/3], $MachinePrecision], 1/3], $MachinePrecision]), $MachinePrecision], 3.0], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 2 regimes

2. if (pow.f64 a 2) < 5e176

   1. Initial program 60.0%

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

      1. associate-*l* 60.0%

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

      2. associate-*l* 60.0%

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

      3. unpow2 60.0%

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

      4. unpow2 60.0%

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

      5. difference-of-squares 60.0%

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

   3. Simplified 60.0%

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

   4. Taylor expanded in angle around inf 64.7%

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

      1. associate-*r* 64.8%

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

      2. *-commutative 64.8%

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

      3. *-commutative 64.8%

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

   6. Simplified 64.8%

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

      1. associate-*r* 64.9%

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

      2. *-commutative 64.9%

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

      3. metadata-eval 64.9%

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

      4. div-inv 66.4%

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

      5. *-commutative 66.4%

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

      6. add-log-exp 66.4%

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

   8. Applied egg-rr 66.4%

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

   if 5e176 < (pow.f64 a 2)

   1. Initial program 42.9%

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

      1. associate-*l* 42.9%

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

      2. associate-*l* 42.9%

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

      3. unpow2 42.9%

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

      4. unpow2 42.9%

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

      5. difference-of-squares 55.8%

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

   3. Simplified 55.8%

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

   4. Taylor expanded in angle around inf 79.3%

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

      1. associate-*r* 79.3%

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

      2. *-commutative 79.3%

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

      3. *-commutative 79.3%

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

   6. Simplified 79.3%

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

      1. *-commutative 79.3%

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

      2. *-commutative 79.3%

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

      3. associate-*r* 78.3%

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

      4. add-cube-cbrt 81.9%

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

      5. unpow3 80.7%

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

      6. associate-*r* 80.5%

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

      7. *-commutative 80.5%

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

      8. *-commutative 80.5%

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

      9. associate-*r* 80.7%

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

      10. *-commutative 80.7%

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

      11. metadata-eval 80.7%

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

      12. div-inv 82.9%

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

      13. *-commutative 82.9%

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

   8. Applied egg-rr 82.9%

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

      1. add-cube-cbrt 83.0%

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

      2. unpow-prod-down 84.6%

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

      3. div-inv 85.8%

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

      4. metadata-eval 85.8%

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

      5. div-inv 85.8%

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

      6. metadata-eval 85.8%

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

   10. Applied egg-rr 85.8%

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

       1. add-cbrt-cube 85.8%

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

       2. pow3 85.8%

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

   12. Applied egg-rr 85.8%

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

4. Final simplification 72.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;{a}^{2} \leq 5 \cdot 10^{+176}:\\ \;\;\;\;2 \cdot \left(\left(\sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right) \cdot \left(b + a\right)\right) \cdot \left(\left(b - a\right) \cdot \log \left(e^{\cos \left(\frac{angle}{180} \cdot \pi\right)}\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \left(\left(\sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right) \cdot \left(b + a\right)\right) \cdot \left(\left(b - a\right) \cdot \cos \left({\left(\sqrt[3]{\sqrt[3]{\pi \cdot \left(angle \cdot 0.005555555555555556\right)}}\right)}^{3} \cdot {\left(\sqrt[3]{\sqrt[3]{\pi \cdot \left(angle \cdot 0.005555555555555556\right)}} \cdot \sqrt[3]{\sqrt[3]{\left(angle \cdot 0.005555555555555556\right) \cdot \sqrt[3]{{\pi}^{3}}}}\right)}^{3}\right)\right)\right)\\ \end{array} \]

Alternative 3: 67.2% accurate, 0.4× speedup

Math

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

FPCore

(FPCore (a b angle)
 :precision binary64
 (let* ((t_0 (cbrt (cbrt (* PI (* angle 0.005555555555555556))))))
   (*
    2.0
    (*
     (* (+ b a) (sin (* 0.005555555555555556 (* angle (pow (sqrt PI) 2.0)))))
     (* (- b a) (cos (* (pow (* t_0 t_0) 3.0) (pow t_0 3.0))))))))

C

double code(double a, double b, double angle) {
	double t_0 = cbrt(cbrt((((double) M_PI) * (angle * 0.005555555555555556))));
	return 2.0 * (((b + a) * sin((0.005555555555555556 * (angle * pow(sqrt(((double) M_PI)), 2.0))))) * ((b - a) * cos((pow((t_0 * t_0), 3.0) * pow(t_0, 3.0)))));
}

Java

public static double code(double a, double b, double angle) {
	double t_0 = Math.cbrt(Math.cbrt((Math.PI * (angle * 0.005555555555555556))));
	return 2.0 * (((b + a) * Math.sin((0.005555555555555556 * (angle * Math.pow(Math.sqrt(Math.PI), 2.0))))) * ((b - a) * Math.cos((Math.pow((t_0 * t_0), 3.0) * Math.pow(t_0, 3.0)))));
}

Julia

function code(a, b, angle)
	t_0 = cbrt(cbrt(Float64(pi * Float64(angle * 0.005555555555555556))))
	return Float64(2.0 * Float64(Float64(Float64(b + a) * sin(Float64(0.005555555555555556 * Float64(angle * (sqrt(pi) ^ 2.0))))) * Float64(Float64(b - a) * cos(Float64((Float64(t_0 * t_0) ^ 3.0) * (t_0 ^ 3.0))))))
end

Wolfram

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

Derivation

1. Initial program 54.5%

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

   1. associate-*l* 54.5%

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

   2. associate-*l* 54.5%

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

   3. unpow2 54.5%

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

   4. unpow2 54.5%

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

   5. difference-of-squares 58.6%

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

3. Simplified 58.6%

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

4. Taylor expanded in angle around inf 69.4%

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

   1. associate-*r* 69.4%

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

   2. *-commutative 69.4%

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

   3. *-commutative 69.4%

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

6. Simplified 69.4%

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

   1. *-commutative 69.4%

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

   2. *-commutative 69.4%

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

   3. associate-*r* 69.2%

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

   4. add-cube-cbrt 68.7%

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

   5. unpow3 69.1%

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

   6. associate-*r* 69.0%

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

   7. *-commutative 69.0%

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

   8. *-commutative 69.0%

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

   9. associate-*r* 69.1%

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

   10. *-commutative 69.1%

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

   11. metadata-eval 69.1%

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

   12. div-inv 70.1%

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

   13. *-commutative 70.1%

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

8. Applied egg-rr 70.1%

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

   1. add-cube-cbrt 69.2%

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

   2. unpow-prod-down 70.1%

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

   3. div-inv 70.5%

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

   4. metadata-eval 70.5%

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

   5. div-inv 70.2%

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

   6. metadata-eval 70.2%

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

10. Applied egg-rr 70.2%

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

    1. add-sqr-sqrt 71.6%

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

    2. pow2 71.6%

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

12. Applied egg-rr 71.6%

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

13. Final simplification 71.6%

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

Alternative 4: 67.7% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \sqrt[3]{\sqrt[3]{\pi \cdot \left(angle \cdot 0.005555555555555556\right)}}\\ t_1 := \sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right) \cdot \left(b + a\right)\\ \mathbf{if}\;{a}^{2} \leq 5 \cdot 10^{+176}:\\ \;\;\;\;2 \cdot \left(t_1 \cdot \left(\left(b - a\right) \cdot \log \left(e^{\cos \left(\frac{angle}{180} \cdot \pi\right)}\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \left(t_1 \cdot \left(\left(b - a\right) \cdot \cos \left({\left(t_0 \cdot t_0\right)}^{3} \cdot {t_0}^{3}\right)\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (a b angle)
 :precision binary64
 (let* ((t_0 (cbrt (cbrt (* PI (* angle 0.005555555555555556)))))
        (t_1 (* (sin (* 0.005555555555555556 (* angle PI))) (+ b a))))
   (if (<= (pow a 2.0) 5e+176)
     (* 2.0 (* t_1 (* (- b a) (log (exp (cos (* (/ angle 180.0) PI)))))))
     (*
      2.0
      (* t_1 (* (- b a) (cos (* (pow (* t_0 t_0) 3.0) (pow t_0 3.0)))))))))

C

double code(double a, double b, double angle) {
	double t_0 = cbrt(cbrt((((double) M_PI) * (angle * 0.005555555555555556))));
	double t_1 = sin((0.005555555555555556 * (angle * ((double) M_PI)))) * (b + a);
	double tmp;
	if (pow(a, 2.0) <= 5e+176) {
		tmp = 2.0 * (t_1 * ((b - a) * log(exp(cos(((angle / 180.0) * ((double) M_PI)))))));
	} else {
		tmp = 2.0 * (t_1 * ((b - a) * cos((pow((t_0 * t_0), 3.0) * pow(t_0, 3.0)))));
	}
	return tmp;
}

Java

public static double code(double a, double b, double angle) {
	double t_0 = Math.cbrt(Math.cbrt((Math.PI * (angle * 0.005555555555555556))));
	double t_1 = Math.sin((0.005555555555555556 * (angle * Math.PI))) * (b + a);
	double tmp;
	if (Math.pow(a, 2.0) <= 5e+176) {
		tmp = 2.0 * (t_1 * ((b - a) * Math.log(Math.exp(Math.cos(((angle / 180.0) * Math.PI))))));
	} else {
		tmp = 2.0 * (t_1 * ((b - a) * Math.cos((Math.pow((t_0 * t_0), 3.0) * Math.pow(t_0, 3.0)))));
	}
	return tmp;
}

Julia

function code(a, b, angle)
	t_0 = cbrt(cbrt(Float64(pi * Float64(angle * 0.005555555555555556))))
	t_1 = Float64(sin(Float64(0.005555555555555556 * Float64(angle * pi))) * Float64(b + a))
	tmp = 0.0
	if ((a ^ 2.0) <= 5e+176)
		tmp = Float64(2.0 * Float64(t_1 * Float64(Float64(b - a) * log(exp(cos(Float64(Float64(angle / 180.0) * pi)))))));
	else
		tmp = Float64(2.0 * Float64(t_1 * Float64(Float64(b - a) * cos(Float64((Float64(t_0 * t_0) ^ 3.0) * (t_0 ^ 3.0))))));
	end
	return tmp
end

Wolfram

code[a_, b_, angle_] := Block[{t$95$0 = N[Power[N[Power[N[(Pi * N[(angle * 0.005555555555555556), $MachinePrecision]), $MachinePrecision], 1/3], $MachinePrecision], 1/3], $MachinePrecision]}, Block[{t$95$1 = N[(N[Sin[N[(0.005555555555555556 * N[(angle * Pi), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * N[(b + a), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[Power[a, 2.0], $MachinePrecision], 5e+176], N[(2.0 * N[(t$95$1 * N[(N[(b - a), $MachinePrecision] * N[Log[N[Exp[N[Cos[N[(N[(angle / 180.0), $MachinePrecision] * Pi), $MachinePrecision]], $MachinePrecision]], $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(2.0 * N[(t$95$1 * N[(N[(b - a), $MachinePrecision] * N[Cos[N[(N[Power[N[(t$95$0 * t$95$0), $MachinePrecision], 3.0], $MachinePrecision] * N[Power[t$95$0, 3.0], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 2 regimes

