ab-angle->ABCF B

Percentage Accurate: 54.1% → 67.0%

Time: 44.6s

Alternatives: 13

Speedup: 32.2×

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

Specification

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \pi \cdot \frac{angle}{180}\\ \left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin t\_0\right) \cdot \cos t\_0 \end{array} \end{array} \]

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 54.1% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \pi \cdot \frac{angle}{180}\\ \left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin t\_0\right) \cdot \cos t\_0 \end{array} \end{array} \]

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 67.0% accurate, 0.5× speedup

Math

\[\begin{array}{l} angle_m = \left|angle\right| \\ angle_s = \mathsf{copysign}\left(1, angle\right) \\ \begin{array}{l} t_0 := \pi \cdot \left(angle\_m \cdot 0.005555555555555556\right)\\ angle\_s \cdot \begin{array}{l} \mathbf{if}\;\frac{angle\_m}{180} \leq 2 \cdot 10^{+93}:\\ \;\;\;\;2 \cdot \left(\left(\left(\sin \left(0.005555555555555556 \cdot \left(angle\_m \cdot \pi\right)\right) \cdot \left(a + b\right)\right) \cdot \left(b - a\right)\right) \cdot \cos \left({\left({t\_0}^{3}\right)}^{0.3333333333333333}\right)\right)\\ \mathbf{elif}\;\frac{angle\_m}{180} \leq 1.5 \cdot 10^{+190}:\\ \;\;\;\;2 \cdot \left(\left(\left(b - a\right) \cdot \left(\left(a + b\right) \cdot \sin \left(\frac{angle\_m \cdot \pi}{180}\right)\right)\right) \cdot \left|\cos \left(\left(angle\_m \cdot \pi\right) \cdot -0.005555555555555556\right)\right|\right)\\ \mathbf{elif}\;\frac{angle\_m}{180} \leq 5 \cdot 10^{+259}:\\ \;\;\;\;\left(\left(a + b\right) \cdot \left(b - a\right)\right) \cdot \left(2 \cdot \left(\sin \left(\frac{angle\_m}{180} \cdot \pi\right) \cdot \cos \left({\left(\sqrt[3]{{\left(\sqrt{\pi}\right)}^{2} \cdot \left(angle\_m \cdot 0.005555555555555556\right)}\right)}^{3}\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \left(\sqrt{{\left(\sin t\_0 \cdot \left({b}^{2} - {a}^{2}\right)\right)}^{2}} \cdot {\left(\sqrt[3]{\cos t\_0}\right)}^{3}\right)\\ \end{array} \end{array} \end{array} \]

FPCore

angle_m = (fabs.f64 angle)
angle_s = (copysign.f64 1 angle)
(FPCore (angle_s a b angle_m)
 :precision binary64
 (let* ((t_0 (* PI (* angle_m 0.005555555555555556))))
   (*
    angle_s
    (if (<= (/ angle_m 180.0) 2e+93)
      (*
       2.0
       (*
        (* (* (sin (* 0.005555555555555556 (* angle_m PI))) (+ a b)) (- b a))
        (cos (pow (pow t_0 3.0) 0.3333333333333333))))
      (if (<= (/ angle_m 180.0) 1.5e+190)
        (*
         2.0
         (*
          (* (- b a) (* (+ a b) (sin (/ (* angle_m PI) 180.0))))
          (fabs (cos (* (* angle_m PI) -0.005555555555555556)))))
        (if (<= (/ angle_m 180.0) 5e+259)
          (*
           (* (+ a b) (- b a))
           (*
            2.0
            (*
             (sin (* (/ angle_m 180.0) PI))
             (cos
              (pow
               (cbrt (* (pow (sqrt PI) 2.0) (* angle_m 0.005555555555555556)))
               3.0)))))
          (*
           2.0
           (*
            (sqrt (pow (* (sin t_0) (- (pow b 2.0) (pow a 2.0))) 2.0))
            (pow (cbrt (cos t_0)) 3.0)))))))))

C

angle_m = fabs(angle);
angle_s = copysign(1.0, angle);
double code(double angle_s, double a, double b, double angle_m) {
	double t_0 = ((double) M_PI) * (angle_m * 0.005555555555555556);
	double tmp;
	if ((angle_m / 180.0) <= 2e+93) {
		tmp = 2.0 * (((sin((0.005555555555555556 * (angle_m * ((double) M_PI)))) * (a + b)) * (b - a)) * cos(pow(pow(t_0, 3.0), 0.3333333333333333)));
	} else if ((angle_m / 180.0) <= 1.5e+190) {
		tmp = 2.0 * (((b - a) * ((a + b) * sin(((angle_m * ((double) M_PI)) / 180.0)))) * fabs(cos(((angle_m * ((double) M_PI)) * -0.005555555555555556))));
	} else if ((angle_m / 180.0) <= 5e+259) {
		tmp = ((a + b) * (b - a)) * (2.0 * (sin(((angle_m / 180.0) * ((double) M_PI))) * cos(pow(cbrt((pow(sqrt(((double) M_PI)), 2.0) * (angle_m * 0.005555555555555556))), 3.0))));
	} else {
		tmp = 2.0 * (sqrt(pow((sin(t_0) * (pow(b, 2.0) - pow(a, 2.0))), 2.0)) * pow(cbrt(cos(t_0)), 3.0));
	}
	return angle_s * tmp;
}

Java

angle_m = Math.abs(angle);
angle_s = Math.copySign(1.0, angle);
public static double code(double angle_s, double a, double b, double angle_m) {
	double t_0 = Math.PI * (angle_m * 0.005555555555555556);
	double tmp;
	if ((angle_m / 180.0) <= 2e+93) {
		tmp = 2.0 * (((Math.sin((0.005555555555555556 * (angle_m * Math.PI))) * (a + b)) * (b - a)) * Math.cos(Math.pow(Math.pow(t_0, 3.0), 0.3333333333333333)));
	} else if ((angle_m / 180.0) <= 1.5e+190) {
		tmp = 2.0 * (((b - a) * ((a + b) * Math.sin(((angle_m * Math.PI) / 180.0)))) * Math.abs(Math.cos(((angle_m * Math.PI) * -0.005555555555555556))));
	} else if ((angle_m / 180.0) <= 5e+259) {
		tmp = ((a + b) * (b - a)) * (2.0 * (Math.sin(((angle_m / 180.0) * Math.PI)) * Math.cos(Math.pow(Math.cbrt((Math.pow(Math.sqrt(Math.PI), 2.0) * (angle_m * 0.005555555555555556))), 3.0))));
	} else {
		tmp = 2.0 * (Math.sqrt(Math.pow((Math.sin(t_0) * (Math.pow(b, 2.0) - Math.pow(a, 2.0))), 2.0)) * Math.pow(Math.cbrt(Math.cos(t_0)), 3.0));
	}
	return angle_s * tmp;
}

Julia

angle_m = abs(angle)
angle_s = copysign(1.0, angle)
function code(angle_s, a, b, angle_m)
	t_0 = Float64(pi * Float64(angle_m * 0.005555555555555556))
	tmp = 0.0
	if (Float64(angle_m / 180.0) <= 2e+93)
		tmp = Float64(2.0 * Float64(Float64(Float64(sin(Float64(0.005555555555555556 * Float64(angle_m * pi))) * Float64(a + b)) * Float64(b - a)) * cos(((t_0 ^ 3.0) ^ 0.3333333333333333))));
	elseif (Float64(angle_m / 180.0) <= 1.5e+190)
		tmp = Float64(2.0 * Float64(Float64(Float64(b - a) * Float64(Float64(a + b) * sin(Float64(Float64(angle_m * pi) / 180.0)))) * abs(cos(Float64(Float64(angle_m * pi) * -0.005555555555555556)))));
	elseif (Float64(angle_m / 180.0) <= 5e+259)
		tmp = Float64(Float64(Float64(a + b) * Float64(b - a)) * Float64(2.0 * Float64(sin(Float64(Float64(angle_m / 180.0) * pi)) * cos((cbrt(Float64((sqrt(pi) ^ 2.0) * Float64(angle_m * 0.005555555555555556))) ^ 3.0)))));
	else
		tmp = Float64(2.0 * Float64(sqrt((Float64(sin(t_0) * Float64((b ^ 2.0) - (a ^ 2.0))) ^ 2.0)) * (cbrt(cos(t_0)) ^ 3.0)));
	end
	return Float64(angle_s * tmp)
end

Wolfram

angle_m = N[Abs[angle], $MachinePrecision]
angle_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[angle]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[angle$95$s_, a_, b_, angle$95$m_] := Block[{t$95$0 = N[(Pi * N[(angle$95$m * 0.005555555555555556), $MachinePrecision]), $MachinePrecision]}, N[(angle$95$s * If[LessEqual[N[(angle$95$m / 180.0), $MachinePrecision], 2e+93], N[(2.0 * N[(N[(N[(N[Sin[N[(0.005555555555555556 * N[(angle$95$m * Pi), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * N[(a + b), $MachinePrecision]), $MachinePrecision] * N[(b - a), $MachinePrecision]), $MachinePrecision] * N[Cos[N[Power[N[Power[t$95$0, 3.0], $MachinePrecision], 0.3333333333333333], $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[N[(angle$95$m / 180.0), $MachinePrecision], 1.5e+190], N[(2.0 * N[(N[(N[(b - a), $MachinePrecision] * N[(N[(a + b), $MachinePrecision] * N[Sin[N[(N[(angle$95$m * Pi), $MachinePrecision] / 180.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[Abs[N[Cos[N[(N[(angle$95$m * Pi), $MachinePrecision] * -0.005555555555555556), $MachinePrecision]], $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[N[(angle$95$m / 180.0), $MachinePrecision], 5e+259], N[(N[(N[(a + b), $MachinePrecision] * N[(b - a), $MachinePrecision]), $MachinePrecision] * N[(2.0 * N[(N[Sin[N[(N[(angle$95$m / 180.0), $MachinePrecision] * Pi), $MachinePrecision]], $MachinePrecision] * N[Cos[N[Power[N[Power[N[(N[Power[N[Sqrt[Pi], $MachinePrecision], 2.0], $MachinePrecision] * N[(angle$95$m * 0.005555555555555556), $MachinePrecision]), $MachinePrecision], 1/3], $MachinePrecision], 3.0], $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(2.0 * N[(N[Sqrt[N[Power[N[(N[Sin[t$95$0], $MachinePrecision] * N[(N[Power[b, 2.0], $MachinePrecision] - N[Power[a, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]], $MachinePrecision] * N[Power[N[Power[N[Cos[t$95$0], $MachinePrecision], 1/3], $MachinePrecision], 3.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]), $MachinePrecision]]

