ab-angle->ABCF B

Percentage Accurate: 54.0% → 66.6%

Time: 45.2s

Alternatives: 14

Speedup: 5.4×

• 32.9% of points produce a very large (infinite) output. You may want to add a precondition.

Specification

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 54.0% 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: 66.6% accurate, 0.4× speedup

Math

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

FPCore

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

C

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

Java

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

Julia

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

Wolfram

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

Derivation

1. Initial program 53.6%

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

   1. associate-*l* 53.6%

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

   2. unpow2 53.6%

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

   3. unpow2 53.6%

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

   4. difference-of-squares 56.9%

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

3. Simplified 56.9%

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

4. Taylor expanded in angle around inf 66.6%

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

   1. add-cube-cbrt 66.8%

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

6. Applied egg-rr 66.8%

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

   1. associate-*r* 67.7%

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

   2. *-commutative 67.7%

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

   3. metadata-eval 67.7%

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

   4. div-inv 67.8%

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

   5. *-commutative 67.8%

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

   6. div-inv 67.7%

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

   7. metadata-eval 67.7%

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

   8. *-commutative 67.7%

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

   9. add-cube-cbrt 67.7%

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

8. Applied egg-rr 67.7%

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

9. Final simplification 67.7%

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

Alternative 2: 66.7% accurate, 0.6× speedup

Math

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

FPCore

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

C

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

Java

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

Julia

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

Wolfram

code[a_, b_, angle_] := N[(2.0 * N[(N[(b - a), $MachinePrecision] * N[(N[(N[Sin[N[(0.005555555555555556 * N[(angle * N[(N[Power[Pi, 1/3], $MachinePrecision] * N[(N[Power[Pi, 1/3], $MachinePrecision] * N[Power[Pi, 1/3], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * N[(b + a), $MachinePrecision]), $MachinePrecision] * N[Log[N[Exp[N[Cos[N[(Pi * N[(0.005555555555555556 * angle), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]], $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 53.6%

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

   1. associate-*l* 53.6%

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

   2. unpow2 53.6%

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

   3. unpow2 53.6%

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

   4. difference-of-squares 56.9%

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

3. Simplified 56.9%

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

4. Taylor expanded in angle around inf 66.6%

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

   1. add-cube-cbrt 66.8%

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

6. Applied egg-rr 66.8%

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

   1. associate-*r* 67.7%

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

   2. *-commutative 67.7%

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

   3. metadata-eval 67.7%

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

   4. div-inv 67.8%

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

   5. *-commutative 67.8%

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

   6. div-inv 67.7%

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

   7. metadata-eval 67.7%

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

   8. *-commutative 67.7%

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

   9. add-log-exp 67.7%

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

8. Applied egg-rr 67.7%

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

9. Final simplification 67.7%

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

Alternative 3: 66.6% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ 2 \cdot \left(\left(b - a\right) \cdot \left(\left(\sin \left(0.005555555555555556 \cdot \left(angle \cdot \left(\sqrt[3]{\pi} \cdot \left(\sqrt[3]{\pi} \cdot \sqrt[3]{\pi}\right)\right)\right)\right) \cdot \left(b + a\right)\right) \cdot \mathsf{log1p}\left(\mathsf{expm1}\left(\cos \left(\pi \cdot \left(0.005555555555555556 \cdot angle\right)\right)\right)\right)\right)\right) \end{array} \]

FPCore

(FPCore (a b angle)
 :precision binary64
 (*
  2.0
  (*
   (- b a)
   (*
    (*
     (sin
      (* 0.005555555555555556 (* angle (* (cbrt PI) (* (cbrt PI) (cbrt PI))))))
     (+ b a))
    (log1p (expm1 (cos (* PI (* 0.005555555555555556 angle)))))))))

C

double code(double a, double b, double angle) {
	return 2.0 * ((b - a) * ((sin((0.005555555555555556 * (angle * (cbrt(((double) M_PI)) * (cbrt(((double) M_PI)) * cbrt(((double) M_PI))))))) * (b + a)) * log1p(expm1(cos((((double) M_PI) * (0.005555555555555556 * angle)))))));
}