2. if (pow.f64 a 2) < 5e176

   1. Initial program 60.0%

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

      1. associate-*l* 60.0%

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

      2. associate-*l* 60.0%

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

      3. unpow2 60.0%

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

      4. unpow2 60.0%

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

      5. difference-of-squares 60.0%

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

   3. Simplified 60.0%

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

   4. Taylor expanded in angle around inf 64.7%

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

      1. associate-*r* 64.8%

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

      2. *-commutative 64.8%

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

      3. *-commutative 64.8%

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

   6. Simplified 64.8%

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

      1. associate-*r* 64.9%

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

      2. *-commutative 64.9%

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

      3. metadata-eval 64.9%

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

      4. div-inv 66.4%

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

      5. *-commutative 66.4%

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

      6. add-log-exp 66.4%

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

   8. Applied egg-rr 66.4%

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

   if 5e176 < (pow.f64 a 2)

   1. Initial program 42.9%

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

      1. associate-*l* 42.9%

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

      2. associate-*l* 42.9%

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

      3. unpow2 42.9%

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

      4. unpow2 42.9%

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

      5. difference-of-squares 55.8%

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

   3. Simplified 55.8%

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

   4. Taylor expanded in angle around inf 79.3%

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

      1. associate-*r* 79.3%

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

      2. *-commutative 79.3%

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

      3. *-commutative 79.3%

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

   6. Simplified 79.3%

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

      1. *-commutative 79.3%

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

      2. *-commutative 79.3%

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

      3. associate-*r* 78.3%

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

      4. add-cube-cbrt 81.9%

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

      5. unpow3 80.7%

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

      6. associate-*r* 80.5%

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

      7. *-commutative 80.5%

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

      8. *-commutative 80.5%

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

      9. associate-*r* 80.7%

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

      10. *-commutative 80.7%

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

      11. metadata-eval 80.7%

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

      12. div-inv 82.9%

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

      13. *-commutative 82.9%

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

   8. Applied egg-rr 82.9%

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

      1. add-cube-cbrt 83.0%

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

      2. unpow-prod-down 84.6%

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

      3. div-inv 85.8%

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

      4. metadata-eval 85.8%

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

      5. div-inv 85.8%

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

      6. metadata-eval 85.8%

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

   10. Applied egg-rr 85.8%

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

4. Final simplification 72.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;{a}^{2} \leq 5 \cdot 10^{+176}:\\ \;\;\;\;2 \cdot \left(\left(\sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right) \cdot \left(b + a\right)\right) \cdot \left(\left(b - a\right) \cdot \log \left(e^{\cos \left(\frac{angle}{180} \cdot \pi\right)}\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \left(\left(\sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right) \cdot \left(b + a\right)\right) \cdot \left(\left(b - a\right) \cdot \cos \left({\left(\sqrt[3]{\sqrt[3]{\pi \cdot \left(angle \cdot 0.005555555555555556\right)}} \cdot \sqrt[3]{\sqrt[3]{\pi \cdot \left(angle \cdot 0.005555555555555556\right)}}\right)}^{3} \cdot {\left(\sqrt[3]{\sqrt[3]{\pi \cdot \left(angle \cdot 0.005555555555555556\right)}}\right)}^{3}\right)\right)\right)\\ \end{array} \]

Alternative 5: 67.0% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right) \cdot \left(b + a\right)\\ \mathbf{if}\;{a}^{2} \leq 5 \cdot 10^{+176}:\\ \;\;\;\;2 \cdot \left(t_0 \cdot \left(\left(b - a\right) \cdot \log \left(e^{\cos \left(\frac{angle}{180} \cdot \pi\right)}\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \left(t_0 \cdot \left(\left(b - a\right) \cdot \left|\cos \left(angle \cdot \frac{\pi}{180}\right)\right|\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (a b angle)
 :precision binary64
 (let* ((t_0 (* (sin (* 0.005555555555555556 (* angle PI))) (+ b a))))
   (if (<= (pow a 2.0) 5e+176)
     (* 2.0 (* t_0 (* (- b a) (log (exp (cos (* (/ angle 180.0) PI)))))))
     (* 2.0 (* t_0 (* (- b a) (fabs (cos (* angle (/ PI 180.0))))))))))

C

double code(double a, double b, double angle) {
	double t_0 = sin((0.005555555555555556 * (angle * ((double) M_PI)))) * (b + a);
	double tmp;
	if (pow(a, 2.0) <= 5e+176) {
		tmp = 2.0 * (t_0 * ((b - a) * log(exp(cos(((angle / 180.0) * ((double) M_PI)))))));
	} else {
		tmp = 2.0 * (t_0 * ((b - a) * fabs(cos((angle * (((double) M_PI) / 180.0))))));
	}
	return tmp;
}

Java

public static double code(double a, double b, double angle) {
	double t_0 = Math.sin((0.005555555555555556 * (angle * Math.PI))) * (b + a);
	double tmp;
	if (Math.pow(a, 2.0) <= 5e+176) {
		tmp = 2.0 * (t_0 * ((b - a) * Math.log(Math.exp(Math.cos(((angle / 180.0) * Math.PI))))));
	} else {
		tmp = 2.0 * (t_0 * ((b - a) * Math.abs(Math.cos((angle * (Math.PI / 180.0))))));
	}
	return tmp;
}

Python

def code(a, b, angle):
	t_0 = math.sin((0.005555555555555556 * (angle * math.pi))) * (b + a)
	tmp = 0
	if math.pow(a, 2.0) <= 5e+176:
		tmp = 2.0 * (t_0 * ((b - a) * math.log(math.exp(math.cos(((angle / 180.0) * math.pi))))))
	else:
		tmp = 2.0 * (t_0 * ((b - a) * math.fabs(math.cos((angle * (math.pi / 180.0))))))
	return tmp

Julia

function code(a, b, angle)
	t_0 = Float64(sin(Float64(0.005555555555555556 * Float64(angle * pi))) * Float64(b + a))
	tmp = 0.0
	if ((a ^ 2.0) <= 5e+176)
		tmp = Float64(2.0 * Float64(t_0 * Float64(Float64(b - a) * log(exp(cos(Float64(Float64(angle / 180.0) * pi)))))));
	else
		tmp = Float64(2.0 * Float64(t_0 * Float64(Float64(b - a) * abs(cos(Float64(angle * Float64(pi / 180.0)))))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, angle)
	t_0 = sin((0.005555555555555556 * (angle * pi))) * (b + a);
	tmp = 0.0;
	if ((a ^ 2.0) <= 5e+176)
		tmp = 2.0 * (t_0 * ((b - a) * log(exp(cos(((angle / 180.0) * pi))))));
	else
		tmp = 2.0 * (t_0 * ((b - a) * abs(cos((angle * (pi / 180.0))))));
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, angle_] := Block[{t$95$0 = N[(N[Sin[N[(0.005555555555555556 * N[(angle * Pi), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * N[(b + a), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[Power[a, 2.0], $MachinePrecision], 5e+176], N[(2.0 * N[(t$95$0 * N[(N[(b - a), $MachinePrecision] * N[Log[N[Exp[N[Cos[N[(N[(angle / 180.0), $MachinePrecision] * Pi), $MachinePrecision]], $MachinePrecision]], $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(2.0 * N[(t$95$0 * N[(N[(b - a), $MachinePrecision] * N[Abs[N[Cos[N[(angle * N[(Pi / 180.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (pow.f64 a 2) < 5e176

   1. Initial program 60.0%

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

      1. associate-*l* 60.0%

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

      2. associate-*l* 60.0%

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

      3. unpow2 60.0%

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

      4. unpow2 60.0%

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

      5. difference-of-squares 60.0%

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

   3. Simplified 60.0%

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

   4. Taylor expanded in angle around inf 64.7%

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

      1. associate-*r* 64.8%

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

      2. *-commutative 64.8%

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

      3. *-commutative 64.8%

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

   6. Simplified 64.8%

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

      1. associate-*r* 64.9%

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

      2. *-commutative 64.9%

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

      3. metadata-eval 64.9%

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

      4. div-inv 66.4%

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

      5. *-commutative 66.4%

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

      6. add-log-exp 66.4%

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

   8. Applied egg-rr 66.4%

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

   if 5e176 < (pow.f64 a 2)

   1. Initial program 42.9%

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

      1. associate-*l* 42.9%

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

      2. associate-*l* 42.9%

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

      3. unpow2 42.9%

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

      4. unpow2 42.9%

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

      5. difference-of-squares 55.8%

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

   3. Simplified 55.8%

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

   4. Taylor expanded in angle around inf 79.3%

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

      1. associate-*r* 79.3%

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

      2. *-commutative 79.3%

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

      3. *-commutative 79.3%

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

   6. Simplified 79.3%

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

      1. *-commutative 79.3%

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

      2. *-commutative 79.3%

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

      3. associate-*r* 78.3%

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

      4. add-cube-cbrt 81.9%

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

      5. unpow3 80.7%

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

      6. associate-*r* 80.5%

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

      7. *-commutative 80.5%

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

      8. *-commutative 80.5%

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

      9. associate-*r* 80.7%

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

      10. *-commutative 80.7%

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

      11. metadata-eval 80.7%

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

      12. div-inv 82.9%

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

      13. *-commutative 82.9%

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

   8. Applied egg-rr 82.9%

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

      1. add-cube-cbrt 83.0%

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

      2. unpow-prod-down 84.6%

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

      3. div-inv 85.8%

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

      4. metadata-eval 85.8%

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

      5. div-inv 85.8%

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

      6. metadata-eval 85.8%

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

   10. Applied egg-rr 85.8%

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

       1. pow-prod-down 81.7%

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

       2. add-cube-cbrt 80.7%

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

       3. metadata-eval 80.7%

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

       4. div-inv 82.9%

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

       5. rem-cube-cbrt 76.9%

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

       6. add-sqr-sqrt 72.0%

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

       7. sqrt-unprod 84.4%

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

       8. pow2 84.4%

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

   12. Applied egg-rr 84.4%

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

       1. unpow2 84.4%

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

       2. rem-sqrt-square 84.4%

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

       3. associate-*r/ 84.4%

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

       4. associate-*l/ 84.4%

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

       5. *-commutative 84.4%

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

   14. Simplified 84.4%

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

4. Final simplification 72.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;{a}^{2} \leq 5 \cdot 10^{+176}:\\ \;\;\;\;2 \cdot \left(\left(\sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right) \cdot \left(b + a\right)\right) \cdot \left(\left(b - a\right) \cdot \log \left(e^{\cos \left(\frac{angle}{180} \cdot \pi\right)}\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \left(\left(\sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right) \cdot \left(b + a\right)\right) \cdot \left(\left(b - a\right) \cdot \left|\cos \left(angle \cdot \frac{\pi}{180}\right)\right|\right)\right)\\ \end{array} \]

