ab-angle->ABCF B

Percentage Accurate: 53.9% → 67.2%

Time: 41.9s

Alternatives: 20

Speedup: 32.2×

• 33.5% of points produce a very large (infinite) output. You may want to add a precondition.

Specification

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 53.9% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 67.2% accurate, 1.0× speedup

Math

\[\begin{array}{l} a_m = \left|a\right| \\ b_m = \left|b\right| \\ angle\_m = \left|angle\right| \\ angle\_s = \mathsf{copysign}\left(1, angle\right) \\ \begin{array}{l} t_0 := \pi \cdot \left(0.005555555555555556 \cdot angle\_m\right)\\ t_1 := 0.005555555555555556 \cdot \left(angle\_m \cdot \pi\right)\\ angle\_s \cdot \begin{array}{l} \mathbf{if}\;b\_m \leq 1.62 \cdot 10^{-74}:\\ \;\;\;\;2 \cdot \left(\left(\left(\sin t\_0 \cdot \left(a\_m + b\_m\right)\right) \cdot \left(b\_m - a\_m\right)\right) \cdot \cos t\_1\right)\\ \mathbf{elif}\;b\_m \leq 1.25 \cdot 10^{+185}:\\ \;\;\;\;2 \cdot \left(\cos \left({\left(\sqrt[3]{t\_0}\right)}^{3}\right) \cdot \left(\left(b\_m - a\_m\right) \cdot \left(\sin t\_1 \cdot \left(a\_m + b\_m\right)\right)\right)\right)\\ \mathbf{elif}\;b\_m \leq 1.16 \cdot 10^{+197}:\\ \;\;\;\;2 \cdot \left(\left(b\_m - a\_m\right) \cdot \left(b\_m \cdot \sin \left(\mathsf{expm1}\left(\mathsf{log1p}\left(t\_0\right)\right)\right)\right)\right)\\ \mathbf{elif}\;b\_m \leq 4.4 \cdot 10^{+237}:\\ \;\;\;\;\left(a\_m + b\_m\right) \cdot \left(\left(b\_m - a\_m\right) \cdot \sin \left(\left(angle\_m \cdot \sqrt[3]{{\pi}^{3}}\right) \cdot 0.011111111111111112\right)\right)\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \left(b\_m \cdot \left(\left(b\_m - a\_m\right) \cdot \sin \left(angle\_m \cdot \left(0.005555555555555556 \cdot \pi\right)\right)\right)\right)\\ \end{array} \end{array} \end{array} \]

FPCore

a_m = (fabs.f64 a)
b_m = (fabs.f64 b)
angle\_m = (fabs.f64 angle)
angle\_s = (copysign.f64 #s(literal 1 binary64) angle)
(FPCore (angle_s a_m b_m angle_m)
 :precision binary64
 (let* ((t_0 (* PI (* 0.005555555555555556 angle_m)))
        (t_1 (* 0.005555555555555556 (* angle_m PI))))
   (*
    angle_s
    (if (<= b_m 1.62e-74)
      (* 2.0 (* (* (* (sin t_0) (+ a_m b_m)) (- b_m a_m)) (cos t_1)))
      (if (<= b_m 1.25e+185)
        (*
         2.0
         (*
          (cos (pow (cbrt t_0) 3.0))
          (* (- b_m a_m) (* (sin t_1) (+ a_m b_m)))))
        (if (<= b_m 1.16e+197)
          (* 2.0 (* (- b_m a_m) (* b_m (sin (expm1 (log1p t_0))))))
          (if (<= b_m 4.4e+237)
            (*
             (+ a_m b_m)
             (*
              (- b_m a_m)
              (sin (* (* angle_m (cbrt (pow PI 3.0))) 0.011111111111111112))))
            (*
             2.0
             (*
              b_m
              (*
               (- b_m a_m)
               (sin (* angle_m (* 0.005555555555555556 PI)))))))))))))

C

a_m = fabs(a);
b_m = fabs(b);
angle\_m = fabs(angle);
angle\_s = copysign(1.0, angle);
double code(double angle_s, double a_m, double b_m, double angle_m) {
	double t_0 = ((double) M_PI) * (0.005555555555555556 * angle_m);
	double t_1 = 0.005555555555555556 * (angle_m * ((double) M_PI));
	double tmp;
	if (b_m <= 1.62e-74) {
		tmp = 2.0 * (((sin(t_0) * (a_m + b_m)) * (b_m - a_m)) * cos(t_1));
	} else if (b_m <= 1.25e+185) {
		tmp = 2.0 * (cos(pow(cbrt(t_0), 3.0)) * ((b_m - a_m) * (sin(t_1) * (a_m + b_m))));
	} else if (b_m <= 1.16e+197) {
		tmp = 2.0 * ((b_m - a_m) * (b_m * sin(expm1(log1p(t_0)))));
	} else if (b_m <= 4.4e+237) {
		tmp = (a_m + b_m) * ((b_m - a_m) * sin(((angle_m * cbrt(pow(((double) M_PI), 3.0))) * 0.011111111111111112)));
	} else {
		tmp = 2.0 * (b_m * ((b_m - a_m) * sin((angle_m * (0.005555555555555556 * ((double) M_PI))))));
	}
	return angle_s * tmp;
}

Java

a_m = Math.abs(a);
b_m = Math.abs(b);
angle\_m = Math.abs(angle);
angle\_s = Math.copySign(1.0, angle);
public static double code(double angle_s, double a_m, double b_m, double angle_m) {
	double t_0 = Math.PI * (0.005555555555555556 * angle_m);
	double t_1 = 0.005555555555555556 * (angle_m * Math.PI);
	double tmp;
	if (b_m <= 1.62e-74) {
		tmp = 2.0 * (((Math.sin(t_0) * (a_m + b_m)) * (b_m - a_m)) * Math.cos(t_1));
	} else if (b_m <= 1.25e+185) {
		tmp = 2.0 * (Math.cos(Math.pow(Math.cbrt(t_0), 3.0)) * ((b_m - a_m) * (Math.sin(t_1) * (a_m + b_m))));
	} else if (b_m <= 1.16e+197) {
		tmp = 2.0 * ((b_m - a_m) * (b_m * Math.sin(Math.expm1(Math.log1p(t_0)))));
	} else if (b_m <= 4.4e+237) {
		tmp = (a_m + b_m) * ((b_m - a_m) * Math.sin(((angle_m * Math.cbrt(Math.pow(Math.PI, 3.0))) * 0.011111111111111112)));
	} else {
		tmp = 2.0 * (b_m * ((b_m - a_m) * Math.sin((angle_m * (0.005555555555555556 * Math.PI)))));
	}
	return angle_s * tmp;
}

Julia

a_m = abs(a)
b_m = abs(b)
angle\_m = abs(angle)
angle\_s = copysign(1.0, angle)
function code(angle_s, a_m, b_m, angle_m)
	t_0 = Float64(pi * Float64(0.005555555555555556 * angle_m))
	t_1 = Float64(0.005555555555555556 * Float64(angle_m * pi))
	tmp = 0.0
	if (b_m <= 1.62e-74)
		tmp = Float64(2.0 * Float64(Float64(Float64(sin(t_0) * Float64(a_m + b_m)) * Float64(b_m - a_m)) * cos(t_1)));
	elseif (b_m <= 1.25e+185)
		tmp = Float64(2.0 * Float64(cos((cbrt(t_0) ^ 3.0)) * Float64(Float64(b_m - a_m) * Float64(sin(t_1) * Float64(a_m + b_m)))));
	elseif (b_m <= 1.16e+197)
		tmp = Float64(2.0 * Float64(Float64(b_m - a_m) * Float64(b_m * sin(expm1(log1p(t_0))))));
	elseif (b_m <= 4.4e+237)
		tmp = Float64(Float64(a_m + b_m) * Float64(Float64(b_m - a_m) * sin(Float64(Float64(angle_m * cbrt((pi ^ 3.0))) * 0.011111111111111112))));
	else
		tmp = Float64(2.0 * Float64(b_m * Float64(Float64(b_m - a_m) * sin(Float64(angle_m * Float64(0.005555555555555556 * pi))))));
	end
	return Float64(angle_s * tmp)
end

Wolfram

a_m = N[Abs[a], $MachinePrecision]
b_m = N[Abs[b], $MachinePrecision]
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$95$m_, b$95$m_, angle$95$m_] := Block[{t$95$0 = N[(Pi * N[(0.005555555555555556 * angle$95$m), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(0.005555555555555556 * N[(angle$95$m * Pi), $MachinePrecision]), $MachinePrecision]}, N[(angle$95$s * If[LessEqual[b$95$m, 1.62e-74], N[(2.0 * N[(N[(N[(N[Sin[t$95$0], $MachinePrecision] * N[(a$95$m + b$95$m), $MachinePrecision]), $MachinePrecision] * N[(b$95$m - a$95$m), $MachinePrecision]), $MachinePrecision] * N[Cos[t$95$1], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[b$95$m, 1.25e+185], N[(2.0 * N[(N[Cos[N[Power[N[Power[t$95$0, 1/3], $MachinePrecision], 3.0], $MachinePrecision]], $MachinePrecision] * N[(N[(b$95$m - a$95$m), $MachinePrecision] * N[(N[Sin[t$95$1], $MachinePrecision] * N[(a$95$m + b$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[b$95$m, 1.16e+197], N[(2.0 * N[(N[(b$95$m - a$95$m), $MachinePrecision] * N[(b$95$m * N[Sin[N[(Exp[N[Log[1 + t$95$0], $MachinePrecision]] - 1), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[b$95$m, 4.4e+237], N[(N[(a$95$m + b$95$m), $MachinePrecision] * N[(N[(b$95$m - a$95$m), $MachinePrecision] * N[Sin[N[(N[(angle$95$m * N[Power[N[Power[Pi, 3.0], $MachinePrecision], 1/3], $MachinePrecision]), $MachinePrecision] * 0.011111111111111112), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(2.0 * N[(b$95$m * N[(N[(b$95$m - a$95$m), $MachinePrecision] * N[Sin[N[(angle$95$m * N[(0.005555555555555556 * Pi), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]), $MachinePrecision]]]

Derivation

1. Split input into 5 regimes

2. if b < 1.62000000000000007e-74

   1. Initial program 62.1%

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

      1. associate-*l* 62.1%

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

      2. *-commutative 62.1%

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

      3. associate-*l* 62.1%

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

   3. Simplified 62.1%

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

      1. unpow2 62.1%

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

      2. unpow2 62.1%

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

      3. difference-of-squares 64.3%

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

   6. Applied egg-rr 64.3%

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

      1. add-cube-cbrt 63.0%

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

      2. pow2 63.0%

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

   8. Applied egg-rr 63.0%

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

   9. Taylor expanded in angle around inf 63.6%

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

       1. associate-*r* 62.9%

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

       2. associate-*r* 69.0%

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

       3. associate-*r* 68.4%

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

   11. Simplified 68.4%

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

   12. Taylor expanded in angle around inf 68.5%

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

   if 1.62000000000000007e-74 < b < 1.24999999999999997e185

   1. Initial program 50.8%

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

      1. associate-*l* 50.8%

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

      2. *-commutative 50.8%

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

      3. associate-*l* 50.8%

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

   3. Simplified 50.8%

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

      1. unpow2 50.8%

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

      2. unpow2 50.8%

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

      3. difference-of-squares 53.3%

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

   6. Applied egg-rr 53.3%

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

      1. add-cube-cbrt 49.4%

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

      2. pow2 49.4%

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

   8. Applied egg-rr 49.4%

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

   9. Taylor expanded in angle around inf 51.3%

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

       1. associate-*r* 50.8%

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

       2. associate-*r* 59.7%

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

       3. associate-*r* 59.7%

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

   11. Simplified 59.7%

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

       1. *-commutative 59.7%

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

       2. *-commutative 59.7%

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

       3. add-cube-cbrt 62.9%

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

       4. pow3 65.3%

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

   13. Applied egg-rr 65.3%

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

   14. Taylor expanded in angle around inf 65.3%

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

   if 1.24999999999999997e185 < b < 1.16e197

   1. Initial program 51.5%

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

      1. associate-*l* 51.5%

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

      2. *-commutative 51.5%

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

      3. associate-*l* 51.5%

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

   3. Simplified 51.5%

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

      1. unpow2 51.5%

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

      2. unpow2 51.5%

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

      3. difference-of-squares 51.5%

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

   6. Applied egg-rr 51.5%

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

      1. add-cube-cbrt 76.5%

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

      2. pow2 76.5%

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

   8. Applied egg-rr 76.5%

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

   9. Taylor expanded in angle around inf 51.5%

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

       1. associate-*r* 51.5%

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

       2. associate-*r* 75.0%

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

       3. associate-*r* 74.6%

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

   11. Simplified 74.6%

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

   12. Taylor expanded in angle around 0 99.6%

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

   13. Taylor expanded in a around 0 100.0%

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

       1. associate-*r* 99.6%

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

   15. Simplified 99.6%

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

       1. *-commutative 99.6%

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

       2. *-commutative 99.6%

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

       3. expm1-log1p-u 74.6%

          \[\leadsto 2 \cdot \left(1 \cdot \left(\left(b \cdot \sin \color{blue}{\left(\mathsf{expm1}\left(\mathsf{log1p}\left(\pi \cdot \left(angle \cdot 0.005555555555555556\right)\right)\right)\right)}\right) \cdot \left(b - a\right)\right)\right) \]

       4. expm1-undefine 25.9%

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

   17. Applied egg-rr 25.9%

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

       1. expm1-define 74.6%

          \[\leadsto 2 \cdot \left(1 \cdot \left(\left(b \cdot \sin \color{blue}{\left(\mathsf{expm1}\left(\mathsf{log1p}\left(\pi \cdot \left(angle \cdot 0.005555555555555556\right)\right)\right)\right)}\right) \cdot \left(b - a\right)\right)\right) \]

   19. Simplified 74.6%

       \[\leadsto 2 \cdot \left(1 \cdot \left(\left(b \cdot \sin \color{blue}{\left(\mathsf{expm1}\left(\mathsf{log1p}\left(\pi \cdot \left(angle \cdot 0.005555555555555556\right)\right)\right)\right)}\right) \cdot \left(b - a\right)\right)\right) \]

   if 1.16e197 < b < 4.4e237

   1. Initial program 11.1%

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

      1. associate-*l* 11.1%

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

      2. *-commutative 11.1%

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

      3. associate-*l* 11.1%

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

   3. Simplified 11.1%

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

      1. unpow2 11.1%

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

      2. unpow2 11.1%

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

      3. difference-of-squares 38.3%

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

   6. Applied egg-rr 38.3%

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

      1. add-exp-log 18.2%

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

      2. 2-sin 18.2%

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

      3. div-inv 27.3%

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

      4. metadata-eval 27.3%

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

   8. Applied egg-rr 27.3%

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

      1. rem-exp-log 47.4%

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

      2. associate-*l* 81.7%

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

      3. +-commutative 81.7%

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

      4. count-2 81.7%

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

      5. add-cube-cbrt 81.3%

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

      6. unpow2 81.3%

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

      7. metadata-eval 81.3%

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

      8. div-inv 81.3%

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

      9. metadata-eval 81.3%

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

      10. div-inv 81.3%

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

   10. Applied egg-rr 81.7%

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

       1. add-cbrt-cube 99.6%

          \[\leadsto \left(a + b\right) \cdot \left(\left(b - a\right) \cdot \sin \left(\left(angle \cdot \color{blue}{\sqrt[3]{\left(\pi \cdot \pi\right) \cdot \pi}}\right) \cdot 0.011111111111111112\right)\right) \]

       2. pow3 99.6%

          \[\leadsto \left(a + b\right) \cdot \left(\left(b - a\right) \cdot \sin \left(\left(angle \cdot \sqrt[3]{\color{blue}{{\pi}^{3}}}\right) \cdot 0.011111111111111112\right)\right) \]

   12. Applied egg-rr 99.6%

       \[\leadsto \left(a + b\right) \cdot \left(\left(b - a\right) \cdot \sin \left(\left(angle \cdot \color{blue}{\sqrt[3]{{\pi}^{3}}}\right) \cdot 0.011111111111111112\right)\right) \]

   if 4.4e237 < b

   1. Initial program 39.8%

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

      1. associate-*l* 39.8%

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

      2. *-commutative 39.8%

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

      3. associate-*l* 39.8%

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

   3. Simplified 39.8%

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

      1. unpow2 39.8%

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

      2. unpow2 39.8%

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

      3. difference-of-squares 62.0%

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

   6. Applied egg-rr 62.0%

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

      1. add-cube-cbrt 50.9%

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

      2. pow2 50.9%

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

   8. Applied egg-rr 50.9%

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

   9. Taylor expanded in angle around inf 62.0%

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

       1. associate-*r* 67.6%

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

       2. associate-*r* 77.8%

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

       3. associate-*r* 72.2%

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

   11. Simplified 72.2%

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

   12. Taylor expanded in angle around 0 83.3%

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

   13. Taylor expanded in a around 0 83.3%

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

       1. associate-*r* 77.8%

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

   15. Simplified 77.8%

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

   16. Taylor expanded in b around 0 83.3%

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

       1. associate-*r* 83.3%

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

       2. neg-mul-1 83.3%

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

       3. distribute-rgt-out 83.3%

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

       4. *-commutative 83.3%

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

       5. associate-*r* 77.8%

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

       6. +-commutative 77.8%

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

       7. sub-neg 77.8%

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

   18. Simplified 77.8%

       \[\leadsto 2 \cdot \left(1 \cdot \color{blue}{\left(b \cdot \left(\sin \left(angle \cdot \left(\pi \cdot 0.005555555555555556\right)\right) \cdot \left(b - a\right)\right)\right)}\right) \]
3. Recombined 5 regimes into one program.

4. Final simplification 70.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq 1.62 \cdot 10^{-74}:\\ \;\;\;\;2 \cdot \left(\left(\left(\sin \left(\pi \cdot \left(0.005555555555555556 \cdot angle\right)\right) \cdot \left(a + b\right)\right) \cdot \left(b - a\right)\right) \cdot \cos \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)\right)\\ \mathbf{elif}\;b \leq 1.25 \cdot 10^{+185}:\\ \;\;\;\;2 \cdot \left(\cos \left({\left(\sqrt[3]{\pi \cdot \left(0.005555555555555556 \cdot angle\right)}\right)}^{3}\right) \cdot \left(\left(b - a\right) \cdot \left(\sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right) \cdot \left(a + b\right)\right)\right)\right)\\ \mathbf{elif}\;b \leq 1.16 \cdot 10^{+197}:\\ \;\;\;\;2 \cdot \left(\left(b - a\right) \cdot \left(b \cdot \sin \left(\mathsf{expm1}\left(\mathsf{log1p}\left(\pi \cdot \left(0.005555555555555556 \cdot angle\right)\right)\right)\right)\right)\right)\\ \mathbf{elif}\;b \leq 4.4 \cdot 10^{+237}:\\ \;\;\;\;\left(a + b\right) \cdot \left(\left(b - a\right) \cdot \sin \left(\left(angle \cdot \sqrt[3]{{\pi}^{3}}\right) \cdot 0.011111111111111112\right)\right)\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \left(b \cdot \left(\left(b - a\right) \cdot \sin \left(angle \cdot \left(0.005555555555555556 \cdot \pi\right)\right)\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 67.0% accurate, 0.4× speedup

Math

\[\begin{array}{l} a_m = \left|a\right| \\ b_m = \left|b\right| \\ angle\_m = \left|angle\right| \\ angle\_s = \mathsf{copysign}\left(1, angle\right) \\ \begin{array}{l} t_0 := \sin \left(0.005555555555555556 \cdot \left(angle\_m \cdot \pi\right)\right)\\ t_1 := \sqrt[3]{angle\_m \cdot \left(0.005555555555555556 \cdot \pi\right)}\\ t_2 := \pi \cdot \left(0.005555555555555556 \cdot angle\_m\right)\\ t_3 := \left(\sin t\_2 \cdot \left(a\_m + b\_m\right)\right) \cdot \left(b\_m - a\_m\right)\\ angle\_s \cdot \begin{array}{l} \mathbf{if}\;{a\_m}^{2} \leq 2 \cdot 10^{-186}:\\ \;\;\;\;\mathsf{fma}\left(b\_m, 2 \cdot \left(b\_m \cdot t\_0\right), -2 \cdot \left({a\_m}^{2} \cdot t\_0\right)\right) \cdot \sqrt{{\cos t\_2}^{2}}\\ \mathbf{elif}\;{a\_m}^{2} \leq 10^{+252}:\\ \;\;\;\;2 \cdot \left(\cos \left(\sqrt[3]{t\_1} \cdot {\left(\sqrt[3]{{t\_1}^{2}}\right)}^{4}\right) \cdot t\_3\right)\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \left(t\_3 \cdot \cos \left({t\_2}^{0.16666666666666666} \cdot \left(\left({\left(angle\_m \cdot \pi\right)}^{0.16666666666666666} \cdot {0.005555555555555556}^{0.16666666666666666}\right) \cdot {\left(\sqrt[3]{t\_2}\right)}^{2}\right)\right)\right)\\ \end{array} \end{array} \end{array} \]

FPCore

a_m = (fabs.f64 a)
b_m = (fabs.f64 b)
angle\_m = (fabs.f64 angle)
angle\_s = (copysign.f64 #s(literal 1 binary64) angle)
(FPCore (angle_s a_m b_m angle_m)
 :precision binary64
 (let* ((t_0 (sin (* 0.005555555555555556 (* angle_m PI))))
        (t_1 (cbrt (* angle_m (* 0.005555555555555556 PI))))
        (t_2 (* PI (* 0.005555555555555556 angle_m)))
        (t_3 (* (* (sin t_2) (+ a_m b_m)) (- b_m a_m))))
   (*
    angle_s
    (if (<= (pow a_m 2.0) 2e-186)
      (*
       (fma b_m (* 2.0 (* b_m t_0)) (* -2.0 (* (pow a_m 2.0) t_0)))
       (sqrt (pow (cos t_2) 2.0)))
      (if (<= (pow a_m 2.0) 1e+252)
        (* 2.0 (* (cos (* (cbrt t_1) (pow (cbrt (pow t_1 2.0)) 4.0))) t_3))
        (*
         2.0
         (*
          t_3
          (cos
           (*
            (pow t_2 0.16666666666666666)
            (*
             (*
              (pow (* angle_m PI) 0.16666666666666666)
              (pow 0.005555555555555556 0.16666666666666666))
             (pow (cbrt t_2) 2.0)))))))))))

C

a_m = fabs(a);
b_m = fabs(b);
angle\_m = fabs(angle);
angle\_s = copysign(1.0, angle);
double code(double angle_s, double a_m, double b_m, double angle_m) {
	double t_0 = sin((0.005555555555555556 * (angle_m * ((double) M_PI))));
	double t_1 = cbrt((angle_m * (0.005555555555555556 * ((double) M_PI))));
	double t_2 = ((double) M_PI) * (0.005555555555555556 * angle_m);
	double t_3 = (sin(t_2) * (a_m + b_m)) * (b_m - a_m);
	double tmp;
	if (pow(a_m, 2.0) <= 2e-186) {
		tmp = fma(b_m, (2.0 * (b_m * t_0)), (-2.0 * (pow(a_m, 2.0) * t_0))) * sqrt(pow(cos(t_2), 2.0));
	} else if (pow(a_m, 2.0) <= 1e+252) {
		tmp = 2.0 * (cos((cbrt(t_1) * pow(cbrt(pow(t_1, 2.0)), 4.0))) * t_3);
	} else {
		tmp = 2.0 * (t_3 * cos((pow(t_2, 0.16666666666666666) * ((pow((angle_m * ((double) M_PI)), 0.16666666666666666) * pow(0.005555555555555556, 0.16666666666666666)) * pow(cbrt(t_2), 2.0)))));
	}
	return angle_s * tmp;
}

Julia

a_m = abs(a)
b_m = abs(b)
angle\_m = abs(angle)
angle\_s = copysign(1.0, angle)
function code(angle_s, a_m, b_m, angle_m)
	t_0 = sin(Float64(0.005555555555555556 * Float64(angle_m * pi)))
	t_1 = cbrt(Float64(angle_m * Float64(0.005555555555555556 * pi)))
	t_2 = Float64(pi * Float64(0.005555555555555556 * angle_m))
	t_3 = Float64(Float64(sin(t_2) * Float64(a_m + b_m)) * Float64(b_m - a_m))
	tmp = 0.0
	if ((a_m ^ 2.0) <= 2e-186)
		tmp = Float64(fma(b_m, Float64(2.0 * Float64(b_m * t_0)), Float64(-2.0 * Float64((a_m ^ 2.0) * t_0))) * sqrt((cos(t_2) ^ 2.0)));
	elseif ((a_m ^ 2.0) <= 1e+252)
		tmp = Float64(2.0 * Float64(cos(Float64(cbrt(t_1) * (cbrt((t_1 ^ 2.0)) ^ 4.0))) * t_3));
	else
		tmp = Float64(2.0 * Float64(t_3 * cos(Float64((t_2 ^ 0.16666666666666666) * Float64(Float64((Float64(angle_m * pi) ^ 0.16666666666666666) * (0.005555555555555556 ^ 0.16666666666666666)) * (cbrt(t_2) ^ 2.0))))));
	end
	return Float64(angle_s * tmp)
end

Wolfram

a_m = N[Abs[a], $MachinePrecision]
b_m = N[Abs[b], $MachinePrecision]
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$95$m_, b$95$m_, angle$95$m_] := Block[{t$95$0 = N[Sin[N[(0.005555555555555556 * N[(angle$95$m * Pi), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$1 = N[Power[N[(angle$95$m * N[(0.005555555555555556 * Pi), $MachinePrecision]), $MachinePrecision], 1/3], $MachinePrecision]}, Block[{t$95$2 = N[(Pi * N[(0.005555555555555556 * angle$95$m), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$3 = N[(N[(N[Sin[t$95$2], $MachinePrecision] * N[(a$95$m + b$95$m), $MachinePrecision]), $MachinePrecision] * N[(b$95$m - a$95$m), $MachinePrecision]), $MachinePrecision]}, N[(angle$95$s * If[LessEqual[N[Power[a$95$m, 2.0], $MachinePrecision], 2e-186], N[(N[(b$95$m * N[(2.0 * N[(b$95$m * t$95$0), $MachinePrecision]), $MachinePrecision] + N[(-2.0 * N[(N[Power[a$95$m, 2.0], $MachinePrecision] * t$95$0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[Sqrt[N[Power[N[Cos[t$95$2], $MachinePrecision], 2.0], $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[N[Power[a$95$m, 2.0], $MachinePrecision], 1e+252], N[(2.0 * N[(N[Cos[N[(N[Power[t$95$1, 1/3], $MachinePrecision] * N[Power[N[Power[N[Power[t$95$1, 2.0], $MachinePrecision], 1/3], $MachinePrecision], 4.0], $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * t$95$3), $MachinePrecision]), $MachinePrecision], N[(2.0 * N[(t$95$3 * N[Cos[N[(N[Power[t$95$2, 0.16666666666666666], $MachinePrecision] * N[(N[(N[Power[N[(angle$95$m * Pi), $MachinePrecision], 0.16666666666666666], $MachinePrecision] * N[Power[0.005555555555555556, 0.16666666666666666], $MachinePrecision]), $MachinePrecision] * N[Power[N[Power[t$95$2, 1/3], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

2. if (pow.f64 a #s(literal 2 binary64)) < 1.9999999999999998e-186

   1. Initial program 64.7%

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

      1. unpow2 64.7%

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

      2. unpow2 64.7%

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

      3. difference-of-squares 64.7%

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

   4. Applied egg-rr 64.7%

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

   5. Taylor expanded in b around 0 70.8%

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

      1. +-commutative 70.8%

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

      2. fma-define 70.8%

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

   7. Simplified 70.8%

      \[\leadsto \color{blue}{\mathsf{fma}\left(b, 2 \cdot \left(\sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right) \cdot \left(b + 0\right)\right), -2 \cdot \left({a}^{2} \cdot \sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)\right)\right)} \cdot \cos \left(\pi \cdot \frac{angle}{180}\right) \]
   8. Step-by-step derivation

      1. add-sqr-sqrt 62.8%

         \[\leadsto \mathsf{fma}\left(b, 2 \cdot \left(\sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right) \cdot \left(b + 0\right)\right), -2 \cdot \left({a}^{2} \cdot \sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)\right)\right) \cdot \color{blue}{\left(\sqrt{\cos \left(\pi \cdot \frac{angle}{180}\right)} \cdot \sqrt{\cos \left(\pi \cdot \frac{angle}{180}\right)}\right)} \]

      2. sqrt-unprod 74.6%

         \[\leadsto \mathsf{fma}\left(b, 2 \cdot \left(\sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right) \cdot \left(b + 0\right)\right), -2 \cdot \left({a}^{2} \cdot \sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)\right)\right) \cdot \color{blue}{\sqrt{\cos \left(\pi \cdot \frac{angle}{180}\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right)}} \]

      3. pow2 74.6%

         \[\leadsto \mathsf{fma}\left(b, 2 \cdot \left(\sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right) \cdot \left(b + 0\right)\right), -2 \cdot \left({a}^{2} \cdot \sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)\right)\right) \cdot \sqrt{\color{blue}{{\cos \left(\pi \cdot \frac{angle}{180}\right)}^{2}}} \]

      4. div-inv 74.6%

         \[\leadsto \mathsf{fma}\left(b, 2 \cdot \left(\sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right) \cdot \left(b + 0\right)\right), -2 \cdot \left({a}^{2} \cdot \sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)\right)\right) \cdot \sqrt{{\cos \left(\pi \cdot \color{blue}{\left(angle \cdot \frac{1}{180}\right)}\right)}^{2}} \]

      5. metadata-eval 74.6%

         \[\leadsto \mathsf{fma}\left(b, 2 \cdot \left(\sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right) \cdot \left(b + 0\right)\right), -2 \cdot \left({a}^{2} \cdot \sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)\right)\right) \cdot \sqrt{{\cos \left(\pi \cdot \left(angle \cdot \color{blue}{0.005555555555555556}\right)\right)}^{2}} \]

   9. Applied egg-rr 74.6%

      \[\leadsto \mathsf{fma}\left(b, 2 \cdot \left(\sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right) \cdot \left(b + 0\right)\right), -2 \cdot \left({a}^{2} \cdot \sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)\right)\right) \cdot \color{blue}{\sqrt{{\cos \left(\pi \cdot \left(angle \cdot 0.005555555555555556\right)\right)}^{2}}} \]

   if 1.9999999999999998e-186 < (pow.f64 a #s(literal 2 binary64)) < 1.0000000000000001e252

   1. Initial program 55.8%

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

      1. associate-*l* 55.8%

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

      2. *-commutative 55.8%

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

      3. associate-*l* 55.8%

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

   3. Simplified 55.8%

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

      1. unpow2 55.8%

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

      2. unpow2 55.8%

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

      3. difference-of-squares 55.8%

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

   6. Applied egg-rr 55.8%

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

      1. add-cube-cbrt 54.9%

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

      2. pow2 54.9%

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

   8. Applied egg-rr 54.9%

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

   9. Taylor expanded in angle around inf 56.6%

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

       1. associate-*r* 54.9%

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

       2. associate-*r* 58.4%

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

       3. associate-*r* 58.3%

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

   11. Simplified 58.3%

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

       1. *-commutative 58.3%

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

       2. *-commutative 58.3%

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

       3. add-cube-cbrt 57.1%

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

       4. pow3 59.4%

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

   13. Applied egg-rr 59.4%

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

       1. unpow3 57.1%

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

       2. add-cube-cbrt 58.0%

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

       3. associate-*r* 58.9%

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

       4. pow2 58.9%

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

       5. cbrt-unprod 59.8%

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

       6. pow2 59.8%

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

   15. Applied egg-rr 59.8%

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

       1. *-commutative 59.8%

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

       2. associate-*r* 59.7%

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

       3. *-commutative 59.7%

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

       4. associate-*r* 59.7%

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

       5. rem-cube-cbrt 59.4%

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

       6. pow-plus 60.2%

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

       7. associate-*r* 60.9%

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

       8. *-commutative 60.9%

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

       9. associate-*r* 61.3%

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

       10. metadata-eval 61.3%

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

   17. Simplified 61.3%

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

   if 1.0000000000000001e252 < (pow.f64 a #s(literal 2 binary64))

   1. Initial program 44.9%

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

      1. associate-*l* 44.9%

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

      2. *-commutative 44.9%

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

      3. associate-*l* 44.9%

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

   3. Simplified 44.9%

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

      1. unpow2 44.9%

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

      2. unpow2 44.9%

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

      3. difference-of-squares 61.5%

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

   6. Applied egg-rr 61.5%

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

      1. add-cube-cbrt 57.2%

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

      2. pow2 57.2%

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

   8. Applied egg-rr 57.2%

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

   9. Taylor expanded in angle around inf 61.4%

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

       1. associate-*r* 61.7%

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

       2. associate-*r* 76.9%

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

       3. associate-*r* 72.8%

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

   11. Simplified 72.8%

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

       1. *-commutative 72.8%

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

       2. *-commutative 72.8%

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

       3. add-cube-cbrt 81.0%

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

       4. pow3 79.7%

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

   13. Applied egg-rr 79.7%

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

       1. cube-mult 81.0%

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

       2. add-sqr-sqrt 39.7%

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

       3. associate-*l* 39.7%

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

       4. pow1/3 35.6%

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

       5. sqrt-pow1 37.0%

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

       6. metadata-eval 37.0%

          \[\leadsto 2 \cdot \left(\cos \left({\left(\pi \cdot \left(angle \cdot 0.005555555555555556\right)\right)}^{\color{blue}{0.16666666666666666}} \cdot \left(\sqrt{\sqrt[3]{\pi \cdot \left(angle \cdot 0.005555555555555556\right)}} \cdot \left(\sqrt[3]{\pi \cdot \left(angle \cdot 0.005555555555555556\right)} \cdot \sqrt[3]{\pi \cdot \left(angle \cdot 0.005555555555555556\right)}\right)\right)\right) \cdot \left(\left(\sin \left(\left(0.005555555555555556 \cdot angle\right) \cdot \pi\right) \cdot \left(a + b\right)\right) \cdot \left(b - a\right)\right)\right) \]

       7. pow1/3 37.2%

          \[\leadsto 2 \cdot \left(\cos \left({\left(\pi \cdot \left(angle \cdot 0.005555555555555556\right)\right)}^{0.16666666666666666} \cdot \left(\sqrt{\color{blue}{{\left(\pi \cdot \left(angle \cdot 0.005555555555555556\right)\right)}^{0.3333333333333333}}} \cdot \left(\sqrt[3]{\pi \cdot \left(angle \cdot 0.005555555555555556\right)} \cdot \sqrt[3]{\pi \cdot \left(angle \cdot 0.005555555555555556\right)}\right)\right)\right) \cdot \left(\left(\sin \left(\left(0.005555555555555556 \cdot angle\right) \cdot \pi\right) \cdot \left(a + b\right)\right) \cdot \left(b - a\right)\right)\right) \]