Java

public static double code(double a, double b, double angle) {
	return 2.0 * ((b - a) * ((Math.sin((0.005555555555555556 * (angle * (Math.cbrt(Math.PI) * (Math.cbrt(Math.PI) * Math.cbrt(Math.PI)))))) * (b + a)) * Math.log1p(Math.expm1(Math.cos((Math.PI * (0.005555555555555556 * angle)))))));
}

Julia

function code(a, b, angle)
	return Float64(2.0 * Float64(Float64(b - a) * Float64(Float64(sin(Float64(0.005555555555555556 * Float64(angle * Float64(cbrt(pi) * Float64(cbrt(pi) * cbrt(pi)))))) * Float64(b + a)) * log1p(expm1(cos(Float64(pi * Float64(0.005555555555555556 * angle))))))))
end

Wolfram

code[a_, b_, angle_] := N[(2.0 * N[(N[(b - a), $MachinePrecision] * N[(N[(N[Sin[N[(0.005555555555555556 * N[(angle * N[(N[Power[Pi, 1/3], $MachinePrecision] * N[(N[Power[Pi, 1/3], $MachinePrecision] * N[Power[Pi, 1/3], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * N[(b + a), $MachinePrecision]), $MachinePrecision] * N[Log[1 + N[(Exp[N[Cos[N[(Pi * N[(0.005555555555555556 * angle), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]] - 1), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 53.6%

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

   1. associate-*l* 53.6%

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

   2. unpow2 53.6%

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

   3. unpow2 53.6%

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

   4. difference-of-squares 56.9%

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

3. Simplified 56.9%

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

4. Taylor expanded in angle around inf 66.6%

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

   1. add-cube-cbrt 66.8%

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

6. Applied egg-rr 66.8%

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

   1. associate-*r* 67.7%

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

   2. *-commutative 67.7%

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

   3. metadata-eval 67.7%

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

   4. div-inv 67.8%

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

   5. *-commutative 67.8%

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

   6. div-inv 67.7%

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

   7. metadata-eval 67.7%

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

   8. *-commutative 67.7%

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

   9. log1p-expm1-u 67.7%

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

8. Applied egg-rr 67.7%

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

9. Final simplification 67.7%

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

Alternative 4: 66.7% accurate, 0.7× speedup

Math

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

FPCore

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

C

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

Java

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

Julia

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

Wolfram

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

Derivation

1. Initial program 53.6%

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

   1. associate-*l* 53.6%

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

   2. unpow2 53.6%

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

   3. unpow2 53.6%

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

   4. difference-of-squares 56.9%

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

3. Simplified 56.9%

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

4. Taylor expanded in angle around inf 66.6%

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

   1. add-cube-cbrt 66.8%

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

6. Applied egg-rr 66.8%

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

7. Final simplification 66.8%

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

Alternative 5: 67.0% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := 0.005555555555555556 \cdot \left(\pi \cdot angle\right)\\ \mathbf{if}\;a \leq -1.3 \cdot 10^{+111}:\\ \;\;\;\;2 \cdot \left(\left(b - a\right) \cdot \left(\left(b + a\right) \cdot \sin t_0\right)\right)\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \left(\left(\left(b - a\right) \cdot \cos t_0\right) \cdot \left(\left(b + a\right) \cdot \sin \left(\pi \cdot \left(0.005555555555555556 \cdot angle\right)\right)\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (a b angle)
 :precision binary64
 (let* ((t_0 (* 0.005555555555555556 (* PI angle))))
   (if (<= a -1.3e+111)
     (* 2.0 (* (- b a) (* (+ b a) (sin t_0))))
     (*
      2.0
      (*
       (* (- b a) (cos t_0))
       (* (+ b a) (sin (* PI (* 0.005555555555555556 angle)))))))))

C

double code(double a, double b, double angle) {
	double t_0 = 0.005555555555555556 * (((double) M_PI) * angle);
	double tmp;
	if (a <= -1.3e+111) {
		tmp = 2.0 * ((b - a) * ((b + a) * sin(t_0)));
	} else {
		tmp = 2.0 * (((b - a) * cos(t_0)) * ((b + a) * sin((((double) M_PI) * (0.005555555555555556 * angle)))));
	}
	return tmp;
}

