ab-angle->ABCF B

Percentage Accurate: 53.6% → 66.4%

Time: 39.8s

Alternatives: 22

Speedup: 5.5×

• 32.7% 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: 53.6% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 66.4% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := 2 \cdot \left(b - a\right)\\ t_1 := \pi \cdot \left(0.005555555555555556 \cdot angle\right)\\ t_2 := \sin t_1\\ \mathbf{if}\;b \leq -5 \cdot 10^{+218}:\\ \;\;\;\;t_0 \cdot \left(\left(t_2 \cdot \cos \left({\left(\sqrt[3]{t_1}\right)}^{3}\right)\right) \cdot \left(b + a\right)\right)\\ \mathbf{else}:\\ \;\;\;\;t_0 \cdot \left(\left(b + a\right) \cdot \left(t_2 \cdot \left(\sqrt{\sqrt[3]{{\cos t_1}^{4}}} \cdot \left|\sqrt[3]{\cos \left(0.005555555555555556 \cdot \left(angle \cdot \sqrt[3]{{\pi}^{3}}\right)\right)}\right|\right)\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (a b angle)
 :precision binary64
 (let* ((t_0 (* 2.0 (- b a)))
        (t_1 (* PI (* 0.005555555555555556 angle)))
        (t_2 (sin t_1)))
   (if (<= b -5e+218)
     (* t_0 (* (* t_2 (cos (pow (cbrt t_1) 3.0))) (+ b a)))
     (*
      t_0
      (*
       (+ b a)
       (*
        t_2
        (*
         (sqrt (cbrt (pow (cos t_1) 4.0)))
         (fabs
          (cbrt
           (cos
            (* 0.005555555555555556 (* angle (cbrt (pow PI 3.0))))))))))))))

C

double code(double a, double b, double angle) {
	double t_0 = 2.0 * (b - a);
	double t_1 = ((double) M_PI) * (0.005555555555555556 * angle);
	double t_2 = sin(t_1);
	double tmp;
	if (b <= -5e+218) {
		tmp = t_0 * ((t_2 * cos(pow(cbrt(t_1), 3.0))) * (b + a));
	} else {
		tmp = t_0 * ((b + a) * (t_2 * (sqrt(cbrt(pow(cos(t_1), 4.0))) * fabs(cbrt(cos((0.005555555555555556 * (angle * cbrt(pow(((double) M_PI), 3.0))))))))));
	}
	return tmp;
}

Java

public static double code(double a, double b, double angle) {
	double t_0 = 2.0 * (b - a);
	double t_1 = Math.PI * (0.005555555555555556 * angle);
	double t_2 = Math.sin(t_1);
	double tmp;
	if (b <= -5e+218) {
		tmp = t_0 * ((t_2 * Math.cos(Math.pow(Math.cbrt(t_1), 3.0))) * (b + a));
	} else {
		tmp = t_0 * ((b + a) * (t_2 * (Math.sqrt(Math.cbrt(Math.pow(Math.cos(t_1), 4.0))) * Math.abs(Math.cbrt(Math.cos((0.005555555555555556 * (angle * Math.cbrt(Math.pow(Math.PI, 3.0))))))))));
	}
	return tmp;
}

Julia

function code(a, b, angle)
	t_0 = Float64(2.0 * Float64(b - a))
	t_1 = Float64(pi * Float64(0.005555555555555556 * angle))
	t_2 = sin(t_1)
	tmp = 0.0
	if (b <= -5e+218)
		tmp = Float64(t_0 * Float64(Float64(t_2 * cos((cbrt(t_1) ^ 3.0))) * Float64(b + a)));
	else
		tmp = Float64(t_0 * Float64(Float64(b + a) * Float64(t_2 * Float64(sqrt(cbrt((cos(t_1) ^ 4.0))) * abs(cbrt(cos(Float64(0.005555555555555556 * Float64(angle * cbrt((pi ^ 3.0)))))))))));
	end
	return tmp
end

Wolfram

code[a_, b_, angle_] := Block[{t$95$0 = N[(2.0 * N[(b - a), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(Pi * N[(0.005555555555555556 * angle), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[Sin[t$95$1], $MachinePrecision]}, If[LessEqual[b, -5e+218], N[(t$95$0 * N[(N[(t$95$2 * N[Cos[N[Power[N[Power[t$95$1, 1/3], $MachinePrecision], 3.0], $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[(b + a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(t$95$0 * N[(N[(b + a), $MachinePrecision] * N[(t$95$2 * N[(N[Sqrt[N[Power[N[Power[N[Cos[t$95$1], $MachinePrecision], 4.0], $MachinePrecision], 1/3], $MachinePrecision]], $MachinePrecision] * N[Abs[N[Power[N[Cos[N[(0.005555555555555556 * N[(angle * N[Power[N[Power[Pi, 3.0], $MachinePrecision], 1/3], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], 1/3], $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 2 regimes

2. if b < -4.99999999999999983e218

   1. Initial program 44.8%

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

      1. associate-*l* 44.8%

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

      2. unpow2 44.8%

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

      3. unpow2 44.8%

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

      4. difference-of-squares 57.3%

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

   3. Simplified 57.3%

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

   4. Taylor expanded in angle around inf 68.7%

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

      1. associate-*r* 68.7%

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

      2. associate-*r* 68.7%

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

      3. *-commutative 68.7%

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

      4. associate-*r* 62.4%

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

      5. *-commutative 62.4%

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

      6. *-commutative 62.4%

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

      7. *-commutative 62.4%

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

      8. associate-*r* 74.9%

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

      9. *-commutative 74.9%

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

      10. *-commutative 74.9%

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

      11. *-commutative 74.9%

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

   6. Simplified 74.9%

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

      1. *-commutative 74.9%

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

      2. add-cube-cbrt 93.7%

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

      3. pow3 99.9%

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

      4. *-commutative 99.9%

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

   8. Applied egg-rr 99.9%

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

   if -4.99999999999999983e218 < b

   1. Initial program 52.9%

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

      1. associate-*l* 52.9%

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

      2. unpow2 52.9%

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

      3. unpow2 52.9%

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

      4. difference-of-squares 57.2%

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

   3. Simplified 57.2%

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

   4. Taylor expanded in angle around inf 65.8%

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

      1. associate-*r* 65.8%

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

      2. associate-*r* 65.7%

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

      3. *-commutative 65.7%

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

      4. associate-*r* 67.2%

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

      5. *-commutative 67.2%

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

      6. *-commutative 67.2%

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

      7. *-commutative 67.2%

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

      8. associate-*r* 67.0%

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

      9. *-commutative 67.0%

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

      10. *-commutative 67.0%

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

      11. *-commutative 67.0%

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

   6. Simplified 67.0%

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

      1. *-commutative 67.0%

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

      2. pow1 67.0%

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

      3. metadata-eval 67.0%

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

      4. sqrt-pow1 71.6%

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

      5. pow1/2 71.6%

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

      6. add-cube-cbrt 71.6%

         \[\leadsto \left(2 \cdot \left(b - a\right)\right) \cdot \left(\left(\sin \left(\pi \cdot \left(0.005555555555555556 \cdot angle\right)\right) \cdot {\color{blue}{\left(\left(\sqrt[3]{{\cos \left(\pi \cdot \left(angle \cdot 0.005555555555555556\right)\right)}^{2}} \cdot \sqrt[3]{{\cos \left(\pi \cdot \left(angle \cdot 0.005555555555555556\right)\right)}^{2}}\right) \cdot \sqrt[3]{{\cos \left(\pi \cdot \left(angle \cdot 0.005555555555555556\right)\right)}^{2}}\right)}}^{0.5}\right) \cdot \left(a + b\right)\right) \]

      7. unpow-prod-down 71.6%

         \[\leadsto \left(2 \cdot \left(b - a\right)\right) \cdot \left(\left(\sin \left(\pi \cdot \left(0.005555555555555556 \cdot angle\right)\right) \cdot \color{blue}{\left({\left(\sqrt[3]{{\cos \left(\pi \cdot \left(angle \cdot 0.005555555555555556\right)\right)}^{2}} \cdot \sqrt[3]{{\cos \left(\pi \cdot \left(angle \cdot 0.005555555555555556\right)\right)}^{2}}\right)}^{0.5} \cdot {\left(\sqrt[3]{{\cos \left(\pi \cdot \left(angle \cdot 0.005555555555555556\right)\right)}^{2}}\right)}^{0.5}\right)}\right) \cdot \left(a + b\right)\right) \]

   8. Applied egg-rr 71.6%

      \[\leadsto \left(2 \cdot \left(b - a\right)\right) \cdot \left(\left(\sin \left(\pi \cdot \left(0.005555555555555556 \cdot angle\right)\right) \cdot \color{blue}{\left({\left(\sqrt[3]{{\cos \left(\pi \cdot \left(0.005555555555555556 \cdot angle\right)\right)}^{4}}\right)}^{0.5} \cdot {\left({\left(\sqrt[3]{\cos \left(\pi \cdot \left(0.005555555555555556 \cdot angle\right)\right)}\right)}^{2}\right)}^{0.5}\right)}\right) \cdot \left(a + b\right)\right) \]
   9. Step-by-step derivation

      1. unpow1/2 71.6%

         \[\leadsto \left(2 \cdot \left(b - a\right)\right) \cdot \left(\left(\sin \left(\pi \cdot \left(0.005555555555555556 \cdot angle\right)\right) \cdot \left(\color{blue}{\sqrt{\sqrt[3]{{\cos \left(\pi \cdot \left(0.005555555555555556 \cdot angle\right)\right)}^{4}}}} \cdot {\left({\left(\sqrt[3]{\cos \left(\pi \cdot \left(0.005555555555555556 \cdot angle\right)\right)}\right)}^{2}\right)}^{0.5}\right)\right) \cdot \left(a + b\right)\right) \]

      2. *-commutative 71.6%

         \[\leadsto \left(2 \cdot \left(b - a\right)\right) \cdot \left(\left(\sin \left(\pi \cdot \left(0.005555555555555556 \cdot angle\right)\right) \cdot \left(\sqrt{\sqrt[3]{{\cos \color{blue}{\left(\left(0.005555555555555556 \cdot angle\right) \cdot \pi\right)}}^{4}}} \cdot {\left({\left(\sqrt[3]{\cos \left(\pi \cdot \left(0.005555555555555556 \cdot angle\right)\right)}\right)}^{2}\right)}^{0.5}\right)\right) \cdot \left(a + b\right)\right) \]

      3. associate-*r* 71.3%

         \[\leadsto \left(2 \cdot \left(b - a\right)\right) \cdot \left(\left(\sin \left(\pi \cdot \left(0.005555555555555556 \cdot angle\right)\right) \cdot \left(\sqrt{\sqrt[3]{{\cos \color{blue}{\left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}}^{4}}} \cdot {\left({\left(\sqrt[3]{\cos \left(\pi \cdot \left(0.005555555555555556 \cdot angle\right)\right)}\right)}^{2}\right)}^{0.5}\right)\right) \cdot \left(a + b\right)\right) \]

      4. unpow1/2 71.3%

         \[\leadsto \left(2 \cdot \left(b - a\right)\right) \cdot \left(\left(\sin \left(\pi \cdot \left(0.005555555555555556 \cdot angle\right)\right) \cdot \left(\sqrt{\sqrt[3]{{\cos \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{4}}} \cdot \color{blue}{\sqrt{{\left(\sqrt[3]{\cos \left(\pi \cdot \left(0.005555555555555556 \cdot angle\right)\right)}\right)}^{2}}}\right)\right) \cdot \left(a + b\right)\right) \]

      5. unpow2 71.3%

         \[\leadsto \left(2 \cdot \left(b - a\right)\right) \cdot \left(\left(\sin \left(\pi \cdot \left(0.005555555555555556 \cdot angle\right)\right) \cdot \left(\sqrt{\sqrt[3]{{\cos \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{4}}} \cdot \sqrt{\color{blue}{\sqrt[3]{\cos \left(\pi \cdot \left(0.005555555555555556 \cdot angle\right)\right)} \cdot \sqrt[3]{\cos \left(\pi \cdot \left(0.005555555555555556 \cdot angle\right)\right)}}}\right)\right) \cdot \left(a + b\right)\right) \]

      6. rem-sqrt-square 71.3%

         \[\leadsto \left(2 \cdot \left(b - a\right)\right) \cdot \left(\left(\sin \left(\pi \cdot \left(0.005555555555555556 \cdot angle\right)\right) \cdot \left(\sqrt{\sqrt[3]{{\cos \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{4}}} \cdot \color{blue}{\left|\sqrt[3]{\cos \left(\pi \cdot \left(0.005555555555555556 \cdot angle\right)\right)}\right|}\right)\right) \cdot \left(a + b\right)\right) \]

      7. *-commutative 71.3%

         \[\leadsto \left(2 \cdot \left(b - a\right)\right) \cdot \left(\left(\sin \left(\pi \cdot \left(0.005555555555555556 \cdot angle\right)\right) \cdot \left(\sqrt{\sqrt[3]{{\cos \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{4}}} \cdot \left|\sqrt[3]{\cos \color{blue}{\left(\left(0.005555555555555556 \cdot angle\right) \cdot \pi\right)}}\right|\right)\right) \cdot \left(a + b\right)\right) \]

      8. associate-*r* 71.2%

         \[\leadsto \left(2 \cdot \left(b - a\right)\right) \cdot \left(\left(\sin \left(\pi \cdot \left(0.005555555555555556 \cdot angle\right)\right) \cdot \left(\sqrt{\sqrt[3]{{\cos \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{4}}} \cdot \left|\sqrt[3]{\cos \color{blue}{\left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}}\right|\right)\right) \cdot \left(a + b\right)\right) \]

   10. Simplified 71.2%

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

       1. expm1-log1p-u 71.2%

          \[\leadsto \left(2 \cdot \left(b - a\right)\right) \cdot \left(\left(\sin \left(\pi \cdot \left(0.005555555555555556 \cdot angle\right)\right) \cdot \left(\sqrt{\sqrt[3]{{\color{blue}{\left(\mathsf{expm1}\left(\mathsf{log1p}\left(\cos \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)\right)\right)\right)}}^{4}}} \cdot \left|\sqrt[3]{\cos \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}\right|\right)\right) \cdot \left(a + b\right)\right) \]

       2. expm1-udef 71.2%

          \[\leadsto \left(2 \cdot \left(b - a\right)\right) \cdot \left(\left(\sin \left(\pi \cdot \left(0.005555555555555556 \cdot angle\right)\right) \cdot \left(\sqrt{\sqrt[3]{{\color{blue}{\left(e^{\mathsf{log1p}\left(\cos \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)\right)} - 1\right)}}^{4}}} \cdot \left|\sqrt[3]{\cos \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}\right|\right)\right) \cdot \left(a + b\right)\right) \]

       3. associate-*r* 71.3%

          \[\leadsto \left(2 \cdot \left(b - a\right)\right) \cdot \left(\left(\sin \left(\pi \cdot \left(0.005555555555555556 \cdot angle\right)\right) \cdot \left(\sqrt{\sqrt[3]{{\left(e^{\mathsf{log1p}\left(\cos \color{blue}{\left(\left(0.005555555555555556 \cdot angle\right) \cdot \pi\right)}\right)} - 1\right)}^{4}}} \cdot \left|\sqrt[3]{\cos \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}\right|\right)\right) \cdot \left(a + b\right)\right) \]

       4. *-commutative 71.3%

          \[\leadsto \left(2 \cdot \left(b - a\right)\right) \cdot \left(\left(\sin \left(\pi \cdot \left(0.005555555555555556 \cdot angle\right)\right) \cdot \left(\sqrt{\sqrt[3]{{\left(e^{\mathsf{log1p}\left(\cos \color{blue}{\left(\pi \cdot \left(0.005555555555555556 \cdot angle\right)\right)}\right)} - 1\right)}^{4}}} \cdot \left|\sqrt[3]{\cos \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}\right|\right)\right) \cdot \left(a + b\right)\right) \]

   12. Applied egg-rr 71.3%

       \[\leadsto \left(2 \cdot \left(b - a\right)\right) \cdot \left(\left(\sin \left(\pi \cdot \left(0.005555555555555556 \cdot angle\right)\right) \cdot \left(\sqrt{\sqrt[3]{{\color{blue}{\left(e^{\mathsf{log1p}\left(\cos \left(\pi \cdot \left(0.005555555555555556 \cdot angle\right)\right)\right)} - 1\right)}}^{4}}} \cdot \left|\sqrt[3]{\cos \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}\right|\right)\right) \cdot \left(a + b\right)\right) \]
   13. Step-by-step derivation

       1. expm1-def 71.3%

          \[\leadsto \left(2 \cdot \left(b - a\right)\right) \cdot \left(\left(\sin \left(\pi \cdot \left(0.005555555555555556 \cdot angle\right)\right) \cdot \left(\sqrt{\sqrt[3]{{\color{blue}{\left(\mathsf{expm1}\left(\mathsf{log1p}\left(\cos \left(\pi \cdot \left(0.005555555555555556 \cdot angle\right)\right)\right)\right)\right)}}^{4}}} \cdot \left|\sqrt[3]{\cos \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}\right|\right)\right) \cdot \left(a + b\right)\right) \]

       2. expm1-log1p 71.3%

          \[\leadsto \left(2 \cdot \left(b - a\right)\right) \cdot \left(\left(\sin \left(\pi \cdot \left(0.005555555555555556 \cdot angle\right)\right) \cdot \left(\sqrt{\sqrt[3]{{\color{blue}{\cos \left(\pi \cdot \left(0.005555555555555556 \cdot angle\right)\right)}}^{4}}} \cdot \left|\sqrt[3]{\cos \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}\right|\right)\right) \cdot \left(a + b\right)\right) \]

       3. *-commutative 71.3%

          \[\leadsto \left(2 \cdot \left(b - a\right)\right) \cdot \left(\left(\sin \left(\pi \cdot \left(0.005555555555555556 \cdot angle\right)\right) \cdot \left(\sqrt{\sqrt[3]{{\cos \left(\pi \cdot \color{blue}{\left(angle \cdot 0.005555555555555556\right)}\right)}^{4}}} \cdot \left|\sqrt[3]{\cos \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}\right|\right)\right) \cdot \left(a + b\right)\right) \]

   14. Simplified 71.3%

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

       1. add-cbrt-cube 71.3%

          \[\leadsto \left(2 \cdot \left(b - a\right)\right) \cdot \left(\left(\sin \left(\pi \cdot \left(0.005555555555555556 \cdot angle\right)\right) \cdot \left(\sqrt{\sqrt[3]{{\cos \left(\pi \cdot \left(angle \cdot 0.005555555555555556\right)\right)}^{4}}} \cdot \left|\sqrt[3]{\cos \left(0.005555555555555556 \cdot \left(angle \cdot \color{blue}{\sqrt[3]{\left(\pi \cdot \pi\right) \cdot \pi}}\right)\right)}\right|\right)\right) \cdot \left(a + b\right)\right) \]

       2. pow3 71.3%

          \[\leadsto \left(2 \cdot \left(b - a\right)\right) \cdot \left(\left(\sin \left(\pi \cdot \left(0.005555555555555556 \cdot angle\right)\right) \cdot \left(\sqrt{\sqrt[3]{{\cos \left(\pi \cdot \left(angle \cdot 0.005555555555555556\right)\right)}^{4}}} \cdot \left|\sqrt[3]{\cos \left(0.005555555555555556 \cdot \left(angle \cdot \sqrt[3]{\color{blue}{{\pi}^{3}}}\right)\right)}\right|\right)\right) \cdot \left(a + b\right)\right) \]

   16. Applied egg-rr 71.3%

       \[\leadsto \left(2 \cdot \left(b - a\right)\right) \cdot \left(\left(\sin \left(\pi \cdot \left(0.005555555555555556 \cdot angle\right)\right) \cdot \left(\sqrt{\sqrt[3]{{\cos \left(\pi \cdot \left(angle \cdot 0.005555555555555556\right)\right)}^{4}}} \cdot \left|\sqrt[3]{\cos \left(0.005555555555555556 \cdot \left(angle \cdot \color{blue}{\sqrt[3]{{\pi}^{3}}}\right)\right)}\right|\right)\right) \cdot \left(a + b\right)\right) \]
3. Recombined 2 regimes into one program.