       8. sqrt-pow1 35.9%

          \[\leadsto 2 \cdot \left(\cos \left({\left(\pi \cdot \left(angle \cdot 0.005555555555555556\right)\right)}^{0.16666666666666666} \cdot \left(\color{blue}{{\left(\pi \cdot \left(angle \cdot 0.005555555555555556\right)\right)}^{\left(\frac{0.3333333333333333}{2}\right)}} \cdot \left(\sqrt[3]{\pi \cdot \left(angle \cdot 0.005555555555555556\right)} \cdot \sqrt[3]{\pi \cdot \left(angle \cdot 0.005555555555555556\right)}\right)\right)\right) \cdot \left(\left(\sin \left(\left(0.005555555555555556 \cdot angle\right) \cdot \pi\right) \cdot \left(a + b\right)\right) \cdot \left(b - a\right)\right)\right) \]

       9. metadata-eval 35.9%

          \[\leadsto 2 \cdot \left(\cos \left({\left(\pi \cdot \left(angle \cdot 0.005555555555555556\right)\right)}^{0.16666666666666666} \cdot \left({\left(\pi \cdot \left(angle \cdot 0.005555555555555556\right)\right)}^{\color{blue}{0.16666666666666666}} \cdot \left(\sqrt[3]{\pi \cdot \left(angle \cdot 0.005555555555555556\right)} \cdot \sqrt[3]{\pi \cdot \left(angle \cdot 0.005555555555555556\right)}\right)\right)\right) \cdot \left(\left(\sin \left(\left(0.005555555555555556 \cdot angle\right) \cdot \pi\right) \cdot \left(a + b\right)\right) \cdot \left(b - a\right)\right)\right) \]

       10. pow2 35.9%

           \[\leadsto 2 \cdot \left(\cos \left({\left(\pi \cdot \left(angle \cdot 0.005555555555555556\right)\right)}^{0.16666666666666666} \cdot \left({\left(\pi \cdot \left(angle \cdot 0.005555555555555556\right)\right)}^{0.16666666666666666} \cdot \color{blue}{{\left(\sqrt[3]{\pi \cdot \left(angle \cdot 0.005555555555555556\right)}\right)}^{2}}\right)\right) \cdot \left(\left(\sin \left(\left(0.005555555555555556 \cdot angle\right) \cdot \pi\right) \cdot \left(a + b\right)\right) \cdot \left(b - a\right)\right)\right) \]

   15. Applied egg-rr 35.9%

       \[\leadsto 2 \cdot \left(\cos \color{blue}{\left({\left(\pi \cdot \left(angle \cdot 0.005555555555555556\right)\right)}^{0.16666666666666666} \cdot \left({\left(\pi \cdot \left(angle \cdot 0.005555555555555556\right)\right)}^{0.16666666666666666} \cdot {\left(\sqrt[3]{\pi \cdot \left(angle \cdot 0.005555555555555556\right)}\right)}^{2}\right)\right)} \cdot \left(\left(\sin \left(\left(0.005555555555555556 \cdot angle\right) \cdot \pi\right) \cdot \left(a + b\right)\right) \cdot \left(b - a\right)\right)\right) \]
   16. Step-by-step derivation

       1. associate-*r* 37.2%

          \[\leadsto 2 \cdot \left(\cos \left({\left(\pi \cdot \left(angle \cdot 0.005555555555555556\right)\right)}^{0.16666666666666666} \cdot \left({\color{blue}{\left(\left(\pi \cdot angle\right) \cdot 0.005555555555555556\right)}}^{0.16666666666666666} \cdot {\left(\sqrt[3]{\pi \cdot \left(angle \cdot 0.005555555555555556\right)}\right)}^{2}\right)\right) \cdot \left(\left(\sin \left(\left(0.005555555555555556 \cdot angle\right) \cdot \pi\right) \cdot \left(a + b\right)\right) \cdot \left(b - a\right)\right)\right) \]

       2. *-commutative 37.2%

          \[\leadsto 2 \cdot \left(\cos \left({\left(\pi \cdot \left(angle \cdot 0.005555555555555556\right)\right)}^{0.16666666666666666} \cdot \left({\left(\color{blue}{\left(angle \cdot \pi\right)} \cdot 0.005555555555555556\right)}^{0.16666666666666666} \cdot {\left(\sqrt[3]{\pi \cdot \left(angle \cdot 0.005555555555555556\right)}\right)}^{2}\right)\right) \cdot \left(\left(\sin \left(\left(0.005555555555555556 \cdot angle\right) \cdot \pi\right) \cdot \left(a + b\right)\right) \cdot \left(b - a\right)\right)\right) \]

       3. unpow-prod-down 38.6%

          \[\leadsto 2 \cdot \left(\cos \left({\left(\pi \cdot \left(angle \cdot 0.005555555555555556\right)\right)}^{0.16666666666666666} \cdot \left(\color{blue}{\left({\left(angle \cdot \pi\right)}^{0.16666666666666666} \cdot {0.005555555555555556}^{0.16666666666666666}\right)} \cdot {\left(\sqrt[3]{\pi \cdot \left(angle \cdot 0.005555555555555556\right)}\right)}^{2}\right)\right) \cdot \left(\left(\sin \left(\left(0.005555555555555556 \cdot angle\right) \cdot \pi\right) \cdot \left(a + b\right)\right) \cdot \left(b - a\right)\right)\right) \]

       4. *-commutative 38.6%

          \[\leadsto 2 \cdot \left(\cos \left({\left(\pi \cdot \left(angle \cdot 0.005555555555555556\right)\right)}^{0.16666666666666666} \cdot \left(\left({\color{blue}{\left(\pi \cdot angle\right)}}^{0.16666666666666666} \cdot {0.005555555555555556}^{0.16666666666666666}\right) \cdot {\left(\sqrt[3]{\pi \cdot \left(angle \cdot 0.005555555555555556\right)}\right)}^{2}\right)\right) \cdot \left(\left(\sin \left(\left(0.005555555555555556 \cdot angle\right) \cdot \pi\right) \cdot \left(a + b\right)\right) \cdot \left(b - a\right)\right)\right) \]

   17. Applied egg-rr 38.6%

       \[\leadsto 2 \cdot \left(\cos \left({\left(\pi \cdot \left(angle \cdot 0.005555555555555556\right)\right)}^{0.16666666666666666} \cdot \left(\color{blue}{\left({\left(\pi \cdot angle\right)}^{0.16666666666666666} \cdot {0.005555555555555556}^{0.16666666666666666}\right)} \cdot {\left(\sqrt[3]{\pi \cdot \left(angle \cdot 0.005555555555555556\right)}\right)}^{2}\right)\right) \cdot \left(\left(\sin \left(\left(0.005555555555555556 \cdot angle\right) \cdot \pi\right) \cdot \left(a + b\right)\right) \cdot \left(b - a\right)\right)\right) \]
3. Recombined 3 regimes into one program.

4. Final simplification 60.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;{a}^{2} \leq 2 \cdot 10^{-186}:\\ \;\;\;\;\mathsf{fma}\left(b, 2 \cdot \left(b \cdot \sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)\right), -2 \cdot \left({a}^{2} \cdot \sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)\right)\right) \cdot \sqrt{{\cos \left(\pi \cdot \left(0.005555555555555556 \cdot angle\right)\right)}^{2}}\\ \mathbf{elif}\;{a}^{2} \leq 10^{+252}:\\ \;\;\;\;2 \cdot \left(\cos \left(\sqrt[3]{\sqrt[3]{angle \cdot \left(0.005555555555555556 \cdot \pi\right)}} \cdot {\left(\sqrt[3]{{\left(\sqrt[3]{angle \cdot \left(0.005555555555555556 \cdot \pi\right)}\right)}^{2}}\right)}^{4}\right) \cdot \left(\left(\sin \left(\pi \cdot \left(0.005555555555555556 \cdot angle\right)\right) \cdot \left(a + b\right)\right) \cdot \left(b - a\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \left(\left(\left(\sin \left(\pi \cdot \left(0.005555555555555556 \cdot angle\right)\right) \cdot \left(a + b\right)\right) \cdot \left(b - a\right)\right) \cdot \cos \left({\left(\pi \cdot \left(0.005555555555555556 \cdot angle\right)\right)}^{0.16666666666666666} \cdot \left(\left({\left(angle \cdot \pi\right)}^{0.16666666666666666} \cdot {0.005555555555555556}^{0.16666666666666666}\right) \cdot {\left(\sqrt[3]{\pi \cdot \left(0.005555555555555556 \cdot angle\right)}\right)}^{2}\right)\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 66.7% accurate, 1.0× speedup

Math

\[\begin{array}{l} a_m = \left|a\right| \\ b_m = \left|b\right| \\ angle\_m = \left|angle\right| \\ angle\_s = \mathsf{copysign}\left(1, angle\right) \\ angle\_s \cdot \begin{array}{l} \mathbf{if}\;b\_m \leq 1.2 \cdot 10^{+198}:\\ \;\;\;\;\left(a\_m + b\_m\right) \cdot \left(\left(b\_m - a\_m\right) \cdot \sin \left(0.011111111111111112 \cdot \left(angle\_m \cdot \left(\sqrt[3]{\pi} \cdot {\left(\sqrt[3]{\pi}\right)}^{2}\right)\right)\right)\right)\\ \mathbf{elif}\;b\_m \leq 8.5 \cdot 10^{+246}:\\ \;\;\;\;\left(a\_m + b\_m\right) \cdot \left(\left(b\_m - a\_m\right) \cdot \sin \left(\left(angle\_m \cdot \sqrt[3]{{\pi}^{3}}\right) \cdot 0.011111111111111112\right)\right)\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \left(b\_m \cdot \left(\left(b\_m - a\_m\right) \cdot \sin \left(angle\_m \cdot \left(0.005555555555555556 \cdot \pi\right)\right)\right)\right)\\ \end{array} \end{array} \]

FPCore

a_m = (fabs.f64 a)
b_m = (fabs.f64 b)
angle\_m = (fabs.f64 angle)
angle\_s = (copysign.f64 #s(literal 1 binary64) angle)
(FPCore (angle_s a_m b_m angle_m)
 :precision binary64
 (*
  angle_s
  (if (<= b_m 1.2e+198)
    (*
     (+ a_m b_m)
     (*
      (- b_m a_m)
      (sin
       (*
        0.011111111111111112
        (* angle_m (* (cbrt PI) (pow (cbrt PI) 2.0)))))))
    (if (<= b_m 8.5e+246)
      (*
       (+ a_m b_m)
       (*
        (- b_m a_m)
        (sin (* (* angle_m (cbrt (pow PI 3.0))) 0.011111111111111112))))
      (*
       2.0
       (*
        b_m
        (* (- b_m a_m) (sin (* angle_m (* 0.005555555555555556 PI))))))))))

C

a_m = fabs(a);
b_m = fabs(b);
angle\_m = fabs(angle);
angle\_s = copysign(1.0, angle);
double code(double angle_s, double a_m, double b_m, double angle_m) {
	double tmp;
	if (b_m <= 1.2e+198) {
		tmp = (a_m + b_m) * ((b_m - a_m) * sin((0.011111111111111112 * (angle_m * (cbrt(((double) M_PI)) * pow(cbrt(((double) M_PI)), 2.0))))));
	} else if (b_m <= 8.5e+246) {
		tmp = (a_m + b_m) * ((b_m - a_m) * sin(((angle_m * cbrt(pow(((double) M_PI), 3.0))) * 0.011111111111111112)));
	} else {
		tmp = 2.0 * (b_m * ((b_m - a_m) * sin((angle_m * (0.005555555555555556 * ((double) M_PI))))));
	}
	return angle_s * tmp;
}

Java

a_m = Math.abs(a);
b_m = Math.abs(b);
angle\_m = Math.abs(angle);
angle\_s = Math.copySign(1.0, angle);
public static double code(double angle_s, double a_m, double b_m, double angle_m) {
	double tmp;
	if (b_m <= 1.2e+198) {
		tmp = (a_m + b_m) * ((b_m - a_m) * Math.sin((0.011111111111111112 * (angle_m * (Math.cbrt(Math.PI) * Math.pow(Math.cbrt(Math.PI), 2.0))))));
	} else if (b_m <= 8.5e+246) {
		tmp = (a_m + b_m) * ((b_m - a_m) * Math.sin(((angle_m * Math.cbrt(Math.pow(Math.PI, 3.0))) * 0.011111111111111112)));
	} else {
		tmp = 2.0 * (b_m * ((b_m - a_m) * Math.sin((angle_m * (0.005555555555555556 * Math.PI)))));
	}
	return angle_s * tmp;
}

Julia

a_m = abs(a)
b_m = abs(b)
angle\_m = abs(angle)
angle\_s = copysign(1.0, angle)
function code(angle_s, a_m, b_m, angle_m)
	tmp = 0.0
	if (b_m <= 1.2e+198)
		tmp = Float64(Float64(a_m + b_m) * Float64(Float64(b_m - a_m) * sin(Float64(0.011111111111111112 * Float64(angle_m * Float64(cbrt(pi) * (cbrt(pi) ^ 2.0)))))));
	elseif (b_m <= 8.5e+246)
		tmp = Float64(Float64(a_m + b_m) * Float64(Float64(b_m - a_m) * sin(Float64(Float64(angle_m * cbrt((pi ^ 3.0))) * 0.011111111111111112))));
	else
		tmp = Float64(2.0 * Float64(b_m * Float64(Float64(b_m - a_m) * sin(Float64(angle_m * Float64(0.005555555555555556 * pi))))));
	end
	return Float64(angle_s * tmp)
end

Wolfram

a_m = N[Abs[a], $MachinePrecision]
b_m = N[Abs[b], $MachinePrecision]
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$95$m_, b$95$m_, angle$95$m_] := N[(angle$95$s * If[LessEqual[b$95$m, 1.2e+198], N[(N[(a$95$m + b$95$m), $MachinePrecision] * N[(N[(b$95$m - a$95$m), $MachinePrecision] * N[Sin[N[(0.011111111111111112 * N[(angle$95$m * N[(N[Power[Pi, 1/3], $MachinePrecision] * N[Power[N[Power[Pi, 1/3], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[b$95$m, 8.5e+246], N[(N[(a$95$m + b$95$m), $MachinePrecision] * N[(N[(b$95$m - a$95$m), $MachinePrecision] * N[Sin[N[(N[(angle$95$m * N[Power[N[Power[Pi, 3.0], $MachinePrecision], 1/3], $MachinePrecision]), $MachinePrecision] * 0.011111111111111112), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(2.0 * N[(b$95$m * N[(N[(b$95$m - a$95$m), $MachinePrecision] * N[Sin[N[(angle$95$m * N[(0.005555555555555556 * Pi), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]), $MachinePrecision]

Derivation

1. Split input into 3 regimes

2. if b < 1.2000000000000001e198

   1. Initial program 59.8%

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

      1. associate-*l* 59.8%

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

      2. *-commutative 59.8%

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

      3. associate-*l* 59.8%

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

   3. Simplified 59.8%

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

      1. unpow2 59.8%

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

      2. unpow2 59.8%

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

      3. difference-of-squares 62.0%

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

   6. Applied egg-rr 62.0%

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

      1. add-exp-log 35.6%

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

      2. 2-sin 35.6%

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

      3. div-inv 34.9%

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

      4. metadata-eval 34.9%

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

   8. Applied egg-rr 34.9%

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

      1. rem-exp-log 60.0%

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

      2. associate-*l* 66.9%

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

      3. +-commutative 66.9%

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

      4. count-2 66.9%

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

      5. add-cube-cbrt 66.1%

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

      6. unpow2 66.1%

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

      7. metadata-eval 66.1%

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

      8. div-inv 67.4%

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

      9. metadata-eval 67.4%

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

      10. div-inv 67.0%

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

   10. Applied egg-rr 68.0%

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

       1. add-cube-cbrt 60.7%

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

       2. pow2 60.7%

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

   12. Applied egg-rr 69.0%

       \[\leadsto \left(a + b\right) \cdot \left(\left(b - a\right) \cdot \sin \left(\left(angle \cdot \color{blue}{\left({\left(\sqrt[3]{\pi}\right)}^{2} \cdot \sqrt[3]{\pi}\right)}\right) \cdot 0.011111111111111112\right)\right) \]

   if 1.2000000000000001e198 < b < 8.49999999999999952e246

   1. Initial program 10.1%

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

      1. associate-*l* 10.1%

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

      2. *-commutative 10.1%

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

      3. associate-*l* 10.1%

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

   3. Simplified 10.1%

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

      1. unpow2 10.1%

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

      2. unpow2 10.1%

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

      3. difference-of-squares 35.1%

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

   6. Applied egg-rr 35.1%

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

      1. add-exp-log 16.7%

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

      2. 2-sin 16.7%

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

      3. div-inv 25.0%

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

      4. metadata-eval 25.0%

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

   8. Applied egg-rr 25.0%

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

      1. rem-exp-log 43.5%

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

      2. associate-*l* 74.9%

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

      3. +-commutative 74.9%

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

      4. count-2 74.9%

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

      5. add-cube-cbrt 74.5%

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

      6. unpow2 74.5%

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

      7. metadata-eval 74.5%

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

      8. div-inv 74.5%

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

      9. metadata-eval 74.5%

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

      10. div-inv 74.5%

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

   10. Applied egg-rr 74.9%

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

       1. add-cbrt-cube 99.6%

          \[\leadsto \left(a + b\right) \cdot \left(\left(b - a\right) \cdot \sin \left(\left(angle \cdot \color{blue}{\sqrt[3]{\left(\pi \cdot \pi\right) \cdot \pi}}\right) \cdot 0.011111111111111112\right)\right) \]

       2. pow3 99.6%

          \[\leadsto \left(a + b\right) \cdot \left(\left(b - a\right) \cdot \sin \left(\left(angle \cdot \sqrt[3]{\color{blue}{{\pi}^{3}}}\right) \cdot 0.011111111111111112\right)\right) \]

   12. Applied egg-rr 99.6%

       \[\leadsto \left(a + b\right) \cdot \left(\left(b - a\right) \cdot \sin \left(\left(angle \cdot \color{blue}{\sqrt[3]{{\pi}^{3}}}\right) \cdot 0.011111111111111112\right)\right) \]

   if 8.49999999999999952e246 < b

   1. Initial program 42.1%

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

      1. associate-*l* 42.1%

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

      2. *-commutative 42.1%

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

      3. associate-*l* 42.1%

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

   3. Simplified 42.1%

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

      1. unpow2 42.1%

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

      2. unpow2 42.1%

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

      3. difference-of-squares 65.7%

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

   6. Applied egg-rr 65.7%

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

      1. add-cube-cbrt 53.9%

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

      2. pow2 53.9%

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

   8. Applied egg-rr 53.9%

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

   9. Taylor expanded in angle around inf 65.7%

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

       1. associate-*r* 65.7%

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

       2. associate-*r* 76.5%

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

       3. associate-*r* 76.5%

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

   11. Simplified 76.5%

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

   12. Taylor expanded in angle around 0 88.2%

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

   13. Taylor expanded in a around 0 82.4%

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

       1. associate-*r* 82.4%

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

   15. Simplified 82.4%

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

   16. Taylor expanded in b around 0 82.4%

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

       1. associate-*r* 82.4%

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

       2. neg-mul-1 82.4%

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

       3. distribute-rgt-out 82.4%

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

       4. *-commutative 82.4%

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

       5. associate-*r* 82.4%

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

       6. +-commutative 82.4%

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

       7. sub-neg 82.4%

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

   18. Simplified 82.4%

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

4. Final simplification 71.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq 1.2 \cdot 10^{+198}:\\ \;\;\;\;\left(a + b\right) \cdot \left(\left(b - a\right) \cdot \sin \left(0.011111111111111112 \cdot \left(angle \cdot \left(\sqrt[3]{\pi} \cdot {\left(\sqrt[3]{\pi}\right)}^{2}\right)\right)\right)\right)\\ \mathbf{elif}\;b \leq 8.5 \cdot 10^{+246}:\\ \;\;\;\;\left(a + b\right) \cdot \left(\left(b - a\right) \cdot \sin \left(\left(angle \cdot \sqrt[3]{{\pi}^{3}}\right) \cdot 0.011111111111111112\right)\right)\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \left(b \cdot \left(\left(b - a\right) \cdot \sin \left(angle \cdot \left(0.005555555555555556 \cdot \pi\right)\right)\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 64.9% accurate, 1.3× speedup

Math

\[\begin{array}{l} a_m = \left|a\right| \\ b_m = \left|b\right| \\ angle\_m = \left|angle\right| \\ angle\_s = \mathsf{copysign}\left(1, angle\right) \\ \begin{array}{l} t_0 := \sin \left(\left(angle\_m \cdot \pi\right) \cdot 0.011111111111111112\right)\\ angle\_s \cdot \begin{array}{l} \mathbf{if}\;{b\_m}^{2} \leq 2 \cdot 10^{-260}:\\ \;\;\;\;\left(a\_m \cdot t\_0\right) \cdot \left(\left(-b\_m\right) - a\_m\right)\\ \mathbf{elif}\;{b\_m}^{2} \leq 5 \cdot 10^{+128}:\\ \;\;\;\;t\_0 \cdot \left(\left(a\_m + b\_m\right) \cdot \left(b\_m - a\_m\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(a\_m + b\_m\right) \cdot \left(angle\_m \cdot \left(\left(b\_m - a\_m\right) \cdot \left(\pi \cdot 0.011111111111111112\right)\right)\right)\\ \end{array} \end{array} \end{array} \]

FPCore

a_m = (fabs.f64 a)
b_m = (fabs.f64 b)
angle\_m = (fabs.f64 angle)
angle\_s = (copysign.f64 #s(literal 1 binary64) angle)
(FPCore (angle_s a_m b_m angle_m)
 :precision binary64
 (let* ((t_0 (sin (* (* angle_m PI) 0.011111111111111112))))
   (*
    angle_s
    (if (<= (pow b_m 2.0) 2e-260)
      (* (* a_m t_0) (- (- b_m) a_m))
      (if (<= (pow b_m 2.0) 5e+128)
        (* t_0 (* (+ a_m b_m) (- b_m a_m)))
        (*
         (+ a_m b_m)
         (* angle_m (* (- b_m a_m) (* PI 0.011111111111111112)))))))))

C

a_m = fabs(a);
b_m = fabs(b);
angle\_m = fabs(angle);
angle\_s = copysign(1.0, angle);
double code(double angle_s, double a_m, double b_m, double angle_m) {
	double t_0 = sin(((angle_m * ((double) M_PI)) * 0.011111111111111112));
	double tmp;
	if (pow(b_m, 2.0) <= 2e-260) {
		tmp = (a_m * t_0) * (-b_m - a_m);
	} else if (pow(b_m, 2.0) <= 5e+128) {
		tmp = t_0 * ((a_m + b_m) * (b_m - a_m));
	} else {
		tmp = (a_m + b_m) * (angle_m * ((b_m - a_m) * (((double) M_PI) * 0.011111111111111112)));
	}
	return angle_s * tmp;
}

Java

a_m = Math.abs(a);
b_m = Math.abs(b);
angle\_m = Math.abs(angle);
angle\_s = Math.copySign(1.0, angle);
public static double code(double angle_s, double a_m, double b_m, double angle_m) {
	double t_0 = Math.sin(((angle_m * Math.PI) * 0.011111111111111112));
	double tmp;
	if (Math.pow(b_m, 2.0) <= 2e-260) {
		tmp = (a_m * t_0) * (-b_m - a_m);
	} else if (Math.pow(b_m, 2.0) <= 5e+128) {
		tmp = t_0 * ((a_m + b_m) * (b_m - a_m));
	} else {
		tmp = (a_m + b_m) * (angle_m * ((b_m - a_m) * (Math.PI * 0.011111111111111112)));
	}
	return angle_s * tmp;
}

Python

a_m = math.fabs(a)
b_m = math.fabs(b)
angle\_m = math.fabs(angle)
angle\_s = math.copysign(1.0, angle)
def code(angle_s, a_m, b_m, angle_m):
	t_0 = math.sin(((angle_m * math.pi) * 0.011111111111111112))
	tmp = 0
	if math.pow(b_m, 2.0) <= 2e-260:
		tmp = (a_m * t_0) * (-b_m - a_m)
	elif math.pow(b_m, 2.0) <= 5e+128:
		tmp = t_0 * ((a_m + b_m) * (b_m - a_m))
	else:
		tmp = (a_m + b_m) * (angle_m * ((b_m - a_m) * (math.pi * 0.011111111111111112)))
	return angle_s * tmp

Julia

a_m = abs(a)
b_m = abs(b)
angle\_m = abs(angle)
angle\_s = copysign(1.0, angle)
function code(angle_s, a_m, b_m, angle_m)
	t_0 = sin(Float64(Float64(angle_m * pi) * 0.011111111111111112))
	tmp = 0.0
	if ((b_m ^ 2.0) <= 2e-260)
		tmp = Float64(Float64(a_m * t_0) * Float64(Float64(-b_m) - a_m));
	elseif ((b_m ^ 2.0) <= 5e+128)
		tmp = Float64(t_0 * Float64(Float64(a_m + b_m) * Float64(b_m - a_m)));
	else
		tmp = Float64(Float64(a_m + b_m) * Float64(angle_m * Float64(Float64(b_m - a_m) * Float64(pi * 0.011111111111111112))));
	end
	return Float64(angle_s * tmp)
end

MATLAB

a_m = abs(a);
b_m = abs(b);
angle\_m = abs(angle);
angle\_s = sign(angle) * abs(1.0);
function tmp_2 = code(angle_s, a_m, b_m, angle_m)
	t_0 = sin(((angle_m * pi) * 0.011111111111111112));
	tmp = 0.0;
	if ((b_m ^ 2.0) <= 2e-260)
		tmp = (a_m * t_0) * (-b_m - a_m);
	elseif ((b_m ^ 2.0) <= 5e+128)
		tmp = t_0 * ((a_m + b_m) * (b_m - a_m));
	else
		tmp = (a_m + b_m) * (angle_m * ((b_m - a_m) * (pi * 0.011111111111111112)));
	end
	tmp_2 = angle_s * tmp;
end

Wolfram

a_m = N[Abs[a], $MachinePrecision]
b_m = N[Abs[b], $MachinePrecision]
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$95$m_, b$95$m_, angle$95$m_] := Block[{t$95$0 = N[Sin[N[(N[(angle$95$m * Pi), $MachinePrecision] * 0.011111111111111112), $MachinePrecision]], $MachinePrecision]}, N[(angle$95$s * If[LessEqual[N[Power[b$95$m, 2.0], $MachinePrecision], 2e-260], N[(N[(a$95$m * t$95$0), $MachinePrecision] * N[((-b$95$m) - a$95$m), $MachinePrecision]), $MachinePrecision], If[LessEqual[N[Power[b$95$m, 2.0], $MachinePrecision], 5e+128], N[(t$95$0 * N[(N[(a$95$m + b$95$m), $MachinePrecision] * N[(b$95$m - a$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(a$95$m + b$95$m), $MachinePrecision] * N[(angle$95$m * N[(N[(b$95$m - a$95$m), $MachinePrecision] * N[(Pi * 0.011111111111111112), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]), $MachinePrecision]]

Derivation

1. Split input into 3 regimes

2. if (pow.f64 b #s(literal 2 binary64)) < 1.99999999999999992e-260

   1. Initial program 74.0%

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

      1. associate-*l* 74.0%

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

      2. *-commutative 74.0%

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

      3. associate-*l* 74.0%

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

   3. Simplified 74.0%

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

      1. unpow2 74.0%

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

      2. unpow2 74.0%

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

      3. difference-of-squares 74.0%

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

   6. Applied egg-rr 74.0%

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

      1. add-exp-log 52.8%

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

      2. 2-sin 52.8%

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

      3. div-inv 52.0%

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

      4. metadata-eval 52.0%

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

   8. Applied egg-rr 52.0%

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

      1. rem-exp-log 72.9%

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

      2. associate-*l* 79.2%

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

      3. +-commutative 79.2%

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

      4. count-2 79.2%

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

      5. add-cube-cbrt 79.8%

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

      6. unpow2 79.8%

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

      7. metadata-eval 79.8%

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

      8. div-inv 79.7%

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

      9. metadata-eval 79.7%

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

      10. div-inv 78.3%

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

   10. Applied egg-rr 81.1%

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

   11. Taylor expanded in b around 0 80.2%

       \[\leadsto \left(a + b\right) \cdot \color{blue}{\left(-1 \cdot \left(a \cdot \sin \left(0.011111111111111112 \cdot \left(angle \cdot \pi\right)\right)\right)\right)} \]
   12. Step-by-step derivation

       1. mul-1-neg 80.2%

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

       2. *-commutative 80.2%

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

       3. distribute-rgt-neg-in 80.2%

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

   13. Simplified 80.2%

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

   if 1.99999999999999992e-260 < (pow.f64 b #s(literal 2 binary64)) < 5e128

   1. Initial program 53.5%

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

      1. associate-*l* 53.5%

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

      2. *-commutative 53.5%

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

      3. associate-*l* 53.5%

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

   3. Simplified 53.5%

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

      1. unpow2 53.5%

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

      2. unpow2 53.5%

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

      3. difference-of-squares 53.6%

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

   6. Applied egg-rr 53.6%

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

      1. add-exp-log 29.8%

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

      2. 2-sin 29.8%

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

      3. div-inv 30.1%

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

      4. metadata-eval 30.1%

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

   8. Applied egg-rr 30.1%

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

      1. rem-exp-log 52.5%

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

      2. difference-of-squares 52.5%

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

      3. pow2 52.5%

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

      4. unpow2 52.5%

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

      5. count-2 52.5%

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

      6. add-cube-cbrt 51.2%

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

      7. unpow2 51.2%

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

      8. metadata-eval 51.2%

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

      9. div-inv 50.9%

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

      10. metadata-eval 50.9%

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

      11. div-inv 51.1%

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

   10. Applied egg-rr 52.6%

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

       1. unpow2 53.5%

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

       2. unpow2 53.5%

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

       3. difference-of-squares 53.6%

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

   12. Applied egg-rr 52.6%

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

   if 5e128 < (pow.f64 b #s(literal 2 binary64))

   1. Initial program 46.0%

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

      1. associate-*l* 46.0%

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

      2. *-commutative 46.0%

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

      3. associate-*l* 46.0%

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

   3. Simplified 46.0%

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

      1. unpow2 46.0%

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

      2. unpow2 46.0%

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

      3. difference-of-squares 57.6%

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

   6. Applied egg-rr 57.6%

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

      1. add-exp-log 26.7%

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

      2. 2-sin 26.7%

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

      3. div-inv 26.7%

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

      4. metadata-eval 26.7%

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

   8. Applied egg-rr 26.7%

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

      1. rem-exp-log 55.7%

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

      2. associate-*l* 69.1%

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

      3. +-commutative 69.1%

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

      4. count-2 69.1%

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

      5. add-cube-cbrt 66.8%

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

      6. unpow2 66.8%

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

      7. metadata-eval 66.8%

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

      8. div-inv 70.0%

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

      9. metadata-eval 70.0%

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

      10. div-inv 69.9%

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

   10. Applied egg-rr 70.1%

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

       1. expm1-log1p-u 70.1%

          \[\leadsto \left(a + b\right) \cdot \left(\left(b - a\right) \cdot \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\sin \left(\left(angle \cdot \pi\right) \cdot 0.011111111111111112\right)\right)\right)}\right) \]

       2. expm1-undefine 24.4%

          \[\leadsto \left(a + b\right) \cdot \left(\left(b - a\right) \cdot \color{blue}{\left(e^{\mathsf{log1p}\left(\sin \left(\left(angle \cdot \pi\right) \cdot 0.011111111111111112\right)\right)} - 1\right)}\right) \]

       3. associate-*l* 22.6%

          \[\leadsto \left(a + b\right) \cdot \left(\left(b - a\right) \cdot \left(e^{\mathsf{log1p}\left(\sin \color{blue}{\left(angle \cdot \left(\pi \cdot 0.011111111111111112\right)\right)}\right)} - 1\right)\right) \]

   12. Applied egg-rr 22.6%

       \[\leadsto \left(a + b\right) \cdot \left(\left(b - a\right) \cdot \color{blue}{\left(e^{\mathsf{log1p}\left(\sin \left(angle \cdot \left(\pi \cdot 0.011111111111111112\right)\right)\right)} - 1\right)}\right) \]

   13. Taylor expanded in angle around 0 74.4%

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

       1. *-commutative 74.4%

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

       2. associate-*r* 74.5%

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

       3. *-commutative 74.5%

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

       4. associate-*l* 74.5%

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

   15. Simplified 74.5%

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

4. Final simplification 69.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;{b}^{2} \leq 2 \cdot 10^{-260}:\\ \;\;\;\;\left(a \cdot \sin \left(\left(angle \cdot \pi\right) \cdot 0.011111111111111112\right)\right) \cdot \left(\left(-b\right) - a\right)\\ \mathbf{elif}\;{b}^{2} \leq 5 \cdot 10^{+128}:\\ \;\;\;\;\sin \left(\left(angle \cdot \pi\right) \cdot 0.011111111111111112\right) \cdot \left(\left(a + b\right) \cdot \left(b - a\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(a + b\right) \cdot \left(angle \cdot \left(\left(b - a\right) \cdot \left(\pi \cdot 0.011111111111111112\right)\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 67.1% accurate, 1.3× speedup

Math

\[\begin{array}{l} a_m = \left|a\right| \\ b_m = \left|b\right| \\ angle\_m = \left|angle\right| \\ angle\_s = \mathsf{copysign}\left(1, angle\right) \\ angle\_s \cdot \begin{array}{l} \mathbf{if}\;b\_m \leq 6.7 \cdot 10^{+193}:\\ \;\;\;\;\left(a\_m + b\_m\right) \cdot \left(\left(b\_m - a\_m\right) \cdot \sin \left(\left(angle\_m \cdot \pi\right) \cdot 0.011111111111111112\right)\right)\\ \mathbf{elif}\;b\_m \leq 10^{+247}:\\ \;\;\;\;2 \cdot \left(\left(b\_m - a\_m\right) \cdot \left(b\_m \cdot \sin \left(\left(0.005555555555555556 \cdot angle\_m\right) \cdot \sqrt[3]{{\pi}^{3}}\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \left(b\_m \cdot \left(\left(b\_m - a\_m\right) \cdot \sin \left(angle\_m \cdot \left(0.005555555555555556 \cdot \pi\right)\right)\right)\right)\\ \end{array} \end{array} \]

FPCore

a_m = (fabs.f64 a)
b_m = (fabs.f64 b)
angle\_m = (fabs.f64 angle)
angle\_s = (copysign.f64 #s(literal 1 binary64) angle)
(FPCore (angle_s a_m b_m angle_m)
 :precision binary64
 (*
  angle_s
  (if (<= b_m 6.7e+193)
    (*
     (+ a_m b_m)
     (* (- b_m a_m) (sin (* (* angle_m PI) 0.011111111111111112))))
    (if (<= b_m 1e+247)
      (*
       2.0
       (*
        (- b_m a_m)
        (*
         b_m
         (sin (* (* 0.005555555555555556 angle_m) (cbrt (pow PI 3.0)))))))
      (*
       2.0
       (*
        b_m
        (* (- b_m a_m) (sin (* angle_m (* 0.005555555555555556 PI))))))))))