Java

public static double code(double a, double b, double angle) {
	double t_0 = 0.005555555555555556 * (Math.PI * angle);
	double tmp;
	if (a <= -1.3e+111) {
		tmp = 2.0 * ((b - a) * ((b + a) * Math.sin(t_0)));
	} else {
		tmp = 2.0 * (((b - a) * Math.cos(t_0)) * ((b + a) * Math.sin((Math.PI * (0.005555555555555556 * angle)))));
	}
	return tmp;
}

Python

def code(a, b, angle):
	t_0 = 0.005555555555555556 * (math.pi * angle)
	tmp = 0
	if a <= -1.3e+111:
		tmp = 2.0 * ((b - a) * ((b + a) * math.sin(t_0)))
	else:
		tmp = 2.0 * (((b - a) * math.cos(t_0)) * ((b + a) * math.sin((math.pi * (0.005555555555555556 * angle)))))
	return tmp

Julia

function code(a, b, angle)
	t_0 = Float64(0.005555555555555556 * Float64(pi * angle))
	tmp = 0.0
	if (a <= -1.3e+111)
		tmp = Float64(2.0 * Float64(Float64(b - a) * Float64(Float64(b + a) * sin(t_0))));
	else
		tmp = Float64(2.0 * Float64(Float64(Float64(b - a) * cos(t_0)) * Float64(Float64(b + a) * sin(Float64(pi * Float64(0.005555555555555556 * angle))))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, angle)
	t_0 = 0.005555555555555556 * (pi * angle);
	tmp = 0.0;
	if (a <= -1.3e+111)
		tmp = 2.0 * ((b - a) * ((b + a) * sin(t_0)));
	else
		tmp = 2.0 * (((b - a) * cos(t_0)) * ((b + a) * sin((pi * (0.005555555555555556 * angle)))));
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, angle_] := Block[{t$95$0 = N[(0.005555555555555556 * N[(Pi * angle), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[a, -1.3e+111], N[(2.0 * N[(N[(b - a), $MachinePrecision] * N[(N[(b + a), $MachinePrecision] * N[Sin[t$95$0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(2.0 * N[(N[(N[(b - a), $MachinePrecision] * N[Cos[t$95$0], $MachinePrecision]), $MachinePrecision] * N[(N[(b + a), $MachinePrecision] * N[Sin[N[(Pi * N[(0.005555555555555556 * angle), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if a < -1.2999999999999999e111

   1. Initial program 25.5%

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

      1. associate-*l* 25.5%

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

      2. unpow2 25.5%

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

      3. unpow2 25.5%

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

      4. difference-of-squares 41.9%

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

   3. Simplified 41.9%

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

   4. Taylor expanded in angle around inf 70.8%

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

   5. Taylor expanded in angle around 0 77.0%

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

   if -1.2999999999999999e111 < a

   1. Initial program 58.5%

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

      1. associate-*l* 58.5%

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

      2. unpow2 58.5%

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

      3. unpow2 58.5%

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

      4. difference-of-squares 59.5%

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

   3. Simplified 59.5%

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

   4. Taylor expanded in angle around inf 65.9%

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

      1. associate-*r* 65.9%

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

      2. *-commutative 65.9%

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

      3. *-commutative 65.9%

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

      4. associate-*r* 64.5%

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

      5. *-commutative 64.5%

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

      6. *-commutative 64.5%

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

      7. *-commutative 64.5%

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

      8. associate-*r* 65.5%

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

      9. *-commutative 65.5%

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

      10. +-commutative 65.5%

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

   6. Simplified 65.5%

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

   7. Taylor expanded in angle around inf 66.1%

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

4. Final simplification 67.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \leq -1.3 \cdot 10^{+111}:\\ \;\;\;\;2 \cdot \left(\left(b - a\right) \cdot \left(\left(b + a\right) \cdot \sin \left(0.005555555555555556 \cdot \left(\pi \cdot angle\right)\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \left(\left(\left(b - a\right) \cdot \cos \left(0.005555555555555556 \cdot \left(\pi \cdot angle\right)\right)\right) \cdot \left(\left(b + a\right) \cdot \sin \left(\pi \cdot \left(0.005555555555555556 \cdot angle\right)\right)\right)\right)\\ \end{array} \]