Alternative 6: 67.0% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right) \cdot \left(b + a\right)\\ \mathbf{if}\;{a}^{2} \leq 5 \cdot 10^{+141}:\\ \;\;\;\;2 \cdot \left(t_0 \cdot \left(\left(b - a\right) \cdot \cos \left(\frac{\pi}{\frac{180}{angle}}\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \left(t_0 \cdot \left(\left(b - a\right) \cdot \left|\cos \left(angle \cdot \frac{\pi}{180}\right)\right|\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (a b angle)
 :precision binary64
 (let* ((t_0 (* (sin (* 0.005555555555555556 (* angle PI))) (+ b a))))
   (if (<= (pow a 2.0) 5e+141)
     (* 2.0 (* t_0 (* (- b a) (cos (/ PI (/ 180.0 angle))))))
     (* 2.0 (* t_0 (* (- b a) (fabs (cos (* angle (/ PI 180.0))))))))))

C

double code(double a, double b, double angle) {
	double t_0 = sin((0.005555555555555556 * (angle * ((double) M_PI)))) * (b + a);
	double tmp;
	if (pow(a, 2.0) <= 5e+141) {
		tmp = 2.0 * (t_0 * ((b - a) * cos((((double) M_PI) / (180.0 / angle)))));
	} else {
		tmp = 2.0 * (t_0 * ((b - a) * fabs(cos((angle * (((double) M_PI) / 180.0))))));
	}
	return tmp;
}

Java

public static double code(double a, double b, double angle) {
	double t_0 = Math.sin((0.005555555555555556 * (angle * Math.PI))) * (b + a);
	double tmp;
	if (Math.pow(a, 2.0) <= 5e+141) {
		tmp = 2.0 * (t_0 * ((b - a) * Math.cos((Math.PI / (180.0 / angle)))));
	} else {
		tmp = 2.0 * (t_0 * ((b - a) * Math.abs(Math.cos((angle * (Math.PI / 180.0))))));
	}
	return tmp;
}

Python

def code(a, b, angle):
	t_0 = math.sin((0.005555555555555556 * (angle * math.pi))) * (b + a)
	tmp = 0
	if math.pow(a, 2.0) <= 5e+141:
		tmp = 2.0 * (t_0 * ((b - a) * math.cos((math.pi / (180.0 / angle)))))
	else:
		tmp = 2.0 * (t_0 * ((b - a) * math.fabs(math.cos((angle * (math.pi / 180.0))))))
	return tmp

Julia

function code(a, b, angle)
	t_0 = Float64(sin(Float64(0.005555555555555556 * Float64(angle * pi))) * Float64(b + a))
	tmp = 0.0
	if ((a ^ 2.0) <= 5e+141)
		tmp = Float64(2.0 * Float64(t_0 * Float64(Float64(b - a) * cos(Float64(pi / Float64(180.0 / angle))))));
	else
		tmp = Float64(2.0 * Float64(t_0 * Float64(Float64(b - a) * abs(cos(Float64(angle * Float64(pi / 180.0)))))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, angle)
	t_0 = sin((0.005555555555555556 * (angle * pi))) * (b + a);
	tmp = 0.0;
	if ((a ^ 2.0) <= 5e+141)
		tmp = 2.0 * (t_0 * ((b - a) * cos((pi / (180.0 / angle)))));
	else
		tmp = 2.0 * (t_0 * ((b - a) * abs(cos((angle * (pi / 180.0))))));
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, angle_] := Block[{t$95$0 = N[(N[Sin[N[(0.005555555555555556 * N[(angle * Pi), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * N[(b + a), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[Power[a, 2.0], $MachinePrecision], 5e+141], N[(2.0 * N[(t$95$0 * N[(N[(b - a), $MachinePrecision] * N[Cos[N[(Pi / N[(180.0 / angle), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(2.0 * N[(t$95$0 * N[(N[(b - a), $MachinePrecision] * N[Abs[N[Cos[N[(angle * N[(Pi / 180.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (pow.f64 a 2) < 5.00000000000000025e141

   1. Initial program 60.5%

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

      1. associate-*l* 60.5%

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

      2. associate-*l* 60.5%

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

      3. unpow2 60.5%

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

      4. unpow2 60.5%

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

      5. difference-of-squares 60.5%

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

   3. Simplified 60.5%

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

   4. Taylor expanded in angle around inf 65.2%

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

      1. associate-*r* 65.2%

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

      2. *-commutative 65.2%

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

      3. *-commutative 65.2%

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

   6. Simplified 65.2%

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

      1. *-commutative 65.2%

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

      2. *-commutative 65.2%

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

      3. associate-*r* 65.3%

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

      4. add-cube-cbrt 62.8%

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

      5. unpow3 64.0%

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

      6. associate-*r* 64.1%

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

      7. *-commutative 64.1%

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

      8. *-commutative 64.1%

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

      9. associate-*r* 64.0%

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

      10. *-commutative 64.0%

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

      11. metadata-eval 64.0%

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

      12. div-inv 64.5%

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

      13. *-commutative 64.5%

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

   8. Applied egg-rr 64.5%

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

      1. rem-cube-cbrt 66.9%

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

      2. associate-*r/ 65.3%

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

   10. Applied egg-rr 65.3%

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

       1. associate-/l* 67.1%

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

   12. Simplified 67.1%

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

   if 5.00000000000000025e141 < (pow.f64 a 2)

   1. Initial program 43.6%

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

      1. associate-*l* 43.6%

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

      2. associate-*l* 43.6%

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

      3. unpow2 43.6%

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

      4. unpow2 43.6%

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

      5. difference-of-squares 55.1%

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

   3. Simplified 55.1%

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

   4. Taylor expanded in angle around inf 77.1%

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

      1. associate-*r* 77.1%

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

      2. *-commutative 77.1%

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

      3. *-commutative 77.1%

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

   6. Simplified 77.1%

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

      1. *-commutative 77.1%

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

      2. *-commutative 77.1%

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

      3. associate-*r* 76.2%

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

      4. add-cube-cbrt 79.5%

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

      5. unpow3 78.4%

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

      6. associate-*r* 78.0%

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

      7. *-commutative 78.0%

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

      8. *-commutative 78.0%

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

      9. associate-*r* 78.4%

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

      10. *-commutative 78.4%

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

      11. metadata-eval 78.4%

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

      12. div-inv 80.2%

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

      13. *-commutative 80.2%

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

   8. Applied egg-rr 80.2%

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

      1. add-cube-cbrt 79.3%

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

      2. unpow-prod-down 80.4%

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

      3. div-inv 81.7%

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

      4. metadata-eval 81.7%

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

      5. div-inv 81.7%

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

      6. metadata-eval 81.7%

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

   10. Applied egg-rr 81.7%

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

       1. pow-prod-down 78.2%

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

       2. add-cube-cbrt 78.4%

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

       3. metadata-eval 78.4%

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

       4. div-inv 80.2%

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

       5. rem-cube-cbrt 74.9%

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

       6. add-sqr-sqrt 69.5%

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

       7. sqrt-unprod 80.9%

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

       8. pow2 80.9%

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

   12. Applied egg-rr 80.9%

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

       1. unpow2 80.9%

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

       2. rem-sqrt-square 80.9%

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

       3. associate-*r/ 80.9%

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

       4. associate-*l/ 80.9%

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

       5. *-commutative 80.9%

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

   14. Simplified 80.9%

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

4. Final simplification 72.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;{a}^{2} \leq 5 \cdot 10^{+141}:\\ \;\;\;\;2 \cdot \left(\left(\sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right) \cdot \left(b + a\right)\right) \cdot \left(\left(b - a\right) \cdot \cos \left(\frac{\pi}{\frac{180}{angle}}\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \left(\left(\sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right) \cdot \left(b + a\right)\right) \cdot \left(\left(b - a\right) \cdot \left|\cos \left(angle \cdot \frac{\pi}{180}\right)\right|\right)\right)\\ \end{array} \]

Alternative 7: 67.3% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := 0.005555555555555556 \cdot \left(angle \cdot \pi\right)\\ \mathbf{if}\;\frac{angle}{180} \leq 200000000000:\\ \;\;\;\;2 \cdot \left(\left(\sin t_0 \cdot \left(b + a\right)\right) \cdot \left(\left(b - a\right) \cdot \cos t_0\right)\right)\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \left(\left(\left(b - a\right) \cdot \left(b + a\right)\right) \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (a b angle)
 :precision binary64
 (let* ((t_0 (* 0.005555555555555556 (* angle PI))))
   (if (<= (/ angle 180.0) 200000000000.0)
     (* 2.0 (* (* (sin t_0) (+ b a)) (* (- b a) (cos t_0))))
     (* 2.0 (* (* (- b a) (+ b a)) (sin (* (/ angle 180.0) PI)))))))

C

double code(double a, double b, double angle) {
	double t_0 = 0.005555555555555556 * (angle * ((double) M_PI));
	double tmp;
	if ((angle / 180.0) <= 200000000000.0) {
		tmp = 2.0 * ((sin(t_0) * (b + a)) * ((b - a) * cos(t_0)));
	} else {
		tmp = 2.0 * (((b - a) * (b + a)) * sin(((angle / 180.0) * ((double) M_PI))));
	}
	return tmp;
}

Java

public static double code(double a, double b, double angle) {
	double t_0 = 0.005555555555555556 * (angle * Math.PI);
	double tmp;
	if ((angle / 180.0) <= 200000000000.0) {
		tmp = 2.0 * ((Math.sin(t_0) * (b + a)) * ((b - a) * Math.cos(t_0)));
	} else {
		tmp = 2.0 * (((b - a) * (b + a)) * Math.sin(((angle / 180.0) * Math.PI)));
	}
	return tmp;
}

Python

def code(a, b, angle):
	t_0 = 0.005555555555555556 * (angle * math.pi)
	tmp = 0
	if (angle / 180.0) <= 200000000000.0:
		tmp = 2.0 * ((math.sin(t_0) * (b + a)) * ((b - a) * math.cos(t_0)))
	else:
		tmp = 2.0 * (((b - a) * (b + a)) * math.sin(((angle / 180.0) * math.pi)))
	return tmp

Julia

function code(a, b, angle)
	t_0 = Float64(0.005555555555555556 * Float64(angle * pi))
	tmp = 0.0
	if (Float64(angle / 180.0) <= 200000000000.0)
		tmp = Float64(2.0 * Float64(Float64(sin(t_0) * Float64(b + a)) * Float64(Float64(b - a) * cos(t_0))));
	else
		tmp = Float64(2.0 * Float64(Float64(Float64(b - a) * Float64(b + a)) * sin(Float64(Float64(angle / 180.0) * pi))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, angle)
	t_0 = 0.005555555555555556 * (angle * pi);
	tmp = 0.0;
	if ((angle / 180.0) <= 200000000000.0)
		tmp = 2.0 * ((sin(t_0) * (b + a)) * ((b - a) * cos(t_0)));
	else
		tmp = 2.0 * (((b - a) * (b + a)) * sin(((angle / 180.0) * pi)));
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, angle_] := Block[{t$95$0 = N[(0.005555555555555556 * N[(angle * Pi), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[(angle / 180.0), $MachinePrecision], 200000000000.0], N[(2.0 * N[(N[(N[Sin[t$95$0], $MachinePrecision] * N[(b + a), $MachinePrecision]), $MachinePrecision] * N[(N[(b - a), $MachinePrecision] * N[Cos[t$95$0], $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 2 regimes