C

a_m = fabs(a);
b_m = fabs(b);
angle\_m = fabs(angle);
angle\_s = copysign(1.0, angle);
double code(double angle_s, double a_m, double b_m, double angle_m) {
	double tmp;
	if (b_m <= 6.7e+193) {
		tmp = (a_m + b_m) * ((b_m - a_m) * sin(((angle_m * ((double) M_PI)) * 0.011111111111111112)));
	} else if (b_m <= 1e+247) {
		tmp = 2.0 * ((b_m - a_m) * (b_m * sin(((0.005555555555555556 * angle_m) * cbrt(pow(((double) M_PI), 3.0))))));
	} else {
		tmp = 2.0 * (b_m * ((b_m - a_m) * sin((angle_m * (0.005555555555555556 * ((double) M_PI))))));
	}
	return angle_s * tmp;
}

Java

a_m = Math.abs(a);
b_m = Math.abs(b);
angle\_m = Math.abs(angle);
angle\_s = Math.copySign(1.0, angle);
public static double code(double angle_s, double a_m, double b_m, double angle_m) {
	double tmp;
	if (b_m <= 6.7e+193) {
		tmp = (a_m + b_m) * ((b_m - a_m) * Math.sin(((angle_m * Math.PI) * 0.011111111111111112)));
	} else if (b_m <= 1e+247) {
		tmp = 2.0 * ((b_m - a_m) * (b_m * Math.sin(((0.005555555555555556 * angle_m) * Math.cbrt(Math.pow(Math.PI, 3.0))))));
	} else {
		tmp = 2.0 * (b_m * ((b_m - a_m) * Math.sin((angle_m * (0.005555555555555556 * Math.PI)))));
	}
	return angle_s * tmp;
}

Julia

a_m = abs(a)
b_m = abs(b)
angle\_m = abs(angle)
angle\_s = copysign(1.0, angle)
function code(angle_s, a_m, b_m, angle_m)
	tmp = 0.0
	if (b_m <= 6.7e+193)
		tmp = Float64(Float64(a_m + b_m) * Float64(Float64(b_m - a_m) * sin(Float64(Float64(angle_m * pi) * 0.011111111111111112))));
	elseif (b_m <= 1e+247)
		tmp = Float64(2.0 * Float64(Float64(b_m - a_m) * Float64(b_m * sin(Float64(Float64(0.005555555555555556 * angle_m) * cbrt((pi ^ 3.0)))))));
	else
		tmp = Float64(2.0 * Float64(b_m * Float64(Float64(b_m - a_m) * sin(Float64(angle_m * Float64(0.005555555555555556 * pi))))));
	end
	return Float64(angle_s * tmp)
end

Wolfram

a_m = N[Abs[a], $MachinePrecision]
b_m = N[Abs[b], $MachinePrecision]
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$95$m_, b$95$m_, angle$95$m_] := N[(angle$95$s * If[LessEqual[b$95$m, 6.7e+193], N[(N[(a$95$m + b$95$m), $MachinePrecision] * N[(N[(b$95$m - a$95$m), $MachinePrecision] * N[Sin[N[(N[(angle$95$m * Pi), $MachinePrecision] * 0.011111111111111112), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[b$95$m, 1e+247], N[(2.0 * N[(N[(b$95$m - a$95$m), $MachinePrecision] * N[(b$95$m * N[Sin[N[(N[(0.005555555555555556 * angle$95$m), $MachinePrecision] * N[Power[N[Power[Pi, 3.0], $MachinePrecision], 1/3], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(2.0 * N[(b$95$m * N[(N[(b$95$m - a$95$m), $MachinePrecision] * N[Sin[N[(angle$95$m * N[(0.005555555555555556 * Pi), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]), $MachinePrecision]

Derivation

1. Split input into 3 regimes

2. if b < 6.7000000000000003e193

   1. Initial program 59.6%

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

      1. associate-*l* 59.6%

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

      2. *-commutative 59.6%

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

      3. associate-*l* 59.6%

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

   3. Simplified 59.6%

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

      1. unpow2 59.6%

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

      2. unpow2 59.6%

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

      3. difference-of-squares 61.9%

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

   6. Applied egg-rr 61.9%

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

      1. add-exp-log 35.7%

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

      2. 2-sin 35.7%

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

      3. div-inv 35.1%

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

      4. metadata-eval 35.1%

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

   8. Applied egg-rr 35.1%

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

      1. rem-exp-log 59.8%

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

      2. associate-*l* 66.8%

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

      3. +-commutative 66.8%

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

      4. count-2 66.8%

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

      5. add-cube-cbrt 65.9%

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

      6. unpow2 65.9%

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

      7. metadata-eval 65.9%

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

      8. div-inv 67.2%

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

      9. metadata-eval 67.2%

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

      10. div-inv 66.8%

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

   10. Applied egg-rr 67.9%

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

   if 6.7000000000000003e193 < b < 9.99999999999999952e246

   1. Initial program 17.1%

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

      1. associate-*l* 17.1%

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

      2. *-commutative 17.1%

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

      3. associate-*l* 17.1%

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

   3. Simplified 17.1%

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

      1. unpow2 17.1%

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

      2. unpow2 17.1%

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

      3. difference-of-squares 40.1%

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

   6. Applied egg-rr 40.1%

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

      1. add-cube-cbrt 47.8%

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

      2. pow2 47.8%

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

   8. Applied egg-rr 47.8%

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

   9. Taylor expanded in angle around inf 47.8%

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

       1. associate-*r* 55.5%

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

       2. associate-*r* 84.5%

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

       3. associate-*r* 76.8%

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

   11. Simplified 76.8%

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

   12. Taylor expanded in angle around 0 76.8%

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

   13. Taylor expanded in a around 0 76.8%

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

       1. associate-*r* 69.1%

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

   15. Simplified 69.1%

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

       1. add-cbrt-cube 99.6%

          \[\leadsto \left(a + b\right) \cdot \left(\left(b - a\right) \cdot \sin \left(\left(angle \cdot \color{blue}{\sqrt[3]{\left(\pi \cdot \pi\right) \cdot \pi}}\right) \cdot 0.011111111111111112\right)\right) \]

       2. pow3 99.6%

          \[\leadsto \left(a + b\right) \cdot \left(\left(b - a\right) \cdot \sin \left(\left(angle \cdot \sqrt[3]{\color{blue}{{\pi}^{3}}}\right) \cdot 0.011111111111111112\right)\right) \]

   17. Applied egg-rr 84.4%

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

   if 9.99999999999999952e246 < b

   1. Initial program 42.1%

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

      1. associate-*l* 42.1%

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

      2. *-commutative 42.1%

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

      3. associate-*l* 42.1%

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

   3. Simplified 42.1%

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

      1. unpow2 42.1%

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

      2. unpow2 42.1%

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

      3. difference-of-squares 65.7%

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

   6. Applied egg-rr 65.7%

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

      1. add-cube-cbrt 53.9%

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

      2. pow2 53.9%

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

   8. Applied egg-rr 53.9%

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

   9. Taylor expanded in angle around inf 65.7%

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

       1. associate-*r* 65.7%

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

       2. associate-*r* 76.5%

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

       3. associate-*r* 76.5%

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

   11. Simplified 76.5%

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

   12. Taylor expanded in angle around 0 88.2%

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

   13. Taylor expanded in a around 0 82.4%

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

       1. associate-*r* 82.4%

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

   15. Simplified 82.4%

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

   16. Taylor expanded in b around 0 82.4%

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

       1. associate-*r* 82.4%

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

       2. neg-mul-1 82.4%

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

       3. distribute-rgt-out 82.4%

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

       4. *-commutative 82.4%

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

       5. associate-*r* 82.4%

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

       6. +-commutative 82.4%

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

       7. sub-neg 82.4%

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

   18. Simplified 82.4%

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

4. Final simplification 69.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq 6.7 \cdot 10^{+193}:\\ \;\;\;\;\left(a + b\right) \cdot \left(\left(b - a\right) \cdot \sin \left(\left(angle \cdot \pi\right) \cdot 0.011111111111111112\right)\right)\\ \mathbf{elif}\;b \leq 10^{+247}:\\ \;\;\;\;2 \cdot \left(\left(b - a\right) \cdot \left(b \cdot \sin \left(\left(0.005555555555555556 \cdot angle\right) \cdot \sqrt[3]{{\pi}^{3}}\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \left(b \cdot \left(\left(b - a\right) \cdot \sin \left(angle \cdot \left(0.005555555555555556 \cdot \pi\right)\right)\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 66.7% accurate, 1.9× speedup

Math

\[\begin{array}{l} a_m = \left|a\right| \\ b_m = \left|b\right| \\ angle\_m = \left|angle\right| \\ angle\_s = \mathsf{copysign}\left(1, angle\right) \\ \begin{array}{l} t_0 := \pi \cdot \left(0.005555555555555556 \cdot angle\_m\right)\\ angle\_s \cdot \begin{array}{l} \mathbf{if}\;b\_m \leq 6 \cdot 10^{+84}:\\ \;\;\;\;2 \cdot \left(\left(\left(\sin t\_0 \cdot \left(a\_m + b\_m\right)\right) \cdot \left(b\_m - a\_m\right)\right) \cdot \cos \left(0.005555555555555556 \cdot \left(angle\_m \cdot \pi\right)\right)\right)\\ \mathbf{elif}\;b\_m \leq 1.45 \cdot 10^{+264}:\\ \;\;\;\;2 \cdot \left(\left(b\_m - a\_m\right) \cdot \left(t\_0 \cdot \left(a\_m + b\_m\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \left(b\_m \cdot \left(\left(b\_m - a\_m\right) \cdot \sin \left(angle\_m \cdot \left(0.005555555555555556 \cdot \pi\right)\right)\right)\right)\\ \end{array} \end{array} \end{array} \]

FPCore

a_m = (fabs.f64 a)
b_m = (fabs.f64 b)
angle\_m = (fabs.f64 angle)
angle\_s = (copysign.f64 #s(literal 1 binary64) angle)
(FPCore (angle_s a_m b_m angle_m)
 :precision binary64
 (let* ((t_0 (* PI (* 0.005555555555555556 angle_m))))
   (*
    angle_s
    (if (<= b_m 6e+84)
      (*
       2.0
       (*
        (* (* (sin t_0) (+ a_m b_m)) (- b_m a_m))
        (cos (* 0.005555555555555556 (* angle_m PI)))))
      (if (<= b_m 1.45e+264)
        (* 2.0 (* (- b_m a_m) (* t_0 (+ a_m b_m))))
        (*
         2.0
         (*
          b_m
          (* (- b_m a_m) (sin (* angle_m (* 0.005555555555555556 PI)))))))))))

C

a_m = fabs(a);
b_m = fabs(b);
angle\_m = fabs(angle);
angle\_s = copysign(1.0, angle);
double code(double angle_s, double a_m, double b_m, double angle_m) {
	double t_0 = ((double) M_PI) * (0.005555555555555556 * angle_m);
	double tmp;
	if (b_m <= 6e+84) {
		tmp = 2.0 * (((sin(t_0) * (a_m + b_m)) * (b_m - a_m)) * cos((0.005555555555555556 * (angle_m * ((double) M_PI)))));
	} else if (b_m <= 1.45e+264) {
		tmp = 2.0 * ((b_m - a_m) * (t_0 * (a_m + b_m)));
	} else {
		tmp = 2.0 * (b_m * ((b_m - a_m) * sin((angle_m * (0.005555555555555556 * ((double) M_PI))))));
	}
	return angle_s * tmp;
}

Java

a_m = Math.abs(a);
b_m = Math.abs(b);
angle\_m = Math.abs(angle);
angle\_s = Math.copySign(1.0, angle);
public static double code(double angle_s, double a_m, double b_m, double angle_m) {
	double t_0 = Math.PI * (0.005555555555555556 * angle_m);
	double tmp;
	if (b_m <= 6e+84) {
		tmp = 2.0 * (((Math.sin(t_0) * (a_m + b_m)) * (b_m - a_m)) * Math.cos((0.005555555555555556 * (angle_m * Math.PI))));
	} else if (b_m <= 1.45e+264) {
		tmp = 2.0 * ((b_m - a_m) * (t_0 * (a_m + b_m)));
	} else {
		tmp = 2.0 * (b_m * ((b_m - a_m) * Math.sin((angle_m * (0.005555555555555556 * Math.PI)))));
	}
	return angle_s * tmp;
}

Python

a_m = math.fabs(a)
b_m = math.fabs(b)
angle\_m = math.fabs(angle)
angle\_s = math.copysign(1.0, angle)
def code(angle_s, a_m, b_m, angle_m):
	t_0 = math.pi * (0.005555555555555556 * angle_m)
	tmp = 0
	if b_m <= 6e+84:
		tmp = 2.0 * (((math.sin(t_0) * (a_m + b_m)) * (b_m - a_m)) * math.cos((0.005555555555555556 * (angle_m * math.pi))))
	elif b_m <= 1.45e+264:
		tmp = 2.0 * ((b_m - a_m) * (t_0 * (a_m + b_m)))
	else:
		tmp = 2.0 * (b_m * ((b_m - a_m) * math.sin((angle_m * (0.005555555555555556 * math.pi)))))
	return angle_s * tmp

Julia

a_m = abs(a)
b_m = abs(b)
angle\_m = abs(angle)
angle\_s = copysign(1.0, angle)
function code(angle_s, a_m, b_m, angle_m)
	t_0 = Float64(pi * Float64(0.005555555555555556 * angle_m))
	tmp = 0.0
	if (b_m <= 6e+84)
		tmp = Float64(2.0 * Float64(Float64(Float64(sin(t_0) * Float64(a_m + b_m)) * Float64(b_m - a_m)) * cos(Float64(0.005555555555555556 * Float64(angle_m * pi)))));
	elseif (b_m <= 1.45e+264)
		tmp = Float64(2.0 * Float64(Float64(b_m - a_m) * Float64(t_0 * Float64(a_m + b_m))));
	else
		tmp = Float64(2.0 * Float64(b_m * Float64(Float64(b_m - a_m) * sin(Float64(angle_m * Float64(0.005555555555555556 * pi))))));
	end
	return Float64(angle_s * tmp)
end

MATLAB

a_m = abs(a);
b_m = abs(b);
angle\_m = abs(angle);
angle\_s = sign(angle) * abs(1.0);
function tmp_2 = code(angle_s, a_m, b_m, angle_m)
	t_0 = pi * (0.005555555555555556 * angle_m);
	tmp = 0.0;
	if (b_m <= 6e+84)
		tmp = 2.0 * (((sin(t_0) * (a_m + b_m)) * (b_m - a_m)) * cos((0.005555555555555556 * (angle_m * pi))));
	elseif (b_m <= 1.45e+264)
		tmp = 2.0 * ((b_m - a_m) * (t_0 * (a_m + b_m)));
	else
		tmp = 2.0 * (b_m * ((b_m - a_m) * sin((angle_m * (0.005555555555555556 * pi)))));
	end
	tmp_2 = angle_s * tmp;
end

Wolfram

a_m = N[Abs[a], $MachinePrecision]
b_m = N[Abs[b], $MachinePrecision]
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$95$m_, b$95$m_, angle$95$m_] := Block[{t$95$0 = N[(Pi * N[(0.005555555555555556 * angle$95$m), $MachinePrecision]), $MachinePrecision]}, N[(angle$95$s * If[LessEqual[b$95$m, 6e+84], N[(2.0 * N[(N[(N[(N[Sin[t$95$0], $MachinePrecision] * N[(a$95$m + b$95$m), $MachinePrecision]), $MachinePrecision] * N[(b$95$m - a$95$m), $MachinePrecision]), $MachinePrecision] * N[Cos[N[(0.005555555555555556 * N[(angle$95$m * Pi), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[b$95$m, 1.45e+264], N[(2.0 * N[(N[(b$95$m - a$95$m), $MachinePrecision] * N[(t$95$0 * N[(a$95$m + b$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(2.0 * N[(b$95$m * N[(N[(b$95$m - a$95$m), $MachinePrecision] * N[Sin[N[(angle$95$m * N[(0.005555555555555556 * Pi), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]), $MachinePrecision]]

Derivation

1. Split input into 3 regimes

2. if b < 5.99999999999999992e84

   1. Initial program 61.2%

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

      1. associate-*l* 61.2%

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

      2. *-commutative 61.2%

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

      3. associate-*l* 61.2%

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

   3. Simplified 61.2%

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

      1. unpow2 61.2%

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

      2. unpow2 61.2%

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

      3. difference-of-squares 63.1%

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

   6. Applied egg-rr 63.1%

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

      1. add-cube-cbrt 61.0%

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

      2. pow2 61.0%

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

   8. Applied egg-rr 61.0%

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

   9. Taylor expanded in angle around inf 62.1%

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

       1. associate-*r* 61.4%

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

       2. associate-*r* 67.6%

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

       3. associate-*r* 67.1%

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

   11. Simplified 67.1%

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

   12. Taylor expanded in angle around inf 67.2%

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

   if 5.99999999999999992e84 < b < 1.4499999999999999e264

   1. Initial program 33.7%

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

      1. associate-*l* 33.7%

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

      2. *-commutative 33.7%

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

      3. associate-*l* 33.7%

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

   3. Simplified 33.7%

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

      1. unpow2 33.7%

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

      2. unpow2 33.7%

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

      3. difference-of-squares 48.1%

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

   6. Applied egg-rr 48.1%

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

      1. add-cube-cbrt 54.5%

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

      2. pow2 54.5%

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

   8. Applied egg-rr 54.5%

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

   9. Taylor expanded in angle around inf 50.9%

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

       1. associate-*r* 53.7%

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

       2. associate-*r* 75.2%

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

       3. associate-*r* 72.4%

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

   11. Simplified 72.4%

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

   12. Taylor expanded in angle around 0 74.7%

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

   13. Taylor expanded in angle around 0 80.3%

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

       1. associate-*r* 80.4%

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

       2. *-commutative 80.4%

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

       3. *-commutative 80.4%

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

   15. Simplified 80.4%

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

   if 1.4499999999999999e264 < b

   1. Initial program 39.3%

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

      1. associate-*l* 39.3%

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

      2. *-commutative 39.3%

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

      3. associate-*l* 39.3%

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

   3. Simplified 39.3%

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

      1. unpow2 39.3%

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

      2. unpow2 39.3%

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

      3. difference-of-squares 62.3%

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

   6. Applied egg-rr 62.3%

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

      1. add-cube-cbrt 47.0%

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

      2. pow2 47.0%

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

   8. Applied egg-rr 47.0%

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

   9. Taylor expanded in angle around inf 62.3%

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

       1. associate-*r* 62.3%

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

       2. associate-*r* 69.4%

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

       3. associate-*r* 69.4%

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

   11. Simplified 69.4%

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

   12. Taylor expanded in angle around 0 84.7%

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

   13. Taylor expanded in a around 0 84.7%

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

       1. associate-*r* 84.7%

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

   15. Simplified 84.7%

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

   16. Taylor expanded in b around 0 84.7%

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

       1. associate-*r* 84.7%

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

       2. neg-mul-1 84.7%

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

       3. distribute-rgt-out 84.7%

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

       4. *-commutative 84.7%

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

       5. associate-*r* 84.7%

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

       6. +-commutative 84.7%

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

       7. sub-neg 84.7%

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

   18. Simplified 84.7%

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

4. Final simplification 69.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq 6 \cdot 10^{+84}:\\ \;\;\;\;2 \cdot \left(\left(\left(\sin \left(\pi \cdot \left(0.005555555555555556 \cdot angle\right)\right) \cdot \left(a + b\right)\right) \cdot \left(b - a\right)\right) \cdot \cos \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)\right)\\ \mathbf{elif}\;b \leq 1.45 \cdot 10^{+264}:\\ \;\;\;\;2 \cdot \left(\left(b - a\right) \cdot \left(\left(\pi \cdot \left(0.005555555555555556 \cdot angle\right)\right) \cdot \left(a + b\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \left(b \cdot \left(\left(b - a\right) \cdot \sin \left(angle \cdot \left(0.005555555555555556 \cdot \pi\right)\right)\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 66.9% accurate, 1.9× speedup

Math

\[\begin{array}{l} a_m = \left|a\right| \\ b_m = \left|b\right| \\ angle\_m = \left|angle\right| \\ angle\_s = \mathsf{copysign}\left(1, angle\right) \\ angle\_s \cdot \begin{array}{l} \mathbf{if}\;{b\_m}^{2} \leq 5 \cdot 10^{+128}:\\ \;\;\;\;\left(a\_m + b\_m\right) \cdot \left(\left(b\_m - a\_m\right) \cdot \sin \left(\left(angle\_m \cdot \pi\right) \cdot 0.011111111111111112\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(a\_m + b\_m\right) \cdot \left(angle\_m \cdot \left(\left(b\_m - a\_m\right) \cdot \left(\pi \cdot 0.011111111111111112\right)\right)\right)\\ \end{array} \end{array} \]

FPCore

a_m = (fabs.f64 a)
b_m = (fabs.f64 b)
angle\_m = (fabs.f64 angle)
angle\_s = (copysign.f64 #s(literal 1 binary64) angle)
(FPCore (angle_s a_m b_m angle_m)
 :precision binary64
 (*
  angle_s
  (if (<= (pow b_m 2.0) 5e+128)
    (*
     (+ a_m b_m)
     (* (- b_m a_m) (sin (* (* angle_m PI) 0.011111111111111112))))
    (* (+ a_m b_m) (* angle_m (* (- b_m a_m) (* PI 0.011111111111111112)))))))

C

a_m = fabs(a);
b_m = fabs(b);
angle\_m = fabs(angle);
angle\_s = copysign(1.0, angle);
double code(double angle_s, double a_m, double b_m, double angle_m) {
	double tmp;
	if (pow(b_m, 2.0) <= 5e+128) {
		tmp = (a_m + b_m) * ((b_m - a_m) * sin(((angle_m * ((double) M_PI)) * 0.011111111111111112)));
	} else {
		tmp = (a_m + b_m) * (angle_m * ((b_m - a_m) * (((double) M_PI) * 0.011111111111111112)));
	}
	return angle_s * tmp;
}

Java

a_m = Math.abs(a);
b_m = Math.abs(b);
angle\_m = Math.abs(angle);
angle\_s = Math.copySign(1.0, angle);
public static double code(double angle_s, double a_m, double b_m, double angle_m) {
	double tmp;
	if (Math.pow(b_m, 2.0) <= 5e+128) {
		tmp = (a_m + b_m) * ((b_m - a_m) * Math.sin(((angle_m * Math.PI) * 0.011111111111111112)));
	} else {
		tmp = (a_m + b_m) * (angle_m * ((b_m - a_m) * (Math.PI * 0.011111111111111112)));
	}
	return angle_s * tmp;
}

Python

a_m = math.fabs(a)
b_m = math.fabs(b)
angle\_m = math.fabs(angle)
angle\_s = math.copysign(1.0, angle)
def code(angle_s, a_m, b_m, angle_m):
	tmp = 0
	if math.pow(b_m, 2.0) <= 5e+128:
		tmp = (a_m + b_m) * ((b_m - a_m) * math.sin(((angle_m * math.pi) * 0.011111111111111112)))
	else:
		tmp = (a_m + b_m) * (angle_m * ((b_m - a_m) * (math.pi * 0.011111111111111112)))
	return angle_s * tmp

Julia

a_m = abs(a)
b_m = abs(b)
angle\_m = abs(angle)
angle\_s = copysign(1.0, angle)
function code(angle_s, a_m, b_m, angle_m)
	tmp = 0.0
	if ((b_m ^ 2.0) <= 5e+128)
		tmp = Float64(Float64(a_m + b_m) * Float64(Float64(b_m - a_m) * sin(Float64(Float64(angle_m * pi) * 0.011111111111111112))));
	else
		tmp = Float64(Float64(a_m + b_m) * Float64(angle_m * Float64(Float64(b_m - a_m) * Float64(pi * 0.011111111111111112))));
	end
	return Float64(angle_s * tmp)
end

MATLAB

a_m = abs(a);
b_m = abs(b);
angle\_m = abs(angle);
angle\_s = sign(angle) * abs(1.0);
function tmp_2 = code(angle_s, a_m, b_m, angle_m)
	tmp = 0.0;
	if ((b_m ^ 2.0) <= 5e+128)
		tmp = (a_m + b_m) * ((b_m - a_m) * sin(((angle_m * pi) * 0.011111111111111112)));
	else
		tmp = (a_m + b_m) * (angle_m * ((b_m - a_m) * (pi * 0.011111111111111112)));
	end
	tmp_2 = angle_s * tmp;
end

Wolfram

a_m = N[Abs[a], $MachinePrecision]
b_m = N[Abs[b], $MachinePrecision]
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$95$m_, b$95$m_, angle$95$m_] := N[(angle$95$s * If[LessEqual[N[Power[b$95$m, 2.0], $MachinePrecision], 5e+128], N[(N[(a$95$m + b$95$m), $MachinePrecision] * N[(N[(b$95$m - a$95$m), $MachinePrecision] * N[Sin[N[(N[(angle$95$m * Pi), $MachinePrecision] * 0.011111111111111112), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(a$95$m + b$95$m), $MachinePrecision] * N[(angle$95$m * N[(N[(b$95$m - a$95$m), $MachinePrecision] * N[(Pi * 0.011111111111111112), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if (pow.f64 b #s(literal 2 binary64)) < 5e128

   1. Initial program 63.4%

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

      1. associate-*l* 63.4%

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

      2. *-commutative 63.4%

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

      3. associate-*l* 63.4%

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

   3. Simplified 63.4%

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

      1. unpow2 63.4%

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

      2. unpow2 63.4%

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

      3. difference-of-squares 63.4%

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

   6. Applied egg-rr 63.4%

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

      1. add-exp-log 40.9%

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

      2. 2-sin 40.9%

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

      3. div-inv 40.6%

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

      4. metadata-eval 40.6%

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

   8. Applied egg-rr 40.6%

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

      1. rem-exp-log 62.3%

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

      2. associate-*l* 67.1%

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

      3. +-commutative 67.1%

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

      4. count-2 67.1%

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

      5. add-cube-cbrt 66.7%

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

      6. unpow2 66.7%

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

      7. metadata-eval 66.7%

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

      8. div-inv 66.5%

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

      9. metadata-eval 66.5%

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

      10. div-inv 66.0%

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

   10. Applied egg-rr 68.1%

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

   if 5e128 < (pow.f64 b #s(literal 2 binary64))

   1. Initial program 46.0%

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

      1. associate-*l* 46.0%

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

      2. *-commutative 46.0%

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

      3. associate-*l* 46.0%

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

   3. Simplified 46.0%

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

      1. unpow2 46.0%

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

      2. unpow2 46.0%

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

      3. difference-of-squares 57.6%

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

   6. Applied egg-rr 57.6%

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

      1. add-exp-log 26.7%

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

      2. 2-sin 26.7%

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

      3. div-inv 26.7%

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

      4. metadata-eval 26.7%

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

   8. Applied egg-rr 26.7%

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

      1. rem-exp-log 55.7%

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

      2. associate-*l* 69.1%

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

      3. +-commutative 69.1%

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

      4. count-2 69.1%

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

      5. add-cube-cbrt 66.8%

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

      6. unpow2 66.8%

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

      7. metadata-eval 66.8%

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

      8. div-inv 70.0%

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

      9. metadata-eval 70.0%

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

      10. div-inv 69.9%

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

   10. Applied egg-rr 70.1%

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

       1. expm1-log1p-u 70.1%

          \[\leadsto \left(a + b\right) \cdot \left(\left(b - a\right) \cdot \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\sin \left(\left(angle \cdot \pi\right) \cdot 0.011111111111111112\right)\right)\right)}\right) \]

       2. expm1-undefine 24.4%

          \[\leadsto \left(a + b\right) \cdot \left(\left(b - a\right) \cdot \color{blue}{\left(e^{\mathsf{log1p}\left(\sin \left(\left(angle \cdot \pi\right) \cdot 0.011111111111111112\right)\right)} - 1\right)}\right) \]

       3. associate-*l* 22.6%

          \[\leadsto \left(a + b\right) \cdot \left(\left(b - a\right) \cdot \left(e^{\mathsf{log1p}\left(\sin \color{blue}{\left(angle \cdot \left(\pi \cdot 0.011111111111111112\right)\right)}\right)} - 1\right)\right) \]

   12. Applied egg-rr 22.6%

       \[\leadsto \left(a + b\right) \cdot \left(\left(b - a\right) \cdot \color{blue}{\left(e^{\mathsf{log1p}\left(\sin \left(angle \cdot \left(\pi \cdot 0.011111111111111112\right)\right)\right)} - 1\right)}\right) \]

   13. Taylor expanded in angle around 0 74.4%

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

       1. *-commutative 74.4%

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

       2. associate-*r* 74.5%

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

       3. *-commutative 74.5%

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

       4. associate-*l* 74.5%

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

   15. Simplified 74.5%

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

Alternative 8: 64.9% accurate, 1.9× speedup

Math

\[\begin{array}{l} a_m = \left|a\right| \\ b_m = \left|b\right| \\ angle\_m = \left|angle\right| \\ angle\_s = \mathsf{copysign}\left(1, angle\right) \\ angle\_s \cdot \begin{array}{l} \mathbf{if}\;{b\_m}^{2} \leq 0:\\ \;\;\;\;\left(a\_m \cdot \sin \left(\left(angle\_m \cdot \pi\right) \cdot 0.011111111111111112\right)\right) \cdot \left(\left(-b\_m\right) - a\_m\right)\\ \mathbf{else}:\\ \;\;\;\;\left(a\_m + b\_m\right) \cdot \left(angle\_m \cdot \left(\left(b\_m - a\_m\right) \cdot \left(\pi \cdot 0.011111111111111112\right)\right)\right)\\ \end{array} \end{array} \]

FPCore

a_m = (fabs.f64 a)
b_m = (fabs.f64 b)
angle\_m = (fabs.f64 angle)
angle\_s = (copysign.f64 #s(literal 1 binary64) angle)
(FPCore (angle_s a_m b_m angle_m)
 :precision binary64
 (*
  angle_s
  (if (<= (pow b_m 2.0) 0.0)
    (* (* a_m (sin (* (* angle_m PI) 0.011111111111111112))) (- (- b_m) a_m))
    (* (+ a_m b_m) (* angle_m (* (- b_m a_m) (* PI 0.011111111111111112)))))))

C

a_m = fabs(a);
b_m = fabs(b);
angle\_m = fabs(angle);
angle\_s = copysign(1.0, angle);
double code(double angle_s, double a_m, double b_m, double angle_m) {
	double tmp;
	if (pow(b_m, 2.0) <= 0.0) {
		tmp = (a_m * sin(((angle_m * ((double) M_PI)) * 0.011111111111111112))) * (-b_m - a_m);
	} else {
		tmp = (a_m + b_m) * (angle_m * ((b_m - a_m) * (((double) M_PI) * 0.011111111111111112)));
	}
	return angle_s * tmp;
}

Java

a_m = Math.abs(a);
b_m = Math.abs(b);
angle\_m = Math.abs(angle);
angle\_s = Math.copySign(1.0, angle);
public static double code(double angle_s, double a_m, double b_m, double angle_m) {
	double tmp;
	if (Math.pow(b_m, 2.0) <= 0.0) {
		tmp = (a_m * Math.sin(((angle_m * Math.PI) * 0.011111111111111112))) * (-b_m - a_m);
	} else {
		tmp = (a_m + b_m) * (angle_m * ((b_m - a_m) * (Math.PI * 0.011111111111111112)));
	}
	return angle_s * tmp;
}

Python

a_m = math.fabs(a)
b_m = math.fabs(b)
angle\_m = math.fabs(angle)
angle\_s = math.copysign(1.0, angle)
def code(angle_s, a_m, b_m, angle_m):
	tmp = 0
	if math.pow(b_m, 2.0) <= 0.0:
		tmp = (a_m * math.sin(((angle_m * math.pi) * 0.011111111111111112))) * (-b_m - a_m)
	else:
		tmp = (a_m + b_m) * (angle_m * ((b_m - a_m) * (math.pi * 0.011111111111111112)))
	return angle_s * tmp

Julia

a_m = abs(a)
b_m = abs(b)
angle\_m = abs(angle)
angle\_s = copysign(1.0, angle)
function code(angle_s, a_m, b_m, angle_m)
	tmp = 0.0
	if ((b_m ^ 2.0) <= 0.0)
		tmp = Float64(Float64(a_m * sin(Float64(Float64(angle_m * pi) * 0.011111111111111112))) * Float64(Float64(-b_m) - a_m));
	else
		tmp = Float64(Float64(a_m + b_m) * Float64(angle_m * Float64(Float64(b_m - a_m) * Float64(pi * 0.011111111111111112))));
	end
	return Float64(angle_s * tmp)
end

MATLAB

a_m = abs(a);
b_m = abs(b);
angle\_m = abs(angle);
angle\_s = sign(angle) * abs(1.0);
function tmp_2 = code(angle_s, a_m, b_m, angle_m)
	tmp = 0.0;
	if ((b_m ^ 2.0) <= 0.0)
		tmp = (a_m * sin(((angle_m * pi) * 0.011111111111111112))) * (-b_m - a_m);
	else
		tmp = (a_m + b_m) * (angle_m * ((b_m - a_m) * (pi * 0.011111111111111112)));
	end
	tmp_2 = angle_s * tmp;
end

Wolfram

a_m = N[Abs[a], $MachinePrecision]
b_m = N[Abs[b], $MachinePrecision]
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$95$m_, b$95$m_, angle$95$m_] := N[(angle$95$s * If[LessEqual[N[Power[b$95$m, 2.0], $MachinePrecision], 0.0], N[(N[(a$95$m * N[Sin[N[(N[(angle$95$m * Pi), $MachinePrecision] * 0.011111111111111112), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[((-b$95$m) - a$95$m), $MachinePrecision]), $MachinePrecision], N[(N[(a$95$m + b$95$m), $MachinePrecision] * N[(angle$95$m * N[(N[(b$95$m - a$95$m), $MachinePrecision] * N[(Pi * 0.011111111111111112), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if (pow.f64 b #s(literal 2 binary64)) < 0.0

   1. Initial program 73.3%

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

      1. associate-*l* 73.3%

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

      2. *-commutative 73.3%

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

      3. associate-*l* 73.3%

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

   3. Simplified 73.3%

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

      1. unpow2 73.3%

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

      2. unpow2 73.3%

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

      3. difference-of-squares 73.3%

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

   6. Applied egg-rr 73.3%

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

      1. add-exp-log 51.7%

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

      2. 2-sin 51.7%

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

      3. div-inv 50.6%

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

      4. metadata-eval 50.6%

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

   8. Applied egg-rr 50.6%

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

      1. rem-exp-log 72.1%

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

      2. associate-*l* 78.2%

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

      3. +-commutative 78.2%

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

      4. count-2 78.2%

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

      5. add-cube-cbrt 79.2%

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

      6. unpow2 79.2%

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

      7. metadata-eval 79.2%

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

      8. div-inv 78.9%

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

      9. metadata-eval 78.9%

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

      10. div-inv 77.2%

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

   10. Applied egg-rr 80.6%

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

   11. Taylor expanded in b around 0 80.6%

       \[\leadsto \left(a + b\right) \cdot \color{blue}{\left(-1 \cdot \left(a \cdot \sin \left(0.011111111111111112 \cdot \left(angle \cdot \pi\right)\right)\right)\right)} \]
   12. Step-by-step derivation