Alternative 6: 67.1% accurate, 1.5× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 53.6%

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

   1. associate-*l* 53.6%

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

   2. unpow2 53.6%

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

   3. unpow2 53.6%

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

   4. difference-of-squares 56.9%

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

3. Simplified 56.9%

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

4. Taylor expanded in angle around inf 66.6%

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

5. Final simplification 66.6%

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

Alternative 7: 67.2% accurate, 1.5× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 53.6%

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

   1. associate-*l* 53.6%

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

   2. unpow2 53.6%

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

   3. unpow2 53.6%

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

   4. difference-of-squares 56.9%

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

3. Simplified 56.9%

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

4. Taylor expanded in angle around inf 66.6%

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

   1. associate-*r* 65.1%

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

   2. *-commutative 65.1%

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

   3. metadata-eval 65.1%

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

   4. div-inv 65.2%

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

   5. *-commutative 65.2%

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

   6. associate-*r/ 66.6%

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

6. Applied egg-rr 66.6%

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

7. Final simplification 66.6%

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

Alternative 8: 66.2% accurate, 2.9× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 53.6%

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

   1. associate-*l* 53.6%

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

   2. unpow2 53.6%

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

   3. unpow2 53.6%

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

   4. difference-of-squares 56.9%

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

3. Simplified 56.9%

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

4. Taylor expanded in angle around inf 66.6%

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

5. Taylor expanded in angle around 0 65.1%

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

6. Final simplification 65.1%

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

Alternative 9: 50.5% accurate, 5.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq -2.9 \cdot 10^{-77} \lor \neg \left(b \leq 2.2 \cdot 10^{+31}\right):\\ \;\;\;\;0.011111111111111112 \cdot \left(\pi \cdot \left(b \cdot \left(b \cdot angle\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;0.011111111111111112 \cdot \left(angle \cdot \left(\left(b - a\right) \cdot \left(a \cdot \pi\right)\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (a b angle)
 :precision binary64
 (if (or (<= b -2.9e-77) (not (<= b 2.2e+31)))
   (* 0.011111111111111112 (* PI (* b (* b angle))))
   (* 0.011111111111111112 (* angle (* (- b a) (* a PI))))))

C

double code(double a, double b, double angle) {
	double tmp;
	if ((b <= -2.9e-77) || !(b <= 2.2e+31)) {
		tmp = 0.011111111111111112 * (((double) M_PI) * (b * (b * angle)));
	} else {
		tmp = 0.011111111111111112 * (angle * ((b - a) * (a * ((double) M_PI))));
	}
	return tmp;
}

Java

public static double code(double a, double b, double angle) {
	double tmp;
	if ((b <= -2.9e-77) || !(b <= 2.2e+31)) {
		tmp = 0.011111111111111112 * (Math.PI * (b * (b * angle)));
	} else {
		tmp = 0.011111111111111112 * (angle * ((b - a) * (a * Math.PI)));
	}
	return tmp;
}

Python

def code(a, b, angle):
	tmp = 0
	if (b <= -2.9e-77) or not (b <= 2.2e+31):
		tmp = 0.011111111111111112 * (math.pi * (b * (b * angle)))
	else:
		tmp = 0.011111111111111112 * (angle * ((b - a) * (a * math.pi)))
	return tmp

Julia

function code(a, b, angle)
	tmp = 0.0
	if ((b <= -2.9e-77) || !(b <= 2.2e+31))
		tmp = Float64(0.011111111111111112 * Float64(pi * Float64(b * Float64(b * angle))));
	else
		tmp = Float64(0.011111111111111112 * Float64(angle * Float64(Float64(b - a) * Float64(a * pi))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, angle)
	tmp = 0.0;
	if ((b <= -2.9e-77) || ~((b <= 2.2e+31)))
		tmp = 0.011111111111111112 * (pi * (b * (b * angle)));
	else
		tmp = 0.011111111111111112 * (angle * ((b - a) * (a * pi)));
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, angle_] := If[Or[LessEqual[b, -2.9e-77], N[Not[LessEqual[b, 2.2e+31]], $MachinePrecision]], N[(0.011111111111111112 * N[(Pi * N[(b * N[(b * angle), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(0.011111111111111112 * N[(angle * N[(N[(b - a), $MachinePrecision] * N[(a * Pi), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b < -2.8999999999999999e-77 or 2.2000000000000001e31 < b