2. if (/.f64 angle 180) < 2e11

   1. Initial program 63.9%

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

      1. associate-*l* 63.9%

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

      2. associate-*l* 63.9%

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

      3. unpow2 63.9%

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

      4. unpow2 63.9%

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

      5. difference-of-squares 69.2%

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

   3. Simplified 69.2%

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

   4. Taylor expanded in angle around inf 81.1%

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

      1. associate-*r* 81.1%

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

      2. *-commutative 81.1%

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

      3. *-commutative 81.1%

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

   6. Simplified 81.1%

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

   if 2e11 < (/.f64 angle 180)

   1. Initial program 19.2%

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

      1. associate-*l* 19.2%

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

      2. associate-*l* 19.2%

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

      3. unpow2 19.2%

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

      4. unpow2 19.2%

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

      5. difference-of-squares 19.2%

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

   3. Simplified 19.2%

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

   4. Taylor expanded in angle around 0 36.1%

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

4. Final simplification 71.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{angle}{180} \leq 200000000000:\\ \;\;\;\;2 \cdot \left(\left(\sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right) \cdot \left(b + a\right)\right) \cdot \left(\left(b - a\right) \cdot \cos \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \left(\left(\left(b - a\right) \cdot \left(b + a\right)\right) \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right)\\ \end{array} \]

Alternative 8: 67.4% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{angle}{180} \leq 5 \cdot 10^{+169}:\\ \;\;\;\;2 \cdot \left(\left(\sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right) \cdot \left(b + a\right)\right) \cdot \left(\left(b - a\right) \cdot \cos \left(\frac{\pi}{\frac{180}{angle}}\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \left(\left(\left(b - a\right) \cdot \left(b + a\right)\right) \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (a b angle)
 :precision binary64
 (if (<= (/ angle 180.0) 5e+169)
   (*
    2.0
    (*
     (* (sin (* 0.005555555555555556 (* angle PI))) (+ b a))
     (* (- b a) (cos (/ PI (/ 180.0 angle))))))
   (* 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) <= 5e+169) {
		tmp = 2.0 * ((sin((0.005555555555555556 * (angle * ((double) M_PI)))) * (b + a)) * ((b - a) * cos((((double) M_PI) / (180.0 / angle)))));
	} else {
		tmp = 2.0 * (((b - a) * (b + a)) * sin(((angle / 180.0) * ((double) M_PI))));
	}
	return tmp;
}

Java

public static double code(double a, double b, double angle) {
	double tmp;
	if ((angle / 180.0) <= 5e+169) {
		tmp = 2.0 * ((Math.sin((0.005555555555555556 * (angle * Math.PI))) * (b + a)) * ((b - a) * Math.cos((Math.PI / (180.0 / angle)))));
	} else {
		tmp = 2.0 * (((b - a) * (b + a)) * Math.sin(((angle / 180.0) * Math.PI)));
	}
	return tmp;
}

Python

def code(a, b, angle):
	tmp = 0
	if (angle / 180.0) <= 5e+169:
		tmp = 2.0 * ((math.sin((0.005555555555555556 * (angle * math.pi))) * (b + a)) * ((b - a) * math.cos((math.pi / (180.0 / angle)))))
	else:
		tmp = 2.0 * (((b - a) * (b + a)) * math.sin(((angle / 180.0) * math.pi)))
	return tmp

Julia

function code(a, b, angle)
	tmp = 0.0
	if (Float64(angle / 180.0) <= 5e+169)
		tmp = Float64(2.0 * Float64(Float64(sin(Float64(0.005555555555555556 * Float64(angle * pi))) * Float64(b + a)) * Float64(Float64(b - a) * cos(Float64(pi / Float64(180.0 / angle))))));
	else
		tmp = Float64(2.0 * Float64(Float64(Float64(b - a) * Float64(b + a)) * sin(Float64(Float64(angle / 180.0) * pi))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, angle)
	tmp = 0.0;
	if ((angle / 180.0) <= 5e+169)
		tmp = 2.0 * ((sin((0.005555555555555556 * (angle * pi))) * (b + a)) * ((b - a) * cos((pi / (180.0 / angle)))));
	else
		tmp = 2.0 * (((b - a) * (b + a)) * sin(((angle / 180.0) * pi)));
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, angle_] := If[LessEqual[N[(angle / 180.0), $MachinePrecision], 5e+169], N[(2.0 * N[(N[(N[Sin[N[(0.005555555555555556 * N[(angle * Pi), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * N[(b + a), $MachinePrecision]), $MachinePrecision] * N[(N[(b - a), $MachinePrecision] * N[Cos[N[(Pi / N[(180.0 / angle), $MachinePrecision]), $MachinePrecision]], $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 2 regimes

2. if (/.f64 angle 180) < 5.00000000000000017e169

   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* 59.2%

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

      3. unpow2 59.2%

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

      4. unpow2 59.2%

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

      5. difference-of-squares 63.8%

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

   3. Simplified 63.8%

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

   4. Taylor expanded in angle around inf 75.3%

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

      1. associate-*r* 75.3%

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

      2. *-commutative 75.3%

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

      3. *-commutative 75.3%

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

   6. Simplified 75.3%

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

      1. *-commutative 75.3%

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

      2. *-commutative 75.3%

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

      3. associate-*r* 75.8%

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

      4. add-cube-cbrt 73.1%

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

      5. unpow3 73.9%

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

      6. associate-*r* 73.9%

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

      7. *-commutative 73.9%

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

      8. *-commutative 73.9%

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

      9. associate-*r* 73.9%

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

      10. *-commutative 73.9%

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

      11. metadata-eval 73.9%

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

      12. div-inv 75.2%

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

      13. *-commutative 75.2%

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

   8. Applied egg-rr 75.2%

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

      1. rem-cube-cbrt 76.1%

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

      2. associate-*r/ 76.3%

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

   10. Applied egg-rr 76.3%

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

       1. associate-/l* 77.0%

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

   12. Simplified 77.0%

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

   if 5.00000000000000017e169 < (/.f64 angle 180)

   1. Initial program 19.5%

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

      1. associate-*l* 19.5%

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

      2. associate-*l* 19.5%

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

      3. unpow2 19.5%

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

      4. unpow2 19.5%

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

      5. difference-of-squares 19.5%

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

   3. Simplified 19.5%

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

   4. Taylor expanded in angle around 0 39.4%

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

4. Final simplification 72.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{angle}{180} \leq 5 \cdot 10^{+169}:\\ \;\;\;\;2 \cdot \left(\left(\sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right) \cdot \left(b + a\right)\right) \cdot \left(\left(b - a\right) \cdot \cos \left(\frac{\pi}{\frac{180}{angle}}\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \left(\left(\left(b - a\right) \cdot \left(b + a\right)\right) \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right)\\ \end{array} \]

Alternative 9: 65.9% accurate, 2.9× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

code[a_, b_, angle_] := 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]

Derivation

1. Initial program 54.5%

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

   1. associate-*l* 54.5%

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

   2. associate-*l* 54.5%

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

   3. unpow2 54.5%

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

   4. unpow2 54.5%

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

   5. difference-of-squares 58.6%

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

3. Simplified 58.6%

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

4. Taylor expanded in angle around inf 69.4%

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

   1. associate-*r* 69.4%

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

   2. *-commutative 69.4%

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

   3. *-commutative 69.4%

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

6. Simplified 69.4%

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

7. Taylor expanded in angle around 0 68.1%

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

8. Final simplification 68.1%

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

Alternative 10: 63.0% accurate, 2.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;angle \leq -1.6 \cdot 10^{+56}:\\ \;\;\;\;-0.011111111111111112 \cdot \left|\pi \cdot \left(angle \cdot \left(a \cdot a\right)\right)\right|\\ \mathbf{elif}\;angle \leq 44000000000:\\ \;\;\;\;\left(\left(angle \cdot \left(b - a\right)\right) \cdot \left(\pi \cdot \left(b + a\right)\right)\right) \cdot 0.011111111111111112\\ \mathbf{else}:\\ \;\;\;\;\left(angle \cdot 0.011111111111111112\right) \cdot \left(\pi \cdot \left(a \cdot a + b \cdot b\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (a b angle)
 :precision binary64
 (if (<= angle -1.6e+56)
   (* -0.011111111111111112 (fabs (* PI (* angle (* a a)))))
   (if (<= angle 44000000000.0)
     (* (* (* angle (- b a)) (* PI (+ b a))) 0.011111111111111112)
     (* (* angle 0.011111111111111112) (* PI (+ (* a a) (* b b)))))))

C

double code(double a, double b, double angle) {
	double tmp;
	if (angle <= -1.6e+56) {
		tmp = -0.011111111111111112 * fabs((((double) M_PI) * (angle * (a * a))));
	} else if (angle <= 44000000000.0) {
		tmp = ((angle * (b - a)) * (((double) M_PI) * (b + a))) * 0.011111111111111112;
	} else {
		tmp = (angle * 0.011111111111111112) * (((double) M_PI) * ((a * a) + (b * b)));
	}
	return tmp;
}

Java

public static double code(double a, double b, double angle) {
	double tmp;
	if (angle <= -1.6e+56) {
		tmp = -0.011111111111111112 * Math.abs((Math.PI * (angle * (a * a))));
	} else if (angle <= 44000000000.0) {
		tmp = ((angle * (b - a)) * (Math.PI * (b + a))) * 0.011111111111111112;
	} else {
		tmp = (angle * 0.011111111111111112) * (Math.PI * ((a * a) + (b * b)));
	}
	return tmp;
}

Python

def code(a, b, angle):
	tmp = 0
	if angle <= -1.6e+56:
		tmp = -0.011111111111111112 * math.fabs((math.pi * (angle * (a * a))))
	elif angle <= 44000000000.0:
		tmp = ((angle * (b - a)) * (math.pi * (b + a))) * 0.011111111111111112
	else:
		tmp = (angle * 0.011111111111111112) * (math.pi * ((a * a) + (b * b)))
	return tmp