4. Final simplification 73.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -5 \cdot 10^{+218}:\\ \;\;\;\;\left(2 \cdot \left(b - a\right)\right) \cdot \left(\left(\sin \left(\pi \cdot \left(0.005555555555555556 \cdot angle\right)\right) \cdot \cos \left({\left(\sqrt[3]{\pi \cdot \left(0.005555555555555556 \cdot angle\right)}\right)}^{3}\right)\right) \cdot \left(b + a\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(2 \cdot \left(b - a\right)\right) \cdot \left(\left(b + a\right) \cdot \left(\sin \left(\pi \cdot \left(0.005555555555555556 \cdot angle\right)\right) \cdot \left(\sqrt{\sqrt[3]{{\cos \left(\pi \cdot \left(0.005555555555555556 \cdot angle\right)\right)}^{4}}} \cdot \left|\sqrt[3]{\cos \left(0.005555555555555556 \cdot \left(angle \cdot \sqrt[3]{{\pi}^{3}}\right)\right)}\right|\right)\right)\right)\\ \end{array} \]

Alternative 2: 66.4% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := 2 \cdot \left(b - a\right)\\ t_1 := \pi \cdot \left(0.005555555555555556 \cdot angle\right)\\ t_2 := \sin t_1\\ t_3 := \cos \left(0.005555555555555556 \cdot \left(\pi \cdot angle\right)\right)\\ \mathbf{if}\;b \leq -2 \cdot 10^{+214}:\\ \;\;\;\;t_0 \cdot \left(\left(t_2 \cdot \cos \left({\left(\sqrt[3]{t_1}\right)}^{3}\right)\right) \cdot \left(b + a\right)\right)\\ \mathbf{else}:\\ \;\;\;\;t_0 \cdot \left(\left(b + a\right) \cdot \left(t_2 \cdot \left(\left|\sqrt[3]{t_3}\right| \cdot \sqrt{\sqrt[3]{{t_3}^{4}}}\right)\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (a b angle)
 :precision binary64
 (let* ((t_0 (* 2.0 (- b a)))
        (t_1 (* PI (* 0.005555555555555556 angle)))
        (t_2 (sin t_1))
        (t_3 (cos (* 0.005555555555555556 (* PI angle)))))
   (if (<= b -2e+214)
     (* t_0 (* (* t_2 (cos (pow (cbrt t_1) 3.0))) (+ b a)))
     (*
      t_0
      (* (+ b a) (* t_2 (* (fabs (cbrt t_3)) (sqrt (cbrt (pow t_3 4.0))))))))))

C

double code(double a, double b, double angle) {
	double t_0 = 2.0 * (b - a);
	double t_1 = ((double) M_PI) * (0.005555555555555556 * angle);
	double t_2 = sin(t_1);
	double t_3 = cos((0.005555555555555556 * (((double) M_PI) * angle)));
	double tmp;
	if (b <= -2e+214) {
		tmp = t_0 * ((t_2 * cos(pow(cbrt(t_1), 3.0))) * (b + a));
	} else {
		tmp = t_0 * ((b + a) * (t_2 * (fabs(cbrt(t_3)) * sqrt(cbrt(pow(t_3, 4.0))))));
	}
	return tmp;
}

Java

public static double code(double a, double b, double angle) {
	double t_0 = 2.0 * (b - a);
	double t_1 = Math.PI * (0.005555555555555556 * angle);
	double t_2 = Math.sin(t_1);
	double t_3 = Math.cos((0.005555555555555556 * (Math.PI * angle)));
	double tmp;
	if (b <= -2e+214) {
		tmp = t_0 * ((t_2 * Math.cos(Math.pow(Math.cbrt(t_1), 3.0))) * (b + a));
	} else {
		tmp = t_0 * ((b + a) * (t_2 * (Math.abs(Math.cbrt(t_3)) * Math.sqrt(Math.cbrt(Math.pow(t_3, 4.0))))));
	}
	return tmp;
}

Julia

function code(a, b, angle)
	t_0 = Float64(2.0 * Float64(b - a))
	t_1 = Float64(pi * Float64(0.005555555555555556 * angle))
	t_2 = sin(t_1)
	t_3 = cos(Float64(0.005555555555555556 * Float64(pi * angle)))
	tmp = 0.0
	if (b <= -2e+214)
		tmp = Float64(t_0 * Float64(Float64(t_2 * cos((cbrt(t_1) ^ 3.0))) * Float64(b + a)));
	else
		tmp = Float64(t_0 * Float64(Float64(b + a) * Float64(t_2 * Float64(abs(cbrt(t_3)) * sqrt(cbrt((t_3 ^ 4.0)))))));
	end
	return tmp
end

Wolfram

code[a_, b_, angle_] := Block[{t$95$0 = N[(2.0 * N[(b - a), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(Pi * N[(0.005555555555555556 * angle), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[Sin[t$95$1], $MachinePrecision]}, Block[{t$95$3 = N[Cos[N[(0.005555555555555556 * N[(Pi * angle), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[b, -2e+214], N[(t$95$0 * N[(N[(t$95$2 * N[Cos[N[Power[N[Power[t$95$1, 1/3], $MachinePrecision], 3.0], $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[(b + a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(t$95$0 * N[(N[(b + a), $MachinePrecision] * N[(t$95$2 * N[(N[Abs[N[Power[t$95$3, 1/3], $MachinePrecision]], $MachinePrecision] * N[Sqrt[N[Power[N[Power[t$95$3, 4.0], $MachinePrecision], 1/3], $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]]

Derivation

1. Split input into 2 regimes

2. if b < -1.9999999999999999e214

   1. Initial program 44.8%

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

      1. associate-*l* 44.8%

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

      2. unpow2 44.8%

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

      3. unpow2 44.8%

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

      4. difference-of-squares 57.3%

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

   3. Simplified 57.3%

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

   4. Taylor expanded in angle around inf 68.7%

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

      1. associate-*r* 68.7%

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

      2. associate-*r* 68.7%

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

      3. *-commutative 68.7%

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

      4. associate-*r* 62.4%

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

      5. *-commutative 62.4%

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

      6. *-commutative 62.4%

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

      7. *-commutative 62.4%

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

      8. associate-*r* 74.9%

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

      9. *-commutative 74.9%

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

      10. *-commutative 74.9%

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

      11. *-commutative 74.9%

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

   6. Simplified 74.9%

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

      1. *-commutative 74.9%

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

      2. add-cube-cbrt 93.7%

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

      3. pow3 99.9%

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

      4. *-commutative 99.9%

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

   8. Applied egg-rr 99.9%

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

   if -1.9999999999999999e214 < b

   1. Initial program 52.9%

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

      1. associate-*l* 52.9%

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

      2. unpow2 52.9%

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

      3. unpow2 52.9%

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

      4. difference-of-squares 57.2%

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

   3. Simplified 57.2%

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

   4. Taylor expanded in angle around inf 65.8%

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

      1. associate-*r* 65.8%

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

      2. associate-*r* 65.7%

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

      3. *-commutative 65.7%

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

      4. associate-*r* 67.2%

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

      5. *-commutative 67.2%

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

      6. *-commutative 67.2%

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

      7. *-commutative 67.2%

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

      8. associate-*r* 67.0%

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

      9. *-commutative 67.0%

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

      10. *-commutative 67.0%

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

      11. *-commutative 67.0%

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

   6. Simplified 67.0%

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

      1. *-commutative 67.0%

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

      2. pow1 67.0%

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

      3. metadata-eval 67.0%

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

      4. sqrt-pow1 71.6%

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

      5. pow1/2 71.6%

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

      6. add-cube-cbrt 71.6%

         \[\leadsto \left(2 \cdot \left(b - a\right)\right) \cdot \left(\left(\sin \left(\pi \cdot \left(0.005555555555555556 \cdot angle\right)\right) \cdot {\color{blue}{\left(\left(\sqrt[3]{{\cos \left(\pi \cdot \left(angle \cdot 0.005555555555555556\right)\right)}^{2}} \cdot \sqrt[3]{{\cos \left(\pi \cdot \left(angle \cdot 0.005555555555555556\right)\right)}^{2}}\right) \cdot \sqrt[3]{{\cos \left(\pi \cdot \left(angle \cdot 0.005555555555555556\right)\right)}^{2}}\right)}}^{0.5}\right) \cdot \left(a + b\right)\right) \]

      7. unpow-prod-down 71.6%

         \[\leadsto \left(2 \cdot \left(b - a\right)\right) \cdot \left(\left(\sin \left(\pi \cdot \left(0.005555555555555556 \cdot angle\right)\right) \cdot \color{blue}{\left({\left(\sqrt[3]{{\cos \left(\pi \cdot \left(angle \cdot 0.005555555555555556\right)\right)}^{2}} \cdot \sqrt[3]{{\cos \left(\pi \cdot \left(angle \cdot 0.005555555555555556\right)\right)}^{2}}\right)}^{0.5} \cdot {\left(\sqrt[3]{{\cos \left(\pi \cdot \left(angle \cdot 0.005555555555555556\right)\right)}^{2}}\right)}^{0.5}\right)}\right) \cdot \left(a + b\right)\right) \]

   8. Applied egg-rr 71.6%

      \[\leadsto \left(2 \cdot \left(b - a\right)\right) \cdot \left(\left(\sin \left(\pi \cdot \left(0.005555555555555556 \cdot angle\right)\right) \cdot \color{blue}{\left({\left(\sqrt[3]{{\cos \left(\pi \cdot \left(0.005555555555555556 \cdot angle\right)\right)}^{4}}\right)}^{0.5} \cdot {\left({\left(\sqrt[3]{\cos \left(\pi \cdot \left(0.005555555555555556 \cdot angle\right)\right)}\right)}^{2}\right)}^{0.5}\right)}\right) \cdot \left(a + b\right)\right) \]
   9. Step-by-step derivation

      1. unpow1/2 71.6%

         \[\leadsto \left(2 \cdot \left(b - a\right)\right) \cdot \left(\left(\sin \left(\pi \cdot \left(0.005555555555555556 \cdot angle\right)\right) \cdot \left(\color{blue}{\sqrt{\sqrt[3]{{\cos \left(\pi \cdot \left(0.005555555555555556 \cdot angle\right)\right)}^{4}}}} \cdot {\left({\left(\sqrt[3]{\cos \left(\pi \cdot \left(0.005555555555555556 \cdot angle\right)\right)}\right)}^{2}\right)}^{0.5}\right)\right) \cdot \left(a + b\right)\right) \]

      2. *-commutative 71.6%

         \[\leadsto \left(2 \cdot \left(b - a\right)\right) \cdot \left(\left(\sin \left(\pi \cdot \left(0.005555555555555556 \cdot angle\right)\right) \cdot \left(\sqrt{\sqrt[3]{{\cos \color{blue}{\left(\left(0.005555555555555556 \cdot angle\right) \cdot \pi\right)}}^{4}}} \cdot {\left({\left(\sqrt[3]{\cos \left(\pi \cdot \left(0.005555555555555556 \cdot angle\right)\right)}\right)}^{2}\right)}^{0.5}\right)\right) \cdot \left(a + b\right)\right) \]

      3. associate-*r* 71.3%

         \[\leadsto \left(2 \cdot \left(b - a\right)\right) \cdot \left(\left(\sin \left(\pi \cdot \left(0.005555555555555556 \cdot angle\right)\right) \cdot \left(\sqrt{\sqrt[3]{{\cos \color{blue}{\left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}}^{4}}} \cdot {\left({\left(\sqrt[3]{\cos \left(\pi \cdot \left(0.005555555555555556 \cdot angle\right)\right)}\right)}^{2}\right)}^{0.5}\right)\right) \cdot \left(a + b\right)\right) \]

      4. unpow1/2 71.3%

         \[\leadsto \left(2 \cdot \left(b - a\right)\right) \cdot \left(\left(\sin \left(\pi \cdot \left(0.005555555555555556 \cdot angle\right)\right) \cdot \left(\sqrt{\sqrt[3]{{\cos \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{4}}} \cdot \color{blue}{\sqrt{{\left(\sqrt[3]{\cos \left(\pi \cdot \left(0.005555555555555556 \cdot angle\right)\right)}\right)}^{2}}}\right)\right) \cdot \left(a + b\right)\right) \]

      5. unpow2 71.3%

         \[\leadsto \left(2 \cdot \left(b - a\right)\right) \cdot \left(\left(\sin \left(\pi \cdot \left(0.005555555555555556 \cdot angle\right)\right) \cdot \left(\sqrt{\sqrt[3]{{\cos \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{4}}} \cdot \sqrt{\color{blue}{\sqrt[3]{\cos \left(\pi \cdot \left(0.005555555555555556 \cdot angle\right)\right)} \cdot \sqrt[3]{\cos \left(\pi \cdot \left(0.005555555555555556 \cdot angle\right)\right)}}}\right)\right) \cdot \left(a + b\right)\right) \]

      6. rem-sqrt-square 71.3%

         \[\leadsto \left(2 \cdot \left(b - a\right)\right) \cdot \left(\left(\sin \left(\pi \cdot \left(0.005555555555555556 \cdot angle\right)\right) \cdot \left(\sqrt{\sqrt[3]{{\cos \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{4}}} \cdot \color{blue}{\left|\sqrt[3]{\cos \left(\pi \cdot \left(0.005555555555555556 \cdot angle\right)\right)}\right|}\right)\right) \cdot \left(a + b\right)\right) \]

      7. *-commutative 71.3%

         \[\leadsto \left(2 \cdot \left(b - a\right)\right) \cdot \left(\left(\sin \left(\pi \cdot \left(0.005555555555555556 \cdot angle\right)\right) \cdot \left(\sqrt{\sqrt[3]{{\cos \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{4}}} \cdot \left|\sqrt[3]{\cos \color{blue}{\left(\left(0.005555555555555556 \cdot angle\right) \cdot \pi\right)}}\right|\right)\right) \cdot \left(a + b\right)\right) \]

      8. associate-*r* 71.2%

         \[\leadsto \left(2 \cdot \left(b - a\right)\right) \cdot \left(\left(\sin \left(\pi \cdot \left(0.005555555555555556 \cdot angle\right)\right) \cdot \left(\sqrt{\sqrt[3]{{\cos \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{4}}} \cdot \left|\sqrt[3]{\cos \color{blue}{\left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}}\right|\right)\right) \cdot \left(a + b\right)\right) \]

   10. Simplified 71.2%

       \[\leadsto \left(2 \cdot \left(b - a\right)\right) \cdot \left(\left(\sin \left(\pi \cdot \left(0.005555555555555556 \cdot angle\right)\right) \cdot \color{blue}{\left(\sqrt{\sqrt[3]{{\cos \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{4}}} \cdot \left|\sqrt[3]{\cos \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}\right|\right)}\right) \cdot \left(a + b\right)\right) \]
3. Recombined 2 regimes into one program.

4. Final simplification 73.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -2 \cdot 10^{+214}:\\ \;\;\;\;\left(2 \cdot \left(b - a\right)\right) \cdot \left(\left(\sin \left(\pi \cdot \left(0.005555555555555556 \cdot angle\right)\right) \cdot \cos \left({\left(\sqrt[3]{\pi \cdot \left(0.005555555555555556 \cdot angle\right)}\right)}^{3}\right)\right) \cdot \left(b + a\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(2 \cdot \left(b - a\right)\right) \cdot \left(\left(b + a\right) \cdot \left(\sin \left(\pi \cdot \left(0.005555555555555556 \cdot angle\right)\right) \cdot \left(\left|\sqrt[3]{\cos \left(0.005555555555555556 \cdot \left(\pi \cdot angle\right)\right)}\right| \cdot \sqrt{\sqrt[3]{{\cos \left(0.005555555555555556 \cdot \left(\pi \cdot angle\right)\right)}^{4}}}\right)\right)\right)\\ \end{array} \]

Alternative 3: 66.4% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := 2 \cdot \left(b - a\right)\\ t_1 := \pi \cdot \left(0.005555555555555556 \cdot angle\right)\\ t_2 := \sin t_1\\ \mathbf{if}\;b \leq -7.6 \cdot 10^{+216}:\\ \;\;\;\;t_0 \cdot \left(\left(t_2 \cdot \cos \left({\left(\sqrt[3]{t_1}\right)}^{3}\right)\right) \cdot \left(b + a\right)\right)\\ \mathbf{else}:\\ \;\;\;\;t_0 \cdot \left(\left(b + a\right) \cdot \left(t_2 \cdot \left(\sqrt{\sqrt[3]{{\cos \left(0.005555555555555556 \cdot \left(\pi \cdot angle\right)\right)}^{4}}} \cdot \left|\sqrt[3]{\cos t_1}\right|\right)\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (a b angle)
 :precision binary64
 (let* ((t_0 (* 2.0 (- b a)))
        (t_1 (* PI (* 0.005555555555555556 angle)))
        (t_2 (sin t_1)))
   (if (<= b -7.6e+216)
     (* t_0 (* (* t_2 (cos (pow (cbrt t_1) 3.0))) (+ b a)))
     (*
      t_0
      (*
       (+ b a)
       (*
        t_2
        (*
         (sqrt (cbrt (pow (cos (* 0.005555555555555556 (* PI angle))) 4.0)))
         (fabs (cbrt (cos t_1))))))))))

C

double code(double a, double b, double angle) {
	double t_0 = 2.0 * (b - a);
	double t_1 = ((double) M_PI) * (0.005555555555555556 * angle);
	double t_2 = sin(t_1);
	double tmp;
	if (b <= -7.6e+216) {
		tmp = t_0 * ((t_2 * cos(pow(cbrt(t_1), 3.0))) * (b + a));
	} else {
		tmp = t_0 * ((b + a) * (t_2 * (sqrt(cbrt(pow(cos((0.005555555555555556 * (((double) M_PI) * angle))), 4.0))) * fabs(cbrt(cos(t_1))))));
	}
	return tmp;
}

Java

public static double code(double a, double b, double angle) {
	double t_0 = 2.0 * (b - a);
	double t_1 = Math.PI * (0.005555555555555556 * angle);
	double t_2 = Math.sin(t_1);
	double tmp;
	if (b <= -7.6e+216) {
		tmp = t_0 * ((t_2 * Math.cos(Math.pow(Math.cbrt(t_1), 3.0))) * (b + a));
	} else {
		tmp = t_0 * ((b + a) * (t_2 * (Math.sqrt(Math.cbrt(Math.pow(Math.cos((0.005555555555555556 * (Math.PI * angle))), 4.0))) * Math.abs(Math.cbrt(Math.cos(t_1))))));
	}
	return tmp;
}

Julia

function code(a, b, angle)
	t_0 = Float64(2.0 * Float64(b - a))
	t_1 = Float64(pi * Float64(0.005555555555555556 * angle))
	t_2 = sin(t_1)
	tmp = 0.0
	if (b <= -7.6e+216)
		tmp = Float64(t_0 * Float64(Float64(t_2 * cos((cbrt(t_1) ^ 3.0))) * Float64(b + a)));
	else
		tmp = Float64(t_0 * Float64(Float64(b + a) * Float64(t_2 * Float64(sqrt(cbrt((cos(Float64(0.005555555555555556 * Float64(pi * angle))) ^ 4.0))) * abs(cbrt(cos(t_1)))))));
	end
	return tmp
end

Wolfram

code[a_, b_, angle_] := Block[{t$95$0 = N[(2.0 * N[(b - a), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(Pi * N[(0.005555555555555556 * angle), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[Sin[t$95$1], $MachinePrecision]}, If[LessEqual[b, -7.6e+216], N[(t$95$0 * N[(N[(t$95$2 * N[Cos[N[Power[N[Power[t$95$1, 1/3], $MachinePrecision], 3.0], $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[(b + a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(t$95$0 * N[(N[(b + a), $MachinePrecision] * N[(t$95$2 * N[(N[Sqrt[N[Power[N[Power[N[Cos[N[(0.005555555555555556 * N[(Pi * angle), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], 4.0], $MachinePrecision], 1/3], $MachinePrecision]], $MachinePrecision] * N[Abs[N[Power[N[Cos[t$95$1], $MachinePrecision], 1/3], $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 2 regimes

2. if b < -7.60000000000000029e216

   1. Initial program 44.8%

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

      1. associate-*l* 44.8%

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

      2. unpow2 44.8%

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

      3. unpow2 44.8%

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

      4. difference-of-squares 57.3%

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

   3. Simplified 57.3%

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

   4. Taylor expanded in angle around inf 68.7%

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

      1. associate-*r* 68.7%

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

      2. associate-*r* 68.7%

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

      3. *-commutative 68.7%

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

      4. associate-*r* 62.4%

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

      5. *-commutative 62.4%

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

      6. *-commutative 62.4%

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

      7. *-commutative 62.4%

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

      8. associate-*r* 74.9%

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

      9. *-commutative 74.9%

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

      10. *-commutative 74.9%

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

      11. *-commutative 74.9%

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

   6. Simplified 74.9%

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

      1. *-commutative 74.9%

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

      2. add-cube-cbrt 93.7%

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

      3. pow3 99.9%

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

      4. *-commutative 99.9%

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

   8. Applied egg-rr 99.9%

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

   if -7.60000000000000029e216 < b

   1. Initial program 52.9%

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

      1. associate-*l* 52.9%

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

      2. unpow2 52.9%

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

      3. unpow2 52.9%

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

      4. difference-of-squares 57.2%

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

   3. Simplified 57.2%

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

   4. Taylor expanded in angle around inf 65.8%

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

      1. associate-*r* 65.8%

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

      2. associate-*r* 65.7%

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

      3. *-commutative 65.7%

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

      4. associate-*r* 67.2%

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

      5. *-commutative 67.2%

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

      6. *-commutative 67.2%

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

      7. *-commutative 67.2%

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

      8. associate-*r* 67.0%

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

      9. *-commutative 67.0%

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

      10. *-commutative 67.0%

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

      11. *-commutative 67.0%

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

   6. Simplified 67.0%

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

      1. *-commutative 67.0%

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

      2. pow1 67.0%

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

      3. metadata-eval 67.0%

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

      4. sqrt-pow1 71.6%

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

      5. pow1/2 71.6%

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

      6. add-cube-cbrt 71.6%

         \[\leadsto \left(2 \cdot \left(b - a\right)\right) \cdot \left(\left(\sin \left(\pi \cdot \left(0.005555555555555556 \cdot angle\right)\right) \cdot {\color{blue}{\left(\left(\sqrt[3]{{\cos \left(\pi \cdot \left(angle \cdot 0.005555555555555556\right)\right)}^{2}} \cdot \sqrt[3]{{\cos \left(\pi \cdot \left(angle \cdot 0.005555555555555556\right)\right)}^{2}}\right) \cdot \sqrt[3]{{\cos \left(\pi \cdot \left(angle \cdot 0.005555555555555556\right)\right)}^{2}}\right)}}^{0.5}\right) \cdot \left(a + b\right)\right) \]