   1. Initial program 53.0%

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

      1. associate-*l* 53.0%

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

      2. unpow2 53.0%

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

      3. unpow2 53.0%

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

      4. difference-of-squares 59.8%

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

   3. Simplified 59.8%

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

   4. Taylor expanded in angle around 0 56.5%

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

   5. Taylor expanded in b around inf 48.9%

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

      1. *-commutative 48.9%

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

      2. unpow2 48.9%

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

   7. Simplified 48.9%

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

   8. Taylor expanded in angle around 0 48.9%

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

      1. *-commutative 48.9%

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

      2. associate-*l* 48.9%

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

      3. unpow2 48.9%

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

   10. Simplified 48.9%

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

   11. Taylor expanded in angle around 0 48.9%

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

       1. unpow2 48.9%

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

       2. associate-*r* 48.8%

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

       3. *-commutative 48.8%

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

       4. associate-*r* 56.5%

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

   13. Simplified 56.5%

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

   if -2.8999999999999999e-77 < b < 2.2000000000000001e31

   1. Initial program 54.2%

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

      1. associate-*l* 54.2%

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

      2. unpow2 54.2%

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

      3. unpow2 54.2%

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

      4. difference-of-squares 54.2%

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

   3. Simplified 54.2%

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

   4. Taylor expanded in angle around 0 51.7%

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

   5. Taylor expanded in a around inf 47.1%

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

      1. *-commutative 47.1%

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

   7. Simplified 47.1%

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

4. Final simplification 51.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -2.9 \cdot 10^{-77} \lor \neg \left(b \leq 2.2 \cdot 10^{+31}\right):\\ \;\;\;\;0.011111111111111112 \cdot \left(\pi \cdot \left(b \cdot \left(b \cdot angle\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;0.011111111111111112 \cdot \left(angle \cdot \left(\left(b - a\right) \cdot \left(a \cdot \pi\right)\right)\right)\\ \end{array} \]

Alternative 10: 55.9% accurate, 5.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq 3.8 \cdot 10^{+171}:\\ \;\;\;\;0.011111111111111112 \cdot \left(angle \cdot \left(\left(b - a\right) \cdot \left(\pi \cdot \left(b + a\right)\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;0.011111111111111112 \cdot \left(\pi \cdot \left(b \cdot \left(b \cdot angle\right)\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (a b angle)
 :precision binary64
 (if (<= b 3.8e+171)
   (* 0.011111111111111112 (* angle (* (- b a) (* PI (+ b a)))))
   (* 0.011111111111111112 (* PI (* b (* b angle))))))

C

double code(double a, double b, double angle) {
	double tmp;
	if (b <= 3.8e+171) {
		tmp = 0.011111111111111112 * (angle * ((b - a) * (((double) M_PI) * (b + a))));
	} else {
		tmp = 0.011111111111111112 * (((double) M_PI) * (b * (b * angle)));
	}
	return tmp;
}

Java

public static double code(double a, double b, double angle) {
	double tmp;
	if (b <= 3.8e+171) {
		tmp = 0.011111111111111112 * (angle * ((b - a) * (Math.PI * (b + a))));
	} else {
		tmp = 0.011111111111111112 * (Math.PI * (b * (b * angle)));
	}
	return tmp;
}

Python

def code(a, b, angle):
	tmp = 0
	if b <= 3.8e+171:
		tmp = 0.011111111111111112 * (angle * ((b - a) * (math.pi * (b + a))))
	else:
		tmp = 0.011111111111111112 * (math.pi * (b * (b * angle)))
	return tmp