       1. mul-1-neg 80.6%

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

       2. *-commutative 80.6%

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

       3. distribute-rgt-neg-in 80.6%

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

   13. Simplified 80.6%

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

   if 0.0 < (pow.f64 b #s(literal 2 binary64))

   1. Initial program 51.0%

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

      1. associate-*l* 51.0%

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

      2. *-commutative 51.0%

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

      3. associate-*l* 51.0%

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

   3. Simplified 51.0%

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

      1. unpow2 51.0%

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

      2. unpow2 51.0%

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

      3. difference-of-squares 57.2%

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

   6. Applied egg-rr 57.2%

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

      1. add-exp-log 29.9%

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

      2. 2-sin 29.9%

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

      3. div-inv 30.0%

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

      4. metadata-eval 30.0%

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

   8. Applied egg-rr 30.0%

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

      1. rem-exp-log 55.7%

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

      2. associate-*l* 64.7%

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

      3. +-commutative 64.7%

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

      4. count-2 64.7%

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

      5. add-cube-cbrt 62.9%

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

      6. unpow2 62.9%

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

      7. metadata-eval 62.9%

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

      8. div-inv 64.5%

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

      9. metadata-eval 64.5%

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

      10. div-inv 64.5%

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

   10. Applied egg-rr 65.2%

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

       1. expm1-log1p-u 65.2%

          \[\leadsto \left(a + b\right) \cdot \left(\left(b - a\right) \cdot \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\sin \left(\left(angle \cdot \pi\right) \cdot 0.011111111111111112\right)\right)\right)}\right) \]

       2. expm1-undefine 27.1%

          \[\leadsto \left(a + b\right) \cdot \left(\left(b - a\right) \cdot \color{blue}{\left(e^{\mathsf{log1p}\left(\sin \left(\left(angle \cdot \pi\right) \cdot 0.011111111111111112\right)\right)} - 1\right)}\right) \]

       3. associate-*l* 25.5%

          \[\leadsto \left(a + b\right) \cdot \left(\left(b - a\right) \cdot \left(e^{\mathsf{log1p}\left(\sin \color{blue}{\left(angle \cdot \left(\pi \cdot 0.011111111111111112\right)\right)}\right)} - 1\right)\right) \]

   12. Applied egg-rr 25.5%

       \[\leadsto \left(a + b\right) \cdot \left(\left(b - a\right) \cdot \color{blue}{\left(e^{\mathsf{log1p}\left(\sin \left(angle \cdot \left(\pi \cdot 0.011111111111111112\right)\right)\right)} - 1\right)}\right) \]

   13. Taylor expanded in angle around 0 63.7%

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

       1. *-commutative 63.7%

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

       2. associate-*r* 63.7%

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

       3. *-commutative 63.7%

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

       4. associate-*l* 63.8%

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

   15. Simplified 63.8%

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

4. Final simplification 67.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;{b}^{2} \leq 0:\\ \;\;\;\;\left(a \cdot \sin \left(\left(angle \cdot \pi\right) \cdot 0.011111111111111112\right)\right) \cdot \left(\left(-b\right) - a\right)\\ \mathbf{else}:\\ \;\;\;\;\left(a + b\right) \cdot \left(angle \cdot \left(\left(b - a\right) \cdot \left(\pi \cdot 0.011111111111111112\right)\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 64.7% accurate, 1.9× speedup

Math

\[\begin{array}{l} a_m = \left|a\right| \\ b_m = \left|b\right| \\ angle\_m = \left|angle\right| \\ angle\_s = \mathsf{copysign}\left(1, angle\right) \\ angle\_s \cdot \begin{array}{l} \mathbf{if}\;{a\_m}^{2} \leq 0:\\ \;\;\;\;\left(a\_m + b\_m\right) \cdot \left(b\_m \cdot \sin \left(\pi \cdot \left(angle\_m \cdot 0.011111111111111112\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(a\_m + b\_m\right) \cdot \left(angle\_m \cdot \left(\left(b\_m - a\_m\right) \cdot \left(\pi \cdot 0.011111111111111112\right)\right)\right)\\ \end{array} \end{array} \]

FPCore

a_m = (fabs.f64 a)
b_m = (fabs.f64 b)
angle\_m = (fabs.f64 angle)
angle\_s = (copysign.f64 #s(literal 1 binary64) angle)
(FPCore (angle_s a_m b_m angle_m)
 :precision binary64
 (*
  angle_s
  (if (<= (pow a_m 2.0) 0.0)
    (* (+ a_m b_m) (* b_m (sin (* PI (* angle_m 0.011111111111111112)))))
    (* (+ a_m b_m) (* angle_m (* (- b_m a_m) (* PI 0.011111111111111112)))))))

C

a_m = fabs(a);
b_m = fabs(b);
angle\_m = fabs(angle);
angle\_s = copysign(1.0, angle);
double code(double angle_s, double a_m, double b_m, double angle_m) {
	double tmp;
	if (pow(a_m, 2.0) <= 0.0) {
		tmp = (a_m + b_m) * (b_m * sin((((double) M_PI) * (angle_m * 0.011111111111111112))));
	} else {
		tmp = (a_m + b_m) * (angle_m * ((b_m - a_m) * (((double) M_PI) * 0.011111111111111112)));
	}
	return angle_s * tmp;
}

Java

a_m = Math.abs(a);
b_m = Math.abs(b);
angle\_m = Math.abs(angle);
angle\_s = Math.copySign(1.0, angle);
public static double code(double angle_s, double a_m, double b_m, double angle_m) {
	double tmp;
	if (Math.pow(a_m, 2.0) <= 0.0) {
		tmp = (a_m + b_m) * (b_m * Math.sin((Math.PI * (angle_m * 0.011111111111111112))));
	} else {
		tmp = (a_m + b_m) * (angle_m * ((b_m - a_m) * (Math.PI * 0.011111111111111112)));
	}
	return angle_s * tmp;
}

Python

a_m = math.fabs(a)
b_m = math.fabs(b)
angle\_m = math.fabs(angle)
angle\_s = math.copysign(1.0, angle)
def code(angle_s, a_m, b_m, angle_m):
	tmp = 0
	if math.pow(a_m, 2.0) <= 0.0:
		tmp = (a_m + b_m) * (b_m * math.sin((math.pi * (angle_m * 0.011111111111111112))))
	else:
		tmp = (a_m + b_m) * (angle_m * ((b_m - a_m) * (math.pi * 0.011111111111111112)))
	return angle_s * tmp

Julia

a_m = abs(a)
b_m = abs(b)
angle\_m = abs(angle)
angle\_s = copysign(1.0, angle)
function code(angle_s, a_m, b_m, angle_m)
	tmp = 0.0
	if ((a_m ^ 2.0) <= 0.0)
		tmp = Float64(Float64(a_m + b_m) * Float64(b_m * sin(Float64(pi * Float64(angle_m * 0.011111111111111112)))));
	else
		tmp = Float64(Float64(a_m + b_m) * Float64(angle_m * Float64(Float64(b_m - a_m) * Float64(pi * 0.011111111111111112))));
	end
	return Float64(angle_s * tmp)
end

MATLAB

a_m = abs(a);
b_m = abs(b);
angle\_m = abs(angle);
angle\_s = sign(angle) * abs(1.0);
function tmp_2 = code(angle_s, a_m, b_m, angle_m)
	tmp = 0.0;
	if ((a_m ^ 2.0) <= 0.0)
		tmp = (a_m + b_m) * (b_m * sin((pi * (angle_m * 0.011111111111111112))));
	else
		tmp = (a_m + b_m) * (angle_m * ((b_m - a_m) * (pi * 0.011111111111111112)));
	end
	tmp_2 = angle_s * tmp;
end

Wolfram

a_m = N[Abs[a], $MachinePrecision]
b_m = N[Abs[b], $MachinePrecision]
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$95$m_, b$95$m_, angle$95$m_] := N[(angle$95$s * If[LessEqual[N[Power[a$95$m, 2.0], $MachinePrecision], 0.0], N[(N[(a$95$m + b$95$m), $MachinePrecision] * N[(b$95$m * N[Sin[N[(Pi * N[(angle$95$m * 0.011111111111111112), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(a$95$m + b$95$m), $MachinePrecision] * N[(angle$95$m * N[(N[(b$95$m - a$95$m), $MachinePrecision] * N[(Pi * 0.011111111111111112), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if (pow.f64 a #s(literal 2 binary64)) < 0.0

   1. Initial program 71.1%

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

      1. associate-*l* 71.1%

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

      2. *-commutative 71.1%

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

      3. associate-*l* 71.1%

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

   3. Simplified 71.1%

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

      1. unpow2 71.1%

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

      2. unpow2 71.1%

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

      3. difference-of-squares 71.1%

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

   6. Applied egg-rr 71.1%

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

      1. add-exp-log 48.9%

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

      2. 2-sin 48.9%

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

      3. div-inv 50.3%

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

      4. metadata-eval 50.3%

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

   8. Applied egg-rr 50.3%

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

      1. rem-exp-log 70.9%

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

      2. associate-*l* 78.7%

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

      3. +-commutative 78.7%

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

      4. count-2 78.7%

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

      5. add-cube-cbrt 74.5%

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

      6. unpow2 74.5%

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

      7. metadata-eval 74.5%

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

      8. div-inv 77.4%

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

      9. metadata-eval 77.4%

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

      10. div-inv 77.3%

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

   10. Applied egg-rr 76.2%

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

       1. add-cbrt-cube 69.8%

          \[\leadsto \left(a + b\right) \cdot \color{blue}{\sqrt[3]{\left(\left(\left(b - a\right) \cdot \sin \left(\left(angle \cdot \pi\right) \cdot 0.011111111111111112\right)\right) \cdot \left(\left(b - a\right) \cdot \sin \left(\left(angle \cdot \pi\right) \cdot 0.011111111111111112\right)\right)\right) \cdot \left(\left(b - a\right) \cdot \sin \left(\left(angle \cdot \pi\right) \cdot 0.011111111111111112\right)\right)}} \]

       2. pow1/3 59.8%

          \[\leadsto \left(a + b\right) \cdot \color{blue}{{\left(\left(\left(\left(b - a\right) \cdot \sin \left(\left(angle \cdot \pi\right) \cdot 0.011111111111111112\right)\right) \cdot \left(\left(b - a\right) \cdot \sin \left(\left(angle \cdot \pi\right) \cdot 0.011111111111111112\right)\right)\right) \cdot \left(\left(b - a\right) \cdot \sin \left(\left(angle \cdot \pi\right) \cdot 0.011111111111111112\right)\right)\right)}^{0.3333333333333333}} \]

       3. pow3 59.8%

          \[\leadsto \left(a + b\right) \cdot {\color{blue}{\left({\left(\left(b - a\right) \cdot \sin \left(\left(angle \cdot \pi\right) \cdot 0.011111111111111112\right)\right)}^{3}\right)}}^{0.3333333333333333} \]

       4. *-commutative 59.8%

          \[\leadsto \left(a + b\right) \cdot {\left({\color{blue}{\left(\sin \left(\left(angle \cdot \pi\right) \cdot 0.011111111111111112\right) \cdot \left(b - a\right)\right)}}^{3}\right)}^{0.3333333333333333} \]

       5. associate-*l* 59.5%

          \[\leadsto \left(a + b\right) \cdot {\left({\left(\sin \color{blue}{\left(angle \cdot \left(\pi \cdot 0.011111111111111112\right)\right)} \cdot \left(b - a\right)\right)}^{3}\right)}^{0.3333333333333333} \]

       6. sub-neg 59.5%

          \[\leadsto \left(a + b\right) \cdot {\left({\left(\sin \left(angle \cdot \left(\pi \cdot 0.011111111111111112\right)\right) \cdot \color{blue}{\left(b + \left(-a\right)\right)}\right)}^{3}\right)}^{0.3333333333333333} \]

       7. add-sqr-sqrt 26.3%

          \[\leadsto \left(a + b\right) \cdot {\left({\left(\sin \left(angle \cdot \left(\pi \cdot 0.011111111111111112\right)\right) \cdot \left(b + \color{blue}{\sqrt{-a} \cdot \sqrt{-a}}\right)\right)}^{3}\right)}^{0.3333333333333333} \]

       8. sqrt-unprod 59.5%

          \[\leadsto \left(a + b\right) \cdot {\left({\left(\sin \left(angle \cdot \left(\pi \cdot 0.011111111111111112\right)\right) \cdot \left(b + \color{blue}{\sqrt{\left(-a\right) \cdot \left(-a\right)}}\right)\right)}^{3}\right)}^{0.3333333333333333} \]

       9. sqr-neg 59.5%

          \[\leadsto \left(a + b\right) \cdot {\left({\left(\sin \left(angle \cdot \left(\pi \cdot 0.011111111111111112\right)\right) \cdot \left(b + \sqrt{\color{blue}{a \cdot a}}\right)\right)}^{3}\right)}^{0.3333333333333333} \]

       10. sqrt-prod 33.2%

           \[\leadsto \left(a + b\right) \cdot {\left({\left(\sin \left(angle \cdot \left(\pi \cdot 0.011111111111111112\right)\right) \cdot \left(b + \color{blue}{\sqrt{a} \cdot \sqrt{a}}\right)\right)}^{3}\right)}^{0.3333333333333333} \]

       11. add-sqr-sqrt 59.5%

           \[\leadsto \left(a + b\right) \cdot {\left({\left(\sin \left(angle \cdot \left(\pi \cdot 0.011111111111111112\right)\right) \cdot \left(b + \color{blue}{a}\right)\right)}^{3}\right)}^{0.3333333333333333} \]

   12. Applied egg-rr 59.5%

       \[\leadsto \left(a + b\right) \cdot \color{blue}{{\left({\left(\sin \left(angle \cdot \left(\pi \cdot 0.011111111111111112\right)\right) \cdot \left(b + a\right)\right)}^{3}\right)}^{0.3333333333333333}} \]

   13. Taylor expanded in b around inf 76.2%

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

       1. associate-*r* 78.7%

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

       2. *-commutative 78.7%

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

       3. *-commutative 78.7%

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

   15. Simplified 78.7%

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

   if 0.0 < (pow.f64 a #s(literal 2 binary64))

   1. Initial program 50.4%

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

      1. associate-*l* 50.4%

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

      2. *-commutative 50.4%

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

      3. associate-*l* 50.4%

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

   3. Simplified 50.4%

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

      1. unpow2 50.4%

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

      2. unpow2 50.4%

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

      3. difference-of-squares 57.0%

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

   6. Applied egg-rr 57.0%

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

      1. add-exp-log 29.6%

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

      2. 2-sin 29.6%

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

      3. div-inv 28.8%

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

      4. metadata-eval 28.8%

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

   8. Applied egg-rr 28.8%

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

      1. rem-exp-log 55.1%

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

      2. associate-*l* 63.6%

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

      3. +-commutative 63.6%

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

      4. count-2 63.6%

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

      5. add-cube-cbrt 63.7%

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

      6. unpow2 63.7%

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

      7. metadata-eval 63.7%

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

      8. div-inv 64.2%

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

      9. metadata-eval 64.2%

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

      10. div-inv 63.7%

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

   10. Applied egg-rr 66.0%

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

       1. expm1-log1p-u 66.0%

          \[\leadsto \left(a + b\right) \cdot \left(\left(b - a\right) \cdot \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\sin \left(\left(angle \cdot \pi\right) \cdot 0.011111111111111112\right)\right)\right)}\right) \]

       2. expm1-undefine 26.5%

          \[\leadsto \left(a + b\right) \cdot \left(\left(b - a\right) \cdot \color{blue}{\left(e^{\mathsf{log1p}\left(\sin \left(\left(angle \cdot \pi\right) \cdot 0.011111111111111112\right)\right)} - 1\right)}\right) \]

       3. associate-*l* 24.6%

          \[\leadsto \left(a + b\right) \cdot \left(\left(b - a\right) \cdot \left(e^{\mathsf{log1p}\left(\sin \color{blue}{\left(angle \cdot \left(\pi \cdot 0.011111111111111112\right)\right)}\right)} - 1\right)\right) \]

   12. Applied egg-rr 24.6%

       \[\leadsto \left(a + b\right) \cdot \left(\left(b - a\right) \cdot \color{blue}{\left(e^{\mathsf{log1p}\left(\sin \left(angle \cdot \left(\pi \cdot 0.011111111111111112\right)\right)\right)} - 1\right)}\right) \]

   13. Taylor expanded in angle around 0 61.5%

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

       1. *-commutative 61.5%

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

       2. associate-*r* 61.5%

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

       3. *-commutative 61.5%

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

       4. associate-*l* 61.6%

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

   15. Simplified 61.6%

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

Alternative 10: 64.8% accurate, 1.9× speedup

Math

\[\begin{array}{l} a_m = \left|a\right| \\ b_m = \left|b\right| \\ angle\_m = \left|angle\right| \\ angle\_s = \mathsf{copysign}\left(1, angle\right) \\ angle\_s \cdot \begin{array}{l} \mathbf{if}\;{a\_m}^{2} \leq 4 \cdot 10^{-310}:\\ \;\;\;\;\left(a\_m + b\_m\right) \cdot \left(b\_m \cdot \sin \left(angle\_m \cdot \left(\pi \cdot 0.011111111111111112\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(a\_m + b\_m\right) \cdot \left(angle\_m \cdot \left(\left(b\_m - a\_m\right) \cdot \left(\pi \cdot 0.011111111111111112\right)\right)\right)\\ \end{array} \end{array} \]

FPCore

a_m = (fabs.f64 a)
b_m = (fabs.f64 b)
angle\_m = (fabs.f64 angle)
angle\_s = (copysign.f64 #s(literal 1 binary64) angle)
(FPCore (angle_s a_m b_m angle_m)
 :precision binary64
 (*
  angle_s
  (if (<= (pow a_m 2.0) 4e-310)
    (* (+ a_m b_m) (* b_m (sin (* angle_m (* PI 0.011111111111111112)))))
    (* (+ a_m b_m) (* angle_m (* (- b_m a_m) (* PI 0.011111111111111112)))))))

C

a_m = fabs(a);
b_m = fabs(b);
angle\_m = fabs(angle);
angle\_s = copysign(1.0, angle);
double code(double angle_s, double a_m, double b_m, double angle_m) {
	double tmp;
	if (pow(a_m, 2.0) <= 4e-310) {
		tmp = (a_m + b_m) * (b_m * sin((angle_m * (((double) M_PI) * 0.011111111111111112))));
	} else {
		tmp = (a_m + b_m) * (angle_m * ((b_m - a_m) * (((double) M_PI) * 0.011111111111111112)));
	}
	return angle_s * tmp;
}

Java

a_m = Math.abs(a);
b_m = Math.abs(b);
angle\_m = Math.abs(angle);
angle\_s = Math.copySign(1.0, angle);
public static double code(double angle_s, double a_m, double b_m, double angle_m) {
	double tmp;
	if (Math.pow(a_m, 2.0) <= 4e-310) {
		tmp = (a_m + b_m) * (b_m * Math.sin((angle_m * (Math.PI * 0.011111111111111112))));
	} else {
		tmp = (a_m + b_m) * (angle_m * ((b_m - a_m) * (Math.PI * 0.011111111111111112)));
	}
	return angle_s * tmp;
}

Python

a_m = math.fabs(a)
b_m = math.fabs(b)
angle\_m = math.fabs(angle)
angle\_s = math.copysign(1.0, angle)
def code(angle_s, a_m, b_m, angle_m):
	tmp = 0
	if math.pow(a_m, 2.0) <= 4e-310:
		tmp = (a_m + b_m) * (b_m * math.sin((angle_m * (math.pi * 0.011111111111111112))))
	else:
		tmp = (a_m + b_m) * (angle_m * ((b_m - a_m) * (math.pi * 0.011111111111111112)))
	return angle_s * tmp

Julia

a_m = abs(a)
b_m = abs(b)
angle\_m = abs(angle)
angle\_s = copysign(1.0, angle)
function code(angle_s, a_m, b_m, angle_m)
	tmp = 0.0
	if ((a_m ^ 2.0) <= 4e-310)
		tmp = Float64(Float64(a_m + b_m) * Float64(b_m * sin(Float64(angle_m * Float64(pi * 0.011111111111111112)))));
	else
		tmp = Float64(Float64(a_m + b_m) * Float64(angle_m * Float64(Float64(b_m - a_m) * Float64(pi * 0.011111111111111112))));
	end
	return Float64(angle_s * tmp)
end

MATLAB

a_m = abs(a);
b_m = abs(b);
angle\_m = abs(angle);
angle\_s = sign(angle) * abs(1.0);
function tmp_2 = code(angle_s, a_m, b_m, angle_m)
	tmp = 0.0;
	if ((a_m ^ 2.0) <= 4e-310)
		tmp = (a_m + b_m) * (b_m * sin((angle_m * (pi * 0.011111111111111112))));
	else
		tmp = (a_m + b_m) * (angle_m * ((b_m - a_m) * (pi * 0.011111111111111112)));
	end
	tmp_2 = angle_s * tmp;
end

Wolfram

a_m = N[Abs[a], $MachinePrecision]
b_m = N[Abs[b], $MachinePrecision]
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$95$m_, b$95$m_, angle$95$m_] := N[(angle$95$s * If[LessEqual[N[Power[a$95$m, 2.0], $MachinePrecision], 4e-310], N[(N[(a$95$m + b$95$m), $MachinePrecision] * N[(b$95$m * N[Sin[N[(angle$95$m * N[(Pi * 0.011111111111111112), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(a$95$m + b$95$m), $MachinePrecision] * N[(angle$95$m * N[(N[(b$95$m - a$95$m), $MachinePrecision] * N[(Pi * 0.011111111111111112), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if (pow.f64 a #s(literal 2 binary64)) < 3.999999999999988e-310

   1. Initial program 71.7%

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

      1. associate-*l* 71.7%

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

      2. *-commutative 71.7%

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

      3. associate-*l* 71.7%

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

   3. Simplified 71.7%

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

      1. unpow2 71.7%

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

      2. unpow2 71.7%

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

      3. difference-of-squares 71.7%

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

   6. Applied egg-rr 71.7%

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

      1. add-exp-log 48.5%

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

      2. 2-sin 48.5%

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

      3. div-inv 49.9%

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

      4. metadata-eval 49.9%

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

   8. Applied egg-rr 49.9%

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

      1. rem-exp-log 71.6%

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

      2. associate-*l* 78.8%

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

      3. +-commutative 78.8%

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

      4. count-2 78.8%

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

      5. add-cube-cbrt 74.9%

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

      6. unpow2 74.9%

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

      7. metadata-eval 74.9%

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

      8. div-inv 77.6%

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

      9. metadata-eval 77.6%

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

      10. div-inv 77.6%

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

   10. Applied egg-rr 76.5%

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

       1. expm1-log1p-u 76.5%

          \[\leadsto \left(a + b\right) \cdot \left(\left(b - a\right) \cdot \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\sin \left(\left(angle \cdot \pi\right) \cdot 0.011111111111111112\right)\right)\right)}\right) \]

       2. expm1-undefine 47.9%

          \[\leadsto \left(a + b\right) \cdot \left(\left(b - a\right) \cdot \color{blue}{\left(e^{\mathsf{log1p}\left(\sin \left(\left(angle \cdot \pi\right) \cdot 0.011111111111111112\right)\right)} - 1\right)}\right) \]

       3. associate-*l* 46.4%

          \[\leadsto \left(a + b\right) \cdot \left(\left(b - a\right) \cdot \left(e^{\mathsf{log1p}\left(\sin \color{blue}{\left(angle \cdot \left(\pi \cdot 0.011111111111111112\right)\right)}\right)} - 1\right)\right) \]

   12. Applied egg-rr 46.4%

       \[\leadsto \left(a + b\right) \cdot \left(\left(b - a\right) \cdot \color{blue}{\left(e^{\mathsf{log1p}\left(\sin \left(angle \cdot \left(\pi \cdot 0.011111111111111112\right)\right)\right)} - 1\right)}\right) \]

   13. Taylor expanded in b around inf 76.6%

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

       1. *-commutative 76.6%

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

       2. associate-*r* 75.0%

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

   15. Simplified 75.0%

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

   if 3.999999999999988e-310 < (pow.f64 a #s(literal 2 binary64))

   1. Initial program 49.5%

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

      1. associate-*l* 49.5%

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

      2. *-commutative 49.5%

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

      3. associate-*l* 49.5%

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

   3. Simplified 49.5%

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

      1. unpow2 49.5%

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

      2. unpow2 49.5%

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

      3. difference-of-squares 56.3%

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

   6. Applied egg-rr 56.3%

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

      1. add-exp-log 29.2%

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

      2. 2-sin 29.2%

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

      3. div-inv 28.4%

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

      4. metadata-eval 28.4%

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

   8. Applied egg-rr 28.4%

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

      1. rem-exp-log 54.3%

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

      2. associate-*l* 63.1%

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

      3. +-commutative 63.1%

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

      4. count-2 63.1%

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

      5. add-cube-cbrt 63.2%

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

      6. unpow2 63.2%

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

      7. metadata-eval 63.2%

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

      8. div-inv 63.7%

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

      9. metadata-eval 63.7%

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

      10. div-inv 63.2%

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

   10. Applied egg-rr 65.6%

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

       1. expm1-log1p-u 65.6%

          \[\leadsto \left(a + b\right) \cdot \left(\left(b - a\right) \cdot \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\sin \left(\left(angle \cdot \pi\right) \cdot 0.011111111111111112\right)\right)\right)}\right) \]

       2. expm1-undefine 26.1%

          \[\leadsto \left(a + b\right) \cdot \left(\left(b - a\right) \cdot \color{blue}{\left(e^{\mathsf{log1p}\left(\sin \left(\left(angle \cdot \pi\right) \cdot 0.011111111111111112\right)\right)} - 1\right)}\right) \]

       3. associate-*l* 24.1%

          \[\leadsto \left(a + b\right) \cdot \left(\left(b - a\right) \cdot \left(e^{\mathsf{log1p}\left(\sin \color{blue}{\left(angle \cdot \left(\pi \cdot 0.011111111111111112\right)\right)}\right)} - 1\right)\right) \]

   12. Applied egg-rr 24.1%

       \[\leadsto \left(a + b\right) \cdot \left(\left(b - a\right) \cdot \color{blue}{\left(e^{\mathsf{log1p}\left(\sin \left(angle \cdot \left(\pi \cdot 0.011111111111111112\right)\right)\right)} - 1\right)}\right) \]

   13. Taylor expanded in angle around 0 61.5%

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

       1. *-commutative 61.5%

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

       2. associate-*r* 61.5%

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

       3. *-commutative 61.5%

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

       4. associate-*l* 61.6%

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

   15. Simplified 61.6%

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

Alternative 11: 67.0% accurate, 3.5× speedup

Math

\[\begin{array}{l} a_m = \left|a\right| \\ b_m = \left|b\right| \\ angle\_m = \left|angle\right| \\ angle\_s = \mathsf{copysign}\left(1, angle\right) \\ angle\_s \cdot \begin{array}{l} \mathbf{if}\;b\_m \leq 9 \cdot 10^{+50}:\\ \;\;\;\;\left(a\_m + b\_m\right) \cdot \left(\left(b\_m - a\_m\right) \cdot \sin \left(\left(angle\_m \cdot \pi\right) \cdot 0.011111111111111112\right)\right)\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \left(\left(\sin \left(\pi \cdot \left(0.005555555555555556 \cdot angle\_m\right)\right) \cdot \left(a\_m + b\_m\right)\right) \cdot \left(b\_m - a\_m\right)\right)\\ \end{array} \end{array} \]

FPCore

a_m = (fabs.f64 a)
b_m = (fabs.f64 b)
angle\_m = (fabs.f64 angle)
angle\_s = (copysign.f64 #s(literal 1 binary64) angle)
(FPCore (angle_s a_m b_m angle_m)
 :precision binary64
 (*
  angle_s
  (if (<= b_m 9e+50)
    (*
     (+ a_m b_m)
     (* (- b_m a_m) (sin (* (* angle_m PI) 0.011111111111111112))))
    (*
     2.0
     (*
      (* (sin (* PI (* 0.005555555555555556 angle_m))) (+ a_m b_m))
      (- b_m a_m))))))

C

a_m = fabs(a);
b_m = fabs(b);
angle\_m = fabs(angle);
angle\_s = copysign(1.0, angle);
double code(double angle_s, double a_m, double b_m, double angle_m) {
	double tmp;
	if (b_m <= 9e+50) {
		tmp = (a_m + b_m) * ((b_m - a_m) * sin(((angle_m * ((double) M_PI)) * 0.011111111111111112)));
	} else {
		tmp = 2.0 * ((sin((((double) M_PI) * (0.005555555555555556 * angle_m))) * (a_m + b_m)) * (b_m - a_m));
	}
	return angle_s * tmp;
}

Java

a_m = Math.abs(a);
b_m = Math.abs(b);
angle\_m = Math.abs(angle);
angle\_s = Math.copySign(1.0, angle);
public static double code(double angle_s, double a_m, double b_m, double angle_m) {
	double tmp;
	if (b_m <= 9e+50) {
		tmp = (a_m + b_m) * ((b_m - a_m) * Math.sin(((angle_m * Math.PI) * 0.011111111111111112)));
	} else {
		tmp = 2.0 * ((Math.sin((Math.PI * (0.005555555555555556 * angle_m))) * (a_m + b_m)) * (b_m - a_m));
	}
	return angle_s * tmp;
}

Python

a_m = math.fabs(a)
b_m = math.fabs(b)
angle\_m = math.fabs(angle)
angle\_s = math.copysign(1.0, angle)
def code(angle_s, a_m, b_m, angle_m):
	tmp = 0
	if b_m <= 9e+50:
		tmp = (a_m + b_m) * ((b_m - a_m) * math.sin(((angle_m * math.pi) * 0.011111111111111112)))
	else:
		tmp = 2.0 * ((math.sin((math.pi * (0.005555555555555556 * angle_m))) * (a_m + b_m)) * (b_m - a_m))
	return angle_s * tmp

Julia

a_m = abs(a)
b_m = abs(b)
angle\_m = abs(angle)
angle\_s = copysign(1.0, angle)
function code(angle_s, a_m, b_m, angle_m)
	tmp = 0.0
	if (b_m <= 9e+50)
		tmp = Float64(Float64(a_m + b_m) * Float64(Float64(b_m - a_m) * sin(Float64(Float64(angle_m * pi) * 0.011111111111111112))));
	else
		tmp = Float64(2.0 * Float64(Float64(sin(Float64(pi * Float64(0.005555555555555556 * angle_m))) * Float64(a_m + b_m)) * Float64(b_m - a_m)));
	end
	return Float64(angle_s * tmp)
end

MATLAB

a_m = abs(a);
b_m = abs(b);
angle\_m = abs(angle);
angle\_s = sign(angle) * abs(1.0);
function tmp_2 = code(angle_s, a_m, b_m, angle_m)
	tmp = 0.0;
	if (b_m <= 9e+50)
		tmp = (a_m + b_m) * ((b_m - a_m) * sin(((angle_m * pi) * 0.011111111111111112)));
	else
		tmp = 2.0 * ((sin((pi * (0.005555555555555556 * angle_m))) * (a_m + b_m)) * (b_m - a_m));
	end
	tmp_2 = angle_s * tmp;
end

Wolfram

a_m = N[Abs[a], $MachinePrecision]
b_m = N[Abs[b], $MachinePrecision]
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$95$m_, b$95$m_, angle$95$m_] := N[(angle$95$s * If[LessEqual[b$95$m, 9e+50], N[(N[(a$95$m + b$95$m), $MachinePrecision] * N[(N[(b$95$m - a$95$m), $MachinePrecision] * N[Sin[N[(N[(angle$95$m * Pi), $MachinePrecision] * 0.011111111111111112), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(2.0 * N[(N[(N[Sin[N[(Pi * N[(0.005555555555555556 * angle$95$m), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * N[(a$95$m + b$95$m), $MachinePrecision]), $MachinePrecision] * N[(b$95$m - a$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if b < 9.00000000000000027e50

   1. Initial program 61.3%

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

      1. associate-*l* 61.3%

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

      2. *-commutative 61.3%

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

      3. associate-*l* 61.3%

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

   3. Simplified 61.3%

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

      1. unpow2 61.3%

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

      2. unpow2 61.3%

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

      3. difference-of-squares 63.4%

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

   6. Applied egg-rr 63.4%

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

      1. add-exp-log 36.3%

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

      2. 2-sin 36.3%

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

      3. div-inv 35.6%

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

      4. metadata-eval 35.6%

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

   8. Applied egg-rr 35.6%

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

      1. rem-exp-log 61.0%

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

      2. associate-*l* 67.0%

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

      3. +-commutative 67.0%

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

      4. count-2 67.0%

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

      5. add-cube-cbrt 65.1%

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

      6. unpow2 65.1%

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

      7. metadata-eval 65.1%

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

      8. div-inv 66.6%

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

      9. metadata-eval 66.6%

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

      10. div-inv 66.1%

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

   10. Applied egg-rr 68.2%

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

   if 9.00000000000000027e50 < b

   1. Initial program 37.4%

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

      1. associate-*l* 37.4%

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

      2. *-commutative 37.4%

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

      3. associate-*l* 37.4%

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

   3. Simplified 37.4%

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

      1. unpow2 37.4%

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

      2. unpow2 37.4%

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

      3. difference-of-squares 52.3%

         \[\leadsto \left(\left(2 \cdot \color{blue}{\left(\left(b + a\right) \cdot \left(b - a\right)\right)}\right) \cdot \sin \left(\pi \cdot \frac{angle}{180}\right)\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right) \]

   6. Applied egg-rr 52.3%

      \[\leadsto \color{blue}{\left(\left(b + a\right) \cdot \left(b - a\right)\right)} \cdot \left(2 \cdot \left(\sin \left(\pi \cdot \frac{angle}{180}\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right)\right)\right) \]
   7. Step-by-step derivation

      1. add-cube-cbrt 52.6%

         \[\leadsto \left(\left(b + a\right) \cdot \left(b - a\right)\right) \cdot \left(2 \cdot \left(\sin \left(\color{blue}{\left(\left(\sqrt[3]{\pi} \cdot \sqrt[3]{\pi}\right) \cdot \sqrt[3]{\pi}\right)} \cdot \frac{angle}{180}\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right)\right)\right) \]

      2. pow2 52.6%

         \[\leadsto \left(\left(b + a\right) \cdot \left(b - a\right)\right) \cdot \left(2 \cdot \left(\sin \left(\left(\color{blue}{{\left(\sqrt[3]{\pi}\right)}^{2}} \cdot \sqrt[3]{\pi}\right) \cdot \frac{angle}{180}\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right)\right)\right) \]

   8. Applied egg-rr 52.6%

      \[\leadsto \left(\left(b + a\right) \cdot \left(b - a\right)\right) \cdot \left(2 \cdot \left(\sin \left(\color{blue}{\left({\left(\sqrt[3]{\pi}\right)}^{2} \cdot \sqrt[3]{\pi}\right)} \cdot \frac{angle}{180}\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right)\right)\right) \]

   9. Taylor expanded in angle around inf 54.1%

      \[\leadsto \color{blue}{2 \cdot \left(\cos \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right) \cdot \left(\sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right) \cdot \left(\left(a + b\right) \cdot \left(b - a\right)\right)\right)\right)} \]
   10. Step-by-step derivation

       1. associate-*r* 56.0%

          \[\leadsto 2 \cdot \left(\cos \color{blue}{\left(\left(0.005555555555555556 \cdot angle\right) \cdot \pi\right)} \cdot \left(\sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right) \cdot \left(\left(a + b\right) \cdot \left(b - a\right)\right)\right)\right) \]

       2. associate-*r* 73.2%

          \[\leadsto 2 \cdot \left(\cos \left(\left(0.005555555555555556 \cdot angle\right) \cdot \pi\right) \cdot \color{blue}{\left(\left(\sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right) \cdot \left(a + b\right)\right) \cdot \left(b - a\right)\right)}\right) \]

       3. associate-*r* 71.4%

          \[\leadsto 2 \cdot \left(\cos \left(\left(0.005555555555555556 \cdot angle\right) \cdot \pi\right) \cdot \left(\left(\sin \color{blue}{\left(\left(0.005555555555555556 \cdot angle\right) \cdot \pi\right)} \cdot \left(a + b\right)\right) \cdot \left(b - a\right)\right)\right) \]

   11. Simplified 71.4%

       \[\leadsto \color{blue}{2 \cdot \left(\cos \left(\left(0.005555555555555556 \cdot angle\right) \cdot \pi\right) \cdot \left(\left(\sin \left(\left(0.005555555555555556 \cdot angle\right) \cdot \pi\right) \cdot \left(a + b\right)\right) \cdot \left(b - a\right)\right)\right)} \]

   12. Taylor expanded in angle around 0 76.5%

       \[\leadsto 2 \cdot \left(\color{blue}{1} \cdot \left(\left(\sin \left(\left(0.005555555555555556 \cdot angle\right) \cdot \pi\right) \cdot \left(a + b\right)\right) \cdot \left(b - a\right)\right)\right) \]
3. Recombined 2 regimes into one program.