Julia

function code(a, b, angle)
	tmp = 0.0
	if (angle <= -1.6e+56)
		tmp = Float64(-0.011111111111111112 * abs(Float64(pi * Float64(angle * Float64(a * a)))));
	elseif (angle <= 44000000000.0)
		tmp = Float64(Float64(Float64(angle * Float64(b - a)) * Float64(pi * Float64(b + a))) * 0.011111111111111112);
	else
		tmp = Float64(Float64(angle * 0.011111111111111112) * Float64(pi * Float64(Float64(a * a) + Float64(b * b))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, angle)
	tmp = 0.0;
	if (angle <= -1.6e+56)
		tmp = -0.011111111111111112 * abs((pi * (angle * (a * a))));
	elseif (angle <= 44000000000.0)
		tmp = ((angle * (b - a)) * (pi * (b + a))) * 0.011111111111111112;
	else
		tmp = (angle * 0.011111111111111112) * (pi * ((a * a) + (b * b)));
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, angle_] := If[LessEqual[angle, -1.6e+56], N[(-0.011111111111111112 * N[Abs[N[(Pi * N[(angle * N[(a * a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[angle, 44000000000.0], N[(N[(N[(angle * N[(b - a), $MachinePrecision]), $MachinePrecision] * N[(Pi * N[(b + a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * 0.011111111111111112), $MachinePrecision], N[(N[(angle * 0.011111111111111112), $MachinePrecision] * N[(Pi * N[(N[(a * a), $MachinePrecision] + N[(b * b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if angle < -1.60000000000000002e56

   1. Initial program 38.2%

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

      1. *-commutative 38.2%

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

      2. associate-*l* 38.2%

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

      3. unpow2 38.2%

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

      4. fma-neg 38.2%

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

      5. unpow2 38.2%

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

   3. Simplified 38.2%

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

   4. Taylor expanded in angle around 0 20.3%

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

      1. associate-*r* 20.3%

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

      2. unpow2 20.3%

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

      3. unpow2 20.3%

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

      4. *-lft-identity 20.3%

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

      5. cancel-sign-sub-inv 20.3%

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

      6. metadata-eval 20.3%

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

      7. unpow2 20.3%

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

      8. +-commutative 20.3%

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

      9. unpow2 20.3%

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

      10. fma-def 20.3%

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

      11. unpow2 20.3%

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

      12. unpow2 20.3%

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

      13. fma-def 20.3%

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

      14. mul-1-neg 20.3%

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

      15. +-commutative 20.3%

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

      16. sub-neg 20.3%

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

   6. Simplified 20.3%

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

   7. Taylor expanded in b around 0 17.5%

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

      1. unpow2 17.5%

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

      2. *-commutative 17.5%

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

   9. Simplified 17.5%

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

   10. Taylor expanded in a around 0 17.5%

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

       1. unpow2 17.5%

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

       2. associate-*l* 17.5%

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

   12. Simplified 17.5%

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

       1. add-sqr-sqrt 7.0%

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

       2. sqrt-unprod 31.1%

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

       3. pow2 31.1%

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

       4. *-commutative 31.1%

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

       5. *-commutative 31.1%

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

       6. associate-*l* 31.1%

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

   14. Applied egg-rr 31.1%

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

       1. unpow2 31.1%

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

       2. rem-sqrt-square 29.1%

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

       3. *-commutative 29.1%

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

       4. unpow2 29.1%

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

       5. associate-*l* 29.1%

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

       6. *-commutative 29.1%

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

       7. unpow2 29.1%

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

   16. Simplified 29.1%

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

   if -1.60000000000000002e56 < angle < 4.4e10

   1. Initial program 71.1%

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

      1. associate-*l* 71.1%

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

      2. unpow2 71.1%

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

      3. unpow2 71.1%

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

      4. difference-of-squares 77.8%

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

   3. Simplified 77.8%

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

   4. Taylor expanded in angle around 0 73.6%

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

      1. *-commutative 73.6%

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

      2. associate-*r* 88.8%

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

   6. Simplified 88.8%

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

   if 4.4e10 < angle

   1. Initial program 20.1%

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

      1. *-commutative 20.1%

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

      2. associate-*l* 20.1%

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

      3. unpow2 20.1%

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

      4. fma-neg 23.8%

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

      5. unpow2 23.8%

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

   3. Simplified 23.8%

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

      1. fma-udef 20.1%

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

      2. distribute-rgt-neg-in 20.1%

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

      3. add-sqr-sqrt 13.5%

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

      4. sqrt-unprod 25.6%

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

      5. sqr-neg 25.6%

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

      6. sqrt-prod 12.1%

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

      7. add-sqr-sqrt 29.5%

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

   5. Applied egg-rr 29.5%

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

      1. add-exp-log 13.9%

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

      2. associate-*l* 13.9%

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

      3. fma-def 13.9%

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

   7. Applied egg-rr 13.9%

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

   8. Taylor expanded in angle around 0 29.2%

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

      1. associate-*r* 29.2%

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

      2. *-commutative 29.2%

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

      3. +-commutative 29.2%

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

      4. unpow2 29.2%

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

      5. unpow2 29.2%

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

   10. Simplified 29.2%

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

4. Final simplification 65.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;angle \leq -1.6 \cdot 10^{+56}:\\ \;\;\;\;-0.011111111111111112 \cdot \left|\pi \cdot \left(angle \cdot \left(a \cdot a\right)\right)\right|\\ \mathbf{elif}\;angle \leq 44000000000:\\ \;\;\;\;\left(\left(angle \cdot \left(b - a\right)\right) \cdot \left(\pi \cdot \left(b + a\right)\right)\right) \cdot 0.011111111111111112\\ \mathbf{else}:\\ \;\;\;\;\left(angle \cdot 0.011111111111111112\right) \cdot \left(\pi \cdot \left(a \cdot a + b \cdot b\right)\right)\\ \end{array} \]

Alternative 11: 57.2% accurate, 5.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;a \leq -3.6 \cdot 10^{+164} \lor \neg \left(a \leq 2.5 \cdot 10^{+118}\right):\\ \;\;\;\;-0.011111111111111112 \cdot \left(\pi \cdot \left(a \cdot \left(angle \cdot a\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;0.011111111111111112 \cdot \left(\pi \cdot \left(angle \cdot \left(b \cdot b - a \cdot a\right)\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (a b angle)
 :precision binary64
 (if (or (<= a -3.6e+164) (not (<= a 2.5e+118)))
   (* -0.011111111111111112 (* PI (* a (* angle a))))
   (* 0.011111111111111112 (* PI (* angle (- (* b b) (* a a)))))))

C

double code(double a, double b, double angle) {
	double tmp;
	if ((a <= -3.6e+164) || !(a <= 2.5e+118)) {
		tmp = -0.011111111111111112 * (((double) M_PI) * (a * (angle * a)));
	} else {
		tmp = 0.011111111111111112 * (((double) M_PI) * (angle * ((b * b) - (a * a))));
	}
	return tmp;
}

Java

public static double code(double a, double b, double angle) {
	double tmp;
	if ((a <= -3.6e+164) || !(a <= 2.5e+118)) {
		tmp = -0.011111111111111112 * (Math.PI * (a * (angle * a)));
	} else {
		tmp = 0.011111111111111112 * (Math.PI * (angle * ((b * b) - (a * a))));
	}
	return tmp;
}

Python

def code(a, b, angle):
	tmp = 0
	if (a <= -3.6e+164) or not (a <= 2.5e+118):
		tmp = -0.011111111111111112 * (math.pi * (a * (angle * a)))
	else:
		tmp = 0.011111111111111112 * (math.pi * (angle * ((b * b) - (a * a))))
	return tmp

Julia

function code(a, b, angle)
	tmp = 0.0
	if ((a <= -3.6e+164) || !(a <= 2.5e+118))
		tmp = Float64(-0.011111111111111112 * Float64(pi * Float64(a * Float64(angle * a))));
	else
		tmp = Float64(0.011111111111111112 * Float64(pi * Float64(angle * Float64(Float64(b * b) - Float64(a * a)))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, angle)
	tmp = 0.0;
	if ((a <= -3.6e+164) || ~((a <= 2.5e+118)))
		tmp = -0.011111111111111112 * (pi * (a * (angle * a)));
	else
		tmp = 0.011111111111111112 * (pi * (angle * ((b * b) - (a * a))));
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, angle_] := If[Or[LessEqual[a, -3.6e+164], N[Not[LessEqual[a, 2.5e+118]], $MachinePrecision]], N[(-0.011111111111111112 * N[(Pi * N[(a * N[(angle * a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(0.011111111111111112 * N[(Pi * N[(angle * N[(N[(b * b), $MachinePrecision] - N[(a * a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if a < -3.5999999999999999e164 or 2.49999999999999986e118 < a

   1. Initial program 38.2%

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

      1. *-commutative 38.2%

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

      2. associate-*l* 38.2%

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

      3. unpow2 38.2%

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

      4. fma-neg 51.7%

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

      5. unpow2 51.7%

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

   3. Simplified 51.7%

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

   4. Taylor expanded in angle around 0 36.8%

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

      1. associate-*r* 36.8%

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

      2. unpow2 36.8%

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

      3. unpow2 36.8%

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

      4. *-lft-identity 36.8%

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

      5. cancel-sign-sub-inv 36.8%

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

      6. metadata-eval 36.8%

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

      7. unpow2 36.8%

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

      8. +-commutative 36.8%

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

      9. unpow2 36.8%

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

      10. fma-def 36.8%

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

      11. unpow2 36.8%

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

      12. unpow2 36.8%

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

      13. fma-def 36.8%

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

      14. mul-1-neg 36.8%

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

      15. +-commutative 36.8%

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

      16. sub-neg 36.8%

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

   6. Simplified 36.8%

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

   7. Taylor expanded in b around 0 48.9%

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

      1. unpow2 48.9%

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

      2. *-commutative 48.9%

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

   9. Simplified 48.9%

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

   10. Taylor expanded in angle around 0 48.9%

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

       1. associate-*r* 48.9%

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

       2. *-commutative 48.9%

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

       3. unpow2 48.9%

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

       4. associate-*r* 69.2%

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

   12. Simplified 69.2%

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

   if -3.5999999999999999e164 < a < 2.49999999999999986e118

   1. Initial program 60.4%

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

      1. *-commutative 60.4%

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

      2. associate-*l* 60.4%

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

      3. unpow2 60.4%

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

      4. fma-neg 60.4%

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

      5. unpow2 60.4%

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

   3. Simplified 60.4%

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

   4. Taylor expanded in angle around 0 54.1%

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

      1. associate-*r* 54.1%

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

      2. unpow2 54.1%

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

      3. unpow2 54.1%

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

      4. *-lft-identity 54.1%

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

      5. cancel-sign-sub-inv 54.1%

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

      6. metadata-eval 54.1%

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

      7. unpow2 54.1%

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

      8. +-commutative 54.1%

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

      9. unpow2 54.1%

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

      10. fma-def 54.1%

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

      11. unpow2 54.1%

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

      12. unpow2 54.1%

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

      13. fma-def 54.1%

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

      14. mul-1-neg 54.1%

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

      15. +-commutative 54.1%

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

      16. sub-neg 54.1%

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

   6. Simplified 54.1%

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

4. Final simplification 58.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \leq -3.6 \cdot 10^{+164} \lor \neg \left(a \leq 2.5 \cdot 10^{+118}\right):\\ \;\;\;\;-0.011111111111111112 \cdot \left(\pi \cdot \left(a \cdot \left(angle \cdot a\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;0.011111111111111112 \cdot \left(\pi \cdot \left(angle \cdot \left(b \cdot b - a \cdot a\right)\right)\right)\\ \end{array} \]

