ab-angle->ABCF B

Percentage Accurate: 53.7% → 67.2%

Time: 46.9s

Alternatives: 19

Speedup: 5.5×

• 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: 53.7% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 67.2% accurate, 0.6× speedup

Math

\[\begin{array}{l} a = |a|\\ b = |b|\\ \\ \begin{array}{l} t_0 := \pi \cdot \left(angle \cdot \left(b - a\right)\right)\\ t_1 := \left(b - a\right) \cdot \left(b + a\right)\\ t_2 := t_1 \cdot \cos \left(angle \cdot \frac{\pi}{-180}\right)\\ t_3 := \sqrt[3]{angle \cdot \pi}\\ \mathbf{if}\;\frac{angle}{180} \leq -5 \cdot 10^{+282}:\\ \;\;\;\;2 \cdot \left(\sin \left(\frac{angle \cdot \pi}{-180}\right) \cdot t_2\right)\\ \mathbf{elif}\;\frac{angle}{180} \leq -5 \cdot 10^{+117}:\\ \;\;\;\;2 \cdot \left(t_2 \cdot \sin \left(\frac{{t_3}^{2}}{\frac{180}{t_3}}\right)\right)\\ \mathbf{elif}\;\frac{angle}{180} \leq -1 \cdot 10^{-115}:\\ \;\;\;\;2 \cdot \left(\sin \left(\pi \cdot \left(angle \cdot 0.005555555555555556\right)\right) \cdot \left(t_1 \cdot {\left(\sqrt[3]{\cos \left(\left(\sqrt[3]{\pi} \cdot {\left(\sqrt[3]{\pi}\right)}^{2}\right) \cdot \left(angle \cdot 0.005555555555555556\right)\right)}\right)}^{3}\right)\right)\\ \mathbf{elif}\;\frac{angle}{180} \leq 10^{-39}:\\ \;\;\;\;\left(b \cdot t_0 + a \cdot t_0\right) \cdot 0.011111111111111112\\ \mathbf{elif}\;\frac{angle}{180} \leq 10^{+87}:\\ \;\;\;\;2 \cdot \left(\sin \left({\left(\sqrt{angle \cdot \left(\pi \cdot 0.005555555555555556\right)}\right)}^{2}\right) \cdot \left(t_1 \cdot \cos \left(angle \cdot \frac{\sqrt[3]{\pi \cdot {\pi}^{2}}}{-180}\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \left(\sin \left(\frac{\pi}{\frac{180}{angle}}\right) \cdot \left(t_1 \cdot \sqrt{{\cos \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{2}}\right)\right)\\ \end{array} \end{array} \]

FPCore

NOTE: a should be positive before calling this function
NOTE: b should be positive before calling this function
(FPCore (a b angle)
 :precision binary64
 (let* ((t_0 (* PI (* angle (- b a))))
        (t_1 (* (- b a) (+ b a)))
        (t_2 (* t_1 (cos (* angle (/ PI -180.0)))))
        (t_3 (cbrt (* angle PI))))
   (if (<= (/ angle 180.0) -5e+282)
     (* 2.0 (* (sin (/ (* angle PI) -180.0)) t_2))
     (if (<= (/ angle 180.0) -5e+117)
       (* 2.0 (* t_2 (sin (/ (pow t_3 2.0) (/ 180.0 t_3)))))
       (if (<= (/ angle 180.0) -1e-115)
         (*
          2.0
          (*
           (sin (* PI (* angle 0.005555555555555556)))
           (*
            t_1
            (pow
             (cbrt
              (cos
               (*
                (* (cbrt PI) (pow (cbrt PI) 2.0))
                (* angle 0.005555555555555556))))
             3.0))))
         (if (<= (/ angle 180.0) 1e-39)
           (* (+ (* b t_0) (* a t_0)) 0.011111111111111112)
           (if (<= (/ angle 180.0) 1e+87)
             (*
              2.0
              (*
               (sin (pow (sqrt (* angle (* PI 0.005555555555555556))) 2.0))
               (* t_1 (cos (* angle (/ (cbrt (* PI (pow PI 2.0))) -180.0))))))
             (*
              2.0
              (*
               (sin (/ PI (/ 180.0 angle)))
               (*
                t_1
                (sqrt
                 (pow
                  (cos (* 0.005555555555555556 (* angle PI)))
                  2.0))))))))))))

C

a = abs(a);
b = abs(b);
double code(double a, double b, double angle) {
	double t_0 = ((double) M_PI) * (angle * (b - a));
	double t_1 = (b - a) * (b + a);
	double t_2 = t_1 * cos((angle * (((double) M_PI) / -180.0)));
	double t_3 = cbrt((angle * ((double) M_PI)));
	double tmp;
	if ((angle / 180.0) <= -5e+282) {
		tmp = 2.0 * (sin(((angle * ((double) M_PI)) / -180.0)) * t_2);
	} else if ((angle / 180.0) <= -5e+117) {
		tmp = 2.0 * (t_2 * sin((pow(t_3, 2.0) / (180.0 / t_3))));
	} else if ((angle / 180.0) <= -1e-115) {
		tmp = 2.0 * (sin((((double) M_PI) * (angle * 0.005555555555555556))) * (t_1 * pow(cbrt(cos(((cbrt(((double) M_PI)) * pow(cbrt(((double) M_PI)), 2.0)) * (angle * 0.005555555555555556)))), 3.0)));
	} else if ((angle / 180.0) <= 1e-39) {
		tmp = ((b * t_0) + (a * t_0)) * 0.011111111111111112;
	} else if ((angle / 180.0) <= 1e+87) {
		tmp = 2.0 * (sin(pow(sqrt((angle * (((double) M_PI) * 0.005555555555555556))), 2.0)) * (t_1 * cos((angle * (cbrt((((double) M_PI) * pow(((double) M_PI), 2.0))) / -180.0)))));
	} else {
		tmp = 2.0 * (sin((((double) M_PI) / (180.0 / angle))) * (t_1 * sqrt(pow(cos((0.005555555555555556 * (angle * ((double) M_PI)))), 2.0))));
	}
	return tmp;
}

Java

a = Math.abs(a);
b = Math.abs(b);
public static double code(double a, double b, double angle) {
	double t_0 = Math.PI * (angle * (b - a));
	double t_1 = (b - a) * (b + a);
	double t_2 = t_1 * Math.cos((angle * (Math.PI / -180.0)));
	double t_3 = Math.cbrt((angle * Math.PI));
	double tmp;
	if ((angle / 180.0) <= -5e+282) {
		tmp = 2.0 * (Math.sin(((angle * Math.PI) / -180.0)) * t_2);
	} else if ((angle / 180.0) <= -5e+117) {
		tmp = 2.0 * (t_2 * Math.sin((Math.pow(t_3, 2.0) / (180.0 / t_3))));
	} else if ((angle / 180.0) <= -1e-115) {
		tmp = 2.0 * (Math.sin((Math.PI * (angle * 0.005555555555555556))) * (t_1 * Math.pow(Math.cbrt(Math.cos(((Math.cbrt(Math.PI) * Math.pow(Math.cbrt(Math.PI), 2.0)) * (angle * 0.005555555555555556)))), 3.0)));
	} else if ((angle / 180.0) <= 1e-39) {
		tmp = ((b * t_0) + (a * t_0)) * 0.011111111111111112;
	} else if ((angle / 180.0) <= 1e+87) {
		tmp = 2.0 * (Math.sin(Math.pow(Math.sqrt((angle * (Math.PI * 0.005555555555555556))), 2.0)) * (t_1 * Math.cos((angle * (Math.cbrt((Math.PI * Math.pow(Math.PI, 2.0))) / -180.0)))));
	} else {
		tmp = 2.0 * (Math.sin((Math.PI / (180.0 / angle))) * (t_1 * Math.sqrt(Math.pow(Math.cos((0.005555555555555556 * (angle * Math.PI))), 2.0))));
	}
	return tmp;
}

Julia

a = abs(a)
b = abs(b)
function code(a, b, angle)
	t_0 = Float64(pi * Float64(angle * Float64(b - a)))
	t_1 = Float64(Float64(b - a) * Float64(b + a))
	t_2 = Float64(t_1 * cos(Float64(angle * Float64(pi / -180.0))))
	t_3 = cbrt(Float64(angle * pi))
	tmp = 0.0
	if (Float64(angle / 180.0) <= -5e+282)
		tmp = Float64(2.0 * Float64(sin(Float64(Float64(angle * pi) / -180.0)) * t_2));
	elseif (Float64(angle / 180.0) <= -5e+117)
		tmp = Float64(2.0 * Float64(t_2 * sin(Float64((t_3 ^ 2.0) / Float64(180.0 / t_3)))));
	elseif (Float64(angle / 180.0) <= -1e-115)
		tmp = Float64(2.0 * Float64(sin(Float64(pi * Float64(angle * 0.005555555555555556))) * Float64(t_1 * (cbrt(cos(Float64(Float64(cbrt(pi) * (cbrt(pi) ^ 2.0)) * Float64(angle * 0.005555555555555556)))) ^ 3.0))));
	elseif (Float64(angle / 180.0) <= 1e-39)
		tmp = Float64(Float64(Float64(b * t_0) + Float64(a * t_0)) * 0.011111111111111112);
	elseif (Float64(angle / 180.0) <= 1e+87)
		tmp = Float64(2.0 * Float64(sin((sqrt(Float64(angle * Float64(pi * 0.005555555555555556))) ^ 2.0)) * Float64(t_1 * cos(Float64(angle * Float64(cbrt(Float64(pi * (pi ^ 2.0))) / -180.0))))));
	else
		tmp = Float64(2.0 * Float64(sin(Float64(pi / Float64(180.0 / angle))) * Float64(t_1 * sqrt((cos(Float64(0.005555555555555556 * Float64(angle * pi))) ^ 2.0)))));
	end
	return tmp
end

Wolfram

NOTE: a should be positive before calling this function
NOTE: b should be positive before calling this function
code[a_, b_, angle_] := Block[{t$95$0 = N[(Pi * N[(angle * N[(b - a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[(b - a), $MachinePrecision] * N[(b + a), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(t$95$1 * N[Cos[N[(angle * N[(Pi / -180.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$3 = N[Power[N[(angle * Pi), $MachinePrecision], 1/3], $MachinePrecision]}, If[LessEqual[N[(angle / 180.0), $MachinePrecision], -5e+282], N[(2.0 * N[(N[Sin[N[(N[(angle * Pi), $MachinePrecision] / -180.0), $MachinePrecision]], $MachinePrecision] * t$95$2), $MachinePrecision]), $MachinePrecision], If[LessEqual[N[(angle / 180.0), $MachinePrecision], -5e+117], N[(2.0 * N[(t$95$2 * N[Sin[N[(N[Power[t$95$3, 2.0], $MachinePrecision] / N[(180.0 / t$95$3), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[N[(angle / 180.0), $MachinePrecision], -1e-115], N[(2.0 * N[(N[Sin[N[(Pi * N[(angle * 0.005555555555555556), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * N[(t$95$1 * N[Power[N[Power[N[Cos[N[(N[(N[Power[Pi, 1/3], $MachinePrecision] * N[Power[N[Power[Pi, 1/3], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] * N[(angle * 0.005555555555555556), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], 1/3], $MachinePrecision], 3.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[N[(angle / 180.0), $MachinePrecision], 1e-39], N[(N[(N[(b * t$95$0), $MachinePrecision] + N[(a * t$95$0), $MachinePrecision]), $MachinePrecision] * 0.011111111111111112), $MachinePrecision], If[LessEqual[N[(angle / 180.0), $MachinePrecision], 1e+87], N[(2.0 * N[(N[Sin[N[Power[N[Sqrt[N[(angle * N[(Pi * 0.005555555555555556), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], 2.0], $MachinePrecision]], $MachinePrecision] * N[(t$95$1 * N[Cos[N[(angle * N[(N[Power[N[(Pi * N[Power[Pi, 2.0], $MachinePrecision]), $MachinePrecision], 1/3], $MachinePrecision] / -180.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(2.0 * N[(N[Sin[N[(Pi / N[(180.0 / angle), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * N[(t$95$1 * N[Sqrt[N[Power[N[Cos[N[(0.005555555555555556 * N[(angle * Pi), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], 2.0], $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]]]]]]

Derivation

1. Split input into 6 regimes

2. if (/.f64 angle 180) < -4.99999999999999978e282

   1. Initial program 40.1%

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

   3. Simplified 40.1%

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

      1. unpow2 40.1%

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

      2. unpow2 40.1%

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

      3. difference-of-squares 40.1%

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

   5. Applied egg-rr 40.1%

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

      1. *-commutative 40.1%

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

      2. div-inv 40.1%

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

      3. metadata-eval 40.1%

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

      4. *-commutative 40.1%

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

      5. associate-*r* 40.1%

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

      6. add-sqr-sqrt 0.0%

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

      7. sqrt-unprod 0.0%

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

      8. associate-*r* 0.0%

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

      9. associate-*r* 0.0%

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

      10. *-commutative 0.0%

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

      11. metadata-eval 0.0%

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

      12. div-inv 0.0%

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

      13. *-commutative 0.0%

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

      14. metadata-eval 0.0%

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

      15. div-inv 0.0%

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

      16. frac-times 0.0%

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

      17. metadata-eval 0.0%

          \[\leadsto 2 \cdot \left(\sin \left(\sqrt{\frac{\left(angle \cdot \pi\right) \cdot \left(angle \cdot \pi\right)}{\color{blue}{32400}}}\right) \cdot \left(\left(\left(b + a\right) \cdot \left(b - a\right)\right) \cdot \cos \left(\frac{\pi}{-180} \cdot angle\right)\right)\right) \]

      18. metadata-eval 0.0%

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

      19. frac-times 0.0%

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

      20. *-commutative 0.0%

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

      21. associate-*l/ 0.0%

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

   7. Applied egg-rr 83.7%

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

   if -4.99999999999999978e282 < (/.f64 angle 180) < -4.99999999999999983e117

   1. Initial program 22.2%

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

   3. Simplified 22.5%

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

      1. unpow2 22.5%

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

      2. unpow2 22.5%

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

      3. difference-of-squares 29.6%

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

   5. Applied egg-rr 29.6%

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

      1. associate-*r/ 25.4%

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

      2. *-commutative 25.4%

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

      3. add-cube-cbrt 31.0%

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

      4. associate-/l* 41.3%

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

      5. pow2 41.3%

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

      6. *-commutative 41.3%

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

      7. *-commutative 41.3%

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

   7. Applied egg-rr 41.3%

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

   if -4.99999999999999983e117 < (/.f64 angle 180) < -1.0000000000000001e-115

   1. Initial program 61.7%

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

   3. Simplified 60.0%

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

      1. unpow2 60.0%

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

      2. unpow2 60.0%

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

      3. difference-of-squares 73.7%

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

   5. Applied egg-rr 73.7%

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

      1. *-commutative 73.7%

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

      2. add-sqr-sqrt 73.1%

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

      3. sqrt-unprod 73.7%

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

      4. *-commutative 73.7%

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

      5. *-commutative 73.7%

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

      6. associate-*l/ 73.0%

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

      7. associate-*l/ 73.1%

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

      8. frac-times 75.0%

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

      9. metadata-eval 75.0%

         \[\leadsto 2 \cdot \left(\sin \left(\pi \cdot \frac{angle}{180}\right) \cdot \left(\left(\left(b + a\right) \cdot \left(b - a\right)\right) \cdot \cos \left(\sqrt{\frac{\left(\pi \cdot angle\right) \cdot \left(\pi \cdot angle\right)}{\color{blue}{32400}}}\right)\right)\right) \]

      10. metadata-eval 75.0%

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

      11. frac-times 73.1%

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

      12. associate-*r/ 75.1%

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

      13. associate-*r/ 75.4%

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

      14. sqrt-unprod 0.0%

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

      15. add-sqr-sqrt 75.4%

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

      16. add-cube-cbrt 75.4%

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

   7. Applied egg-rr 75.3%

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

      1. add-cube-cbrt 75.3%

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

      2. pow2 75.3%

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

   9. Applied egg-rr 75.3%

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

   10. Taylor expanded in angle around inf 75.7%

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

       1. *-commutative 75.7%

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

       2. *-commutative 75.7%

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

       3. associate-*r* 79.3%

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

       4. *-commutative 79.3%

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

   12. Simplified 79.3%

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

   if -1.0000000000000001e-115 < (/.f64 angle 180) < 9.99999999999999929e-40

   1. Initial program 62.9%

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

   3. Simplified 63.0%

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

   4. Taylor expanded in angle around 0 63.0%

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

      1. unpow2 62.9%

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

      2. unpow2 62.9%

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

      3. difference-of-squares 68.0%

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

   6. Applied egg-rr 68.1%

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

   7. Taylor expanded in angle around 0 68.1%

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

      1. *-commutative 68.1%

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

      2. associate-*r* 68.1%

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

      3. +-commutative 68.1%

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

      4. *-commutative 68.1%

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

      5. +-commutative 68.1%

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

   9. Simplified 68.1%

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

       1. associate-*r* 99.4%

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

       2. +-commutative 99.4%

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

       3. distribute-lft-in 97.0%

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

       4. *-commutative 97.0%

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

       5. associate-*l* 96.9%

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

       6. *-commutative 96.9%

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

       7. associate-*l* 97.0%

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

   11. Applied egg-rr 97.0%

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

   if 9.99999999999999929e-40 < (/.f64 angle 180) < 9.9999999999999996e86

   1. Initial program 48.0%

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

   3. Simplified 47.8%

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

      1. unpow2 47.8%

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

      2. unpow2 47.8%

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

      3. difference-of-squares 62.6%

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

   5. Applied egg-rr 62.6%

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

      1. *-commutative 62.6%

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

      2. div-inv 55.2%

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

      3. metadata-eval 55.2%

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

      4. *-commutative 55.2%

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

      5. associate-*r* 58.9%

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

      6. add-sqr-sqrt 50.9%

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

      7. sqrt-unprod 58.9%

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

      8. associate-*r* 55.2%

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

      9. associate-*r* 58.9%

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

      10. *-commutative 58.9%

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

      11. metadata-eval 58.9%

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

      12. div-inv 58.9%

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

      13. *-commutative 58.9%

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

      14. metadata-eval 58.9%

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

      15. div-inv 58.4%

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

      16. frac-times 62.7%

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

      17. metadata-eval 62.7%

          \[\leadsto 2 \cdot \left(\sin \left(\sqrt{\frac{\left(angle \cdot \pi\right) \cdot \left(angle \cdot \pi\right)}{\color{blue}{32400}}}\right) \cdot \left(\left(\left(b + a\right) \cdot \left(b - a\right)\right) \cdot \cos \left(\frac{\pi}{-180} \cdot angle\right)\right)\right) \]

      18. metadata-eval 62.7%

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

      19. frac-times 58.4%

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

      20. *-commutative 58.4%

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

      21. associate-*l/ 58.3%

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

   7. Applied egg-rr 50.9%

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

      1. add-cbrt-cube 66.2%

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

      2. *-commutative 66.2%

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

      3. pow2 66.2%

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

   9. Applied egg-rr 66.2%

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

   if 9.9999999999999996e86 < (/.f64 angle 180)

   1. Initial program 23.1%

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

   3. Simplified 26.1%

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

      1. unpow2 26.1%

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

      2. unpow2 26.1%

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

      3. difference-of-squares 32.2%

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

   5. Applied egg-rr 32.2%

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

      1. clear-num 28.2%

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

      2. un-div-inv 28.2%

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

   7. Applied egg-rr 28.2%

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

   9. Applied egg-rr 44.6%

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

       1. associate-*r* 44.8%

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

       2. *-commutative 44.8%

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

       3. *-commutative 44.8%

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

   11. Simplified 44.8%

       \[\leadsto 2 \cdot \left(\sin \left(\frac{\pi}{\frac{180}{angle}}\right) \cdot \left(\left(\left(b + a\right) \cdot \left(b - a\right)\right) \cdot \color{blue}{\sqrt{{\cos \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{2}}}\right)\right) \]
3. Recombined 6 regimes into one program.