Derivation

1. Split input into 4 regimes

2. if (/.f64 angle 180) < 2.00000000000000009e93

   1. Initial program 57.6%

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

      1. associate-*l* 57.6%

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

      2. *-commutative 57.6%

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

      3. associate-*l* 57.6%

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

   3. Simplified 57.6%

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

      1. unpow2 57.6%

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

      2. unpow2 57.6%

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

      3. difference-of-squares 62.2%

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

   6. Applied egg-rr 62.2%

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

   7. Taylor expanded in angle around inf 63.1%

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

      1. *-commutative 63.1%

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

      2. associate-*r* 73.3%

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

   9. Simplified 73.3%

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

       1. *-commutative 73.3%

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

       2. *-commutative 73.3%

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

       3. associate-*r* 74.1%

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

       4. add-cube-cbrt 73.1%

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

       5. unpow3 73.0%

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

       6. add-cbrt-cube 64.9%

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

       7. pow1/3 51.9%

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

       8. pow3 52.0%

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

       9. unpow3 52.0%

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

       10. add-cube-cbrt 51.4%

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

       11. *-commutative 51.4%

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

   11. Applied egg-rr 51.4%

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

   if 2.00000000000000009e93 < (/.f64 angle 180) < 1.49999999999999991e190

   1. Initial program 34.4%

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

      1. associate-*l* 34.4%

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

      2. *-commutative 34.4%

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

      3. associate-*l* 34.4%

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

   3. Simplified 34.4%

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

      1. unpow2 34.4%

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

      2. unpow2 34.4%

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

      3. difference-of-squares 34.4%

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

   6. Applied egg-rr 34.4%

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

   7. Taylor expanded in angle around inf 33.6%

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

      1. *-commutative 33.6%

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

      2. associate-*r* 33.6%

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

   9. Simplified 33.6%

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

       1. associate-*r* 41.2%

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

       2. *-commutative 41.2%

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

       3. metadata-eval 41.2%

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

       4. div-inv 34.0%

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

       5. *-commutative 34.0%

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

       6. add-sqr-sqrt 25.5%

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

       7. sqrt-unprod 56.8%

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

       8. pow2 56.8%

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

       9. div-inv 56.8%

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

       10. metadata-eval 56.8%

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

       11. *-commutative 56.8%

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

   11. Applied egg-rr 56.8%

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

       1. unpow2 56.8%

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

       2. rem-sqrt-square 56.8%

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

       3. *-commutative 56.8%

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

       4. associate-*r* 57.6%

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

       5. *-commutative 57.6%

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

       6. metadata-eval 57.6%

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

       7. rem-cube-cbrt 56.8%

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

       8. distribute-rgt-neg-in 56.8%

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

       9. associate-*r* 56.9%

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

       10. cos-neg 56.9%

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

       11. associate-*r* 56.8%

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

       12. rem-cube-cbrt 57.6%

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

   13. Simplified 57.6%

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

       1. metadata-eval 57.6%

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

       2. *-commutative 57.6%

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

       3. associate-/r/ 59.8%

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

       4. clear-num 58.0%

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

   15. Applied egg-rr 58.0%

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

   if 1.49999999999999991e190 < (/.f64 angle 180) < 5.00000000000000033e259

   1. Initial program 46.4%

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

      1. associate-*l* 46.4%

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

      2. *-commutative 46.4%

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

      3. associate-*l* 46.4%

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

   3. Simplified 46.4%

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

      1. unpow2 46.4%

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

      2. unpow2 46.4%

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

      3. difference-of-squares 54.0%

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

   6. Applied egg-rr 54.0%

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

      1. add-cube-cbrt 20.3%

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

      2. pow3 28.1%

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

      3. div-inv 29.2%

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

      4. metadata-eval 29.2%

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

   8. Applied egg-rr 29.2%

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

      1. add-sqr-sqrt 49.7%

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

      2. pow2 49.7%

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

   10. Applied egg-rr 49.7%

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

   if 5.00000000000000033e259 < (/.f64 angle 180)

   1. Initial program 15.2%

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

      1. associate-*l* 15.2%

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

      2. associate-*l* 15.2%

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

   3. Simplified 15.2%

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

      1. add-cube-cbrt 15.2%

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

      2. pow3 15.2%

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

      3. div-inv 15.5%

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

      4. metadata-eval 15.5%

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

   6. Applied egg-rr 15.5%

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

      1. add-sqr-sqrt 14.8%

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

      2. sqrt-unprod 49.8%

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

      3. pow2 49.8%

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

      4. *-commutative 49.8%

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

      5. div-inv 49.8%

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

      6. metadata-eval 49.8%

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

   8. Applied egg-rr 49.8%

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

4. Final simplification 51.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{angle}{180} \leq 2 \cdot 10^{+93}:\\ \;\;\;\;2 \cdot \left(\left(\left(\sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right) \cdot \left(a + b\right)\right) \cdot \left(b - a\right)\right) \cdot \cos \left({\left({\left(\pi \cdot \left(angle \cdot 0.005555555555555556\right)\right)}^{3}\right)}^{0.3333333333333333}\right)\right)\\ \mathbf{elif}\;\frac{angle}{180} \leq 1.5 \cdot 10^{+190}:\\ \;\;\;\;2 \cdot \left(\left(\left(b - a\right) \cdot \left(\left(a + b\right) \cdot \sin \left(\frac{angle \cdot \pi}{180}\right)\right)\right) \cdot \left|\cos \left(\left(angle \cdot \pi\right) \cdot -0.005555555555555556\right)\right|\right)\\ \mathbf{elif}\;\frac{angle}{180} \leq 5 \cdot 10^{+259}:\\ \;\;\;\;\left(\left(a + b\right) \cdot \left(b - a\right)\right) \cdot \left(2 \cdot \left(\sin \left(\frac{angle}{180} \cdot \pi\right) \cdot \cos \left({\left(\sqrt[3]{{\left(\sqrt{\pi}\right)}^{2} \cdot \left(angle \cdot 0.005555555555555556\right)}\right)}^{3}\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \left(\sqrt{{\left(\sin \left(\pi \cdot \left(angle \cdot 0.005555555555555556\right)\right) \cdot \left({b}^{2} - {a}^{2}\right)\right)}^{2}} \cdot {\left(\sqrt[3]{\cos \left(\pi \cdot \left(angle \cdot 0.005555555555555556\right)\right)}\right)}^{3}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 66.9% accurate, 0.7× speedup

Math

\[\begin{array}{l} angle_m = \left|angle\right| \\ angle_s = \mathsf{copysign}\left(1, angle\right) \\ \begin{array}{l} t_0 := \frac{angle\_m}{180} \cdot \pi\\ t_1 := \pi \cdot \left(angle\_m \cdot 0.005555555555555556\right)\\ angle\_s \cdot \begin{array}{l} \mathbf{if}\;\frac{angle\_m}{180} \leq 2 \cdot 10^{+93}:\\ \;\;\;\;2 \cdot \left(\left(\left(\sin \left(0.005555555555555556 \cdot \left(angle\_m \cdot \pi\right)\right) \cdot \left(a + b\right)\right) \cdot \left(b - a\right)\right) \cdot \cos \left({\left({t\_1}^{3}\right)}^{0.3333333333333333}\right)\right)\\ \mathbf{elif}\;\frac{angle\_m}{180} \leq 1.5 \cdot 10^{+190}:\\ \;\;\;\;2 \cdot \left(\left(\left(b - a\right) \cdot \left(\left(a + b\right) \cdot \sin \left(\frac{angle\_m \cdot \pi}{180}\right)\right)\right) \cdot \left|\cos \left(\left(angle\_m \cdot \pi\right) \cdot -0.005555555555555556\right)\right|\right)\\ \mathbf{elif}\;\frac{angle\_m}{180} \leq 5 \cdot 10^{+259}:\\ \;\;\;\;\left(\left(a + b\right) \cdot \left(b - a\right)\right) \cdot \left(2 \cdot \left(\sin t\_0 \cdot \cos \left({\left(\sqrt[3]{{\left(\sqrt{\pi}\right)}^{2} \cdot \left(angle\_m \cdot 0.005555555555555556\right)}\right)}^{3}\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \left({\left({\left(\sin t\_1 \cdot \left({b}^{2} - {a}^{2}\right)\right)}^{3}\right)}^{0.3333333333333333} \cdot \cos t\_0\right)\\ \end{array} \end{array} \end{array} \]

FPCore

angle_m = (fabs.f64 angle)
angle_s = (copysign.f64 1 angle)
(FPCore (angle_s a b angle_m)
 :precision binary64
 (let* ((t_0 (* (/ angle_m 180.0) PI))
        (t_1 (* PI (* angle_m 0.005555555555555556))))
   (*
    angle_s
    (if (<= (/ angle_m 180.0) 2e+93)
      (*
       2.0
       (*
        (* (* (sin (* 0.005555555555555556 (* angle_m PI))) (+ a b)) (- b a))
        (cos (pow (pow t_1 3.0) 0.3333333333333333))))
      (if (<= (/ angle_m 180.0) 1.5e+190)
        (*
         2.0
         (*
          (* (- b a) (* (+ a b) (sin (/ (* angle_m PI) 180.0))))
          (fabs (cos (* (* angle_m PI) -0.005555555555555556)))))
        (if (<= (/ angle_m 180.0) 5e+259)
          (*
           (* (+ a b) (- b a))
           (*
            2.0
            (*
             (sin t_0)
             (cos
              (pow
               (cbrt (* (pow (sqrt PI) 2.0) (* angle_m 0.005555555555555556)))
               3.0)))))
          (*
           2.0
           (*
            (pow
             (pow (* (sin t_1) (- (pow b 2.0) (pow a 2.0))) 3.0)
             0.3333333333333333)
            (cos t_0)))))))))

C

angle_m = fabs(angle);
angle_s = copysign(1.0, angle);
double code(double angle_s, double a, double b, double angle_m) {
	double t_0 = (angle_m / 180.0) * ((double) M_PI);
	double t_1 = ((double) M_PI) * (angle_m * 0.005555555555555556);
	double tmp;
	if ((angle_m / 180.0) <= 2e+93) {
		tmp = 2.0 * (((sin((0.005555555555555556 * (angle_m * ((double) M_PI)))) * (a + b)) * (b - a)) * cos(pow(pow(t_1, 3.0), 0.3333333333333333)));
	} else if ((angle_m / 180.0) <= 1.5e+190) {
		tmp = 2.0 * (((b - a) * ((a + b) * sin(((angle_m * ((double) M_PI)) / 180.0)))) * fabs(cos(((angle_m * ((double) M_PI)) * -0.005555555555555556))));
	} else if ((angle_m / 180.0) <= 5e+259) {
		tmp = ((a + b) * (b - a)) * (2.0 * (sin(t_0) * cos(pow(cbrt((pow(sqrt(((double) M_PI)), 2.0) * (angle_m * 0.005555555555555556))), 3.0))));
	} else {
		tmp = 2.0 * (pow(pow((sin(t_1) * (pow(b, 2.0) - pow(a, 2.0))), 3.0), 0.3333333333333333) * cos(t_0));
	}
	return angle_s * tmp;
}

Java

angle_m = Math.abs(angle);
angle_s = Math.copySign(1.0, angle);
public static double code(double angle_s, double a, double b, double angle_m) {
	double t_0 = (angle_m / 180.0) * Math.PI;
	double t_1 = Math.PI * (angle_m * 0.005555555555555556);
	double tmp;
	if ((angle_m / 180.0) <= 2e+93) {
		tmp = 2.0 * (((Math.sin((0.005555555555555556 * (angle_m * Math.PI))) * (a + b)) * (b - a)) * Math.cos(Math.pow(Math.pow(t_1, 3.0), 0.3333333333333333)));
	} else if ((angle_m / 180.0) <= 1.5e+190) {
		tmp = 2.0 * (((b - a) * ((a + b) * Math.sin(((angle_m * Math.PI) / 180.0)))) * Math.abs(Math.cos(((angle_m * Math.PI) * -0.005555555555555556))));
	} else if ((angle_m / 180.0) <= 5e+259) {
		tmp = ((a + b) * (b - a)) * (2.0 * (Math.sin(t_0) * Math.cos(Math.pow(Math.cbrt((Math.pow(Math.sqrt(Math.PI), 2.0) * (angle_m * 0.005555555555555556))), 3.0))));
	} else {
		tmp = 2.0 * (Math.pow(Math.pow((Math.sin(t_1) * (Math.pow(b, 2.0) - Math.pow(a, 2.0))), 3.0), 0.3333333333333333) * Math.cos(t_0));
	}
	return angle_s * tmp;
}

Julia

angle_m = abs(angle)
angle_s = copysign(1.0, angle)
function code(angle_s, a, b, angle_m)
	t_0 = Float64(Float64(angle_m / 180.0) * pi)
	t_1 = Float64(pi * Float64(angle_m * 0.005555555555555556))
	tmp = 0.0
	if (Float64(angle_m / 180.0) <= 2e+93)
		tmp = Float64(2.0 * Float64(Float64(Float64(sin(Float64(0.005555555555555556 * Float64(angle_m * pi))) * Float64(a + b)) * Float64(b - a)) * cos(((t_1 ^ 3.0) ^ 0.3333333333333333))));
	elseif (Float64(angle_m / 180.0) <= 1.5e+190)
		tmp = Float64(2.0 * Float64(Float64(Float64(b - a) * Float64(Float64(a + b) * sin(Float64(Float64(angle_m * pi) / 180.0)))) * abs(cos(Float64(Float64(angle_m * pi) * -0.005555555555555556)))));
	elseif (Float64(angle_m / 180.0) <= 5e+259)
		tmp = Float64(Float64(Float64(a + b) * Float64(b - a)) * Float64(2.0 * Float64(sin(t_0) * cos((cbrt(Float64((sqrt(pi) ^ 2.0) * Float64(angle_m * 0.005555555555555556))) ^ 3.0)))));
	else
		tmp = Float64(2.0 * Float64(((Float64(sin(t_1) * Float64((b ^ 2.0) - (a ^ 2.0))) ^ 3.0) ^ 0.3333333333333333) * cos(t_0)));
	end
	return Float64(angle_s * tmp)
end