4. Final simplification 70.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq 9 \cdot 10^{+50}:\\ \;\;\;\;\left(a + b\right) \cdot \left(\left(b - a\right) \cdot \sin \left(\left(angle \cdot \pi\right) \cdot 0.011111111111111112\right)\right)\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \left(\left(\sin \left(\pi \cdot \left(0.005555555555555556 \cdot angle\right)\right) \cdot \left(a + b\right)\right) \cdot \left(b - a\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 12: 65.1% accurate, 3.6× speedup

Math

\[\begin{array}{l} a_m = \left|a\right| \\ b_m = \left|b\right| \\ angle\_m = \left|angle\right| \\ angle\_s = \mathsf{copysign}\left(1, angle\right) \\ angle\_s \cdot \begin{array}{l} \mathbf{if}\;b\_m \leq 8.8 \cdot 10^{-148}:\\ \;\;\;\;\left(a\_m + b\_m\right) \cdot \left(a\_m \cdot \left(-\sin \left(angle\_m \cdot \left(\pi \cdot 0.011111111111111112\right)\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(a\_m + b\_m\right) \cdot \left(angle\_m \cdot \left(\left(b\_m - a\_m\right) \cdot \left(\pi \cdot 0.011111111111111112\right)\right)\right)\\ \end{array} \end{array} \]

FPCore

a_m = (fabs.f64 a)
b_m = (fabs.f64 b)
angle\_m = (fabs.f64 angle)
angle\_s = (copysign.f64 #s(literal 1 binary64) angle)
(FPCore (angle_s a_m b_m angle_m)
 :precision binary64
 (*
  angle_s
  (if (<= b_m 8.8e-148)
    (* (+ a_m b_m) (* a_m (- (sin (* angle_m (* PI 0.011111111111111112))))))
    (* (+ a_m b_m) (* angle_m (* (- b_m a_m) (* PI 0.011111111111111112)))))))

C

a_m = fabs(a);
b_m = fabs(b);
angle\_m = fabs(angle);
angle\_s = copysign(1.0, angle);
double code(double angle_s, double a_m, double b_m, double angle_m) {
	double tmp;
	if (b_m <= 8.8e-148) {
		tmp = (a_m + b_m) * (a_m * -sin((angle_m * (((double) M_PI) * 0.011111111111111112))));
	} else {
		tmp = (a_m + b_m) * (angle_m * ((b_m - a_m) * (((double) M_PI) * 0.011111111111111112)));
	}
	return angle_s * tmp;
}

Java

a_m = Math.abs(a);
b_m = Math.abs(b);
angle\_m = Math.abs(angle);
angle\_s = Math.copySign(1.0, angle);
public static double code(double angle_s, double a_m, double b_m, double angle_m) {
	double tmp;
	if (b_m <= 8.8e-148) {
		tmp = (a_m + b_m) * (a_m * -Math.sin((angle_m * (Math.PI * 0.011111111111111112))));
	} else {
		tmp = (a_m + b_m) * (angle_m * ((b_m - a_m) * (Math.PI * 0.011111111111111112)));
	}
	return angle_s * tmp;
}

Python

a_m = math.fabs(a)
b_m = math.fabs(b)
angle\_m = math.fabs(angle)
angle\_s = math.copysign(1.0, angle)
def code(angle_s, a_m, b_m, angle_m):
	tmp = 0
	if b_m <= 8.8e-148:
		tmp = (a_m + b_m) * (a_m * -math.sin((angle_m * (math.pi * 0.011111111111111112))))
	else:
		tmp = (a_m + b_m) * (angle_m * ((b_m - a_m) * (math.pi * 0.011111111111111112)))
	return angle_s * tmp

Julia

a_m = abs(a)
b_m = abs(b)
angle\_m = abs(angle)
angle\_s = copysign(1.0, angle)
function code(angle_s, a_m, b_m, angle_m)
	tmp = 0.0
	if (b_m <= 8.8e-148)
		tmp = Float64(Float64(a_m + b_m) * Float64(a_m * Float64(-sin(Float64(angle_m * Float64(pi * 0.011111111111111112))))));
	else
		tmp = Float64(Float64(a_m + b_m) * Float64(angle_m * Float64(Float64(b_m - a_m) * Float64(pi * 0.011111111111111112))));
	end
	return Float64(angle_s * tmp)
end

MATLAB

a_m = abs(a);
b_m = abs(b);
angle\_m = abs(angle);
angle\_s = sign(angle) * abs(1.0);
function tmp_2 = code(angle_s, a_m, b_m, angle_m)
	tmp = 0.0;
	if (b_m <= 8.8e-148)
		tmp = (a_m + b_m) * (a_m * -sin((angle_m * (pi * 0.011111111111111112))));
	else
		tmp = (a_m + b_m) * (angle_m * ((b_m - a_m) * (pi * 0.011111111111111112)));
	end
	tmp_2 = angle_s * tmp;
end

Wolfram

a_m = N[Abs[a], $MachinePrecision]
b_m = N[Abs[b], $MachinePrecision]
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$95$m_, b$95$m_, angle$95$m_] := N[(angle$95$s * If[LessEqual[b$95$m, 8.8e-148], N[(N[(a$95$m + b$95$m), $MachinePrecision] * N[(a$95$m * (-N[Sin[N[(angle$95$m * N[(Pi * 0.011111111111111112), $MachinePrecision]), $MachinePrecision]], $MachinePrecision])), $MachinePrecision]), $MachinePrecision], N[(N[(a$95$m + b$95$m), $MachinePrecision] * N[(angle$95$m * N[(N[(b$95$m - a$95$m), $MachinePrecision] * N[(Pi * 0.011111111111111112), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if b < 8.80000000000000068e-148

   1. Initial program 62.7%

      \[\left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin \left(\pi \cdot \frac{angle}{180}\right)\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right) \]
   2. Step-by-step derivation

      1. associate-*l* 62.7%

         \[\leadsto \color{blue}{\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \left(\sin \left(\pi \cdot \frac{angle}{180}\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right)\right)} \]

      2. *-commutative 62.7%

         \[\leadsto \color{blue}{\left(\left({b}^{2} - {a}^{2}\right) \cdot 2\right)} \cdot \left(\sin \left(\pi \cdot \frac{angle}{180}\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right)\right) \]

      3. associate-*l* 62.7%

         \[\leadsto \color{blue}{\left({b}^{2} - {a}^{2}\right) \cdot \left(2 \cdot \left(\sin \left(\pi \cdot \frac{angle}{180}\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right)\right)\right)} \]

   3. Simplified 62.7%

      \[\leadsto \color{blue}{\left({b}^{2} - {a}^{2}\right) \cdot \left(2 \cdot \left(\sin \left(\pi \cdot \frac{angle}{180}\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right)\right)\right)} \]
   4. Add Preprocessing
   5. Step-by-step derivation

      1. unpow2 62.7%

         \[\leadsto \left(\left(2 \cdot \left(\color{blue}{b \cdot b} - {a}^{2}\right)\right) \cdot \sin \left(\pi \cdot \frac{angle}{180}\right)\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right) \]

      2. unpow2 62.7%

         \[\leadsto \left(\left(2 \cdot \left(b \cdot b - \color{blue}{a \cdot a}\right)\right) \cdot \sin \left(\pi \cdot \frac{angle}{180}\right)\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right) \]

      3. difference-of-squares 65.1%

         \[\leadsto \left(\left(2 \cdot \color{blue}{\left(\left(b + a\right) \cdot \left(b - a\right)\right)}\right) \cdot \sin \left(\pi \cdot \frac{angle}{180}\right)\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right) \]

   6. Applied egg-rr 65.1%

      \[\leadsto \color{blue}{\left(\left(b + a\right) \cdot \left(b - a\right)\right)} \cdot \left(2 \cdot \left(\sin \left(\pi \cdot \frac{angle}{180}\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right)\right)\right) \]
   7. Step-by-step derivation

      1. add-exp-log 36.7%

         \[\leadsto \color{blue}{e^{\log \left(\left(\left(b + a\right) \cdot \left(b - a\right)\right) \cdot \left(2 \cdot \left(\sin \left(\pi \cdot \frac{angle}{180}\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right)\right)\right)\right)}} \]

      2. 2-sin 36.7%

         \[\leadsto e^{\log \left(\left(\left(b + a\right) \cdot \left(b - a\right)\right) \cdot \color{blue}{\sin \left(2 \cdot \left(\pi \cdot \frac{angle}{180}\right)\right)}\right)} \]

      3. div-inv 35.8%

         \[\leadsto e^{\log \left(\left(\left(b + a\right) \cdot \left(b - a\right)\right) \cdot \sin \left(2 \cdot \left(\pi \cdot \color{blue}{\left(angle \cdot \frac{1}{180}\right)}\right)\right)\right)} \]

      4. metadata-eval 35.8%

         \[\leadsto e^{\log \left(\left(\left(b + a\right) \cdot \left(b - a\right)\right) \cdot \sin \left(2 \cdot \left(\pi \cdot \left(angle \cdot \color{blue}{0.005555555555555556}\right)\right)\right)\right)} \]

   8. Applied egg-rr 35.8%

      \[\leadsto \color{blue}{e^{\log \left(\left(\left(b + a\right) \cdot \left(b - a\right)\right) \cdot \sin \left(2 \cdot \left(\pi \cdot \left(angle \cdot 0.005555555555555556\right)\right)\right)\right)}} \]
   9. Step-by-step derivation

      1. rem-exp-log 62.9%

         \[\leadsto \color{blue}{\left(\left(b + a\right) \cdot \left(b - a\right)\right) \cdot \sin \left(2 \cdot \left(\pi \cdot \left(angle \cdot 0.005555555555555556\right)\right)\right)} \]

      2. associate-*l* 69.5%

         \[\leadsto \color{blue}{\left(b + a\right) \cdot \left(\left(b - a\right) \cdot \sin \left(2 \cdot \left(\pi \cdot \left(angle \cdot 0.005555555555555556\right)\right)\right)\right)} \]

      3. +-commutative 69.5%

         \[\leadsto \color{blue}{\left(a + b\right)} \cdot \left(\left(b - a\right) \cdot \sin \left(2 \cdot \left(\pi \cdot \left(angle \cdot 0.005555555555555556\right)\right)\right)\right) \]

      4. count-2 69.5%

         \[\leadsto \left(a + b\right) \cdot \left(\left(b - a\right) \cdot \sin \color{blue}{\left(\pi \cdot \left(angle \cdot 0.005555555555555556\right) + \pi \cdot \left(angle \cdot 0.005555555555555556\right)\right)}\right) \]

      5. add-cube-cbrt 67.5%

         \[\leadsto \left(a + b\right) \cdot \left(\left(b - a\right) \cdot \sin \left(\color{blue}{\left(\left(\sqrt[3]{\pi} \cdot \sqrt[3]{\pi}\right) \cdot \sqrt[3]{\pi}\right)} \cdot \left(angle \cdot 0.005555555555555556\right) + \pi \cdot \left(angle \cdot 0.005555555555555556\right)\right)\right) \]

      6. unpow2 67.5%

         \[\leadsto \left(a + b\right) \cdot \left(\left(b - a\right) \cdot \sin \left(\left(\color{blue}{{\left(\sqrt[3]{\pi}\right)}^{2}} \cdot \sqrt[3]{\pi}\right) \cdot \left(angle \cdot 0.005555555555555556\right) + \pi \cdot \left(angle \cdot 0.005555555555555556\right)\right)\right) \]

      7. metadata-eval 67.5%

         \[\leadsto \left(a + b\right) \cdot \left(\left(b - a\right) \cdot \sin \left(\left({\left(\sqrt[3]{\pi}\right)}^{2} \cdot \sqrt[3]{\pi}\right) \cdot \left(angle \cdot \color{blue}{\frac{1}{180}}\right) + \pi \cdot \left(angle \cdot 0.005555555555555556\right)\right)\right) \]

      8. div-inv 69.4%

         \[\leadsto \left(a + b\right) \cdot \left(\left(b - a\right) \cdot \sin \left(\left({\left(\sqrt[3]{\pi}\right)}^{2} \cdot \sqrt[3]{\pi}\right) \cdot \color{blue}{\frac{angle}{180}} + \pi \cdot \left(angle \cdot 0.005555555555555556\right)\right)\right) \]

      9. metadata-eval 69.4%

         \[\leadsto \left(a + b\right) \cdot \left(\left(b - a\right) \cdot \sin \left(\left({\left(\sqrt[3]{\pi}\right)}^{2} \cdot \sqrt[3]{\pi}\right) \cdot \frac{angle}{180} + \pi \cdot \left(angle \cdot \color{blue}{\frac{1}{180}}\right)\right)\right) \]

      10. div-inv 68.8%

          \[\leadsto \left(a + b\right) \cdot \left(\left(b - a\right) \cdot \sin \left(\left({\left(\sqrt[3]{\pi}\right)}^{2} \cdot \sqrt[3]{\pi}\right) \cdot \frac{angle}{180} + \pi \cdot \color{blue}{\frac{angle}{180}}\right)\right) \]

   10. Applied egg-rr 70.9%

       \[\leadsto \color{blue}{\left(a + b\right) \cdot \left(\left(b - a\right) \cdot \sin \left(\left(angle \cdot \pi\right) \cdot 0.011111111111111112\right)\right)} \]
   11. Step-by-step derivation

       1. expm1-log1p-u 70.9%

          \[\leadsto \left(a + b\right) \cdot \left(\left(b - a\right) \cdot \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\sin \left(\left(angle \cdot \pi\right) \cdot 0.011111111111111112\right)\right)\right)}\right) \]

       2. expm1-undefine 37.4%

          \[\leadsto \left(a + b\right) \cdot \left(\left(b - a\right) \cdot \color{blue}{\left(e^{\mathsf{log1p}\left(\sin \left(\left(angle \cdot \pi\right) \cdot 0.011111111111111112\right)\right)} - 1\right)}\right) \]

       3. associate-*l* 35.9%

          \[\leadsto \left(a + b\right) \cdot \left(\left(b - a\right) \cdot \left(e^{\mathsf{log1p}\left(\sin \color{blue}{\left(angle \cdot \left(\pi \cdot 0.011111111111111112\right)\right)}\right)} - 1\right)\right) \]

   12. Applied egg-rr 35.9%

       \[\leadsto \left(a + b\right) \cdot \left(\left(b - a\right) \cdot \color{blue}{\left(e^{\mathsf{log1p}\left(\sin \left(angle \cdot \left(\pi \cdot 0.011111111111111112\right)\right)\right)} - 1\right)}\right) \]

   13. Taylor expanded in b around 0 49.2%

       \[\leadsto \left(a + b\right) \cdot \color{blue}{\left(-1 \cdot \left(a \cdot \sin \left(0.011111111111111112 \cdot \left(angle \cdot \pi\right)\right)\right)\right)} \]
   14. Step-by-step derivation

       1. associate-*r* 49.2%

          \[\leadsto \left(a + b\right) \cdot \color{blue}{\left(\left(-1 \cdot a\right) \cdot \sin \left(0.011111111111111112 \cdot \left(angle \cdot \pi\right)\right)\right)} \]

       2. neg-mul-1 49.2%

          \[\leadsto \left(a + b\right) \cdot \left(\color{blue}{\left(-a\right)} \cdot \sin \left(0.011111111111111112 \cdot \left(angle \cdot \pi\right)\right)\right) \]

       3. *-commutative 49.2%

          \[\leadsto \left(a + b\right) \cdot \left(\left(-a\right) \cdot \sin \color{blue}{\left(\left(angle \cdot \pi\right) \cdot 0.011111111111111112\right)}\right) \]

       4. associate-*r* 48.4%

          \[\leadsto \left(a + b\right) \cdot \left(\left(-a\right) \cdot \sin \color{blue}{\left(angle \cdot \left(\pi \cdot 0.011111111111111112\right)\right)}\right) \]

   15. Simplified 48.4%

       \[\leadsto \left(a + b\right) \cdot \color{blue}{\left(\left(-a\right) \cdot \sin \left(angle \cdot \left(\pi \cdot 0.011111111111111112\right)\right)\right)} \]

   if 8.80000000000000068e-148 < b

   1. Initial program 44.4%

      \[\left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin \left(\pi \cdot \frac{angle}{180}\right)\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right) \]
   2. Step-by-step derivation

      1. associate-*l* 44.4%

         \[\leadsto \color{blue}{\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \left(\sin \left(\pi \cdot \frac{angle}{180}\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right)\right)} \]

      2. *-commutative 44.4%

         \[\leadsto \color{blue}{\left(\left({b}^{2} - {a}^{2}\right) \cdot 2\right)} \cdot \left(\sin \left(\pi \cdot \frac{angle}{180}\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right)\right) \]

      3. associate-*l* 44.4%

         \[\leadsto \color{blue}{\left({b}^{2} - {a}^{2}\right) \cdot \left(2 \cdot \left(\sin \left(\pi \cdot \frac{angle}{180}\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right)\right)\right)} \]

   3. Simplified 44.4%

      \[\leadsto \color{blue}{\left({b}^{2} - {a}^{2}\right) \cdot \left(2 \cdot \left(\sin \left(\pi \cdot \frac{angle}{180}\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right)\right)\right)} \]
   4. Add Preprocessing
   5. Step-by-step derivation

      1. unpow2 44.4%

         \[\leadsto \left(\left(2 \cdot \left(\color{blue}{b \cdot b} - {a}^{2}\right)\right) \cdot \sin \left(\pi \cdot \frac{angle}{180}\right)\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right) \]

      2. unpow2 44.4%

         \[\leadsto \left(\left(2 \cdot \left(b \cdot b - \color{blue}{a \cdot a}\right)\right) \cdot \sin \left(\pi \cdot \frac{angle}{180}\right)\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right) \]

      3. difference-of-squares 53.4%

         \[\leadsto \left(\left(2 \cdot \color{blue}{\left(\left(b + a\right) \cdot \left(b - a\right)\right)}\right) \cdot \sin \left(\pi \cdot \frac{angle}{180}\right)\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right) \]

   6. Applied egg-rr 53.4%

      \[\leadsto \color{blue}{\left(\left(b + a\right) \cdot \left(b - a\right)\right)} \cdot \left(2 \cdot \left(\sin \left(\pi \cdot \frac{angle}{180}\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right)\right)\right) \]
   7. Step-by-step derivation

      1. add-exp-log 32.1%

         \[\leadsto \color{blue}{e^{\log \left(\left(\left(b + a\right) \cdot \left(b - a\right)\right) \cdot \left(2 \cdot \left(\sin \left(\pi \cdot \frac{angle}{180}\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right)\right)\right)\right)}} \]

      2. 2-sin 32.1%

         \[\leadsto e^{\log \left(\left(\left(b + a\right) \cdot \left(b - a\right)\right) \cdot \color{blue}{\sin \left(2 \cdot \left(\pi \cdot \frac{angle}{180}\right)\right)}\right)} \]

      3. div-inv 33.2%

         \[\leadsto e^{\log \left(\left(\left(b + a\right) \cdot \left(b - a\right)\right) \cdot \sin \left(2 \cdot \left(\pi \cdot \color{blue}{\left(angle \cdot \frac{1}{180}\right)}\right)\right)\right)} \]

      4. metadata-eval 33.2%

         \[\leadsto e^{\log \left(\left(\left(b + a\right) \cdot \left(b - a\right)\right) \cdot \sin \left(2 \cdot \left(\pi \cdot \left(angle \cdot \color{blue}{0.005555555555555556}\right)\right)\right)\right)} \]

   8. Applied egg-rr 33.2%

      \[\leadsto \color{blue}{e^{\log \left(\left(\left(b + a\right) \cdot \left(b - a\right)\right) \cdot \sin \left(2 \cdot \left(\pi \cdot \left(angle \cdot 0.005555555555555556\right)\right)\right)\right)}} \]
   9. Step-by-step derivation

      1. rem-exp-log 53.3%

         \[\leadsto \color{blue}{\left(\left(b + a\right) \cdot \left(b - a\right)\right) \cdot \sin \left(2 \cdot \left(\pi \cdot \left(angle \cdot 0.005555555555555556\right)\right)\right)} \]

      2. associate-*l* 64.9%

         \[\leadsto \color{blue}{\left(b + a\right) \cdot \left(\left(b - a\right) \cdot \sin \left(2 \cdot \left(\pi \cdot \left(angle \cdot 0.005555555555555556\right)\right)\right)\right)} \]

      3. +-commutative 64.9%

         \[\leadsto \color{blue}{\left(a + b\right)} \cdot \left(\left(b - a\right) \cdot \sin \left(2 \cdot \left(\pi \cdot \left(angle \cdot 0.005555555555555556\right)\right)\right)\right) \]

      4. count-2 64.9%

         \[\leadsto \left(a + b\right) \cdot \left(\left(b - a\right) \cdot \sin \color{blue}{\left(\pi \cdot \left(angle \cdot 0.005555555555555556\right) + \pi \cdot \left(angle \cdot 0.005555555555555556\right)\right)}\right) \]

      5. add-cube-cbrt 65.5%

         \[\leadsto \left(a + b\right) \cdot \left(\left(b - a\right) \cdot \sin \left(\color{blue}{\left(\left(\sqrt[3]{\pi} \cdot \sqrt[3]{\pi}\right) \cdot \sqrt[3]{\pi}\right)} \cdot \left(angle \cdot 0.005555555555555556\right) + \pi \cdot \left(angle \cdot 0.005555555555555556\right)\right)\right) \]

      6. unpow2 65.5%

         \[\leadsto \left(a + b\right) \cdot \left(\left(b - a\right) \cdot \sin \left(\left(\color{blue}{{\left(\sqrt[3]{\pi}\right)}^{2}} \cdot \sqrt[3]{\pi}\right) \cdot \left(angle \cdot 0.005555555555555556\right) + \pi \cdot \left(angle \cdot 0.005555555555555556\right)\right)\right) \]

      7. metadata-eval 65.5%

         \[\leadsto \left(a + b\right) \cdot \left(\left(b - a\right) \cdot \sin \left(\left({\left(\sqrt[3]{\pi}\right)}^{2} \cdot \sqrt[3]{\pi}\right) \cdot \left(angle \cdot \color{blue}{\frac{1}{180}}\right) + \pi \cdot \left(angle \cdot 0.005555555555555556\right)\right)\right) \]

      8. div-inv 65.2%

         \[\leadsto \left(a + b\right) \cdot \left(\left(b - a\right) \cdot \sin \left(\left({\left(\sqrt[3]{\pi}\right)}^{2} \cdot \sqrt[3]{\pi}\right) \cdot \color{blue}{\frac{angle}{180}} + \pi \cdot \left(angle \cdot 0.005555555555555556\right)\right)\right) \]

      9. metadata-eval 65.2%

         \[\leadsto \left(a + b\right) \cdot \left(\left(b - a\right) \cdot \sin \left(\left({\left(\sqrt[3]{\pi}\right)}^{2} \cdot \sqrt[3]{\pi}\right) \cdot \frac{angle}{180} + \pi \cdot \left(angle \cdot \color{blue}{\frac{1}{180}}\right)\right)\right) \]

      10. div-inv 65.2%

          \[\leadsto \left(a + b\right) \cdot \left(\left(b - a\right) \cdot \sin \left(\left({\left(\sqrt[3]{\pi}\right)}^{2} \cdot \sqrt[3]{\pi}\right) \cdot \frac{angle}{180} + \pi \cdot \color{blue}{\frac{angle}{180}}\right)\right) \]

   10. Applied egg-rr 65.2%

       \[\leadsto \color{blue}{\left(a + b\right) \cdot \left(\left(b - a\right) \cdot \sin \left(\left(angle \cdot \pi\right) \cdot 0.011111111111111112\right)\right)} \]
   11. Step-by-step derivation

       1. expm1-log1p-u 65.2%

          \[\leadsto \left(a + b\right) \cdot \left(\left(b - a\right) \cdot \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\sin \left(\left(angle \cdot \pi\right) \cdot 0.011111111111111112\right)\right)\right)}\right) \]

       2. expm1-undefine 24.1%

          \[\leadsto \left(a + b\right) \cdot \left(\left(b - a\right) \cdot \color{blue}{\left(e^{\mathsf{log1p}\left(\sin \left(\left(angle \cdot \pi\right) \cdot 0.011111111111111112\right)\right)} - 1\right)}\right) \]

       3. associate-*l* 21.5%

          \[\leadsto \left(a + b\right) \cdot \left(\left(b - a\right) \cdot \left(e^{\mathsf{log1p}\left(\sin \color{blue}{\left(angle \cdot \left(\pi \cdot 0.011111111111111112\right)\right)}\right)} - 1\right)\right) \]

   12. Applied egg-rr 21.5%

       \[\leadsto \left(a + b\right) \cdot \left(\left(b - a\right) \cdot \color{blue}{\left(e^{\mathsf{log1p}\left(\sin \left(angle \cdot \left(\pi \cdot 0.011111111111111112\right)\right)\right)} - 1\right)}\right) \]

   13. Taylor expanded in angle around 0 65.8%

       \[\leadsto \left(a + b\right) \cdot \color{blue}{\left(0.011111111111111112 \cdot \left(angle \cdot \left(\pi \cdot \left(b - a\right)\right)\right)\right)} \]
   14. Step-by-step derivation

       1. *-commutative 65.8%

          \[\leadsto \left(a + b\right) \cdot \color{blue}{\left(\left(angle \cdot \left(\pi \cdot \left(b - a\right)\right)\right) \cdot 0.011111111111111112\right)} \]

       2. associate-*r* 65.8%

          \[\leadsto \left(a + b\right) \cdot \color{blue}{\left(angle \cdot \left(\left(\pi \cdot \left(b - a\right)\right) \cdot 0.011111111111111112\right)\right)} \]

       3. *-commutative 65.8%

          \[\leadsto \left(a + b\right) \cdot \left(angle \cdot \left(\color{blue}{\left(\left(b - a\right) \cdot \pi\right)} \cdot 0.011111111111111112\right)\right) \]

       4. associate-*l* 65.8%

          \[\leadsto \left(a + b\right) \cdot \left(angle \cdot \color{blue}{\left(\left(b - a\right) \cdot \left(\pi \cdot 0.011111111111111112\right)\right)}\right) \]

   15. Simplified 65.8%

       \[\leadsto \left(a + b\right) \cdot \color{blue}{\left(angle \cdot \left(\left(b - a\right) \cdot \left(\pi \cdot 0.011111111111111112\right)\right)\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 54.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq 8.8 \cdot 10^{-148}:\\ \;\;\;\;\left(a + b\right) \cdot \left(a \cdot \left(-\sin \left(angle \cdot \left(\pi \cdot 0.011111111111111112\right)\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(a + b\right) \cdot \left(angle \cdot \left(\left(b - a\right) \cdot \left(\pi \cdot 0.011111111111111112\right)\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 13: 38.9% accurate, 26.2× speedup

Math

\[\begin{array}{l} a_m = \left|a\right| \\ b_m = \left|b\right| \\ angle\_m = \left|angle\right| \\ angle\_s = \mathsf{copysign}\left(1, angle\right) \\ angle\_s \cdot \begin{array}{l} \mathbf{if}\;a\_m \leq 5.5 \cdot 10^{+86}:\\ \;\;\;\;0.011111111111111112 \cdot \left(angle\_m \cdot \left(\left(a\_m + b\_m\right) \cdot \left(b\_m \cdot \pi\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;0.011111111111111112 \cdot \left(\left(angle\_m \cdot \pi\right) \cdot \left(a\_m \cdot \left(a\_m + b\_m\right)\right)\right)\\ \end{array} \end{array} \]

FPCore

a_m = (fabs.f64 a)
b_m = (fabs.f64 b)
angle\_m = (fabs.f64 angle)
angle\_s = (copysign.f64 #s(literal 1 binary64) angle)
(FPCore (angle_s a_m b_m angle_m)
 :precision binary64
 (*
  angle_s
  (if (<= a_m 5.5e+86)
    (* 0.011111111111111112 (* angle_m (* (+ a_m b_m) (* b_m PI))))
    (* 0.011111111111111112 (* (* angle_m PI) (* a_m (+ a_m b_m)))))))

C

a_m = fabs(a);
b_m = fabs(b);
angle\_m = fabs(angle);
angle\_s = copysign(1.0, angle);
double code(double angle_s, double a_m, double b_m, double angle_m) {
	double tmp;
	if (a_m <= 5.5e+86) {
		tmp = 0.011111111111111112 * (angle_m * ((a_m + b_m) * (b_m * ((double) M_PI))));
	} else {
		tmp = 0.011111111111111112 * ((angle_m * ((double) M_PI)) * (a_m * (a_m + b_m)));
	}
	return angle_s * tmp;
}

Java

a_m = Math.abs(a);
b_m = Math.abs(b);
angle\_m = Math.abs(angle);
angle\_s = Math.copySign(1.0, angle);
public static double code(double angle_s, double a_m, double b_m, double angle_m) {
	double tmp;
	if (a_m <= 5.5e+86) {
		tmp = 0.011111111111111112 * (angle_m * ((a_m + b_m) * (b_m * Math.PI)));
	} else {
		tmp = 0.011111111111111112 * ((angle_m * Math.PI) * (a_m * (a_m + b_m)));
	}
	return angle_s * tmp;
}

Python

a_m = math.fabs(a)
b_m = math.fabs(b)
angle\_m = math.fabs(angle)
angle\_s = math.copysign(1.0, angle)
def code(angle_s, a_m, b_m, angle_m):
	tmp = 0
	if a_m <= 5.5e+86:
		tmp = 0.011111111111111112 * (angle_m * ((a_m + b_m) * (b_m * math.pi)))
	else:
		tmp = 0.011111111111111112 * ((angle_m * math.pi) * (a_m * (a_m + b_m)))
	return angle_s * tmp

Julia

a_m = abs(a)
b_m = abs(b)
angle\_m = abs(angle)
angle\_s = copysign(1.0, angle)
function code(angle_s, a_m, b_m, angle_m)
	tmp = 0.0
	if (a_m <= 5.5e+86)
		tmp = Float64(0.011111111111111112 * Float64(angle_m * Float64(Float64(a_m + b_m) * Float64(b_m * pi))));
	else
		tmp = Float64(0.011111111111111112 * Float64(Float64(angle_m * pi) * Float64(a_m * Float64(a_m + b_m))));
	end
	return Float64(angle_s * tmp)
end

MATLAB

a_m = abs(a);
b_m = abs(b);
angle\_m = abs(angle);
angle\_s = sign(angle) * abs(1.0);
function tmp_2 = code(angle_s, a_m, b_m, angle_m)
	tmp = 0.0;
	if (a_m <= 5.5e+86)
		tmp = 0.011111111111111112 * (angle_m * ((a_m + b_m) * (b_m * pi)));
	else
		tmp = 0.011111111111111112 * ((angle_m * pi) * (a_m * (a_m + b_m)));
	end
	tmp_2 = angle_s * tmp;
end

Wolfram

a_m = N[Abs[a], $MachinePrecision]
b_m = N[Abs[b], $MachinePrecision]
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$95$m_, b$95$m_, angle$95$m_] := N[(angle$95$s * If[LessEqual[a$95$m, 5.5e+86], N[(0.011111111111111112 * N[(angle$95$m * N[(N[(a$95$m + b$95$m), $MachinePrecision] * N[(b$95$m * Pi), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(0.011111111111111112 * N[(N[(angle$95$m * Pi), $MachinePrecision] * N[(a$95$m * N[(a$95$m + b$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if a < 5.5000000000000002e86

   1. Initial program 59.8%

      \[\left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin \left(\pi \cdot \frac{angle}{180}\right)\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right) \]
   2. Step-by-step derivation

      1. associate-*l* 59.8%

         \[\leadsto \color{blue}{\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \left(\sin \left(\pi \cdot \frac{angle}{180}\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right)\right)} \]

      2. *-commutative 59.8%

         \[\leadsto \color{blue}{\left(\left({b}^{2} - {a}^{2}\right) \cdot 2\right)} \cdot \left(\sin \left(\pi \cdot \frac{angle}{180}\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right)\right) \]

      3. associate-*l* 59.8%

         \[\leadsto \color{blue}{\left({b}^{2} - {a}^{2}\right) \cdot \left(2 \cdot \left(\sin \left(\pi \cdot \frac{angle}{180}\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right)\right)\right)} \]

   3. Simplified 59.8%

      \[\leadsto \color{blue}{\left({b}^{2} - {a}^{2}\right) \cdot \left(2 \cdot \left(\sin \left(\pi \cdot \frac{angle}{180}\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right)\right)\right)} \]
   4. Add Preprocessing
   5. Step-by-step derivation

      1. unpow2 59.8%

         \[\leadsto \left(\left(2 \cdot \left(\color{blue}{b \cdot b} - {a}^{2}\right)\right) \cdot \sin \left(\pi \cdot \frac{angle}{180}\right)\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right) \]

      2. unpow2 59.8%

         \[\leadsto \left(\left(2 \cdot \left(b \cdot b - \color{blue}{a \cdot a}\right)\right) \cdot \sin \left(\pi \cdot \frac{angle}{180}\right)\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right) \]