Julia

function code(a, b, angle)
	tmp = 0.0
	if (b <= 3.8e+171)
		tmp = Float64(0.011111111111111112 * Float64(angle * Float64(Float64(b - a) * Float64(pi * Float64(b + a)))));
	else
		tmp = Float64(0.011111111111111112 * Float64(pi * Float64(b * Float64(b * angle))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, angle)
	tmp = 0.0;
	if (b <= 3.8e+171)
		tmp = 0.011111111111111112 * (angle * ((b - a) * (pi * (b + a))));
	else
		tmp = 0.011111111111111112 * (pi * (b * (b * angle)));
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, angle_] := If[LessEqual[b, 3.8e+171], N[(0.011111111111111112 * N[(angle * N[(N[(b - a), $MachinePrecision] * N[(Pi * N[(b + a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(0.011111111111111112 * N[(Pi * N[(b * N[(b * angle), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b < 3.8000000000000002e171

   1. Initial program 57.2%

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

      1. associate-*l* 57.2%

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

      2. unpow2 57.2%

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

      3. unpow2 57.2%

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

      4. difference-of-squares 58.1%

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

   3. Simplified 58.1%

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

   4. Taylor expanded in angle around 0 53.6%

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

   if 3.8000000000000002e171 < b

   1. Initial program 23.3%

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

      1. associate-*l* 23.3%

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

      2. unpow2 23.3%

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

      3. unpow2 23.3%

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

      4. difference-of-squares 46.3%

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

   3. Simplified 46.3%

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

   4. Taylor expanded in angle around 0 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. Taylor expanded in b around inf 57.1%

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

      1. *-commutative 57.1%

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

      2. unpow2 57.1%

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

   7. Simplified 57.1%

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

   8. Taylor expanded in angle around 0 57.1%

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

      1. *-commutative 57.1%

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

      2. associate-*l* 57.1%

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

      3. unpow2 57.1%

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

   10. Simplified 57.1%

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

   11. Taylor expanded in angle around 0 57.1%

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

       1. unpow2 57.1%

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

       2. associate-*r* 57.1%

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

       3. *-commutative 57.1%

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

       4. associate-*r* 81.3%

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

   13. Simplified 81.3%

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

4. Final simplification 56.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq 3.8 \cdot 10^{+171}:\\ \;\;\;\;0.011111111111111112 \cdot \left(angle \cdot \left(\left(b - a\right) \cdot \left(\pi \cdot \left(b + a\right)\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;0.011111111111111112 \cdot \left(\pi \cdot \left(b \cdot \left(b \cdot angle\right)\right)\right)\\ \end{array} \]

Alternative 11: 62.5% accurate, 5.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;angle \leq 9.5 \cdot 10^{+142}:\\ \;\;\;\;0.011111111111111112 \cdot \left(\left(\left(b - a\right) \cdot angle\right) \cdot \left(\pi \cdot \left(b + a\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;0.011111111111111112 \cdot \left(angle \cdot \left(\left(b - a\right) \cdot \left(b \cdot \pi\right)\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (a b angle)
 :precision binary64
 (if (<= angle 9.5e+142)
   (* 0.011111111111111112 (* (* (- b a) angle) (* PI (+ b a))))
   (* 0.011111111111111112 (* angle (* (- b a) (* b PI))))))

C

double code(double a, double b, double angle) {
	double tmp;
	if (angle <= 9.5e+142) {
		tmp = 0.011111111111111112 * (((b - a) * angle) * (((double) M_PI) * (b + a)));
	} else {
		tmp = 0.011111111111111112 * (angle * ((b - a) * (b * ((double) M_PI))));
	}
	return tmp;
}

Java

public static double code(double a, double b, double angle) {
	double tmp;
	if (angle <= 9.5e+142) {
		tmp = 0.011111111111111112 * (((b - a) * angle) * (Math.PI * (b + a)));
	} else {
		tmp = 0.011111111111111112 * (angle * ((b - a) * (b * Math.PI)));
	}
	return tmp;
}

Python

def code(a, b, angle):
	tmp = 0
	if angle <= 9.5e+142:
		tmp = 0.011111111111111112 * (((b - a) * angle) * (math.pi * (b + a)))
	else:
		tmp = 0.011111111111111112 * (angle * ((b - a) * (b * math.pi)))
	return tmp