Wolfram

angle_m = N[Abs[angle], $MachinePrecision]
angle_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[angle]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[angle$95$s_, a_, b_, angle$95$m_] := Block[{t$95$0 = N[(N[(angle$95$m / 180.0), $MachinePrecision] * Pi), $MachinePrecision]}, Block[{t$95$1 = N[(Pi * N[(angle$95$m * 0.005555555555555556), $MachinePrecision]), $MachinePrecision]}, N[(angle$95$s * If[LessEqual[N[(angle$95$m / 180.0), $MachinePrecision], 2e+93], N[(2.0 * N[(N[(N[(N[Sin[N[(0.005555555555555556 * N[(angle$95$m * Pi), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * N[(a + b), $MachinePrecision]), $MachinePrecision] * N[(b - a), $MachinePrecision]), $MachinePrecision] * N[Cos[N[Power[N[Power[t$95$1, 3.0], $MachinePrecision], 0.3333333333333333], $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[N[(angle$95$m / 180.0), $MachinePrecision], 1.5e+190], N[(2.0 * N[(N[(N[(b - a), $MachinePrecision] * N[(N[(a + b), $MachinePrecision] * N[Sin[N[(N[(angle$95$m * Pi), $MachinePrecision] / 180.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[Abs[N[Cos[N[(N[(angle$95$m * Pi), $MachinePrecision] * -0.005555555555555556), $MachinePrecision]], $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[N[(angle$95$m / 180.0), $MachinePrecision], 5e+259], N[(N[(N[(a + b), $MachinePrecision] * N[(b - a), $MachinePrecision]), $MachinePrecision] * N[(2.0 * N[(N[Sin[t$95$0], $MachinePrecision] * N[Cos[N[Power[N[Power[N[(N[Power[N[Sqrt[Pi], $MachinePrecision], 2.0], $MachinePrecision] * N[(angle$95$m * 0.005555555555555556), $MachinePrecision]), $MachinePrecision], 1/3], $MachinePrecision], 3.0], $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(2.0 * N[(N[Power[N[Power[N[(N[Sin[t$95$1], $MachinePrecision] * N[(N[Power[b, 2.0], $MachinePrecision] - N[Power[a, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 3.0], $MachinePrecision], 0.3333333333333333], $MachinePrecision] * N[Cos[t$95$0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]), $MachinePrecision]]]

Derivation

1. Split input into 4 regimes

2. if (/.f64 angle 180) < 2.00000000000000009e93

   1. Initial program 57.6%

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

      1. associate-*l* 57.6%

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

      2. *-commutative 57.6%

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

      3. associate-*l* 57.6%

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

   3. Simplified 57.6%

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

      1. unpow2 57.6%

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

      2. unpow2 57.6%

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

      3. difference-of-squares 62.2%

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

   6. Applied egg-rr 62.2%

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

   7. Taylor expanded in angle around inf 63.1%

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

      1. *-commutative 63.1%

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

      2. associate-*r* 73.3%

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

   9. Simplified 73.3%

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

       1. *-commutative 73.3%

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

       2. *-commutative 73.3%

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

       3. associate-*r* 74.1%

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

       4. add-cube-cbrt 73.1%

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

       5. unpow3 73.0%

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

       6. add-cbrt-cube 64.9%

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

       7. pow1/3 51.9%

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

       8. pow3 52.0%

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

       9. unpow3 52.0%

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

       10. add-cube-cbrt 51.4%

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

       11. *-commutative 51.4%

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

   11. Applied egg-rr 51.4%

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

   if 2.00000000000000009e93 < (/.f64 angle 180) < 1.49999999999999991e190

   1. Initial program 34.4%

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

      1. associate-*l* 34.4%

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

      2. *-commutative 34.4%

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

      3. associate-*l* 34.4%

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

   3. Simplified 34.4%

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

      1. unpow2 34.4%

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

      2. unpow2 34.4%

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

      3. difference-of-squares 34.4%

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

   6. Applied egg-rr 34.4%

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

   7. Taylor expanded in angle around inf 33.6%

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

      1. *-commutative 33.6%

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

      2. associate-*r* 33.6%

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

   9. Simplified 33.6%

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

       1. associate-*r* 41.2%

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

       2. *-commutative 41.2%

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

       3. metadata-eval 41.2%

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

       4. div-inv 34.0%

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

       5. *-commutative 34.0%

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

       6. add-sqr-sqrt 25.5%

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

       7. sqrt-unprod 56.8%

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

       8. pow2 56.8%

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

       9. div-inv 56.8%

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

       10. metadata-eval 56.8%

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

       11. *-commutative 56.8%

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

   11. Applied egg-rr 56.8%

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

       1. unpow2 56.8%

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

       2. rem-sqrt-square 56.8%

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

       3. *-commutative 56.8%

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

       4. associate-*r* 57.6%

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

       5. *-commutative 57.6%

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

       6. metadata-eval 57.6%

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

       7. rem-cube-cbrt 56.8%

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

       8. distribute-rgt-neg-in 56.8%

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

       9. associate-*r* 56.9%

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

       10. cos-neg 56.9%

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

       11. associate-*r* 56.8%

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

       12. rem-cube-cbrt 57.6%

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

   13. Simplified 57.6%

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

       1. metadata-eval 57.6%

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

       2. *-commutative 57.6%

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

       3. associate-/r/ 59.8%

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

       4. clear-num 58.0%

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

   15. Applied egg-rr 58.0%

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

   if 1.49999999999999991e190 < (/.f64 angle 180) < 5.00000000000000033e259

   1. Initial program 46.4%

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

      1. associate-*l* 46.4%

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

      2. *-commutative 46.4%

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

      3. associate-*l* 46.4%

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

   3. Simplified 46.4%

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

      1. unpow2 46.4%

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

      2. unpow2 46.4%

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

      3. difference-of-squares 54.0%

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

   6. Applied egg-rr 54.0%

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

      1. add-cube-cbrt 20.3%

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

      2. pow3 28.1%

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

      3. div-inv 29.2%

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

      4. metadata-eval 29.2%

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

   8. Applied egg-rr 29.2%

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

      1. add-sqr-sqrt 49.7%

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

      2. pow2 49.7%

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

   10. Applied egg-rr 49.7%

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

   if 5.00000000000000033e259 < (/.f64 angle 180)

   1. Initial program 15.2%

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

      1. associate-*l* 15.2%

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

      2. associate-*l* 15.2%

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

   3. Simplified 15.2%

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

      1. add-cbrt-cube 14.3%

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

      2. pow1/3 48.1%

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

      3. pow3 48.1%

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

      4. *-commutative 48.1%

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

      5. div-inv 48.1%

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

      6. metadata-eval 48.1%

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

   6. Applied egg-rr 48.1%

      \[\leadsto 2 \cdot \left(\color{blue}{{\left({\left(\sin \left(\pi \cdot \left(angle \cdot 0.005555555555555556\right)\right) \cdot \left({b}^{2} - {a}^{2}\right)\right)}^{3}\right)}^{0.3333333333333333}} \cdot \cos \left(\pi \cdot \frac{angle}{180}\right)\right) \]
3. Recombined 4 regimes into one program.

4. Final simplification 51.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{angle}{180} \leq 2 \cdot 10^{+93}:\\ \;\;\;\;2 \cdot \left(\left(\left(\sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right) \cdot \left(a + b\right)\right) \cdot \left(b - a\right)\right) \cdot \cos \left({\left({\left(\pi \cdot \left(angle \cdot 0.005555555555555556\right)\right)}^{3}\right)}^{0.3333333333333333}\right)\right)\\ \mathbf{elif}\;\frac{angle}{180} \leq 1.5 \cdot 10^{+190}:\\ \;\;\;\;2 \cdot \left(\left(\left(b - a\right) \cdot \left(\left(a + b\right) \cdot \sin \left(\frac{angle \cdot \pi}{180}\right)\right)\right) \cdot \left|\cos \left(\left(angle \cdot \pi\right) \cdot -0.005555555555555556\right)\right|\right)\\ \mathbf{elif}\;\frac{angle}{180} \leq 5 \cdot 10^{+259}:\\ \;\;\;\;\left(\left(a + b\right) \cdot \left(b - a\right)\right) \cdot \left(2 \cdot \left(\sin \left(\frac{angle}{180} \cdot \pi\right) \cdot \cos \left({\left(\sqrt[3]{{\left(\sqrt{\pi}\right)}^{2} \cdot \left(angle \cdot 0.005555555555555556\right)}\right)}^{3}\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \left({\left({\left(\sin \left(\pi \cdot \left(angle \cdot 0.005555555555555556\right)\right) \cdot \left({b}^{2} - {a}^{2}\right)\right)}^{3}\right)}^{0.3333333333333333} \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 67.4% accurate, 1.0× speedup

Math

\[\begin{array}{l} angle_m = \left|angle\right| \\ angle_s = \mathsf{copysign}\left(1, angle\right) \\ angle\_s \cdot \begin{array}{l} \mathbf{if}\;\frac{angle\_m}{180} \leq 10^{+87}:\\ \;\;\;\;2 \cdot \left(\left(\left(\sin \left(0.005555555555555556 \cdot \left(angle\_m \cdot \pi\right)\right) \cdot \left(a + b\right)\right) \cdot \left(b - a\right)\right) \cdot \cos \left({\left({\left(\pi \cdot \left(angle\_m \cdot 0.005555555555555556\right)\right)}^{3}\right)}^{0.3333333333333333}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \left(\left|\cos \left(\left(angle\_m \cdot \pi\right) \cdot -0.005555555555555556\right)\right| \cdot \left(\left(b - a\right) \cdot \left(\left(a + b\right) \cdot \sin \left(\frac{-1}{\frac{-180}{angle\_m \cdot \pi}}\right)\right)\right)\right)\\ \end{array} \end{array} \]

FPCore

angle_m = (fabs.f64 angle)
angle_s = (copysign.f64 1 angle)
(FPCore (angle_s a b angle_m)
 :precision binary64
 (*
  angle_s
  (if (<= (/ angle_m 180.0) 1e+87)
    (*
     2.0
     (*
      (* (* (sin (* 0.005555555555555556 (* angle_m PI))) (+ a b)) (- b a))
      (cos
       (pow
        (pow (* PI (* angle_m 0.005555555555555556)) 3.0)
        0.3333333333333333))))
    (*
     2.0
     (*
      (fabs (cos (* (* angle_m PI) -0.005555555555555556)))
      (* (- b a) (* (+ a b) (sin (/ -1.0 (/ -180.0 (* angle_m PI)))))))))))

C

angle_m = fabs(angle);
angle_s = copysign(1.0, angle);
double code(double angle_s, double a, double b, double angle_m) {
	double tmp;
	if ((angle_m / 180.0) <= 1e+87) {
		tmp = 2.0 * (((sin((0.005555555555555556 * (angle_m * ((double) M_PI)))) * (a + b)) * (b - a)) * cos(pow(pow((((double) M_PI) * (angle_m * 0.005555555555555556)), 3.0), 0.3333333333333333)));
	} else {
		tmp = 2.0 * (fabs(cos(((angle_m * ((double) M_PI)) * -0.005555555555555556))) * ((b - a) * ((a + b) * sin((-1.0 / (-180.0 / (angle_m * ((double) M_PI))))))));
	}
	return angle_s * tmp;
}

Java

angle_m = Math.abs(angle);
angle_s = Math.copySign(1.0, angle);
public static double code(double angle_s, double a, double b, double angle_m) {
	double tmp;
	if ((angle_m / 180.0) <= 1e+87) {
		tmp = 2.0 * (((Math.sin((0.005555555555555556 * (angle_m * Math.PI))) * (a + b)) * (b - a)) * Math.cos(Math.pow(Math.pow((Math.PI * (angle_m * 0.005555555555555556)), 3.0), 0.3333333333333333)));
	} else {
		tmp = 2.0 * (Math.abs(Math.cos(((angle_m * Math.PI) * -0.005555555555555556))) * ((b - a) * ((a + b) * Math.sin((-1.0 / (-180.0 / (angle_m * Math.PI)))))));
	}
	return angle_s * tmp;
}