Alternative 12: 57.1% accurate, 5.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;a \leq -3.6 \cdot 10^{+164} \lor \neg \left(a \leq 5.4 \cdot 10^{+109}\right):\\ \;\;\;\;-0.011111111111111112 \cdot \left(\pi \cdot \left(a \cdot \left(angle \cdot a\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(angle \cdot 0.011111111111111112\right) \cdot \left(\pi \cdot \left(b \cdot b - a \cdot a\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (a b angle)
 :precision binary64
 (if (or (<= a -3.6e+164) (not (<= a 5.4e+109)))
   (* -0.011111111111111112 (* PI (* a (* angle a))))
   (* (* angle 0.011111111111111112) (* PI (- (* b b) (* a a))))))

C

double code(double a, double b, double angle) {
	double tmp;
	if ((a <= -3.6e+164) || !(a <= 5.4e+109)) {
		tmp = -0.011111111111111112 * (((double) M_PI) * (a * (angle * a)));
	} else {
		tmp = (angle * 0.011111111111111112) * (((double) M_PI) * ((b * b) - (a * a)));
	}
	return tmp;
}

Java

public static double code(double a, double b, double angle) {
	double tmp;
	if ((a <= -3.6e+164) || !(a <= 5.4e+109)) {
		tmp = -0.011111111111111112 * (Math.PI * (a * (angle * a)));
	} else {
		tmp = (angle * 0.011111111111111112) * (Math.PI * ((b * b) - (a * a)));
	}
	return tmp;
}

Python

def code(a, b, angle):
	tmp = 0
	if (a <= -3.6e+164) or not (a <= 5.4e+109):
		tmp = -0.011111111111111112 * (math.pi * (a * (angle * a)))
	else:
		tmp = (angle * 0.011111111111111112) * (math.pi * ((b * b) - (a * a)))
	return tmp

Julia

function code(a, b, angle)
	tmp = 0.0
	if ((a <= -3.6e+164) || !(a <= 5.4e+109))
		tmp = Float64(-0.011111111111111112 * Float64(pi * Float64(a * Float64(angle * a))));
	else
		tmp = Float64(Float64(angle * 0.011111111111111112) * Float64(pi * Float64(Float64(b * b) - Float64(a * a))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, angle)
	tmp = 0.0;
	if ((a <= -3.6e+164) || ~((a <= 5.4e+109)))
		tmp = -0.011111111111111112 * (pi * (a * (angle * a)));
	else
		tmp = (angle * 0.011111111111111112) * (pi * ((b * b) - (a * a)));
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, angle_] := If[Or[LessEqual[a, -3.6e+164], N[Not[LessEqual[a, 5.4e+109]], $MachinePrecision]], N[(-0.011111111111111112 * N[(Pi * N[(a * N[(angle * a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(angle * 0.011111111111111112), $MachinePrecision] * N[(Pi * N[(N[(b * b), $MachinePrecision] - N[(a * a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if a < -3.5999999999999999e164 or 5.40000000000000003e109 < a

   1. Initial program 40.0%

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

      1. *-commutative 40.0%

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

      2. associate-*l* 40.0%

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

      3. unpow2 40.0%

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

      4. fma-neg 53.1%

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

      5. unpow2 53.1%

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

   3. Simplified 53.1%

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

   4. Taylor expanded in angle around 0 38.5%

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

      1. associate-*r* 38.6%

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

      2. unpow2 38.6%

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

      3. unpow2 38.6%

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

      4. *-lft-identity 38.6%

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

      5. cancel-sign-sub-inv 38.6%

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

      6. metadata-eval 38.6%

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

      7. unpow2 38.6%

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

      8. +-commutative 38.6%

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

      9. unpow2 38.6%

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

      10. fma-def 38.6%

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

      11. unpow2 38.6%

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

      12. unpow2 38.6%

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

      13. fma-def 38.6%

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

      14. mul-1-neg 38.6%

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

      15. +-commutative 38.6%

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

      16. sub-neg 38.6%

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

   6. Simplified 38.6%

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

   7. Taylor expanded in b around 0 50.3%

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

      1. unpow2 50.3%

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

      2. *-commutative 50.3%

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

   9. Simplified 50.3%

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

   10. Taylor expanded in angle around 0 50.3%

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

       1. associate-*r* 50.3%

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

       2. *-commutative 50.3%

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

       3. unpow2 50.3%

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

       4. associate-*r* 70.0%

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

   12. Simplified 70.0%

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

   if -3.5999999999999999e164 < a < 5.40000000000000003e109

   1. Initial program 60.0%

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

      1. *-commutative 60.0%

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

      2. associate-*l* 60.0%

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

      3. unpow2 60.0%

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

      4. fma-neg 60.0%

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

      5. unpow2 60.0%

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

   3. Simplified 60.0%

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

   4. Taylor expanded in angle around 0 53.7%

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

      1. associate-*r* 53.7%

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

      2. *-commutative 53.7%

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

      3. *-commutative 53.7%

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

      4. unpow2 53.7%

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

      5. unpow2 53.7%

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

      6. *-commutative 53.7%

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

   6. Simplified 53.7%

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

4. Final simplification 58.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \leq -3.6 \cdot 10^{+164} \lor \neg \left(a \leq 5.4 \cdot 10^{+109}\right):\\ \;\;\;\;-0.011111111111111112 \cdot \left(\pi \cdot \left(a \cdot \left(angle \cdot a\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(angle \cdot 0.011111111111111112\right) \cdot \left(\pi \cdot \left(b \cdot b - a \cdot a\right)\right)\\ \end{array} \]

Alternative 13: 64.2% accurate, 5.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;angle \leq -9.8 \cdot 10^{+64} \lor \neg \left(angle \leq 44000000000\right):\\ \;\;\;\;\left(angle \cdot 0.011111111111111112\right) \cdot \left(\pi \cdot \left(a \cdot a + b \cdot b\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\left(angle \cdot \left(b - a\right)\right) \cdot \left(\pi \cdot \left(b + a\right)\right)\right) \cdot 0.011111111111111112\\ \end{array} \end{array} \]

FPCore

(FPCore (a b angle)
 :precision binary64
 (if (or (<= angle -9.8e+64) (not (<= angle 44000000000.0)))
   (* (* angle 0.011111111111111112) (* PI (+ (* a a) (* b b))))
   (* (* (* angle (- b a)) (* PI (+ b a))) 0.011111111111111112)))

C

double code(double a, double b, double angle) {
	double tmp;
	if ((angle <= -9.8e+64) || !(angle <= 44000000000.0)) {
		tmp = (angle * 0.011111111111111112) * (((double) M_PI) * ((a * a) + (b * b)));
	} else {
		tmp = ((angle * (b - a)) * (((double) M_PI) * (b + a))) * 0.011111111111111112;
	}
	return tmp;
}

Java

public static double code(double a, double b, double angle) {
	double tmp;
	if ((angle <= -9.8e+64) || !(angle <= 44000000000.0)) {
		tmp = (angle * 0.011111111111111112) * (Math.PI * ((a * a) + (b * b)));
	} else {
		tmp = ((angle * (b - a)) * (Math.PI * (b + a))) * 0.011111111111111112;
	}
	return tmp;
}

Python

def code(a, b, angle):
	tmp = 0
	if (angle <= -9.8e+64) or not (angle <= 44000000000.0):
		tmp = (angle * 0.011111111111111112) * (math.pi * ((a * a) + (b * b)))
	else:
		tmp = ((angle * (b - a)) * (math.pi * (b + a))) * 0.011111111111111112
	return tmp

Julia

function code(a, b, angle)
	tmp = 0.0
	if ((angle <= -9.8e+64) || !(angle <= 44000000000.0))
		tmp = Float64(Float64(angle * 0.011111111111111112) * Float64(pi * Float64(Float64(a * a) + Float64(b * b))));
	else
		tmp = Float64(Float64(Float64(angle * Float64(b - a)) * Float64(pi * Float64(b + a))) * 0.011111111111111112);
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, angle)
	tmp = 0.0;
	if ((angle <= -9.8e+64) || ~((angle <= 44000000000.0)))
		tmp = (angle * 0.011111111111111112) * (pi * ((a * a) + (b * b)));
	else
		tmp = ((angle * (b - a)) * (pi * (b + a))) * 0.011111111111111112;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, angle_] := If[Or[LessEqual[angle, -9.8e+64], N[Not[LessEqual[angle, 44000000000.0]], $MachinePrecision]], N[(N[(angle * 0.011111111111111112), $MachinePrecision] * N[(Pi * N[(N[(a * a), $MachinePrecision] + N[(b * b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(angle * N[(b - a), $MachinePrecision]), $MachinePrecision] * N[(Pi * N[(b + a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * 0.011111111111111112), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if angle < -9.8000000000000005e64 or 4.4e10 < angle

   1. Initial program 26.7%

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

      1. *-commutative 26.7%

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

      2. associate-*l* 26.7%

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

      3. unpow2 26.7%

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

      4. fma-neg 28.8%

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

      5. unpow2 28.8%

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

   3. Simplified 28.8%

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

      1. fma-udef 26.7%

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

      2. distribute-rgt-neg-in 26.7%

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

      3. add-sqr-sqrt 16.4%

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

      4. sqrt-unprod 30.4%

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

      5. sqr-neg 30.4%

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

      6. sqrt-prod 14.0%

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

      7. add-sqr-sqrt 31.0%

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

   5. Applied egg-rr 31.0%

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

      1. add-exp-log 18.6%

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

      2. associate-*l* 18.6%

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

      3. fma-def 18.6%

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

   7. Applied egg-rr 18.6%

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

   8. Taylor expanded in angle around 0 29.0%

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

      1. associate-*r* 29.0%

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

      2. *-commutative 29.0%

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

      3. +-commutative 29.0%

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

      4. unpow2 29.0%

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

      5. unpow2 29.0%

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

   10. Simplified 29.0%

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

   if -9.8000000000000005e64 < angle < 4.4e10

   1. Initial program 71.5%

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

      1. associate-*l* 71.5%

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

      2. unpow2 71.5%

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

      3. unpow2 71.5%

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

      4. difference-of-squares 78.1%

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

   3. Simplified 78.1%

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

   4. Taylor expanded in angle around 0 72.7%

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

      1. *-commutative 72.7%

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

      2. associate-*r* 87.7%

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

   6. Simplified 87.7%

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

4. Final simplification 65.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;angle \leq -9.8 \cdot 10^{+64} \lor \neg \left(angle \leq 44000000000\right):\\ \;\;\;\;\left(angle \cdot 0.011111111111111112\right) \cdot \left(\pi \cdot \left(a \cdot a + b \cdot b\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\left(angle \cdot \left(b - a\right)\right) \cdot \left(\pi \cdot \left(b + a\right)\right)\right) \cdot 0.011111111111111112\\ \end{array} \]