4. Final simplification 70.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{angle}{180} \leq -5 \cdot 10^{+282}:\\ \;\;\;\;2 \cdot \left(\sin \left(\frac{angle \cdot \pi}{-180}\right) \cdot \left(\left(\left(b - a\right) \cdot \left(b + a\right)\right) \cdot \cos \left(angle \cdot \frac{\pi}{-180}\right)\right)\right)\\ \mathbf{elif}\;\frac{angle}{180} \leq -5 \cdot 10^{+117}:\\ \;\;\;\;2 \cdot \left(\left(\left(\left(b - a\right) \cdot \left(b + a\right)\right) \cdot \cos \left(angle \cdot \frac{\pi}{-180}\right)\right) \cdot \sin \left(\frac{{\left(\sqrt[3]{angle \cdot \pi}\right)}^{2}}{\frac{180}{\sqrt[3]{angle \cdot \pi}}}\right)\right)\\ \mathbf{elif}\;\frac{angle}{180} \leq -1 \cdot 10^{-115}:\\ \;\;\;\;2 \cdot \left(\sin \left(\pi \cdot \left(angle \cdot 0.005555555555555556\right)\right) \cdot \left(\left(\left(b - a\right) \cdot \left(b + a\right)\right) \cdot {\left(\sqrt[3]{\cos \left(\left(\sqrt[3]{\pi} \cdot {\left(\sqrt[3]{\pi}\right)}^{2}\right) \cdot \left(angle \cdot 0.005555555555555556\right)\right)}\right)}^{3}\right)\right)\\ \mathbf{elif}\;\frac{angle}{180} \leq 10^{-39}:\\ \;\;\;\;\left(b \cdot \left(\pi \cdot \left(angle \cdot \left(b - a\right)\right)\right) + a \cdot \left(\pi \cdot \left(angle \cdot \left(b - a\right)\right)\right)\right) \cdot 0.011111111111111112\\ \mathbf{elif}\;\frac{angle}{180} \leq 10^{+87}:\\ \;\;\;\;2 \cdot \left(\sin \left({\left(\sqrt{angle \cdot \left(\pi \cdot 0.005555555555555556\right)}\right)}^{2}\right) \cdot \left(\left(\left(b - a\right) \cdot \left(b + a\right)\right) \cdot \cos \left(angle \cdot \frac{\sqrt[3]{\pi \cdot {\pi}^{2}}}{-180}\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \left(\sin \left(\frac{\pi}{\frac{180}{angle}}\right) \cdot \left(\left(\left(b - a\right) \cdot \left(b + a\right)\right) \cdot \sqrt{{\cos \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{2}}\right)\right)\\ \end{array} \]

Alternative 2: 67.4% accurate, 0.7× speedup

Math

\[\begin{array}{l} a = |a|\\ b = |b|\\ \\ \begin{array}{l} t_0 := \left(b - a\right) \cdot \left(b + a\right)\\ t_1 := t_0 \cdot \cos \left(angle \cdot \frac{\pi}{-180}\right)\\ t_2 := \sqrt[3]{angle \cdot \pi}\\ t_3 := \sin \left(\frac{\pi}{\frac{180}{angle}}\right)\\ t_4 := \pi \cdot \left(angle \cdot \left(b - a\right)\right)\\ \mathbf{if}\;\frac{angle}{180} \leq -5 \cdot 10^{+282}:\\ \;\;\;\;2 \cdot \left(\sin \left(\frac{angle \cdot \pi}{-180}\right) \cdot t_1\right)\\ \mathbf{elif}\;\frac{angle}{180} \leq -2 \cdot 10^{+133}:\\ \;\;\;\;2 \cdot \left(t_1 \cdot \sin \left(\frac{{t_2}^{2}}{\frac{180}{t_2}}\right)\right)\\ \mathbf{elif}\;\frac{angle}{180} \leq -5 \cdot 10^{-83}:\\ \;\;\;\;2 \cdot \left(t_3 \cdot \left(t_0 \cdot \cos \left(angle \cdot \frac{\sqrt[3]{\pi} \cdot {\left(\sqrt[3]{\pi}\right)}^{2}}{-180}\right)\right)\right)\\ \mathbf{elif}\;\frac{angle}{180} \leq 10^{-39}:\\ \;\;\;\;\left(b \cdot t_4 + a \cdot t_4\right) \cdot 0.011111111111111112\\ \mathbf{elif}\;\frac{angle}{180} \leq 10^{+87}:\\ \;\;\;\;2 \cdot \left(\sin \left({\left(\sqrt{angle \cdot \left(\pi \cdot 0.005555555555555556\right)}\right)}^{2}\right) \cdot \left(t_0 \cdot \cos \left(angle \cdot \frac{\sqrt[3]{\pi \cdot {\pi}^{2}}}{-180}\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \left(t_3 \cdot \left(t_0 \cdot \sqrt{{\cos \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{2}}\right)\right)\\ \end{array} \end{array} \]

FPCore

NOTE: a should be positive before calling this function
NOTE: b should be positive before calling this function
(FPCore (a b angle)
 :precision binary64
 (let* ((t_0 (* (- b a) (+ b a)))
        (t_1 (* t_0 (cos (* angle (/ PI -180.0)))))
        (t_2 (cbrt (* angle PI)))
        (t_3 (sin (/ PI (/ 180.0 angle))))
        (t_4 (* PI (* angle (- b a)))))
   (if (<= (/ angle 180.0) -5e+282)
     (* 2.0 (* (sin (/ (* angle PI) -180.0)) t_1))
     (if (<= (/ angle 180.0) -2e+133)
       (* 2.0 (* t_1 (sin (/ (pow t_2 2.0) (/ 180.0 t_2)))))
       (if (<= (/ angle 180.0) -5e-83)
         (*
          2.0
          (*
           t_3
           (*
            t_0
            (cos (* angle (/ (* (cbrt PI) (pow (cbrt PI) 2.0)) -180.0))))))
         (if (<= (/ angle 180.0) 1e-39)
           (* (+ (* b t_4) (* a t_4)) 0.011111111111111112)
           (if (<= (/ angle 180.0) 1e+87)
             (*
              2.0
              (*
               (sin (pow (sqrt (* angle (* PI 0.005555555555555556))) 2.0))
               (* t_0 (cos (* angle (/ (cbrt (* PI (pow PI 2.0))) -180.0))))))
             (*
              2.0
              (*
               t_3
               (*
                t_0
                (sqrt
                 (pow
                  (cos (* 0.005555555555555556 (* angle PI)))
                  2.0))))))))))))

C

a = abs(a);
b = abs(b);
double code(double a, double b, double angle) {
	double t_0 = (b - a) * (b + a);
	double t_1 = t_0 * cos((angle * (((double) M_PI) / -180.0)));
	double t_2 = cbrt((angle * ((double) M_PI)));
	double t_3 = sin((((double) M_PI) / (180.0 / angle)));
	double t_4 = ((double) M_PI) * (angle * (b - a));
	double tmp;
	if ((angle / 180.0) <= -5e+282) {
		tmp = 2.0 * (sin(((angle * ((double) M_PI)) / -180.0)) * t_1);
	} else if ((angle / 180.0) <= -2e+133) {
		tmp = 2.0 * (t_1 * sin((pow(t_2, 2.0) / (180.0 / t_2))));
	} else if ((angle / 180.0) <= -5e-83) {
		tmp = 2.0 * (t_3 * (t_0 * cos((angle * ((cbrt(((double) M_PI)) * pow(cbrt(((double) M_PI)), 2.0)) / -180.0)))));
	} else if ((angle / 180.0) <= 1e-39) {
		tmp = ((b * t_4) + (a * t_4)) * 0.011111111111111112;
	} else if ((angle / 180.0) <= 1e+87) {
		tmp = 2.0 * (sin(pow(sqrt((angle * (((double) M_PI) * 0.005555555555555556))), 2.0)) * (t_0 * cos((angle * (cbrt((((double) M_PI) * pow(((double) M_PI), 2.0))) / -180.0)))));
	} else {
		tmp = 2.0 * (t_3 * (t_0 * sqrt(pow(cos((0.005555555555555556 * (angle * ((double) M_PI)))), 2.0))));
	}
	return tmp;
}

Java

a = Math.abs(a);
b = Math.abs(b);
public static double code(double a, double b, double angle) {
	double t_0 = (b - a) * (b + a);
	double t_1 = t_0 * Math.cos((angle * (Math.PI / -180.0)));
	double t_2 = Math.cbrt((angle * Math.PI));
	double t_3 = Math.sin((Math.PI / (180.0 / angle)));
	double t_4 = Math.PI * (angle * (b - a));
	double tmp;
	if ((angle / 180.0) <= -5e+282) {
		tmp = 2.0 * (Math.sin(((angle * Math.PI) / -180.0)) * t_1);
	} else if ((angle / 180.0) <= -2e+133) {
		tmp = 2.0 * (t_1 * Math.sin((Math.pow(t_2, 2.0) / (180.0 / t_2))));
	} else if ((angle / 180.0) <= -5e-83) {
		tmp = 2.0 * (t_3 * (t_0 * Math.cos((angle * ((Math.cbrt(Math.PI) * Math.pow(Math.cbrt(Math.PI), 2.0)) / -180.0)))));
	} else if ((angle / 180.0) <= 1e-39) {
		tmp = ((b * t_4) + (a * t_4)) * 0.011111111111111112;
	} else if ((angle / 180.0) <= 1e+87) {
		tmp = 2.0 * (Math.sin(Math.pow(Math.sqrt((angle * (Math.PI * 0.005555555555555556))), 2.0)) * (t_0 * Math.cos((angle * (Math.cbrt((Math.PI * Math.pow(Math.PI, 2.0))) / -180.0)))));
	} else {
		tmp = 2.0 * (t_3 * (t_0 * Math.sqrt(Math.pow(Math.cos((0.005555555555555556 * (angle * Math.PI))), 2.0))));
	}
	return tmp;
}

Julia

a = abs(a)
b = abs(b)
function code(a, b, angle)
	t_0 = Float64(Float64(b - a) * Float64(b + a))
	t_1 = Float64(t_0 * cos(Float64(angle * Float64(pi / -180.0))))
	t_2 = cbrt(Float64(angle * pi))
	t_3 = sin(Float64(pi / Float64(180.0 / angle)))
	t_4 = Float64(pi * Float64(angle * Float64(b - a)))
	tmp = 0.0
	if (Float64(angle / 180.0) <= -5e+282)
		tmp = Float64(2.0 * Float64(sin(Float64(Float64(angle * pi) / -180.0)) * t_1));
	elseif (Float64(angle / 180.0) <= -2e+133)
		tmp = Float64(2.0 * Float64(t_1 * sin(Float64((t_2 ^ 2.0) / Float64(180.0 / t_2)))));
	elseif (Float64(angle / 180.0) <= -5e-83)
		tmp = Float64(2.0 * Float64(t_3 * Float64(t_0 * cos(Float64(angle * Float64(Float64(cbrt(pi) * (cbrt(pi) ^ 2.0)) / -180.0))))));
	elseif (Float64(angle / 180.0) <= 1e-39)
		tmp = Float64(Float64(Float64(b * t_4) + Float64(a * t_4)) * 0.011111111111111112);
	elseif (Float64(angle / 180.0) <= 1e+87)
		tmp = Float64(2.0 * Float64(sin((sqrt(Float64(angle * Float64(pi * 0.005555555555555556))) ^ 2.0)) * Float64(t_0 * cos(Float64(angle * Float64(cbrt(Float64(pi * (pi ^ 2.0))) / -180.0))))));
	else
		tmp = Float64(2.0 * Float64(t_3 * Float64(t_0 * sqrt((cos(Float64(0.005555555555555556 * Float64(angle * pi))) ^ 2.0)))));
	end
	return tmp
end

Wolfram

NOTE: a should be positive before calling this function
NOTE: b should be positive before calling this function
code[a_, b_, angle_] := Block[{t$95$0 = N[(N[(b - a), $MachinePrecision] * N[(b + a), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(t$95$0 * N[Cos[N[(angle * N[(Pi / -180.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[Power[N[(angle * Pi), $MachinePrecision], 1/3], $MachinePrecision]}, Block[{t$95$3 = N[Sin[N[(Pi / N[(180.0 / angle), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$4 = N[(Pi * N[(angle * N[(b - a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[(angle / 180.0), $MachinePrecision], -5e+282], N[(2.0 * N[(N[Sin[N[(N[(angle * Pi), $MachinePrecision] / -180.0), $MachinePrecision]], $MachinePrecision] * t$95$1), $MachinePrecision]), $MachinePrecision], If[LessEqual[N[(angle / 180.0), $MachinePrecision], -2e+133], N[(2.0 * N[(t$95$1 * N[Sin[N[(N[Power[t$95$2, 2.0], $MachinePrecision] / N[(180.0 / t$95$2), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[N[(angle / 180.0), $MachinePrecision], -5e-83], N[(2.0 * N[(t$95$3 * N[(t$95$0 * N[Cos[N[(angle * N[(N[(N[Power[Pi, 1/3], $MachinePrecision] * N[Power[N[Power[Pi, 1/3], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] / -180.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[N[(angle / 180.0), $MachinePrecision], 1e-39], N[(N[(N[(b * t$95$4), $MachinePrecision] + N[(a * t$95$4), $MachinePrecision]), $MachinePrecision] * 0.011111111111111112), $MachinePrecision], If[LessEqual[N[(angle / 180.0), $MachinePrecision], 1e+87], N[(2.0 * N[(N[Sin[N[Power[N[Sqrt[N[(angle * N[(Pi * 0.005555555555555556), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], 2.0], $MachinePrecision]], $MachinePrecision] * N[(t$95$0 * N[Cos[N[(angle * N[(N[Power[N[(Pi * N[Power[Pi, 2.0], $MachinePrecision]), $MachinePrecision], 1/3], $MachinePrecision] / -180.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(2.0 * N[(t$95$3 * N[(t$95$0 * N[Sqrt[N[Power[N[Cos[N[(0.005555555555555556 * N[(angle * Pi), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], 2.0], $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]]]]]]]

Derivation

1. Split input into 6 regimes

2. if (/.f64 angle 180) < -4.99999999999999978e282

   1. Initial program 40.1%

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

   3. Simplified 40.1%

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

      1. unpow2 40.1%

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

      2. unpow2 40.1%

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

      3. difference-of-squares 40.1%

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

   5. Applied egg-rr 40.1%

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

      1. *-commutative 40.1%

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

      2. div-inv 40.1%

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

      3. metadata-eval 40.1%

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

      4. *-commutative 40.1%

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

      5. associate-*r* 40.1%

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

      6. add-sqr-sqrt 0.0%

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

      7. sqrt-unprod 0.0%

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

      8. associate-*r* 0.0%

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

      9. associate-*r* 0.0%

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

      10. *-commutative 0.0%

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

      11. metadata-eval 0.0%

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

      12. div-inv 0.0%

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

      13. *-commutative 0.0%

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

      14. metadata-eval 0.0%

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

      15. div-inv 0.0%

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

      16. frac-times 0.0%

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

      17. metadata-eval 0.0%

          \[\leadsto 2 \cdot \left(\sin \left(\sqrt{\frac{\left(angle \cdot \pi\right) \cdot \left(angle \cdot \pi\right)}{\color{blue}{32400}}}\right) \cdot \left(\left(\left(b + a\right) \cdot \left(b - a\right)\right) \cdot \cos \left(\frac{\pi}{-180} \cdot angle\right)\right)\right) \]

      18. metadata-eval 0.0%

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

      19. frac-times 0.0%

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

      20. *-commutative 0.0%

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

      21. associate-*l/ 0.0%

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

   7. Applied egg-rr 83.7%

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

   if -4.99999999999999978e282 < (/.f64 angle 180) < -2e133

   1. Initial program 22.8%

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

   3. Simplified 23.1%

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

      1. unpow2 23.1%

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

      2. unpow2 23.1%

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

      3. difference-of-squares 31.0%

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

   5. Applied egg-rr 31.0%

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

      1. associate-*r/ 26.4%

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

      2. *-commutative 26.4%

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

      3. add-cube-cbrt 33.8%

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

      4. associate-/l* 42.9%

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

      5. pow2 42.9%

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

      6. *-commutative 42.9%

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

      7. *-commutative 42.9%

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

   7. Applied egg-rr 42.9%

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

   if -2e133 < (/.f64 angle 180) < -5e-83

   1. Initial program 55.7%

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

   3. Simplified 53.8%

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

      1. unpow2 53.8%

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

      2. unpow2 53.8%

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

      3. difference-of-squares 64.4%

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

   5. Applied egg-rr 64.4%

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

      1. clear-num 62.7%

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

      2. un-div-inv 62.6%

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

   7. Applied egg-rr 62.6%

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

      1. add-cube-cbrt 65.3%

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

      2. pow2 65.3%

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

   9. Applied egg-rr 69.7%

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

   if -5e-83 < (/.f64 angle 180) < 9.99999999999999929e-40

   1. Initial program 64.0%

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

   3. Simplified 64.0%

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

   4. Taylor expanded in angle around 0 64.0%

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

      1. unpow2 63.9%

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

      2. unpow2 63.9%

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

      3. difference-of-squares 70.7%

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

   6. Applied egg-rr 70.8%

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

   7. Taylor expanded in angle around 0 70.8%

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

      1. *-commutative 70.8%

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

      2. associate-*r* 70.9%

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

      3. +-commutative 70.9%

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

      4. *-commutative 70.9%

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

      5. +-commutative 70.9%

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

   9. Simplified 70.9%

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

       1. associate-*r* 99.4%

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

       2. +-commutative 99.4%

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

       3. distribute-lft-in 96.1%

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

       4. *-commutative 96.1%

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

       5. associate-*l* 96.1%

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

       6. *-commutative 96.1%

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

       7. associate-*l* 96.1%

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

   11. Applied egg-rr 96.1%

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

   if 9.99999999999999929e-40 < (/.f64 angle 180) < 9.9999999999999996e86

   1. Initial program 48.0%

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

   3. Simplified 47.8%

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

      1. unpow2 47.8%

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

      2. unpow2 47.8%

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

      3. difference-of-squares 62.6%

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

   5. Applied egg-rr 62.6%

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

      1. *-commutative 62.6%

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

      2. div-inv 55.2%

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

      3. metadata-eval 55.2%

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

      4. *-commutative 55.2%

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

      5. associate-*r* 58.9%

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

      6. add-sqr-sqrt 50.9%

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

      7. sqrt-unprod 58.9%

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

      8. associate-*r* 55.2%

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

      9. associate-*r* 58.9%

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

      10. *-commutative 58.9%

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

      11. metadata-eval 58.9%

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

      12. div-inv 58.9%

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

      13. *-commutative 58.9%

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

      14. metadata-eval 58.9%

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

      15. div-inv 58.4%

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

      16. frac-times 62.7%

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

      17. metadata-eval 62.7%

          \[\leadsto 2 \cdot \left(\sin \left(\sqrt{\frac{\left(angle \cdot \pi\right) \cdot \left(angle \cdot \pi\right)}{\color{blue}{32400}}}\right) \cdot \left(\left(\left(b + a\right) \cdot \left(b - a\right)\right) \cdot \cos \left(\frac{\pi}{-180} \cdot angle\right)\right)\right) \]

      18. metadata-eval 62.7%

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

      19. frac-times 58.4%

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

      20. *-commutative 58.4%

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

      21. associate-*l/ 58.3%

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

   7. Applied egg-rr 50.9%

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

      1. add-cbrt-cube 66.2%

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

      2. *-commutative 66.2%

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

      3. pow2 66.2%

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

   9. Applied egg-rr 66.2%

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

   if 9.9999999999999996e86 < (/.f64 angle 180)

   1. Initial program 23.1%

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

   3. Simplified 26.1%

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

      1. unpow2 26.1%

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

      2. unpow2 26.1%

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

      3. difference-of-squares 32.2%

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

   5. Applied egg-rr 32.2%

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

      1. clear-num 28.2%

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

      2. un-div-inv 28.2%

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

   7. Applied egg-rr 28.2%

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

   9. Applied egg-rr 44.6%

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

       1. associate-*r* 44.8%

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

       2. *-commutative 44.8%

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

       3. *-commutative 44.8%

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

   11. Simplified 44.8%

       \[\leadsto 2 \cdot \left(\sin \left(\frac{\pi}{\frac{180}{angle}}\right) \cdot \left(\left(\left(b + a\right) \cdot \left(b - a\right)\right) \cdot \color{blue}{\sqrt{{\cos \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{2}}}\right)\right) \]
3. Recombined 6 regimes into one program.