      7. unpow-prod-down 71.6%

         \[\leadsto \left(2 \cdot \left(b - a\right)\right) \cdot \left(\left(\sin \left(\pi \cdot \left(0.005555555555555556 \cdot angle\right)\right) \cdot \color{blue}{\left({\left(\sqrt[3]{{\cos \left(\pi \cdot \left(angle \cdot 0.005555555555555556\right)\right)}^{2}} \cdot \sqrt[3]{{\cos \left(\pi \cdot \left(angle \cdot 0.005555555555555556\right)\right)}^{2}}\right)}^{0.5} \cdot {\left(\sqrt[3]{{\cos \left(\pi \cdot \left(angle \cdot 0.005555555555555556\right)\right)}^{2}}\right)}^{0.5}\right)}\right) \cdot \left(a + b\right)\right) \]

   8. Applied egg-rr 71.6%

      \[\leadsto \left(2 \cdot \left(b - a\right)\right) \cdot \left(\left(\sin \left(\pi \cdot \left(0.005555555555555556 \cdot angle\right)\right) \cdot \color{blue}{\left({\left(\sqrt[3]{{\cos \left(\pi \cdot \left(0.005555555555555556 \cdot angle\right)\right)}^{4}}\right)}^{0.5} \cdot {\left({\left(\sqrt[3]{\cos \left(\pi \cdot \left(0.005555555555555556 \cdot angle\right)\right)}\right)}^{2}\right)}^{0.5}\right)}\right) \cdot \left(a + b\right)\right) \]
   9. Step-by-step derivation

      1. unpow1/2 71.6%

         \[\leadsto \left(2 \cdot \left(b - a\right)\right) \cdot \left(\left(\sin \left(\pi \cdot \left(0.005555555555555556 \cdot angle\right)\right) \cdot \left(\color{blue}{\sqrt{\sqrt[3]{{\cos \left(\pi \cdot \left(0.005555555555555556 \cdot angle\right)\right)}^{4}}}} \cdot {\left({\left(\sqrt[3]{\cos \left(\pi \cdot \left(0.005555555555555556 \cdot angle\right)\right)}\right)}^{2}\right)}^{0.5}\right)\right) \cdot \left(a + b\right)\right) \]

      2. *-commutative 71.6%

         \[\leadsto \left(2 \cdot \left(b - a\right)\right) \cdot \left(\left(\sin \left(\pi \cdot \left(0.005555555555555556 \cdot angle\right)\right) \cdot \left(\sqrt{\sqrt[3]{{\cos \color{blue}{\left(\left(0.005555555555555556 \cdot angle\right) \cdot \pi\right)}}^{4}}} \cdot {\left({\left(\sqrt[3]{\cos \left(\pi \cdot \left(0.005555555555555556 \cdot angle\right)\right)}\right)}^{2}\right)}^{0.5}\right)\right) \cdot \left(a + b\right)\right) \]

      3. associate-*r* 71.3%

         \[\leadsto \left(2 \cdot \left(b - a\right)\right) \cdot \left(\left(\sin \left(\pi \cdot \left(0.005555555555555556 \cdot angle\right)\right) \cdot \left(\sqrt{\sqrt[3]{{\cos \color{blue}{\left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}}^{4}}} \cdot {\left({\left(\sqrt[3]{\cos \left(\pi \cdot \left(0.005555555555555556 \cdot angle\right)\right)}\right)}^{2}\right)}^{0.5}\right)\right) \cdot \left(a + b\right)\right) \]

      4. unpow1/2 71.3%

         \[\leadsto \left(2 \cdot \left(b - a\right)\right) \cdot \left(\left(\sin \left(\pi \cdot \left(0.005555555555555556 \cdot angle\right)\right) \cdot \left(\sqrt{\sqrt[3]{{\cos \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{4}}} \cdot \color{blue}{\sqrt{{\left(\sqrt[3]{\cos \left(\pi \cdot \left(0.005555555555555556 \cdot angle\right)\right)}\right)}^{2}}}\right)\right) \cdot \left(a + b\right)\right) \]

      5. unpow2 71.3%

         \[\leadsto \left(2 \cdot \left(b - a\right)\right) \cdot \left(\left(\sin \left(\pi \cdot \left(0.005555555555555556 \cdot angle\right)\right) \cdot \left(\sqrt{\sqrt[3]{{\cos \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{4}}} \cdot \sqrt{\color{blue}{\sqrt[3]{\cos \left(\pi \cdot \left(0.005555555555555556 \cdot angle\right)\right)} \cdot \sqrt[3]{\cos \left(\pi \cdot \left(0.005555555555555556 \cdot angle\right)\right)}}}\right)\right) \cdot \left(a + b\right)\right) \]

      6. rem-sqrt-square 71.3%

         \[\leadsto \left(2 \cdot \left(b - a\right)\right) \cdot \left(\left(\sin \left(\pi \cdot \left(0.005555555555555556 \cdot angle\right)\right) \cdot \left(\sqrt{\sqrt[3]{{\cos \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{4}}} \cdot \color{blue}{\left|\sqrt[3]{\cos \left(\pi \cdot \left(0.005555555555555556 \cdot angle\right)\right)}\right|}\right)\right) \cdot \left(a + b\right)\right) \]

      7. *-commutative 71.3%

         \[\leadsto \left(2 \cdot \left(b - a\right)\right) \cdot \left(\left(\sin \left(\pi \cdot \left(0.005555555555555556 \cdot angle\right)\right) \cdot \left(\sqrt{\sqrt[3]{{\cos \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{4}}} \cdot \left|\sqrt[3]{\cos \color{blue}{\left(\left(0.005555555555555556 \cdot angle\right) \cdot \pi\right)}}\right|\right)\right) \cdot \left(a + b\right)\right) \]

      8. associate-*r* 71.2%

         \[\leadsto \left(2 \cdot \left(b - a\right)\right) \cdot \left(\left(\sin \left(\pi \cdot \left(0.005555555555555556 \cdot angle\right)\right) \cdot \left(\sqrt{\sqrt[3]{{\cos \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{4}}} \cdot \left|\sqrt[3]{\cos \color{blue}{\left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}}\right|\right)\right) \cdot \left(a + b\right)\right) \]

   10. Simplified 71.2%

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

       1. expm1-log1p-u 71.2%

          \[\leadsto \left(2 \cdot \left(b - a\right)\right) \cdot \left(\left(\sin \left(\pi \cdot \left(0.005555555555555556 \cdot angle\right)\right) \cdot \left(\sqrt{\sqrt[3]{{\color{blue}{\left(\mathsf{expm1}\left(\mathsf{log1p}\left(\cos \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)\right)\right)\right)}}^{4}}} \cdot \left|\sqrt[3]{\cos \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}\right|\right)\right) \cdot \left(a + b\right)\right) \]

       2. expm1-udef 71.2%

          \[\leadsto \left(2 \cdot \left(b - a\right)\right) \cdot \left(\left(\sin \left(\pi \cdot \left(0.005555555555555556 \cdot angle\right)\right) \cdot \left(\sqrt{\sqrt[3]{{\color{blue}{\left(e^{\mathsf{log1p}\left(\cos \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)\right)} - 1\right)}}^{4}}} \cdot \left|\sqrt[3]{\cos \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}\right|\right)\right) \cdot \left(a + b\right)\right) \]

       3. associate-*r* 71.3%

          \[\leadsto \left(2 \cdot \left(b - a\right)\right) \cdot \left(\left(\sin \left(\pi \cdot \left(0.005555555555555556 \cdot angle\right)\right) \cdot \left(\sqrt{\sqrt[3]{{\left(e^{\mathsf{log1p}\left(\cos \color{blue}{\left(\left(0.005555555555555556 \cdot angle\right) \cdot \pi\right)}\right)} - 1\right)}^{4}}} \cdot \left|\sqrt[3]{\cos \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}\right|\right)\right) \cdot \left(a + b\right)\right) \]

       4. *-commutative 71.3%

          \[\leadsto \left(2 \cdot \left(b - a\right)\right) \cdot \left(\left(\sin \left(\pi \cdot \left(0.005555555555555556 \cdot angle\right)\right) \cdot \left(\sqrt{\sqrt[3]{{\left(e^{\mathsf{log1p}\left(\cos \color{blue}{\left(\pi \cdot \left(0.005555555555555556 \cdot angle\right)\right)}\right)} - 1\right)}^{4}}} \cdot \left|\sqrt[3]{\cos \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}\right|\right)\right) \cdot \left(a + b\right)\right) \]

   12. Applied egg-rr 71.3%

       \[\leadsto \left(2 \cdot \left(b - a\right)\right) \cdot \left(\left(\sin \left(\pi \cdot \left(0.005555555555555556 \cdot angle\right)\right) \cdot \left(\sqrt{\sqrt[3]{{\cos \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{4}}} \cdot \left|\sqrt[3]{\color{blue}{e^{\mathsf{log1p}\left(\cos \left(\pi \cdot \left(0.005555555555555556 \cdot angle\right)\right)\right)} - 1}}\right|\right)\right) \cdot \left(a + b\right)\right) \]
   13. Step-by-step derivation

       1. expm1-def 71.3%

          \[\leadsto \left(2 \cdot \left(b - a\right)\right) \cdot \left(\left(\sin \left(\pi \cdot \left(0.005555555555555556 \cdot angle\right)\right) \cdot \left(\sqrt{\sqrt[3]{{\color{blue}{\left(\mathsf{expm1}\left(\mathsf{log1p}\left(\cos \left(\pi \cdot \left(0.005555555555555556 \cdot angle\right)\right)\right)\right)\right)}}^{4}}} \cdot \left|\sqrt[3]{\cos \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}\right|\right)\right) \cdot \left(a + b\right)\right) \]

       2. expm1-log1p 71.3%

          \[\leadsto \left(2 \cdot \left(b - a\right)\right) \cdot \left(\left(\sin \left(\pi \cdot \left(0.005555555555555556 \cdot angle\right)\right) \cdot \left(\sqrt{\sqrt[3]{{\color{blue}{\cos \left(\pi \cdot \left(0.005555555555555556 \cdot angle\right)\right)}}^{4}}} \cdot \left|\sqrt[3]{\cos \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}\right|\right)\right) \cdot \left(a + b\right)\right) \]

       3. *-commutative 71.3%

          \[\leadsto \left(2 \cdot \left(b - a\right)\right) \cdot \left(\left(\sin \left(\pi \cdot \left(0.005555555555555556 \cdot angle\right)\right) \cdot \left(\sqrt{\sqrt[3]{{\cos \left(\pi \cdot \color{blue}{\left(angle \cdot 0.005555555555555556\right)}\right)}^{4}}} \cdot \left|\sqrt[3]{\cos \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}\right|\right)\right) \cdot \left(a + b\right)\right) \]

   14. Simplified 71.3%

       \[\leadsto \left(2 \cdot \left(b - a\right)\right) \cdot \left(\left(\sin \left(\pi \cdot \left(0.005555555555555556 \cdot angle\right)\right) \cdot \left(\sqrt{\sqrt[3]{{\cos \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{4}}} \cdot \left|\sqrt[3]{\color{blue}{\cos \left(\pi \cdot \left(angle \cdot 0.005555555555555556\right)\right)}}\right|\right)\right) \cdot \left(a + b\right)\right) \]
3. Recombined 2 regimes into one program.

4. Final simplification 73.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -7.6 \cdot 10^{+216}:\\ \;\;\;\;\left(2 \cdot \left(b - a\right)\right) \cdot \left(\left(\sin \left(\pi \cdot \left(0.005555555555555556 \cdot angle\right)\right) \cdot \cos \left({\left(\sqrt[3]{\pi \cdot \left(0.005555555555555556 \cdot angle\right)}\right)}^{3}\right)\right) \cdot \left(b + a\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(2 \cdot \left(b - a\right)\right) \cdot \left(\left(b + a\right) \cdot \left(\sin \left(\pi \cdot \left(0.005555555555555556 \cdot angle\right)\right) \cdot \left(\sqrt{\sqrt[3]{{\cos \left(0.005555555555555556 \cdot \left(\pi \cdot angle\right)\right)}^{4}}} \cdot \left|\sqrt[3]{\cos \left(\pi \cdot \left(0.005555555555555556 \cdot angle\right)\right)}\right|\right)\right)\right)\\ \end{array} \]

Alternative 4: 66.4% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := 2 \cdot \left(b - a\right)\\ t_1 := \pi \cdot \left(0.005555555555555556 \cdot angle\right)\\ t_2 := \sin t_1\\ \mathbf{if}\;b \leq -1 \cdot 10^{+211}:\\ \;\;\;\;t_0 \cdot \left(\left(t_2 \cdot \cos \left({\left(\sqrt[3]{t_1}\right)}^{3}\right)\right) \cdot \left(b + a\right)\right)\\ \mathbf{else}:\\ \;\;\;\;t_0 \cdot \left(\left(b + a\right) \cdot \left(t_2 \cdot \left(\sqrt{\sqrt[3]{{\cos t_1}^{4}}} \cdot \left|\sqrt[3]{\cos \left(0.005555555555555556 \cdot \left(\pi \cdot angle\right)\right)}\right|\right)\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (a b angle)
 :precision binary64
 (let* ((t_0 (* 2.0 (- b a)))
        (t_1 (* PI (* 0.005555555555555556 angle)))
        (t_2 (sin t_1)))
   (if (<= b -1e+211)
     (* t_0 (* (* t_2 (cos (pow (cbrt t_1) 3.0))) (+ b a)))
     (*
      t_0
      (*
       (+ b a)
       (*
        t_2
        (*
         (sqrt (cbrt (pow (cos t_1) 4.0)))
         (fabs (cbrt (cos (* 0.005555555555555556 (* PI angle))))))))))))

C

double code(double a, double b, double angle) {
	double t_0 = 2.0 * (b - a);
	double t_1 = ((double) M_PI) * (0.005555555555555556 * angle);
	double t_2 = sin(t_1);
	double tmp;
	if (b <= -1e+211) {
		tmp = t_0 * ((t_2 * cos(pow(cbrt(t_1), 3.0))) * (b + a));
	} else {
		tmp = t_0 * ((b + a) * (t_2 * (sqrt(cbrt(pow(cos(t_1), 4.0))) * fabs(cbrt(cos((0.005555555555555556 * (((double) M_PI) * angle))))))));
	}
	return tmp;
}

Java

public static double code(double a, double b, double angle) {
	double t_0 = 2.0 * (b - a);
	double t_1 = Math.PI * (0.005555555555555556 * angle);
	double t_2 = Math.sin(t_1);
	double tmp;
	if (b <= -1e+211) {
		tmp = t_0 * ((t_2 * Math.cos(Math.pow(Math.cbrt(t_1), 3.0))) * (b + a));
	} else {
		tmp = t_0 * ((b + a) * (t_2 * (Math.sqrt(Math.cbrt(Math.pow(Math.cos(t_1), 4.0))) * Math.abs(Math.cbrt(Math.cos((0.005555555555555556 * (Math.PI * angle))))))));
	}
	return tmp;
}

Julia

function code(a, b, angle)
	t_0 = Float64(2.0 * Float64(b - a))
	t_1 = Float64(pi * Float64(0.005555555555555556 * angle))
	t_2 = sin(t_1)
	tmp = 0.0
	if (b <= -1e+211)
		tmp = Float64(t_0 * Float64(Float64(t_2 * cos((cbrt(t_1) ^ 3.0))) * Float64(b + a)));
	else
		tmp = Float64(t_0 * Float64(Float64(b + a) * Float64(t_2 * Float64(sqrt(cbrt((cos(t_1) ^ 4.0))) * abs(cbrt(cos(Float64(0.005555555555555556 * Float64(pi * angle)))))))));
	end
	return tmp
end

Wolfram

code[a_, b_, angle_] := Block[{t$95$0 = N[(2.0 * N[(b - a), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(Pi * N[(0.005555555555555556 * angle), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[Sin[t$95$1], $MachinePrecision]}, If[LessEqual[b, -1e+211], N[(t$95$0 * N[(N[(t$95$2 * N[Cos[N[Power[N[Power[t$95$1, 1/3], $MachinePrecision], 3.0], $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[(b + a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(t$95$0 * N[(N[(b + a), $MachinePrecision] * N[(t$95$2 * N[(N[Sqrt[N[Power[N[Power[N[Cos[t$95$1], $MachinePrecision], 4.0], $MachinePrecision], 1/3], $MachinePrecision]], $MachinePrecision] * N[Abs[N[Power[N[Cos[N[(0.005555555555555556 * N[(Pi * angle), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], 1/3], $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 2 regimes

2. if b < -9.9999999999999996e210

   1. Initial program 44.8%

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

      1. associate-*l* 44.8%

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

      2. unpow2 44.8%

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

      3. unpow2 44.8%

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

      4. difference-of-squares 57.3%

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

   3. Simplified 57.3%

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

   4. Taylor expanded in angle around inf 68.7%

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

      1. associate-*r* 68.7%

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

      2. associate-*r* 68.7%

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

      3. *-commutative 68.7%

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

      4. associate-*r* 62.4%

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

      5. *-commutative 62.4%

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

      6. *-commutative 62.4%

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

      7. *-commutative 62.4%

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

      8. associate-*r* 74.9%

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

      9. *-commutative 74.9%

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

      10. *-commutative 74.9%

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

      11. *-commutative 74.9%

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

   6. Simplified 74.9%

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

      1. *-commutative 74.9%

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

      2. add-cube-cbrt 93.7%

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

      3. pow3 99.9%

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

      4. *-commutative 99.9%

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

   8. Applied egg-rr 99.9%

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

   if -9.9999999999999996e210 < b

   1. Initial program 52.9%

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

      1. associate-*l* 52.9%

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

      2. unpow2 52.9%

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

      3. unpow2 52.9%

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

      4. difference-of-squares 57.2%

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

   3. Simplified 57.2%

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

   4. Taylor expanded in angle around inf 65.8%

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

      1. associate-*r* 65.8%

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

      2. associate-*r* 65.7%

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

      3. *-commutative 65.7%

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

      4. associate-*r* 67.2%

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

      5. *-commutative 67.2%

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

      6. *-commutative 67.2%

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

      7. *-commutative 67.2%

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

      8. associate-*r* 67.0%

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

      9. *-commutative 67.0%

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

      10. *-commutative 67.0%

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

      11. *-commutative 67.0%

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

   6. Simplified 67.0%

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

      1. *-commutative 67.0%

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

      2. pow1 67.0%

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

      3. metadata-eval 67.0%

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

      4. sqrt-pow1 71.6%

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

      5. pow1/2 71.6%

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

      6. add-cube-cbrt 71.6%

         \[\leadsto \left(2 \cdot \left(b - a\right)\right) \cdot \left(\left(\sin \left(\pi \cdot \left(0.005555555555555556 \cdot angle\right)\right) \cdot {\color{blue}{\left(\left(\sqrt[3]{{\cos \left(\pi \cdot \left(angle \cdot 0.005555555555555556\right)\right)}^{2}} \cdot \sqrt[3]{{\cos \left(\pi \cdot \left(angle \cdot 0.005555555555555556\right)\right)}^{2}}\right) \cdot \sqrt[3]{{\cos \left(\pi \cdot \left(angle \cdot 0.005555555555555556\right)\right)}^{2}}\right)}}^{0.5}\right) \cdot \left(a + b\right)\right) \]

      7. unpow-prod-down 71.6%

         \[\leadsto \left(2 \cdot \left(b - a\right)\right) \cdot \left(\left(\sin \left(\pi \cdot \left(0.005555555555555556 \cdot angle\right)\right) \cdot \color{blue}{\left({\left(\sqrt[3]{{\cos \left(\pi \cdot \left(angle \cdot 0.005555555555555556\right)\right)}^{2}} \cdot \sqrt[3]{{\cos \left(\pi \cdot \left(angle \cdot 0.005555555555555556\right)\right)}^{2}}\right)}^{0.5} \cdot {\left(\sqrt[3]{{\cos \left(\pi \cdot \left(angle \cdot 0.005555555555555556\right)\right)}^{2}}\right)}^{0.5}\right)}\right) \cdot \left(a + b\right)\right) \]

   8. Applied egg-rr 71.6%

      \[\leadsto \left(2 \cdot \left(b - a\right)\right) \cdot \left(\left(\sin \left(\pi \cdot \left(0.005555555555555556 \cdot angle\right)\right) \cdot \color{blue}{\left({\left(\sqrt[3]{{\cos \left(\pi \cdot \left(0.005555555555555556 \cdot angle\right)\right)}^{4}}\right)}^{0.5} \cdot {\left({\left(\sqrt[3]{\cos \left(\pi \cdot \left(0.005555555555555556 \cdot angle\right)\right)}\right)}^{2}\right)}^{0.5}\right)}\right) \cdot \left(a + b\right)\right) \]
   9. Step-by-step derivation

      1. unpow1/2 71.6%

         \[\leadsto \left(2 \cdot \left(b - a\right)\right) \cdot \left(\left(\sin \left(\pi \cdot \left(0.005555555555555556 \cdot angle\right)\right) \cdot \left(\color{blue}{\sqrt{\sqrt[3]{{\cos \left(\pi \cdot \left(0.005555555555555556 \cdot angle\right)\right)}^{4}}}} \cdot {\left({\left(\sqrt[3]{\cos \left(\pi \cdot \left(0.005555555555555556 \cdot angle\right)\right)}\right)}^{2}\right)}^{0.5}\right)\right) \cdot \left(a + b\right)\right) \]