Julia

function code(a, b, angle)
	tmp = 0.0
	if (angle <= 9.5e+142)
		tmp = Float64(0.011111111111111112 * Float64(Float64(Float64(b - a) * angle) * Float64(pi * Float64(b + a))));
	else
		tmp = Float64(0.011111111111111112 * Float64(angle * Float64(Float64(b - a) * Float64(b * pi))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, angle)
	tmp = 0.0;
	if (angle <= 9.5e+142)
		tmp = 0.011111111111111112 * (((b - a) * angle) * (pi * (b + a)));
	else
		tmp = 0.011111111111111112 * (angle * ((b - a) * (b * pi)));
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, angle_] := If[LessEqual[angle, 9.5e+142], N[(0.011111111111111112 * N[(N[(N[(b - a), $MachinePrecision] * angle), $MachinePrecision] * N[(Pi * N[(b + a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(0.011111111111111112 * N[(angle * N[(N[(b - a), $MachinePrecision] * N[(b * Pi), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if angle < 9.50000000000000001e142

   1. Initial program 57.2%

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

      1. associate-*l* 57.2%

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

      2. unpow2 57.2%

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

      3. unpow2 57.2%

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

      4. difference-of-squares 60.5%

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

   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 0 58.3%

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

      1. associate-*r* 66.5%

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

      2. *-commutative 66.5%

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

      3. +-commutative 66.5%

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

   6. Simplified 66.5%

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

   if 9.50000000000000001e142 < angle

   1. Initial program 28.2%

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

      1. associate-*l* 28.2%

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

      2. unpow2 28.2%

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

      3. unpow2 28.2%

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

      4. difference-of-squares 31.4%

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

   3. Simplified 31.4%

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

   4. Taylor expanded in angle around 0 23.8%

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

   5. Taylor expanded in a around 0 27.4%

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

      1. *-commutative 27.4%

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

   7. Simplified 27.4%

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

4. Final simplification 61.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;angle \leq 9.5 \cdot 10^{+142}:\\ \;\;\;\;0.011111111111111112 \cdot \left(\left(\left(b - a\right) \cdot angle\right) \cdot \left(\pi \cdot \left(b + a\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;0.011111111111111112 \cdot \left(angle \cdot \left(\left(b - a\right) \cdot \left(b \cdot \pi\right)\right)\right)\\ \end{array} \]

Alternative 12: 48.8% accurate, 5.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq -9.2 \cdot 10^{-136} \lor \neg \left(b \leq 4.5 \cdot 10^{+51}\right):\\ \;\;\;\;0.011111111111111112 \cdot \left(\pi \cdot \left(b \cdot \left(b \cdot angle\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;0.011111111111111112 \cdot \left(angle \cdot \left(\pi \cdot \left(a \cdot \left(-a\right)\right)\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (a b angle)
 :precision binary64
 (if (or (<= b -9.2e-136) (not (<= b 4.5e+51)))
   (* 0.011111111111111112 (* PI (* b (* b angle))))
   (* 0.011111111111111112 (* angle (* PI (* a (- a)))))))

C

double code(double a, double b, double angle) {
	double tmp;
	if ((b <= -9.2e-136) || !(b <= 4.5e+51)) {
		tmp = 0.011111111111111112 * (((double) M_PI) * (b * (b * angle)));
	} 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 <= -9.2e-136) || !(b <= 4.5e+51)) {
		tmp = 0.011111111111111112 * (Math.PI * (b * (b * angle)));
	} else {
		tmp = 0.011111111111111112 * (angle * (Math.PI * (a * -a)));
	}
	return tmp;
}

Python

def code(a, b, angle):
	tmp = 0
	if (b <= -9.2e-136) or not (b <= 4.5e+51):
		tmp = 0.011111111111111112 * (math.pi * (b * (b * angle)))
	else:
		tmp = 0.011111111111111112 * (angle * (math.pi * (a * -a)))
	return tmp

Julia

function code(a, b, angle)
	tmp = 0.0
	if ((b <= -9.2e-136) || !(b <= 4.5e+51))
		tmp = Float64(0.011111111111111112 * Float64(pi * Float64(b * Float64(b * angle))));
	else
		tmp = Float64(0.011111111111111112 * Float64(angle * Float64(pi * Float64(a * Float64(-a)))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, angle)
	tmp = 0.0;
	if ((b <= -9.2e-136) || ~((b <= 4.5e+51)))
		tmp = 0.011111111111111112 * (pi * (b * (b * angle)));
	else
		tmp = 0.011111111111111112 * (angle * (pi * (a * -a)));
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, angle_] := If[Or[LessEqual[b, -9.2e-136], N[Not[LessEqual[b, 4.5e+51]], $MachinePrecision]], N[(0.011111111111111112 * N[(Pi * N[(b * N[(b * angle), $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 < -9.19999999999999994e-136 or 4.5e51 < b