      3. difference-of-squares 63.2%

         \[\leadsto \left(\left(2 \cdot \color{blue}{\left(\left(b + a\right) \cdot \left(b - a\right)\right)}\right) \cdot \sin \left(\pi \cdot \frac{angle}{180}\right)\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right) \]

   6. Applied egg-rr 63.2%

      \[\leadsto \color{blue}{\left(\left(b + a\right) \cdot \left(b - a\right)\right)} \cdot \left(2 \cdot \left(\sin \left(\pi \cdot \frac{angle}{180}\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right)\right)\right) \]

   7. Taylor expanded in angle around 0 58.0%

      \[\leadsto \color{blue}{0.011111111111111112 \cdot \left(angle \cdot \left(\pi \cdot \left(\left(a + b\right) \cdot \left(b - a\right)\right)\right)\right)} \]
   8. Step-by-step derivation

      1. associate-*r* 58.0%

         \[\leadsto 0.011111111111111112 \cdot \left(angle \cdot \color{blue}{\left(\left(\pi \cdot \left(a + b\right)\right) \cdot \left(b - a\right)\right)}\right) \]

      2. sub-neg 58.0%

         \[\leadsto 0.011111111111111112 \cdot \left(angle \cdot \left(\left(\pi \cdot \left(a + b\right)\right) \cdot \color{blue}{\left(b + \left(-a\right)\right)}\right)\right) \]

      3. distribute-lft-in 56.1%

         \[\leadsto 0.011111111111111112 \cdot \left(angle \cdot \color{blue}{\left(\left(\pi \cdot \left(a + b\right)\right) \cdot b + \left(\pi \cdot \left(a + b\right)\right) \cdot \left(-a\right)\right)}\right) \]

   9. Applied egg-rr 56.1%

      \[\leadsto 0.011111111111111112 \cdot \left(angle \cdot \color{blue}{\left(\left(\pi \cdot \left(a + b\right)\right) \cdot b + \left(\pi \cdot \left(a + b\right)\right) \cdot \left(-a\right)\right)}\right) \]
   10. Step-by-step derivation

       1. distribute-lft-out 58.0%

          \[\leadsto 0.011111111111111112 \cdot \left(angle \cdot \color{blue}{\left(\left(\pi \cdot \left(a + b\right)\right) \cdot \left(b + \left(-a\right)\right)\right)}\right) \]

       2. +-commutative 58.0%

          \[\leadsto 0.011111111111111112 \cdot \left(angle \cdot \left(\left(\pi \cdot \color{blue}{\left(b + a\right)}\right) \cdot \left(b + \left(-a\right)\right)\right)\right) \]

       3. add-sqr-sqrt 32.5%

          \[\leadsto 0.011111111111111112 \cdot \left(angle \cdot \left(\left(\pi \cdot \left(b + a\right)\right) \cdot \left(b + \color{blue}{\sqrt{-a} \cdot \sqrt{-a}}\right)\right)\right) \]

       4. sqrt-unprod 53.3%

          \[\leadsto 0.011111111111111112 \cdot \left(angle \cdot \left(\left(\pi \cdot \left(b + a\right)\right) \cdot \left(b + \color{blue}{\sqrt{\left(-a\right) \cdot \left(-a\right)}}\right)\right)\right) \]

       5. sqr-neg 53.3%

          \[\leadsto 0.011111111111111112 \cdot \left(angle \cdot \left(\left(\pi \cdot \left(b + a\right)\right) \cdot \left(b + \sqrt{\color{blue}{a \cdot a}}\right)\right)\right) \]

       6. sqrt-prod 21.7%

          \[\leadsto 0.011111111111111112 \cdot \left(angle \cdot \left(\left(\pi \cdot \left(b + a\right)\right) \cdot \left(b + \color{blue}{\sqrt{a} \cdot \sqrt{a}}\right)\right)\right) \]

       7. add-sqr-sqrt 48.9%

          \[\leadsto 0.011111111111111112 \cdot \left(angle \cdot \left(\left(\pi \cdot \left(b + a\right)\right) \cdot \left(b + \color{blue}{a}\right)\right)\right) \]

   11. Applied egg-rr 48.9%

       \[\leadsto 0.011111111111111112 \cdot \left(angle \cdot \color{blue}{\left(\left(\pi \cdot \left(b + a\right)\right) \cdot \left(b + a\right)\right)}\right) \]

   12. Taylor expanded in b around inf 48.9%

       \[\leadsto 0.011111111111111112 \cdot \left(angle \cdot \left(\color{blue}{\left(b \cdot \pi\right)} \cdot \left(b + a\right)\right)\right) \]
   13. Step-by-step derivation

       1. *-commutative 48.9%

          \[\leadsto 0.011111111111111112 \cdot \left(angle \cdot \left(\color{blue}{\left(\pi \cdot b\right)} \cdot \left(b + a\right)\right)\right) \]

   14. Simplified 48.9%

       \[\leadsto 0.011111111111111112 \cdot \left(angle \cdot \left(\color{blue}{\left(\pi \cdot b\right)} \cdot \left(b + a\right)\right)\right) \]

   if 5.5000000000000002e86 < a

   1. Initial program 41.9%

      \[\left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin \left(\pi \cdot \frac{angle}{180}\right)\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right) \]
   2. Step-by-step derivation

      1. associate-*l* 41.9%

         \[\leadsto \color{blue}{\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \left(\sin \left(\pi \cdot \frac{angle}{180}\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right)\right)} \]

      2. *-commutative 41.9%

         \[\leadsto \color{blue}{\left(\left({b}^{2} - {a}^{2}\right) \cdot 2\right)} \cdot \left(\sin \left(\pi \cdot \frac{angle}{180}\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right)\right) \]

      3. associate-*l* 41.9%

         \[\leadsto \color{blue}{\left({b}^{2} - {a}^{2}\right) \cdot \left(2 \cdot \left(\sin \left(\pi \cdot \frac{angle}{180}\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right)\right)\right)} \]

   3. Simplified 41.9%

      \[\leadsto \color{blue}{\left({b}^{2} - {a}^{2}\right) \cdot \left(2 \cdot \left(\sin \left(\pi \cdot \frac{angle}{180}\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right)\right)\right)} \]
   4. Add Preprocessing
   5. Step-by-step derivation

      1. unpow2 41.9%

         \[\leadsto \left(\left(2 \cdot \left(\color{blue}{b \cdot b} - {a}^{2}\right)\right) \cdot \sin \left(\pi \cdot \frac{angle}{180}\right)\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right) \]

      2. unpow2 41.9%

         \[\leadsto \left(\left(2 \cdot \left(b \cdot b - \color{blue}{a \cdot a}\right)\right) \cdot \sin \left(\pi \cdot \frac{angle}{180}\right)\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right) \]

      3. difference-of-squares 52.0%

         \[\leadsto \left(\left(2 \cdot \color{blue}{\left(\left(b + a\right) \cdot \left(b - a\right)\right)}\right) \cdot \sin \left(\pi \cdot \frac{angle}{180}\right)\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right) \]

   6. Applied egg-rr 52.0%

      \[\leadsto \color{blue}{\left(\left(b + a\right) \cdot \left(b - a\right)\right)} \cdot \left(2 \cdot \left(\sin \left(\pi \cdot \frac{angle}{180}\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right)\right)\right) \]

   7. Taylor expanded in angle around 0 55.4%

      \[\leadsto \color{blue}{0.011111111111111112 \cdot \left(angle \cdot \left(\pi \cdot \left(\left(a + b\right) \cdot \left(b - a\right)\right)\right)\right)} \]
   8. Step-by-step derivation

      1. associate-*r* 55.4%

         \[\leadsto 0.011111111111111112 \cdot \left(angle \cdot \color{blue}{\left(\left(\pi \cdot \left(a + b\right)\right) \cdot \left(b - a\right)\right)}\right) \]

      2. sub-neg 55.4%

         \[\leadsto 0.011111111111111112 \cdot \left(angle \cdot \left(\left(\pi \cdot \left(a + b\right)\right) \cdot \color{blue}{\left(b + \left(-a\right)\right)}\right)\right) \]

      3. distribute-lft-in 46.9%

         \[\leadsto 0.011111111111111112 \cdot \left(angle \cdot \color{blue}{\left(\left(\pi \cdot \left(a + b\right)\right) \cdot b + \left(\pi \cdot \left(a + b\right)\right) \cdot \left(-a\right)\right)}\right) \]

   9. Applied egg-rr 46.9%

      \[\leadsto 0.011111111111111112 \cdot \left(angle \cdot \color{blue}{\left(\left(\pi \cdot \left(a + b\right)\right) \cdot b + \left(\pi \cdot \left(a + b\right)\right) \cdot \left(-a\right)\right)}\right) \]
   10. Step-by-step derivation

       1. distribute-lft-out 55.4%

          \[\leadsto 0.011111111111111112 \cdot \left(angle \cdot \color{blue}{\left(\left(\pi \cdot \left(a + b\right)\right) \cdot \left(b + \left(-a\right)\right)\right)}\right) \]

       2. +-commutative 55.4%

          \[\leadsto 0.011111111111111112 \cdot \left(angle \cdot \left(\left(\pi \cdot \color{blue}{\left(b + a\right)}\right) \cdot \left(b + \left(-a\right)\right)\right)\right) \]

       3. add-sqr-sqrt 0.0%

          \[\leadsto 0.011111111111111112 \cdot \left(angle \cdot \left(\left(\pi \cdot \left(b + a\right)\right) \cdot \left(b + \color{blue}{\sqrt{-a} \cdot \sqrt{-a}}\right)\right)\right) \]

       4. sqrt-unprod 18.9%

          \[\leadsto 0.011111111111111112 \cdot \left(angle \cdot \left(\left(\pi \cdot \left(b + a\right)\right) \cdot \left(b + \color{blue}{\sqrt{\left(-a\right) \cdot \left(-a\right)}}\right)\right)\right) \]

       5. sqr-neg 18.9%

          \[\leadsto 0.011111111111111112 \cdot \left(angle \cdot \left(\left(\pi \cdot \left(b + a\right)\right) \cdot \left(b + \sqrt{\color{blue}{a \cdot a}}\right)\right)\right) \]

       6. sqrt-prod 20.9%

          \[\leadsto 0.011111111111111112 \cdot \left(angle \cdot \left(\left(\pi \cdot \left(b + a\right)\right) \cdot \left(b + \color{blue}{\sqrt{a} \cdot \sqrt{a}}\right)\right)\right) \]

       7. add-sqr-sqrt 20.9%

          \[\leadsto 0.011111111111111112 \cdot \left(angle \cdot \left(\left(\pi \cdot \left(b + a\right)\right) \cdot \left(b + \color{blue}{a}\right)\right)\right) \]

   11. Applied egg-rr 20.9%

       \[\leadsto 0.011111111111111112 \cdot \left(angle \cdot \color{blue}{\left(\left(\pi \cdot \left(b + a\right)\right) \cdot \left(b + a\right)\right)}\right) \]

   12. Taylor expanded in b around 0 16.9%

       \[\leadsto 0.011111111111111112 \cdot \left(angle \cdot \left(\color{blue}{\left(a \cdot \pi\right)} \cdot \left(b + a\right)\right)\right) \]
   13. Step-by-step derivation

       1. *-commutative 16.9%

          \[\leadsto 0.011111111111111112 \cdot \left(angle \cdot \left(\color{blue}{\left(\pi \cdot a\right)} \cdot \left(b + a\right)\right)\right) \]

   14. Simplified 16.9%

       \[\leadsto 0.011111111111111112 \cdot \left(angle \cdot \left(\color{blue}{\left(\pi \cdot a\right)} \cdot \left(b + a\right)\right)\right) \]

   15. Taylor expanded in angle around 0 16.9%

       \[\leadsto 0.011111111111111112 \cdot \color{blue}{\left(a \cdot \left(angle \cdot \left(\pi \cdot \left(a + b\right)\right)\right)\right)} \]
   16. Step-by-step derivation

       1. associate-*r* 16.9%

          \[\leadsto 0.011111111111111112 \cdot \left(a \cdot \color{blue}{\left(\left(angle \cdot \pi\right) \cdot \left(a + b\right)\right)}\right) \]

       2. distribute-rgt-in 12.9%

          \[\leadsto 0.011111111111111112 \cdot \left(a \cdot \color{blue}{\left(a \cdot \left(angle \cdot \pi\right) + b \cdot \left(angle \cdot \pi\right)\right)}\right) \]

       3. distribute-rgt-out 10.8%

          \[\leadsto 0.011111111111111112 \cdot \color{blue}{\left(\left(a \cdot \left(angle \cdot \pi\right)\right) \cdot a + \left(b \cdot \left(angle \cdot \pi\right)\right) \cdot a\right)} \]

       4. associate-*r* 10.8%

          \[\leadsto 0.011111111111111112 \cdot \left(\color{blue}{a \cdot \left(\left(angle \cdot \pi\right) \cdot a\right)} + \left(b \cdot \left(angle \cdot \pi\right)\right) \cdot a\right) \]

       5. associate-*r* 8.7%

          \[\leadsto 0.011111111111111112 \cdot \left(a \cdot \left(\left(angle \cdot \pi\right) \cdot a\right) + \color{blue}{b \cdot \left(\left(angle \cdot \pi\right) \cdot a\right)}\right) \]

       6. distribute-rgt-out 16.9%

          \[\leadsto 0.011111111111111112 \cdot \color{blue}{\left(\left(\left(angle \cdot \pi\right) \cdot a\right) \cdot \left(a + b\right)\right)} \]

       7. distribute-lft-out 8.7%

          \[\leadsto 0.011111111111111112 \cdot \color{blue}{\left(\left(\left(angle \cdot \pi\right) \cdot a\right) \cdot a + \left(\left(angle \cdot \pi\right) \cdot a\right) \cdot b\right)} \]

       8. associate-*l* 8.7%

          \[\leadsto 0.011111111111111112 \cdot \left(\color{blue}{\left(angle \cdot \pi\right) \cdot \left(a \cdot a\right)} + \left(\left(angle \cdot \pi\right) \cdot a\right) \cdot b\right) \]

       9. unpow2 8.7%

          \[\leadsto 0.011111111111111112 \cdot \left(\left(angle \cdot \pi\right) \cdot \color{blue}{{a}^{2}} + \left(\left(angle \cdot \pi\right) \cdot a\right) \cdot b\right) \]

       10. associate-*l* 10.9%

           \[\leadsto 0.011111111111111112 \cdot \left(\left(angle \cdot \pi\right) \cdot {a}^{2} + \color{blue}{\left(angle \cdot \pi\right) \cdot \left(a \cdot b\right)}\right) \]

       11. *-commutative 10.9%

           \[\leadsto 0.011111111111111112 \cdot \left(\left(angle \cdot \pi\right) \cdot {a}^{2} + \left(angle \cdot \pi\right) \cdot \color{blue}{\left(b \cdot a\right)}\right) \]

       12. distribute-lft-out 12.9%

           \[\leadsto 0.011111111111111112 \cdot \color{blue}{\left(\left(angle \cdot \pi\right) \cdot \left({a}^{2} + b \cdot a\right)\right)} \]

       13. unpow2 12.9%

           \[\leadsto 0.011111111111111112 \cdot \left(\left(angle \cdot \pi\right) \cdot \left(\color{blue}{a \cdot a} + b \cdot a\right)\right) \]

       14. distribute-rgt-in 16.9%

           \[\leadsto 0.011111111111111112 \cdot \left(\left(angle \cdot \pi\right) \cdot \color{blue}{\left(a \cdot \left(a + b\right)\right)}\right) \]

   17. Simplified 16.9%

       \[\leadsto 0.011111111111111112 \cdot \color{blue}{\left(\left(angle \cdot \pi\right) \cdot \left(a \cdot \left(a + b\right)\right)\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 42.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \leq 5.5 \cdot 10^{+86}:\\ \;\;\;\;0.011111111111111112 \cdot \left(angle \cdot \left(\left(a + b\right) \cdot \left(b \cdot \pi\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;0.011111111111111112 \cdot \left(\left(angle \cdot \pi\right) \cdot \left(a \cdot \left(a + b\right)\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 14: 38.9% accurate, 26.2× speedup

Math

\[\begin{array}{l} a_m = \left|a\right| \\ b_m = \left|b\right| \\ angle\_m = \left|angle\right| \\ angle\_s = \mathsf{copysign}\left(1, angle\right) \\ angle\_s \cdot \begin{array}{l} \mathbf{if}\;a\_m \leq 2.7 \cdot 10^{+87}:\\ \;\;\;\;0.011111111111111112 \cdot \left(angle\_m \cdot \left(\left(a\_m + b\_m\right) \cdot \left(b\_m \cdot \pi\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;0.011111111111111112 \cdot \left(angle\_m \cdot \left(\left(a\_m + b\_m\right) \cdot \left(a\_m \cdot \pi\right)\right)\right)\\ \end{array} \end{array} \]

FPCore

a_m = (fabs.f64 a)
b_m = (fabs.f64 b)
angle\_m = (fabs.f64 angle)
angle\_s = (copysign.f64 #s(literal 1 binary64) angle)
(FPCore (angle_s a_m b_m angle_m)
 :precision binary64
 (*
  angle_s
  (if (<= a_m 2.7e+87)
    (* 0.011111111111111112 (* angle_m (* (+ a_m b_m) (* b_m PI))))
    (* 0.011111111111111112 (* angle_m (* (+ a_m b_m) (* a_m PI)))))))

C

a_m = fabs(a);
b_m = fabs(b);
angle\_m = fabs(angle);
angle\_s = copysign(1.0, angle);
double code(double angle_s, double a_m, double b_m, double angle_m) {
	double tmp;
	if (a_m <= 2.7e+87) {
		tmp = 0.011111111111111112 * (angle_m * ((a_m + b_m) * (b_m * ((double) M_PI))));
	} else {
		tmp = 0.011111111111111112 * (angle_m * ((a_m + b_m) * (a_m * ((double) M_PI))));
	}
	return angle_s * tmp;
}

Java

a_m = Math.abs(a);
b_m = Math.abs(b);
angle\_m = Math.abs(angle);
angle\_s = Math.copySign(1.0, angle);
public static double code(double angle_s, double a_m, double b_m, double angle_m) {
	double tmp;
	if (a_m <= 2.7e+87) {
		tmp = 0.011111111111111112 * (angle_m * ((a_m + b_m) * (b_m * Math.PI)));
	} else {
		tmp = 0.011111111111111112 * (angle_m * ((a_m + b_m) * (a_m * Math.PI)));
	}
	return angle_s * tmp;
}

Python

a_m = math.fabs(a)
b_m = math.fabs(b)
angle\_m = math.fabs(angle)
angle\_s = math.copysign(1.0, angle)
def code(angle_s, a_m, b_m, angle_m):
	tmp = 0
	if a_m <= 2.7e+87:
		tmp = 0.011111111111111112 * (angle_m * ((a_m + b_m) * (b_m * math.pi)))
	else:
		tmp = 0.011111111111111112 * (angle_m * ((a_m + b_m) * (a_m * math.pi)))
	return angle_s * tmp

Julia

a_m = abs(a)
b_m = abs(b)
angle\_m = abs(angle)
angle\_s = copysign(1.0, angle)
function code(angle_s, a_m, b_m, angle_m)
	tmp = 0.0
	if (a_m <= 2.7e+87)
		tmp = Float64(0.011111111111111112 * Float64(angle_m * Float64(Float64(a_m + b_m) * Float64(b_m * pi))));
	else
		tmp = Float64(0.011111111111111112 * Float64(angle_m * Float64(Float64(a_m + b_m) * Float64(a_m * pi))));
	end
	return Float64(angle_s * tmp)
end

MATLAB

a_m = abs(a);
b_m = abs(b);
angle\_m = abs(angle);
angle\_s = sign(angle) * abs(1.0);
function tmp_2 = code(angle_s, a_m, b_m, angle_m)
	tmp = 0.0;
	if (a_m <= 2.7e+87)
		tmp = 0.011111111111111112 * (angle_m * ((a_m + b_m) * (b_m * pi)));
	else
		tmp = 0.011111111111111112 * (angle_m * ((a_m + b_m) * (a_m * pi)));
	end
	tmp_2 = angle_s * tmp;
end

Wolfram

a_m = N[Abs[a], $MachinePrecision]
b_m = N[Abs[b], $MachinePrecision]
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$95$m_, b$95$m_, angle$95$m_] := N[(angle$95$s * If[LessEqual[a$95$m, 2.7e+87], N[(0.011111111111111112 * N[(angle$95$m * N[(N[(a$95$m + b$95$m), $MachinePrecision] * N[(b$95$m * Pi), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(0.011111111111111112 * N[(angle$95$m * N[(N[(a$95$m + b$95$m), $MachinePrecision] * N[(a$95$m * Pi), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if a < 2.70000000000000007e87

   1. Initial program 59.8%

      \[\left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin \left(\pi \cdot \frac{angle}{180}\right)\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right) \]
   2. Step-by-step derivation

      1. associate-*l* 59.8%

         \[\leadsto \color{blue}{\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \left(\sin \left(\pi \cdot \frac{angle}{180}\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right)\right)} \]

      2. *-commutative 59.8%

         \[\leadsto \color{blue}{\left(\left({b}^{2} - {a}^{2}\right) \cdot 2\right)} \cdot \left(\sin \left(\pi \cdot \frac{angle}{180}\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right)\right) \]

      3. associate-*l* 59.8%

         \[\leadsto \color{blue}{\left({b}^{2} - {a}^{2}\right) \cdot \left(2 \cdot \left(\sin \left(\pi \cdot \frac{angle}{180}\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right)\right)\right)} \]

   3. Simplified 59.8%

      \[\leadsto \color{blue}{\left({b}^{2} - {a}^{2}\right) \cdot \left(2 \cdot \left(\sin \left(\pi \cdot \frac{angle}{180}\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right)\right)\right)} \]
   4. Add Preprocessing
   5. Step-by-step derivation

      1. unpow2 59.8%

         \[\leadsto \left(\left(2 \cdot \left(\color{blue}{b \cdot b} - {a}^{2}\right)\right) \cdot \sin \left(\pi \cdot \frac{angle}{180}\right)\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right) \]

      2. unpow2 59.8%

         \[\leadsto \left(\left(2 \cdot \left(b \cdot b - \color{blue}{a \cdot a}\right)\right) \cdot \sin \left(\pi \cdot \frac{angle}{180}\right)\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right) \]

      3. difference-of-squares 63.2%

         \[\leadsto \left(\left(2 \cdot \color{blue}{\left(\left(b + a\right) \cdot \left(b - a\right)\right)}\right) \cdot \sin \left(\pi \cdot \frac{angle}{180}\right)\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right) \]

   6. Applied egg-rr 63.2%

      \[\leadsto \color{blue}{\left(\left(b + a\right) \cdot \left(b - a\right)\right)} \cdot \left(2 \cdot \left(\sin \left(\pi \cdot \frac{angle}{180}\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right)\right)\right) \]

   7. Taylor expanded in angle around 0 58.0%

      \[\leadsto \color{blue}{0.011111111111111112 \cdot \left(angle \cdot \left(\pi \cdot \left(\left(a + b\right) \cdot \left(b - a\right)\right)\right)\right)} \]
   8. Step-by-step derivation

      1. associate-*r* 58.0%

         \[\leadsto 0.011111111111111112 \cdot \left(angle \cdot \color{blue}{\left(\left(\pi \cdot \left(a + b\right)\right) \cdot \left(b - a\right)\right)}\right) \]

      2. sub-neg 58.0%

         \[\leadsto 0.011111111111111112 \cdot \left(angle \cdot \left(\left(\pi \cdot \left(a + b\right)\right) \cdot \color{blue}{\left(b + \left(-a\right)\right)}\right)\right) \]

      3. distribute-lft-in 56.1%

         \[\leadsto 0.011111111111111112 \cdot \left(angle \cdot \color{blue}{\left(\left(\pi \cdot \left(a + b\right)\right) \cdot b + \left(\pi \cdot \left(a + b\right)\right) \cdot \left(-a\right)\right)}\right) \]

   9. Applied egg-rr 56.1%

      \[\leadsto 0.011111111111111112 \cdot \left(angle \cdot \color{blue}{\left(\left(\pi \cdot \left(a + b\right)\right) \cdot b + \left(\pi \cdot \left(a + b\right)\right) \cdot \left(-a\right)\right)}\right) \]
   10. Step-by-step derivation

       1. distribute-lft-out 58.0%

          \[\leadsto 0.011111111111111112 \cdot \left(angle \cdot \color{blue}{\left(\left(\pi \cdot \left(a + b\right)\right) \cdot \left(b + \left(-a\right)\right)\right)}\right) \]

       2. +-commutative 58.0%

          \[\leadsto 0.011111111111111112 \cdot \left(angle \cdot \left(\left(\pi \cdot \color{blue}{\left(b + a\right)}\right) \cdot \left(b + \left(-a\right)\right)\right)\right) \]

       3. add-sqr-sqrt 32.5%

          \[\leadsto 0.011111111111111112 \cdot \left(angle \cdot \left(\left(\pi \cdot \left(b + a\right)\right) \cdot \left(b + \color{blue}{\sqrt{-a} \cdot \sqrt{-a}}\right)\right)\right) \]

       4. sqrt-unprod 53.3%

          \[\leadsto 0.011111111111111112 \cdot \left(angle \cdot \left(\left(\pi \cdot \left(b + a\right)\right) \cdot \left(b + \color{blue}{\sqrt{\left(-a\right) \cdot \left(-a\right)}}\right)\right)\right) \]

       5. sqr-neg 53.3%

          \[\leadsto 0.011111111111111112 \cdot \left(angle \cdot \left(\left(\pi \cdot \left(b + a\right)\right) \cdot \left(b + \sqrt{\color{blue}{a \cdot a}}\right)\right)\right) \]

       6. sqrt-prod 21.7%

          \[\leadsto 0.011111111111111112 \cdot \left(angle \cdot \left(\left(\pi \cdot \left(b + a\right)\right) \cdot \left(b + \color{blue}{\sqrt{a} \cdot \sqrt{a}}\right)\right)\right) \]

       7. add-sqr-sqrt 48.9%

          \[\leadsto 0.011111111111111112 \cdot \left(angle \cdot \left(\left(\pi \cdot \left(b + a\right)\right) \cdot \left(b + \color{blue}{a}\right)\right)\right) \]

   11. Applied egg-rr 48.9%

       \[\leadsto 0.011111111111111112 \cdot \left(angle \cdot \color{blue}{\left(\left(\pi \cdot \left(b + a\right)\right) \cdot \left(b + a\right)\right)}\right) \]

   12. Taylor expanded in b around inf 48.9%

       \[\leadsto 0.011111111111111112 \cdot \left(angle \cdot \left(\color{blue}{\left(b \cdot \pi\right)} \cdot \left(b + a\right)\right)\right) \]
   13. Step-by-step derivation

       1. *-commutative 48.9%

          \[\leadsto 0.011111111111111112 \cdot \left(angle \cdot \left(\color{blue}{\left(\pi \cdot b\right)} \cdot \left(b + a\right)\right)\right) \]

   14. Simplified 48.9%

       \[\leadsto 0.011111111111111112 \cdot \left(angle \cdot \left(\color{blue}{\left(\pi \cdot b\right)} \cdot \left(b + a\right)\right)\right) \]

   if 2.70000000000000007e87 < a

   1. Initial program 41.9%

      \[\left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin \left(\pi \cdot \frac{angle}{180}\right)\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right) \]
   2. Step-by-step derivation

      1. associate-*l* 41.9%

         \[\leadsto \color{blue}{\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \left(\sin \left(\pi \cdot \frac{angle}{180}\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right)\right)} \]

      2. *-commutative 41.9%

         \[\leadsto \color{blue}{\left(\left({b}^{2} - {a}^{2}\right) \cdot 2\right)} \cdot \left(\sin \left(\pi \cdot \frac{angle}{180}\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right)\right) \]

      3. associate-*l* 41.9%

         \[\leadsto \color{blue}{\left({b}^{2} - {a}^{2}\right) \cdot \left(2 \cdot \left(\sin \left(\pi \cdot \frac{angle}{180}\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right)\right)\right)} \]

   3. Simplified 41.9%

      \[\leadsto \color{blue}{\left({b}^{2} - {a}^{2}\right) \cdot \left(2 \cdot \left(\sin \left(\pi \cdot \frac{angle}{180}\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right)\right)\right)} \]
   4. Add Preprocessing
   5. Step-by-step derivation

      1. unpow2 41.9%

         \[\leadsto \left(\left(2 \cdot \left(\color{blue}{b \cdot b} - {a}^{2}\right)\right) \cdot \sin \left(\pi \cdot \frac{angle}{180}\right)\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right) \]

      2. unpow2 41.9%

         \[\leadsto \left(\left(2 \cdot \left(b \cdot b - \color{blue}{a \cdot a}\right)\right) \cdot \sin \left(\pi \cdot \frac{angle}{180}\right)\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right) \]

      3. difference-of-squares 52.0%

         \[\leadsto \left(\left(2 \cdot \color{blue}{\left(\left(b + a\right) \cdot \left(b - a\right)\right)}\right) \cdot \sin \left(\pi \cdot \frac{angle}{180}\right)\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right) \]

   6. Applied egg-rr 52.0%

      \[\leadsto \color{blue}{\left(\left(b + a\right) \cdot \left(b - a\right)\right)} \cdot \left(2 \cdot \left(\sin \left(\pi \cdot \frac{angle}{180}\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right)\right)\right) \]

   7. Taylor expanded in angle around 0 55.4%

      \[\leadsto \color{blue}{0.011111111111111112 \cdot \left(angle \cdot \left(\pi \cdot \left(\left(a + b\right) \cdot \left(b - a\right)\right)\right)\right)} \]
   8. Step-by-step derivation

      1. associate-*r* 55.4%

         \[\leadsto 0.011111111111111112 \cdot \left(angle \cdot \color{blue}{\left(\left(\pi \cdot \left(a + b\right)\right) \cdot \left(b - a\right)\right)}\right) \]

      2. sub-neg 55.4%

         \[\leadsto 0.011111111111111112 \cdot \left(angle \cdot \left(\left(\pi \cdot \left(a + b\right)\right) \cdot \color{blue}{\left(b + \left(-a\right)\right)}\right)\right) \]

      3. distribute-lft-in 46.9%

         \[\leadsto 0.011111111111111112 \cdot \left(angle \cdot \color{blue}{\left(\left(\pi \cdot \left(a + b\right)\right) \cdot b + \left(\pi \cdot \left(a + b\right)\right) \cdot \left(-a\right)\right)}\right) \]

   9. Applied egg-rr 46.9%

      \[\leadsto 0.011111111111111112 \cdot \left(angle \cdot \color{blue}{\left(\left(\pi \cdot \left(a + b\right)\right) \cdot b + \left(\pi \cdot \left(a + b\right)\right) \cdot \left(-a\right)\right)}\right) \]
   10. Step-by-step derivation

       1. distribute-lft-out 55.4%

          \[\leadsto 0.011111111111111112 \cdot \left(angle \cdot \color{blue}{\left(\left(\pi \cdot \left(a + b\right)\right) \cdot \left(b + \left(-a\right)\right)\right)}\right) \]

       2. +-commutative 55.4%

          \[\leadsto 0.011111111111111112 \cdot \left(angle \cdot \left(\left(\pi \cdot \color{blue}{\left(b + a\right)}\right) \cdot \left(b + \left(-a\right)\right)\right)\right) \]

       3. add-sqr-sqrt 0.0%

          \[\leadsto 0.011111111111111112 \cdot \left(angle \cdot \left(\left(\pi \cdot \left(b + a\right)\right) \cdot \left(b + \color{blue}{\sqrt{-a} \cdot \sqrt{-a}}\right)\right)\right) \]

       4. sqrt-unprod 18.9%

          \[\leadsto 0.011111111111111112 \cdot \left(angle \cdot \left(\left(\pi \cdot \left(b + a\right)\right) \cdot \left(b + \color{blue}{\sqrt{\left(-a\right) \cdot \left(-a\right)}}\right)\right)\right) \]

       5. sqr-neg 18.9%

          \[\leadsto 0.011111111111111112 \cdot \left(angle \cdot \left(\left(\pi \cdot \left(b + a\right)\right) \cdot \left(b + \sqrt{\color{blue}{a \cdot a}}\right)\right)\right) \]

       6. sqrt-prod 20.9%

          \[\leadsto 0.011111111111111112 \cdot \left(angle \cdot \left(\left(\pi \cdot \left(b + a\right)\right) \cdot \left(b + \color{blue}{\sqrt{a} \cdot \sqrt{a}}\right)\right)\right) \]

       7. add-sqr-sqrt 20.9%

          \[\leadsto 0.011111111111111112 \cdot \left(angle \cdot \left(\left(\pi \cdot \left(b + a\right)\right) \cdot \left(b + \color{blue}{a}\right)\right)\right) \]

   11. Applied egg-rr 20.9%

       \[\leadsto 0.011111111111111112 \cdot \left(angle \cdot \color{blue}{\left(\left(\pi \cdot \left(b + a\right)\right) \cdot \left(b + a\right)\right)}\right) \]

   12. Taylor expanded in b around 0 16.9%

       \[\leadsto 0.011111111111111112 \cdot \left(angle \cdot \left(\color{blue}{\left(a \cdot \pi\right)} \cdot \left(b + a\right)\right)\right) \]
   13. Step-by-step derivation

       1. *-commutative 16.9%

          \[\leadsto 0.011111111111111112 \cdot \left(angle \cdot \left(\color{blue}{\left(\pi \cdot a\right)} \cdot \left(b + a\right)\right)\right) \]

   14. Simplified 16.9%

       \[\leadsto 0.011111111111111112 \cdot \left(angle \cdot \left(\color{blue}{\left(\pi \cdot a\right)} \cdot \left(b + a\right)\right)\right) \]
3. Recombined 2 regimes into one program.