4. Final simplification 70.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{angle}{180} \leq -5 \cdot 10^{+282}:\\ \;\;\;\;2 \cdot \left(\sin \left(\frac{angle \cdot \pi}{-180}\right) \cdot \left(\left(\left(b - a\right) \cdot \left(b + a\right)\right) \cdot \cos \left(angle \cdot \frac{\pi}{-180}\right)\right)\right)\\ \mathbf{elif}\;\frac{angle}{180} \leq -2 \cdot 10^{+133}:\\ \;\;\;\;2 \cdot \left(\left(\left(\left(b - a\right) \cdot \left(b + a\right)\right) \cdot \cos \left(angle \cdot \frac{\pi}{-180}\right)\right) \cdot \sin \left(\frac{{\left(\sqrt[3]{angle \cdot \pi}\right)}^{2}}{\frac{180}{\sqrt[3]{angle \cdot \pi}}}\right)\right)\\ \mathbf{elif}\;\frac{angle}{180} \leq -5 \cdot 10^{-83}:\\ \;\;\;\;2 \cdot \left(\sin \left(\frac{\pi}{\frac{180}{angle}}\right) \cdot \left(\left(\left(b - a\right) \cdot \left(b + a\right)\right) \cdot \cos \left(angle \cdot \frac{\sqrt[3]{\pi} \cdot {\left(\sqrt[3]{\pi}\right)}^{2}}{-180}\right)\right)\right)\\ \mathbf{elif}\;\frac{angle}{180} \leq 10^{-39}:\\ \;\;\;\;\left(b \cdot \left(\pi \cdot \left(angle \cdot \left(b - a\right)\right)\right) + a \cdot \left(\pi \cdot \left(angle \cdot \left(b - a\right)\right)\right)\right) \cdot 0.011111111111111112\\ \mathbf{elif}\;\frac{angle}{180} \leq 10^{+87}:\\ \;\;\;\;2 \cdot \left(\sin \left({\left(\sqrt{angle \cdot \left(\pi \cdot 0.005555555555555556\right)}\right)}^{2}\right) \cdot \left(\left(\left(b - a\right) \cdot \left(b + a\right)\right) \cdot \cos \left(angle \cdot \frac{\sqrt[3]{\pi \cdot {\pi}^{2}}}{-180}\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \left(\sin \left(\frac{\pi}{\frac{180}{angle}}\right) \cdot \left(\left(\left(b - a\right) \cdot \left(b + a\right)\right) \cdot \sqrt{{\cos \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{2}}\right)\right)\\ \end{array} \]

Alternative 3: 67.4% accurate, 0.7× speedup

Math

\[\begin{array}{l} a = |a|\\ b = |b|\\ \\ \begin{array}{l} t_0 := \left(b - a\right) \cdot \left(b + a\right)\\ t_1 := t_0 \cdot \cos \left(angle \cdot \frac{\pi}{-180}\right)\\ t_2 := \sqrt[3]{angle \cdot \pi}\\ t_3 := \sin \left(\frac{\pi}{\frac{180}{angle}}\right)\\ t_4 := \pi \cdot \left(angle \cdot \left(b - a\right)\right)\\ \mathbf{if}\;\frac{angle}{180} \leq -5 \cdot 10^{+282}:\\ \;\;\;\;2 \cdot \left(\sin \left(\frac{angle \cdot \pi}{-180}\right) \cdot t_1\right)\\ \mathbf{elif}\;\frac{angle}{180} \leq -2 \cdot 10^{+133}:\\ \;\;\;\;2 \cdot \left(t_1 \cdot \sin \left(\frac{{t_2}^{2}}{\frac{180}{t_2}}\right)\right)\\ \mathbf{elif}\;\frac{angle}{180} \leq -5 \cdot 10^{-83}:\\ \;\;\;\;2 \cdot \left(t_3 \cdot \left(t_0 \cdot \cos \left(angle \cdot \frac{\sqrt[3]{\pi} \cdot {\left(\sqrt[3]{\pi}\right)}^{2}}{-180}\right)\right)\right)\\ \mathbf{elif}\;\frac{angle}{180} \leq 10^{-39}:\\ \;\;\;\;\left(b \cdot t_4 + a \cdot t_4\right) \cdot 0.011111111111111112\\ \mathbf{elif}\;\frac{angle}{180} \leq 10^{+87}:\\ \;\;\;\;2 \cdot \left(\left(t_0 \cdot \cos \left(angle \cdot \frac{\sqrt[3]{\pi \cdot {\pi}^{2}}}{-180}\right)\right) \cdot t_3\right)\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \left(t_3 \cdot \left(t_0 \cdot \sqrt{{\cos \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{2}}\right)\right)\\ \end{array} \end{array} \]

FPCore

NOTE: a should be positive before calling this function
NOTE: b should be positive before calling this function
(FPCore (a b angle)
 :precision binary64
 (let* ((t_0 (* (- b a) (+ b a)))
        (t_1 (* t_0 (cos (* angle (/ PI -180.0)))))
        (t_2 (cbrt (* angle PI)))
        (t_3 (sin (/ PI (/ 180.0 angle))))
        (t_4 (* PI (* angle (- b a)))))
   (if (<= (/ angle 180.0) -5e+282)
     (* 2.0 (* (sin (/ (* angle PI) -180.0)) t_1))
     (if (<= (/ angle 180.0) -2e+133)
       (* 2.0 (* t_1 (sin (/ (pow t_2 2.0) (/ 180.0 t_2)))))
       (if (<= (/ angle 180.0) -5e-83)
         (*
          2.0
          (*
           t_3
           (*
            t_0
            (cos (* angle (/ (* (cbrt PI) (pow (cbrt PI) 2.0)) -180.0))))))
         (if (<= (/ angle 180.0) 1e-39)
           (* (+ (* b t_4) (* a t_4)) 0.011111111111111112)
           (if (<= (/ angle 180.0) 1e+87)
             (*
              2.0
              (*
               (* t_0 (cos (* angle (/ (cbrt (* PI (pow PI 2.0))) -180.0))))
               t_3))
             (*
              2.0
              (*
               t_3
               (*
                t_0
                (sqrt
                 (pow
                  (cos (* 0.005555555555555556 (* angle PI)))
                  2.0))))))))))))

C

a = abs(a);
b = abs(b);
double code(double a, double b, double angle) {
	double t_0 = (b - a) * (b + a);
	double t_1 = t_0 * cos((angle * (((double) M_PI) / -180.0)));
	double t_2 = cbrt((angle * ((double) M_PI)));
	double t_3 = sin((((double) M_PI) / (180.0 / angle)));
	double t_4 = ((double) M_PI) * (angle * (b - a));
	double tmp;
	if ((angle / 180.0) <= -5e+282) {
		tmp = 2.0 * (sin(((angle * ((double) M_PI)) / -180.0)) * t_1);
	} else if ((angle / 180.0) <= -2e+133) {
		tmp = 2.0 * (t_1 * sin((pow(t_2, 2.0) / (180.0 / t_2))));
	} else if ((angle / 180.0) <= -5e-83) {
		tmp = 2.0 * (t_3 * (t_0 * cos((angle * ((cbrt(((double) M_PI)) * pow(cbrt(((double) M_PI)), 2.0)) / -180.0)))));
	} else if ((angle / 180.0) <= 1e-39) {
		tmp = ((b * t_4) + (a * t_4)) * 0.011111111111111112;
	} else if ((angle / 180.0) <= 1e+87) {
		tmp = 2.0 * ((t_0 * cos((angle * (cbrt((((double) M_PI) * pow(((double) M_PI), 2.0))) / -180.0)))) * t_3);
	} else {
		tmp = 2.0 * (t_3 * (t_0 * sqrt(pow(cos((0.005555555555555556 * (angle * ((double) M_PI)))), 2.0))));
	}
	return tmp;
}

Java

a = Math.abs(a);
b = Math.abs(b);
public static double code(double a, double b, double angle) {
	double t_0 = (b - a) * (b + a);
	double t_1 = t_0 * Math.cos((angle * (Math.PI / -180.0)));
	double t_2 = Math.cbrt((angle * Math.PI));
	double t_3 = Math.sin((Math.PI / (180.0 / angle)));
	double t_4 = Math.PI * (angle * (b - a));
	double tmp;
	if ((angle / 180.0) <= -5e+282) {
		tmp = 2.0 * (Math.sin(((angle * Math.PI) / -180.0)) * t_1);
	} else if ((angle / 180.0) <= -2e+133) {
		tmp = 2.0 * (t_1 * Math.sin((Math.pow(t_2, 2.0) / (180.0 / t_2))));
	} else if ((angle / 180.0) <= -5e-83) {
		tmp = 2.0 * (t_3 * (t_0 * Math.cos((angle * ((Math.cbrt(Math.PI) * Math.pow(Math.cbrt(Math.PI), 2.0)) / -180.0)))));
	} else if ((angle / 180.0) <= 1e-39) {
		tmp = ((b * t_4) + (a * t_4)) * 0.011111111111111112;
	} else if ((angle / 180.0) <= 1e+87) {
		tmp = 2.0 * ((t_0 * Math.cos((angle * (Math.cbrt((Math.PI * Math.pow(Math.PI, 2.0))) / -180.0)))) * t_3);
	} else {
		tmp = 2.0 * (t_3 * (t_0 * Math.sqrt(Math.pow(Math.cos((0.005555555555555556 * (angle * Math.PI))), 2.0))));
	}
	return tmp;
}

Julia

a = abs(a)
b = abs(b)
function code(a, b, angle)
	t_0 = Float64(Float64(b - a) * Float64(b + a))
	t_1 = Float64(t_0 * cos(Float64(angle * Float64(pi / -180.0))))
	t_2 = cbrt(Float64(angle * pi))
	t_3 = sin(Float64(pi / Float64(180.0 / angle)))
	t_4 = Float64(pi * Float64(angle * Float64(b - a)))
	tmp = 0.0
	if (Float64(angle / 180.0) <= -5e+282)
		tmp = Float64(2.0 * Float64(sin(Float64(Float64(angle * pi) / -180.0)) * t_1));
	elseif (Float64(angle / 180.0) <= -2e+133)
		tmp = Float64(2.0 * Float64(t_1 * sin(Float64((t_2 ^ 2.0) / Float64(180.0 / t_2)))));
	elseif (Float64(angle / 180.0) <= -5e-83)
		tmp = Float64(2.0 * Float64(t_3 * Float64(t_0 * cos(Float64(angle * Float64(Float64(cbrt(pi) * (cbrt(pi) ^ 2.0)) / -180.0))))));
	elseif (Float64(angle / 180.0) <= 1e-39)
		tmp = Float64(Float64(Float64(b * t_4) + Float64(a * t_4)) * 0.011111111111111112);
	elseif (Float64(angle / 180.0) <= 1e+87)
		tmp = Float64(2.0 * Float64(Float64(t_0 * cos(Float64(angle * Float64(cbrt(Float64(pi * (pi ^ 2.0))) / -180.0)))) * t_3));
	else
		tmp = Float64(2.0 * Float64(t_3 * Float64(t_0 * sqrt((cos(Float64(0.005555555555555556 * Float64(angle * pi))) ^ 2.0)))));
	end
	return tmp
end

Wolfram

NOTE: a should be positive before calling this function
NOTE: b should be positive before calling this function
code[a_, b_, angle_] := Block[{t$95$0 = N[(N[(b - a), $MachinePrecision] * N[(b + a), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(t$95$0 * N[Cos[N[(angle * N[(Pi / -180.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[Power[N[(angle * Pi), $MachinePrecision], 1/3], $MachinePrecision]}, Block[{t$95$3 = N[Sin[N[(Pi / N[(180.0 / angle), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$4 = N[(Pi * N[(angle * N[(b - a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[(angle / 180.0), $MachinePrecision], -5e+282], N[(2.0 * N[(N[Sin[N[(N[(angle * Pi), $MachinePrecision] / -180.0), $MachinePrecision]], $MachinePrecision] * t$95$1), $MachinePrecision]), $MachinePrecision], If[LessEqual[N[(angle / 180.0), $MachinePrecision], -2e+133], N[(2.0 * N[(t$95$1 * N[Sin[N[(N[Power[t$95$2, 2.0], $MachinePrecision] / N[(180.0 / t$95$2), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[N[(angle / 180.0), $MachinePrecision], -5e-83], N[(2.0 * N[(t$95$3 * N[(t$95$0 * N[Cos[N[(angle * N[(N[(N[Power[Pi, 1/3], $MachinePrecision] * N[Power[N[Power[Pi, 1/3], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] / -180.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[N[(angle / 180.0), $MachinePrecision], 1e-39], N[(N[(N[(b * t$95$4), $MachinePrecision] + N[(a * t$95$4), $MachinePrecision]), $MachinePrecision] * 0.011111111111111112), $MachinePrecision], If[LessEqual[N[(angle / 180.0), $MachinePrecision], 1e+87], N[(2.0 * N[(N[(t$95$0 * N[Cos[N[(angle * N[(N[Power[N[(Pi * N[Power[Pi, 2.0], $MachinePrecision]), $MachinePrecision], 1/3], $MachinePrecision] / -180.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * t$95$3), $MachinePrecision]), $MachinePrecision], N[(2.0 * N[(t$95$3 * N[(t$95$0 * N[Sqrt[N[Power[N[Cos[N[(0.005555555555555556 * N[(angle * Pi), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], 2.0], $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]]]]]]]

Derivation

1. Split input into 6 regimes

2. if (/.f64 angle 180) < -4.99999999999999978e282

   1. Initial program 40.1%

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

   3. Simplified 40.1%

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

      1. unpow2 40.1%

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

      2. unpow2 40.1%

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

      3. difference-of-squares 40.1%

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

   5. Applied egg-rr 40.1%

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

      1. *-commutative 40.1%

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

      2. div-inv 40.1%

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

      3. metadata-eval 40.1%

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

      4. *-commutative 40.1%

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

      5. associate-*r* 40.1%

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

      6. add-sqr-sqrt 0.0%

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

      7. sqrt-unprod 0.0%

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

      8. associate-*r* 0.0%

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

      9. associate-*r* 0.0%

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

      10. *-commutative 0.0%

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

      11. metadata-eval 0.0%

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

      12. div-inv 0.0%

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

      13. *-commutative 0.0%

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

      14. metadata-eval 0.0%

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

      15. div-inv 0.0%

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

      16. frac-times 0.0%

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

      17. metadata-eval 0.0%

          \[\leadsto 2 \cdot \left(\sin \left(\sqrt{\frac{\left(angle \cdot \pi\right) \cdot \left(angle \cdot \pi\right)}{\color{blue}{32400}}}\right) \cdot \left(\left(\left(b + a\right) \cdot \left(b - a\right)\right) \cdot \cos \left(\frac{\pi}{-180} \cdot angle\right)\right)\right) \]

      18. metadata-eval 0.0%

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

      19. frac-times 0.0%

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

      20. *-commutative 0.0%

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

      21. associate-*l/ 0.0%

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

   7. Applied egg-rr 83.7%

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

   if -4.99999999999999978e282 < (/.f64 angle 180) < -2e133

   1. Initial program 22.8%

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

   3. Simplified 23.1%

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

      1. unpow2 23.1%

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

      2. unpow2 23.1%

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

      3. difference-of-squares 31.0%

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

   5. Applied egg-rr 31.0%

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

      1. associate-*r/ 26.4%

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

      2. *-commutative 26.4%

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

      3. add-cube-cbrt 33.8%

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

      4. associate-/l* 42.9%

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

      5. pow2 42.9%

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

      6. *-commutative 42.9%

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

      7. *-commutative 42.9%

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

   7. Applied egg-rr 42.9%

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

   if -2e133 < (/.f64 angle 180) < -5e-83

   1. Initial program 55.7%

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

   3. Simplified 53.8%

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

      1. unpow2 53.8%

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

      2. unpow2 53.8%

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

      3. difference-of-squares 64.4%

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

   5. Applied egg-rr 64.4%

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

      1. clear-num 62.7%

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

      2. un-div-inv 62.6%

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

   7. Applied egg-rr 62.6%

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

      1. add-cube-cbrt 65.3%

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

      2. pow2 65.3%

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

   9. Applied egg-rr 69.7%

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

   if -5e-83 < (/.f64 angle 180) < 9.99999999999999929e-40

   1. Initial program 64.0%

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

   3. Simplified 64.0%

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

   4. Taylor expanded in angle around 0 64.0%

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

      1. unpow2 63.9%

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

      2. unpow2 63.9%

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

      3. difference-of-squares 70.7%

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

   6. Applied egg-rr 70.8%

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

   7. Taylor expanded in angle around 0 70.8%

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

      1. *-commutative 70.8%

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

      2. associate-*r* 70.9%

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

      3. +-commutative 70.9%

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

      4. *-commutative 70.9%

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

      5. +-commutative 70.9%

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

   9. Simplified 70.9%

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

       1. associate-*r* 99.4%

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

       2. +-commutative 99.4%

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

       3. distribute-lft-in 96.1%

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

       4. *-commutative 96.1%

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

       5. associate-*l* 96.1%

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

       6. *-commutative 96.1%

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

       7. associate-*l* 96.1%

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

   11. Applied egg-rr 96.1%

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

   if 9.99999999999999929e-40 < (/.f64 angle 180) < 9.9999999999999996e86

   1. Initial program 48.0%

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

   3. Simplified 47.8%

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

      1. unpow2 47.8%

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

      2. unpow2 47.8%

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

      3. difference-of-squares 62.6%

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

   5. Applied egg-rr 62.6%

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

      1. clear-num 58.9%

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

      2. un-div-inv 59.1%

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

   7. Applied egg-rr 59.1%

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

      1. add-cbrt-cube 66.2%

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

      2. *-commutative 66.2%

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

      3. pow2 66.2%

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

   9. Applied egg-rr 65.5%

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

   if 9.9999999999999996e86 < (/.f64 angle 180)

   1. Initial program 23.1%

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

   3. Simplified 26.1%

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

      1. unpow2 26.1%

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

      2. unpow2 26.1%

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

      3. difference-of-squares 32.2%

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

   5. Applied egg-rr 32.2%

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

      1. clear-num 28.2%

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

      2. un-div-inv 28.2%

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

   7. Applied egg-rr 28.2%

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

   9. Applied egg-rr 44.6%

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

       1. associate-*r* 44.8%

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

       2. *-commutative 44.8%

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

       3. *-commutative 44.8%

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

   11. Simplified 44.8%

       \[\leadsto 2 \cdot \left(\sin \left(\frac{\pi}{\frac{180}{angle}}\right) \cdot \left(\left(\left(b + a\right) \cdot \left(b - a\right)\right) \cdot \color{blue}{\sqrt{{\cos \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{2}}}\right)\right) \]
3. Recombined 6 regimes into one program.