      2. *-commutative 71.6%

         \[\leadsto \left(2 \cdot \left(b - a\right)\right) \cdot \left(\left(\sin \left(\pi \cdot \left(0.005555555555555556 \cdot angle\right)\right) \cdot \left(\sqrt{\sqrt[3]{{\cos \color{blue}{\left(\left(0.005555555555555556 \cdot angle\right) \cdot \pi\right)}}^{4}}} \cdot {\left({\left(\sqrt[3]{\cos \left(\pi \cdot \left(0.005555555555555556 \cdot angle\right)\right)}\right)}^{2}\right)}^{0.5}\right)\right) \cdot \left(a + b\right)\right) \]

      3. associate-*r* 71.3%

         \[\leadsto \left(2 \cdot \left(b - a\right)\right) \cdot \left(\left(\sin \left(\pi \cdot \left(0.005555555555555556 \cdot angle\right)\right) \cdot \left(\sqrt{\sqrt[3]{{\cos \color{blue}{\left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}}^{4}}} \cdot {\left({\left(\sqrt[3]{\cos \left(\pi \cdot \left(0.005555555555555556 \cdot angle\right)\right)}\right)}^{2}\right)}^{0.5}\right)\right) \cdot \left(a + b\right)\right) \]

      4. unpow1/2 71.3%

         \[\leadsto \left(2 \cdot \left(b - a\right)\right) \cdot \left(\left(\sin \left(\pi \cdot \left(0.005555555555555556 \cdot angle\right)\right) \cdot \left(\sqrt{\sqrt[3]{{\cos \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{4}}} \cdot \color{blue}{\sqrt{{\left(\sqrt[3]{\cos \left(\pi \cdot \left(0.005555555555555556 \cdot angle\right)\right)}\right)}^{2}}}\right)\right) \cdot \left(a + b\right)\right) \]

      5. unpow2 71.3%

         \[\leadsto \left(2 \cdot \left(b - a\right)\right) \cdot \left(\left(\sin \left(\pi \cdot \left(0.005555555555555556 \cdot angle\right)\right) \cdot \left(\sqrt{\sqrt[3]{{\cos \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{4}}} \cdot \sqrt{\color{blue}{\sqrt[3]{\cos \left(\pi \cdot \left(0.005555555555555556 \cdot angle\right)\right)} \cdot \sqrt[3]{\cos \left(\pi \cdot \left(0.005555555555555556 \cdot angle\right)\right)}}}\right)\right) \cdot \left(a + b\right)\right) \]

      6. rem-sqrt-square 71.3%

         \[\leadsto \left(2 \cdot \left(b - a\right)\right) \cdot \left(\left(\sin \left(\pi \cdot \left(0.005555555555555556 \cdot angle\right)\right) \cdot \left(\sqrt{\sqrt[3]{{\cos \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{4}}} \cdot \color{blue}{\left|\sqrt[3]{\cos \left(\pi \cdot \left(0.005555555555555556 \cdot angle\right)\right)}\right|}\right)\right) \cdot \left(a + b\right)\right) \]

      7. *-commutative 71.3%

         \[\leadsto \left(2 \cdot \left(b - a\right)\right) \cdot \left(\left(\sin \left(\pi \cdot \left(0.005555555555555556 \cdot angle\right)\right) \cdot \left(\sqrt{\sqrt[3]{{\cos \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{4}}} \cdot \left|\sqrt[3]{\cos \color{blue}{\left(\left(0.005555555555555556 \cdot angle\right) \cdot \pi\right)}}\right|\right)\right) \cdot \left(a + b\right)\right) \]

      8. associate-*r* 71.2%

         \[\leadsto \left(2 \cdot \left(b - a\right)\right) \cdot \left(\left(\sin \left(\pi \cdot \left(0.005555555555555556 \cdot angle\right)\right) \cdot \left(\sqrt{\sqrt[3]{{\cos \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{4}}} \cdot \left|\sqrt[3]{\cos \color{blue}{\left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}}\right|\right)\right) \cdot \left(a + b\right)\right) \]

   10. Simplified 71.2%

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

       1. expm1-log1p-u 71.2%

          \[\leadsto \left(2 \cdot \left(b - a\right)\right) \cdot \left(\left(\sin \left(\pi \cdot \left(0.005555555555555556 \cdot angle\right)\right) \cdot \left(\sqrt{\sqrt[3]{{\color{blue}{\left(\mathsf{expm1}\left(\mathsf{log1p}\left(\cos \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)\right)\right)\right)}}^{4}}} \cdot \left|\sqrt[3]{\cos \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}\right|\right)\right) \cdot \left(a + b\right)\right) \]

       2. expm1-udef 71.2%

          \[\leadsto \left(2 \cdot \left(b - a\right)\right) \cdot \left(\left(\sin \left(\pi \cdot \left(0.005555555555555556 \cdot angle\right)\right) \cdot \left(\sqrt{\sqrt[3]{{\color{blue}{\left(e^{\mathsf{log1p}\left(\cos \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)\right)} - 1\right)}}^{4}}} \cdot \left|\sqrt[3]{\cos \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}\right|\right)\right) \cdot \left(a + b\right)\right) \]

       3. associate-*r* 71.3%

          \[\leadsto \left(2 \cdot \left(b - a\right)\right) \cdot \left(\left(\sin \left(\pi \cdot \left(0.005555555555555556 \cdot angle\right)\right) \cdot \left(\sqrt{\sqrt[3]{{\left(e^{\mathsf{log1p}\left(\cos \color{blue}{\left(\left(0.005555555555555556 \cdot angle\right) \cdot \pi\right)}\right)} - 1\right)}^{4}}} \cdot \left|\sqrt[3]{\cos \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}\right|\right)\right) \cdot \left(a + b\right)\right) \]

       4. *-commutative 71.3%

          \[\leadsto \left(2 \cdot \left(b - a\right)\right) \cdot \left(\left(\sin \left(\pi \cdot \left(0.005555555555555556 \cdot angle\right)\right) \cdot \left(\sqrt{\sqrt[3]{{\left(e^{\mathsf{log1p}\left(\cos \color{blue}{\left(\pi \cdot \left(0.005555555555555556 \cdot angle\right)\right)}\right)} - 1\right)}^{4}}} \cdot \left|\sqrt[3]{\cos \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}\right|\right)\right) \cdot \left(a + b\right)\right) \]

   12. Applied egg-rr 71.3%

       \[\leadsto \left(2 \cdot \left(b - a\right)\right) \cdot \left(\left(\sin \left(\pi \cdot \left(0.005555555555555556 \cdot angle\right)\right) \cdot \left(\sqrt{\sqrt[3]{{\color{blue}{\left(e^{\mathsf{log1p}\left(\cos \left(\pi \cdot \left(0.005555555555555556 \cdot angle\right)\right)\right)} - 1\right)}}^{4}}} \cdot \left|\sqrt[3]{\cos \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}\right|\right)\right) \cdot \left(a + b\right)\right) \]
   13. Step-by-step derivation

       1. expm1-def 71.3%

          \[\leadsto \left(2 \cdot \left(b - a\right)\right) \cdot \left(\left(\sin \left(\pi \cdot \left(0.005555555555555556 \cdot angle\right)\right) \cdot \left(\sqrt{\sqrt[3]{{\color{blue}{\left(\mathsf{expm1}\left(\mathsf{log1p}\left(\cos \left(\pi \cdot \left(0.005555555555555556 \cdot angle\right)\right)\right)\right)\right)}}^{4}}} \cdot \left|\sqrt[3]{\cos \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}\right|\right)\right) \cdot \left(a + b\right)\right) \]

       2. expm1-log1p 71.3%

          \[\leadsto \left(2 \cdot \left(b - a\right)\right) \cdot \left(\left(\sin \left(\pi \cdot \left(0.005555555555555556 \cdot angle\right)\right) \cdot \left(\sqrt{\sqrt[3]{{\color{blue}{\cos \left(\pi \cdot \left(0.005555555555555556 \cdot angle\right)\right)}}^{4}}} \cdot \left|\sqrt[3]{\cos \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}\right|\right)\right) \cdot \left(a + b\right)\right) \]

       3. *-commutative 71.3%

          \[\leadsto \left(2 \cdot \left(b - a\right)\right) \cdot \left(\left(\sin \left(\pi \cdot \left(0.005555555555555556 \cdot angle\right)\right) \cdot \left(\sqrt{\sqrt[3]{{\cos \left(\pi \cdot \color{blue}{\left(angle \cdot 0.005555555555555556\right)}\right)}^{4}}} \cdot \left|\sqrt[3]{\cos \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}\right|\right)\right) \cdot \left(a + b\right)\right) \]

   14. Simplified 71.3%

       \[\leadsto \left(2 \cdot \left(b - a\right)\right) \cdot \left(\left(\sin \left(\pi \cdot \left(0.005555555555555556 \cdot angle\right)\right) \cdot \left(\sqrt{\sqrt[3]{{\color{blue}{\cos \left(\pi \cdot \left(angle \cdot 0.005555555555555556\right)\right)}}^{4}}} \cdot \left|\sqrt[3]{\cos \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}\right|\right)\right) \cdot \left(a + b\right)\right) \]
3. Recombined 2 regimes into one program.

4. Final simplification 73.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -1 \cdot 10^{+211}:\\ \;\;\;\;\left(2 \cdot \left(b - a\right)\right) \cdot \left(\left(\sin \left(\pi \cdot \left(0.005555555555555556 \cdot angle\right)\right) \cdot \cos \left({\left(\sqrt[3]{\pi \cdot \left(0.005555555555555556 \cdot angle\right)}\right)}^{3}\right)\right) \cdot \left(b + a\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(2 \cdot \left(b - a\right)\right) \cdot \left(\left(b + a\right) \cdot \left(\sin \left(\pi \cdot \left(0.005555555555555556 \cdot angle\right)\right) \cdot \left(\sqrt{\sqrt[3]{{\cos \left(\pi \cdot \left(0.005555555555555556 \cdot angle\right)\right)}^{4}}} \cdot \left|\sqrt[3]{\cos \left(0.005555555555555556 \cdot \left(\pi \cdot angle\right)\right)}\right|\right)\right)\right)\\ \end{array} \]

Alternative 5: 65.9% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := 2 \cdot \left(b - a\right)\\ t_1 := \pi \cdot \left(0.005555555555555556 \cdot angle\right)\\ t_2 := \sin t_1\\ \mathbf{if}\;b \leq -9.8 \cdot 10^{+214}:\\ \;\;\;\;t_0 \cdot \left(\left(t_2 \cdot \cos \left({\left(\sqrt[3]{t_1}\right)}^{3}\right)\right) \cdot \left(b + a\right)\right)\\ \mathbf{else}:\\ \;\;\;\;t_0 \cdot \left(\left(b + a\right) \cdot \left(t_2 \cdot \sqrt{\sqrt[3]{{\cos t_1}^{4}}}\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (a b angle)
 :precision binary64
 (let* ((t_0 (* 2.0 (- b a)))
        (t_1 (* PI (* 0.005555555555555556 angle)))
        (t_2 (sin t_1)))
   (if (<= b -9.8e+214)
     (* t_0 (* (* t_2 (cos (pow (cbrt t_1) 3.0))) (+ b a)))
     (* t_0 (* (+ b a) (* t_2 (sqrt (cbrt (pow (cos t_1) 4.0)))))))))

C

double code(double a, double b, double angle) {
	double t_0 = 2.0 * (b - a);
	double t_1 = ((double) M_PI) * (0.005555555555555556 * angle);
	double t_2 = sin(t_1);
	double tmp;
	if (b <= -9.8e+214) {
		tmp = t_0 * ((t_2 * cos(pow(cbrt(t_1), 3.0))) * (b + a));
	} else {
		tmp = t_0 * ((b + a) * (t_2 * sqrt(cbrt(pow(cos(t_1), 4.0)))));
	}
	return tmp;
}

Java

public static double code(double a, double b, double angle) {
	double t_0 = 2.0 * (b - a);
	double t_1 = Math.PI * (0.005555555555555556 * angle);
	double t_2 = Math.sin(t_1);
	double tmp;
	if (b <= -9.8e+214) {
		tmp = t_0 * ((t_2 * Math.cos(Math.pow(Math.cbrt(t_1), 3.0))) * (b + a));
	} else {
		tmp = t_0 * ((b + a) * (t_2 * Math.sqrt(Math.cbrt(Math.pow(Math.cos(t_1), 4.0)))));
	}
	return tmp;
}

Julia

function code(a, b, angle)
	t_0 = Float64(2.0 * Float64(b - a))
	t_1 = Float64(pi * Float64(0.005555555555555556 * angle))
	t_2 = sin(t_1)
	tmp = 0.0
	if (b <= -9.8e+214)
		tmp = Float64(t_0 * Float64(Float64(t_2 * cos((cbrt(t_1) ^ 3.0))) * Float64(b + a)));
	else
		tmp = Float64(t_0 * Float64(Float64(b + a) * Float64(t_2 * sqrt(cbrt((cos(t_1) ^ 4.0))))));
	end
	return tmp
end

Wolfram

code[a_, b_, angle_] := Block[{t$95$0 = N[(2.0 * N[(b - a), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(Pi * N[(0.005555555555555556 * angle), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[Sin[t$95$1], $MachinePrecision]}, If[LessEqual[b, -9.8e+214], N[(t$95$0 * N[(N[(t$95$2 * N[Cos[N[Power[N[Power[t$95$1, 1/3], $MachinePrecision], 3.0], $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[(b + a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(t$95$0 * N[(N[(b + a), $MachinePrecision] * N[(t$95$2 * N[Sqrt[N[Power[N[Power[N[Cos[t$95$1], $MachinePrecision], 4.0], $MachinePrecision], 1/3], $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 2 regimes

2. if b < -9.80000000000000065e214

   1. Initial program 44.8%

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

      1. associate-*l* 44.8%

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

      2. unpow2 44.8%

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

      3. unpow2 44.8%

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

      4. difference-of-squares 57.3%

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

   3. Simplified 57.3%

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

   4. Taylor expanded in angle around inf 68.7%

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

      1. associate-*r* 68.7%

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

      2. associate-*r* 68.7%

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

      3. *-commutative 68.7%

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

      4. associate-*r* 62.4%

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

      5. *-commutative 62.4%

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

      6. *-commutative 62.4%

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

      7. *-commutative 62.4%

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

      8. associate-*r* 74.9%

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

      9. *-commutative 74.9%

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

      10. *-commutative 74.9%

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

      11. *-commutative 74.9%

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

   6. Simplified 74.9%

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

      1. *-commutative 74.9%

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

      2. add-cube-cbrt 93.7%

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

      3. pow3 99.9%

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

      4. *-commutative 99.9%

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

   8. Applied egg-rr 99.9%

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

   if -9.80000000000000065e214 < b

   1. Initial program 52.9%

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

      1. associate-*l* 52.9%

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

      2. unpow2 52.9%

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

      3. unpow2 52.9%

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

      4. difference-of-squares 57.2%

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

   3. Simplified 57.2%

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

   4. Taylor expanded in angle around inf 65.8%

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

      1. associate-*r* 65.8%

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

      2. associate-*r* 65.7%

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

      3. *-commutative 65.7%

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

      4. associate-*r* 67.2%

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

      5. *-commutative 67.2%

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

      6. *-commutative 67.2%

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

      7. *-commutative 67.2%

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

      8. associate-*r* 67.0%

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

      9. *-commutative 67.0%

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

      10. *-commutative 67.0%

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

      11. *-commutative 67.0%

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

   6. Simplified 67.0%

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

      1. *-commutative 67.0%

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

      2. pow1 67.0%

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

      3. metadata-eval 67.0%

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

      4. sqrt-pow1 71.6%

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

      5. pow1/2 71.6%

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

      6. add-cube-cbrt 71.6%

         \[\leadsto \left(2 \cdot \left(b - a\right)\right) \cdot \left(\left(\sin \left(\pi \cdot \left(0.005555555555555556 \cdot angle\right)\right) \cdot {\color{blue}{\left(\left(\sqrt[3]{{\cos \left(\pi \cdot \left(angle \cdot 0.005555555555555556\right)\right)}^{2}} \cdot \sqrt[3]{{\cos \left(\pi \cdot \left(angle \cdot 0.005555555555555556\right)\right)}^{2}}\right) \cdot \sqrt[3]{{\cos \left(\pi \cdot \left(angle \cdot 0.005555555555555556\right)\right)}^{2}}\right)}}^{0.5}\right) \cdot \left(a + b\right)\right) \]

      7. unpow-prod-down 71.6%

         \[\leadsto \left(2 \cdot \left(b - a\right)\right) \cdot \left(\left(\sin \left(\pi \cdot \left(0.005555555555555556 \cdot angle\right)\right) \cdot \color{blue}{\left({\left(\sqrt[3]{{\cos \left(\pi \cdot \left(angle \cdot 0.005555555555555556\right)\right)}^{2}} \cdot \sqrt[3]{{\cos \left(\pi \cdot \left(angle \cdot 0.005555555555555556\right)\right)}^{2}}\right)}^{0.5} \cdot {\left(\sqrt[3]{{\cos \left(\pi \cdot \left(angle \cdot 0.005555555555555556\right)\right)}^{2}}\right)}^{0.5}\right)}\right) \cdot \left(a + b\right)\right) \]

   8. Applied egg-rr 71.6%

      \[\leadsto \left(2 \cdot \left(b - a\right)\right) \cdot \left(\left(\sin \left(\pi \cdot \left(0.005555555555555556 \cdot angle\right)\right) \cdot \color{blue}{\left({\left(\sqrt[3]{{\cos \left(\pi \cdot \left(0.005555555555555556 \cdot angle\right)\right)}^{4}}\right)}^{0.5} \cdot {\left({\left(\sqrt[3]{\cos \left(\pi \cdot \left(0.005555555555555556 \cdot angle\right)\right)}\right)}^{2}\right)}^{0.5}\right)}\right) \cdot \left(a + b\right)\right) \]
   9. Step-by-step derivation

      1. unpow1/2 71.6%

         \[\leadsto \left(2 \cdot \left(b - a\right)\right) \cdot \left(\left(\sin \left(\pi \cdot \left(0.005555555555555556 \cdot angle\right)\right) \cdot \left(\color{blue}{\sqrt{\sqrt[3]{{\cos \left(\pi \cdot \left(0.005555555555555556 \cdot angle\right)\right)}^{4}}}} \cdot {\left({\left(\sqrt[3]{\cos \left(\pi \cdot \left(0.005555555555555556 \cdot angle\right)\right)}\right)}^{2}\right)}^{0.5}\right)\right) \cdot \left(a + b\right)\right) \]

      2. *-commutative 71.6%

         \[\leadsto \left(2 \cdot \left(b - a\right)\right) \cdot \left(\left(\sin \left(\pi \cdot \left(0.005555555555555556 \cdot angle\right)\right) \cdot \left(\sqrt{\sqrt[3]{{\cos \color{blue}{\left(\left(0.005555555555555556 \cdot angle\right) \cdot \pi\right)}}^{4}}} \cdot {\left({\left(\sqrt[3]{\cos \left(\pi \cdot \left(0.005555555555555556 \cdot angle\right)\right)}\right)}^{2}\right)}^{0.5}\right)\right) \cdot \left(a + b\right)\right) \]

      3. associate-*r* 71.3%

         \[\leadsto \left(2 \cdot \left(b - a\right)\right) \cdot \left(\left(\sin \left(\pi \cdot \left(0.005555555555555556 \cdot angle\right)\right) \cdot \left(\sqrt{\sqrt[3]{{\cos \color{blue}{\left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}}^{4}}} \cdot {\left({\left(\sqrt[3]{\cos \left(\pi \cdot \left(0.005555555555555556 \cdot angle\right)\right)}\right)}^{2}\right)}^{0.5}\right)\right) \cdot \left(a + b\right)\right) \]

      4. unpow1/2 71.3%

         \[\leadsto \left(2 \cdot \left(b - a\right)\right) \cdot \left(\left(\sin \left(\pi \cdot \left(0.005555555555555556 \cdot angle\right)\right) \cdot \left(\sqrt{\sqrt[3]{{\cos \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{4}}} \cdot \color{blue}{\sqrt{{\left(\sqrt[3]{\cos \left(\pi \cdot \left(0.005555555555555556 \cdot angle\right)\right)}\right)}^{2}}}\right)\right) \cdot \left(a + b\right)\right) \]

      5. unpow2 71.3%

         \[\leadsto \left(2 \cdot \left(b - a\right)\right) \cdot \left(\left(\sin \left(\pi \cdot \left(0.005555555555555556 \cdot angle\right)\right) \cdot \left(\sqrt{\sqrt[3]{{\cos \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{4}}} \cdot \sqrt{\color{blue}{\sqrt[3]{\cos \left(\pi \cdot \left(0.005555555555555556 \cdot angle\right)\right)} \cdot \sqrt[3]{\cos \left(\pi \cdot \left(0.005555555555555556 \cdot angle\right)\right)}}}\right)\right) \cdot \left(a + b\right)\right) \]

      6. rem-sqrt-square 71.3%

         \[\leadsto \left(2 \cdot \left(b - a\right)\right) \cdot \left(\left(\sin \left(\pi \cdot \left(0.005555555555555556 \cdot angle\right)\right) \cdot \left(\sqrt{\sqrt[3]{{\cos \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{4}}} \cdot \color{blue}{\left|\sqrt[3]{\cos \left(\pi \cdot \left(0.005555555555555556 \cdot angle\right)\right)}\right|}\right)\right) \cdot \left(a + b\right)\right) \]

      7. *-commutative 71.3%

         \[\leadsto \left(2 \cdot \left(b - a\right)\right) \cdot \left(\left(\sin \left(\pi \cdot \left(0.005555555555555556 \cdot angle\right)\right) \cdot \left(\sqrt{\sqrt[3]{{\cos \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{4}}} \cdot \left|\sqrt[3]{\cos \color{blue}{\left(\left(0.005555555555555556 \cdot angle\right) \cdot \pi\right)}}\right|\right)\right) \cdot \left(a + b\right)\right) \]