   1. Initial program 51.8%

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

      1. associate-*l* 51.8%

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

      2. unpow2 51.8%

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

      3. unpow2 51.8%

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

      4. difference-of-squares 57.9%

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

   3. Simplified 57.9%

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

   4. Taylor expanded in angle around 0 53.8%

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

   5. Taylor expanded in b around inf 47.1%

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

      1. *-commutative 47.1%

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

      2. unpow2 47.1%

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

   7. Simplified 47.1%

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

   8. Taylor expanded in angle around 0 47.1%

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

      1. *-commutative 47.1%

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

      2. associate-*l* 47.1%

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

      3. unpow2 47.1%

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

   10. Simplified 47.1%

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

   11. Taylor expanded in angle around 0 47.1%

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

       1. unpow2 47.1%

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

       2. associate-*r* 47.1%

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

       3. *-commutative 47.1%

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

       4. associate-*r* 54.0%

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

   13. Simplified 54.0%

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

   if -9.19999999999999994e-136 < b < 4.5e51

   1. Initial program 55.6%

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

      1. associate-*l* 55.6%

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

      2. unpow2 55.6%

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

      3. unpow2 55.6%

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

      4. difference-of-squares 55.6%

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

   3. Simplified 55.6%

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

   4. Taylor expanded in angle around 0 54.2%

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

   5. Taylor expanded in b around 0 48.8%

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

      1. mul-1-neg 48.8%

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

      2. distribute-rgt-neg-in 48.8%

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

      3. *-commutative 48.8%

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

      4. distribute-rgt-neg-in 48.8%

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

      5. unpow2 48.8%

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

   7. Simplified 48.8%

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

4. Final simplification 51.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -9.2 \cdot 10^{-136} \lor \neg \left(b \leq 4.5 \cdot 10^{+51}\right):\\ \;\;\;\;0.011111111111111112 \cdot \left(\pi \cdot \left(b \cdot \left(b \cdot angle\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;0.011111111111111112 \cdot \left(angle \cdot \left(\pi \cdot \left(a \cdot \left(-a\right)\right)\right)\right)\\ \end{array} \]

Alternative 13: 34.2% 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 53.6%

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

   1. associate-*l* 53.6%

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

   2. unpow2 53.6%

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

   3. unpow2 53.6%

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

   4. difference-of-squares 56.9%

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

3. Simplified 56.9%

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

4. Taylor expanded in angle around 0 54.0%

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

5. Taylor expanded in b around inf 37.9%

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

   1. *-commutative 37.9%

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

   2. unpow2 37.9%

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

7. Simplified 37.9%

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

8. Final simplification 37.9%

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

Alternative 14: 37.5% accurate, 5.7× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 53.6%

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

   1. associate-*l* 53.6%

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

   2. unpow2 53.6%

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

   3. unpow2 53.6%

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

   4. difference-of-squares 56.9%

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

3. Simplified 56.9%

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

4. Taylor expanded in angle around 0 54.0%

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

5. Taylor expanded in b around inf 37.9%

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

   1. *-commutative 37.9%

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

   2. unpow2 37.9%

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

7. Simplified 37.9%

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

8. Taylor expanded in angle around 0 37.9%

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

   1. *-commutative 37.9%

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

   2. associate-*l* 37.9%

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

   3. unpow2 37.9%

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

10. Simplified 37.9%

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

11. Taylor expanded in angle around 0 37.9%

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

    1. unpow2 37.9%

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

    2. associate-*r* 37.9%

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

    3. *-commutative 37.9%

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

    4. associate-*r* 40.4%

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

13. Simplified 40.4%

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

14. Final simplification 40.4%

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

Reproduce

herbie shell --seed 2023178 
(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)))))