4. Final simplification 70.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{angle}{180} \leq -5 \cdot 10^{+282}:\\ \;\;\;\;2 \cdot \left(\sin \left(\frac{angle \cdot \pi}{-180}\right) \cdot \left(\left(\left(b - a\right) \cdot \left(b + a\right)\right) \cdot \cos \left(angle \cdot \frac{\pi}{-180}\right)\right)\right)\\ \mathbf{elif}\;\frac{angle}{180} \leq -2 \cdot 10^{+133}:\\ \;\;\;\;2 \cdot \left(\left(\left(\left(b - a\right) \cdot \left(b + a\right)\right) \cdot \cos \left(angle \cdot \frac{\pi}{-180}\right)\right) \cdot \sin \left(\frac{{\left(\sqrt[3]{angle \cdot \pi}\right)}^{2}}{\frac{180}{\sqrt[3]{angle \cdot \pi}}}\right)\right)\\ \mathbf{elif}\;\frac{angle}{180} \leq -5 \cdot 10^{-83}:\\ \;\;\;\;2 \cdot \left(\sin \left(\frac{\pi}{\frac{180}{angle}}\right) \cdot \left(\left(\left(b - a\right) \cdot \left(b + a\right)\right) \cdot \cos \left(angle \cdot \frac{\sqrt[3]{\pi} \cdot {\left(\sqrt[3]{\pi}\right)}^{2}}{-180}\right)\right)\right)\\ \mathbf{elif}\;\frac{angle}{180} \leq 10^{-39}:\\ \;\;\;\;\left(b \cdot \left(\pi \cdot \left(angle \cdot \left(b - a\right)\right)\right) + a \cdot \left(\pi \cdot \left(angle \cdot \left(b - a\right)\right)\right)\right) \cdot 0.011111111111111112\\ \mathbf{elif}\;\frac{angle}{180} \leq 10^{+87}:\\ \;\;\;\;2 \cdot \left(\left(\left(\left(b - a\right) \cdot \left(b + a\right)\right) \cdot \cos \left(angle \cdot \frac{\sqrt[3]{\pi \cdot {\pi}^{2}}}{-180}\right)\right) \cdot \sin \left(\frac{\pi}{\frac{180}{angle}}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \left(\sin \left(\frac{\pi}{\frac{180}{angle}}\right) \cdot \left(\left(\left(b - a\right) \cdot \left(b + a\right)\right) \cdot \sqrt{{\cos \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{2}}\right)\right)\\ \end{array} \]

Alternative 4: 67.1% accurate, 0.7× speedup

Math

\[\begin{array}{l} a = |a|\\ b = |b|\\ \\ \begin{array}{l} t_0 := \left(b - a\right) \cdot \left(b + a\right)\\ t_1 := \pi \cdot \left(angle \cdot \left(b - a\right)\right)\\ t_2 := \sin \left(\frac{\pi}{\frac{180}{angle}}\right)\\ \mathbf{if}\;\frac{angle}{180} \leq -5 \cdot 10^{-83}:\\ \;\;\;\;2 \cdot \left(t_2 \cdot \left(t_0 \cdot \cos \left(angle \cdot \frac{\sqrt[3]{\pi} \cdot {\left(\sqrt[3]{\pi}\right)}^{2}}{-180}\right)\right)\right)\\ \mathbf{elif}\;\frac{angle}{180} \leq 10^{-39}:\\ \;\;\;\;\left(b \cdot t_1 + a \cdot t_1\right) \cdot 0.011111111111111112\\ \mathbf{elif}\;\frac{angle}{180} \leq 10^{+87}:\\ \;\;\;\;2 \cdot \left(\left(t_0 \cdot \cos \left(angle \cdot \frac{\sqrt[3]{\pi \cdot {\pi}^{2}}}{-180}\right)\right) \cdot t_2\right)\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \left(t_2 \cdot \left(t_0 \cdot \sqrt{{\cos \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{2}}\right)\right)\\ \end{array} \end{array} \]

FPCore

NOTE: a should be positive before calling this function
NOTE: b should be positive before calling this function
(FPCore (a b angle)
 :precision binary64
 (let* ((t_0 (* (- b a) (+ b a)))
        (t_1 (* PI (* angle (- b a))))
        (t_2 (sin (/ PI (/ 180.0 angle)))))
   (if (<= (/ angle 180.0) -5e-83)
     (*
      2.0
      (*
       t_2
       (* t_0 (cos (* angle (/ (* (cbrt PI) (pow (cbrt PI) 2.0)) -180.0))))))
     (if (<= (/ angle 180.0) 1e-39)
       (* (+ (* b t_1) (* a t_1)) 0.011111111111111112)
       (if (<= (/ angle 180.0) 1e+87)
         (*
          2.0
          (*
           (* t_0 (cos (* angle (/ (cbrt (* PI (pow PI 2.0))) -180.0))))
           t_2))
         (*
          2.0
          (*
           t_2
           (*
            t_0
            (sqrt
             (pow (cos (* 0.005555555555555556 (* angle PI))) 2.0))))))))))

C

a = abs(a);
b = abs(b);
double code(double a, double b, double angle) {
	double t_0 = (b - a) * (b + a);
	double t_1 = ((double) M_PI) * (angle * (b - a));
	double t_2 = sin((((double) M_PI) / (180.0 / angle)));
	double tmp;
	if ((angle / 180.0) <= -5e-83) {
		tmp = 2.0 * (t_2 * (t_0 * cos((angle * ((cbrt(((double) M_PI)) * pow(cbrt(((double) M_PI)), 2.0)) / -180.0)))));
	} else if ((angle / 180.0) <= 1e-39) {
		tmp = ((b * t_1) + (a * t_1)) * 0.011111111111111112;
	} else if ((angle / 180.0) <= 1e+87) {
		tmp = 2.0 * ((t_0 * cos((angle * (cbrt((((double) M_PI) * pow(((double) M_PI), 2.0))) / -180.0)))) * t_2);
	} else {
		tmp = 2.0 * (t_2 * (t_0 * sqrt(pow(cos((0.005555555555555556 * (angle * ((double) M_PI)))), 2.0))));
	}
	return tmp;
}

Java

a = Math.abs(a);
b = Math.abs(b);
public static double code(double a, double b, double angle) {
	double t_0 = (b - a) * (b + a);
	double t_1 = Math.PI * (angle * (b - a));
	double t_2 = Math.sin((Math.PI / (180.0 / angle)));
	double tmp;
	if ((angle / 180.0) <= -5e-83) {
		tmp = 2.0 * (t_2 * (t_0 * Math.cos((angle * ((Math.cbrt(Math.PI) * Math.pow(Math.cbrt(Math.PI), 2.0)) / -180.0)))));
	} else if ((angle / 180.0) <= 1e-39) {
		tmp = ((b * t_1) + (a * t_1)) * 0.011111111111111112;
	} else if ((angle / 180.0) <= 1e+87) {
		tmp = 2.0 * ((t_0 * Math.cos((angle * (Math.cbrt((Math.PI * Math.pow(Math.PI, 2.0))) / -180.0)))) * t_2);
	} else {
		tmp = 2.0 * (t_2 * (t_0 * Math.sqrt(Math.pow(Math.cos((0.005555555555555556 * (angle * Math.PI))), 2.0))));
	}
	return tmp;
}

Julia

a = abs(a)
b = abs(b)
function code(a, b, angle)
	t_0 = Float64(Float64(b - a) * Float64(b + a))
	t_1 = Float64(pi * Float64(angle * Float64(b - a)))
	t_2 = sin(Float64(pi / Float64(180.0 / angle)))
	tmp = 0.0
	if (Float64(angle / 180.0) <= -5e-83)
		tmp = Float64(2.0 * Float64(t_2 * Float64(t_0 * cos(Float64(angle * Float64(Float64(cbrt(pi) * (cbrt(pi) ^ 2.0)) / -180.0))))));
	elseif (Float64(angle / 180.0) <= 1e-39)
		tmp = Float64(Float64(Float64(b * t_1) + Float64(a * t_1)) * 0.011111111111111112);
	elseif (Float64(angle / 180.0) <= 1e+87)
		tmp = Float64(2.0 * Float64(Float64(t_0 * cos(Float64(angle * Float64(cbrt(Float64(pi * (pi ^ 2.0))) / -180.0)))) * t_2));
	else
		tmp = Float64(2.0 * Float64(t_2 * Float64(t_0 * sqrt((cos(Float64(0.005555555555555556 * Float64(angle * pi))) ^ 2.0)))));
	end
	return tmp
end

Wolfram

NOTE: a should be positive before calling this function
NOTE: b should be positive before calling this function
code[a_, b_, angle_] := Block[{t$95$0 = N[(N[(b - a), $MachinePrecision] * N[(b + a), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(Pi * N[(angle * N[(b - a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[Sin[N[(Pi / N[(180.0 / angle), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[N[(angle / 180.0), $MachinePrecision], -5e-83], N[(2.0 * N[(t$95$2 * N[(t$95$0 * N[Cos[N[(angle * N[(N[(N[Power[Pi, 1/3], $MachinePrecision] * N[Power[N[Power[Pi, 1/3], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] / -180.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[N[(angle / 180.0), $MachinePrecision], 1e-39], N[(N[(N[(b * t$95$1), $MachinePrecision] + N[(a * t$95$1), $MachinePrecision]), $MachinePrecision] * 0.011111111111111112), $MachinePrecision], If[LessEqual[N[(angle / 180.0), $MachinePrecision], 1e+87], N[(2.0 * N[(N[(t$95$0 * N[Cos[N[(angle * N[(N[Power[N[(Pi * N[Power[Pi, 2.0], $MachinePrecision]), $MachinePrecision], 1/3], $MachinePrecision] / -180.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * t$95$2), $MachinePrecision]), $MachinePrecision], N[(2.0 * N[(t$95$2 * N[(t$95$0 * N[Sqrt[N[Power[N[Cos[N[(0.005555555555555556 * N[(angle * Pi), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], 2.0], $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]]]

Derivation

1. Split input into 4 regimes

2. if (/.f64 angle 180) < -5e-83

   1. Initial program 40.9%

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

   3. Simplified 40.1%

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

      1. unpow2 40.1%

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

      2. unpow2 40.1%

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

      3. difference-of-squares 49.0%

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

   5. Applied egg-rr 49.0%

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

      1. clear-num 47.9%

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

      2. un-div-inv 47.7%

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

   7. Applied egg-rr 47.7%

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

      1. add-cube-cbrt 50.7%

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

      2. pow2 50.7%

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

   9. Applied egg-rr 53.2%

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

   if -5e-83 < (/.f64 angle 180) < 9.99999999999999929e-40

   1. Initial program 64.0%

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

   3. Simplified 64.0%

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

   4. Taylor expanded in angle around 0 64.0%

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

      1. unpow2 63.9%

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

      2. unpow2 63.9%

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

      3. difference-of-squares 70.7%

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

   6. Applied egg-rr 70.8%

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

   7. Taylor expanded in angle around 0 70.8%

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

      1. *-commutative 70.8%

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

      2. associate-*r* 70.9%

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

      3. +-commutative 70.9%

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

      4. *-commutative 70.9%

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

      5. +-commutative 70.9%

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

   9. Simplified 70.9%

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

       1. associate-*r* 99.4%

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

       2. +-commutative 99.4%

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

       3. distribute-lft-in 96.1%

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

       4. *-commutative 96.1%

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

       5. associate-*l* 96.1%

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

       6. *-commutative 96.1%

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

       7. associate-*l* 96.1%

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

   11. Applied egg-rr 96.1%

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

   if 9.99999999999999929e-40 < (/.f64 angle 180) < 9.9999999999999996e86

   1. Initial program 48.0%

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

   3. Simplified 47.8%

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

      1. unpow2 47.8%

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

      2. unpow2 47.8%

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

      3. difference-of-squares 62.6%

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

   5. Applied egg-rr 62.6%

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

      1. clear-num 58.9%

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

      2. un-div-inv 59.1%

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

   7. Applied egg-rr 59.1%

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

      1. add-cbrt-cube 66.2%

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

      2. *-commutative 66.2%

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

      3. pow2 66.2%

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

   9. Applied egg-rr 65.5%

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

   if 9.9999999999999996e86 < (/.f64 angle 180)

   1. Initial program 23.1%

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

   3. Simplified 26.1%

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

      1. unpow2 26.1%

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

      2. unpow2 26.1%

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

      3. difference-of-squares 32.2%

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

   5. Applied egg-rr 32.2%

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

      1. clear-num 28.2%

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

      2. un-div-inv 28.2%

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

   7. Applied egg-rr 28.2%

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

   9. Applied egg-rr 44.6%

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

       1. associate-*r* 44.8%

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

       2. *-commutative 44.8%

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

       3. *-commutative 44.8%

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

   11. Simplified 44.8%

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

4. Final simplification 67.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{angle}{180} \leq -5 \cdot 10^{-83}:\\ \;\;\;\;2 \cdot \left(\sin \left(\frac{\pi}{\frac{180}{angle}}\right) \cdot \left(\left(\left(b - a\right) \cdot \left(b + a\right)\right) \cdot \cos \left(angle \cdot \frac{\sqrt[3]{\pi} \cdot {\left(\sqrt[3]{\pi}\right)}^{2}}{-180}\right)\right)\right)\\ \mathbf{elif}\;\frac{angle}{180} \leq 10^{-39}:\\ \;\;\;\;\left(b \cdot \left(\pi \cdot \left(angle \cdot \left(b - a\right)\right)\right) + a \cdot \left(\pi \cdot \left(angle \cdot \left(b - a\right)\right)\right)\right) \cdot 0.011111111111111112\\ \mathbf{elif}\;\frac{angle}{180} \leq 10^{+87}:\\ \;\;\;\;2 \cdot \left(\left(\left(\left(b - a\right) \cdot \left(b + a\right)\right) \cdot \cos \left(angle \cdot \frac{\sqrt[3]{\pi \cdot {\pi}^{2}}}{-180}\right)\right) \cdot \sin \left(\frac{\pi}{\frac{180}{angle}}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \left(\sin \left(\frac{\pi}{\frac{180}{angle}}\right) \cdot \left(\left(\left(b - a\right) \cdot \left(b + a\right)\right) \cdot \sqrt{{\cos \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{2}}\right)\right)\\ \end{array} \]

Alternative 5: 66.9% accurate, 0.8× speedup

Math

\[\begin{array}{l} a = |a|\\ b = |b|\\ \\ \begin{array}{l} t_0 := \left(b - a\right) \cdot \left(b + a\right)\\ t_1 := \pi \cdot \left(angle \cdot \left(b - a\right)\right)\\ t_2 := \sin \left(\frac{\pi}{\frac{180}{angle}}\right)\\ \mathbf{if}\;\frac{angle}{180} \leq -5 \cdot 10^{-83}:\\ \;\;\;\;2 \cdot \left(\sin \left(\frac{angle}{180} \cdot \pi\right) \cdot \left(t_0 \cdot {\left(\sqrt[3]{\cos \left(\pi \cdot \left(angle \cdot 0.005555555555555556\right)\right)}\right)}^{3}\right)\right)\\ \mathbf{elif}\;\frac{angle}{180} \leq 10^{-39}:\\ \;\;\;\;\left(b \cdot t_1 + a \cdot t_1\right) \cdot 0.011111111111111112\\ \mathbf{elif}\;\frac{angle}{180} \leq 10^{+87}:\\ \;\;\;\;2 \cdot \left(\left(t_0 \cdot \cos \left(angle \cdot \frac{\sqrt[3]{\pi \cdot {\pi}^{2}}}{-180}\right)\right) \cdot t_2\right)\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \left(t_2 \cdot \left(t_0 \cdot \sqrt{{\cos \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{2}}\right)\right)\\ \end{array} \end{array} \]

FPCore

NOTE: a should be positive before calling this function
NOTE: b should be positive before calling this function
(FPCore (a b angle)
 :precision binary64
 (let* ((t_0 (* (- b a) (+ b a)))
        (t_1 (* PI (* angle (- b a))))
        (t_2 (sin (/ PI (/ 180.0 angle)))))
   (if (<= (/ angle 180.0) -5e-83)
     (*
      2.0
      (*
       (sin (* (/ angle 180.0) PI))
       (* t_0 (pow (cbrt (cos (* PI (* angle 0.005555555555555556)))) 3.0))))
     (if (<= (/ angle 180.0) 1e-39)
       (* (+ (* b t_1) (* a t_1)) 0.011111111111111112)
       (if (<= (/ angle 180.0) 1e+87)
         (*
          2.0
          (*
           (* t_0 (cos (* angle (/ (cbrt (* PI (pow PI 2.0))) -180.0))))
           t_2))
         (*
          2.0
          (*
           t_2
           (*
            t_0
            (sqrt
             (pow (cos (* 0.005555555555555556 (* angle PI))) 2.0))))))))))

C

a = abs(a);
b = abs(b);
double code(double a, double b, double angle) {
	double t_0 = (b - a) * (b + a);
	double t_1 = ((double) M_PI) * (angle * (b - a));
	double t_2 = sin((((double) M_PI) / (180.0 / angle)));
	double tmp;
	if ((angle / 180.0) <= -5e-83) {
		tmp = 2.0 * (sin(((angle / 180.0) * ((double) M_PI))) * (t_0 * pow(cbrt(cos((((double) M_PI) * (angle * 0.005555555555555556)))), 3.0)));
	} else if ((angle / 180.0) <= 1e-39) {
		tmp = ((b * t_1) + (a * t_1)) * 0.011111111111111112;
	} else if ((angle / 180.0) <= 1e+87) {
		tmp = 2.0 * ((t_0 * cos((angle * (cbrt((((double) M_PI) * pow(((double) M_PI), 2.0))) / -180.0)))) * t_2);
	} else {
		tmp = 2.0 * (t_2 * (t_0 * sqrt(pow(cos((0.005555555555555556 * (angle * ((double) M_PI)))), 2.0))));
	}
	return tmp;
}

Java

a = Math.abs(a);
b = Math.abs(b);
public static double code(double a, double b, double angle) {
	double t_0 = (b - a) * (b + a);
	double t_1 = Math.PI * (angle * (b - a));
	double t_2 = Math.sin((Math.PI / (180.0 / angle)));
	double tmp;
	if ((angle / 180.0) <= -5e-83) {
		tmp = 2.0 * (Math.sin(((angle / 180.0) * Math.PI)) * (t_0 * Math.pow(Math.cbrt(Math.cos((Math.PI * (angle * 0.005555555555555556)))), 3.0)));
	} else if ((angle / 180.0) <= 1e-39) {
		tmp = ((b * t_1) + (a * t_1)) * 0.011111111111111112;
	} else if ((angle / 180.0) <= 1e+87) {
		tmp = 2.0 * ((t_0 * Math.cos((angle * (Math.cbrt((Math.PI * Math.pow(Math.PI, 2.0))) / -180.0)))) * t_2);
	} else {
		tmp = 2.0 * (t_2 * (t_0 * Math.sqrt(Math.pow(Math.cos((0.005555555555555556 * (angle * Math.PI))), 2.0))));
	}
	return tmp;
}

Julia

a = abs(a)
b = abs(b)
function code(a, b, angle)
	t_0 = Float64(Float64(b - a) * Float64(b + a))
	t_1 = Float64(pi * Float64(angle * Float64(b - a)))
	t_2 = sin(Float64(pi / Float64(180.0 / angle)))
	tmp = 0.0
	if (Float64(angle / 180.0) <= -5e-83)
		tmp = Float64(2.0 * Float64(sin(Float64(Float64(angle / 180.0) * pi)) * Float64(t_0 * (cbrt(cos(Float64(pi * Float64(angle * 0.005555555555555556)))) ^ 3.0))));
	elseif (Float64(angle / 180.0) <= 1e-39)
		tmp = Float64(Float64(Float64(b * t_1) + Float64(a * t_1)) * 0.011111111111111112);
	elseif (Float64(angle / 180.0) <= 1e+87)
		tmp = Float64(2.0 * Float64(Float64(t_0 * cos(Float64(angle * Float64(cbrt(Float64(pi * (pi ^ 2.0))) / -180.0)))) * t_2));
	else
		tmp = Float64(2.0 * Float64(t_2 * Float64(t_0 * sqrt((cos(Float64(0.005555555555555556 * Float64(angle * pi))) ^ 2.0)))));
	end
	return tmp
end