      8. associate-*r* 71.2%

         \[\leadsto \left(2 \cdot \left(b - a\right)\right) \cdot \left(\left(\sin \left(\pi \cdot \left(0.005555555555555556 \cdot angle\right)\right) \cdot \left(\sqrt{\sqrt[3]{{\cos \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{4}}} \cdot \left|\sqrt[3]{\cos \color{blue}{\left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}}\right|\right)\right) \cdot \left(a + b\right)\right) \]

   10. Simplified 71.2%

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

       1. expm1-log1p-u 71.2%

          \[\leadsto \left(2 \cdot \left(b - a\right)\right) \cdot \left(\left(\sin \left(\pi \cdot \left(0.005555555555555556 \cdot angle\right)\right) \cdot \left(\sqrt{\sqrt[3]{{\color{blue}{\left(\mathsf{expm1}\left(\mathsf{log1p}\left(\cos \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)\right)\right)\right)}}^{4}}} \cdot \left|\sqrt[3]{\cos \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}\right|\right)\right) \cdot \left(a + b\right)\right) \]

       2. expm1-udef 71.2%

          \[\leadsto \left(2 \cdot \left(b - a\right)\right) \cdot \left(\left(\sin \left(\pi \cdot \left(0.005555555555555556 \cdot angle\right)\right) \cdot \left(\sqrt{\sqrt[3]{{\color{blue}{\left(e^{\mathsf{log1p}\left(\cos \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)\right)} - 1\right)}}^{4}}} \cdot \left|\sqrt[3]{\cos \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}\right|\right)\right) \cdot \left(a + b\right)\right) \]

       3. associate-*r* 71.3%

          \[\leadsto \left(2 \cdot \left(b - a\right)\right) \cdot \left(\left(\sin \left(\pi \cdot \left(0.005555555555555556 \cdot angle\right)\right) \cdot \left(\sqrt{\sqrt[3]{{\left(e^{\mathsf{log1p}\left(\cos \color{blue}{\left(\left(0.005555555555555556 \cdot angle\right) \cdot \pi\right)}\right)} - 1\right)}^{4}}} \cdot \left|\sqrt[3]{\cos \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}\right|\right)\right) \cdot \left(a + b\right)\right) \]

       4. *-commutative 71.3%

          \[\leadsto \left(2 \cdot \left(b - a\right)\right) \cdot \left(\left(\sin \left(\pi \cdot \left(0.005555555555555556 \cdot angle\right)\right) \cdot \left(\sqrt{\sqrt[3]{{\left(e^{\mathsf{log1p}\left(\cos \color{blue}{\left(\pi \cdot \left(0.005555555555555556 \cdot angle\right)\right)}\right)} - 1\right)}^{4}}} \cdot \left|\sqrt[3]{\cos \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}\right|\right)\right) \cdot \left(a + b\right)\right) \]

   12. Applied egg-rr 71.3%

       \[\leadsto \left(2 \cdot \left(b - a\right)\right) \cdot \left(\left(\sin \left(\pi \cdot \left(0.005555555555555556 \cdot angle\right)\right) \cdot \left(\sqrt{\sqrt[3]{{\color{blue}{\left(e^{\mathsf{log1p}\left(\cos \left(\pi \cdot \left(0.005555555555555556 \cdot angle\right)\right)\right)} - 1\right)}}^{4}}} \cdot \left|\sqrt[3]{\cos \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}\right|\right)\right) \cdot \left(a + b\right)\right) \]
   13. Step-by-step derivation

       1. expm1-def 71.3%

          \[\leadsto \left(2 \cdot \left(b - a\right)\right) \cdot \left(\left(\sin \left(\pi \cdot \left(0.005555555555555556 \cdot angle\right)\right) \cdot \left(\sqrt{\sqrt[3]{{\color{blue}{\left(\mathsf{expm1}\left(\mathsf{log1p}\left(\cos \left(\pi \cdot \left(0.005555555555555556 \cdot angle\right)\right)\right)\right)\right)}}^{4}}} \cdot \left|\sqrt[3]{\cos \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}\right|\right)\right) \cdot \left(a + b\right)\right) \]

       2. expm1-log1p 71.3%

          \[\leadsto \left(2 \cdot \left(b - a\right)\right) \cdot \left(\left(\sin \left(\pi \cdot \left(0.005555555555555556 \cdot angle\right)\right) \cdot \left(\sqrt{\sqrt[3]{{\color{blue}{\cos \left(\pi \cdot \left(0.005555555555555556 \cdot angle\right)\right)}}^{4}}} \cdot \left|\sqrt[3]{\cos \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}\right|\right)\right) \cdot \left(a + b\right)\right) \]

       3. *-commutative 71.3%

          \[\leadsto \left(2 \cdot \left(b - a\right)\right) \cdot \left(\left(\sin \left(\pi \cdot \left(0.005555555555555556 \cdot angle\right)\right) \cdot \left(\sqrt{\sqrt[3]{{\cos \left(\pi \cdot \color{blue}{\left(angle \cdot 0.005555555555555556\right)}\right)}^{4}}} \cdot \left|\sqrt[3]{\cos \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}\right|\right)\right) \cdot \left(a + b\right)\right) \]

   14. Simplified 71.3%

       \[\leadsto \left(2 \cdot \left(b - a\right)\right) \cdot \left(\left(\sin \left(\pi \cdot \left(0.005555555555555556 \cdot angle\right)\right) \cdot \left(\sqrt{\sqrt[3]{{\color{blue}{\cos \left(\pi \cdot \left(angle \cdot 0.005555555555555556\right)\right)}}^{4}}} \cdot \left|\sqrt[3]{\cos \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}\right|\right)\right) \cdot \left(a + b\right)\right) \]

   15. Taylor expanded in angle around 0 71.0%

       \[\leadsto \left(2 \cdot \left(b - a\right)\right) \cdot \left(\left(\sin \left(\pi \cdot \left(0.005555555555555556 \cdot angle\right)\right) \cdot \left(\sqrt{\sqrt[3]{{\cos \left(\pi \cdot \left(angle \cdot 0.005555555555555556\right)\right)}^{4}}} \cdot \left|\sqrt[3]{\color{blue}{1}}\right|\right)\right) \cdot \left(a + b\right)\right) \]
3. Recombined 2 regimes into one program.

4. Final simplification 72.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -9.8 \cdot 10^{+214}:\\ \;\;\;\;\left(2 \cdot \left(b - a\right)\right) \cdot \left(\left(\sin \left(\pi \cdot \left(0.005555555555555556 \cdot angle\right)\right) \cdot \cos \left({\left(\sqrt[3]{\pi \cdot \left(0.005555555555555556 \cdot angle\right)}\right)}^{3}\right)\right) \cdot \left(b + a\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(2 \cdot \left(b - a\right)\right) \cdot \left(\left(b + a\right) \cdot \left(\sin \left(\pi \cdot \left(0.005555555555555556 \cdot angle\right)\right) \cdot \sqrt{\sqrt[3]{{\cos \left(\pi \cdot \left(0.005555555555555556 \cdot angle\right)\right)}^{4}}}\right)\right)\\ \end{array} \]

Alternative 6: 65.8% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := 2 \cdot \left(b - a\right)\\ t_1 := \pi \cdot \left(0.005555555555555556 \cdot angle\right)\\ t_2 := \sin t_1\\ \mathbf{if}\;b \leq -5 \cdot 10^{+218}:\\ \;\;\;\;t_0 \cdot \left(\left(t_2 \cdot \cos \left({\left(\sqrt[3]{t_1}\right)}^{3}\right)\right) \cdot \left(b + a\right)\right)\\ \mathbf{else}:\\ \;\;\;\;t_0 \cdot \left(t_2 \cdot \left(b + a\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (a b angle)
 :precision binary64
 (let* ((t_0 (* 2.0 (- b a)))
        (t_1 (* PI (* 0.005555555555555556 angle)))
        (t_2 (sin t_1)))
   (if (<= b -5e+218)
     (* t_0 (* (* t_2 (cos (pow (cbrt t_1) 3.0))) (+ b a)))
     (* t_0 (* t_2 (+ b a))))))

C

double code(double a, double b, double angle) {
	double t_0 = 2.0 * (b - a);
	double t_1 = ((double) M_PI) * (0.005555555555555556 * angle);
	double t_2 = sin(t_1);
	double tmp;
	if (b <= -5e+218) {
		tmp = t_0 * ((t_2 * cos(pow(cbrt(t_1), 3.0))) * (b + a));
	} else {
		tmp = t_0 * (t_2 * (b + a));
	}
	return tmp;
}

Java

public static double code(double a, double b, double angle) {
	double t_0 = 2.0 * (b - a);
	double t_1 = Math.PI * (0.005555555555555556 * angle);
	double t_2 = Math.sin(t_1);
	double tmp;
	if (b <= -5e+218) {
		tmp = t_0 * ((t_2 * Math.cos(Math.pow(Math.cbrt(t_1), 3.0))) * (b + a));
	} else {
		tmp = t_0 * (t_2 * (b + a));
	}
	return tmp;
}

Julia

function code(a, b, angle)
	t_0 = Float64(2.0 * Float64(b - a))
	t_1 = Float64(pi * Float64(0.005555555555555556 * angle))
	t_2 = sin(t_1)
	tmp = 0.0
	if (b <= -5e+218)
		tmp = Float64(t_0 * Float64(Float64(t_2 * cos((cbrt(t_1) ^ 3.0))) * Float64(b + a)));
	else
		tmp = Float64(t_0 * Float64(t_2 * Float64(b + a)));
	end
	return tmp
end

Wolfram

code[a_, b_, angle_] := Block[{t$95$0 = N[(2.0 * N[(b - a), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(Pi * N[(0.005555555555555556 * angle), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[Sin[t$95$1], $MachinePrecision]}, If[LessEqual[b, -5e+218], N[(t$95$0 * N[(N[(t$95$2 * N[Cos[N[Power[N[Power[t$95$1, 1/3], $MachinePrecision], 3.0], $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[(b + a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(t$95$0 * N[(t$95$2 * N[(b + a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 2 regimes

2. if b < -4.99999999999999983e218

   1. Initial program 44.8%

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

      1. associate-*l* 44.8%

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

      2. unpow2 44.8%

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

      3. unpow2 44.8%

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

      4. difference-of-squares 57.3%

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

   3. Simplified 57.3%

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

   4. Taylor expanded in angle around inf 68.7%

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

      1. associate-*r* 68.7%

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

      2. associate-*r* 68.7%

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

      3. *-commutative 68.7%

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

      4. associate-*r* 62.4%

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

      5. *-commutative 62.4%

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

      6. *-commutative 62.4%

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

      7. *-commutative 62.4%

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

      8. associate-*r* 74.9%

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

      9. *-commutative 74.9%

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

      10. *-commutative 74.9%

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

      11. *-commutative 74.9%

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

   6. Simplified 74.9%

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

      1. *-commutative 74.9%

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

      2. add-cube-cbrt 93.7%

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

      3. pow3 99.9%

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

      4. *-commutative 99.9%

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

   8. Applied egg-rr 99.9%

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

   if -4.99999999999999983e218 < b

   1. Initial program 52.9%

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

      1. associate-*l* 52.9%

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

      2. unpow2 52.9%

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

      3. unpow2 52.9%

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

      4. difference-of-squares 57.2%

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

   3. Simplified 57.2%

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

   4. Taylor expanded in angle around inf 65.8%

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

      1. associate-*r* 65.8%

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

      2. associate-*r* 65.7%

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

      3. *-commutative 65.7%

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

      4. associate-*r* 67.2%

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

      5. *-commutative 67.2%

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

      6. *-commutative 67.2%

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

      7. *-commutative 67.2%

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

      8. associate-*r* 67.0%

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

      9. *-commutative 67.0%

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

      10. *-commutative 67.0%

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

      11. *-commutative 67.0%

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

   6. Simplified 67.0%

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

   7. Taylor expanded in angle around 0 70.8%

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

4. Final simplification 72.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -5 \cdot 10^{+218}:\\ \;\;\;\;\left(2 \cdot \left(b - a\right)\right) \cdot \left(\left(\sin \left(\pi \cdot \left(0.005555555555555556 \cdot angle\right)\right) \cdot \cos \left({\left(\sqrt[3]{\pi \cdot \left(0.005555555555555556 \cdot angle\right)}\right)}^{3}\right)\right) \cdot \left(b + a\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(2 \cdot \left(b - a\right)\right) \cdot \left(\sin \left(\pi \cdot \left(0.005555555555555556 \cdot angle\right)\right) \cdot \left(b + a\right)\right)\\ \end{array} \]

Alternative 7: 64.9% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq -8 \cdot 10^{+213}:\\ \;\;\;\;\left(2 \cdot \left(\left(b - a\right) \cdot \left(b + a\right)\right)\right) \cdot \left(\sin \left(\pi \cdot \frac{angle}{180}\right) \cdot \left(1 + -1.54320987654321 \cdot 10^{-5} \cdot \left(\left(angle \cdot angle\right) \cdot {\pi}^{2}\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(2 \cdot \left(b - a\right)\right) \cdot \left(\sin \left(\pi \cdot \left(0.005555555555555556 \cdot angle\right)\right) \cdot \left(b + a\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (a b angle)
 :precision binary64
 (if (<= b -8e+213)
   (*
    (* 2.0 (* (- b a) (+ b a)))
    (*
     (sin (* PI (/ angle 180.0)))
     (+ 1.0 (* -1.54320987654321e-5 (* (* angle angle) (pow PI 2.0))))))
   (*
    (* 2.0 (- b a))
    (* (sin (* PI (* 0.005555555555555556 angle))) (+ b a)))))

C

double code(double a, double b, double angle) {
	double tmp;
	if (b <= -8e+213) {
		tmp = (2.0 * ((b - a) * (b + a))) * (sin((((double) M_PI) * (angle / 180.0))) * (1.0 + (-1.54320987654321e-5 * ((angle * angle) * pow(((double) M_PI), 2.0)))));
	} else {
		tmp = (2.0 * (b - a)) * (sin((((double) M_PI) * (0.005555555555555556 * angle))) * (b + a));
	}
	return tmp;
}

Java

public static double code(double a, double b, double angle) {
	double tmp;
	if (b <= -8e+213) {
		tmp = (2.0 * ((b - a) * (b + a))) * (Math.sin((Math.PI * (angle / 180.0))) * (1.0 + (-1.54320987654321e-5 * ((angle * angle) * Math.pow(Math.PI, 2.0)))));
	} else {
		tmp = (2.0 * (b - a)) * (Math.sin((Math.PI * (0.005555555555555556 * angle))) * (b + a));
	}
	return tmp;
}

Python

def code(a, b, angle):
	tmp = 0
	if b <= -8e+213:
		tmp = (2.0 * ((b - a) * (b + a))) * (math.sin((math.pi * (angle / 180.0))) * (1.0 + (-1.54320987654321e-5 * ((angle * angle) * math.pow(math.pi, 2.0)))))
	else:
		tmp = (2.0 * (b - a)) * (math.sin((math.pi * (0.005555555555555556 * angle))) * (b + a))
	return tmp

Julia

function code(a, b, angle)
	tmp = 0.0
	if (b <= -8e+213)
		tmp = Float64(Float64(2.0 * Float64(Float64(b - a) * Float64(b + a))) * Float64(sin(Float64(pi * Float64(angle / 180.0))) * Float64(1.0 + Float64(-1.54320987654321e-5 * Float64(Float64(angle * angle) * (pi ^ 2.0))))));
	else
		tmp = Float64(Float64(2.0 * Float64(b - a)) * Float64(sin(Float64(pi * Float64(0.005555555555555556 * angle))) * Float64(b + a)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, angle)
	tmp = 0.0;
	if (b <= -8e+213)
		tmp = (2.0 * ((b - a) * (b + a))) * (sin((pi * (angle / 180.0))) * (1.0 + (-1.54320987654321e-5 * ((angle * angle) * (pi ^ 2.0)))));
	else
		tmp = (2.0 * (b - a)) * (sin((pi * (0.005555555555555556 * angle))) * (b + a));
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, angle_] := If[LessEqual[b, -8e+213], N[(N[(2.0 * N[(N[(b - a), $MachinePrecision] * N[(b + a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(N[Sin[N[(Pi * N[(angle / 180.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * N[(1.0 + N[(-1.54320987654321e-5 * N[(N[(angle * angle), $MachinePrecision] * N[Power[Pi, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(2.0 * N[(b - a), $MachinePrecision]), $MachinePrecision] * N[(N[Sin[N[(Pi * N[(0.005555555555555556 * angle), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * N[(b + a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b < -7.99999999999999987e213

   1. Initial program 44.8%

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

      1. associate-*l* 44.8%

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

      2. unpow2 44.8%

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

      3. unpow2 44.8%

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

      4. difference-of-squares 57.3%

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

   3. Simplified 57.3%

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

   4. Taylor expanded in angle around 0 88.5%

      \[\leadsto \left(2 \cdot \left(\left(b + a\right) \cdot \left(b - a\right)\right)\right) \cdot \left(\sin \left(\pi \cdot \frac{angle}{180}\right) \cdot \color{blue}{\left(1 + -1.54320987654321 \cdot 10^{-5} \cdot \left({angle}^{2} \cdot {\pi}^{2}\right)\right)}\right) \]
   5. Step-by-step derivation

      1. unpow2 88.5%

         \[\leadsto \left(2 \cdot \left(\left(b + a\right) \cdot \left(b - a\right)\right)\right) \cdot \left(\sin \left(\pi \cdot \frac{angle}{180}\right) \cdot \left(1 + -1.54320987654321 \cdot 10^{-5} \cdot \left(\color{blue}{\left(angle \cdot angle\right)} \cdot {\pi}^{2}\right)\right)\right) \]

   6. Simplified 88.5%

      \[\leadsto \left(2 \cdot \left(\left(b + a\right) \cdot \left(b - a\right)\right)\right) \cdot \left(\sin \left(\pi \cdot \frac{angle}{180}\right) \cdot \color{blue}{\left(1 + -1.54320987654321 \cdot 10^{-5} \cdot \left(\left(angle \cdot angle\right) \cdot {\pi}^{2}\right)\right)}\right) \]

   if -7.99999999999999987e213 < b

   1. Initial program 52.9%

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

      1. associate-*l* 52.9%

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

      2. unpow2 52.9%

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

      3. unpow2 52.9%

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

      4. difference-of-squares 57.2%

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

   3. Simplified 57.2%

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

   4. Taylor expanded in angle around inf 65.8%

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

      1. associate-*r* 65.8%

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

      2. associate-*r* 65.7%

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

      3. *-commutative 65.7%

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

      4. associate-*r* 67.2%

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

      5. *-commutative 67.2%

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

      6. *-commutative 67.2%

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

      7. *-commutative 67.2%

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

      8. associate-*r* 67.0%

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

      9. *-commutative 67.0%

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

      10. *-commutative 67.0%

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

      11. *-commutative 67.0%

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

   6. Simplified 67.0%

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

   7. Taylor expanded in angle around 0 70.8%

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

4. Final simplification 71.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -8 \cdot 10^{+213}:\\ \;\;\;\;\left(2 \cdot \left(\left(b - a\right) \cdot \left(b + a\right)\right)\right) \cdot \left(\sin \left(\pi \cdot \frac{angle}{180}\right) \cdot \left(1 + -1.54320987654321 \cdot 10^{-5} \cdot \left(\left(angle \cdot angle\right) \cdot {\pi}^{2}\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(2 \cdot \left(b - a\right)\right) \cdot \left(\sin \left(\pi \cdot \left(0.005555555555555556 \cdot angle\right)\right) \cdot \left(b + a\right)\right)\\ \end{array} \]

Alternative 8: 65.4% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;{b}^{2} \leq 4 \cdot 10^{+200}:\\ \;\;\;\;\left(2 \cdot \left(b - a\right)\right) \cdot \left(\sin \left(\pi \cdot \left(0.005555555555555556 \cdot angle\right)\right) \cdot \left(b + a\right)\right)\\ \mathbf{else}:\\ \;\;\;\;0.011111111111111112 \cdot \left(\left(b + a\right) \cdot \left(\pi \cdot \left(\left(b - a\right) \cdot angle\right)\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (a b angle)
 :precision binary64
 (if (<= (pow b 2.0) 4e+200)
   (* (* 2.0 (- b a)) (* (sin (* PI (* 0.005555555555555556 angle))) (+ b a)))
   (* 0.011111111111111112 (* (+ b a) (* PI (* (- b a) angle))))))

C

double code(double a, double b, double angle) {
	double tmp;
	if (pow(b, 2.0) <= 4e+200) {
		tmp = (2.0 * (b - a)) * (sin((((double) M_PI) * (0.005555555555555556 * angle))) * (b + a));
	} else {
		tmp = 0.011111111111111112 * ((b + a) * (((double) M_PI) * ((b - a) * angle)));
	}
	return tmp;
}

Java

public static double code(double a, double b, double angle) {
	double tmp;
	if (Math.pow(b, 2.0) <= 4e+200) {
		tmp = (2.0 * (b - a)) * (Math.sin((Math.PI * (0.005555555555555556 * angle))) * (b + a));
	} else {
		tmp = 0.011111111111111112 * ((b + a) * (Math.PI * ((b - a) * angle)));
	}
	return tmp;
}

Python

def code(a, b, angle):
	tmp = 0
	if math.pow(b, 2.0) <= 4e+200:
		tmp = (2.0 * (b - a)) * (math.sin((math.pi * (0.005555555555555556 * angle))) * (b + a))
	else:
		tmp = 0.011111111111111112 * ((b + a) * (math.pi * ((b - a) * angle)))
	return tmp