Python

angle_m = math.fabs(angle)
angle_s = math.copysign(1.0, angle)
def code(angle_s, a, b, angle_m):
	tmp = 0
	if (angle_m / 180.0) <= 1e+87:
		tmp = 2.0 * (((math.sin((0.005555555555555556 * (angle_m * math.pi))) * (a + b)) * (b - a)) * math.cos(math.pow(math.pow((math.pi * (angle_m * 0.005555555555555556)), 3.0), 0.3333333333333333)))
	else:
		tmp = 2.0 * (math.fabs(math.cos(((angle_m * math.pi) * -0.005555555555555556))) * ((b - a) * ((a + b) * math.sin((-1.0 / (-180.0 / (angle_m * math.pi)))))))
	return angle_s * tmp

Julia

angle_m = abs(angle)
angle_s = copysign(1.0, angle)
function code(angle_s, a, b, angle_m)
	tmp = 0.0
	if (Float64(angle_m / 180.0) <= 1e+87)
		tmp = Float64(2.0 * Float64(Float64(Float64(sin(Float64(0.005555555555555556 * Float64(angle_m * pi))) * Float64(a + b)) * Float64(b - a)) * cos(((Float64(pi * Float64(angle_m * 0.005555555555555556)) ^ 3.0) ^ 0.3333333333333333))));
	else
		tmp = Float64(2.0 * Float64(abs(cos(Float64(Float64(angle_m * pi) * -0.005555555555555556))) * Float64(Float64(b - a) * Float64(Float64(a + b) * sin(Float64(-1.0 / Float64(-180.0 / Float64(angle_m * pi))))))));
	end
	return Float64(angle_s * tmp)
end

MATLAB

angle_m = abs(angle);
angle_s = sign(angle) * abs(1.0);
function tmp_2 = code(angle_s, a, b, angle_m)
	tmp = 0.0;
	if ((angle_m / 180.0) <= 1e+87)
		tmp = 2.0 * (((sin((0.005555555555555556 * (angle_m * pi))) * (a + b)) * (b - a)) * cos((((pi * (angle_m * 0.005555555555555556)) ^ 3.0) ^ 0.3333333333333333)));
	else
		tmp = 2.0 * (abs(cos(((angle_m * pi) * -0.005555555555555556))) * ((b - a) * ((a + b) * sin((-1.0 / (-180.0 / (angle_m * pi)))))));
	end
	tmp_2 = angle_s * tmp;
end

Wolfram

angle_m = N[Abs[angle], $MachinePrecision]
angle_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[angle]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[angle$95$s_, a_, b_, angle$95$m_] := N[(angle$95$s * If[LessEqual[N[(angle$95$m / 180.0), $MachinePrecision], 1e+87], N[(2.0 * N[(N[(N[(N[Sin[N[(0.005555555555555556 * N[(angle$95$m * Pi), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * N[(a + b), $MachinePrecision]), $MachinePrecision] * N[(b - a), $MachinePrecision]), $MachinePrecision] * N[Cos[N[Power[N[Power[N[(Pi * N[(angle$95$m * 0.005555555555555556), $MachinePrecision]), $MachinePrecision], 3.0], $MachinePrecision], 0.3333333333333333], $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(2.0 * N[(N[Abs[N[Cos[N[(N[(angle$95$m * Pi), $MachinePrecision] * -0.005555555555555556), $MachinePrecision]], $MachinePrecision]], $MachinePrecision] * N[(N[(b - a), $MachinePrecision] * N[(N[(a + b), $MachinePrecision] * N[Sin[N[(-1.0 / N[(-180.0 / N[(angle$95$m * Pi), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if (/.f64 angle 180) < 9.9999999999999996e86

   1. Initial program 57.6%

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

      1. associate-*l* 57.6%

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

      2. *-commutative 57.6%

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

      3. associate-*l* 57.6%

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

   3. Simplified 57.6%

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

      1. unpow2 57.6%

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

      2. unpow2 57.6%

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

      3. difference-of-squares 62.2%

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

   6. Applied egg-rr 62.2%

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

   7. Taylor expanded in angle around inf 63.1%

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

      1. *-commutative 63.1%

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

      2. associate-*r* 73.3%

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

   9. Simplified 73.3%

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

       1. *-commutative 73.3%

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

       2. *-commutative 73.3%

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

       3. associate-*r* 74.1%

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

       4. add-cube-cbrt 73.1%

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

       5. unpow3 73.0%

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

       6. add-cbrt-cube 64.9%

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

       7. pow1/3 51.9%

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

       8. pow3 52.0%

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

       9. unpow3 52.0%

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

       10. add-cube-cbrt 51.4%

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

       11. *-commutative 51.4%

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

   11. Applied egg-rr 51.4%

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

   if 9.9999999999999996e86 < (/.f64 angle 180)

   1. Initial program 33.9%

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

      1. associate-*l* 33.9%

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

      2. *-commutative 33.9%

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

      3. associate-*l* 33.9%

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

   3. Simplified 33.9%

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

      1. unpow2 33.9%

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

      2. unpow2 33.9%

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

      3. difference-of-squares 36.7%

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

   6. Applied egg-rr 36.7%

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

   7. Taylor expanded in angle around inf 36.4%

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

      1. *-commutative 36.4%

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

      2. associate-*r* 36.4%

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

   9. Simplified 36.4%

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

       1. associate-*r* 34.1%

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

       2. *-commutative 34.1%

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

       3. metadata-eval 34.1%

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

       4. div-inv 31.3%

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

       5. *-commutative 31.3%

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

       6. add-sqr-sqrt 18.5%

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

       7. sqrt-unprod 51.5%

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

       8. pow2 51.5%

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

       9. div-inv 51.4%

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

       10. metadata-eval 51.4%

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

       11. *-commutative 51.4%

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

   11. Applied egg-rr 51.4%

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

       1. unpow2 51.4%

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

       2. rem-sqrt-square 51.4%

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

       3. *-commutative 51.4%

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

       4. associate-*r* 52.0%

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

       5. *-commutative 52.0%

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

       6. metadata-eval 52.0%

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

       7. rem-cube-cbrt 51.4%

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

       8. distribute-rgt-neg-in 51.4%

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

       9. associate-*r* 51.4%

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

       10. cos-neg 51.4%

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

       11. associate-*r* 51.4%

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

       12. rem-cube-cbrt 52.0%

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

   13. Simplified 52.0%

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

       1. metadata-eval 52.0%

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

       2. *-commutative 52.0%

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

       3. associate-/r/ 50.1%

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

       4. frac-2neg 50.1%

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

       5. metadata-eval 50.1%

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

       6. distribute-neg-frac 50.1%

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

       7. metadata-eval 50.1%

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

   15. Applied egg-rr 50.1%

       \[\leadsto 2 \cdot \left(\left(\left(\sin \color{blue}{\left(\frac{-1}{\frac{-180}{\pi \cdot angle}}\right)} \cdot \left(a + b\right)\right) \cdot \left(b - a\right)\right) \cdot \left|\cos \left(\left(angle \cdot \pi\right) \cdot -0.005555555555555556\right)\right|\right) \]
3. Recombined 2 regimes into one program.

4. Final simplification 51.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{angle}{180} \leq 10^{+87}:\\ \;\;\;\;2 \cdot \left(\left(\left(\sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right) \cdot \left(a + b\right)\right) \cdot \left(b - a\right)\right) \cdot \cos \left({\left({\left(\pi \cdot \left(angle \cdot 0.005555555555555556\right)\right)}^{3}\right)}^{0.3333333333333333}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \left(\left|\cos \left(\left(angle \cdot \pi\right) \cdot -0.005555555555555556\right)\right| \cdot \left(\left(b - a\right) \cdot \left(\left(a + b\right) \cdot \sin \left(\frac{-1}{\frac{-180}{angle \cdot \pi}}\right)\right)\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 67.5% accurate, 1.3× speedup

Math

\[\begin{array}{l} angle_m = \left|angle\right| \\ angle_s = \mathsf{copysign}\left(1, angle\right) \\ angle\_s \cdot \begin{array}{l} \mathbf{if}\;\frac{angle\_m}{180} \leq 10^{+107}:\\ \;\;\;\;2 \cdot \left(\left(\left(\sin \left(0.005555555555555556 \cdot \left(angle\_m \cdot \pi\right)\right) \cdot \left(a + b\right)\right) \cdot \left(b - a\right)\right) \cdot \cos \left(\frac{\pi}{\frac{180}{angle\_m}}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \left(\left|\cos \left(\left(angle\_m \cdot \pi\right) \cdot -0.005555555555555556\right)\right| \cdot \left(\left(b - a\right) \cdot \left(\left(a + b\right) \cdot \sin \left(\frac{-1}{\frac{-180}{angle\_m \cdot \pi}}\right)\right)\right)\right)\\ \end{array} \end{array} \]

FPCore

angle_m = (fabs.f64 angle)
angle_s = (copysign.f64 1 angle)
(FPCore (angle_s a b angle_m)
 :precision binary64
 (*
  angle_s
  (if (<= (/ angle_m 180.0) 1e+107)
    (*
     2.0
     (*
      (* (* (sin (* 0.005555555555555556 (* angle_m PI))) (+ a b)) (- b a))
      (cos (/ PI (/ 180.0 angle_m)))))
    (*
     2.0
     (*
      (fabs (cos (* (* angle_m PI) -0.005555555555555556)))
      (* (- b a) (* (+ a b) (sin (/ -1.0 (/ -180.0 (* angle_m PI)))))))))))

C

angle_m = fabs(angle);
angle_s = copysign(1.0, angle);
double code(double angle_s, double a, double b, double angle_m) {
	double tmp;
	if ((angle_m / 180.0) <= 1e+107) {
		tmp = 2.0 * (((sin((0.005555555555555556 * (angle_m * ((double) M_PI)))) * (a + b)) * (b - a)) * cos((((double) M_PI) / (180.0 / angle_m))));
	} else {
		tmp = 2.0 * (fabs(cos(((angle_m * ((double) M_PI)) * -0.005555555555555556))) * ((b - a) * ((a + b) * sin((-1.0 / (-180.0 / (angle_m * ((double) M_PI))))))));
	}
	return angle_s * tmp;
}

Java

angle_m = Math.abs(angle);
angle_s = Math.copySign(1.0, angle);
public static double code(double angle_s, double a, double b, double angle_m) {
	double tmp;
	if ((angle_m / 180.0) <= 1e+107) {
		tmp = 2.0 * (((Math.sin((0.005555555555555556 * (angle_m * Math.PI))) * (a + b)) * (b - a)) * Math.cos((Math.PI / (180.0 / angle_m))));
	} else {
		tmp = 2.0 * (Math.abs(Math.cos(((angle_m * Math.PI) * -0.005555555555555556))) * ((b - a) * ((a + b) * Math.sin((-1.0 / (-180.0 / (angle_m * Math.PI)))))));
	}
	return angle_s * tmp;
}

Python

angle_m = math.fabs(angle)
angle_s = math.copysign(1.0, angle)
def code(angle_s, a, b, angle_m):
	tmp = 0
	if (angle_m / 180.0) <= 1e+107:
		tmp = 2.0 * (((math.sin((0.005555555555555556 * (angle_m * math.pi))) * (a + b)) * (b - a)) * math.cos((math.pi / (180.0 / angle_m))))
	else:
		tmp = 2.0 * (math.fabs(math.cos(((angle_m * math.pi) * -0.005555555555555556))) * ((b - a) * ((a + b) * math.sin((-1.0 / (-180.0 / (angle_m * math.pi)))))))
	return angle_s * tmp