Wolfram

NOTE: a should be positive before calling this function
NOTE: b should be positive before calling this function
code[a_, b_, angle_] := Block[{t$95$0 = N[(N[(b - a), $MachinePrecision] * N[(b + a), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(Pi * N[(angle * N[(b - a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[Sin[N[(Pi / N[(180.0 / angle), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[N[(angle / 180.0), $MachinePrecision], -5e-83], N[(2.0 * N[(N[Sin[N[(N[(angle / 180.0), $MachinePrecision] * Pi), $MachinePrecision]], $MachinePrecision] * N[(t$95$0 * N[Power[N[Power[N[Cos[N[(Pi * N[(angle * 0.005555555555555556), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], 1/3], $MachinePrecision], 3.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[N[(angle / 180.0), $MachinePrecision], 1e-39], N[(N[(N[(b * t$95$1), $MachinePrecision] + N[(a * t$95$1), $MachinePrecision]), $MachinePrecision] * 0.011111111111111112), $MachinePrecision], If[LessEqual[N[(angle / 180.0), $MachinePrecision], 1e+87], N[(2.0 * N[(N[(t$95$0 * N[Cos[N[(angle * N[(N[Power[N[(Pi * N[Power[Pi, 2.0], $MachinePrecision]), $MachinePrecision], 1/3], $MachinePrecision] / -180.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * t$95$2), $MachinePrecision]), $MachinePrecision], N[(2.0 * N[(t$95$2 * N[(t$95$0 * N[Sqrt[N[Power[N[Cos[N[(0.005555555555555556 * N[(angle * Pi), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], 2.0], $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]]]

Derivation

1. Split input into 4 regimes

2. if (/.f64 angle 180) < -5e-83

   1. Initial program 40.9%

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

   3. Simplified 40.1%

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

      1. unpow2 40.1%

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

      2. unpow2 40.1%

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

      3. difference-of-squares 49.0%

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

   5. Applied egg-rr 49.0%

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

      1. *-commutative 49.0%

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

      2. add-sqr-sqrt 47.3%

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

      3. sqrt-unprod 33.9%

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

      4. *-commutative 33.9%

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

      5. *-commutative 33.9%

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

      6. associate-*l/ 33.3%

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

      7. associate-*l/ 33.3%

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

      8. frac-times 34.4%

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

      9. metadata-eval 34.4%

         \[\leadsto 2 \cdot \left(\sin \left(\pi \cdot \frac{angle}{180}\right) \cdot \left(\left(\left(b + a\right) \cdot \left(b - a\right)\right) \cdot \cos \left(\sqrt{\frac{\left(\pi \cdot angle\right) \cdot \left(\pi \cdot angle\right)}{\color{blue}{32400}}}\right)\right)\right) \]

      10. metadata-eval 34.4%

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

      11. frac-times 33.3%

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

      12. associate-*r/ 34.5%

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

      13. associate-*r/ 34.9%

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

      14. sqrt-unprod 0.0%

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

      15. add-sqr-sqrt 49.8%

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

      16. add-cube-cbrt 49.8%

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

   7. Applied egg-rr 52.0%

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

   if -5e-83 < (/.f64 angle 180) < 9.99999999999999929e-40

   1. Initial program 64.0%

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

   3. Simplified 64.0%

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

   4. Taylor expanded in angle around 0 64.0%

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

      1. unpow2 63.9%

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

      2. unpow2 63.9%

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

      3. difference-of-squares 70.7%

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

   6. Applied egg-rr 70.8%

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

   7. Taylor expanded in angle around 0 70.8%

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

      1. *-commutative 70.8%

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

      2. associate-*r* 70.9%

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

      3. +-commutative 70.9%

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

      4. *-commutative 70.9%

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

      5. +-commutative 70.9%

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

   9. Simplified 70.9%

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

       1. associate-*r* 99.4%

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

       2. +-commutative 99.4%

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

       3. distribute-lft-in 96.1%

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

       4. *-commutative 96.1%

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

       5. associate-*l* 96.1%

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

       6. *-commutative 96.1%

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

       7. associate-*l* 96.1%

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

   11. Applied egg-rr 96.1%

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

   if 9.99999999999999929e-40 < (/.f64 angle 180) < 9.9999999999999996e86

   1. Initial program 48.0%

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

   3. Simplified 47.8%

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

      1. unpow2 47.8%

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

      2. unpow2 47.8%

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

      3. difference-of-squares 62.6%

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

   5. Applied egg-rr 62.6%

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

      1. clear-num 58.9%

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

      2. un-div-inv 59.1%

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

   7. Applied egg-rr 59.1%

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

      1. add-cbrt-cube 66.2%

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

      2. *-commutative 66.2%

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

      3. pow2 66.2%

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

   9. Applied egg-rr 65.5%

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

   if 9.9999999999999996e86 < (/.f64 angle 180)

   1. Initial program 23.1%

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

   3. Simplified 26.1%

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

      1. unpow2 26.1%

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

      2. unpow2 26.1%

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

      3. difference-of-squares 32.2%

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

   5. Applied egg-rr 32.2%

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

      1. clear-num 28.2%

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

      2. un-div-inv 28.2%

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

   7. Applied egg-rr 28.2%

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

   9. Applied egg-rr 44.6%

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

       1. associate-*r* 44.8%

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

       2. *-commutative 44.8%

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

       3. *-commutative 44.8%

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

   11. Simplified 44.8%

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

4. Final simplification 67.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{angle}{180} \leq -5 \cdot 10^{-83}:\\ \;\;\;\;2 \cdot \left(\sin \left(\frac{angle}{180} \cdot \pi\right) \cdot \left(\left(\left(b - a\right) \cdot \left(b + a\right)\right) \cdot {\left(\sqrt[3]{\cos \left(\pi \cdot \left(angle \cdot 0.005555555555555556\right)\right)}\right)}^{3}\right)\right)\\ \mathbf{elif}\;\frac{angle}{180} \leq 10^{-39}:\\ \;\;\;\;\left(b \cdot \left(\pi \cdot \left(angle \cdot \left(b - a\right)\right)\right) + a \cdot \left(\pi \cdot \left(angle \cdot \left(b - a\right)\right)\right)\right) \cdot 0.011111111111111112\\ \mathbf{elif}\;\frac{angle}{180} \leq 10^{+87}:\\ \;\;\;\;2 \cdot \left(\left(\left(\left(b - a\right) \cdot \left(b + a\right)\right) \cdot \cos \left(angle \cdot \frac{\sqrt[3]{\pi \cdot {\pi}^{2}}}{-180}\right)\right) \cdot \sin \left(\frac{\pi}{\frac{180}{angle}}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \left(\sin \left(\frac{\pi}{\frac{180}{angle}}\right) \cdot \left(\left(\left(b - a\right) \cdot \left(b + a\right)\right) \cdot \sqrt{{\cos \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{2}}\right)\right)\\ \end{array} \]

Alternative 6: 66.4% accurate, 1.0× speedup

Math

\[\begin{array}{l} a = |a|\\ b = |b|\\ \\ \begin{array}{l} t_0 := \left(b - a\right) \cdot \left(b + a\right)\\ t_1 := \pi \cdot \left(angle \cdot \left(b - a\right)\right)\\ \mathbf{if}\;\frac{angle}{180} \leq -5 \cdot 10^{-83}:\\ \;\;\;\;2 \cdot \left(\sin \left(\frac{angle}{180} \cdot \pi\right) \cdot \left(t_0 \cdot {\left(\sqrt[3]{\cos \left(\pi \cdot \left(angle \cdot 0.005555555555555556\right)\right)}\right)}^{3}\right)\right)\\ \mathbf{elif}\;\frac{angle}{180} \leq 400000000000:\\ \;\;\;\;\left(b \cdot t_1 + a \cdot t_1\right) \cdot 0.011111111111111112\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \left(\sin \left(\frac{\pi}{\frac{180}{angle}}\right) \cdot \left(t_0 \cdot \sqrt{{\cos \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{2}}\right)\right)\\ \end{array} \end{array} \]

FPCore

NOTE: a should be positive before calling this function
NOTE: b should be positive before calling this function
(FPCore (a b angle)
 :precision binary64
 (let* ((t_0 (* (- b a) (+ b a))) (t_1 (* PI (* angle (- b a)))))
   (if (<= (/ angle 180.0) -5e-83)
     (*
      2.0
      (*
       (sin (* (/ angle 180.0) PI))
       (* t_0 (pow (cbrt (cos (* PI (* angle 0.005555555555555556)))) 3.0))))
     (if (<= (/ angle 180.0) 400000000000.0)
       (* (+ (* b t_1) (* a t_1)) 0.011111111111111112)
       (*
        2.0
        (*
         (sin (/ PI (/ 180.0 angle)))
         (*
          t_0
          (sqrt (pow (cos (* 0.005555555555555556 (* angle PI))) 2.0)))))))))

C

a = abs(a);
b = abs(b);
double code(double a, double b, double angle) {
	double t_0 = (b - a) * (b + a);
	double t_1 = ((double) M_PI) * (angle * (b - a));
	double tmp;
	if ((angle / 180.0) <= -5e-83) {
		tmp = 2.0 * (sin(((angle / 180.0) * ((double) M_PI))) * (t_0 * pow(cbrt(cos((((double) M_PI) * (angle * 0.005555555555555556)))), 3.0)));
	} else if ((angle / 180.0) <= 400000000000.0) {
		tmp = ((b * t_1) + (a * t_1)) * 0.011111111111111112;
	} else {
		tmp = 2.0 * (sin((((double) M_PI) / (180.0 / angle))) * (t_0 * sqrt(pow(cos((0.005555555555555556 * (angle * ((double) M_PI)))), 2.0))));
	}
	return tmp;
}

Java

a = Math.abs(a);
b = Math.abs(b);
public static double code(double a, double b, double angle) {
	double t_0 = (b - a) * (b + a);
	double t_1 = Math.PI * (angle * (b - a));
	double tmp;
	if ((angle / 180.0) <= -5e-83) {
		tmp = 2.0 * (Math.sin(((angle / 180.0) * Math.PI)) * (t_0 * Math.pow(Math.cbrt(Math.cos((Math.PI * (angle * 0.005555555555555556)))), 3.0)));
	} else if ((angle / 180.0) <= 400000000000.0) {
		tmp = ((b * t_1) + (a * t_1)) * 0.011111111111111112;
	} else {
		tmp = 2.0 * (Math.sin((Math.PI / (180.0 / angle))) * (t_0 * Math.sqrt(Math.pow(Math.cos((0.005555555555555556 * (angle * Math.PI))), 2.0))));
	}
	return tmp;
}

Julia

a = abs(a)
b = abs(b)
function code(a, b, angle)
	t_0 = Float64(Float64(b - a) * Float64(b + a))
	t_1 = Float64(pi * Float64(angle * Float64(b - a)))
	tmp = 0.0
	if (Float64(angle / 180.0) <= -5e-83)
		tmp = Float64(2.0 * Float64(sin(Float64(Float64(angle / 180.0) * pi)) * Float64(t_0 * (cbrt(cos(Float64(pi * Float64(angle * 0.005555555555555556)))) ^ 3.0))));
	elseif (Float64(angle / 180.0) <= 400000000000.0)
		tmp = Float64(Float64(Float64(b * t_1) + Float64(a * t_1)) * 0.011111111111111112);
	else
		tmp = Float64(2.0 * Float64(sin(Float64(pi / Float64(180.0 / angle))) * Float64(t_0 * sqrt((cos(Float64(0.005555555555555556 * Float64(angle * pi))) ^ 2.0)))));
	end
	return tmp
end

Wolfram

NOTE: a should be positive before calling this function
NOTE: b should be positive before calling this function
code[a_, b_, angle_] := Block[{t$95$0 = N[(N[(b - a), $MachinePrecision] * N[(b + a), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(Pi * N[(angle * N[(b - a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[(angle / 180.0), $MachinePrecision], -5e-83], N[(2.0 * N[(N[Sin[N[(N[(angle / 180.0), $MachinePrecision] * Pi), $MachinePrecision]], $MachinePrecision] * N[(t$95$0 * N[Power[N[Power[N[Cos[N[(Pi * N[(angle * 0.005555555555555556), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], 1/3], $MachinePrecision], 3.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[N[(angle / 180.0), $MachinePrecision], 400000000000.0], N[(N[(N[(b * t$95$1), $MachinePrecision] + N[(a * t$95$1), $MachinePrecision]), $MachinePrecision] * 0.011111111111111112), $MachinePrecision], N[(2.0 * N[(N[Sin[N[(Pi / N[(180.0 / angle), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * N[(t$95$0 * N[Sqrt[N[Power[N[Cos[N[(0.005555555555555556 * N[(angle * Pi), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], 2.0], $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

2. if (/.f64 angle 180) < -5e-83

   1. Initial program 40.9%

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

   3. Simplified 40.1%

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

      1. unpow2 40.1%

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

      2. unpow2 40.1%

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

      3. difference-of-squares 49.0%

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

   5. Applied egg-rr 49.0%

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

      1. *-commutative 49.0%

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

      2. add-sqr-sqrt 47.3%

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

      3. sqrt-unprod 33.9%

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

      4. *-commutative 33.9%

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

      5. *-commutative 33.9%

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

      6. associate-*l/ 33.3%

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

      7. associate-*l/ 33.3%

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

      8. frac-times 34.4%

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

      9. metadata-eval 34.4%

         \[\leadsto 2 \cdot \left(\sin \left(\pi \cdot \frac{angle}{180}\right) \cdot \left(\left(\left(b + a\right) \cdot \left(b - a\right)\right) \cdot \cos \left(\sqrt{\frac{\left(\pi \cdot angle\right) \cdot \left(\pi \cdot angle\right)}{\color{blue}{32400}}}\right)\right)\right) \]

      10. metadata-eval 34.4%

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

      11. frac-times 33.3%

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

      12. associate-*r/ 34.5%

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

      13. associate-*r/ 34.9%

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

      14. sqrt-unprod 0.0%

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

      15. add-sqr-sqrt 49.8%

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

      16. add-cube-cbrt 49.8%

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

   7. Applied egg-rr 52.0%

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

   if -5e-83 < (/.f64 angle 180) < 4e11

   1. Initial program 66.9%

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

   3. Simplified 66.9%

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

   4. Taylor expanded in angle around 0 66.2%

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

      1. unpow2 66.9%

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

      2. unpow2 66.9%

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

      3. difference-of-squares 73.1%

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

   6. Applied egg-rr 72.4%

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

   7. Taylor expanded in angle around 0 72.4%

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

      1. *-commutative 72.4%

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

      2. associate-*r* 72.5%

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

      3. +-commutative 72.5%

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

      4. *-commutative 72.5%

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

      5. +-commutative 72.5%

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

   9. Simplified 72.5%

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

       1. associate-*r* 98.7%

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

       2. +-commutative 98.7%

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

       3. distribute-lft-in 95.6%

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

       4. *-commutative 95.6%

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

       5. associate-*l* 95.6%

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

       6. *-commutative 95.6%

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

       7. associate-*l* 95.6%

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

   11. Applied egg-rr 95.6%

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

   if 4e11 < (/.f64 angle 180)

   1. Initial program 24.0%

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

   3. Simplified 26.0%

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

      1. unpow2 26.0%

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

      2. unpow2 26.0%

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

      3. difference-of-squares 36.3%

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

   5. Applied egg-rr 36.3%

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

      1. clear-num 32.0%

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

      2. un-div-inv 32.1%

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

   7. Applied egg-rr 32.1%

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

   9. Applied egg-rr 41.6%

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

       1. associate-*r* 41.6%

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

       2. *-commutative 41.6%

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

       3. *-commutative 41.6%

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

   11. Simplified 41.6%

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

4. Final simplification 65.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{angle}{180} \leq -5 \cdot 10^{-83}:\\ \;\;\;\;2 \cdot \left(\sin \left(\frac{angle}{180} \cdot \pi\right) \cdot \left(\left(\left(b - a\right) \cdot \left(b + a\right)\right) \cdot {\left(\sqrt[3]{\cos \left(\pi \cdot \left(angle \cdot 0.005555555555555556\right)\right)}\right)}^{3}\right)\right)\\ \mathbf{elif}\;\frac{angle}{180} \leq 400000000000:\\ \;\;\;\;\left(b \cdot \left(\pi \cdot \left(angle \cdot \left(b - a\right)\right)\right) + a \cdot \left(\pi \cdot \left(angle \cdot \left(b - a\right)\right)\right)\right) \cdot 0.011111111111111112\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \left(\sin \left(\frac{\pi}{\frac{180}{angle}}\right) \cdot \left(\left(\left(b - a\right) \cdot \left(b + a\right)\right) \cdot \sqrt{{\cos \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{2}}\right)\right)\\ \end{array} \]

Alternative 7: 66.4% accurate, 1.0× speedup

Math

\[\begin{array}{l} a = |a|\\ b = |b|\\ \\ \begin{array}{l} t_0 := \left(b - a\right) \cdot \left(b + a\right)\\ t_1 := \pi \cdot \left(angle \cdot \left(b - a\right)\right)\\ t_2 := \cos \left(\pi \cdot \left(angle \cdot 0.005555555555555556\right)\right)\\ \mathbf{if}\;\frac{angle}{180} \leq -5 \cdot 10^{-83}:\\ \;\;\;\;2 \cdot \left(\sin \left(\frac{angle}{180} \cdot \pi\right) \cdot \left(t_0 \cdot {\left(\sqrt[3]{t_2}\right)}^{3}\right)\right)\\ \mathbf{elif}\;\frac{angle}{180} \leq 400000000000:\\ \;\;\;\;\left(b \cdot t_1 + a \cdot t_1\right) \cdot 0.011111111111111112\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \left(\sin \left(\frac{\pi}{\frac{180}{angle}}\right) \cdot \left(t_0 \cdot \left|t_2\right|\right)\right)\\ \end{array} \end{array} \]

FPCore

NOTE: a should be positive before calling this function
NOTE: b should be positive before calling this function
(FPCore (a b angle)
 :precision binary64
 (let* ((t_0 (* (- b a) (+ b a)))
        (t_1 (* PI (* angle (- b a))))
        (t_2 (cos (* PI (* angle 0.005555555555555556)))))
   (if (<= (/ angle 180.0) -5e-83)
     (* 2.0 (* (sin (* (/ angle 180.0) PI)) (* t_0 (pow (cbrt t_2) 3.0))))
     (if (<= (/ angle 180.0) 400000000000.0)
       (* (+ (* b t_1) (* a t_1)) 0.011111111111111112)
       (* 2.0 (* (sin (/ PI (/ 180.0 angle))) (* t_0 (fabs t_2))))))))

C

a = abs(a);
b = abs(b);
double code(double a, double b, double angle) {
	double t_0 = (b - a) * (b + a);
	double t_1 = ((double) M_PI) * (angle * (b - a));
	double t_2 = cos((((double) M_PI) * (angle * 0.005555555555555556)));
	double tmp;
	if ((angle / 180.0) <= -5e-83) {
		tmp = 2.0 * (sin(((angle / 180.0) * ((double) M_PI))) * (t_0 * pow(cbrt(t_2), 3.0)));
	} else if ((angle / 180.0) <= 400000000000.0) {
		tmp = ((b * t_1) + (a * t_1)) * 0.011111111111111112;
	} else {
		tmp = 2.0 * (sin((((double) M_PI) / (180.0 / angle))) * (t_0 * fabs(t_2)));
	}
	return tmp;
}