Julia

function code(a, b, angle)
	tmp = 0.0
	if ((b ^ 2.0) <= 4e+200)
		tmp = Float64(Float64(2.0 * Float64(b - a)) * Float64(sin(Float64(pi * Float64(0.005555555555555556 * angle))) * Float64(b + a)));
	else
		tmp = Float64(0.011111111111111112 * Float64(Float64(b + a) * Float64(pi * Float64(Float64(b - a) * angle))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, angle)
	tmp = 0.0;
	if ((b ^ 2.0) <= 4e+200)
		tmp = (2.0 * (b - a)) * (sin((pi * (0.005555555555555556 * angle))) * (b + a));
	else
		tmp = 0.011111111111111112 * ((b + a) * (pi * ((b - a) * angle)));
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, angle_] := If[LessEqual[N[Power[b, 2.0], $MachinePrecision], 4e+200], N[(N[(2.0 * N[(b - a), $MachinePrecision]), $MachinePrecision] * N[(N[Sin[N[(Pi * N[(0.005555555555555556 * angle), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * N[(b + a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(0.011111111111111112 * N[(N[(b + a), $MachinePrecision] * N[(Pi * N[(N[(b - a), $MachinePrecision] * angle), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (pow.f64 b 2) < 3.9999999999999999e200

   1. Initial program 60.5%

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

      1. associate-*l* 60.5%

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

      2. unpow2 60.5%

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

      3. unpow2 60.5%

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

      4. difference-of-squares 60.5%

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

   3. Simplified 60.5%

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

   4. Taylor expanded in angle around inf 62.1%

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

      1. associate-*r* 62.1%

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

      2. associate-*r* 62.1%

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

      3. *-commutative 62.1%

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

      4. associate-*r* 64.5%

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

      5. *-commutative 64.5%

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

      6. *-commutative 64.5%

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

      7. *-commutative 64.5%

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

      8. associate-*r* 63.6%

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

      9. *-commutative 63.6%

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

      10. *-commutative 63.6%

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

      11. *-commutative 63.6%

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

   6. Simplified 63.6%

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

   7. Taylor expanded in angle around 0 65.9%

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

   if 3.9999999999999999e200 < (pow.f64 b 2)

   1. Initial program 37.4%

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

      1. associate-*l* 37.4%

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

      2. unpow2 37.4%

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

      3. unpow2 37.4%

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

      4. difference-of-squares 51.0%

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

   3. Simplified 51.0%

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

   4. Taylor expanded in angle around 0 58.7%

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

   5. Taylor expanded in angle around 0 58.7%

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

      1. associate-*r* 81.4%

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

      2. *-commutative 81.4%

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

      3. associate-*r* 81.4%

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

      4. +-commutative 81.4%

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

   7. Simplified 81.4%

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

4. Final simplification 71.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;{b}^{2} \leq 4 \cdot 10^{+200}:\\ \;\;\;\;\left(2 \cdot \left(b - a\right)\right) \cdot \left(\sin \left(\pi \cdot \left(0.005555555555555556 \cdot angle\right)\right) \cdot \left(b + a\right)\right)\\ \mathbf{else}:\\ \;\;\;\;0.011111111111111112 \cdot \left(\left(b + a\right) \cdot \left(\pi \cdot \left(\left(b - a\right) \cdot angle\right)\right)\right)\\ \end{array} \]

Alternative 9: 62.3% accurate, 2.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(b - a\right) \cdot angle\\ \mathbf{if}\;b \leq -1.25 \cdot 10^{+101}:\\ \;\;\;\;0.011111111111111112 \cdot \left(t_0 \cdot \left(\pi \cdot \left(b + a\right)\right)\right)\\ \mathbf{elif}\;b \leq 4.4 \cdot 10^{-177}:\\ \;\;\;\;\left(2 \cdot \left(\left(b - a\right) \cdot \left(b + a\right)\right)\right) \cdot \sin \left(\pi \cdot \frac{angle}{180}\right)\\ \mathbf{else}:\\ \;\;\;\;0.011111111111111112 \cdot \left(\left(b + a\right) \cdot \left(\pi \cdot t_0\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (a b angle)
 :precision binary64
 (let* ((t_0 (* (- b a) angle)))
   (if (<= b -1.25e+101)
     (* 0.011111111111111112 (* t_0 (* PI (+ b a))))
     (if (<= b 4.4e-177)
       (* (* 2.0 (* (- b a) (+ b a))) (sin (* PI (/ angle 180.0))))
       (* 0.011111111111111112 (* (+ b a) (* PI t_0)))))))

C

double code(double a, double b, double angle) {
	double t_0 = (b - a) * angle;
	double tmp;
	if (b <= -1.25e+101) {
		tmp = 0.011111111111111112 * (t_0 * (((double) M_PI) * (b + a)));
	} else if (b <= 4.4e-177) {
		tmp = (2.0 * ((b - a) * (b + a))) * sin((((double) M_PI) * (angle / 180.0)));
	} else {
		tmp = 0.011111111111111112 * ((b + a) * (((double) M_PI) * t_0));
	}
	return tmp;
}

Java

public static double code(double a, double b, double angle) {
	double t_0 = (b - a) * angle;
	double tmp;
	if (b <= -1.25e+101) {
		tmp = 0.011111111111111112 * (t_0 * (Math.PI * (b + a)));
	} else if (b <= 4.4e-177) {
		tmp = (2.0 * ((b - a) * (b + a))) * Math.sin((Math.PI * (angle / 180.0)));
	} else {
		tmp = 0.011111111111111112 * ((b + a) * (Math.PI * t_0));
	}
	return tmp;
}

Python

def code(a, b, angle):
	t_0 = (b - a) * angle
	tmp = 0
	if b <= -1.25e+101:
		tmp = 0.011111111111111112 * (t_0 * (math.pi * (b + a)))
	elif b <= 4.4e-177:
		tmp = (2.0 * ((b - a) * (b + a))) * math.sin((math.pi * (angle / 180.0)))
	else:
		tmp = 0.011111111111111112 * ((b + a) * (math.pi * t_0))
	return tmp

Julia

function code(a, b, angle)
	t_0 = Float64(Float64(b - a) * angle)
	tmp = 0.0
	if (b <= -1.25e+101)
		tmp = Float64(0.011111111111111112 * Float64(t_0 * Float64(pi * Float64(b + a))));
	elseif (b <= 4.4e-177)
		tmp = Float64(Float64(2.0 * Float64(Float64(b - a) * Float64(b + a))) * sin(Float64(pi * Float64(angle / 180.0))));
	else
		tmp = Float64(0.011111111111111112 * Float64(Float64(b + a) * Float64(pi * t_0)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, angle)
	t_0 = (b - a) * angle;
	tmp = 0.0;
	if (b <= -1.25e+101)
		tmp = 0.011111111111111112 * (t_0 * (pi * (b + a)));
	elseif (b <= 4.4e-177)
		tmp = (2.0 * ((b - a) * (b + a))) * sin((pi * (angle / 180.0)));
	else
		tmp = 0.011111111111111112 * ((b + a) * (pi * t_0));
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, angle_] := Block[{t$95$0 = N[(N[(b - a), $MachinePrecision] * angle), $MachinePrecision]}, If[LessEqual[b, -1.25e+101], N[(0.011111111111111112 * N[(t$95$0 * N[(Pi * N[(b + a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[b, 4.4e-177], N[(N[(2.0 * N[(N[(b - a), $MachinePrecision] * N[(b + a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[Sin[N[(Pi * N[(angle / 180.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(0.011111111111111112 * N[(N[(b + a), $MachinePrecision] * N[(Pi * t$95$0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if b < -1.24999999999999997e101

   1. Initial program 37.0%

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

      1. associate-*l* 37.0%

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

      2. unpow2 37.0%

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

      3. unpow2 37.0%

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

      4. difference-of-squares 52.2%

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

   3. Simplified 52.2%

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

   4. Taylor expanded in angle around 0 57.4%

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

      1. *-commutative 57.4%

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

      2. associate-*r* 83.1%

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

   6. Simplified 83.1%

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

   if -1.24999999999999997e101 < b < 4.40000000000000023e-177

   1. Initial program 64.3%

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

      1. associate-*l* 64.3%

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

      2. unpow2 64.3%

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

      3. unpow2 64.3%

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

      4. difference-of-squares 64.3%

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

   3. Simplified 64.3%

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

   4. Taylor expanded in angle around 0 67.0%

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

   if 4.40000000000000023e-177 < b

   1. Initial program 45.0%

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

      1. associate-*l* 45.0%

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

      2. unpow2 45.0%

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

      3. unpow2 45.0%

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

      4. difference-of-squares 51.1%

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

   3. Simplified 51.1%

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

   4. Taylor expanded in angle around 0 52.9%

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

   5. Taylor expanded in angle around 0 52.9%

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

      1. associate-*r* 66.6%

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

      2. *-commutative 66.6%

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

      3. associate-*r* 66.7%

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

      4. +-commutative 66.7%

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

   7. Simplified 66.7%

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

4. Final simplification 69.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -1.25 \cdot 10^{+101}:\\ \;\;\;\;0.011111111111111112 \cdot \left(\left(\left(b - a\right) \cdot angle\right) \cdot \left(\pi \cdot \left(b + a\right)\right)\right)\\ \mathbf{elif}\;b \leq 4.4 \cdot 10^{-177}:\\ \;\;\;\;\left(2 \cdot \left(\left(b - a\right) \cdot \left(b + a\right)\right)\right) \cdot \sin \left(\pi \cdot \frac{angle}{180}\right)\\ \mathbf{else}:\\ \;\;\;\;0.011111111111111112 \cdot \left(\left(b + a\right) \cdot \left(\pi \cdot \left(\left(b - a\right) \cdot angle\right)\right)\right)\\ \end{array} \]

Alternative 10: 63.0% accurate, 2.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;angle \leq -8.8 \cdot 10^{-128}:\\ \;\;\;\;\left(\left(b - a\right) \cdot \left(\pi \cdot \left(b + a\right)\right)\right) \cdot \left(angle \cdot 0.011111111111111112\right)\\ \mathbf{elif}\;angle \leq 48000000000000:\\ \;\;\;\;0.011111111111111112 \cdot \left(\left(b + a\right) \cdot \left(\pi \cdot \left(\left(b - a\right) \cdot angle\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;0.011111111111111112 \cdot \left(angle \cdot \left(\pi \cdot {\left(b + a\right)}^{2}\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (a b angle)
 :precision binary64
 (if (<= angle -8.8e-128)
   (* (* (- b a) (* PI (+ b a))) (* angle 0.011111111111111112))
   (if (<= angle 48000000000000.0)
     (* 0.011111111111111112 (* (+ b a) (* PI (* (- b a) angle))))
     (* 0.011111111111111112 (* angle (* PI (pow (+ b a) 2.0)))))))

C

double code(double a, double b, double angle) {
	double tmp;
	if (angle <= -8.8e-128) {
		tmp = ((b - a) * (((double) M_PI) * (b + a))) * (angle * 0.011111111111111112);
	} else if (angle <= 48000000000000.0) {
		tmp = 0.011111111111111112 * ((b + a) * (((double) M_PI) * ((b - a) * angle)));
	} else {
		tmp = 0.011111111111111112 * (angle * (((double) M_PI) * pow((b + a), 2.0)));
	}
	return tmp;
}

Java

public static double code(double a, double b, double angle) {
	double tmp;
	if (angle <= -8.8e-128) {
		tmp = ((b - a) * (Math.PI * (b + a))) * (angle * 0.011111111111111112);
	} else if (angle <= 48000000000000.0) {
		tmp = 0.011111111111111112 * ((b + a) * (Math.PI * ((b - a) * angle)));
	} else {
		tmp = 0.011111111111111112 * (angle * (Math.PI * Math.pow((b + a), 2.0)));
	}
	return tmp;
}

Python

def code(a, b, angle):
	tmp = 0
	if angle <= -8.8e-128:
		tmp = ((b - a) * (math.pi * (b + a))) * (angle * 0.011111111111111112)
	elif angle <= 48000000000000.0:
		tmp = 0.011111111111111112 * ((b + a) * (math.pi * ((b - a) * angle)))
	else:
		tmp = 0.011111111111111112 * (angle * (math.pi * math.pow((b + a), 2.0)))
	return tmp

Julia

function code(a, b, angle)
	tmp = 0.0
	if (angle <= -8.8e-128)
		tmp = Float64(Float64(Float64(b - a) * Float64(pi * Float64(b + a))) * Float64(angle * 0.011111111111111112));
	elseif (angle <= 48000000000000.0)
		tmp = Float64(0.011111111111111112 * Float64(Float64(b + a) * Float64(pi * Float64(Float64(b - a) * angle))));
	else
		tmp = Float64(0.011111111111111112 * Float64(angle * Float64(pi * (Float64(b + a) ^ 2.0))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, angle)
	tmp = 0.0;
	if (angle <= -8.8e-128)
		tmp = ((b - a) * (pi * (b + a))) * (angle * 0.011111111111111112);
	elseif (angle <= 48000000000000.0)
		tmp = 0.011111111111111112 * ((b + a) * (pi * ((b - a) * angle)));
	else
		tmp = 0.011111111111111112 * (angle * (pi * ((b + a) ^ 2.0)));
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, angle_] := If[LessEqual[angle, -8.8e-128], N[(N[(N[(b - a), $MachinePrecision] * N[(Pi * N[(b + a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(angle * 0.011111111111111112), $MachinePrecision]), $MachinePrecision], If[LessEqual[angle, 48000000000000.0], N[(0.011111111111111112 * N[(N[(b + a), $MachinePrecision] * N[(Pi * N[(N[(b - a), $MachinePrecision] * angle), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(0.011111111111111112 * N[(angle * N[(Pi * N[Power[N[(b + a), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if angle < -8.80000000000000037e-128

   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. associate-*l* 58.3%

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

      2. unpow2 58.3%

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

      3. unpow2 58.3%

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

      4. difference-of-squares 62.4%

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

   3. Simplified 62.4%

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

   4. Taylor expanded in angle around 0 59.4%

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

      1. associate-*r* 59.4%

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

      2. *-commutative 59.4%

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

      3. *-commutative 59.4%

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

   6. Simplified 59.4%

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

   if -8.80000000000000037e-128 < angle < 4.8e13

   1. Initial program 65.7%

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

      1. associate-*l* 65.7%

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

      2. unpow2 65.7%

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

      3. unpow2 65.7%

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

      4. difference-of-squares 72.3%

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

   3. Simplified 72.3%

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

   4. Taylor expanded in angle around 0 72.0%

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

   5. Taylor expanded in angle around 0 72.0%

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

      1. associate-*r* 99.2%

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

      2. *-commutative 99.2%

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

      3. associate-*r* 99.3%

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

      4. +-commutative 99.3%

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

   7. Simplified 99.3%

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

   if 4.8e13 < angle

   1. Initial program 23.1%

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

      1. *-commutative 23.1%

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

      2. associate-*l* 23.1%

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

      3. unpow2 23.1%

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

      4. fma-neg 27.8%

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

      5. unpow2 27.8%

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

   3. Simplified 27.8%

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

   5. Applied egg-rr 21.4%

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

      1. expm1-def 22.7%

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

      2. expm1-log1p 30.1%

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

      3. associate-*l* 30.1%

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

      4. sin-0 30.1%

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

      5. +-lft-identity 30.1%

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

      6. associate-*l* 30.1%

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

      7. *-commutative 30.1%

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

      8. *-commutative 30.1%

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

      9. +-commutative 30.1%

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

   7. Simplified 30.1%

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

   8. Taylor expanded in angle around 0 35.5%

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

4. Final simplification 68.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;angle \leq -8.8 \cdot 10^{-128}:\\ \;\;\;\;\left(\left(b - a\right) \cdot \left(\pi \cdot \left(b + a\right)\right)\right) \cdot \left(angle \cdot 0.011111111111111112\right)\\ \mathbf{elif}\;angle \leq 48000000000000:\\ \;\;\;\;0.011111111111111112 \cdot \left(\left(b + a\right) \cdot \left(\pi \cdot \left(\left(b - a\right) \cdot angle\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;0.011111111111111112 \cdot \left(angle \cdot \left(\pi \cdot {\left(b + a\right)}^{2}\right)\right)\\ \end{array} \]

Alternative 11: 64.6% accurate, 2.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(b - a\right) \cdot angle\\ \mathbf{if}\;a \leq -1 \cdot 10^{-55}:\\ \;\;\;\;0.011111111111111112 \cdot \left(\left(b + a\right) \cdot \left(\pi \cdot t_0\right)\right)\\ \mathbf{elif}\;a \leq 1.36 \cdot 10^{-160}:\\ \;\;\;\;b \cdot \left(b \cdot \sin \left(angle \cdot \left(\pi \cdot 0.011111111111111112\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;0.011111111111111112 \cdot \left(t_0 \cdot \left(\pi \cdot \left(b + a\right)\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (a b angle)
 :precision binary64
 (let* ((t_0 (* (- b a) angle)))
   (if (<= a -1e-55)
     (* 0.011111111111111112 (* (+ b a) (* PI t_0)))
     (if (<= a 1.36e-160)
       (* b (* b (sin (* angle (* PI 0.011111111111111112)))))
       (* 0.011111111111111112 (* t_0 (* PI (+ b a))))))))

C

double code(double a, double b, double angle) {
	double t_0 = (b - a) * angle;
	double tmp;
	if (a <= -1e-55) {
		tmp = 0.011111111111111112 * ((b + a) * (((double) M_PI) * t_0));
	} else if (a <= 1.36e-160) {
		tmp = b * (b * sin((angle * (((double) M_PI) * 0.011111111111111112))));
	} else {
		tmp = 0.011111111111111112 * (t_0 * (((double) M_PI) * (b + a)));
	}
	return tmp;
}

Java

public static double code(double a, double b, double angle) {
	double t_0 = (b - a) * angle;
	double tmp;
	if (a <= -1e-55) {
		tmp = 0.011111111111111112 * ((b + a) * (Math.PI * t_0));
	} else if (a <= 1.36e-160) {
		tmp = b * (b * Math.sin((angle * (Math.PI * 0.011111111111111112))));
	} else {
		tmp = 0.011111111111111112 * (t_0 * (Math.PI * (b + a)));
	}
	return tmp;
}

Python

def code(a, b, angle):
	t_0 = (b - a) * angle
	tmp = 0
	if a <= -1e-55:
		tmp = 0.011111111111111112 * ((b + a) * (math.pi * t_0))
	elif a <= 1.36e-160:
		tmp = b * (b * math.sin((angle * (math.pi * 0.011111111111111112))))
	else:
		tmp = 0.011111111111111112 * (t_0 * (math.pi * (b + a)))
	return tmp

Julia

function code(a, b, angle)
	t_0 = Float64(Float64(b - a) * angle)
	tmp = 0.0
	if (a <= -1e-55)
		tmp = Float64(0.011111111111111112 * Float64(Float64(b + a) * Float64(pi * t_0)));
	elseif (a <= 1.36e-160)
		tmp = Float64(b * Float64(b * sin(Float64(angle * Float64(pi * 0.011111111111111112)))));
	else
		tmp = Float64(0.011111111111111112 * Float64(t_0 * Float64(pi * Float64(b + a))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, angle)
	t_0 = (b - a) * angle;
	tmp = 0.0;
	if (a <= -1e-55)
		tmp = 0.011111111111111112 * ((b + a) * (pi * t_0));
	elseif (a <= 1.36e-160)
		tmp = b * (b * sin((angle * (pi * 0.011111111111111112))));
	else
		tmp = 0.011111111111111112 * (t_0 * (pi * (b + a)));
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, angle_] := Block[{t$95$0 = N[(N[(b - a), $MachinePrecision] * angle), $MachinePrecision]}, If[LessEqual[a, -1e-55], N[(0.011111111111111112 * N[(N[(b + a), $MachinePrecision] * N[(Pi * t$95$0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[a, 1.36e-160], N[(b * N[(b * N[Sin[N[(angle * N[(Pi * 0.011111111111111112), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(0.011111111111111112 * N[(t$95$0 * N[(Pi * N[(b + a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if a < -9.99999999999999995e-56

   1. Initial program 47.6%

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

      1. associate-*l* 47.6%

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

      2. unpow2 47.6%

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

      3. unpow2 47.6%

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

      4. difference-of-squares 53.4%

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

   3. Simplified 53.4%

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

   4. Taylor expanded in angle around 0 49.0%

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

   5. Taylor expanded in angle around 0 49.0%

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

      1. associate-*r* 61.0%

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

      2. *-commutative 61.0%

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

      3. associate-*r* 61.1%

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

      4. +-commutative 61.1%

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

   7. Simplified 61.1%

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

   if -9.99999999999999995e-56 < a < 1.35999999999999993e-160

   1. Initial program 66.3%

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

      1. *-commutative 66.3%

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

      2. associate-*l* 66.3%

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

      3. unpow2 66.3%

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

      4. fma-neg 66.3%

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

      5. unpow2 66.3%

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

   3. Simplified 66.3%

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

   5. Applied egg-rr 42.1%

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

      1. expm1-def 54.3%

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

      2. expm1-log1p 66.1%

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

      3. associate-*l* 66.1%

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

      4. sin-0 66.1%

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

      5. +-lft-identity 66.1%

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

      6. associate-*l* 66.1%

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

      7. *-commutative 66.1%

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

      8. *-commutative 66.1%

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

      9. +-commutative 66.1%

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

   7. Simplified 66.1%

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

   8. Taylor expanded in a around 0 65.6%

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

      1. *-commutative 65.6%

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

      2. unpow2 65.6%

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

      3. associate-*l* 69.0%

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

      4. *-commutative 69.0%

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

      5. associate-*l* 70.1%

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

   10. Simplified 70.1%

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

   if 1.35999999999999993e-160 < a

   1. Initial program 44.9%

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

      1. associate-*l* 44.9%

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

      2. unpow2 44.9%

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

      3. unpow2 44.9%

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

      4. difference-of-squares 52.7%

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

   3. Simplified 52.7%

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

   4. Taylor expanded in angle around 0 57.6%

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

      1. *-commutative 57.6%

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

      2. associate-*r* 71.5%

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

   6. Simplified 71.5%

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

4. Final simplification 68.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \leq -1 \cdot 10^{-55}:\\ \;\;\;\;0.011111111111111112 \cdot \left(\left(b + a\right) \cdot \left(\pi \cdot \left(\left(b - a\right) \cdot angle\right)\right)\right)\\ \mathbf{elif}\;a \leq 1.36 \cdot 10^{-160}:\\ \;\;\;\;b \cdot \left(b \cdot \sin \left(angle \cdot \left(\pi \cdot 0.011111111111111112\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;0.011111111111111112 \cdot \left(\left(\left(b - a\right) \cdot angle\right) \cdot \left(\pi \cdot \left(b + a\right)\right)\right)\\ \end{array} \]