4. Final simplification 42.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \leq 2.7 \cdot 10^{+87}:\\ \;\;\;\;0.011111111111111112 \cdot \left(angle \cdot \left(\left(a + b\right) \cdot \left(b \cdot \pi\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;0.011111111111111112 \cdot \left(angle \cdot \left(\left(a + b\right) \cdot \left(a \cdot \pi\right)\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 15: 24.1% accurate, 26.2× speedup

Math

\[\begin{array}{l} a_m = \left|a\right| \\ b_m = \left|b\right| \\ angle\_m = \left|angle\right| \\ angle\_s = \mathsf{copysign}\left(1, angle\right) \\ angle\_s \cdot \begin{array}{l} \mathbf{if}\;a\_m \leq 2.7 \cdot 10^{-58}:\\ \;\;\;\;0.011111111111111112 \cdot \left(a\_m \cdot \left(angle\_m \cdot \left(b\_m \cdot \pi\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;0.011111111111111112 \cdot \left(angle\_m \cdot \left(\left(a\_m + b\_m\right) \cdot \left(a\_m \cdot \pi\right)\right)\right)\\ \end{array} \end{array} \]

FPCore

a_m = (fabs.f64 a)
b_m = (fabs.f64 b)
angle\_m = (fabs.f64 angle)
angle\_s = (copysign.f64 #s(literal 1 binary64) angle)
(FPCore (angle_s a_m b_m angle_m)
 :precision binary64
 (*
  angle_s
  (if (<= a_m 2.7e-58)
    (* 0.011111111111111112 (* a_m (* angle_m (* b_m PI))))
    (* 0.011111111111111112 (* angle_m (* (+ a_m b_m) (* a_m PI)))))))

C

a_m = fabs(a);
b_m = fabs(b);
angle\_m = fabs(angle);
angle\_s = copysign(1.0, angle);
double code(double angle_s, double a_m, double b_m, double angle_m) {
	double tmp;
	if (a_m <= 2.7e-58) {
		tmp = 0.011111111111111112 * (a_m * (angle_m * (b_m * ((double) M_PI))));
	} else {
		tmp = 0.011111111111111112 * (angle_m * ((a_m + b_m) * (a_m * ((double) M_PI))));
	}
	return angle_s * tmp;
}

Java

a_m = Math.abs(a);
b_m = Math.abs(b);
angle\_m = Math.abs(angle);
angle\_s = Math.copySign(1.0, angle);
public static double code(double angle_s, double a_m, double b_m, double angle_m) {
	double tmp;
	if (a_m <= 2.7e-58) {
		tmp = 0.011111111111111112 * (a_m * (angle_m * (b_m * Math.PI)));
	} else {
		tmp = 0.011111111111111112 * (angle_m * ((a_m + b_m) * (a_m * Math.PI)));
	}
	return angle_s * tmp;
}

Python

a_m = math.fabs(a)
b_m = math.fabs(b)
angle\_m = math.fabs(angle)
angle\_s = math.copysign(1.0, angle)
def code(angle_s, a_m, b_m, angle_m):
	tmp = 0
	if a_m <= 2.7e-58:
		tmp = 0.011111111111111112 * (a_m * (angle_m * (b_m * math.pi)))
	else:
		tmp = 0.011111111111111112 * (angle_m * ((a_m + b_m) * (a_m * math.pi)))
	return angle_s * tmp

Julia

a_m = abs(a)
b_m = abs(b)
angle\_m = abs(angle)
angle\_s = copysign(1.0, angle)
function code(angle_s, a_m, b_m, angle_m)
	tmp = 0.0
	if (a_m <= 2.7e-58)
		tmp = Float64(0.011111111111111112 * Float64(a_m * Float64(angle_m * Float64(b_m * pi))));
	else
		tmp = Float64(0.011111111111111112 * Float64(angle_m * Float64(Float64(a_m + b_m) * Float64(a_m * pi))));
	end
	return Float64(angle_s * tmp)
end

MATLAB

a_m = abs(a);
b_m = abs(b);
angle\_m = abs(angle);
angle\_s = sign(angle) * abs(1.0);
function tmp_2 = code(angle_s, a_m, b_m, angle_m)
	tmp = 0.0;
	if (a_m <= 2.7e-58)
		tmp = 0.011111111111111112 * (a_m * (angle_m * (b_m * pi)));
	else
		tmp = 0.011111111111111112 * (angle_m * ((a_m + b_m) * (a_m * pi)));
	end
	tmp_2 = angle_s * tmp;
end

Wolfram

a_m = N[Abs[a], $MachinePrecision]
b_m = N[Abs[b], $MachinePrecision]
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$95$m_, b$95$m_, angle$95$m_] := N[(angle$95$s * If[LessEqual[a$95$m, 2.7e-58], N[(0.011111111111111112 * N[(a$95$m * N[(angle$95$m * N[(b$95$m * Pi), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(0.011111111111111112 * N[(angle$95$m * N[(N[(a$95$m + b$95$m), $MachinePrecision] * N[(a$95$m * Pi), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if a < 2.6999999999999999e-58

   1. Initial program 59.9%

      \[\left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin \left(\pi \cdot \frac{angle}{180}\right)\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right) \]
   2. Step-by-step derivation

      1. associate-*l* 59.9%

         \[\leadsto \color{blue}{\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \left(\sin \left(\pi \cdot \frac{angle}{180}\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right)\right)} \]

      2. *-commutative 59.9%

         \[\leadsto \color{blue}{\left(\left({b}^{2} - {a}^{2}\right) \cdot 2\right)} \cdot \left(\sin \left(\pi \cdot \frac{angle}{180}\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right)\right) \]

      3. associate-*l* 59.9%

         \[\leadsto \color{blue}{\left({b}^{2} - {a}^{2}\right) \cdot \left(2 \cdot \left(\sin \left(\pi \cdot \frac{angle}{180}\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right)\right)\right)} \]

   3. Simplified 59.9%

      \[\leadsto \color{blue}{\left({b}^{2} - {a}^{2}\right) \cdot \left(2 \cdot \left(\sin \left(\pi \cdot \frac{angle}{180}\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right)\right)\right)} \]
   4. Add Preprocessing
   5. Step-by-step derivation

      1. unpow2 59.9%

         \[\leadsto \left(\left(2 \cdot \left(\color{blue}{b \cdot b} - {a}^{2}\right)\right) \cdot \sin \left(\pi \cdot \frac{angle}{180}\right)\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right) \]

      2. unpow2 59.9%

         \[\leadsto \left(\left(2 \cdot \left(b \cdot b - \color{blue}{a \cdot a}\right)\right) \cdot \sin \left(\pi \cdot \frac{angle}{180}\right)\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right) \]

      3. difference-of-squares 63.8%

         \[\leadsto \left(\left(2 \cdot \color{blue}{\left(\left(b + a\right) \cdot \left(b - a\right)\right)}\right) \cdot \sin \left(\pi \cdot \frac{angle}{180}\right)\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right) \]

   6. Applied egg-rr 63.8%

      \[\leadsto \color{blue}{\left(\left(b + a\right) \cdot \left(b - a\right)\right)} \cdot \left(2 \cdot \left(\sin \left(\pi \cdot \frac{angle}{180}\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right)\right)\right) \]

   7. Taylor expanded in angle around 0 58.6%

      \[\leadsto \color{blue}{0.011111111111111112 \cdot \left(angle \cdot \left(\pi \cdot \left(\left(a + b\right) \cdot \left(b - a\right)\right)\right)\right)} \]
   8. Step-by-step derivation

      1. associate-*r* 58.6%

         \[\leadsto 0.011111111111111112 \cdot \left(angle \cdot \color{blue}{\left(\left(\pi \cdot \left(a + b\right)\right) \cdot \left(b - a\right)\right)}\right) \]

      2. sub-neg 58.6%

         \[\leadsto 0.011111111111111112 \cdot \left(angle \cdot \left(\left(\pi \cdot \left(a + b\right)\right) \cdot \color{blue}{\left(b + \left(-a\right)\right)}\right)\right) \]

      3. distribute-lft-in 56.4%

         \[\leadsto 0.011111111111111112 \cdot \left(angle \cdot \color{blue}{\left(\left(\pi \cdot \left(a + b\right)\right) \cdot b + \left(\pi \cdot \left(a + b\right)\right) \cdot \left(-a\right)\right)}\right) \]

   9. Applied egg-rr 56.4%

      \[\leadsto 0.011111111111111112 \cdot \left(angle \cdot \color{blue}{\left(\left(\pi \cdot \left(a + b\right)\right) \cdot b + \left(\pi \cdot \left(a + b\right)\right) \cdot \left(-a\right)\right)}\right) \]
   10. Step-by-step derivation

       1. distribute-lft-out 58.6%

          \[\leadsto 0.011111111111111112 \cdot \left(angle \cdot \color{blue}{\left(\left(\pi \cdot \left(a + b\right)\right) \cdot \left(b + \left(-a\right)\right)\right)}\right) \]

       2. +-commutative 58.6%

          \[\leadsto 0.011111111111111112 \cdot \left(angle \cdot \left(\left(\pi \cdot \color{blue}{\left(b + a\right)}\right) \cdot \left(b + \left(-a\right)\right)\right)\right) \]

       3. add-sqr-sqrt 37.0%

          \[\leadsto 0.011111111111111112 \cdot \left(angle \cdot \left(\left(\pi \cdot \left(b + a\right)\right) \cdot \left(b + \color{blue}{\sqrt{-a} \cdot \sqrt{-a}}\right)\right)\right) \]

       4. sqrt-unprod 57.5%

          \[\leadsto 0.011111111111111112 \cdot \left(angle \cdot \left(\left(\pi \cdot \left(b + a\right)\right) \cdot \left(b + \color{blue}{\sqrt{\left(-a\right) \cdot \left(-a\right)}}\right)\right)\right) \]

       5. sqr-neg 57.5%

          \[\leadsto 0.011111111111111112 \cdot \left(angle \cdot \left(\left(\pi \cdot \left(b + a\right)\right) \cdot \left(b + \sqrt{\color{blue}{a \cdot a}}\right)\right)\right) \]

       6. sqrt-prod 21.5%

          \[\leadsto 0.011111111111111112 \cdot \left(angle \cdot \left(\left(\pi \cdot \left(b + a\right)\right) \cdot \left(b + \color{blue}{\sqrt{a} \cdot \sqrt{a}}\right)\right)\right) \]

       7. add-sqr-sqrt 52.5%

          \[\leadsto 0.011111111111111112 \cdot \left(angle \cdot \left(\left(\pi \cdot \left(b + a\right)\right) \cdot \left(b + \color{blue}{a}\right)\right)\right) \]

   11. Applied egg-rr 52.5%

       \[\leadsto 0.011111111111111112 \cdot \left(angle \cdot \color{blue}{\left(\left(\pi \cdot \left(b + a\right)\right) \cdot \left(b + a\right)\right)}\right) \]

   12. Taylor expanded in b around 0 28.0%

       \[\leadsto 0.011111111111111112 \cdot \left(angle \cdot \left(\color{blue}{\left(a \cdot \pi\right)} \cdot \left(b + a\right)\right)\right) \]
   13. Step-by-step derivation

       1. *-commutative 28.0%

          \[\leadsto 0.011111111111111112 \cdot \left(angle \cdot \left(\color{blue}{\left(\pi \cdot a\right)} \cdot \left(b + a\right)\right)\right) \]

   14. Simplified 28.0%

       \[\leadsto 0.011111111111111112 \cdot \left(angle \cdot \left(\color{blue}{\left(\pi \cdot a\right)} \cdot \left(b + a\right)\right)\right) \]

   15. Taylor expanded in a around 0 27.1%

       \[\leadsto \color{blue}{0.011111111111111112 \cdot \left(a \cdot \left(angle \cdot \left(b \cdot \pi\right)\right)\right)} \]
   16. Step-by-step derivation

       1. *-commutative 27.1%

          \[\leadsto \color{blue}{\left(a \cdot \left(angle \cdot \left(b \cdot \pi\right)\right)\right) \cdot 0.011111111111111112} \]

       2. *-commutative 27.1%

          \[\leadsto \left(a \cdot \left(angle \cdot \color{blue}{\left(\pi \cdot b\right)}\right)\right) \cdot 0.011111111111111112 \]

   17. Simplified 27.1%

       \[\leadsto \color{blue}{\left(a \cdot \left(angle \cdot \left(\pi \cdot b\right)\right)\right) \cdot 0.011111111111111112} \]

   if 2.6999999999999999e-58 < a

   1. Initial program 47.7%

      \[\left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin \left(\pi \cdot \frac{angle}{180}\right)\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right) \]
   2. Step-by-step derivation

      1. associate-*l* 47.7%

         \[\leadsto \color{blue}{\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \left(\sin \left(\pi \cdot \frac{angle}{180}\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right)\right)} \]

      2. *-commutative 47.7%

         \[\leadsto \color{blue}{\left(\left({b}^{2} - {a}^{2}\right) \cdot 2\right)} \cdot \left(\sin \left(\pi \cdot \frac{angle}{180}\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right)\right) \]

      3. associate-*l* 47.7%

         \[\leadsto \color{blue}{\left({b}^{2} - {a}^{2}\right) \cdot \left(2 \cdot \left(\sin \left(\pi \cdot \frac{angle}{180}\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right)\right)\right)} \]

   3. Simplified 47.7%

      \[\leadsto \color{blue}{\left({b}^{2} - {a}^{2}\right) \cdot \left(2 \cdot \left(\sin \left(\pi \cdot \frac{angle}{180}\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right)\right)\right)} \]
   4. Add Preprocessing
   5. Step-by-step derivation

      1. unpow2 47.7%

         \[\leadsto \left(\left(2 \cdot \left(\color{blue}{b \cdot b} - {a}^{2}\right)\right) \cdot \sin \left(\pi \cdot \frac{angle}{180}\right)\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right) \]

      2. unpow2 47.7%

         \[\leadsto \left(\left(2 \cdot \left(b \cdot b - \color{blue}{a \cdot a}\right)\right) \cdot \sin \left(\pi \cdot \frac{angle}{180}\right)\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right) \]

      3. difference-of-squares 54.4%

         \[\leadsto \left(\left(2 \cdot \color{blue}{\left(\left(b + a\right) \cdot \left(b - a\right)\right)}\right) \cdot \sin \left(\pi \cdot \frac{angle}{180}\right)\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right) \]

   6. Applied egg-rr 54.5%

      \[\leadsto \color{blue}{\left(\left(b + a\right) \cdot \left(b - a\right)\right)} \cdot \left(2 \cdot \left(\sin \left(\pi \cdot \frac{angle}{180}\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right)\right)\right) \]

   7. Taylor expanded in angle around 0 54.9%

      \[\leadsto \color{blue}{0.011111111111111112 \cdot \left(angle \cdot \left(\pi \cdot \left(\left(a + b\right) \cdot \left(b - a\right)\right)\right)\right)} \]
   8. Step-by-step derivation

      1. associate-*r* 55.0%

         \[\leadsto 0.011111111111111112 \cdot \left(angle \cdot \color{blue}{\left(\left(\pi \cdot \left(a + b\right)\right) \cdot \left(b - a\right)\right)}\right) \]

      2. sub-neg 55.0%

         \[\leadsto 0.011111111111111112 \cdot \left(angle \cdot \left(\left(\pi \cdot \left(a + b\right)\right) \cdot \color{blue}{\left(b + \left(-a\right)\right)}\right)\right) \]

      3. distribute-lft-in 49.4%

         \[\leadsto 0.011111111111111112 \cdot \left(angle \cdot \color{blue}{\left(\left(\pi \cdot \left(a + b\right)\right) \cdot b + \left(\pi \cdot \left(a + b\right)\right) \cdot \left(-a\right)\right)}\right) \]

   9. Applied egg-rr 49.4%

      \[\leadsto 0.011111111111111112 \cdot \left(angle \cdot \color{blue}{\left(\left(\pi \cdot \left(a + b\right)\right) \cdot b + \left(\pi \cdot \left(a + b\right)\right) \cdot \left(-a\right)\right)}\right) \]
   10. Step-by-step derivation

       1. distribute-lft-out 55.0%

          \[\leadsto 0.011111111111111112 \cdot \left(angle \cdot \color{blue}{\left(\left(\pi \cdot \left(a + b\right)\right) \cdot \left(b + \left(-a\right)\right)\right)}\right) \]

       2. +-commutative 55.0%

          \[\leadsto 0.011111111111111112 \cdot \left(angle \cdot \left(\left(\pi \cdot \color{blue}{\left(b + a\right)}\right) \cdot \left(b + \left(-a\right)\right)\right)\right) \]

       3. add-sqr-sqrt 0.0%

          \[\leadsto 0.011111111111111112 \cdot \left(angle \cdot \left(\left(\pi \cdot \left(b + a\right)\right) \cdot \left(b + \color{blue}{\sqrt{-a} \cdot \sqrt{-a}}\right)\right)\right) \]

       4. sqrt-unprod 20.3%

          \[\leadsto 0.011111111111111112 \cdot \left(angle \cdot \left(\left(\pi \cdot \left(b + a\right)\right) \cdot \left(b + \color{blue}{\sqrt{\left(-a\right) \cdot \left(-a\right)}}\right)\right)\right) \]

       5. sqr-neg 20.3%

          \[\leadsto 0.011111111111111112 \cdot \left(angle \cdot \left(\left(\pi \cdot \left(b + a\right)\right) \cdot \left(b + \sqrt{\color{blue}{a \cdot a}}\right)\right)\right) \]

       6. sqrt-prod 21.6%

          \[\leadsto 0.011111111111111112 \cdot \left(angle \cdot \left(\left(\pi \cdot \left(b + a\right)\right) \cdot \left(b + \color{blue}{\sqrt{a} \cdot \sqrt{a}}\right)\right)\right) \]

       7. add-sqr-sqrt 21.6%

          \[\leadsto 0.011111111111111112 \cdot \left(angle \cdot \left(\left(\pi \cdot \left(b + a\right)\right) \cdot \left(b + \color{blue}{a}\right)\right)\right) \]

   11. Applied egg-rr 21.6%

       \[\leadsto 0.011111111111111112 \cdot \left(angle \cdot \color{blue}{\left(\left(\pi \cdot \left(b + a\right)\right) \cdot \left(b + a\right)\right)}\right) \]

   12. Taylor expanded in b around 0 13.5%

       \[\leadsto 0.011111111111111112 \cdot \left(angle \cdot \left(\color{blue}{\left(a \cdot \pi\right)} \cdot \left(b + a\right)\right)\right) \]
   13. Step-by-step derivation

       1. *-commutative 13.5%

          \[\leadsto 0.011111111111111112 \cdot \left(angle \cdot \left(\color{blue}{\left(\pi \cdot a\right)} \cdot \left(b + a\right)\right)\right) \]

   14. Simplified 13.5%

       \[\leadsto 0.011111111111111112 \cdot \left(angle \cdot \left(\color{blue}{\left(\pi \cdot a\right)} \cdot \left(b + a\right)\right)\right) \]
3. Recombined 2 regimes into one program.

4. Final simplification 23.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \leq 2.7 \cdot 10^{-58}:\\ \;\;\;\;0.011111111111111112 \cdot \left(a \cdot \left(angle \cdot \left(b \cdot \pi\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;0.011111111111111112 \cdot \left(angle \cdot \left(\left(a + b\right) \cdot \left(a \cdot \pi\right)\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 16: 62.4% accurate, 32.2× speedup

Math

\[\begin{array}{l} a_m = \left|a\right| \\ b_m = \left|b\right| \\ angle\_m = \left|angle\right| \\ angle\_s = \mathsf{copysign}\left(1, angle\right) \\ angle\_s \cdot \left(\left(a\_m + b\_m\right) \cdot \left(angle\_m \cdot \left(\left(b\_m - a\_m\right) \cdot \left(\pi \cdot 0.011111111111111112\right)\right)\right)\right) \end{array} \]

FPCore

a_m = (fabs.f64 a)
b_m = (fabs.f64 b)
angle\_m = (fabs.f64 angle)
angle\_s = (copysign.f64 #s(literal 1 binary64) angle)
(FPCore (angle_s a_m b_m angle_m)
 :precision binary64
 (*
  angle_s
  (* (+ a_m b_m) (* angle_m (* (- b_m a_m) (* PI 0.011111111111111112))))))

C

a_m = fabs(a);
b_m = fabs(b);
angle\_m = fabs(angle);
angle\_s = copysign(1.0, angle);
double code(double angle_s, double a_m, double b_m, double angle_m) {
	return angle_s * ((a_m + b_m) * (angle_m * ((b_m - a_m) * (((double) M_PI) * 0.011111111111111112))));
}

Java

a_m = Math.abs(a);
b_m = Math.abs(b);
angle\_m = Math.abs(angle);
angle\_s = Math.copySign(1.0, angle);
public static double code(double angle_s, double a_m, double b_m, double angle_m) {
	return angle_s * ((a_m + b_m) * (angle_m * ((b_m - a_m) * (Math.PI * 0.011111111111111112))));
}

Python

a_m = math.fabs(a)
b_m = math.fabs(b)
angle\_m = math.fabs(angle)
angle\_s = math.copysign(1.0, angle)
def code(angle_s, a_m, b_m, angle_m):
	return angle_s * ((a_m + b_m) * (angle_m * ((b_m - a_m) * (math.pi * 0.011111111111111112))))

Julia

a_m = abs(a)
b_m = abs(b)
angle\_m = abs(angle)
angle\_s = copysign(1.0, angle)
function code(angle_s, a_m, b_m, angle_m)
	return Float64(angle_s * Float64(Float64(a_m + b_m) * Float64(angle_m * Float64(Float64(b_m - a_m) * Float64(pi * 0.011111111111111112)))))
end

MATLAB

a_m = abs(a);
b_m = abs(b);
angle\_m = abs(angle);
angle\_s = sign(angle) * abs(1.0);
function tmp = code(angle_s, a_m, b_m, angle_m)
	tmp = angle_s * ((a_m + b_m) * (angle_m * ((b_m - a_m) * (pi * 0.011111111111111112))));
end

Wolfram

a_m = N[Abs[a], $MachinePrecision]
b_m = N[Abs[b], $MachinePrecision]
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$95$m_, b$95$m_, angle$95$m_] := N[(angle$95$s * N[(N[(a$95$m + b$95$m), $MachinePrecision] * N[(angle$95$m * N[(N[(b$95$m - a$95$m), $MachinePrecision] * N[(Pi * 0.011111111111111112), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 56.3%

   \[\left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin \left(\pi \cdot \frac{angle}{180}\right)\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right) \]
2. Step-by-step derivation

   1. associate-*l* 56.3%

      \[\leadsto \color{blue}{\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \left(\sin \left(\pi \cdot \frac{angle}{180}\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right)\right)} \]

   2. *-commutative 56.3%

      \[\leadsto \color{blue}{\left(\left({b}^{2} - {a}^{2}\right) \cdot 2\right)} \cdot \left(\sin \left(\pi \cdot \frac{angle}{180}\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right)\right) \]

   3. associate-*l* 56.3%

      \[\leadsto \color{blue}{\left({b}^{2} - {a}^{2}\right) \cdot \left(2 \cdot \left(\sin \left(\pi \cdot \frac{angle}{180}\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right)\right)\right)} \]

3. Simplified 56.3%

   \[\leadsto \color{blue}{\left({b}^{2} - {a}^{2}\right) \cdot \left(2 \cdot \left(\sin \left(\pi \cdot \frac{angle}{180}\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right)\right)\right)} \]
4. Add Preprocessing
5. Step-by-step derivation

   1. unpow2 56.3%

      \[\leadsto \left(\left(2 \cdot \left(\color{blue}{b \cdot b} - {a}^{2}\right)\right) \cdot \sin \left(\pi \cdot \frac{angle}{180}\right)\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right) \]

   2. unpow2 56.3%

      \[\leadsto \left(\left(2 \cdot \left(b \cdot b - \color{blue}{a \cdot a}\right)\right) \cdot \sin \left(\pi \cdot \frac{angle}{180}\right)\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right) \]

   3. difference-of-squares 61.0%

      \[\leadsto \left(\left(2 \cdot \color{blue}{\left(\left(b + a\right) \cdot \left(b - a\right)\right)}\right) \cdot \sin \left(\pi \cdot \frac{angle}{180}\right)\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right) \]

6. Applied egg-rr 61.0%

   \[\leadsto \color{blue}{\left(\left(b + a\right) \cdot \left(b - a\right)\right)} \cdot \left(2 \cdot \left(\sin \left(\pi \cdot \frac{angle}{180}\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right)\right)\right) \]
7. Step-by-step derivation

   1. add-exp-log 35.1%

      \[\leadsto \color{blue}{e^{\log \left(\left(\left(b + a\right) \cdot \left(b - a\right)\right) \cdot \left(2 \cdot \left(\sin \left(\pi \cdot \frac{angle}{180}\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right)\right)\right)\right)}} \]

   2. 2-sin 35.1%

      \[\leadsto e^{\log \left(\left(\left(b + a\right) \cdot \left(b - a\right)\right) \cdot \color{blue}{\sin \left(2 \cdot \left(\pi \cdot \frac{angle}{180}\right)\right)}\right)} \]

   3. div-inv 34.9%

      \[\leadsto e^{\log \left(\left(\left(b + a\right) \cdot \left(b - a\right)\right) \cdot \sin \left(2 \cdot \left(\pi \cdot \color{blue}{\left(angle \cdot \frac{1}{180}\right)}\right)\right)\right)} \]

   4. metadata-eval 34.9%

      \[\leadsto e^{\log \left(\left(\left(b + a\right) \cdot \left(b - a\right)\right) \cdot \sin \left(2 \cdot \left(\pi \cdot \left(angle \cdot \color{blue}{0.005555555555555556}\right)\right)\right)\right)} \]

8. Applied egg-rr 34.9%

   \[\leadsto \color{blue}{e^{\log \left(\left(\left(b + a\right) \cdot \left(b - a\right)\right) \cdot \sin \left(2 \cdot \left(\pi \cdot \left(angle \cdot 0.005555555555555556\right)\right)\right)\right)}} \]
9. Step-by-step derivation

   1. rem-exp-log 59.6%

      \[\leadsto \color{blue}{\left(\left(b + a\right) \cdot \left(b - a\right)\right) \cdot \sin \left(2 \cdot \left(\pi \cdot \left(angle \cdot 0.005555555555555556\right)\right)\right)} \]

   2. associate-*l* 67.9%

      \[\leadsto \color{blue}{\left(b + a\right) \cdot \left(\left(b - a\right) \cdot \sin \left(2 \cdot \left(\pi \cdot \left(angle \cdot 0.005555555555555556\right)\right)\right)\right)} \]

   3. +-commutative 67.9%

      \[\leadsto \color{blue}{\left(a + b\right)} \cdot \left(\left(b - a\right) \cdot \sin \left(2 \cdot \left(\pi \cdot \left(angle \cdot 0.005555555555555556\right)\right)\right)\right) \]

   4. count-2 67.9%

      \[\leadsto \left(a + b\right) \cdot \left(\left(b - a\right) \cdot \sin \color{blue}{\left(\pi \cdot \left(angle \cdot 0.005555555555555556\right) + \pi \cdot \left(angle \cdot 0.005555555555555556\right)\right)}\right) \]

   5. add-cube-cbrt 66.8%

      \[\leadsto \left(a + b\right) \cdot \left(\left(b - a\right) \cdot \sin \left(\color{blue}{\left(\left(\sqrt[3]{\pi} \cdot \sqrt[3]{\pi}\right) \cdot \sqrt[3]{\pi}\right)} \cdot \left(angle \cdot 0.005555555555555556\right) + \pi \cdot \left(angle \cdot 0.005555555555555556\right)\right)\right) \]

   6. unpow2 66.8%

      \[\leadsto \left(a + b\right) \cdot \left(\left(b - a\right) \cdot \sin \left(\left(\color{blue}{{\left(\sqrt[3]{\pi}\right)}^{2}} \cdot \sqrt[3]{\pi}\right) \cdot \left(angle \cdot 0.005555555555555556\right) + \pi \cdot \left(angle \cdot 0.005555555555555556\right)\right)\right) \]

   7. metadata-eval 66.8%

      \[\leadsto \left(a + b\right) \cdot \left(\left(b - a\right) \cdot \sin \left(\left({\left(\sqrt[3]{\pi}\right)}^{2} \cdot \sqrt[3]{\pi}\right) \cdot \left(angle \cdot \color{blue}{\frac{1}{180}}\right) + \pi \cdot \left(angle \cdot 0.005555555555555556\right)\right)\right) \]

   8. div-inv 67.9%

      \[\leadsto \left(a + b\right) \cdot \left(\left(b - a\right) \cdot \sin \left(\left({\left(\sqrt[3]{\pi}\right)}^{2} \cdot \sqrt[3]{\pi}\right) \cdot \color{blue}{\frac{angle}{180}} + \pi \cdot \left(angle \cdot 0.005555555555555556\right)\right)\right) \]

   9. metadata-eval 67.9%

      \[\leadsto \left(a + b\right) \cdot \left(\left(b - a\right) \cdot \sin \left(\left({\left(\sqrt[3]{\pi}\right)}^{2} \cdot \sqrt[3]{\pi}\right) \cdot \frac{angle}{180} + \pi \cdot \left(angle \cdot \color{blue}{\frac{1}{180}}\right)\right)\right) \]

   10. div-inv 67.6%

       \[\leadsto \left(a + b\right) \cdot \left(\left(b - a\right) \cdot \sin \left(\left({\left(\sqrt[3]{\pi}\right)}^{2} \cdot \sqrt[3]{\pi}\right) \cdot \frac{angle}{180} + \pi \cdot \color{blue}{\frac{angle}{180}}\right)\right) \]

10. Applied egg-rr 68.9%

    \[\leadsto \color{blue}{\left(a + b\right) \cdot \left(\left(b - a\right) \cdot \sin \left(\left(angle \cdot \pi\right) \cdot 0.011111111111111112\right)\right)} \]
11. Step-by-step derivation

    1. expm1-log1p-u 68.9%

       \[\leadsto \left(a + b\right) \cdot \left(\left(b - a\right) \cdot \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\sin \left(\left(angle \cdot \pi\right) \cdot 0.011111111111111112\right)\right)\right)}\right) \]

    2. expm1-undefine 32.7%

       \[\leadsto \left(a + b\right) \cdot \left(\left(b - a\right) \cdot \color{blue}{\left(e^{\mathsf{log1p}\left(\sin \left(\left(angle \cdot \pi\right) \cdot 0.011111111111111112\right)\right)} - 1\right)}\right) \]

    3. associate-*l* 30.9%

       \[\leadsto \left(a + b\right) \cdot \left(\left(b - a\right) \cdot \left(e^{\mathsf{log1p}\left(\sin \color{blue}{\left(angle \cdot \left(\pi \cdot 0.011111111111111112\right)\right)}\right)} - 1\right)\right) \]

12. Applied egg-rr 30.9%

    \[\leadsto \left(a + b\right) \cdot \left(\left(b - a\right) \cdot \color{blue}{\left(e^{\mathsf{log1p}\left(\sin \left(angle \cdot \left(\pi \cdot 0.011111111111111112\right)\right)\right)} - 1\right)}\right) \]

13. Taylor expanded in angle around 0 64.0%

    \[\leadsto \left(a + b\right) \cdot \color{blue}{\left(0.011111111111111112 \cdot \left(angle \cdot \left(\pi \cdot \left(b - a\right)\right)\right)\right)} \]
14. Step-by-step derivation

    1. *-commutative 64.0%

       \[\leadsto \left(a + b\right) \cdot \color{blue}{\left(\left(angle \cdot \left(\pi \cdot \left(b - a\right)\right)\right) \cdot 0.011111111111111112\right)} \]

    2. associate-*r* 64.0%

       \[\leadsto \left(a + b\right) \cdot \color{blue}{\left(angle \cdot \left(\left(\pi \cdot \left(b - a\right)\right) \cdot 0.011111111111111112\right)\right)} \]

    3. *-commutative 64.0%

       \[\leadsto \left(a + b\right) \cdot \left(angle \cdot \left(\color{blue}{\left(\left(b - a\right) \cdot \pi\right)} \cdot 0.011111111111111112\right)\right) \]

    4. associate-*l* 64.1%

       \[\leadsto \left(a + b\right) \cdot \left(angle \cdot \color{blue}{\left(\left(b - a\right) \cdot \left(\pi \cdot 0.011111111111111112\right)\right)}\right) \]

15. Simplified 64.1%

    \[\leadsto \left(a + b\right) \cdot \color{blue}{\left(angle \cdot \left(\left(b - a\right) \cdot \left(\pi \cdot 0.011111111111111112\right)\right)\right)} \]
16. Add Preprocessing

Alternative 17: 62.4% accurate, 32.2× speedup

Math

\[\begin{array}{l} a_m = \left|a\right| \\ b_m = \left|b\right| \\ angle\_m = \left|angle\right| \\ angle\_s = \mathsf{copysign}\left(1, angle\right) \\ angle\_s \cdot \left(\left(a\_m + b\_m\right) \cdot \left(0.011111111111111112 \cdot \left(angle\_m \cdot \left(\pi \cdot \left(b\_m - a\_m\right)\right)\right)\right)\right) \end{array} \]

FPCore

a_m = (fabs.f64 a)
b_m = (fabs.f64 b)
angle\_m = (fabs.f64 angle)
angle\_s = (copysign.f64 #s(literal 1 binary64) angle)
(FPCore (angle_s a_m b_m angle_m)
 :precision binary64
 (*
  angle_s
  (* (+ a_m b_m) (* 0.011111111111111112 (* angle_m (* PI (- b_m a_m)))))))

C

a_m = fabs(a);
b_m = fabs(b);
angle\_m = fabs(angle);
angle\_s = copysign(1.0, angle);
double code(double angle_s, double a_m, double b_m, double angle_m) {
	return angle_s * ((a_m + b_m) * (0.011111111111111112 * (angle_m * (((double) M_PI) * (b_m - a_m)))));
}

Java

a_m = Math.abs(a);
b_m = Math.abs(b);
angle\_m = Math.abs(angle);
angle\_s = Math.copySign(1.0, angle);
public static double code(double angle_s, double a_m, double b_m, double angle_m) {
	return angle_s * ((a_m + b_m) * (0.011111111111111112 * (angle_m * (Math.PI * (b_m - a_m)))));
}

Python

a_m = math.fabs(a)
b_m = math.fabs(b)
angle\_m = math.fabs(angle)
angle\_s = math.copysign(1.0, angle)
def code(angle_s, a_m, b_m, angle_m):
	return angle_s * ((a_m + b_m) * (0.011111111111111112 * (angle_m * (math.pi * (b_m - a_m)))))

Julia

a_m = abs(a)
b_m = abs(b)
angle\_m = abs(angle)
angle\_s = copysign(1.0, angle)
function code(angle_s, a_m, b_m, angle_m)
	return Float64(angle_s * Float64(Float64(a_m + b_m) * Float64(0.011111111111111112 * Float64(angle_m * Float64(pi * Float64(b_m - a_m))))))
end

MATLAB

a_m = abs(a);
b_m = abs(b);
angle\_m = abs(angle);
angle\_s = sign(angle) * abs(1.0);
function tmp = code(angle_s, a_m, b_m, angle_m)
	tmp = angle_s * ((a_m + b_m) * (0.011111111111111112 * (angle_m * (pi * (b_m - a_m)))));
end

Wolfram

a_m = N[Abs[a], $MachinePrecision]
b_m = N[Abs[b], $MachinePrecision]
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$95$m_, b$95$m_, angle$95$m_] := N[(angle$95$s * N[(N[(a$95$m + b$95$m), $MachinePrecision] * N[(0.011111111111111112 * N[(angle$95$m * N[(Pi * N[(b$95$m - a$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 56.3%

   \[\left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin \left(\pi \cdot \frac{angle}{180}\right)\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right) \]
2. Step-by-step derivation

   1. associate-*l* 56.3%

      \[\leadsto \color{blue}{\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \left(\sin \left(\pi \cdot \frac{angle}{180}\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right)\right)} \]

   2. *-commutative 56.3%

      \[\leadsto \color{blue}{\left(\left({b}^{2} - {a}^{2}\right) \cdot 2\right)} \cdot \left(\sin \left(\pi \cdot \frac{angle}{180}\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right)\right) \]

   3. associate-*l* 56.3%

      \[\leadsto \color{blue}{\left({b}^{2} - {a}^{2}\right) \cdot \left(2 \cdot \left(\sin \left(\pi \cdot \frac{angle}{180}\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right)\right)\right)} \]