Java

a = Math.abs(a);
b = Math.abs(b);
public static double code(double a, double b, double angle) {
	double t_0 = (b - a) * (b + a);
	double t_1 = Math.PI * (angle * (b - a));
	double t_2 = Math.cos((Math.PI * (angle * 0.005555555555555556)));
	double tmp;
	if ((angle / 180.0) <= -5e-83) {
		tmp = 2.0 * (Math.sin(((angle / 180.0) * Math.PI)) * (t_0 * Math.pow(Math.cbrt(t_2), 3.0)));
	} else if ((angle / 180.0) <= 400000000000.0) {
		tmp = ((b * t_1) + (a * t_1)) * 0.011111111111111112;
	} else {
		tmp = 2.0 * (Math.sin((Math.PI / (180.0 / angle))) * (t_0 * Math.abs(t_2)));
	}
	return tmp;
}

Julia

a = abs(a)
b = abs(b)
function code(a, b, angle)
	t_0 = Float64(Float64(b - a) * Float64(b + a))
	t_1 = Float64(pi * Float64(angle * Float64(b - a)))
	t_2 = cos(Float64(pi * Float64(angle * 0.005555555555555556)))
	tmp = 0.0
	if (Float64(angle / 180.0) <= -5e-83)
		tmp = Float64(2.0 * Float64(sin(Float64(Float64(angle / 180.0) * pi)) * Float64(t_0 * (cbrt(t_2) ^ 3.0))));
	elseif (Float64(angle / 180.0) <= 400000000000.0)
		tmp = Float64(Float64(Float64(b * t_1) + Float64(a * t_1)) * 0.011111111111111112);
	else
		tmp = Float64(2.0 * Float64(sin(Float64(pi / Float64(180.0 / angle))) * Float64(t_0 * abs(t_2))));
	end
	return tmp
end

Wolfram

NOTE: a should be positive before calling this function
NOTE: b should be positive before calling this function
code[a_, b_, angle_] := Block[{t$95$0 = N[(N[(b - a), $MachinePrecision] * N[(b + a), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(Pi * N[(angle * N[(b - a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[Cos[N[(Pi * N[(angle * 0.005555555555555556), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[N[(angle / 180.0), $MachinePrecision], -5e-83], N[(2.0 * N[(N[Sin[N[(N[(angle / 180.0), $MachinePrecision] * Pi), $MachinePrecision]], $MachinePrecision] * N[(t$95$0 * N[Power[N[Power[t$95$2, 1/3], $MachinePrecision], 3.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[N[(angle / 180.0), $MachinePrecision], 400000000000.0], N[(N[(N[(b * t$95$1), $MachinePrecision] + N[(a * t$95$1), $MachinePrecision]), $MachinePrecision] * 0.011111111111111112), $MachinePrecision], N[(2.0 * N[(N[Sin[N[(Pi / N[(180.0 / angle), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * N[(t$95$0 * N[Abs[t$95$2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]]

Derivation

1. Split input into 3 regimes

2. if (/.f64 angle 180) < -5e-83

   1. Initial program 40.9%

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

   3. Simplified 40.1%

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

      1. unpow2 40.1%

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

      2. unpow2 40.1%

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

      3. difference-of-squares 49.0%

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

   5. Applied egg-rr 49.0%

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

      1. *-commutative 49.0%

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

      2. add-sqr-sqrt 47.3%

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

      3. sqrt-unprod 33.9%

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

      4. *-commutative 33.9%

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

      5. *-commutative 33.9%

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

      6. associate-*l/ 33.3%

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

      7. associate-*l/ 33.3%

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

      8. frac-times 34.4%

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

      9. metadata-eval 34.4%

         \[\leadsto 2 \cdot \left(\sin \left(\pi \cdot \frac{angle}{180}\right) \cdot \left(\left(\left(b + a\right) \cdot \left(b - a\right)\right) \cdot \cos \left(\sqrt{\frac{\left(\pi \cdot angle\right) \cdot \left(\pi \cdot angle\right)}{\color{blue}{32400}}}\right)\right)\right) \]

      10. metadata-eval 34.4%

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

      11. frac-times 33.3%

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

      12. associate-*r/ 34.5%

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

      13. associate-*r/ 34.9%

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

      14. sqrt-unprod 0.0%

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

      15. add-sqr-sqrt 49.8%

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

      16. add-cube-cbrt 49.8%

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

   7. Applied egg-rr 52.0%

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

   if -5e-83 < (/.f64 angle 180) < 4e11

   1. Initial program 66.9%

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

   3. Simplified 66.9%

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

   4. Taylor expanded in angle around 0 66.2%

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

      1. unpow2 66.9%

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

      2. unpow2 66.9%

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

      3. difference-of-squares 73.1%

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

   6. Applied egg-rr 72.4%

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

   7. Taylor expanded in angle around 0 72.4%

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

      1. *-commutative 72.4%

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

      2. associate-*r* 72.5%

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

      3. +-commutative 72.5%

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

      4. *-commutative 72.5%

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

      5. +-commutative 72.5%

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

   9. Simplified 72.5%

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

       1. associate-*r* 98.7%

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

       2. +-commutative 98.7%

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

       3. distribute-lft-in 95.6%

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

       4. *-commutative 95.6%

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

       5. associate-*l* 95.6%

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

       6. *-commutative 95.6%

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

       7. associate-*l* 95.6%

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

   11. Applied egg-rr 95.6%

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

   if 4e11 < (/.f64 angle 180)

   1. Initial program 24.0%

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

   3. Simplified 26.0%

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

      1. unpow2 26.0%

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

      2. unpow2 26.0%

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

      3. difference-of-squares 36.3%

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

   5. Applied egg-rr 36.3%

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

      1. clear-num 32.0%

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

      2. un-div-inv 32.1%

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

   7. Applied egg-rr 32.1%

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

   9. Applied egg-rr 41.6%

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

       1. unpow2 41.6%

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

       2. rem-sqrt-square 41.6%

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

       3. *-commutative 41.6%

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

   11. Simplified 41.6%

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

4. Final simplification 65.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{angle}{180} \leq -5 \cdot 10^{-83}:\\ \;\;\;\;2 \cdot \left(\sin \left(\frac{angle}{180} \cdot \pi\right) \cdot \left(\left(\left(b - a\right) \cdot \left(b + a\right)\right) \cdot {\left(\sqrt[3]{\cos \left(\pi \cdot \left(angle \cdot 0.005555555555555556\right)\right)}\right)}^{3}\right)\right)\\ \mathbf{elif}\;\frac{angle}{180} \leq 400000000000:\\ \;\;\;\;\left(b \cdot \left(\pi \cdot \left(angle \cdot \left(b - a\right)\right)\right) + a \cdot \left(\pi \cdot \left(angle \cdot \left(b - a\right)\right)\right)\right) \cdot 0.011111111111111112\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \left(\sin \left(\frac{\pi}{\frac{180}{angle}}\right) \cdot \left(\left(\left(b - a\right) \cdot \left(b + a\right)\right) \cdot \left|\cos \left(\pi \cdot \left(angle \cdot 0.005555555555555556\right)\right)\right|\right)\right)\\ \end{array} \]

Alternative 8: 66.4% accurate, 1.0× speedup

Math

\[\begin{array}{l} a = |a|\\ b = |b|\\ \\ \begin{array}{l} t_0 := \left(b - a\right) \cdot \left(b + a\right)\\ t_1 := \pi \cdot \left(angle \cdot \left(b - a\right)\right)\\ t_2 := \cos \left(\pi \cdot \left(angle \cdot 0.005555555555555556\right)\right)\\ \mathbf{if}\;\frac{angle}{180} \leq -5 \cdot 10^{-83}:\\ \;\;\;\;2 \cdot \left(\sin \left(\frac{angle}{180} \cdot \pi\right) \cdot \left(t_0 \cdot \mathsf{log1p}\left(\mathsf{expm1}\left(t_2\right)\right)\right)\right)\\ \mathbf{elif}\;\frac{angle}{180} \leq 400000000000:\\ \;\;\;\;\left(b \cdot t_1 + a \cdot t_1\right) \cdot 0.011111111111111112\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \left(\sin \left(\frac{\pi}{\frac{180}{angle}}\right) \cdot \left(t_0 \cdot \left|t_2\right|\right)\right)\\ \end{array} \end{array} \]

FPCore

NOTE: a should be positive before calling this function
NOTE: b should be positive before calling this function
(FPCore (a b angle)
 :precision binary64
 (let* ((t_0 (* (- b a) (+ b a)))
        (t_1 (* PI (* angle (- b a))))
        (t_2 (cos (* PI (* angle 0.005555555555555556)))))
   (if (<= (/ angle 180.0) -5e-83)
     (* 2.0 (* (sin (* (/ angle 180.0) PI)) (* t_0 (log1p (expm1 t_2)))))
     (if (<= (/ angle 180.0) 400000000000.0)
       (* (+ (* b t_1) (* a t_1)) 0.011111111111111112)
       (* 2.0 (* (sin (/ PI (/ 180.0 angle))) (* t_0 (fabs t_2))))))))

C

a = abs(a);
b = abs(b);
double code(double a, double b, double angle) {
	double t_0 = (b - a) * (b + a);
	double t_1 = ((double) M_PI) * (angle * (b - a));
	double t_2 = cos((((double) M_PI) * (angle * 0.005555555555555556)));
	double tmp;
	if ((angle / 180.0) <= -5e-83) {
		tmp = 2.0 * (sin(((angle / 180.0) * ((double) M_PI))) * (t_0 * log1p(expm1(t_2))));
	} else if ((angle / 180.0) <= 400000000000.0) {
		tmp = ((b * t_1) + (a * t_1)) * 0.011111111111111112;
	} else {
		tmp = 2.0 * (sin((((double) M_PI) / (180.0 / angle))) * (t_0 * fabs(t_2)));
	}
	return tmp;
}

Java

a = Math.abs(a);
b = Math.abs(b);
public static double code(double a, double b, double angle) {
	double t_0 = (b - a) * (b + a);
	double t_1 = Math.PI * (angle * (b - a));
	double t_2 = Math.cos((Math.PI * (angle * 0.005555555555555556)));
	double tmp;
	if ((angle / 180.0) <= -5e-83) {
		tmp = 2.0 * (Math.sin(((angle / 180.0) * Math.PI)) * (t_0 * Math.log1p(Math.expm1(t_2))));
	} else if ((angle / 180.0) <= 400000000000.0) {
		tmp = ((b * t_1) + (a * t_1)) * 0.011111111111111112;
	} else {
		tmp = 2.0 * (Math.sin((Math.PI / (180.0 / angle))) * (t_0 * Math.abs(t_2)));
	}
	return tmp;
}

Python

a = abs(a)
b = abs(b)
def code(a, b, angle):
	t_0 = (b - a) * (b + a)
	t_1 = math.pi * (angle * (b - a))
	t_2 = math.cos((math.pi * (angle * 0.005555555555555556)))
	tmp = 0
	if (angle / 180.0) <= -5e-83:
		tmp = 2.0 * (math.sin(((angle / 180.0) * math.pi)) * (t_0 * math.log1p(math.expm1(t_2))))
	elif (angle / 180.0) <= 400000000000.0:
		tmp = ((b * t_1) + (a * t_1)) * 0.011111111111111112
	else:
		tmp = 2.0 * (math.sin((math.pi / (180.0 / angle))) * (t_0 * math.fabs(t_2)))
	return tmp

Julia

a = abs(a)
b = abs(b)
function code(a, b, angle)
	t_0 = Float64(Float64(b - a) * Float64(b + a))
	t_1 = Float64(pi * Float64(angle * Float64(b - a)))
	t_2 = cos(Float64(pi * Float64(angle * 0.005555555555555556)))
	tmp = 0.0
	if (Float64(angle / 180.0) <= -5e-83)
		tmp = Float64(2.0 * Float64(sin(Float64(Float64(angle / 180.0) * pi)) * Float64(t_0 * log1p(expm1(t_2)))));
	elseif (Float64(angle / 180.0) <= 400000000000.0)
		tmp = Float64(Float64(Float64(b * t_1) + Float64(a * t_1)) * 0.011111111111111112);
	else
		tmp = Float64(2.0 * Float64(sin(Float64(pi / Float64(180.0 / angle))) * Float64(t_0 * abs(t_2))));
	end
	return tmp
end

Wolfram

NOTE: a should be positive before calling this function
NOTE: b should be positive before calling this function
code[a_, b_, angle_] := Block[{t$95$0 = N[(N[(b - a), $MachinePrecision] * N[(b + a), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(Pi * N[(angle * N[(b - a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[Cos[N[(Pi * N[(angle * 0.005555555555555556), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[N[(angle / 180.0), $MachinePrecision], -5e-83], N[(2.0 * N[(N[Sin[N[(N[(angle / 180.0), $MachinePrecision] * Pi), $MachinePrecision]], $MachinePrecision] * N[(t$95$0 * N[Log[1 + N[(Exp[t$95$2] - 1), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[N[(angle / 180.0), $MachinePrecision], 400000000000.0], N[(N[(N[(b * t$95$1), $MachinePrecision] + N[(a * t$95$1), $MachinePrecision]), $MachinePrecision] * 0.011111111111111112), $MachinePrecision], N[(2.0 * N[(N[Sin[N[(Pi / N[(180.0 / angle), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * N[(t$95$0 * N[Abs[t$95$2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]]

Derivation

1. Split input into 3 regimes

2. if (/.f64 angle 180) < -5e-83

   1. Initial program 40.9%

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

   3. Simplified 40.1%

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

      1. unpow2 40.1%

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

      2. unpow2 40.1%

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

      3. difference-of-squares 49.0%

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

   5. Applied egg-rr 49.0%

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

      1. *-commutative 49.0%

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

      2. add-sqr-sqrt 47.3%

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

      3. sqrt-unprod 33.9%

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

      4. *-commutative 33.9%

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

      5. *-commutative 33.9%

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

      6. associate-*l/ 33.3%

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

      7. associate-*l/ 33.3%

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

      8. frac-times 34.4%

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

      9. metadata-eval 34.4%

         \[\leadsto 2 \cdot \left(\sin \left(\pi \cdot \frac{angle}{180}\right) \cdot \left(\left(\left(b + a\right) \cdot \left(b - a\right)\right) \cdot \cos \left(\sqrt{\frac{\left(\pi \cdot angle\right) \cdot \left(\pi \cdot angle\right)}{\color{blue}{32400}}}\right)\right)\right) \]

      10. metadata-eval 34.4%

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

      11. frac-times 33.3%

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

      12. associate-*r/ 34.5%

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

      13. associate-*r/ 34.9%

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

      14. sqrt-unprod 0.0%

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

      15. add-sqr-sqrt 49.8%

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

      16. log1p-expm1-u 49.8%

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

   7. Applied egg-rr 52.0%

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

   if -5e-83 < (/.f64 angle 180) < 4e11

   1. Initial program 66.9%

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

   3. Simplified 66.9%

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

   4. Taylor expanded in angle around 0 66.2%

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

      1. unpow2 66.9%

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

      2. unpow2 66.9%

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

      3. difference-of-squares 73.1%

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

   6. Applied egg-rr 72.4%

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

   7. Taylor expanded in angle around 0 72.4%

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

      1. *-commutative 72.4%

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

      2. associate-*r* 72.5%

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

      3. +-commutative 72.5%

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

      4. *-commutative 72.5%

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

      5. +-commutative 72.5%

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

   9. Simplified 72.5%

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

       1. associate-*r* 98.7%

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

       2. +-commutative 98.7%

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

       3. distribute-lft-in 95.6%

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

       4. *-commutative 95.6%

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

       5. associate-*l* 95.6%

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

       6. *-commutative 95.6%

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

       7. associate-*l* 95.6%

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

   11. Applied egg-rr 95.6%

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

   if 4e11 < (/.f64 angle 180)

   1. Initial program 24.0%

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

   3. Simplified 26.0%

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

      1. unpow2 26.0%

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

      2. unpow2 26.0%

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

      3. difference-of-squares 36.3%

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

   5. Applied egg-rr 36.3%

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

      1. clear-num 32.0%

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

      2. un-div-inv 32.1%

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

   7. Applied egg-rr 32.1%

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

   9. Applied egg-rr 41.6%

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

       1. unpow2 41.6%

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

       2. rem-sqrt-square 41.6%

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

       3. *-commutative 41.6%

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

   11. Simplified 41.6%

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

4. Final simplification 65.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{angle}{180} \leq -5 \cdot 10^{-83}:\\ \;\;\;\;2 \cdot \left(\sin \left(\frac{angle}{180} \cdot \pi\right) \cdot \left(\left(\left(b - a\right) \cdot \left(b + a\right)\right) \cdot \mathsf{log1p}\left(\mathsf{expm1}\left(\cos \left(\pi \cdot \left(angle \cdot 0.005555555555555556\right)\right)\right)\right)\right)\right)\\ \mathbf{elif}\;\frac{angle}{180} \leq 400000000000:\\ \;\;\;\;\left(b \cdot \left(\pi \cdot \left(angle \cdot \left(b - a\right)\right)\right) + a \cdot \left(\pi \cdot \left(angle \cdot \left(b - a\right)\right)\right)\right) \cdot 0.011111111111111112\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \left(\sin \left(\frac{\pi}{\frac{180}{angle}}\right) \cdot \left(\left(\left(b - a\right) \cdot \left(b + a\right)\right) \cdot \left|\cos \left(\pi \cdot \left(angle \cdot 0.005555555555555556\right)\right)\right|\right)\right)\\ \end{array} \]

Alternative 9: 66.1% accurate, 1.2× speedup

Math

\[\begin{array}{l} a = |a|\\ b = |b|\\ \\ \begin{array}{l} t_0 := \pi \cdot \left(angle \cdot \left(b - a\right)\right)\\ t_1 := \left(b - a\right) \cdot \left(b + a\right)\\ \mathbf{if}\;\frac{angle}{180} \leq -1 \cdot 10^{+20}:\\ \;\;\;\;2 \cdot \left(\sin \left(\frac{angle \cdot \pi}{-180}\right) \cdot \left(t_1 \cdot \cos \left(angle \cdot \frac{\pi}{-180}\right)\right)\right)\\ \mathbf{elif}\;\frac{angle}{180} \leq 400000000000:\\ \;\;\;\;\left(b \cdot t_0 + a \cdot t_0\right) \cdot 0.011111111111111112\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \left(\sin \left(\frac{\pi}{\frac{180}{angle}}\right) \cdot \left(t_1 \cdot \left|\cos \left(\pi \cdot \left(angle \cdot 0.005555555555555556\right)\right)\right|\right)\right)\\ \end{array} \end{array} \]

FPCore

NOTE: a should be positive before calling this function
NOTE: b should be positive before calling this function
(FPCore (a b angle)
 :precision binary64
 (let* ((t_0 (* PI (* angle (- b a)))) (t_1 (* (- b a) (+ b a))))
   (if (<= (/ angle 180.0) -1e+20)
     (*
      2.0
      (* (sin (/ (* angle PI) -180.0)) (* t_1 (cos (* angle (/ PI -180.0))))))
     (if (<= (/ angle 180.0) 400000000000.0)
       (* (+ (* b t_0) (* a t_0)) 0.011111111111111112)
       (*
        2.0
        (*
         (sin (/ PI (/ 180.0 angle)))
         (* t_1 (fabs (cos (* PI (* angle 0.005555555555555556)))))))))))