Alternative 12: 61.7% accurate, 5.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;a \leq -6 \cdot 10^{-84} \lor \neg \left(a \leq 1.35 \cdot 10^{-171}\right):\\ \;\;\;\;0.011111111111111112 \cdot \left(\left(b + a\right) \cdot \left(\pi \cdot \left(\left(b - a\right) \cdot angle\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;0.011111111111111112 \cdot \left(\pi \cdot \left(angle \cdot \left(b \cdot b\right)\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (a b angle)
 :precision binary64
 (if (or (<= a -6e-84) (not (<= a 1.35e-171)))
   (* 0.011111111111111112 (* (+ b a) (* PI (* (- b a) angle))))
   (* 0.011111111111111112 (* PI (* angle (* b b))))))

C

double code(double a, double b, double angle) {
	double tmp;
	if ((a <= -6e-84) || !(a <= 1.35e-171)) {
		tmp = 0.011111111111111112 * ((b + a) * (((double) M_PI) * ((b - a) * angle)));
	} else {
		tmp = 0.011111111111111112 * (((double) M_PI) * (angle * (b * b)));
	}
	return tmp;
}

Java

public static double code(double a, double b, double angle) {
	double tmp;
	if ((a <= -6e-84) || !(a <= 1.35e-171)) {
		tmp = 0.011111111111111112 * ((b + a) * (Math.PI * ((b - a) * angle)));
	} else {
		tmp = 0.011111111111111112 * (Math.PI * (angle * (b * b)));
	}
	return tmp;
}

Python

def code(a, b, angle):
	tmp = 0
	if (a <= -6e-84) or not (a <= 1.35e-171):
		tmp = 0.011111111111111112 * ((b + a) * (math.pi * ((b - a) * angle)))
	else:
		tmp = 0.011111111111111112 * (math.pi * (angle * (b * b)))
	return tmp

Julia

function code(a, b, angle)
	tmp = 0.0
	if ((a <= -6e-84) || !(a <= 1.35e-171))
		tmp = Float64(0.011111111111111112 * Float64(Float64(b + a) * Float64(pi * Float64(Float64(b - a) * angle))));
	else
		tmp = Float64(0.011111111111111112 * Float64(pi * Float64(angle * Float64(b * b))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, angle)
	tmp = 0.0;
	if ((a <= -6e-84) || ~((a <= 1.35e-171)))
		tmp = 0.011111111111111112 * ((b + a) * (pi * ((b - a) * angle)));
	else
		tmp = 0.011111111111111112 * (pi * (angle * (b * b)));
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, angle_] := If[Or[LessEqual[a, -6e-84], N[Not[LessEqual[a, 1.35e-171]], $MachinePrecision]], N[(0.011111111111111112 * N[(N[(b + a), $MachinePrecision] * N[(Pi * N[(N[(b - a), $MachinePrecision] * angle), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(0.011111111111111112 * N[(Pi * N[(angle * N[(b * b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if a < -6.0000000000000002e-84 or 1.35000000000000007e-171 < a

   1. Initial program 46.4%

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

      1. associate-*l* 46.4%

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

      2. unpow2 46.4%

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

      3. unpow2 46.4%

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

      4. difference-of-squares 52.9%

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

   3. Simplified 52.9%

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

   4. Taylor expanded in angle around 0 53.0%

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

   5. Taylor expanded in angle around 0 53.0%

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

      1. associate-*r* 66.3%

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

      2. *-commutative 66.3%

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

      3. associate-*r* 66.4%

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

      4. +-commutative 66.4%

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

   7. Simplified 66.4%

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

   if -6.0000000000000002e-84 < a < 1.35000000000000007e-171

   1. Initial program 68.7%

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

      1. associate-*l* 68.7%

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

      2. unpow2 68.7%

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

      3. unpow2 68.7%

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

      4. difference-of-squares 68.7%

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

   3. Simplified 68.7%

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

   4. Taylor expanded in angle around 0 63.6%

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

   5. Taylor expanded in b around inf 65.0%

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

      1. associate-*r* 65.0%

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

      2. *-commutative 65.0%

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

      3. unpow2 65.0%

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

   7. Simplified 65.0%

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

4. Final simplification 66.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \leq -6 \cdot 10^{-84} \lor \neg \left(a \leq 1.35 \cdot 10^{-171}\right):\\ \;\;\;\;0.011111111111111112 \cdot \left(\left(b + a\right) \cdot \left(\pi \cdot \left(\left(b - a\right) \cdot angle\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;0.011111111111111112 \cdot \left(\pi \cdot \left(angle \cdot \left(b \cdot b\right)\right)\right)\\ \end{array} \]

Alternative 13: 61.7% accurate, 5.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;a \leq -4.6 \cdot 10^{-84} \lor \neg \left(a \leq 5.5 \cdot 10^{-173}\right):\\ \;\;\;\;0.011111111111111112 \cdot \left(\left(b + a\right) \cdot \left(\pi \cdot \left(\left(b - a\right) \cdot angle\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(0.005555555555555556 \cdot \left(\pi \cdot angle\right)\right) \cdot \left(2 \cdot \left(b \cdot b\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (a b angle)
 :precision binary64
 (if (or (<= a -4.6e-84) (not (<= a 5.5e-173)))
   (* 0.011111111111111112 (* (+ b a) (* PI (* (- b a) angle))))
   (* (* 0.005555555555555556 (* PI angle)) (* 2.0 (* b b)))))

C

double code(double a, double b, double angle) {
	double tmp;
	if ((a <= -4.6e-84) || !(a <= 5.5e-173)) {
		tmp = 0.011111111111111112 * ((b + a) * (((double) M_PI) * ((b - a) * angle)));
	} else {
		tmp = (0.005555555555555556 * (((double) M_PI) * angle)) * (2.0 * (b * b));
	}
	return tmp;
}

Java

public static double code(double a, double b, double angle) {
	double tmp;
	if ((a <= -4.6e-84) || !(a <= 5.5e-173)) {
		tmp = 0.011111111111111112 * ((b + a) * (Math.PI * ((b - a) * angle)));
	} else {
		tmp = (0.005555555555555556 * (Math.PI * angle)) * (2.0 * (b * b));
	}
	return tmp;
}

Python

def code(a, b, angle):
	tmp = 0
	if (a <= -4.6e-84) or not (a <= 5.5e-173):
		tmp = 0.011111111111111112 * ((b + a) * (math.pi * ((b - a) * angle)))
	else:
		tmp = (0.005555555555555556 * (math.pi * angle)) * (2.0 * (b * b))
	return tmp

Julia

function code(a, b, angle)
	tmp = 0.0
	if ((a <= -4.6e-84) || !(a <= 5.5e-173))
		tmp = Float64(0.011111111111111112 * Float64(Float64(b + a) * Float64(pi * Float64(Float64(b - a) * angle))));
	else
		tmp = Float64(Float64(0.005555555555555556 * Float64(pi * angle)) * Float64(2.0 * Float64(b * b)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, angle)
	tmp = 0.0;
	if ((a <= -4.6e-84) || ~((a <= 5.5e-173)))
		tmp = 0.011111111111111112 * ((b + a) * (pi * ((b - a) * angle)));
	else
		tmp = (0.005555555555555556 * (pi * angle)) * (2.0 * (b * b));
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, angle_] := If[Or[LessEqual[a, -4.6e-84], N[Not[LessEqual[a, 5.5e-173]], $MachinePrecision]], N[(0.011111111111111112 * N[(N[(b + a), $MachinePrecision] * N[(Pi * N[(N[(b - a), $MachinePrecision] * angle), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(0.005555555555555556 * N[(Pi * angle), $MachinePrecision]), $MachinePrecision] * N[(2.0 * N[(b * b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if a < -4.59999999999999961e-84 or 5.50000000000000022e-173 < a

   1. Initial program 46.4%

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

      1. associate-*l* 46.4%

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

      2. unpow2 46.4%

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

      3. unpow2 46.4%

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

      4. difference-of-squares 52.9%

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

   3. Simplified 52.9%

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

   4. Taylor expanded in angle around 0 53.0%

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

   5. Taylor expanded in angle around 0 53.0%

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

      1. associate-*r* 66.3%

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

      2. *-commutative 66.3%

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

      3. associate-*r* 66.4%

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

      4. +-commutative 66.4%

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

   7. Simplified 66.4%

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

   if -4.59999999999999961e-84 < a < 5.50000000000000022e-173

   1. Initial program 68.7%

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

      1. *-commutative 68.7%

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

      2. associate-*l* 68.7%

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

      3. unpow2 68.7%

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

      4. fma-neg 68.7%

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

      5. unpow2 68.7%

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

      6. distribute-rgt-neg-in 68.7%

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

   3. Simplified 68.7%

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

   4. Taylor expanded in b around inf 68.5%

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

      1. unpow2 68.5%

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

      2. associate-*r* 68.5%

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

      3. *-commutative 68.5%

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

      4. associate-*r* 68.9%

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

      5. *-commutative 68.9%

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

      6. *-commutative 68.9%

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

      7. *-commutative 68.9%

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

   6. Simplified 68.9%

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

   7. Taylor expanded in angle around 0 64.0%

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

   8. Taylor expanded in angle around 0 65.1%

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

4. Final simplification 66.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \leq -4.6 \cdot 10^{-84} \lor \neg \left(a \leq 5.5 \cdot 10^{-173}\right):\\ \;\;\;\;0.011111111111111112 \cdot \left(\left(b + a\right) \cdot \left(\pi \cdot \left(\left(b - a\right) \cdot angle\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(0.005555555555555556 \cdot \left(\pi \cdot angle\right)\right) \cdot \left(2 \cdot \left(b \cdot b\right)\right)\\ \end{array} \]

Alternative 14: 61.7% accurate, 5.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(b - a\right) \cdot angle\\ \mathbf{if}\;a \leq -4.6 \cdot 10^{-84}:\\ \;\;\;\;0.011111111111111112 \cdot \left(\left(b + a\right) \cdot \left(\pi \cdot t_0\right)\right)\\ \mathbf{elif}\;a \leq 5.8 \cdot 10^{-175}:\\ \;\;\;\;\left(0.005555555555555556 \cdot \left(\pi \cdot angle\right)\right) \cdot \left(2 \cdot \left(b \cdot b\right)\right)\\ \mathbf{else}:\\ \;\;\;\;0.011111111111111112 \cdot \left(t_0 \cdot \left(\pi \cdot \left(b + a\right)\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (a b angle)
 :precision binary64
 (let* ((t_0 (* (- b a) angle)))
   (if (<= a -4.6e-84)
     (* 0.011111111111111112 (* (+ b a) (* PI t_0)))
     (if (<= a 5.8e-175)
       (* (* 0.005555555555555556 (* PI angle)) (* 2.0 (* b b)))
       (* 0.011111111111111112 (* t_0 (* PI (+ b a))))))))

C

double code(double a, double b, double angle) {
	double t_0 = (b - a) * angle;
	double tmp;
	if (a <= -4.6e-84) {
		tmp = 0.011111111111111112 * ((b + a) * (((double) M_PI) * t_0));
	} else if (a <= 5.8e-175) {
		tmp = (0.005555555555555556 * (((double) M_PI) * angle)) * (2.0 * (b * b));
	} else {
		tmp = 0.011111111111111112 * (t_0 * (((double) M_PI) * (b + a)));
	}
	return tmp;
}

Java

public static double code(double a, double b, double angle) {
	double t_0 = (b - a) * angle;
	double tmp;
	if (a <= -4.6e-84) {
		tmp = 0.011111111111111112 * ((b + a) * (Math.PI * t_0));
	} else if (a <= 5.8e-175) {
		tmp = (0.005555555555555556 * (Math.PI * angle)) * (2.0 * (b * b));
	} else {
		tmp = 0.011111111111111112 * (t_0 * (Math.PI * (b + a)));
	}
	return tmp;
}

Python

def code(a, b, angle):
	t_0 = (b - a) * angle
	tmp = 0
	if a <= -4.6e-84:
		tmp = 0.011111111111111112 * ((b + a) * (math.pi * t_0))
	elif a <= 5.8e-175:
		tmp = (0.005555555555555556 * (math.pi * angle)) * (2.0 * (b * b))
	else:
		tmp = 0.011111111111111112 * (t_0 * (math.pi * (b + a)))
	return tmp

Julia

function code(a, b, angle)
	t_0 = Float64(Float64(b - a) * angle)
	tmp = 0.0
	if (a <= -4.6e-84)
		tmp = Float64(0.011111111111111112 * Float64(Float64(b + a) * Float64(pi * t_0)));
	elseif (a <= 5.8e-175)
		tmp = Float64(Float64(0.005555555555555556 * Float64(pi * angle)) * Float64(2.0 * Float64(b * b)));
	else
		tmp = Float64(0.011111111111111112 * Float64(t_0 * Float64(pi * Float64(b + a))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, angle)
	t_0 = (b - a) * angle;
	tmp = 0.0;
	if (a <= -4.6e-84)
		tmp = 0.011111111111111112 * ((b + a) * (pi * t_0));
	elseif (a <= 5.8e-175)
		tmp = (0.005555555555555556 * (pi * angle)) * (2.0 * (b * b));
	else
		tmp = 0.011111111111111112 * (t_0 * (pi * (b + a)));
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, angle_] := Block[{t$95$0 = N[(N[(b - a), $MachinePrecision] * angle), $MachinePrecision]}, If[LessEqual[a, -4.6e-84], N[(0.011111111111111112 * N[(N[(b + a), $MachinePrecision] * N[(Pi * t$95$0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[a, 5.8e-175], N[(N[(0.005555555555555556 * N[(Pi * angle), $MachinePrecision]), $MachinePrecision] * N[(2.0 * N[(b * b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(0.011111111111111112 * N[(t$95$0 * N[(Pi * N[(b + a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if a < -4.59999999999999961e-84

   1. Initial program 47.7%

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

      1. associate-*l* 47.7%

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

      2. unpow2 47.7%

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

      3. unpow2 47.7%

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

      4. difference-of-squares 52.9%

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

   3. Simplified 52.9%

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

   4. Taylor expanded in angle around 0 47.3%

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

   5. Taylor expanded in angle around 0 47.3%

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

      1. associate-*r* 59.4%

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

      2. *-commutative 59.4%

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

      3. associate-*r* 59.5%

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

      4. +-commutative 59.5%

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

   7. Simplified 59.5%

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

   if -4.59999999999999961e-84 < a < 5.79999999999999998e-175

   1. Initial program 68.7%

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

      1. *-commutative 68.7%

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

      2. associate-*l* 68.7%

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

      3. unpow2 68.7%

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

      4. fma-neg 68.7%

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

      5. unpow2 68.7%

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

      6. distribute-rgt-neg-in 68.7%

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

   3. Simplified 68.7%

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

   4. Taylor expanded in b around inf 68.5%

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

      1. unpow2 68.5%

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

      2. associate-*r* 68.5%

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

      3. *-commutative 68.5%

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

      4. associate-*r* 68.9%

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

      5. *-commutative 68.9%

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

      6. *-commutative 68.9%

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

      7. *-commutative 68.9%

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

   6. Simplified 68.9%

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

   7. Taylor expanded in angle around 0 64.0%

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

   8. Taylor expanded in angle around 0 65.1%

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

   if 5.79999999999999998e-175 < a

   1. Initial program 45.5%

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

      1. associate-*l* 45.5%

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

      2. unpow2 45.5%

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

      3. unpow2 45.5%

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

      4. difference-of-squares 52.9%

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

   3. Simplified 52.9%

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

   4. Taylor expanded in angle around 0 56.9%

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

      1. *-commutative 56.9%

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

      2. associate-*r* 71.1%

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

   6. Simplified 71.1%

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

4. Final simplification 66.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \leq -4.6 \cdot 10^{-84}:\\ \;\;\;\;0.011111111111111112 \cdot \left(\left(b + a\right) \cdot \left(\pi \cdot \left(\left(b - a\right) \cdot angle\right)\right)\right)\\ \mathbf{elif}\;a \leq 5.8 \cdot 10^{-175}:\\ \;\;\;\;\left(0.005555555555555556 \cdot \left(\pi \cdot angle\right)\right) \cdot \left(2 \cdot \left(b \cdot b\right)\right)\\ \mathbf{else}:\\ \;\;\;\;0.011111111111111112 \cdot \left(\left(\left(b - a\right) \cdot angle\right) \cdot \left(\pi \cdot \left(b + a\right)\right)\right)\\ \end{array} \]

Alternative 15: 45.6% accurate, 5.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;a \leq -1.25 \cdot 10^{+110} \lor \neg \left(a \leq 8.2 \cdot 10^{+122}\right):\\ \;\;\;\;0.011111111111111112 \cdot \left(angle \cdot \left(\left(b - a\right) \cdot \left(a \cdot \pi\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;angle \cdot \left(0.011111111111111112 \cdot \left(\pi \cdot \left(b \cdot b\right)\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (a b angle)
 :precision binary64
 (if (or (<= a -1.25e+110) (not (<= a 8.2e+122)))
   (* 0.011111111111111112 (* angle (* (- b a) (* a PI))))
   (* angle (* 0.011111111111111112 (* PI (* b b))))))

C

double code(double a, double b, double angle) {
	double tmp;
	if ((a <= -1.25e+110) || !(a <= 8.2e+122)) {
		tmp = 0.011111111111111112 * (angle * ((b - a) * (a * ((double) M_PI))));
	} else {
		tmp = angle * (0.011111111111111112 * (((double) M_PI) * (b * b)));
	}
	return tmp;
}

Java

public static double code(double a, double b, double angle) {
	double tmp;
	if ((a <= -1.25e+110) || !(a <= 8.2e+122)) {
		tmp = 0.011111111111111112 * (angle * ((b - a) * (a * Math.PI)));
	} else {
		tmp = angle * (0.011111111111111112 * (Math.PI * (b * b)));
	}
	return tmp;
}

Python

def code(a, b, angle):
	tmp = 0
	if (a <= -1.25e+110) or not (a <= 8.2e+122):
		tmp = 0.011111111111111112 * (angle * ((b - a) * (a * math.pi)))
	else:
		tmp = angle * (0.011111111111111112 * (math.pi * (b * b)))
	return tmp

Julia

function code(a, b, angle)
	tmp = 0.0
	if ((a <= -1.25e+110) || !(a <= 8.2e+122))
		tmp = Float64(0.011111111111111112 * Float64(angle * Float64(Float64(b - a) * Float64(a * pi))));
	else
		tmp = Float64(angle * Float64(0.011111111111111112 * Float64(pi * Float64(b * b))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, angle)
	tmp = 0.0;
	if ((a <= -1.25e+110) || ~((a <= 8.2e+122)))
		tmp = 0.011111111111111112 * (angle * ((b - a) * (a * pi)));
	else
		tmp = angle * (0.011111111111111112 * (pi * (b * b)));
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, angle_] := If[Or[LessEqual[a, -1.25e+110], N[Not[LessEqual[a, 8.2e+122]], $MachinePrecision]], N[(0.011111111111111112 * N[(angle * N[(N[(b - a), $MachinePrecision] * N[(a * Pi), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(angle * N[(0.011111111111111112 * N[(Pi * N[(b * b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if a < -1.24999999999999995e110 or 8.2000000000000004e122 < a

   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. unpow2 42.9%

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

      3. unpow2 42.9%

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

      4. difference-of-squares 57.0%

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

   3. Simplified 57.0%

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

   4. Taylor expanded in angle around 0 56.8%

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

   5. Taylor expanded in a around inf 53.4%

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

      1. *-commutative 53.4%

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

   7. Simplified 53.4%

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

   if -1.24999999999999995e110 < a < 8.2000000000000004e122

   1. Initial program 57.3%

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

      1. associate-*l* 57.3%

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

      2. unpow2 57.3%

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

      3. unpow2 57.3%

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

      4. difference-of-squares 57.3%

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

   3. Simplified 57.3%

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

   4. Taylor expanded in angle around 0 55.4%

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

   5. Taylor expanded in b around inf 48.0%

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

      1. associate-*r* 48.0%

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

      2. *-commutative 48.0%

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

      3. unpow2 48.0%

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

   7. Simplified 48.0%

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

   8. Taylor expanded in angle around 0 48.0%

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

      1. associate-*r* 48.0%

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

      2. *-commutative 48.0%

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

      3. associate-*l* 48.0%

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

      4. *-commutative 48.0%

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

      5. unpow2 48.0%

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

   10. Simplified 48.0%

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

4. Final simplification 49.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \leq -1.25 \cdot 10^{+110} \lor \neg \left(a \leq 8.2 \cdot 10^{+122}\right):\\ \;\;\;\;0.011111111111111112 \cdot \left(angle \cdot \left(\left(b - a\right) \cdot \left(a \cdot \pi\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;angle \cdot \left(0.011111111111111112 \cdot \left(\pi \cdot \left(b \cdot b\right)\right)\right)\\ \end{array} \]