Julia

angle_m = abs(angle)
angle_s = copysign(1.0, angle)
function code(angle_s, a, b, angle_m)
	tmp = 0.0
	if (Float64(angle_m / 180.0) <= 1e+107)
		tmp = Float64(2.0 * Float64(Float64(Float64(sin(Float64(0.005555555555555556 * Float64(angle_m * pi))) * Float64(a + b)) * Float64(b - a)) * cos(Float64(pi / Float64(180.0 / angle_m)))));
	else
		tmp = Float64(2.0 * Float64(abs(cos(Float64(Float64(angle_m * pi) * -0.005555555555555556))) * Float64(Float64(b - a) * Float64(Float64(a + b) * sin(Float64(-1.0 / Float64(-180.0 / Float64(angle_m * pi))))))));
	end
	return Float64(angle_s * tmp)
end

MATLAB

angle_m = abs(angle);
angle_s = sign(angle) * abs(1.0);
function tmp_2 = code(angle_s, a, b, angle_m)
	tmp = 0.0;
	if ((angle_m / 180.0) <= 1e+107)
		tmp = 2.0 * (((sin((0.005555555555555556 * (angle_m * pi))) * (a + b)) * (b - a)) * cos((pi / (180.0 / angle_m))));
	else
		tmp = 2.0 * (abs(cos(((angle_m * pi) * -0.005555555555555556))) * ((b - a) * ((a + b) * sin((-1.0 / (-180.0 / (angle_m * pi)))))));
	end
	tmp_2 = angle_s * tmp;
end

Wolfram

angle_m = N[Abs[angle], $MachinePrecision]
angle_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[angle]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[angle$95$s_, a_, b_, angle$95$m_] := N[(angle$95$s * If[LessEqual[N[(angle$95$m / 180.0), $MachinePrecision], 1e+107], N[(2.0 * N[(N[(N[(N[Sin[N[(0.005555555555555556 * N[(angle$95$m * Pi), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * N[(a + b), $MachinePrecision]), $MachinePrecision] * N[(b - a), $MachinePrecision]), $MachinePrecision] * N[Cos[N[(Pi / N[(180.0 / angle$95$m), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(2.0 * N[(N[Abs[N[Cos[N[(N[(angle$95$m * Pi), $MachinePrecision] * -0.005555555555555556), $MachinePrecision]], $MachinePrecision]], $MachinePrecision] * N[(N[(b - a), $MachinePrecision] * N[(N[(a + b), $MachinePrecision] * N[Sin[N[(-1.0 / N[(-180.0 / N[(angle$95$m * Pi), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if (/.f64 angle 180) < 9.9999999999999997e106

   1. Initial program 57.8%

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

      1. associate-*l* 57.8%

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

      2. *-commutative 57.8%

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

      3. associate-*l* 57.8%

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

   3. Simplified 57.8%

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

      1. unpow2 57.8%

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

      2. unpow2 57.8%

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

      3. difference-of-squares 62.4%

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

   6. Applied egg-rr 62.4%

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

   7. Taylor expanded in angle around inf 63.3%

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

      1. *-commutative 63.3%

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

      2. associate-*r* 73.4%

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

   9. Simplified 73.4%

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

       1. *-commutative 73.4%

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

       2. *-commutative 73.4%

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

       3. associate-*r* 74.2%

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

       4. metadata-eval 74.2%

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

       5. div-inv 74.9%

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

       6. clear-num 76.0%

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

       7. un-div-inv 74.8%

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

   11. Applied egg-rr 74.8%

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

   if 9.9999999999999997e106 < (/.f64 angle 180)

   1. Initial program 32.0%

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

      1. associate-*l* 32.0%

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

      2. *-commutative 32.0%

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

      3. associate-*l* 32.0%

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

   3. Simplified 32.0%

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

      1. unpow2 32.0%

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

      2. unpow2 32.0%

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

      3. difference-of-squares 34.9%

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

   6. Applied egg-rr 34.9%

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

   7. Taylor expanded in angle around inf 34.6%

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

      1. *-commutative 34.6%

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

      2. associate-*r* 34.6%

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

   9. Simplified 34.6%

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

       1. associate-*r* 32.2%

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

       2. *-commutative 32.2%

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

       3. metadata-eval 32.2%

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

       4. div-inv 29.4%

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

       5. *-commutative 29.4%

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

       6. add-sqr-sqrt 16.2%

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

       7. sqrt-unprod 50.1%

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

       8. pow2 50.1%

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

       9. div-inv 50.0%

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

       10. metadata-eval 50.0%

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

       11. *-commutative 50.0%

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

   11. Applied egg-rr 50.0%

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

       1. unpow2 50.0%

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

       2. rem-sqrt-square 50.0%

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

       3. *-commutative 50.0%

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

       4. associate-*r* 50.6%

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

       5. *-commutative 50.6%

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

       6. metadata-eval 50.6%

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

       7. rem-cube-cbrt 50.0%

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

       8. distribute-rgt-neg-in 50.0%

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

       9. associate-*r* 50.0%

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

       10. cos-neg 50.0%

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

       11. associate-*r* 50.0%

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

       12. rem-cube-cbrt 50.6%

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

   13. Simplified 50.6%

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

       1. metadata-eval 50.6%

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

       2. *-commutative 50.6%

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

       3. associate-/r/ 48.6%

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

       4. frac-2neg 48.6%

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

       5. metadata-eval 48.6%

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

       6. distribute-neg-frac 48.6%

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

       7. metadata-eval 48.6%

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

   15. Applied egg-rr 48.6%

       \[\leadsto 2 \cdot \left(\left(\left(\sin \color{blue}{\left(\frac{-1}{\frac{-180}{\pi \cdot angle}}\right)} \cdot \left(a + b\right)\right) \cdot \left(b - a\right)\right) \cdot \left|\cos \left(\left(angle \cdot \pi\right) \cdot -0.005555555555555556\right)\right|\right) \]
3. Recombined 2 regimes into one program.

4. Final simplification 71.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{angle}{180} \leq 10^{+107}:\\ \;\;\;\;2 \cdot \left(\left(\left(\sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right) \cdot \left(a + b\right)\right) \cdot \left(b - a\right)\right) \cdot \cos \left(\frac{\pi}{\frac{180}{angle}}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \left(\left|\cos \left(\left(angle \cdot \pi\right) \cdot -0.005555555555555556\right)\right| \cdot \left(\left(b - a\right) \cdot \left(\left(a + b\right) \cdot \sin \left(\frac{-1}{\frac{-180}{angle \cdot \pi}}\right)\right)\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 67.4% accurate, 1.8× speedup

Math

\[\begin{array}{l} angle_m = \left|angle\right| \\ angle_s = \mathsf{copysign}\left(1, angle\right) \\ \begin{array}{l} t_0 := 0.005555555555555556 \cdot \left(angle\_m \cdot \pi\right)\\ angle\_s \cdot \begin{array}{l} \mathbf{if}\;\frac{angle\_m}{180} \leq 10^{+84}:\\ \;\;\;\;2 \cdot \left(\left(\left(\sin t\_0 \cdot \left(a + b\right)\right) \cdot \left(b - a\right)\right) \cdot \cos t\_0\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\left(a + b\right) \cdot \left(b - a\right)\right) \cdot \left(2 \cdot \sin \left(\frac{1}{\frac{180}{angle\_m \cdot \pi}}\right)\right)\\ \end{array} \end{array} \end{array} \]

FPCore

angle_m = (fabs.f64 angle)
angle_s = (copysign.f64 1 angle)
(FPCore (angle_s a b angle_m)
 :precision binary64
 (let* ((t_0 (* 0.005555555555555556 (* angle_m PI))))
   (*
    angle_s
    (if (<= (/ angle_m 180.0) 1e+84)
      (* 2.0 (* (* (* (sin t_0) (+ a b)) (- b a)) (cos t_0)))
      (*
       (* (+ a b) (- b a))
       (* 2.0 (sin (/ 1.0 (/ 180.0 (* angle_m PI))))))))))

C

angle_m = fabs(angle);
angle_s = copysign(1.0, angle);
double code(double angle_s, double a, double b, double angle_m) {
	double t_0 = 0.005555555555555556 * (angle_m * ((double) M_PI));
	double tmp;
	if ((angle_m / 180.0) <= 1e+84) {
		tmp = 2.0 * (((sin(t_0) * (a + b)) * (b - a)) * cos(t_0));
	} else {
		tmp = ((a + b) * (b - a)) * (2.0 * sin((1.0 / (180.0 / (angle_m * ((double) M_PI))))));
	}
	return angle_s * tmp;
}

Java

angle_m = Math.abs(angle);
angle_s = Math.copySign(1.0, angle);
public static double code(double angle_s, double a, double b, double angle_m) {
	double t_0 = 0.005555555555555556 * (angle_m * Math.PI);
	double tmp;
	if ((angle_m / 180.0) <= 1e+84) {
		tmp = 2.0 * (((Math.sin(t_0) * (a + b)) * (b - a)) * Math.cos(t_0));
	} else {
		tmp = ((a + b) * (b - a)) * (2.0 * Math.sin((1.0 / (180.0 / (angle_m * Math.PI)))));
	}
	return angle_s * tmp;
}

Python

angle_m = math.fabs(angle)
angle_s = math.copysign(1.0, angle)
def code(angle_s, a, b, angle_m):
	t_0 = 0.005555555555555556 * (angle_m * math.pi)
	tmp = 0
	if (angle_m / 180.0) <= 1e+84:
		tmp = 2.0 * (((math.sin(t_0) * (a + b)) * (b - a)) * math.cos(t_0))
	else:
		tmp = ((a + b) * (b - a)) * (2.0 * math.sin((1.0 / (180.0 / (angle_m * math.pi)))))
	return angle_s * tmp

Julia

angle_m = abs(angle)
angle_s = copysign(1.0, angle)
function code(angle_s, a, b, angle_m)
	t_0 = Float64(0.005555555555555556 * Float64(angle_m * pi))
	tmp = 0.0
	if (Float64(angle_m / 180.0) <= 1e+84)
		tmp = Float64(2.0 * Float64(Float64(Float64(sin(t_0) * Float64(a + b)) * Float64(b - a)) * cos(t_0)));
	else
		tmp = Float64(Float64(Float64(a + b) * Float64(b - a)) * Float64(2.0 * sin(Float64(1.0 / Float64(180.0 / Float64(angle_m * pi))))));
	end
	return Float64(angle_s * tmp)
end

MATLAB

angle_m = abs(angle);
angle_s = sign(angle) * abs(1.0);
function tmp_2 = code(angle_s, a, b, angle_m)
	t_0 = 0.005555555555555556 * (angle_m * pi);
	tmp = 0.0;
	if ((angle_m / 180.0) <= 1e+84)
		tmp = 2.0 * (((sin(t_0) * (a + b)) * (b - a)) * cos(t_0));
	else
		tmp = ((a + b) * (b - a)) * (2.0 * sin((1.0 / (180.0 / (angle_m * pi)))));
	end
	tmp_2 = angle_s * tmp;
end

Wolfram

angle_m = N[Abs[angle], $MachinePrecision]
angle_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[angle]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[angle$95$s_, a_, b_, angle$95$m_] := Block[{t$95$0 = N[(0.005555555555555556 * N[(angle$95$m * Pi), $MachinePrecision]), $MachinePrecision]}, N[(angle$95$s * If[LessEqual[N[(angle$95$m / 180.0), $MachinePrecision], 1e+84], N[(2.0 * N[(N[(N[(N[Sin[t$95$0], $MachinePrecision] * N[(a + b), $MachinePrecision]), $MachinePrecision] * N[(b - a), $MachinePrecision]), $MachinePrecision] * N[Cos[t$95$0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(a + b), $MachinePrecision] * N[(b - a), $MachinePrecision]), $MachinePrecision] * N[(2.0 * N[Sin[N[(1.0 / N[(180.0 / N[(angle$95$m * Pi), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 angle 180) < 1.00000000000000006e84

   1. Initial program 58.6%

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

      1. associate-*l* 58.6%

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

      2. *-commutative 58.6%

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

      3. associate-*l* 58.6%

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

   3. Simplified 58.6%

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

      1. unpow2 58.6%

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

      2. unpow2 58.6%

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

      3. difference-of-squares 63.3%

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

   6. Applied egg-rr 63.3%

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

   7. Taylor expanded in angle around inf 64.2%

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

      1. *-commutative 64.2%

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

      2. associate-*r* 74.6%

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

   9. Simplified 74.6%

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

   if 1.00000000000000006e84 < (/.f64 angle 180)