C

a = abs(a);
b = abs(b);
double code(double a, double b, double angle) {
	double t_0 = ((double) M_PI) * (angle * (b - a));
	double t_1 = (b - a) * (b + a);
	double tmp;
	if ((angle / 180.0) <= -1e+20) {
		tmp = 2.0 * (sin(((angle * ((double) M_PI)) / -180.0)) * (t_1 * cos((angle * (((double) M_PI) / -180.0)))));
	} else if ((angle / 180.0) <= 400000000000.0) {
		tmp = ((b * t_0) + (a * t_0)) * 0.011111111111111112;
	} else {
		tmp = 2.0 * (sin((((double) M_PI) / (180.0 / angle))) * (t_1 * fabs(cos((((double) M_PI) * (angle * 0.005555555555555556))))));
	}
	return tmp;
}

Java

a = Math.abs(a);
b = Math.abs(b);
public static double code(double a, double b, double angle) {
	double t_0 = Math.PI * (angle * (b - a));
	double t_1 = (b - a) * (b + a);
	double tmp;
	if ((angle / 180.0) <= -1e+20) {
		tmp = 2.0 * (Math.sin(((angle * Math.PI) / -180.0)) * (t_1 * Math.cos((angle * (Math.PI / -180.0)))));
	} else if ((angle / 180.0) <= 400000000000.0) {
		tmp = ((b * t_0) + (a * t_0)) * 0.011111111111111112;
	} else {
		tmp = 2.0 * (Math.sin((Math.PI / (180.0 / angle))) * (t_1 * Math.abs(Math.cos((Math.PI * (angle * 0.005555555555555556))))));
	}
	return tmp;
}

Python

a = abs(a)
b = abs(b)
def code(a, b, angle):
	t_0 = math.pi * (angle * (b - a))
	t_1 = (b - a) * (b + a)
	tmp = 0
	if (angle / 180.0) <= -1e+20:
		tmp = 2.0 * (math.sin(((angle * math.pi) / -180.0)) * (t_1 * math.cos((angle * (math.pi / -180.0)))))
	elif (angle / 180.0) <= 400000000000.0:
		tmp = ((b * t_0) + (a * t_0)) * 0.011111111111111112
	else:
		tmp = 2.0 * (math.sin((math.pi / (180.0 / angle))) * (t_1 * math.fabs(math.cos((math.pi * (angle * 0.005555555555555556))))))
	return tmp

Julia

a = abs(a)
b = abs(b)
function code(a, b, angle)
	t_0 = Float64(pi * Float64(angle * Float64(b - a)))
	t_1 = Float64(Float64(b - a) * Float64(b + a))
	tmp = 0.0
	if (Float64(angle / 180.0) <= -1e+20)
		tmp = Float64(2.0 * Float64(sin(Float64(Float64(angle * pi) / -180.0)) * Float64(t_1 * cos(Float64(angle * Float64(pi / -180.0))))));
	elseif (Float64(angle / 180.0) <= 400000000000.0)
		tmp = Float64(Float64(Float64(b * t_0) + Float64(a * t_0)) * 0.011111111111111112);
	else
		tmp = Float64(2.0 * Float64(sin(Float64(pi / Float64(180.0 / angle))) * Float64(t_1 * abs(cos(Float64(pi * Float64(angle * 0.005555555555555556)))))));
	end
	return tmp
end

MATLAB

a = abs(a)
b = abs(b)
function tmp_2 = code(a, b, angle)
	t_0 = pi * (angle * (b - a));
	t_1 = (b - a) * (b + a);
	tmp = 0.0;
	if ((angle / 180.0) <= -1e+20)
		tmp = 2.0 * (sin(((angle * pi) / -180.0)) * (t_1 * cos((angle * (pi / -180.0)))));
	elseif ((angle / 180.0) <= 400000000000.0)
		tmp = ((b * t_0) + (a * t_0)) * 0.011111111111111112;
	else
		tmp = 2.0 * (sin((pi / (180.0 / angle))) * (t_1 * abs(cos((pi * (angle * 0.005555555555555556))))));
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: a should be positive before calling this function
NOTE: b should be positive before calling this function
code[a_, b_, angle_] := Block[{t$95$0 = N[(Pi * N[(angle * N[(b - a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[(b - a), $MachinePrecision] * N[(b + a), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[(angle / 180.0), $MachinePrecision], -1e+20], N[(2.0 * N[(N[Sin[N[(N[(angle * Pi), $MachinePrecision] / -180.0), $MachinePrecision]], $MachinePrecision] * N[(t$95$1 * N[Cos[N[(angle * N[(Pi / -180.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[N[(angle / 180.0), $MachinePrecision], 400000000000.0], N[(N[(N[(b * t$95$0), $MachinePrecision] + N[(a * t$95$0), $MachinePrecision]), $MachinePrecision] * 0.011111111111111112), $MachinePrecision], N[(2.0 * N[(N[Sin[N[(Pi / N[(180.0 / angle), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * N[(t$95$1 * N[Abs[N[Cos[N[(Pi * N[(angle * 0.005555555555555556), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

2. if (/.f64 angle 180) < -1e20

   1. Initial program 24.9%

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

   3. Simplified 23.7%

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

      1. unpow2 23.7%

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

      2. unpow2 23.7%

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

      3. difference-of-squares 28.4%

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

   5. Applied egg-rr 28.4%

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

      1. *-commutative 28.4%

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

      2. div-inv 28.9%

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

      3. metadata-eval 28.9%

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

      4. *-commutative 28.9%

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

      5. associate-*r* 26.4%

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

      6. add-sqr-sqrt 0.0%

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

      7. sqrt-unprod 9.1%

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

      8. associate-*r* 10.6%

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

      9. associate-*r* 9.1%

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

      10. *-commutative 9.1%

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

      11. metadata-eval 9.1%

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

      12. div-inv 8.7%

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

      13. *-commutative 8.7%

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

      14. metadata-eval 8.7%

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

      15. div-inv 10.3%

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

      16. frac-times 10.3%

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

      17. metadata-eval 10.3%

          \[\leadsto 2 \cdot \left(\sin \left(\sqrt{\frac{\left(angle \cdot \pi\right) \cdot \left(angle \cdot \pi\right)}{\color{blue}{32400}}}\right) \cdot \left(\left(\left(b + a\right) \cdot \left(b - a\right)\right) \cdot \cos \left(\frac{\pi}{-180} \cdot angle\right)\right)\right) \]

      18. metadata-eval 10.3%

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

      19. frac-times 10.3%

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

      20. *-commutative 10.3%

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

      21. associate-*l/ 9.1%

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

   7. Applied egg-rr 34.6%

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

   if -1e20 < (/.f64 angle 180) < 4e11

   1. Initial program 69.7%

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

   3. Simplified 69.8%

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

   4. Taylor expanded in angle around 0 67.7%

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

      1. unpow2 69.7%

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

      2. unpow2 69.7%

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

      3. difference-of-squares 78.7%

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

   6. Applied egg-rr 76.7%

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

   7. Taylor expanded in angle around 0 76.7%

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

      1. *-commutative 76.7%

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

      2. associate-*r* 76.7%

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

      3. +-commutative 76.7%

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

      4. *-commutative 76.7%

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

      5. +-commutative 76.7%

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

   9. Simplified 76.7%

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

       1. associate-*r* 97.4%

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

       2. +-commutative 97.4%

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

       3. distribute-lft-in 92.6%

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

       4. *-commutative 92.6%

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

       5. associate-*l* 92.5%

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

       6. *-commutative 92.5%

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

       7. associate-*l* 92.5%

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

   11. Applied egg-rr 92.5%

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

   if 4e11 < (/.f64 angle 180)

   1. Initial program 24.0%

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

   3. Simplified 26.0%

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

      1. unpow2 26.0%

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

      2. unpow2 26.0%

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

      3. difference-of-squares 36.3%

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

   5. Applied egg-rr 36.3%

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

      1. clear-num 32.0%

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

      2. un-div-inv 32.1%

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

   7. Applied egg-rr 32.1%

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

   9. Applied egg-rr 41.6%

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

       1. unpow2 41.6%

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

       2. rem-sqrt-square 41.6%

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

       3. *-commutative 41.6%

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

   11. Simplified 41.6%

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

4. Final simplification 64.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{angle}{180} \leq -1 \cdot 10^{+20}:\\ \;\;\;\;2 \cdot \left(\sin \left(\frac{angle \cdot \pi}{-180}\right) \cdot \left(\left(\left(b - a\right) \cdot \left(b + a\right)\right) \cdot \cos \left(angle \cdot \frac{\pi}{-180}\right)\right)\right)\\ \mathbf{elif}\;\frac{angle}{180} \leq 400000000000:\\ \;\;\;\;\left(b \cdot \left(\pi \cdot \left(angle \cdot \left(b - a\right)\right)\right) + a \cdot \left(\pi \cdot \left(angle \cdot \left(b - a\right)\right)\right)\right) \cdot 0.011111111111111112\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \left(\sin \left(\frac{\pi}{\frac{180}{angle}}\right) \cdot \left(\left(\left(b - a\right) \cdot \left(b + a\right)\right) \cdot \left|\cos \left(\pi \cdot \left(angle \cdot 0.005555555555555556\right)\right)\right|\right)\right)\\ \end{array} \]

Alternative 10: 66.6% accurate, 1.5× speedup

Math

\[\begin{array}{l} a = |a|\\ b = |b|\\ \\ \begin{array}{l} t_0 := \pi \cdot \left(angle \cdot \left(b - a\right)\right)\\ t_1 := \left(b - a\right) \cdot \left(b + a\right)\\ \mathbf{if}\;\frac{angle}{180} \leq -2 \cdot 10^{-97}:\\ \;\;\;\;2 \cdot \left(\cos \left(\left(angle \cdot \pi\right) \cdot -0.005555555555555556\right) \cdot \left(t_1 \cdot \sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)\right)\right)\\ \mathbf{elif}\;\frac{angle}{180} \leq 400000000000:\\ \;\;\;\;\left(b \cdot t_0 + a \cdot t_0\right) \cdot 0.011111111111111112\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \left(t_1 \cdot \sin \left(\frac{\pi}{\frac{180}{angle}}\right)\right)\\ \end{array} \end{array} \]

FPCore

NOTE: a should be positive before calling this function
NOTE: b should be positive before calling this function
(FPCore (a b angle)
 :precision binary64
 (let* ((t_0 (* PI (* angle (- b a)))) (t_1 (* (- b a) (+ b a))))
   (if (<= (/ angle 180.0) -2e-97)
     (*
      2.0
      (*
       (cos (* (* angle PI) -0.005555555555555556))
       (* t_1 (sin (* 0.005555555555555556 (* angle PI))))))
     (if (<= (/ angle 180.0) 400000000000.0)
       (* (+ (* b t_0) (* a t_0)) 0.011111111111111112)
       (* 2.0 (* t_1 (sin (/ PI (/ 180.0 angle)))))))))

C

a = abs(a);
b = abs(b);
double code(double a, double b, double angle) {
	double t_0 = ((double) M_PI) * (angle * (b - a));
	double t_1 = (b - a) * (b + a);
	double tmp;
	if ((angle / 180.0) <= -2e-97) {
		tmp = 2.0 * (cos(((angle * ((double) M_PI)) * -0.005555555555555556)) * (t_1 * sin((0.005555555555555556 * (angle * ((double) M_PI))))));
	} else if ((angle / 180.0) <= 400000000000.0) {
		tmp = ((b * t_0) + (a * t_0)) * 0.011111111111111112;
	} else {
		tmp = 2.0 * (t_1 * sin((((double) M_PI) / (180.0 / angle))));
	}
	return tmp;
}

Java

a = Math.abs(a);
b = Math.abs(b);
public static double code(double a, double b, double angle) {
	double t_0 = Math.PI * (angle * (b - a));
	double t_1 = (b - a) * (b + a);
	double tmp;
	if ((angle / 180.0) <= -2e-97) {
		tmp = 2.0 * (Math.cos(((angle * Math.PI) * -0.005555555555555556)) * (t_1 * Math.sin((0.005555555555555556 * (angle * Math.PI)))));
	} else if ((angle / 180.0) <= 400000000000.0) {
		tmp = ((b * t_0) + (a * t_0)) * 0.011111111111111112;
	} else {
		tmp = 2.0 * (t_1 * Math.sin((Math.PI / (180.0 / angle))));
	}
	return tmp;
}

Python

a = abs(a)
b = abs(b)
def code(a, b, angle):
	t_0 = math.pi * (angle * (b - a))
	t_1 = (b - a) * (b + a)
	tmp = 0
	if (angle / 180.0) <= -2e-97:
		tmp = 2.0 * (math.cos(((angle * math.pi) * -0.005555555555555556)) * (t_1 * math.sin((0.005555555555555556 * (angle * math.pi)))))
	elif (angle / 180.0) <= 400000000000.0:
		tmp = ((b * t_0) + (a * t_0)) * 0.011111111111111112
	else:
		tmp = 2.0 * (t_1 * math.sin((math.pi / (180.0 / angle))))
	return tmp

Julia

a = abs(a)
b = abs(b)
function code(a, b, angle)
	t_0 = Float64(pi * Float64(angle * Float64(b - a)))
	t_1 = Float64(Float64(b - a) * Float64(b + a))
	tmp = 0.0
	if (Float64(angle / 180.0) <= -2e-97)
		tmp = Float64(2.0 * Float64(cos(Float64(Float64(angle * pi) * -0.005555555555555556)) * Float64(t_1 * sin(Float64(0.005555555555555556 * Float64(angle * pi))))));
	elseif (Float64(angle / 180.0) <= 400000000000.0)
		tmp = Float64(Float64(Float64(b * t_0) + Float64(a * t_0)) * 0.011111111111111112);
	else
		tmp = Float64(2.0 * Float64(t_1 * sin(Float64(pi / Float64(180.0 / angle)))));
	end
	return tmp
end

MATLAB

a = abs(a)
b = abs(b)
function tmp_2 = code(a, b, angle)
	t_0 = pi * (angle * (b - a));
	t_1 = (b - a) * (b + a);
	tmp = 0.0;
	if ((angle / 180.0) <= -2e-97)
		tmp = 2.0 * (cos(((angle * pi) * -0.005555555555555556)) * (t_1 * sin((0.005555555555555556 * (angle * pi)))));
	elseif ((angle / 180.0) <= 400000000000.0)
		tmp = ((b * t_0) + (a * t_0)) * 0.011111111111111112;
	else
		tmp = 2.0 * (t_1 * sin((pi / (180.0 / angle))));
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: a should be positive before calling this function
NOTE: b should be positive before calling this function
code[a_, b_, angle_] := Block[{t$95$0 = N[(Pi * N[(angle * N[(b - a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[(b - a), $MachinePrecision] * N[(b + a), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[(angle / 180.0), $MachinePrecision], -2e-97], N[(2.0 * N[(N[Cos[N[(N[(angle * Pi), $MachinePrecision] * -0.005555555555555556), $MachinePrecision]], $MachinePrecision] * N[(t$95$1 * N[Sin[N[(0.005555555555555556 * N[(angle * Pi), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[N[(angle / 180.0), $MachinePrecision], 400000000000.0], N[(N[(N[(b * t$95$0), $MachinePrecision] + N[(a * t$95$0), $MachinePrecision]), $MachinePrecision] * 0.011111111111111112), $MachinePrecision], N[(2.0 * N[(t$95$1 * N[Sin[N[(Pi / N[(180.0 / angle), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

2. if (/.f64 angle 180) < -2.00000000000000007e-97

   1. Initial program 43.4%

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

   3. Simplified 42.6%

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

      1. unpow2 42.6%

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

      2. unpow2 42.6%

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

      3. difference-of-squares 51.1%

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

   5. Applied egg-rr 51.1%

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

   6. Taylor expanded in angle around inf 51.6%

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

   if -2.00000000000000007e-97 < (/.f64 angle 180) < 4e11

   1. Initial program 65.5%

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

   3. Simplified 65.5%

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

   4. Taylor expanded in angle around 0 64.8%

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

      1. unpow2 65.5%

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

      2. unpow2 65.5%

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

      3. difference-of-squares 72.0%

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

   6. Applied egg-rr 71.3%

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

   7. Taylor expanded in angle around 0 71.3%

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

      1. *-commutative 71.3%

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

      2. associate-*r* 71.3%

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

      3. +-commutative 71.3%

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

      4. *-commutative 71.3%

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

      5. +-commutative 71.3%

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

   9. Simplified 71.3%

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

       1. associate-*r* 98.6%

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

       2. +-commutative 98.6%

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

       3. distribute-lft-in 95.5%

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

       4. *-commutative 95.5%

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

       5. associate-*l* 95.4%

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

       6. *-commutative 95.4%

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

       7. associate-*l* 95.4%

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

   11. Applied egg-rr 95.4%

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

   if 4e11 < (/.f64 angle 180)

   1. Initial program 24.0%

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

   3. Simplified 26.0%

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

      1. unpow2 26.0%

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

      2. unpow2 26.0%

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

      3. difference-of-squares 36.3%

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

   5. Applied egg-rr 36.3%

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

      1. clear-num 32.0%

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

      2. un-div-inv 32.1%

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

   7. Applied egg-rr 32.1%

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

   8. Taylor expanded in angle around 0 41.3%

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

4. Final simplification 65.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{angle}{180} \leq -2 \cdot 10^{-97}:\\ \;\;\;\;2 \cdot \left(\cos \left(\left(angle \cdot \pi\right) \cdot -0.005555555555555556\right) \cdot \left(\left(\left(b - a\right) \cdot \left(b + a\right)\right) \cdot \sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)\right)\right)\\ \mathbf{elif}\;\frac{angle}{180} \leq 400000000000:\\ \;\;\;\;\left(b \cdot \left(\pi \cdot \left(angle \cdot \left(b - a\right)\right)\right) + a \cdot \left(\pi \cdot \left(angle \cdot \left(b - a\right)\right)\right)\right) \cdot 0.011111111111111112\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \left(\left(\left(b - a\right) \cdot \left(b + a\right)\right) \cdot \sin \left(\frac{\pi}{\frac{180}{angle}}\right)\right)\\ \end{array} \]

Alternative 11: 66.2% accurate, 1.5× speedup

Math

\[\begin{array}{l} a = |a|\\ b = |b|\\ \\ \begin{array}{l} t_0 := \left(b - a\right) \cdot \left(b + a\right)\\ t_1 := \pi \cdot \left(angle \cdot \left(b - a\right)\right)\\ \mathbf{if}\;\frac{angle}{180} \leq -5 \cdot 10^{-83}:\\ \;\;\;\;2 \cdot \left(\sin \left(\frac{angle}{180} \cdot \pi\right) \cdot \left(t_0 \cdot \cos \left(\left(angle \cdot \pi\right) \cdot -0.005555555555555556\right)\right)\right)\\ \mathbf{elif}\;\frac{angle}{180} \leq 400000000000:\\ \;\;\;\;\left(b \cdot t_1 + a \cdot t_1\right) \cdot 0.011111111111111112\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \left(t_0 \cdot \sin \left(\frac{\pi}{\frac{180}{angle}}\right)\right)\\ \end{array} \end{array} \]

FPCore

NOTE: a should be positive before calling this function
NOTE: b should be positive before calling this function
(FPCore (a b angle)
 :precision binary64
 (let* ((t_0 (* (- b a) (+ b a))) (t_1 (* PI (* angle (- b a)))))
   (if (<= (/ angle 180.0) -5e-83)
     (*
      2.0
      (*
       (sin (* (/ angle 180.0) PI))
       (* t_0 (cos (* (* angle PI) -0.005555555555555556)))))
     (if (<= (/ angle 180.0) 400000000000.0)
       (* (+ (* b t_1) (* a t_1)) 0.011111111111111112)
       (* 2.0 (* t_0 (sin (/ PI (/ 180.0 angle)))))))))