Alternative 14: 50.7% accurate, 5.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;a \leq -1.4 \cdot 10^{-72} \lor \neg \left(a \leq 4 \cdot 10^{-29}\right):\\ \;\;\;\;-0.011111111111111112 \cdot \left(\pi \cdot \left(a \cdot \left(angle \cdot a\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;0.011111111111111112 \cdot \left(angle \cdot \left(\pi \cdot \left(b \cdot b\right)\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (a b angle)
 :precision binary64
 (if (or (<= a -1.4e-72) (not (<= a 4e-29)))
   (* -0.011111111111111112 (* PI (* a (* angle a))))
   (* 0.011111111111111112 (* angle (* PI (* b b))))))

C

double code(double a, double b, double angle) {
	double tmp;
	if ((a <= -1.4e-72) || !(a <= 4e-29)) {
		tmp = -0.011111111111111112 * (((double) M_PI) * (a * (angle * 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 ((a <= -1.4e-72) || !(a <= 4e-29)) {
		tmp = -0.011111111111111112 * (Math.PI * (a * (angle * a)));
	} else {
		tmp = 0.011111111111111112 * (angle * (Math.PI * (b * b)));
	}
	return tmp;
}

Python

def code(a, b, angle):
	tmp = 0
	if (a <= -1.4e-72) or not (a <= 4e-29):
		tmp = -0.011111111111111112 * (math.pi * (a * (angle * a)))
	else:
		tmp = 0.011111111111111112 * (angle * (math.pi * (b * b)))
	return tmp

Julia

function code(a, b, angle)
	tmp = 0.0
	if ((a <= -1.4e-72) || !(a <= 4e-29))
		tmp = Float64(-0.011111111111111112 * Float64(pi * Float64(a * Float64(angle * 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 ((a <= -1.4e-72) || ~((a <= 4e-29)))
		tmp = -0.011111111111111112 * (pi * (a * (angle * a)));
	else
		tmp = 0.011111111111111112 * (angle * (pi * (b * b)));
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, angle_] := If[Or[LessEqual[a, -1.4e-72], N[Not[LessEqual[a, 4e-29]], $MachinePrecision]], N[(-0.011111111111111112 * N[(Pi * N[(a * N[(angle * 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 a < -1.3999999999999999e-72 or 3.99999999999999977e-29 < a

   1. Initial program 51.6%

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

      1. *-commutative 51.6%

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

      2. associate-*l* 51.6%

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

      3. unpow2 51.6%

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

      4. fma-neg 57.9%

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

      5. unpow2 57.9%

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

   3. Simplified 57.9%

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

   4. Taylor expanded in angle around 0 47.6%

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

      1. associate-*r* 47.6%

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

      2. unpow2 47.6%

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

      3. unpow2 47.6%

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

      4. *-lft-identity 47.6%

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

      5. cancel-sign-sub-inv 47.6%

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

      6. metadata-eval 47.6%

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

      7. unpow2 47.6%

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

      8. +-commutative 47.6%

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

      9. unpow2 47.6%

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

      10. fma-def 47.6%

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

      11. unpow2 47.6%

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

      12. unpow2 47.6%

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

      13. fma-def 47.6%

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

      14. mul-1-neg 47.6%

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

      15. +-commutative 47.6%

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

      16. sub-neg 47.6%

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

   6. Simplified 47.6%

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

   7. Taylor expanded in b around 0 44.5%

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

      1. unpow2 44.5%

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

      2. *-commutative 44.5%

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

   9. Simplified 44.5%

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

   10. Taylor expanded in angle around 0 44.5%

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

       1. associate-*r* 44.5%

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

       2. *-commutative 44.5%

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

       3. unpow2 44.5%

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

       4. associate-*r* 54.1%

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

   12. Simplified 54.1%

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

   if -1.3999999999999999e-72 < a < 3.99999999999999977e-29

   1. Initial program 58.3%

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

      1. *-commutative 58.3%

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

      2. associate-*l* 58.3%

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

      3. unpow2 58.3%

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

      4. fma-neg 58.3%

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

      5. unpow2 58.3%

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

   3. Simplified 58.3%

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

   4. Taylor expanded in angle around 0 52.0%

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

      1. associate-*r* 52.0%

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

      2. unpow2 52.0%

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

      3. unpow2 52.0%

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

      4. *-lft-identity 52.0%

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

      5. cancel-sign-sub-inv 52.0%

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

      6. metadata-eval 52.0%

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

      7. unpow2 52.0%

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

      8. +-commutative 52.0%

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

      9. unpow2 52.0%

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

      10. fma-def 52.0%

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

      11. unpow2 52.0%

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

      12. unpow2 52.0%

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

      13. fma-def 52.0%

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

      14. mul-1-neg 52.0%

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

      15. +-commutative 52.0%

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

      16. sub-neg 52.0%

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

   6. Simplified 52.0%

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

   7. Taylor expanded in b around inf 49.2%

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

      1. unpow2 49.2%

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

      2. *-commutative 49.2%

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

   9. Simplified 49.2%

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

4. Final simplification 51.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \leq -1.4 \cdot 10^{-72} \lor \neg \left(a \leq 4 \cdot 10^{-29}\right):\\ \;\;\;\;-0.011111111111111112 \cdot \left(\pi \cdot \left(a \cdot \left(angle \cdot a\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;0.011111111111111112 \cdot \left(angle \cdot \left(\pi \cdot \left(b \cdot b\right)\right)\right)\\ \end{array} \]

Alternative 15: 34.4% accurate, 5.7× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 54.5%

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

   1. *-commutative 54.5%

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

   2. associate-*l* 54.5%

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

   3. unpow2 54.5%

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

   4. fma-neg 58.1%

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

   5. unpow2 58.1%

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

3. Simplified 58.1%

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

4. Taylor expanded in angle around 0 49.5%

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

   1. associate-*r* 49.5%

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

   2. unpow2 49.5%

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

   3. unpow2 49.5%

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

   4. *-lft-identity 49.5%

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

   5. cancel-sign-sub-inv 49.5%

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

   6. metadata-eval 49.5%

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

   7. unpow2 49.5%

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

   8. +-commutative 49.5%

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

   9. unpow2 49.5%

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

   10. fma-def 49.5%

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

   11. unpow2 49.5%

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

   12. unpow2 49.5%

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

   13. fma-def 49.5%

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

   14. mul-1-neg 49.5%

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

   15. +-commutative 49.5%

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

   16. sub-neg 49.5%

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

6. Simplified 49.5%

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

7. Taylor expanded in b around 0 39.3%

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

   1. unpow2 39.3%

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

   2. *-commutative 39.3%

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

9. Simplified 39.3%

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

10. Taylor expanded in a around 0 39.3%

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

    1. unpow2 39.3%

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

    2. associate-*l* 39.2%

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

12. Simplified 39.2%

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

13. Final simplification 39.2%

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

Alternative 16: 34.4% accurate, 5.7× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 54.5%

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

   1. *-commutative 54.5%

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

   2. associate-*l* 54.5%

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

   3. unpow2 54.5%

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

   4. fma-neg 58.1%

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

   5. unpow2 58.1%

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

3. Simplified 58.1%

   \[\leadsto \color{blue}{\sin \left(\pi \cdot \frac{angle}{180}\right) \cdot \left(\left(2 \cdot \mathsf{fma}\left(b, b, -a \cdot a\right)\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right)\right)} \]

4. Taylor expanded in angle around 0 49.5%

   \[\leadsto \color{blue}{0.011111111111111112 \cdot \left(angle \cdot \left(\left({b}^{2} - {a}^{2}\right) \cdot \pi\right)\right)} \]
5. Step-by-step derivation

   1. associate-*r* 49.5%

      \[\leadsto 0.011111111111111112 \cdot \color{blue}{\left(\left(angle \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \pi\right)} \]

   2. unpow2 49.5%

      \[\leadsto 0.011111111111111112 \cdot \left(\left(angle \cdot \left(\color{blue}{b \cdot b} - {a}^{2}\right)\right) \cdot \pi\right) \]

   3. unpow2 49.5%

      \[\leadsto 0.011111111111111112 \cdot \left(\left(angle \cdot \left(b \cdot b - \color{blue}{a \cdot a}\right)\right) \cdot \pi\right) \]

   4. *-lft-identity 49.5%

      \[\leadsto 0.011111111111111112 \cdot \left(\left(angle \cdot \left(b \cdot b - \color{blue}{1 \cdot \left(a \cdot a\right)}\right)\right) \cdot \pi\right) \]

   5. cancel-sign-sub-inv 49.5%

      \[\leadsto 0.011111111111111112 \cdot \left(\left(angle \cdot \color{blue}{\left(b \cdot b + \left(-1\right) \cdot \left(a \cdot a\right)\right)}\right) \cdot \pi\right) \]

   6. metadata-eval 49.5%

      \[\leadsto 0.011111111111111112 \cdot \left(\left(angle \cdot \left(b \cdot b + \color{blue}{-1} \cdot \left(a \cdot a\right)\right)\right) \cdot \pi\right) \]

   7. unpow2 49.5%

      \[\leadsto 0.011111111111111112 \cdot \left(\left(angle \cdot \left(b \cdot b + -1 \cdot \color{blue}{{a}^{2}}\right)\right) \cdot \pi\right) \]

   8. +-commutative 49.5%

      \[\leadsto 0.011111111111111112 \cdot \left(\left(angle \cdot \color{blue}{\left(-1 \cdot {a}^{2} + b \cdot b\right)}\right) \cdot \pi\right) \]

   9. unpow2 49.5%

      \[\leadsto 0.011111111111111112 \cdot \left(\left(angle \cdot \left(-1 \cdot {a}^{2} + \color{blue}{{b}^{2}}\right)\right) \cdot \pi\right) \]

   10. fma-def 49.5%

       \[\leadsto 0.011111111111111112 \cdot \left(\left(angle \cdot \color{blue}{\mathsf{fma}\left(-1, {a}^{2}, {b}^{2}\right)}\right) \cdot \pi\right) \]

   11. unpow2 49.5%

       \[\leadsto 0.011111111111111112 \cdot \left(\left(angle \cdot \mathsf{fma}\left(-1, \color{blue}{a \cdot a}, {b}^{2}\right)\right) \cdot \pi\right) \]

   12. unpow2 49.5%

       \[\leadsto 0.011111111111111112 \cdot \left(\left(angle \cdot \mathsf{fma}\left(-1, a \cdot a, \color{blue}{b \cdot b}\right)\right) \cdot \pi\right) \]

   13. fma-def 49.5%

       \[\leadsto 0.011111111111111112 \cdot \left(\left(angle \cdot \color{blue}{\left(-1 \cdot \left(a \cdot a\right) + b \cdot b\right)}\right) \cdot \pi\right) \]

   14. mul-1-neg 49.5%

       \[\leadsto 0.011111111111111112 \cdot \left(\left(angle \cdot \left(\color{blue}{\left(-a \cdot a\right)} + b \cdot b\right)\right) \cdot \pi\right) \]

   15. +-commutative 49.5%

       \[\leadsto 0.011111111111111112 \cdot \left(\left(angle \cdot \color{blue}{\left(b \cdot b + \left(-a \cdot a\right)\right)}\right) \cdot \pi\right) \]