Alternative 16: 48.7% accurate, 5.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq -1.02 \cdot 10^{-22} \lor \neg \left(b \leq 1.4 \cdot 10^{-11}\right):\\ \;\;\;\;0.011111111111111112 \cdot \left(angle \cdot \left(\left(b - a\right) \cdot \left(b \cdot \pi\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;0.011111111111111112 \cdot \left(\left(-angle\right) \cdot \left(\pi \cdot \left(a \cdot a\right)\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (a b angle)
 :precision binary64
 (if (or (<= b -1.02e-22) (not (<= b 1.4e-11)))
   (* 0.011111111111111112 (* angle (* (- b a) (* b PI))))
   (* 0.011111111111111112 (* (- angle) (* PI (* a a))))))

C

double code(double a, double b, double angle) {
	double tmp;
	if ((b <= -1.02e-22) || !(b <= 1.4e-11)) {
		tmp = 0.011111111111111112 * (angle * ((b - a) * (b * ((double) M_PI))));
	} else {
		tmp = 0.011111111111111112 * (-angle * (((double) M_PI) * (a * a)));
	}
	return tmp;
}

Java

public static double code(double a, double b, double angle) {
	double tmp;
	if ((b <= -1.02e-22) || !(b <= 1.4e-11)) {
		tmp = 0.011111111111111112 * (angle * ((b - a) * (b * Math.PI)));
	} else {
		tmp = 0.011111111111111112 * (-angle * (Math.PI * (a * a)));
	}
	return tmp;
}

Python

def code(a, b, angle):
	tmp = 0
	if (b <= -1.02e-22) or not (b <= 1.4e-11):
		tmp = 0.011111111111111112 * (angle * ((b - a) * (b * math.pi)))
	else:
		tmp = 0.011111111111111112 * (-angle * (math.pi * (a * a)))
	return tmp

Julia

function code(a, b, angle)
	tmp = 0.0
	if ((b <= -1.02e-22) || !(b <= 1.4e-11))
		tmp = Float64(0.011111111111111112 * Float64(angle * Float64(Float64(b - a) * Float64(b * pi))));
	else
		tmp = Float64(0.011111111111111112 * Float64(Float64(-angle) * Float64(pi * Float64(a * a))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, angle)
	tmp = 0.0;
	if ((b <= -1.02e-22) || ~((b <= 1.4e-11)))
		tmp = 0.011111111111111112 * (angle * ((b - a) * (b * pi)));
	else
		tmp = 0.011111111111111112 * (-angle * (pi * (a * a)));
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, angle_] := If[Or[LessEqual[b, -1.02e-22], N[Not[LessEqual[b, 1.4e-11]], $MachinePrecision]], N[(0.011111111111111112 * N[(angle * N[(N[(b - a), $MachinePrecision] * N[(b * Pi), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(0.011111111111111112 * N[((-angle) * N[(Pi * N[(a * a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b < -1.02000000000000002e-22 or 1.4e-11 < b

   1. Initial program 42.3%

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

      1. associate-*l* 42.3%

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

      2. unpow2 42.3%

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

      3. unpow2 42.3%

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

      4. difference-of-squares 52.1%

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

   3. Simplified 52.1%

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

   4. Taylor expanded in angle around 0 55.1%

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

   5. Taylor expanded in a around 0 49.7%

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

      1. *-commutative 49.7%

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

   7. Simplified 49.7%

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

   if -1.02000000000000002e-22 < b < 1.4e-11

   1. Initial program 62.0%

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

      1. associate-*l* 62.0%

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

      2. unpow2 62.0%

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

      3. unpow2 62.0%

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

      4. difference-of-squares 62.0%

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

   3. Simplified 62.0%

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

   4. Taylor expanded in angle around 0 56.6%

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

   5. Taylor expanded in b around 0 54.9%

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

      1. associate-*r* 54.9%

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

      2. neg-mul-1 54.9%

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

      3. *-commutative 54.9%

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

      4. unpow2 54.9%

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

   7. Simplified 54.9%

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

4. Final simplification 52.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -1.02 \cdot 10^{-22} \lor \neg \left(b \leq 1.4 \cdot 10^{-11}\right):\\ \;\;\;\;0.011111111111111112 \cdot \left(angle \cdot \left(\left(b - a\right) \cdot \left(b \cdot \pi\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;0.011111111111111112 \cdot \left(\left(-angle\right) \cdot \left(\pi \cdot \left(a \cdot a\right)\right)\right)\\ \end{array} \]

Alternative 17: 47.0% accurate, 5.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq -1.02 \cdot 10^{-22}:\\ \;\;\;\;0.011111111111111112 \cdot \left(\pi \cdot \left(angle \cdot \left(b \cdot b\right)\right)\right)\\ \mathbf{elif}\;b \leq 1.8 \cdot 10^{-14}:\\ \;\;\;\;0.011111111111111112 \cdot \left(\left(-angle\right) \cdot \left(\pi \cdot \left(a \cdot a\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;angle \cdot \left(0.011111111111111112 \cdot \left(\pi \cdot \left(b \cdot b\right)\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (a b angle)
 :precision binary64
 (if (<= b -1.02e-22)
   (* 0.011111111111111112 (* PI (* angle (* b b))))
   (if (<= b 1.8e-14)
     (* 0.011111111111111112 (* (- angle) (* PI (* a a))))
     (* angle (* 0.011111111111111112 (* PI (* b b)))))))

C

double code(double a, double b, double angle) {
	double tmp;
	if (b <= -1.02e-22) {
		tmp = 0.011111111111111112 * (((double) M_PI) * (angle * (b * b)));
	} else if (b <= 1.8e-14) {
		tmp = 0.011111111111111112 * (-angle * (((double) M_PI) * (a * a)));
	} else {
		tmp = angle * (0.011111111111111112 * (((double) M_PI) * (b * b)));
	}
	return tmp;
}

Java

public static double code(double a, double b, double angle) {
	double tmp;
	if (b <= -1.02e-22) {
		tmp = 0.011111111111111112 * (Math.PI * (angle * (b * b)));
	} else if (b <= 1.8e-14) {
		tmp = 0.011111111111111112 * (-angle * (Math.PI * (a * a)));
	} else {
		tmp = angle * (0.011111111111111112 * (Math.PI * (b * b)));
	}
	return tmp;
}

Python

def code(a, b, angle):
	tmp = 0
	if b <= -1.02e-22:
		tmp = 0.011111111111111112 * (math.pi * (angle * (b * b)))
	elif b <= 1.8e-14:
		tmp = 0.011111111111111112 * (-angle * (math.pi * (a * a)))
	else:
		tmp = angle * (0.011111111111111112 * (math.pi * (b * b)))
	return tmp

Julia

function code(a, b, angle)
	tmp = 0.0
	if (b <= -1.02e-22)
		tmp = Float64(0.011111111111111112 * Float64(pi * Float64(angle * Float64(b * b))));
	elseif (b <= 1.8e-14)
		tmp = Float64(0.011111111111111112 * Float64(Float64(-angle) * Float64(pi * Float64(a * a))));
	else
		tmp = Float64(angle * Float64(0.011111111111111112 * Float64(pi * Float64(b * b))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, angle)
	tmp = 0.0;
	if (b <= -1.02e-22)
		tmp = 0.011111111111111112 * (pi * (angle * (b * b)));
	elseif (b <= 1.8e-14)
		tmp = 0.011111111111111112 * (-angle * (pi * (a * a)));
	else
		tmp = angle * (0.011111111111111112 * (pi * (b * b)));
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, angle_] := If[LessEqual[b, -1.02e-22], N[(0.011111111111111112 * N[(Pi * N[(angle * N[(b * b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[b, 1.8e-14], N[(0.011111111111111112 * N[((-angle) * N[(Pi * N[(a * a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(angle * N[(0.011111111111111112 * N[(Pi * N[(b * b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if b < -1.02000000000000002e-22

   1. Initial program 41.9%

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

      1. associate-*l* 41.9%

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

      2. unpow2 41.9%

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

      3. unpow2 41.9%

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

      4. difference-of-squares 52.2%

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

   3. Simplified 52.2%

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

   4. Taylor expanded in angle around 0 52.0%

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

   5. Taylor expanded in b around inf 36.9%

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

      1. associate-*r* 36.9%

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

      2. *-commutative 36.9%

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

      3. unpow2 36.9%

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

   7. Simplified 36.9%

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

   if -1.02000000000000002e-22 < b < 1.7999999999999999e-14

   1. Initial program 62.0%

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

      1. associate-*l* 62.0%

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

      2. unpow2 62.0%

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

      3. unpow2 62.0%

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

      4. difference-of-squares 62.0%

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

   3. Simplified 62.0%

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

   4. Taylor expanded in angle around 0 56.6%

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

   5. Taylor expanded in b around 0 54.9%

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

      1. associate-*r* 54.9%

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

      2. neg-mul-1 54.9%

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

      3. *-commutative 54.9%

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

      4. unpow2 54.9%

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

   7. Simplified 54.9%

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

   if 1.7999999999999999e-14 < b

   1. Initial program 42.7%

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

      1. associate-*l* 42.7%

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

      2. unpow2 42.7%

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

      3. unpow2 42.7%

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

      4. difference-of-squares 52.1%

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

   3. Simplified 52.1%

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

   4. Taylor expanded in angle around 0 57.8%

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

   5. Taylor expanded in b around inf 50.8%

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

      1. associate-*r* 50.8%

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

      2. *-commutative 50.8%

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

      3. unpow2 50.8%

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

   7. Simplified 50.8%

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

   8. Taylor expanded in angle around 0 50.8%

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

      1. associate-*r* 50.8%

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

      2. *-commutative 50.8%

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

      3. associate-*l* 50.8%

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

      4. *-commutative 50.8%

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

      5. unpow2 50.8%

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

   10. Simplified 50.8%

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

4. Final simplification 49.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -1.02 \cdot 10^{-22}:\\ \;\;\;\;0.011111111111111112 \cdot \left(\pi \cdot \left(angle \cdot \left(b \cdot b\right)\right)\right)\\ \mathbf{elif}\;b \leq 1.8 \cdot 10^{-14}:\\ \;\;\;\;0.011111111111111112 \cdot \left(\left(-angle\right) \cdot \left(\pi \cdot \left(a \cdot a\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;angle \cdot \left(0.011111111111111112 \cdot \left(\pi \cdot \left(b \cdot b\right)\right)\right)\\ \end{array} \]

Alternative 18: 47.1% accurate, 5.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq -1.5 \cdot 10^{-23}:\\ \;\;\;\;0.011111111111111112 \cdot \left(\pi \cdot \left(angle \cdot \left(b \cdot b\right)\right)\right)\\ \mathbf{elif}\;b \leq 1.6 \cdot 10^{-10}:\\ \;\;\;\;\left(\pi \cdot \left(a \cdot a\right)\right) \cdot \left(angle \cdot -0.011111111111111112\right)\\ \mathbf{else}:\\ \;\;\;\;angle \cdot \left(0.011111111111111112 \cdot \left(\pi \cdot \left(b \cdot b\right)\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (a b angle)
 :precision binary64
 (if (<= b -1.5e-23)
   (* 0.011111111111111112 (* PI (* angle (* b b))))
   (if (<= b 1.6e-10)
     (* (* PI (* a a)) (* angle -0.011111111111111112))
     (* angle (* 0.011111111111111112 (* PI (* b b)))))))

C

double code(double a, double b, double angle) {
	double tmp;
	if (b <= -1.5e-23) {
		tmp = 0.011111111111111112 * (((double) M_PI) * (angle * (b * b)));
	} else if (b <= 1.6e-10) {
		tmp = (((double) M_PI) * (a * a)) * (angle * -0.011111111111111112);
	} else {
		tmp = angle * (0.011111111111111112 * (((double) M_PI) * (b * b)));
	}
	return tmp;
}

Java

public static double code(double a, double b, double angle) {
	double tmp;
	if (b <= -1.5e-23) {
		tmp = 0.011111111111111112 * (Math.PI * (angle * (b * b)));
	} else if (b <= 1.6e-10) {
		tmp = (Math.PI * (a * a)) * (angle * -0.011111111111111112);
	} else {
		tmp = angle * (0.011111111111111112 * (Math.PI * (b * b)));
	}
	return tmp;
}

Python

def code(a, b, angle):
	tmp = 0
	if b <= -1.5e-23:
		tmp = 0.011111111111111112 * (math.pi * (angle * (b * b)))
	elif b <= 1.6e-10:
		tmp = (math.pi * (a * a)) * (angle * -0.011111111111111112)
	else:
		tmp = angle * (0.011111111111111112 * (math.pi * (b * b)))
	return tmp

Julia

function code(a, b, angle)
	tmp = 0.0
	if (b <= -1.5e-23)
		tmp = Float64(0.011111111111111112 * Float64(pi * Float64(angle * Float64(b * b))));
	elseif (b <= 1.6e-10)
		tmp = Float64(Float64(pi * Float64(a * a)) * Float64(angle * -0.011111111111111112));
	else
		tmp = Float64(angle * Float64(0.011111111111111112 * Float64(pi * Float64(b * b))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, angle)
	tmp = 0.0;
	if (b <= -1.5e-23)
		tmp = 0.011111111111111112 * (pi * (angle * (b * b)));
	elseif (b <= 1.6e-10)
		tmp = (pi * (a * a)) * (angle * -0.011111111111111112);
	else
		tmp = angle * (0.011111111111111112 * (pi * (b * b)));
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, angle_] := If[LessEqual[b, -1.5e-23], N[(0.011111111111111112 * N[(Pi * N[(angle * N[(b * b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[b, 1.6e-10], N[(N[(Pi * N[(a * a), $MachinePrecision]), $MachinePrecision] * N[(angle * -0.011111111111111112), $MachinePrecision]), $MachinePrecision], N[(angle * N[(0.011111111111111112 * N[(Pi * N[(b * b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if b < -1.50000000000000001e-23

   1. Initial program 41.9%

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

      1. associate-*l* 41.9%

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

      2. unpow2 41.9%

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

      3. unpow2 41.9%

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

      4. difference-of-squares 52.2%

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

   3. Simplified 52.2%

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

   4. Taylor expanded in angle around 0 52.0%

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

   5. Taylor expanded in b around inf 36.9%

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

      1. associate-*r* 36.9%

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

      2. *-commutative 36.9%

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

      3. unpow2 36.9%

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

   7. Simplified 36.9%

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

   if -1.50000000000000001e-23 < b < 1.5999999999999999e-10

   1. Initial program 62.0%

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

      1. associate-*l* 62.0%

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

      2. unpow2 62.0%

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

      3. unpow2 62.0%

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

      4. difference-of-squares 62.0%

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

   3. Simplified 62.0%

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

   4. Taylor expanded in angle around 0 56.6%

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

   5. Taylor expanded in b around 0 54.9%

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

      1. *-commutative 54.9%

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

      2. *-commutative 54.9%

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

      3. associate-*l* 54.9%

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

      4. *-commutative 54.9%

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

      5. unpow2 54.9%

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

   7. Simplified 54.9%

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

   if 1.5999999999999999e-10 < b

   1. Initial program 42.7%

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

      1. associate-*l* 42.7%

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

      2. unpow2 42.7%

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

      3. unpow2 42.7%

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

      4. difference-of-squares 52.1%

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

   3. Simplified 52.1%

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

   4. Taylor expanded in angle around 0 57.8%

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

   5. Taylor expanded in b around inf 50.8%

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

      1. associate-*r* 50.8%

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

      2. *-commutative 50.8%

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

      3. unpow2 50.8%

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

   7. Simplified 50.8%

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

   8. Taylor expanded in angle around 0 50.8%

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

      1. associate-*r* 50.8%

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

      2. *-commutative 50.8%

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

      3. associate-*l* 50.8%

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

      4. *-commutative 50.8%

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

      5. unpow2 50.8%

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

   10. Simplified 50.8%

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

4. Final simplification 49.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -1.5 \cdot 10^{-23}:\\ \;\;\;\;0.011111111111111112 \cdot \left(\pi \cdot \left(angle \cdot \left(b \cdot b\right)\right)\right)\\ \mathbf{elif}\;b \leq 1.6 \cdot 10^{-10}:\\ \;\;\;\;\left(\pi \cdot \left(a \cdot a\right)\right) \cdot \left(angle \cdot -0.011111111111111112\right)\\ \mathbf{else}:\\ \;\;\;\;angle \cdot \left(0.011111111111111112 \cdot \left(\pi \cdot \left(b \cdot b\right)\right)\right)\\ \end{array} \]

Alternative 19: 53.9% accurate, 5.5× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 52.4%

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

   1. associate-*l* 52.4%

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

   2. unpow2 52.4%

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

   3. unpow2 52.4%

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

   4. difference-of-squares 57.2%

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

3. Simplified 57.2%

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

4. Taylor expanded in angle around 0 55.9%

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

5. Final simplification 55.9%

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

Alternative 20: 35.1% accurate, 5.7× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 52.4%

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

   1. associate-*l* 52.4%

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

   2. unpow2 52.4%

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

   3. unpow2 52.4%

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

   4. difference-of-squares 57.2%

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

3. Simplified 57.2%

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

4. Taylor expanded in angle around 0 55.9%

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

5. Taylor expanded in b around inf 36.2%

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

   1. associate-*r* 36.2%

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

   2. *-commutative 36.2%

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

   3. unpow2 36.2%

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

7. Simplified 36.2%

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

8. Taylor expanded in angle around 0 36.2%

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

   1. associate-*r* 36.1%

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

   2. *-commutative 36.1%

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

   3. associate-*l* 36.2%

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

   4. *-commutative 36.2%

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

   5. unpow2 36.2%

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

10. Simplified 36.2%

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

11. Taylor expanded in angle around 0 36.2%

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

    1. unpow2 36.2%

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

    2. *-commutative 36.2%

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

13. Simplified 36.2%

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

14. Final simplification 36.2%

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

Alternative 21: 35.1% accurate, 5.7× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 52.4%

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

   1. associate-*l* 52.4%

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

   2. unpow2 52.4%

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

   3. unpow2 52.4%

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

   4. difference-of-squares 57.2%

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

3. Simplified 57.2%

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

4. Taylor expanded in angle around 0 55.9%

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

5. Taylor expanded in b around inf 36.2%

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

   1. associate-*r* 36.2%

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

   2. *-commutative 36.2%

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

   3. unpow2 36.2%

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

7. Simplified 36.2%

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

8. Final simplification 36.2%

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

Alternative 22: 35.1% accurate, 5.7× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 52.4%

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

   1. associate-*l* 52.4%

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

   2. unpow2 52.4%

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

   3. unpow2 52.4%

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

   4. difference-of-squares 57.2%

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

3. Simplified 57.2%

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

4. Taylor expanded in angle around 0 55.9%

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

5. Taylor expanded in b around inf 36.2%

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

   1. associate-*r* 36.2%

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

   2. *-commutative 36.2%

      \[\leadsto 0.011111111111111112 \cdot \color{blue}{\left(\pi \cdot \left(angle \cdot {b}^{2}\right)\right)} \]

   3. unpow2 36.2%

      \[\leadsto 0.011111111111111112 \cdot \left(\pi \cdot \left(angle \cdot \color{blue}{\left(b \cdot b\right)}\right)\right) \]

7. Simplified 36.2%

   \[\leadsto 0.011111111111111112 \cdot \color{blue}{\left(\pi \cdot \left(angle \cdot \left(b \cdot b\right)\right)\right)} \]

8. Taylor expanded in angle around 0 36.2%

   \[\leadsto \color{blue}{0.011111111111111112 \cdot \left(angle \cdot \left({b}^{2} \cdot \pi\right)\right)} \]
9. Step-by-step derivation

   1. associate-*r* 36.1%

      \[\leadsto \color{blue}{\left(0.011111111111111112 \cdot angle\right) \cdot \left({b}^{2} \cdot \pi\right)} \]

   2. *-commutative 36.1%

      \[\leadsto \color{blue}{\left(angle \cdot 0.011111111111111112\right)} \cdot \left({b}^{2} \cdot \pi\right) \]

   3. associate-*l* 36.2%

      \[\leadsto \color{blue}{angle \cdot \left(0.011111111111111112 \cdot \left({b}^{2} \cdot \pi\right)\right)} \]

   4. *-commutative 36.2%

      \[\leadsto angle \cdot \left(0.011111111111111112 \cdot \color{blue}{\left(\pi \cdot {b}^{2}\right)}\right) \]

   5. unpow2 36.2%

      \[\leadsto angle \cdot \left(0.011111111111111112 \cdot \left(\pi \cdot \color{blue}{\left(b \cdot b\right)}\right)\right) \]

10. Simplified 36.2%

    \[\leadsto \color{blue}{angle \cdot \left(0.011111111111111112 \cdot \left(\pi \cdot \left(b \cdot b\right)\right)\right)} \]

11. Final simplification 36.2%

    \[\leadsto angle \cdot \left(0.011111111111111112 \cdot \left(\pi \cdot \left(b \cdot b\right)\right)\right) \]

Reproduce

herbie shell --seed 2023182 
(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)))))