   1. Initial program 30.9%

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

      1. associate-*l* 30.9%

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

      2. *-commutative 30.9%

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

      3. associate-*l* 30.9%

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

   3. Simplified 30.9%

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

      1. unpow2 30.9%

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

      2. unpow2 30.9%

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

      3. difference-of-squares 33.4%

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

   6. Applied egg-rr 33.4%

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

      1. associate-*r/ 28.1%

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

      2. clear-num 33.7%

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

   8. Applied egg-rr 33.7%

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

   9. Taylor expanded in angle around 0 47.4%

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

4. Final simplification 70.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{angle}{180} \leq 10^{+84}:\\ \;\;\;\;2 \cdot \left(\left(\left(\sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right) \cdot \left(a + b\right)\right) \cdot \left(b - a\right)\right) \cdot \cos \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\left(a + b\right) \cdot \left(b - a\right)\right) \cdot \left(2 \cdot \sin \left(\frac{1}{\frac{180}{angle \cdot \pi}}\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 67.4% accurate, 1.8× speedup

Math

\[\begin{array}{l} angle_m = \left|angle\right| \\ angle_s = \mathsf{copysign}\left(1, angle\right) \\ angle\_s \cdot \begin{array}{l} \mathbf{if}\;\frac{angle\_m}{180} \leq 10^{+107}:\\ \;\;\;\;2 \cdot \left(\left(\left(\sin \left(0.005555555555555556 \cdot \left(angle\_m \cdot \pi\right)\right) \cdot \left(a + b\right)\right) \cdot \left(b - a\right)\right) \cdot \cos \left(\frac{\pi}{\frac{180}{angle\_m}}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\left(a + b\right) \cdot \left(b - a\right)\right) \cdot \left(2 \cdot \sin \left(\frac{1}{\frac{180}{angle\_m \cdot \pi}}\right)\right)\\ \end{array} \end{array} \]

FPCore

angle_m = (fabs.f64 angle)
angle_s = (copysign.f64 1 angle)
(FPCore (angle_s a b angle_m)
 :precision binary64
 (*
  angle_s
  (if (<= (/ angle_m 180.0) 1e+107)
    (*
     2.0
     (*
      (* (* (sin (* 0.005555555555555556 (* angle_m PI))) (+ a b)) (- b a))
      (cos (/ PI (/ 180.0 angle_m)))))
    (* (* (+ a b) (- b a)) (* 2.0 (sin (/ 1.0 (/ 180.0 (* angle_m PI)))))))))

C

angle_m = fabs(angle);
angle_s = copysign(1.0, angle);
double code(double angle_s, double a, double b, double angle_m) {
	double tmp;
	if ((angle_m / 180.0) <= 1e+107) {
		tmp = 2.0 * (((sin((0.005555555555555556 * (angle_m * ((double) M_PI)))) * (a + b)) * (b - a)) * cos((((double) M_PI) / (180.0 / angle_m))));
	} else {
		tmp = ((a + b) * (b - a)) * (2.0 * sin((1.0 / (180.0 / (angle_m * ((double) M_PI))))));
	}
	return angle_s * tmp;
}

Java

angle_m = Math.abs(angle);
angle_s = Math.copySign(1.0, angle);
public static double code(double angle_s, double a, double b, double angle_m) {
	double tmp;
	if ((angle_m / 180.0) <= 1e+107) {
		tmp = 2.0 * (((Math.sin((0.005555555555555556 * (angle_m * Math.PI))) * (a + b)) * (b - a)) * Math.cos((Math.PI / (180.0 / angle_m))));
	} else {
		tmp = ((a + b) * (b - a)) * (2.0 * Math.sin((1.0 / (180.0 / (angle_m * Math.PI)))));
	}
	return angle_s * tmp;
}

Python

angle_m = math.fabs(angle)
angle_s = math.copysign(1.0, angle)
def code(angle_s, a, b, angle_m):
	tmp = 0
	if (angle_m / 180.0) <= 1e+107:
		tmp = 2.0 * (((math.sin((0.005555555555555556 * (angle_m * math.pi))) * (a + b)) * (b - a)) * math.cos((math.pi / (180.0 / angle_m))))
	else:
		tmp = ((a + b) * (b - a)) * (2.0 * math.sin((1.0 / (180.0 / (angle_m * math.pi)))))
	return angle_s * tmp

Julia

angle_m = abs(angle)
angle_s = copysign(1.0, angle)
function code(angle_s, a, b, angle_m)
	tmp = 0.0
	if (Float64(angle_m / 180.0) <= 1e+107)
		tmp = Float64(2.0 * Float64(Float64(Float64(sin(Float64(0.005555555555555556 * Float64(angle_m * pi))) * Float64(a + b)) * Float64(b - a)) * cos(Float64(pi / Float64(180.0 / angle_m)))));
	else
		tmp = Float64(Float64(Float64(a + b) * Float64(b - a)) * Float64(2.0 * sin(Float64(1.0 / Float64(180.0 / Float64(angle_m * pi))))));
	end
	return Float64(angle_s * tmp)
end

MATLAB

angle_m = abs(angle);
angle_s = sign(angle) * abs(1.0);
function tmp_2 = code(angle_s, a, b, angle_m)
	tmp = 0.0;
	if ((angle_m / 180.0) <= 1e+107)
		tmp = 2.0 * (((sin((0.005555555555555556 * (angle_m * pi))) * (a + b)) * (b - a)) * cos((pi / (180.0 / angle_m))));
	else
		tmp = ((a + b) * (b - a)) * (2.0 * sin((1.0 / (180.0 / (angle_m * pi)))));
	end
	tmp_2 = angle_s * tmp;
end

Wolfram

angle_m = N[Abs[angle], $MachinePrecision]
angle_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[angle]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[angle$95$s_, a_, b_, angle$95$m_] := N[(angle$95$s * If[LessEqual[N[(angle$95$m / 180.0), $MachinePrecision], 1e+107], N[(2.0 * N[(N[(N[(N[Sin[N[(0.005555555555555556 * N[(angle$95$m * Pi), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * N[(a + b), $MachinePrecision]), $MachinePrecision] * N[(b - a), $MachinePrecision]), $MachinePrecision] * N[Cos[N[(Pi / N[(180.0 / angle$95$m), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(a + b), $MachinePrecision] * N[(b - a), $MachinePrecision]), $MachinePrecision] * N[(2.0 * N[Sin[N[(1.0 / N[(180.0 / N[(angle$95$m * Pi), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if (/.f64 angle 180) < 9.9999999999999997e106

   1. Initial program 57.8%

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

      1. associate-*l* 57.8%

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

      2. *-commutative 57.8%

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

      3. associate-*l* 57.8%

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

   3. Simplified 57.8%

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

      1. unpow2 57.8%

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

      2. unpow2 57.8%

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

      3. difference-of-squares 62.4%

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

   6. Applied egg-rr 62.4%

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

   7. Taylor expanded in angle around inf 63.3%

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

      1. *-commutative 63.3%

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

      2. associate-*r* 73.4%

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

   9. Simplified 73.4%

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

       1. *-commutative 73.4%

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

       2. *-commutative 73.4%

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

       3. associate-*r* 74.2%

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

       4. metadata-eval 74.2%

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

       5. div-inv 74.9%

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

       6. clear-num 76.0%

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

       7. un-div-inv 74.8%

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

   11. Applied egg-rr 74.8%

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

   if 9.9999999999999997e106 < (/.f64 angle 180)

   1. Initial program 32.0%

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

      1. associate-*l* 32.0%

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

      2. *-commutative 32.0%

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

      3. associate-*l* 32.0%

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

   3. Simplified 32.0%

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

      1. unpow2 32.0%

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

      2. unpow2 32.0%

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

      3. difference-of-squares 34.9%

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

   6. Applied egg-rr 34.9%

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

      1. associate-*r/ 25.9%

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

      2. clear-num 32.4%

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

   8. Applied egg-rr 32.4%

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

   9. Taylor expanded in angle around 0 48.0%

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

4. Final simplification 71.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{angle}{180} \leq 10^{+107}:\\ \;\;\;\;2 \cdot \left(\left(\left(\sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right) \cdot \left(a + b\right)\right) \cdot \left(b - a\right)\right) \cdot \cos \left(\frac{\pi}{\frac{180}{angle}}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\left(a + b\right) \cdot \left(b - a\right)\right) \cdot \left(2 \cdot \sin \left(\frac{1}{\frac{180}{angle \cdot \pi}}\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 65.7% accurate, 3.6× speedup

Math

\[\begin{array}{l} angle_m = \left|angle\right| \\ angle_s = \mathsf{copysign}\left(1, angle\right) \\ angle\_s \cdot \left(2 \cdot \left(\left(\sin \left(0.005555555555555556 \cdot \left(angle\_m \cdot \pi\right)\right) \cdot \left(a + b\right)\right) \cdot \left(b - a\right)\right)\right) \end{array} \]

FPCore

angle_m = (fabs.f64 angle)
angle_s = (copysign.f64 1 angle)
(FPCore (angle_s a b angle_m)
 :precision binary64
 (*
  angle_s
  (*
   2.0
   (* (* (sin (* 0.005555555555555556 (* angle_m PI))) (+ a b)) (- b a)))))

C

angle_m = fabs(angle);
angle_s = copysign(1.0, angle);
double code(double angle_s, double a, double b, double angle_m) {
	return angle_s * (2.0 * ((sin((0.005555555555555556 * (angle_m * ((double) M_PI)))) * (a + b)) * (b - a)));
}

Java

angle_m = Math.abs(angle);
angle_s = Math.copySign(1.0, angle);
public static double code(double angle_s, double a, double b, double angle_m) {
	return angle_s * (2.0 * ((Math.sin((0.005555555555555556 * (angle_m * Math.PI))) * (a + b)) * (b - a)));
}

Python

angle_m = math.fabs(angle)
angle_s = math.copysign(1.0, angle)
def code(angle_s, a, b, angle_m):
	return angle_s * (2.0 * ((math.sin((0.005555555555555556 * (angle_m * math.pi))) * (a + b)) * (b - a)))

Julia

angle_m = abs(angle)
angle_s = copysign(1.0, angle)
function code(angle_s, a, b, angle_m)
	return Float64(angle_s * Float64(2.0 * Float64(Float64(sin(Float64(0.005555555555555556 * Float64(angle_m * pi))) * Float64(a + b)) * Float64(b - a))))
end

MATLAB

angle_m = abs(angle);
angle_s = sign(angle) * abs(1.0);
function tmp = code(angle_s, a, b, angle_m)
	tmp = angle_s * (2.0 * ((sin((0.005555555555555556 * (angle_m * pi))) * (a + b)) * (b - a)));
end

Wolfram

angle_m = N[Abs[angle], $MachinePrecision]
angle_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[angle]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[angle$95$s_, a_, b_, angle$95$m_] := N[(angle$95$s * N[(2.0 * N[(N[(N[Sin[N[(0.005555555555555556 * N[(angle$95$m * Pi), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * N[(a + b), $MachinePrecision]), $MachinePrecision] * N[(b - a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 54.3%

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

   1. associate-*l* 54.3%

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

   2. *-commutative 54.3%

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

   3. associate-*l* 54.3%

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

3. Simplified 54.3%

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

   1. unpow2 54.3%

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

   2. unpow2 54.3%

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

   3. difference-of-squares 58.6%

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

6. Applied egg-rr 58.6%

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

7. Taylor expanded in angle around inf 59.4%

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

   1. *-commutative 59.4%

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

   2. associate-*r* 68.1%

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

9. Simplified 68.1%

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

10. Taylor expanded in angle around 0 70.9%

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

11. Final simplification 70.9%

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

Alternative 8: 63.1% accurate, 15.0× speedup

Math

\[\begin{array}{l} angle_m = \left|angle\right| \\ angle_s = \mathsf{copysign}\left(1, angle\right) \\ \begin{array}{l} t_0 := \left(\pi \cdot \left(a + b\right)\right) \cdot \left(angle\_m \cdot 0.011111111111111112\right)\\ angle\_s \cdot \begin{array}{l} \mathbf{if}\;angle\_m \leq 4.4 \cdot 10^{-90}:\\ \;\;\;\;b \cdot t\_0 - a \cdot t\_0\\ \mathbf{else}:\\ \;\;\;\;0.011111111111111112 \cdot \left(\left(angle\_m \cdot \pi\right) \cdot \left(\left(a + b\right) \cdot \left(b - a\right)\right)\right)\\ \end{array} \end{array} \end{array} \]