   16. sub-neg 49.5%

       \[\leadsto 0.011111111111111112 \cdot \left(\left(angle \cdot \color{blue}{\left(b \cdot b - a \cdot a\right)}\right) \cdot \pi\right) \]

6. Simplified 49.5%

   \[\leadsto \color{blue}{0.011111111111111112 \cdot \left(\left(angle \cdot \left(b \cdot b - a \cdot a\right)\right) \cdot \pi\right)} \]

7. Taylor expanded in b around 0 39.3%

   \[\leadsto \color{blue}{-0.011111111111111112 \cdot \left(angle \cdot \left({a}^{2} \cdot \pi\right)\right)} \]
8. Step-by-step derivation

   1. unpow2 39.3%

      \[\leadsto -0.011111111111111112 \cdot \left(angle \cdot \left(\color{blue}{\left(a \cdot a\right)} \cdot \pi\right)\right) \]

   2. *-commutative 39.3%

      \[\leadsto -0.011111111111111112 \cdot \left(angle \cdot \color{blue}{\left(\pi \cdot \left(a \cdot a\right)\right)}\right) \]

9. Simplified 39.3%

   \[\leadsto \color{blue}{-0.011111111111111112 \cdot \left(angle \cdot \left(\pi \cdot \left(a \cdot a\right)\right)\right)} \]

10. Final simplification 39.3%

    \[\leadsto -0.011111111111111112 \cdot \left(angle \cdot \left(\pi \cdot \left(a \cdot a\right)\right)\right) \]

Alternative 17: 37.6% accurate, 5.7× speedup

Math

\[\begin{array}{l} \\ -0.011111111111111112 \cdot \left(\pi \cdot \left(a \cdot \left(angle \cdot a\right)\right)\right) \end{array} \]

FPCore

(FPCore (a b angle)
 :precision binary64
 (* -0.011111111111111112 (* PI (* a (* angle a)))))

C

double code(double a, double b, double angle) {
	return -0.011111111111111112 * (((double) M_PI) * (a * (angle * a)));
}

Java

public static double code(double a, double b, double angle) {
	return -0.011111111111111112 * (Math.PI * (a * (angle * a)));
}

Python

def code(a, b, angle):
	return -0.011111111111111112 * (math.pi * (a * (angle * a)))

Julia

function code(a, b, angle)
	return Float64(-0.011111111111111112 * Float64(pi * Float64(a * Float64(angle * a))))
end

MATLAB

function tmp = code(a, b, angle)
	tmp = -0.011111111111111112 * (pi * (a * (angle * a)));
end

Wolfram

code[a_, b_, angle_] := N[(-0.011111111111111112 * N[(Pi * N[(a * N[(angle * a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 54.5%

   \[\left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin \left(\pi \cdot \frac{angle}{180}\right)\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right) \]
2. Step-by-step derivation

   1. *-commutative 54.5%

      \[\leadsto \color{blue}{\left(\sin \left(\pi \cdot \frac{angle}{180}\right) \cdot \left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right)\right)} \cdot \cos \left(\pi \cdot \frac{angle}{180}\right) \]

   2. associate-*l* 54.5%

      \[\leadsto \color{blue}{\sin \left(\pi \cdot \frac{angle}{180}\right) \cdot \left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right)\right)} \]

   3. unpow2 54.5%

      \[\leadsto \sin \left(\pi \cdot \frac{angle}{180}\right) \cdot \left(\left(2 \cdot \left(\color{blue}{b \cdot b} - {a}^{2}\right)\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right)\right) \]

   4. fma-neg 58.1%

      \[\leadsto \sin \left(\pi \cdot \frac{angle}{180}\right) \cdot \left(\left(2 \cdot \color{blue}{\mathsf{fma}\left(b, b, -{a}^{2}\right)}\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right)\right) \]

   5. unpow2 58.1%

      \[\leadsto \sin \left(\pi \cdot \frac{angle}{180}\right) \cdot \left(\left(2 \cdot \mathsf{fma}\left(b, b, -\color{blue}{a \cdot a}\right)\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right)\right) \]

3. Simplified 58.1%

   \[\leadsto \color{blue}{\sin \left(\pi \cdot \frac{angle}{180}\right) \cdot \left(\left(2 \cdot \mathsf{fma}\left(b, b, -a \cdot a\right)\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right)\right)} \]

4. Taylor expanded in angle around 0 49.5%

   \[\leadsto \color{blue}{0.011111111111111112 \cdot \left(angle \cdot \left(\left({b}^{2} - {a}^{2}\right) \cdot \pi\right)\right)} \]
5. Step-by-step derivation

   1. associate-*r* 49.5%

      \[\leadsto 0.011111111111111112 \cdot \color{blue}{\left(\left(angle \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \pi\right)} \]

   2. unpow2 49.5%

      \[\leadsto 0.011111111111111112 \cdot \left(\left(angle \cdot \left(\color{blue}{b \cdot b} - {a}^{2}\right)\right) \cdot \pi\right) \]

   3. unpow2 49.5%

      \[\leadsto 0.011111111111111112 \cdot \left(\left(angle \cdot \left(b \cdot b - \color{blue}{a \cdot a}\right)\right) \cdot \pi\right) \]

   4. *-lft-identity 49.5%

      \[\leadsto 0.011111111111111112 \cdot \left(\left(angle \cdot \left(b \cdot b - \color{blue}{1 \cdot \left(a \cdot a\right)}\right)\right) \cdot \pi\right) \]

   5. cancel-sign-sub-inv 49.5%

      \[\leadsto 0.011111111111111112 \cdot \left(\left(angle \cdot \color{blue}{\left(b \cdot b + \left(-1\right) \cdot \left(a \cdot a\right)\right)}\right) \cdot \pi\right) \]

   6. metadata-eval 49.5%

      \[\leadsto 0.011111111111111112 \cdot \left(\left(angle \cdot \left(b \cdot b + \color{blue}{-1} \cdot \left(a \cdot a\right)\right)\right) \cdot \pi\right) \]

   7. unpow2 49.5%

      \[\leadsto 0.011111111111111112 \cdot \left(\left(angle \cdot \left(b \cdot b + -1 \cdot \color{blue}{{a}^{2}}\right)\right) \cdot \pi\right) \]

   8. +-commutative 49.5%

      \[\leadsto 0.011111111111111112 \cdot \left(\left(angle \cdot \color{blue}{\left(-1 \cdot {a}^{2} + b \cdot b\right)}\right) \cdot \pi\right) \]

   9. unpow2 49.5%

      \[\leadsto 0.011111111111111112 \cdot \left(\left(angle \cdot \left(-1 \cdot {a}^{2} + \color{blue}{{b}^{2}}\right)\right) \cdot \pi\right) \]

   10. fma-def 49.5%

       \[\leadsto 0.011111111111111112 \cdot \left(\left(angle \cdot \color{blue}{\mathsf{fma}\left(-1, {a}^{2}, {b}^{2}\right)}\right) \cdot \pi\right) \]

   11. unpow2 49.5%

       \[\leadsto 0.011111111111111112 \cdot \left(\left(angle \cdot \mathsf{fma}\left(-1, \color{blue}{a \cdot a}, {b}^{2}\right)\right) \cdot \pi\right) \]

   12. unpow2 49.5%

       \[\leadsto 0.011111111111111112 \cdot \left(\left(angle \cdot \mathsf{fma}\left(-1, a \cdot a, \color{blue}{b \cdot b}\right)\right) \cdot \pi\right) \]

   13. fma-def 49.5%

       \[\leadsto 0.011111111111111112 \cdot \left(\left(angle \cdot \color{blue}{\left(-1 \cdot \left(a \cdot a\right) + b \cdot b\right)}\right) \cdot \pi\right) \]

   14. mul-1-neg 49.5%

       \[\leadsto 0.011111111111111112 \cdot \left(\left(angle \cdot \left(\color{blue}{\left(-a \cdot a\right)} + b \cdot b\right)\right) \cdot \pi\right) \]

   15. +-commutative 49.5%

       \[\leadsto 0.011111111111111112 \cdot \left(\left(angle \cdot \color{blue}{\left(b \cdot b + \left(-a \cdot a\right)\right)}\right) \cdot \pi\right) \]

   16. sub-neg 49.5%

       \[\leadsto 0.011111111111111112 \cdot \left(\left(angle \cdot \color{blue}{\left(b \cdot b - a \cdot a\right)}\right) \cdot \pi\right) \]

6. Simplified 49.5%

   \[\leadsto \color{blue}{0.011111111111111112 \cdot \left(\left(angle \cdot \left(b \cdot b - a \cdot a\right)\right) \cdot \pi\right)} \]

7. Taylor expanded in b around 0 39.3%

   \[\leadsto \color{blue}{-0.011111111111111112 \cdot \left(angle \cdot \left({a}^{2} \cdot \pi\right)\right)} \]
8. Step-by-step derivation

   1. unpow2 39.3%

      \[\leadsto -0.011111111111111112 \cdot \left(angle \cdot \left(\color{blue}{\left(a \cdot a\right)} \cdot \pi\right)\right) \]

   2. *-commutative 39.3%

      \[\leadsto -0.011111111111111112 \cdot \left(angle \cdot \color{blue}{\left(\pi \cdot \left(a \cdot a\right)\right)}\right) \]

9. Simplified 39.3%

   \[\leadsto \color{blue}{-0.011111111111111112 \cdot \left(angle \cdot \left(\pi \cdot \left(a \cdot a\right)\right)\right)} \]

10. Taylor expanded in angle around 0 39.3%

    \[\leadsto -0.011111111111111112 \cdot \color{blue}{\left(angle \cdot \left({a}^{2} \cdot \pi\right)\right)} \]
11. Step-by-step derivation

    1. associate-*r* 39.3%

       \[\leadsto -0.011111111111111112 \cdot \color{blue}{\left(\left(angle \cdot {a}^{2}\right) \cdot \pi\right)} \]

    2. *-commutative 39.3%

       \[\leadsto -0.011111111111111112 \cdot \color{blue}{\left(\pi \cdot \left(angle \cdot {a}^{2}\right)\right)} \]

    3. unpow2 39.3%

       \[\leadsto -0.011111111111111112 \cdot \left(\pi \cdot \left(angle \cdot \color{blue}{\left(a \cdot a\right)}\right)\right) \]

    4. associate-*r* 42.8%

       \[\leadsto -0.011111111111111112 \cdot \left(\pi \cdot \color{blue}{\left(\left(angle \cdot a\right) \cdot a\right)}\right) \]

12. Simplified 42.8%

    \[\leadsto -0.011111111111111112 \cdot \color{blue}{\left(\pi \cdot \left(\left(angle \cdot a\right) \cdot a\right)\right)} \]

13. Final simplification 42.8%

    \[\leadsto -0.011111111111111112 \cdot \left(\pi \cdot \left(a \cdot \left(angle \cdot a\right)\right)\right) \]

Reproduce

herbie shell --seed 2023174 
(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)))))