3. Simplified 56.3%

   \[\leadsto \color{blue}{\left({b}^{2} - {a}^{2}\right) \cdot \left(2 \cdot \left(\sin \left(\pi \cdot \frac{angle}{180}\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right)\right)\right)} \]
4. Add Preprocessing
5. Step-by-step derivation

   1. unpow2 56.3%

      \[\leadsto \left(\left(2 \cdot \left(\color{blue}{b \cdot b} - {a}^{2}\right)\right) \cdot \sin \left(\pi \cdot \frac{angle}{180}\right)\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right) \]

   2. unpow2 56.3%

      \[\leadsto \left(\left(2 \cdot \left(b \cdot b - \color{blue}{a \cdot a}\right)\right) \cdot \sin \left(\pi \cdot \frac{angle}{180}\right)\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right) \]

   3. difference-of-squares 61.0%

      \[\leadsto \left(\left(2 \cdot \color{blue}{\left(\left(b + a\right) \cdot \left(b - a\right)\right)}\right) \cdot \sin \left(\pi \cdot \frac{angle}{180}\right)\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right) \]

6. Applied egg-rr 61.0%

   \[\leadsto \color{blue}{\left(\left(b + a\right) \cdot \left(b - a\right)\right)} \cdot \left(2 \cdot \left(\sin \left(\pi \cdot \frac{angle}{180}\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right)\right)\right) \]
7. Step-by-step derivation

   1. add-exp-log 35.1%

      \[\leadsto \color{blue}{e^{\log \left(\left(\left(b + a\right) \cdot \left(b - a\right)\right) \cdot \left(2 \cdot \left(\sin \left(\pi \cdot \frac{angle}{180}\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right)\right)\right)\right)}} \]

   2. 2-sin 35.1%

      \[\leadsto e^{\log \left(\left(\left(b + a\right) \cdot \left(b - a\right)\right) \cdot \color{blue}{\sin \left(2 \cdot \left(\pi \cdot \frac{angle}{180}\right)\right)}\right)} \]

   3. div-inv 34.9%

      \[\leadsto e^{\log \left(\left(\left(b + a\right) \cdot \left(b - a\right)\right) \cdot \sin \left(2 \cdot \left(\pi \cdot \color{blue}{\left(angle \cdot \frac{1}{180}\right)}\right)\right)\right)} \]

   4. metadata-eval 34.9%

      \[\leadsto e^{\log \left(\left(\left(b + a\right) \cdot \left(b - a\right)\right) \cdot \sin \left(2 \cdot \left(\pi \cdot \left(angle \cdot \color{blue}{0.005555555555555556}\right)\right)\right)\right)} \]

8. Applied egg-rr 34.9%

   \[\leadsto \color{blue}{e^{\log \left(\left(\left(b + a\right) \cdot \left(b - a\right)\right) \cdot \sin \left(2 \cdot \left(\pi \cdot \left(angle \cdot 0.005555555555555556\right)\right)\right)\right)}} \]
9. Step-by-step derivation

   1. rem-exp-log 59.6%

      \[\leadsto \color{blue}{\left(\left(b + a\right) \cdot \left(b - a\right)\right) \cdot \sin \left(2 \cdot \left(\pi \cdot \left(angle \cdot 0.005555555555555556\right)\right)\right)} \]

   2. associate-*l* 67.9%

      \[\leadsto \color{blue}{\left(b + a\right) \cdot \left(\left(b - a\right) \cdot \sin \left(2 \cdot \left(\pi \cdot \left(angle \cdot 0.005555555555555556\right)\right)\right)\right)} \]

   3. +-commutative 67.9%

      \[\leadsto \color{blue}{\left(a + b\right)} \cdot \left(\left(b - a\right) \cdot \sin \left(2 \cdot \left(\pi \cdot \left(angle \cdot 0.005555555555555556\right)\right)\right)\right) \]

   4. count-2 67.9%

      \[\leadsto \left(a + b\right) \cdot \left(\left(b - a\right) \cdot \sin \color{blue}{\left(\pi \cdot \left(angle \cdot 0.005555555555555556\right) + \pi \cdot \left(angle \cdot 0.005555555555555556\right)\right)}\right) \]

   5. add-cube-cbrt 66.8%

      \[\leadsto \left(a + b\right) \cdot \left(\left(b - a\right) \cdot \sin \left(\color{blue}{\left(\left(\sqrt[3]{\pi} \cdot \sqrt[3]{\pi}\right) \cdot \sqrt[3]{\pi}\right)} \cdot \left(angle \cdot 0.005555555555555556\right) + \pi \cdot \left(angle \cdot 0.005555555555555556\right)\right)\right) \]

   6. unpow2 66.8%

      \[\leadsto \left(a + b\right) \cdot \left(\left(b - a\right) \cdot \sin \left(\left(\color{blue}{{\left(\sqrt[3]{\pi}\right)}^{2}} \cdot \sqrt[3]{\pi}\right) \cdot \left(angle \cdot 0.005555555555555556\right) + \pi \cdot \left(angle \cdot 0.005555555555555556\right)\right)\right) \]

   7. metadata-eval 66.8%

      \[\leadsto \left(a + b\right) \cdot \left(\left(b - a\right) \cdot \sin \left(\left({\left(\sqrt[3]{\pi}\right)}^{2} \cdot \sqrt[3]{\pi}\right) \cdot \left(angle \cdot \color{blue}{\frac{1}{180}}\right) + \pi \cdot \left(angle \cdot 0.005555555555555556\right)\right)\right) \]

   8. div-inv 67.9%

      \[\leadsto \left(a + b\right) \cdot \left(\left(b - a\right) \cdot \sin \left(\left({\left(\sqrt[3]{\pi}\right)}^{2} \cdot \sqrt[3]{\pi}\right) \cdot \color{blue}{\frac{angle}{180}} + \pi \cdot \left(angle \cdot 0.005555555555555556\right)\right)\right) \]

   9. metadata-eval 67.9%

      \[\leadsto \left(a + b\right) \cdot \left(\left(b - a\right) \cdot \sin \left(\left({\left(\sqrt[3]{\pi}\right)}^{2} \cdot \sqrt[3]{\pi}\right) \cdot \frac{angle}{180} + \pi \cdot \left(angle \cdot \color{blue}{\frac{1}{180}}\right)\right)\right) \]

   10. div-inv 67.6%

       \[\leadsto \left(a + b\right) \cdot \left(\left(b - a\right) \cdot \sin \left(\left({\left(\sqrt[3]{\pi}\right)}^{2} \cdot \sqrt[3]{\pi}\right) \cdot \frac{angle}{180} + \pi \cdot \color{blue}{\frac{angle}{180}}\right)\right) \]

10. Applied egg-rr 68.9%

    \[\leadsto \color{blue}{\left(a + b\right) \cdot \left(\left(b - a\right) \cdot \sin \left(\left(angle \cdot \pi\right) \cdot 0.011111111111111112\right)\right)} \]

11. Taylor expanded in angle around 0 64.0%

    \[\leadsto \left(a + b\right) \cdot \color{blue}{\left(0.011111111111111112 \cdot \left(angle \cdot \left(\pi \cdot \left(b - a\right)\right)\right)\right)} \]
12. Add Preprocessing

Alternative 18: 54.4% accurate, 32.2× speedup

Math

\[\begin{array}{l} a_m = \left|a\right| \\ b_m = \left|b\right| \\ angle\_m = \left|angle\right| \\ angle\_s = \mathsf{copysign}\left(1, angle\right) \\ angle\_s \cdot \left(0.011111111111111112 \cdot \left(angle\_m \cdot \left(\pi \cdot \left(\left(a\_m + b\_m\right) \cdot \left(b\_m - a\_m\right)\right)\right)\right)\right) \end{array} \]

FPCore

a_m = (fabs.f64 a)
b_m = (fabs.f64 b)
angle\_m = (fabs.f64 angle)
angle\_s = (copysign.f64 #s(literal 1 binary64) angle)
(FPCore (angle_s a_m b_m angle_m)
 :precision binary64
 (*
  angle_s
  (* 0.011111111111111112 (* angle_m (* PI (* (+ a_m b_m) (- b_m a_m)))))))

C

a_m = fabs(a);
b_m = fabs(b);
angle\_m = fabs(angle);
angle\_s = copysign(1.0, angle);
double code(double angle_s, double a_m, double b_m, double angle_m) {
	return angle_s * (0.011111111111111112 * (angle_m * (((double) M_PI) * ((a_m + b_m) * (b_m - a_m)))));
}

Java

a_m = Math.abs(a);
b_m = Math.abs(b);
angle\_m = Math.abs(angle);
angle\_s = Math.copySign(1.0, angle);
public static double code(double angle_s, double a_m, double b_m, double angle_m) {
	return angle_s * (0.011111111111111112 * (angle_m * (Math.PI * ((a_m + b_m) * (b_m - a_m)))));
}

Python

a_m = math.fabs(a)
b_m = math.fabs(b)
angle\_m = math.fabs(angle)
angle\_s = math.copysign(1.0, angle)
def code(angle_s, a_m, b_m, angle_m):
	return angle_s * (0.011111111111111112 * (angle_m * (math.pi * ((a_m + b_m) * (b_m - a_m)))))

Julia

a_m = abs(a)
b_m = abs(b)
angle\_m = abs(angle)
angle\_s = copysign(1.0, angle)
function code(angle_s, a_m, b_m, angle_m)
	return Float64(angle_s * Float64(0.011111111111111112 * Float64(angle_m * Float64(pi * Float64(Float64(a_m + b_m) * Float64(b_m - a_m))))))
end

MATLAB

a_m = abs(a);
b_m = abs(b);
angle\_m = abs(angle);
angle\_s = sign(angle) * abs(1.0);
function tmp = code(angle_s, a_m, b_m, angle_m)
	tmp = angle_s * (0.011111111111111112 * (angle_m * (pi * ((a_m + b_m) * (b_m - a_m)))));
end

Wolfram

a_m = N[Abs[a], $MachinePrecision]
b_m = N[Abs[b], $MachinePrecision]
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$95$m_, b$95$m_, angle$95$m_] := N[(angle$95$s * N[(0.011111111111111112 * N[(angle$95$m * N[(Pi * N[(N[(a$95$m + b$95$m), $MachinePrecision] * N[(b$95$m - a$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 56.3%

   \[\left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin \left(\pi \cdot \frac{angle}{180}\right)\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right) \]
2. Step-by-step derivation

   1. associate-*l* 56.3%

      \[\leadsto \color{blue}{\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \left(\sin \left(\pi \cdot \frac{angle}{180}\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right)\right)} \]

   2. *-commutative 56.3%

      \[\leadsto \color{blue}{\left(\left({b}^{2} - {a}^{2}\right) \cdot 2\right)} \cdot \left(\sin \left(\pi \cdot \frac{angle}{180}\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right)\right) \]

   3. associate-*l* 56.3%

      \[\leadsto \color{blue}{\left({b}^{2} - {a}^{2}\right) \cdot \left(2 \cdot \left(\sin \left(\pi \cdot \frac{angle}{180}\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right)\right)\right)} \]

3. Simplified 56.3%

   \[\leadsto \color{blue}{\left({b}^{2} - {a}^{2}\right) \cdot \left(2 \cdot \left(\sin \left(\pi \cdot \frac{angle}{180}\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right)\right)\right)} \]
4. Add Preprocessing
5. Step-by-step derivation

   1. unpow2 56.3%

      \[\leadsto \left(\left(2 \cdot \left(\color{blue}{b \cdot b} - {a}^{2}\right)\right) \cdot \sin \left(\pi \cdot \frac{angle}{180}\right)\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right) \]

   2. unpow2 56.3%

      \[\leadsto \left(\left(2 \cdot \left(b \cdot b - \color{blue}{a \cdot a}\right)\right) \cdot \sin \left(\pi \cdot \frac{angle}{180}\right)\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right) \]

   3. difference-of-squares 61.0%

      \[\leadsto \left(\left(2 \cdot \color{blue}{\left(\left(b + a\right) \cdot \left(b - a\right)\right)}\right) \cdot \sin \left(\pi \cdot \frac{angle}{180}\right)\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right) \]

6. Applied egg-rr 61.0%

   \[\leadsto \color{blue}{\left(\left(b + a\right) \cdot \left(b - a\right)\right)} \cdot \left(2 \cdot \left(\sin \left(\pi \cdot \frac{angle}{180}\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right)\right)\right) \]

7. Taylor expanded in angle around 0 57.5%

   \[\leadsto \color{blue}{0.011111111111111112 \cdot \left(angle \cdot \left(\pi \cdot \left(\left(a + b\right) \cdot \left(b - a\right)\right)\right)\right)} \]
8. Add Preprocessing

Alternative 19: 21.4% accurate, 46.6× speedup

Math

\[\begin{array}{l} a_m = \left|a\right| \\ b_m = \left|b\right| \\ angle\_m = \left|angle\right| \\ angle\_s = \mathsf{copysign}\left(1, angle\right) \\ angle\_s \cdot \left(0.011111111111111112 \cdot \left(a\_m \cdot \left(angle\_m \cdot \left(b\_m \cdot \pi\right)\right)\right)\right) \end{array} \]

FPCore

a_m = (fabs.f64 a)
b_m = (fabs.f64 b)
angle\_m = (fabs.f64 angle)
angle\_s = (copysign.f64 #s(literal 1 binary64) angle)
(FPCore (angle_s a_m b_m angle_m)
 :precision binary64
 (* angle_s (* 0.011111111111111112 (* a_m (* angle_m (* b_m PI))))))

C

a_m = fabs(a);
b_m = fabs(b);
angle\_m = fabs(angle);
angle\_s = copysign(1.0, angle);
double code(double angle_s, double a_m, double b_m, double angle_m) {
	return angle_s * (0.011111111111111112 * (a_m * (angle_m * (b_m * ((double) M_PI)))));
}

Java

a_m = Math.abs(a);
b_m = Math.abs(b);
angle\_m = Math.abs(angle);
angle\_s = Math.copySign(1.0, angle);
public static double code(double angle_s, double a_m, double b_m, double angle_m) {
	return angle_s * (0.011111111111111112 * (a_m * (angle_m * (b_m * Math.PI))));
}

Python

a_m = math.fabs(a)
b_m = math.fabs(b)
angle\_m = math.fabs(angle)
angle\_s = math.copysign(1.0, angle)
def code(angle_s, a_m, b_m, angle_m):
	return angle_s * (0.011111111111111112 * (a_m * (angle_m * (b_m * math.pi))))

Julia

a_m = abs(a)
b_m = abs(b)
angle\_m = abs(angle)
angle\_s = copysign(1.0, angle)
function code(angle_s, a_m, b_m, angle_m)
	return Float64(angle_s * Float64(0.011111111111111112 * Float64(a_m * Float64(angle_m * Float64(b_m * pi)))))
end

MATLAB

a_m = abs(a);
b_m = abs(b);
angle\_m = abs(angle);
angle\_s = sign(angle) * abs(1.0);
function tmp = code(angle_s, a_m, b_m, angle_m)
	tmp = angle_s * (0.011111111111111112 * (a_m * (angle_m * (b_m * pi))));
end

Wolfram

a_m = N[Abs[a], $MachinePrecision]
b_m = N[Abs[b], $MachinePrecision]
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$95$m_, b$95$m_, angle$95$m_] := N[(angle$95$s * N[(0.011111111111111112 * N[(a$95$m * N[(angle$95$m * N[(b$95$m * Pi), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 56.3%

   \[\left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin \left(\pi \cdot \frac{angle}{180}\right)\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right) \]
2. Step-by-step derivation

   1. associate-*l* 56.3%

      \[\leadsto \color{blue}{\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \left(\sin \left(\pi \cdot \frac{angle}{180}\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right)\right)} \]

   2. *-commutative 56.3%

      \[\leadsto \color{blue}{\left(\left({b}^{2} - {a}^{2}\right) \cdot 2\right)} \cdot \left(\sin \left(\pi \cdot \frac{angle}{180}\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right)\right) \]

   3. associate-*l* 56.3%

      \[\leadsto \color{blue}{\left({b}^{2} - {a}^{2}\right) \cdot \left(2 \cdot \left(\sin \left(\pi \cdot \frac{angle}{180}\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right)\right)\right)} \]

3. Simplified 56.3%

   \[\leadsto \color{blue}{\left({b}^{2} - {a}^{2}\right) \cdot \left(2 \cdot \left(\sin \left(\pi \cdot \frac{angle}{180}\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right)\right)\right)} \]
4. Add Preprocessing
5. Step-by-step derivation

   1. unpow2 56.3%

      \[\leadsto \left(\left(2 \cdot \left(\color{blue}{b \cdot b} - {a}^{2}\right)\right) \cdot \sin \left(\pi \cdot \frac{angle}{180}\right)\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right) \]

   2. unpow2 56.3%

      \[\leadsto \left(\left(2 \cdot \left(b \cdot b - \color{blue}{a \cdot a}\right)\right) \cdot \sin \left(\pi \cdot \frac{angle}{180}\right)\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right) \]

   3. difference-of-squares 61.0%

      \[\leadsto \left(\left(2 \cdot \color{blue}{\left(\left(b + a\right) \cdot \left(b - a\right)\right)}\right) \cdot \sin \left(\pi \cdot \frac{angle}{180}\right)\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right) \]

6. Applied egg-rr 61.0%

   \[\leadsto \color{blue}{\left(\left(b + a\right) \cdot \left(b - a\right)\right)} \cdot \left(2 \cdot \left(\sin \left(\pi \cdot \frac{angle}{180}\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right)\right)\right) \]

7. Taylor expanded in angle around 0 57.5%

   \[\leadsto \color{blue}{0.011111111111111112 \cdot \left(angle \cdot \left(\pi \cdot \left(\left(a + b\right) \cdot \left(b - a\right)\right)\right)\right)} \]
8. Step-by-step derivation

   1. associate-*r* 57.5%

      \[\leadsto 0.011111111111111112 \cdot \left(angle \cdot \color{blue}{\left(\left(\pi \cdot \left(a + b\right)\right) \cdot \left(b - a\right)\right)}\right) \]

   2. sub-neg 57.5%

      \[\leadsto 0.011111111111111112 \cdot \left(angle \cdot \left(\left(\pi \cdot \left(a + b\right)\right) \cdot \color{blue}{\left(b + \left(-a\right)\right)}\right)\right) \]

   3. distribute-lft-in 54.3%

      \[\leadsto 0.011111111111111112 \cdot \left(angle \cdot \color{blue}{\left(\left(\pi \cdot \left(a + b\right)\right) \cdot b + \left(\pi \cdot \left(a + b\right)\right) \cdot \left(-a\right)\right)}\right) \]

9. Applied egg-rr 54.3%

   \[\leadsto 0.011111111111111112 \cdot \left(angle \cdot \color{blue}{\left(\left(\pi \cdot \left(a + b\right)\right) \cdot b + \left(\pi \cdot \left(a + b\right)\right) \cdot \left(-a\right)\right)}\right) \]
10. Step-by-step derivation

    1. distribute-lft-out 57.5%

       \[\leadsto 0.011111111111111112 \cdot \left(angle \cdot \color{blue}{\left(\left(\pi \cdot \left(a + b\right)\right) \cdot \left(b + \left(-a\right)\right)\right)}\right) \]

    2. +-commutative 57.5%

       \[\leadsto 0.011111111111111112 \cdot \left(angle \cdot \left(\left(\pi \cdot \color{blue}{\left(b + a\right)}\right) \cdot \left(b + \left(-a\right)\right)\right)\right) \]

    3. add-sqr-sqrt 26.2%

       \[\leadsto 0.011111111111111112 \cdot \left(angle \cdot \left(\left(\pi \cdot \left(b + a\right)\right) \cdot \left(b + \color{blue}{\sqrt{-a} \cdot \sqrt{-a}}\right)\right)\right) \]

    4. sqrt-unprod 46.6%

       \[\leadsto 0.011111111111111112 \cdot \left(angle \cdot \left(\left(\pi \cdot \left(b + a\right)\right) \cdot \left(b + \color{blue}{\sqrt{\left(-a\right) \cdot \left(-a\right)}}\right)\right)\right) \]

    5. sqr-neg 46.6%

       \[\leadsto 0.011111111111111112 \cdot \left(angle \cdot \left(\left(\pi \cdot \left(b + a\right)\right) \cdot \left(b + \sqrt{\color{blue}{a \cdot a}}\right)\right)\right) \]

    6. sqrt-prod 21.6%

       \[\leadsto 0.011111111111111112 \cdot \left(angle \cdot \left(\left(\pi \cdot \left(b + a\right)\right) \cdot \left(b + \color{blue}{\sqrt{a} \cdot \sqrt{a}}\right)\right)\right) \]

    7. add-sqr-sqrt 43.4%

       \[\leadsto 0.011111111111111112 \cdot \left(angle \cdot \left(\left(\pi \cdot \left(b + a\right)\right) \cdot \left(b + \color{blue}{a}\right)\right)\right) \]

11. Applied egg-rr 43.4%

    \[\leadsto 0.011111111111111112 \cdot \left(angle \cdot \color{blue}{\left(\left(\pi \cdot \left(b + a\right)\right) \cdot \left(b + a\right)\right)}\right) \]

12. Taylor expanded in b around 0 23.8%

    \[\leadsto 0.011111111111111112 \cdot \left(angle \cdot \left(\color{blue}{\left(a \cdot \pi\right)} \cdot \left(b + a\right)\right)\right) \]
13. Step-by-step derivation

    1. *-commutative 23.8%

       \[\leadsto 0.011111111111111112 \cdot \left(angle \cdot \left(\color{blue}{\left(\pi \cdot a\right)} \cdot \left(b + a\right)\right)\right) \]

14. Simplified 23.8%

    \[\leadsto 0.011111111111111112 \cdot \left(angle \cdot \left(\color{blue}{\left(\pi \cdot a\right)} \cdot \left(b + a\right)\right)\right) \]

15. Taylor expanded in a around 0 23.1%

    \[\leadsto \color{blue}{0.011111111111111112 \cdot \left(a \cdot \left(angle \cdot \left(b \cdot \pi\right)\right)\right)} \]
16. Step-by-step derivation

    1. *-commutative 23.1%

       \[\leadsto \color{blue}{\left(a \cdot \left(angle \cdot \left(b \cdot \pi\right)\right)\right) \cdot 0.011111111111111112} \]

    2. *-commutative 23.1%

       \[\leadsto \left(a \cdot \left(angle \cdot \color{blue}{\left(\pi \cdot b\right)}\right)\right) \cdot 0.011111111111111112 \]

17. Simplified 23.1%

    \[\leadsto \color{blue}{\left(a \cdot \left(angle \cdot \left(\pi \cdot b\right)\right)\right) \cdot 0.011111111111111112} \]

18. Final simplification 23.1%

    \[\leadsto 0.011111111111111112 \cdot \left(a \cdot \left(angle \cdot \left(b \cdot \pi\right)\right)\right) \]
19. Add Preprocessing

Alternative 20: 21.2% accurate, 46.6× speedup

Math

\[\begin{array}{l} a_m = \left|a\right| \\ b_m = \left|b\right| \\ angle\_m = \left|angle\right| \\ angle\_s = \mathsf{copysign}\left(1, angle\right) \\ angle\_s \cdot \left(\pi \cdot \left(b\_m \cdot \left(0.011111111111111112 \cdot \left(a\_m \cdot angle\_m\right)\right)\right)\right) \end{array} \]

FPCore

a_m = (fabs.f64 a)
b_m = (fabs.f64 b)
angle\_m = (fabs.f64 angle)
angle\_s = (copysign.f64 #s(literal 1 binary64) angle)
(FPCore (angle_s a_m b_m angle_m)
 :precision binary64
 (* angle_s (* PI (* b_m (* 0.011111111111111112 (* a_m angle_m))))))

C

a_m = fabs(a);
b_m = fabs(b);
angle\_m = fabs(angle);
angle\_s = copysign(1.0, angle);
double code(double angle_s, double a_m, double b_m, double angle_m) {
	return angle_s * (((double) M_PI) * (b_m * (0.011111111111111112 * (a_m * angle_m))));
}

Java

a_m = Math.abs(a);
b_m = Math.abs(b);
angle\_m = Math.abs(angle);
angle\_s = Math.copySign(1.0, angle);
public static double code(double angle_s, double a_m, double b_m, double angle_m) {
	return angle_s * (Math.PI * (b_m * (0.011111111111111112 * (a_m * angle_m))));
}

Python

a_m = math.fabs(a)
b_m = math.fabs(b)
angle\_m = math.fabs(angle)
angle\_s = math.copysign(1.0, angle)
def code(angle_s, a_m, b_m, angle_m):
	return angle_s * (math.pi * (b_m * (0.011111111111111112 * (a_m * angle_m))))

Julia

a_m = abs(a)
b_m = abs(b)
angle\_m = abs(angle)
angle\_s = copysign(1.0, angle)
function code(angle_s, a_m, b_m, angle_m)
	return Float64(angle_s * Float64(pi * Float64(b_m * Float64(0.011111111111111112 * Float64(a_m * angle_m)))))
end

MATLAB

a_m = abs(a);
b_m = abs(b);
angle\_m = abs(angle);
angle\_s = sign(angle) * abs(1.0);
function tmp = code(angle_s, a_m, b_m, angle_m)
	tmp = angle_s * (pi * (b_m * (0.011111111111111112 * (a_m * angle_m))));
end

Wolfram

a_m = N[Abs[a], $MachinePrecision]
b_m = N[Abs[b], $MachinePrecision]
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$95$m_, b$95$m_, angle$95$m_] := N[(angle$95$s * N[(Pi * N[(b$95$m * N[(0.011111111111111112 * N[(a$95$m * angle$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 56.3%

   \[\left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin \left(\pi \cdot \frac{angle}{180}\right)\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right) \]
2. Step-by-step derivation

   1. associate-*l* 56.3%

      \[\leadsto \color{blue}{\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \left(\sin \left(\pi \cdot \frac{angle}{180}\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right)\right)} \]

   2. *-commutative 56.3%

      \[\leadsto \color{blue}{\left(\left({b}^{2} - {a}^{2}\right) \cdot 2\right)} \cdot \left(\sin \left(\pi \cdot \frac{angle}{180}\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right)\right) \]

   3. associate-*l* 56.3%

      \[\leadsto \color{blue}{\left({b}^{2} - {a}^{2}\right) \cdot \left(2 \cdot \left(\sin \left(\pi \cdot \frac{angle}{180}\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right)\right)\right)} \]

3. Simplified 56.3%

   \[\leadsto \color{blue}{\left({b}^{2} - {a}^{2}\right) \cdot \left(2 \cdot \left(\sin \left(\pi \cdot \frac{angle}{180}\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right)\right)\right)} \]
4. Add Preprocessing
5. Step-by-step derivation

   1. unpow2 56.3%

      \[\leadsto \left(\left(2 \cdot \left(\color{blue}{b \cdot b} - {a}^{2}\right)\right) \cdot \sin \left(\pi \cdot \frac{angle}{180}\right)\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right) \]

   2. unpow2 56.3%

      \[\leadsto \left(\left(2 \cdot \left(b \cdot b - \color{blue}{a \cdot a}\right)\right) \cdot \sin \left(\pi \cdot \frac{angle}{180}\right)\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right) \]

   3. difference-of-squares 61.0%

      \[\leadsto \left(\left(2 \cdot \color{blue}{\left(\left(b + a\right) \cdot \left(b - a\right)\right)}\right) \cdot \sin \left(\pi \cdot \frac{angle}{180}\right)\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right) \]

6. Applied egg-rr 61.0%

   \[\leadsto \color{blue}{\left(\left(b + a\right) \cdot \left(b - a\right)\right)} \cdot \left(2 \cdot \left(\sin \left(\pi \cdot \frac{angle}{180}\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right)\right)\right) \]

7. Taylor expanded in angle around 0 57.5%

   \[\leadsto \color{blue}{0.011111111111111112 \cdot \left(angle \cdot \left(\pi \cdot \left(\left(a + b\right) \cdot \left(b - a\right)\right)\right)\right)} \]
8. Step-by-step derivation

   1. associate-*r* 57.5%

      \[\leadsto 0.011111111111111112 \cdot \left(angle \cdot \color{blue}{\left(\left(\pi \cdot \left(a + b\right)\right) \cdot \left(b - a\right)\right)}\right) \]

   2. sub-neg 57.5%

      \[\leadsto 0.011111111111111112 \cdot \left(angle \cdot \left(\left(\pi \cdot \left(a + b\right)\right) \cdot \color{blue}{\left(b + \left(-a\right)\right)}\right)\right) \]

   3. distribute-lft-in 54.3%

      \[\leadsto 0.011111111111111112 \cdot \left(angle \cdot \color{blue}{\left(\left(\pi \cdot \left(a + b\right)\right) \cdot b + \left(\pi \cdot \left(a + b\right)\right) \cdot \left(-a\right)\right)}\right) \]

9. Applied egg-rr 54.3%

   \[\leadsto 0.011111111111111112 \cdot \left(angle \cdot \color{blue}{\left(\left(\pi \cdot \left(a + b\right)\right) \cdot b + \left(\pi \cdot \left(a + b\right)\right) \cdot \left(-a\right)\right)}\right) \]
10. Step-by-step derivation

    1. distribute-lft-out 57.5%

       \[\leadsto 0.011111111111111112 \cdot \left(angle \cdot \color{blue}{\left(\left(\pi \cdot \left(a + b\right)\right) \cdot \left(b + \left(-a\right)\right)\right)}\right) \]

    2. +-commutative 57.5%

       \[\leadsto 0.011111111111111112 \cdot \left(angle \cdot \left(\left(\pi \cdot \color{blue}{\left(b + a\right)}\right) \cdot \left(b + \left(-a\right)\right)\right)\right) \]

    3. add-sqr-sqrt 26.2%

       \[\leadsto 0.011111111111111112 \cdot \left(angle \cdot \left(\left(\pi \cdot \left(b + a\right)\right) \cdot \left(b + \color{blue}{\sqrt{-a} \cdot \sqrt{-a}}\right)\right)\right) \]

    4. sqrt-unprod 46.6%

       \[\leadsto 0.011111111111111112 \cdot \left(angle \cdot \left(\left(\pi \cdot \left(b + a\right)\right) \cdot \left(b + \color{blue}{\sqrt{\left(-a\right) \cdot \left(-a\right)}}\right)\right)\right) \]

    5. sqr-neg 46.6%

       \[\leadsto 0.011111111111111112 \cdot \left(angle \cdot \left(\left(\pi \cdot \left(b + a\right)\right) \cdot \left(b + \sqrt{\color{blue}{a \cdot a}}\right)\right)\right) \]

    6. sqrt-prod 21.6%

       \[\leadsto 0.011111111111111112 \cdot \left(angle \cdot \left(\left(\pi \cdot \left(b + a\right)\right) \cdot \left(b + \color{blue}{\sqrt{a} \cdot \sqrt{a}}\right)\right)\right) \]

    7. add-sqr-sqrt 43.4%

       \[\leadsto 0.011111111111111112 \cdot \left(angle \cdot \left(\left(\pi \cdot \left(b + a\right)\right) \cdot \left(b + \color{blue}{a}\right)\right)\right) \]

11. Applied egg-rr 43.4%

    \[\leadsto 0.011111111111111112 \cdot \left(angle \cdot \color{blue}{\left(\left(\pi \cdot \left(b + a\right)\right) \cdot \left(b + a\right)\right)}\right) \]

12. Taylor expanded in b around 0 23.8%

    \[\leadsto 0.011111111111111112 \cdot \left(angle \cdot \left(\color{blue}{\left(a \cdot \pi\right)} \cdot \left(b + a\right)\right)\right) \]
13. Step-by-step derivation

    1. *-commutative 23.8%

       \[\leadsto 0.011111111111111112 \cdot \left(angle \cdot \left(\color{blue}{\left(\pi \cdot a\right)} \cdot \left(b + a\right)\right)\right) \]

14. Simplified 23.8%

    \[\leadsto 0.011111111111111112 \cdot \left(angle \cdot \left(\color{blue}{\left(\pi \cdot a\right)} \cdot \left(b + a\right)\right)\right) \]

15. Taylor expanded in a around 0 23.1%

    \[\leadsto \color{blue}{0.011111111111111112 \cdot \left(a \cdot \left(angle \cdot \left(b \cdot \pi\right)\right)\right)} \]
16. Step-by-step derivation

    1. *-commutative 23.1%

       \[\leadsto \color{blue}{\left(a \cdot \left(angle \cdot \left(b \cdot \pi\right)\right)\right) \cdot 0.011111111111111112} \]

    2. associate-*r* 23.1%

       \[\leadsto \left(a \cdot \color{blue}{\left(\left(angle \cdot b\right) \cdot \pi\right)}\right) \cdot 0.011111111111111112 \]

17. Simplified 23.1%

    \[\leadsto \color{blue}{\left(a \cdot \left(\left(angle \cdot b\right) \cdot \pi\right)\right) \cdot 0.011111111111111112} \]

18. Taylor expanded in a around 0 23.1%

    \[\leadsto \color{blue}{0.011111111111111112 \cdot \left(a \cdot \left(angle \cdot \left(b \cdot \pi\right)\right)\right)} \]
19. Step-by-step derivation

    1. associate-*r* 23.1%

       \[\leadsto 0.011111111111111112 \cdot \left(a \cdot \color{blue}{\left(\left(angle \cdot b\right) \cdot \pi\right)}\right) \]

    2. associate-*r* 23.1%

       \[\leadsto 0.011111111111111112 \cdot \color{blue}{\left(\left(a \cdot \left(angle \cdot b\right)\right) \cdot \pi\right)} \]

    3. associate-*l* 23.1%

       \[\leadsto \color{blue}{\left(0.011111111111111112 \cdot \left(a \cdot \left(angle \cdot b\right)\right)\right) \cdot \pi} \]

    4. *-commutative 23.1%

       \[\leadsto \color{blue}{\pi \cdot \left(0.011111111111111112 \cdot \left(a \cdot \left(angle \cdot b\right)\right)\right)} \]

    5. associate-*r* 22.4%

       \[\leadsto \pi \cdot \left(0.011111111111111112 \cdot \color{blue}{\left(\left(a \cdot angle\right) \cdot b\right)}\right) \]

    6. associate-*r* 22.0%

       \[\leadsto \pi \cdot \color{blue}{\left(\left(0.011111111111111112 \cdot \left(a \cdot angle\right)\right) \cdot b\right)} \]

    7. *-commutative 22.0%

       \[\leadsto \pi \cdot \left(\left(0.011111111111111112 \cdot \color{blue}{\left(angle \cdot a\right)}\right) \cdot b\right) \]

20. Simplified 22.0%

    \[\leadsto \color{blue}{\pi \cdot \left(\left(0.011111111111111112 \cdot \left(angle \cdot a\right)\right) \cdot b\right)} \]

21. Final simplification 22.0%

    \[\leadsto \pi \cdot \left(b \cdot \left(0.011111111111111112 \cdot \left(a \cdot angle\right)\right)\right) \]
22. Add Preprocessing

Reproduce

herbie shell --seed 2024107 
(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)))))