FPCore

angle_m = (fabs.f64 angle)
angle_s = (copysign.f64 1 angle)
(FPCore (angle_s a b angle_m)
 :precision binary64
 (let* ((t_0 (* (* PI (+ a b)) (* angle_m 0.011111111111111112))))
   (*
    angle_s
    (if (<= angle_m 4.4e-90)
      (- (* b t_0) (* a t_0))
      (* 0.011111111111111112 (* (* angle_m PI) (* (+ a b) (- b a))))))))

C

angle_m = fabs(angle);
angle_s = copysign(1.0, angle);
double code(double angle_s, double a, double b, double angle_m) {
	double t_0 = (((double) M_PI) * (a + b)) * (angle_m * 0.011111111111111112);
	double tmp;
	if (angle_m <= 4.4e-90) {
		tmp = (b * t_0) - (a * t_0);
	} else {
		tmp = 0.011111111111111112 * ((angle_m * ((double) M_PI)) * ((a + b) * (b - a)));
	}
	return angle_s * tmp;
}

Java

angle_m = Math.abs(angle);
angle_s = Math.copySign(1.0, angle);
public static double code(double angle_s, double a, double b, double angle_m) {
	double t_0 = (Math.PI * (a + b)) * (angle_m * 0.011111111111111112);
	double tmp;
	if (angle_m <= 4.4e-90) {
		tmp = (b * t_0) - (a * t_0);
	} else {
		tmp = 0.011111111111111112 * ((angle_m * Math.PI) * ((a + b) * (b - a)));
	}
	return angle_s * tmp;
}

Python

angle_m = math.fabs(angle)
angle_s = math.copysign(1.0, angle)
def code(angle_s, a, b, angle_m):
	t_0 = (math.pi * (a + b)) * (angle_m * 0.011111111111111112)
	tmp = 0
	if angle_m <= 4.4e-90:
		tmp = (b * t_0) - (a * t_0)
	else:
		tmp = 0.011111111111111112 * ((angle_m * math.pi) * ((a + b) * (b - a)))
	return angle_s * tmp

Julia

angle_m = abs(angle)
angle_s = copysign(1.0, angle)
function code(angle_s, a, b, angle_m)
	t_0 = Float64(Float64(pi * Float64(a + b)) * Float64(angle_m * 0.011111111111111112))
	tmp = 0.0
	if (angle_m <= 4.4e-90)
		tmp = Float64(Float64(b * t_0) - Float64(a * t_0));
	else
		tmp = Float64(0.011111111111111112 * Float64(Float64(angle_m * pi) * Float64(Float64(a + b) * Float64(b - a))));
	end
	return Float64(angle_s * tmp)
end

MATLAB

angle_m = abs(angle);
angle_s = sign(angle) * abs(1.0);
function tmp_2 = code(angle_s, a, b, angle_m)
	t_0 = (pi * (a + b)) * (angle_m * 0.011111111111111112);
	tmp = 0.0;
	if (angle_m <= 4.4e-90)
		tmp = (b * t_0) - (a * t_0);
	else
		tmp = 0.011111111111111112 * ((angle_m * pi) * ((a + b) * (b - a)));
	end
	tmp_2 = angle_s * tmp;
end

Wolfram

angle_m = N[Abs[angle], $MachinePrecision]
angle_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[angle]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[angle$95$s_, a_, b_, angle$95$m_] := Block[{t$95$0 = N[(N[(Pi * N[(a + b), $MachinePrecision]), $MachinePrecision] * N[(angle$95$m * 0.011111111111111112), $MachinePrecision]), $MachinePrecision]}, N[(angle$95$s * If[LessEqual[angle$95$m, 4.4e-90], N[(N[(b * t$95$0), $MachinePrecision] - N[(a * t$95$0), $MachinePrecision]), $MachinePrecision], N[(0.011111111111111112 * N[(N[(angle$95$m * Pi), $MachinePrecision] * N[(N[(a + b), $MachinePrecision] * N[(b - a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if angle < 4.39999999999999972e-90

   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. *-commutative 57.2%

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

      3. associate-*l* 57.2%

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

   3. Simplified 57.2%

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

      1. unpow2 57.2%

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

      2. unpow2 57.2%

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

      3. difference-of-squares 61.8%

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

   6. Applied egg-rr 61.8%

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

   7. Taylor expanded in angle around 0 66.0%

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

      1. associate-*r* 66.0%

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

      2. associate-*r* 66.0%

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

   9. Simplified 66.0%

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

       1. associate-*r* 75.4%

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

       2. sub-neg 75.4%

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

       3. distribute-lft-in 69.3%

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

       4. *-commutative 69.3%

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

       5. *-commutative 69.3%

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

   11. Applied egg-rr 69.3%

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

   if 4.39999999999999972e-90 < angle

   1. Initial program 47.4%

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

      1. associate-*l* 47.4%

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

      2. *-commutative 47.4%

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

      3. associate-*l* 47.4%

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

   3. Simplified 47.4%

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

      1. unpow2 47.4%

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

      2. unpow2 47.4%

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

      3. difference-of-squares 51.3%

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

   6. Applied egg-rr 51.3%

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

   7. Taylor expanded in angle around 0 44.1%

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

      1. associate-*r* 44.1%

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

      2. associate-*r* 44.0%

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

   9. Simplified 44.0%

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

   10. Taylor expanded in angle around 0 44.1%

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

       1. associate-*r* 44.1%

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

   12. Simplified 44.1%

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

4. Final simplification 61.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;angle \leq 4.4 \cdot 10^{-90}:\\ \;\;\;\;b \cdot \left(\left(\pi \cdot \left(a + b\right)\right) \cdot \left(angle \cdot 0.011111111111111112\right)\right) - a \cdot \left(\left(\pi \cdot \left(a + b\right)\right) \cdot \left(angle \cdot 0.011111111111111112\right)\right)\\ \mathbf{else}:\\ \;\;\;\;0.011111111111111112 \cdot \left(\left(angle \cdot \pi\right) \cdot \left(\left(a + b\right) \cdot \left(b - a\right)\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 43.6% accurate, 26.2× speedup

Math

\[\begin{array}{l} angle_m = \left|angle\right| \\ angle_s = \mathsf{copysign}\left(1, angle\right) \\ angle\_s \cdot \begin{array}{l} \mathbf{if}\;a \leq 2.05 \cdot 10^{-45}:\\ \;\;\;\;\left(angle\_m \cdot 0.011111111111111112\right) \cdot \left(\left(b - a\right) \cdot \left(\pi \cdot b\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(angle\_m \cdot 0.011111111111111112\right) \cdot \left(\left(b - a\right) \cdot \left(\pi \cdot a\right)\right)\\ \end{array} \end{array} \]

FPCore

angle_m = (fabs.f64 angle)
angle_s = (copysign.f64 1 angle)
(FPCore (angle_s a b angle_m)
 :precision binary64
 (*
  angle_s
  (if (<= a 2.05e-45)
    (* (* angle_m 0.011111111111111112) (* (- b a) (* PI b)))
    (* (* angle_m 0.011111111111111112) (* (- b a) (* PI a))))))

C

angle_m = fabs(angle);
angle_s = copysign(1.0, angle);
double code(double angle_s, double a, double b, double angle_m) {
	double tmp;
	if (a <= 2.05e-45) {
		tmp = (angle_m * 0.011111111111111112) * ((b - a) * (((double) M_PI) * b));
	} else {
		tmp = (angle_m * 0.011111111111111112) * ((b - a) * (((double) M_PI) * a));
	}
	return angle_s * tmp;
}

Java

angle_m = Math.abs(angle);
angle_s = Math.copySign(1.0, angle);
public static double code(double angle_s, double a, double b, double angle_m) {
	double tmp;
	if (a <= 2.05e-45) {
		tmp = (angle_m * 0.011111111111111112) * ((b - a) * (Math.PI * b));
	} else {
		tmp = (angle_m * 0.011111111111111112) * ((b - a) * (Math.PI * a));
	}
	return angle_s * tmp;
}

Python

angle_m = math.fabs(angle)
angle_s = math.copysign(1.0, angle)
def code(angle_s, a, b, angle_m):
	tmp = 0
	if a <= 2.05e-45:
		tmp = (angle_m * 0.011111111111111112) * ((b - a) * (math.pi * b))
	else:
		tmp = (angle_m * 0.011111111111111112) * ((b - a) * (math.pi * a))
	return angle_s * tmp

Julia

angle_m = abs(angle)
angle_s = copysign(1.0, angle)
function code(angle_s, a, b, angle_m)
	tmp = 0.0
	if (a <= 2.05e-45)
		tmp = Float64(Float64(angle_m * 0.011111111111111112) * Float64(Float64(b - a) * Float64(pi * b)));
	else
		tmp = Float64(Float64(angle_m * 0.011111111111111112) * Float64(Float64(b - a) * Float64(pi * a)));
	end
	return Float64(angle_s * tmp)
end

MATLAB

angle_m = abs(angle);
angle_s = sign(angle) * abs(1.0);
function tmp_2 = code(angle_s, a, b, angle_m)
	tmp = 0.0;
	if (a <= 2.05e-45)
		tmp = (angle_m * 0.011111111111111112) * ((b - a) * (pi * b));
	else
		tmp = (angle_m * 0.011111111111111112) * ((b - a) * (pi * a));
	end
	tmp_2 = angle_s * tmp;
end

Wolfram

angle_m = N[Abs[angle], $MachinePrecision]
angle_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[angle]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[angle$95$s_, a_, b_, angle$95$m_] := N[(angle$95$s * If[LessEqual[a, 2.05e-45], N[(N[(angle$95$m * 0.011111111111111112), $MachinePrecision] * N[(N[(b - a), $MachinePrecision] * N[(Pi * b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(angle$95$m * 0.011111111111111112), $MachinePrecision] * N[(N[(b - a), $MachinePrecision] * N[(Pi * a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if a < 2.05e-45

   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. *-commutative 57.2%

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

      3. associate-*l* 57.2%

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

   3. Simplified 57.2%

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

      1. unpow2 57.2%

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

      2. unpow2 57.2%

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

      3. difference-of-squares 60.5%

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

   6. Applied egg-rr 60.5%

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

   7. Taylor expanded in angle around 0 61.6%

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

      1. associate-*r* 61.5%

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

      2. associate-*r* 61.5%

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

   9. Simplified 61.5%

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

   10. Taylor expanded in a around 0 45.6%

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

       1. *-commutative 45.6%

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

   12. Simplified 45.6%

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

   if 2.05e-45 < a

   1. Initial program 46.8%

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

      1. associate-*l* 46.8%

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

      2. *-commutative 46.8%

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

      3. associate-*l* 46.8%

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

   3. Simplified 46.8%

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

      1. unpow2 46.8%

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

      2. unpow2 46.8%

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

      3. difference-of-squares 53.8%

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

   6. Applied egg-rr 53.8%

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

   7. Taylor expanded in angle around 0 53.9%

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

      1. associate-*r* 53.9%

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

      2. associate-*r* 53.9%

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

   9. Simplified 53.9%

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

   10. Taylor expanded in a around inf 40.3%

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

       1. *-commutative 40.3%

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

   12. Simplified 40.3%

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

4. Final simplification 44.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \leq 2.05 \cdot 10^{-45}:\\ \;\;\;\;\left(angle \cdot 0.011111111111111112\right) \cdot \left(\left(b - a\right) \cdot \left(\pi \cdot b\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(angle \cdot 0.011111111111111112\right) \cdot \left(\left(b - a\right) \cdot \left(\pi \cdot a\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 54.8% accurate, 32.2× speedup

Math

\[\begin{array}{l} angle_m = \left|angle\right| \\ angle_s = \mathsf{copysign}\left(1, angle\right) \\ angle\_s \cdot \left(0.011111111111111112 \cdot \left(angle\_m \cdot \left(\pi \cdot \left(\left(a + b\right) \cdot \left(b - a\right)\right)\right)\right)\right) \end{array} \]

FPCore

angle_m = (fabs.f64 angle)
angle_s = (copysign.f64 1 angle)
(FPCore (angle_s a b angle_m)
 :precision binary64
 (* angle_s (* 0.011111111111111112 (* angle_m (* PI (* (+ a b) (- b a)))))))

C

angle_m = fabs(angle);
angle_s = copysign(1.0, angle);
double code(double angle_s, double a, double b, double angle_m) {
	return angle_s * (0.011111111111111112 * (angle_m * (((double) M_PI) * ((a + b) * (b - a)))));
}

Java

angle_m = Math.abs(angle);
angle_s = Math.copySign(1.0, angle);
public static double code(double angle_s, double a, double b, double angle_m) {
	return angle_s * (0.011111111111111112 * (angle_m * (Math.PI * ((a + b) * (b - a)))));
}

Python

angle_m = math.fabs(angle)
angle_s = math.copysign(1.0, angle)
def code(angle_s, a, b, angle_m):
	return angle_s * (0.011111111111111112 * (angle_m * (math.pi * ((a + b) * (b - a)))))

Julia

angle_m = abs(angle)
angle_s = copysign(1.0, angle)
function code(angle_s, a, b, angle_m)
	return Float64(angle_s * Float64(0.011111111111111112 * Float64(angle_m * Float64(pi * Float64(Float64(a + b) * Float64(b - a))))))
end

MATLAB

angle_m = abs(angle);
angle_s = sign(angle) * abs(1.0);
function tmp = code(angle_s, a, b, angle_m)
	tmp = angle_s * (0.011111111111111112 * (angle_m * (pi * ((a + b) * (b - a)))));
end

Wolfram

angle_m = N[Abs[angle], $MachinePrecision]
angle_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[angle]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[angle$95$s_, a_, b_, angle$95$m_] := N[(angle$95$s * N[(0.011111111111111112 * N[(angle$95$m * N[(Pi * N[(N[(a + b), $MachinePrecision] * N[(b - a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 54.3%

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

   1. associate-*l* 54.3%

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

   2. *-commutative 54.3%

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

   3. associate-*l* 54.3%

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

3. Simplified 54.3%

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

5. Taylor expanded in angle around 0 54.7%

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

   1. unpow2 54.3%

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

   2. unpow2 54.3%

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

   3. difference-of-squares 58.6%

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

7. Applied egg-rr 59.4%

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

8. Final simplification 59.4%

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

Alternative 11: 54.8% accurate, 32.2× speedup

Math

\[\begin{array}{l} angle_m = \left|angle\right| \\ angle_s = \mathsf{copysign}\left(1, angle\right) \\ angle\_s \cdot \left(0.011111111111111112 \cdot \left(\left(angle\_m \cdot \pi\right) \cdot \left(\left(a + b\right) \cdot \left(b - a\right)\right)\right)\right) \end{array} \]

FPCore

angle_m = (fabs.f64 angle)
angle_s = (copysign.f64 1 angle)
(FPCore (angle_s a b angle_m)
 :precision binary64
 (* angle_s (* 0.011111111111111112 (* (* angle_m PI) (* (+ a b) (- b a))))))

C

angle_m = fabs(angle);
angle_s = copysign(1.0, angle);
double code(double angle_s, double a, double b, double angle_m) {
	return angle_s * (0.011111111111111112 * ((angle_m * ((double) M_PI)) * ((a + b) * (b - a))));
}

Java

angle_m = Math.abs(angle);
angle_s = Math.copySign(1.0, angle);
public static double code(double angle_s, double a, double b, double angle_m) {
	return angle_s * (0.011111111111111112 * ((angle_m * Math.PI) * ((a + b) * (b - a))));
}

Python

angle_m = math.fabs(angle)
angle_s = math.copysign(1.0, angle)
def code(angle_s, a, b, angle_m):
	return angle_s * (0.011111111111111112 * ((angle_m * math.pi) * ((a + b) * (b - a))))

Julia

angle_m = abs(angle)
angle_s = copysign(1.0, angle)
function code(angle_s, a, b, angle_m)
	return Float64(angle_s * Float64(0.011111111111111112 * Float64(Float64(angle_m * pi) * Float64(Float64(a + b) * Float64(b - a)))))
end

MATLAB

angle_m = abs(angle);
angle_s = sign(angle) * abs(1.0);
function tmp = code(angle_s, a, b, angle_m)
	tmp = angle_s * (0.011111111111111112 * ((angle_m * pi) * ((a + b) * (b - a))));
end

Wolfram

angle_m = N[Abs[angle], $MachinePrecision]
angle_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[angle]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[angle$95$s_, a_, b_, angle$95$m_] := N[(angle$95$s * N[(0.011111111111111112 * N[(N[(angle$95$m * Pi), $MachinePrecision] * N[(N[(a + b), $MachinePrecision] * N[(b - a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 54.3%

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

   1. associate-*l* 54.3%

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

   2. *-commutative 54.3%

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

   3. associate-*l* 54.3%

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

3. Simplified 54.3%

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

   1. unpow2 54.3%

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

   2. unpow2 54.3%

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

   3. difference-of-squares 58.6%

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

6. Applied egg-rr 58.6%

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

7. Taylor expanded in angle around 0 59.4%

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

   1. associate-*r* 59.4%

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

   2. associate-*r* 59.4%

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

9. Simplified 59.4%

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

10. Taylor expanded in angle around 0 59.4%

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

    1. associate-*r* 59.4%

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

12. Simplified 59.4%

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

13. Final simplification 59.4%

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

Alternative 12: 37.4% accurate, 38.1× speedup

Math

\[\begin{array}{l} angle_m = \left|angle\right| \\ angle_s = \mathsf{copysign}\left(1, angle\right) \\ angle\_s \cdot \left(\left(angle\_m \cdot 0.011111111111111112\right) \cdot \left(\left(b - a\right) \cdot \left(\pi \cdot a\right)\right)\right) \end{array} \]

FPCore

angle_m = (fabs.f64 angle)
angle_s = (copysign.f64 1 angle)
(FPCore (angle_s a b angle_m)
 :precision binary64
 (* angle_s (* (* angle_m 0.011111111111111112) (* (- b a) (* PI a)))))

C

angle_m = fabs(angle);
angle_s = copysign(1.0, angle);
double code(double angle_s, double a, double b, double angle_m) {
	return angle_s * ((angle_m * 0.011111111111111112) * ((b - a) * (((double) M_PI) * a)));
}

Java

angle_m = Math.abs(angle);
angle_s = Math.copySign(1.0, angle);
public static double code(double angle_s, double a, double b, double angle_m) {
	return angle_s * ((angle_m * 0.011111111111111112) * ((b - a) * (Math.PI * a)));
}

Python

angle_m = math.fabs(angle)
angle_s = math.copysign(1.0, angle)
def code(angle_s, a, b, angle_m):
	return angle_s * ((angle_m * 0.011111111111111112) * ((b - a) * (math.pi * a)))

Julia

angle_m = abs(angle)
angle_s = copysign(1.0, angle)
function code(angle_s, a, b, angle_m)
	return Float64(angle_s * Float64(Float64(angle_m * 0.011111111111111112) * Float64(Float64(b - a) * Float64(pi * a))))
end

MATLAB

angle_m = abs(angle);
angle_s = sign(angle) * abs(1.0);
function tmp = code(angle_s, a, b, angle_m)
	tmp = angle_s * ((angle_m * 0.011111111111111112) * ((b - a) * (pi * a)));
end

Wolfram

angle_m = N[Abs[angle], $MachinePrecision]
angle_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[angle]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[angle$95$s_, a_, b_, angle$95$m_] := N[(angle$95$s * N[(N[(angle$95$m * 0.011111111111111112), $MachinePrecision] * N[(N[(b - a), $MachinePrecision] * N[(Pi * a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 54.3%

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

   1. associate-*l* 54.3%

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

   2. *-commutative 54.3%

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

   3. associate-*l* 54.3%

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

3. Simplified 54.3%

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

   1. unpow2 54.3%

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

   2. unpow2 54.3%

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

   3. difference-of-squares 58.6%

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

6. Applied egg-rr 58.6%

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

7. Taylor expanded in angle around 0 59.4%

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

   1. associate-*r* 59.4%

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

   2. associate-*r* 59.4%

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

9. Simplified 59.4%

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

10. Taylor expanded in a around inf 39.5%

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

    1. *-commutative 39.5%

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

12. Simplified 39.5%

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

13. Final simplification 39.5%

    \[\leadsto \left(angle \cdot 0.011111111111111112\right) \cdot \left(\left(b - a\right) \cdot \left(\pi \cdot a\right)\right) \]
14. Add Preprocessing

Alternative 13: 12.8% accurate, 419.0× speedup

Math

\[\begin{array}{l} angle_m = \left|angle\right| \\ angle_s = \mathsf{copysign}\left(1, angle\right) \\ angle\_s \cdot 0 \end{array} \]

FPCore

angle_m = (fabs.f64 angle)
angle_s = (copysign.f64 1 angle)
(FPCore (angle_s a b angle_m) :precision binary64 (* angle_s 0.0))

C

angle_m = fabs(angle);
angle_s = copysign(1.0, angle);
double code(double angle_s, double a, double b, double angle_m) {
	return angle_s * 0.0;
}

Fortran

angle_m = abs(angle)
angle_s = copysign(1.0d0, angle)
real(8) function code(angle_s, a, b, angle_m)
    real(8), intent (in) :: angle_s
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: angle_m
    code = angle_s * 0.0d0
end function

Java

angle_m = Math.abs(angle);
angle_s = Math.copySign(1.0, angle);
public static double code(double angle_s, double a, double b, double angle_m) {
	return angle_s * 0.0;
}

Python

angle_m = math.fabs(angle)
angle_s = math.copysign(1.0, angle)
def code(angle_s, a, b, angle_m):
	return angle_s * 0.0

Julia

angle_m = abs(angle)
angle_s = copysign(1.0, angle)
function code(angle_s, a, b, angle_m)
	return Float64(angle_s * 0.0)
end

MATLAB

angle_m = abs(angle);
angle_s = sign(angle) * abs(1.0);
function tmp = code(angle_s, a, b, angle_m)
	tmp = angle_s * 0.0;
end

Wolfram

angle_m = N[Abs[angle], $MachinePrecision]
angle_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[angle]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[angle$95$s_, a_, b_, angle$95$m_] := N[(angle$95$s * 0.0), $MachinePrecision]

Derivation

1. Initial program 54.3%

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

   1. associate-*l* 54.3%

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

   2. *-commutative 54.3%

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

   3. associate-*l* 54.3%

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

3. Simplified 54.3%

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

   1. unpow2 54.3%

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

   2. unpow2 54.3%

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

   3. difference-of-squares 58.6%

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

6. Applied egg-rr 58.6%

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

7. Taylor expanded in angle around inf 57.4%

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

   1. associate-*r* 57.8%

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

   2. *-commutative 57.8%

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

   3. *-commutative 57.8%

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

   4. *-commutative 57.8%

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

9. Simplified 57.8%

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

11. Applied egg-rr 12.5%

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

    1. log1p-undefine 12.5%

       \[\leadsto e^{\color{blue}{\log \left(1 + \sin 0 \cdot \left({b}^{2} - {a}^{2}\right)\right)}} - 1 \]

    2. rem-exp-log 12.5%

       \[\leadsto \color{blue}{\left(1 + \sin 0 \cdot \left({b}^{2} - {a}^{2}\right)\right)} - 1 \]

    3. sin-0 12.5%

       \[\leadsto \left(1 + \color{blue}{0} \cdot \left({b}^{2} - {a}^{2}\right)\right) - 1 \]

    4. mul0-lft 14.4%

       \[\leadsto \left(1 + \color{blue}{0}\right) - 1 \]

    5. metadata-eval 14.4%

       \[\leadsto \color{blue}{1} - 1 \]

    6. metadata-eval 14.4%

       \[\leadsto \color{blue}{0} \]

13. Simplified 14.4%

    \[\leadsto \color{blue}{0} \]

14. Final simplification 14.4%

    \[\leadsto 0 \]
15. Add Preprocessing

Reproduce

herbie shell --seed 2024046 
(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)))))