C

a = abs(a);
b = abs(b);
double code(double a, double b, double angle) {
	double t_0 = (b - a) * (b + a);
	double t_1 = ((double) M_PI) * (angle * (b - a));
	double tmp;
	if ((angle / 180.0) <= -5e-83) {
		tmp = 2.0 * (sin(((angle / 180.0) * ((double) M_PI))) * (t_0 * cos(((angle * ((double) M_PI)) * -0.005555555555555556))));
	} else if ((angle / 180.0) <= 400000000000.0) {
		tmp = ((b * t_1) + (a * t_1)) * 0.011111111111111112;
	} else {
		tmp = 2.0 * (t_0 * sin((((double) M_PI) / (180.0 / angle))));
	}
	return tmp;
}

Java

a = Math.abs(a);
b = Math.abs(b);
public static double code(double a, double b, double angle) {
	double t_0 = (b - a) * (b + a);
	double t_1 = Math.PI * (angle * (b - a));
	double tmp;
	if ((angle / 180.0) <= -5e-83) {
		tmp = 2.0 * (Math.sin(((angle / 180.0) * Math.PI)) * (t_0 * Math.cos(((angle * Math.PI) * -0.005555555555555556))));
	} else if ((angle / 180.0) <= 400000000000.0) {
		tmp = ((b * t_1) + (a * t_1)) * 0.011111111111111112;
	} else {
		tmp = 2.0 * (t_0 * Math.sin((Math.PI / (180.0 / angle))));
	}
	return tmp;
}

Python

a = abs(a)
b = abs(b)
def code(a, b, angle):
	t_0 = (b - a) * (b + a)
	t_1 = math.pi * (angle * (b - a))
	tmp = 0
	if (angle / 180.0) <= -5e-83:
		tmp = 2.0 * (math.sin(((angle / 180.0) * math.pi)) * (t_0 * math.cos(((angle * math.pi) * -0.005555555555555556))))
	elif (angle / 180.0) <= 400000000000.0:
		tmp = ((b * t_1) + (a * t_1)) * 0.011111111111111112
	else:
		tmp = 2.0 * (t_0 * math.sin((math.pi / (180.0 / angle))))
	return tmp

Julia

a = abs(a)
b = abs(b)
function code(a, b, angle)
	t_0 = Float64(Float64(b - a) * Float64(b + a))
	t_1 = Float64(pi * Float64(angle * Float64(b - a)))
	tmp = 0.0
	if (Float64(angle / 180.0) <= -5e-83)
		tmp = Float64(2.0 * Float64(sin(Float64(Float64(angle / 180.0) * pi)) * Float64(t_0 * cos(Float64(Float64(angle * pi) * -0.005555555555555556)))));
	elseif (Float64(angle / 180.0) <= 400000000000.0)
		tmp = Float64(Float64(Float64(b * t_1) + Float64(a * t_1)) * 0.011111111111111112);
	else
		tmp = Float64(2.0 * Float64(t_0 * sin(Float64(pi / Float64(180.0 / angle)))));
	end
	return tmp
end

MATLAB

a = abs(a)
b = abs(b)
function tmp_2 = code(a, b, angle)
	t_0 = (b - a) * (b + a);
	t_1 = pi * (angle * (b - a));
	tmp = 0.0;
	if ((angle / 180.0) <= -5e-83)
		tmp = 2.0 * (sin(((angle / 180.0) * pi)) * (t_0 * cos(((angle * pi) * -0.005555555555555556))));
	elseif ((angle / 180.0) <= 400000000000.0)
		tmp = ((b * t_1) + (a * t_1)) * 0.011111111111111112;
	else
		tmp = 2.0 * (t_0 * sin((pi / (180.0 / angle))));
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: a should be positive before calling this function
NOTE: b should be positive before calling this function
code[a_, b_, angle_] := Block[{t$95$0 = N[(N[(b - a), $MachinePrecision] * N[(b + a), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(Pi * N[(angle * N[(b - a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[(angle / 180.0), $MachinePrecision], -5e-83], N[(2.0 * N[(N[Sin[N[(N[(angle / 180.0), $MachinePrecision] * Pi), $MachinePrecision]], $MachinePrecision] * N[(t$95$0 * N[Cos[N[(N[(angle * Pi), $MachinePrecision] * -0.005555555555555556), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[N[(angle / 180.0), $MachinePrecision], 400000000000.0], N[(N[(N[(b * t$95$1), $MachinePrecision] + N[(a * t$95$1), $MachinePrecision]), $MachinePrecision] * 0.011111111111111112), $MachinePrecision], N[(2.0 * N[(t$95$0 * N[Sin[N[(Pi / N[(180.0 / angle), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

2. if (/.f64 angle 180) < -5e-83

   1. Initial program 40.9%

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

   3. Simplified 40.1%

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

      1. unpow2 40.1%

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

      2. unpow2 40.1%

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

      3. difference-of-squares 49.0%

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

   5. Applied egg-rr 49.0%

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

   6. Taylor expanded in angle around inf 49.8%

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

      1. *-commutative 49.8%

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

   8. Simplified 49.8%

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

   if -5e-83 < (/.f64 angle 180) < 4e11

   1. Initial program 66.9%

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

   3. Simplified 66.9%

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

   4. Taylor expanded in angle around 0 66.2%

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

      1. unpow2 66.9%

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

      2. unpow2 66.9%

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

      3. difference-of-squares 73.1%

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

   6. Applied egg-rr 72.4%

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

   7. Taylor expanded in angle around 0 72.4%

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

      1. *-commutative 72.4%

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

      2. associate-*r* 72.5%

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

      3. +-commutative 72.5%

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

      4. *-commutative 72.5%

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

      5. +-commutative 72.5%

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

   9. Simplified 72.5%

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

       1. associate-*r* 98.7%

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

       2. +-commutative 98.7%

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

       3. distribute-lft-in 95.6%

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

       4. *-commutative 95.6%

          \[\leadsto \left(\left(\color{blue}{\left(\pi \cdot angle\right)} \cdot \left(b - a\right)\right) \cdot b + \left(\left(angle \cdot \pi\right) \cdot \left(b - a\right)\right) \cdot a\right) \cdot 0.011111111111111112 \]

       5. associate-*l* 95.6%

          \[\leadsto \left(\color{blue}{\left(\pi \cdot \left(angle \cdot \left(b - a\right)\right)\right)} \cdot b + \left(\left(angle \cdot \pi\right) \cdot \left(b - a\right)\right) \cdot a\right) \cdot 0.011111111111111112 \]

       6. *-commutative 95.6%

          \[\leadsto \left(\left(\pi \cdot \left(angle \cdot \left(b - a\right)\right)\right) \cdot b + \left(\color{blue}{\left(\pi \cdot angle\right)} \cdot \left(b - a\right)\right) \cdot a\right) \cdot 0.011111111111111112 \]

       7. associate-*l* 95.6%

          \[\leadsto \left(\left(\pi \cdot \left(angle \cdot \left(b - a\right)\right)\right) \cdot b + \color{blue}{\left(\pi \cdot \left(angle \cdot \left(b - a\right)\right)\right)} \cdot a\right) \cdot 0.011111111111111112 \]

   11. Applied egg-rr 95.6%

       \[\leadsto \color{blue}{\left(\left(\pi \cdot \left(angle \cdot \left(b - a\right)\right)\right) \cdot b + \left(\pi \cdot \left(angle \cdot \left(b - a\right)\right)\right) \cdot a\right)} \cdot 0.011111111111111112 \]

   if 4e11 < (/.f64 angle 180)

   1. Initial program 24.0%

      \[\left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin \left(\pi \cdot \frac{angle}{180}\right)\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right) \]

   3. Simplified 26.0%

      \[\leadsto \color{blue}{2 \cdot \left(\sin \left(\pi \cdot \frac{angle}{180}\right) \cdot \left(\left({b}^{2} - {a}^{2}\right) \cdot \cos \left(\frac{\pi}{-180} \cdot angle\right)\right)\right)} \]
   4. Step-by-step derivation

      1. unpow2 26.0%

         \[\leadsto 2 \cdot \left(\sin \left(\pi \cdot \frac{angle}{180}\right) \cdot \left(\left(\color{blue}{b \cdot b} - {a}^{2}\right) \cdot \cos \left(\frac{\pi}{-180} \cdot angle\right)\right)\right) \]

      2. unpow2 26.0%

         \[\leadsto 2 \cdot \left(\sin \left(\pi \cdot \frac{angle}{180}\right) \cdot \left(\left(b \cdot b - \color{blue}{a \cdot a}\right) \cdot \cos \left(\frac{\pi}{-180} \cdot angle\right)\right)\right) \]

      3. difference-of-squares 36.3%

         \[\leadsto 2 \cdot \left(\sin \left(\pi \cdot \frac{angle}{180}\right) \cdot \left(\color{blue}{\left(\left(b + a\right) \cdot \left(b - a\right)\right)} \cdot \cos \left(\frac{\pi}{-180} \cdot angle\right)\right)\right) \]

   5. Applied egg-rr 36.3%

      \[\leadsto 2 \cdot \left(\sin \left(\pi \cdot \frac{angle}{180}\right) \cdot \left(\color{blue}{\left(\left(b + a\right) \cdot \left(b - a\right)\right)} \cdot \cos \left(\frac{\pi}{-180} \cdot angle\right)\right)\right) \]
   6. Step-by-step derivation

      1. clear-num 32.0%

         \[\leadsto 2 \cdot \left(\sin \left(\pi \cdot \color{blue}{\frac{1}{\frac{180}{angle}}}\right) \cdot \left(\left(\left(b + a\right) \cdot \left(b - a\right)\right) \cdot \cos \left(\frac{\pi}{-180} \cdot angle\right)\right)\right) \]

      2. un-div-inv 32.1%

         \[\leadsto 2 \cdot \left(\sin \color{blue}{\left(\frac{\pi}{\frac{180}{angle}}\right)} \cdot \left(\left(\left(b + a\right) \cdot \left(b - a\right)\right) \cdot \cos \left(\frac{\pi}{-180} \cdot angle\right)\right)\right) \]

   7. Applied egg-rr 32.1%

      \[\leadsto 2 \cdot \left(\sin \color{blue}{\left(\frac{\pi}{\frac{180}{angle}}\right)} \cdot \left(\left(\left(b + a\right) \cdot \left(b - a\right)\right) \cdot \cos \left(\frac{\pi}{-180} \cdot angle\right)\right)\right) \]

   8. Taylor expanded in angle around 0 41.3%

      \[\leadsto 2 \cdot \left(\sin \left(\frac{\pi}{\frac{180}{angle}}\right) \cdot \left(\left(\left(b + a\right) \cdot \left(b - a\right)\right) \cdot \color{blue}{1}\right)\right) \]
3. Recombined 3 regimes into one program.

4. Final simplification 65.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{angle}{180} \leq -5 \cdot 10^{-83}:\\ \;\;\;\;2 \cdot \left(\sin \left(\frac{angle}{180} \cdot \pi\right) \cdot \left(\left(\left(b - a\right) \cdot \left(b + a\right)\right) \cdot \cos \left(\left(angle \cdot \pi\right) \cdot -0.005555555555555556\right)\right)\right)\\ \mathbf{elif}\;\frac{angle}{180} \leq 400000000000:\\ \;\;\;\;\left(b \cdot \left(\pi \cdot \left(angle \cdot \left(b - a\right)\right)\right) + a \cdot \left(\pi \cdot \left(angle \cdot \left(b - a\right)\right)\right)\right) \cdot 0.011111111111111112\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \left(\left(\left(b - a\right) \cdot \left(b + a\right)\right) \cdot \sin \left(\frac{\pi}{\frac{180}{angle}}\right)\right)\\ \end{array} \]

Alternative 12: 66.0% accurate, 1.5× speedup

Math

\[\begin{array}{l} a = |a|\\ b = |b|\\ \\ \begin{array}{l} t_0 := \pi \cdot \left(angle \cdot \left(b - a\right)\right)\\ t_1 := \left(b - a\right) \cdot \left(b + a\right)\\ \mathbf{if}\;\frac{angle}{180} \leq -1 \cdot 10^{+20}:\\ \;\;\;\;2 \cdot \left(\sin \left(\frac{angle \cdot \pi}{-180}\right) \cdot \left(t_1 \cdot \cos \left(angle \cdot \frac{\pi}{-180}\right)\right)\right)\\ \mathbf{elif}\;\frac{angle}{180} \leq 400000000000:\\ \;\;\;\;\left(b \cdot t_0 + a \cdot t_0\right) \cdot 0.011111111111111112\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \left(t_1 \cdot \sin \left(\frac{\pi}{\frac{180}{angle}}\right)\right)\\ \end{array} \end{array} \]

FPCore

NOTE: a should be positive before calling this function
NOTE: b should be positive before calling this function
(FPCore (a b angle)
 :precision binary64
 (let* ((t_0 (* PI (* angle (- b a)))) (t_1 (* (- b a) (+ b a))))
   (if (<= (/ angle 180.0) -1e+20)
     (*
      2.0
      (* (sin (/ (* angle PI) -180.0)) (* t_1 (cos (* angle (/ PI -180.0))))))
     (if (<= (/ angle 180.0) 400000000000.0)
       (* (+ (* b t_0) (* a t_0)) 0.011111111111111112)
       (* 2.0 (* t_1 (sin (/ PI (/ 180.0 angle)))))))))

C

a = abs(a);
b = abs(b);
double code(double a, double b, double angle) {
	double t_0 = ((double) M_PI) * (angle * (b - a));
	double t_1 = (b - a) * (b + a);
	double tmp;
	if ((angle / 180.0) <= -1e+20) {
		tmp = 2.0 * (sin(((angle * ((double) M_PI)) / -180.0)) * (t_1 * cos((angle * (((double) M_PI) / -180.0)))));
	} else if ((angle / 180.0) <= 400000000000.0) {
		tmp = ((b * t_0) + (a * t_0)) * 0.011111111111111112;
	} else {
		tmp = 2.0 * (t_1 * sin((((double) M_PI) / (180.0 / angle))));
	}
	return tmp;
}

Java

a = Math.abs(a);
b = Math.abs(b);
public static double code(double a, double b, double angle) {
	double t_0 = Math.PI * (angle * (b - a));
	double t_1 = (b - a) * (b + a);
	double tmp;
	if ((angle / 180.0) <= -1e+20) {
		tmp = 2.0 * (Math.sin(((angle * Math.PI) / -180.0)) * (t_1 * Math.cos((angle * (Math.PI / -180.0)))));
	} else if ((angle / 180.0) <= 400000000000.0) {
		tmp = ((b * t_0) + (a * t_0)) * 0.011111111111111112;
	} else {
		tmp = 2.0 * (t_1 * Math.sin((Math.PI / (180.0 / angle))));
	}
	return tmp;
}

Python

a = abs(a)
b = abs(b)
def code(a, b, angle):
	t_0 = math.pi * (angle * (b - a))
	t_1 = (b - a) * (b + a)
	tmp = 0
	if (angle / 180.0) <= -1e+20:
		tmp = 2.0 * (math.sin(((angle * math.pi) / -180.0)) * (t_1 * math.cos((angle * (math.pi / -180.0)))))
	elif (angle / 180.0) <= 400000000000.0:
		tmp = ((b * t_0) + (a * t_0)) * 0.011111111111111112
	else:
		tmp = 2.0 * (t_1 * math.sin((math.pi / (180.0 / angle))))
	return tmp

Julia

a = abs(a)
b = abs(b)
function code(a, b, angle)
	t_0 = Float64(pi * Float64(angle * Float64(b - a)))
	t_1 = Float64(Float64(b - a) * Float64(b + a))
	tmp = 0.0
	if (Float64(angle / 180.0) <= -1e+20)
		tmp = Float64(2.0 * Float64(sin(Float64(Float64(angle * pi) / -180.0)) * Float64(t_1 * cos(Float64(angle * Float64(pi / -180.0))))));
	elseif (Float64(angle / 180.0) <= 400000000000.0)
		tmp = Float64(Float64(Float64(b * t_0) + Float64(a * t_0)) * 0.011111111111111112);
	else
		tmp = Float64(2.0 * Float64(t_1 * sin(Float64(pi / Float64(180.0 / angle)))));
	end
	return tmp
end

MATLAB

a = abs(a)
b = abs(b)
function tmp_2 = code(a, b, angle)
	t_0 = pi * (angle * (b - a));
	t_1 = (b - a) * (b + a);
	tmp = 0.0;
	if ((angle / 180.0) <= -1e+20)
		tmp = 2.0 * (sin(((angle * pi) / -180.0)) * (t_1 * cos((angle * (pi / -180.0)))));
	elseif ((angle / 180.0) <= 400000000000.0)
		tmp = ((b * t_0) + (a * t_0)) * 0.011111111111111112;
	else
		tmp = 2.0 * (t_1 * sin((pi / (180.0 / angle))));
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: a should be positive before calling this function
NOTE: b should be positive before calling this function
code[a_, b_, angle_] := Block[{t$95$0 = N[(Pi * N[(angle * N[(b - a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[(b - a), $MachinePrecision] * N[(b + a), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[(angle / 180.0), $MachinePrecision], -1e+20], N[(2.0 * N[(N[Sin[N[(N[(angle * Pi), $MachinePrecision] / -180.0), $MachinePrecision]], $MachinePrecision] * N[(t$95$1 * N[Cos[N[(angle * N[(Pi / -180.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[N[(angle / 180.0), $MachinePrecision], 400000000000.0], N[(N[(N[(b * t$95$0), $MachinePrecision] + N[(a * t$95$0), $MachinePrecision]), $MachinePrecision] * 0.011111111111111112), $MachinePrecision], N[(2.0 * N[(t$95$1 * N[Sin[N[(Pi / N[(180.0 / angle), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

2. if (/.f64 angle 180) < -1e20

   1. Initial program 24.9%

      \[\left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin \left(\pi \cdot \frac{angle}{180}\right)\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right) \]

   3. Simplified 23.7%

      \[\leadsto \color{blue}{2 \cdot \left(\sin \left(\pi \cdot \frac{angle}{180}\right) \cdot \left(\left({b}^{2} - {a}^{2}\right) \cdot \cos \left(\frac{\pi}{-180} \cdot angle\right)\right)\right)} \]
   4. Step-by-step derivation

      1. unpow2 23.7%

         \[\leadsto 2 \cdot \left(\sin \left(\pi \cdot \frac{angle}{180}\right) \cdot \left(\left(\color{blue}{b \cdot b} - {a}^{2}\right) \cdot \cos \left(\frac{\pi}{-180} \cdot angle\right)\right)\right) \]

      2. unpow2 23.7%

         \[\leadsto 2 \cdot \left(\sin \left(\pi \cdot \frac{angle}{180}\right) \cdot \left(\left(b \cdot b - \color{blue}{a \cdot a}\right) \cdot \cos \left(\frac{\pi}{-180} \cdot angle\right)\right)\right) \]

      3. difference-of-squares 28.4%

         \[\leadsto 2 \cdot \left(\sin \left(\pi \cdot \frac{angle}{180}\right) \cdot \left(\color{blue}{\left(\left(b + a\right) \cdot \left(b - a\right)\right)} \cdot \cos \left(\frac{\pi}{-180} \cdot angle\right)\right)\right) \]

   5. Applied egg-rr 28.4%

      \[\leadsto 2 \cdot \left(\sin \left(\pi \cdot \frac{angle}{180}\right) \cdot \left(\color{blue}{\left(\left(b + a\right) \cdot \left(b - a\right)\right)} \cdot \cos \left(\frac{\pi}{-180} \cdot angle\right)\right)\right) \]
   6. Step-by-step derivation

      1. *-commutative 28.4%

         \[\leadsto 2 \cdot \left(\sin \color{blue}{\left(\frac{angle}{180} \cdot \pi\right)} \cdot \left(\left(\left(b + a\right) \cdot \left(b - a\right)\right) \cdot \cos \left(\frac{\pi}{-180} \cdot angle\right)\right)\right) \]

      2. div-inv 28.9%

         \[\leadsto 2 \cdot \left(\sin \left(\color{blue}{\left(angle \cdot \frac{1}{180}\right)} \cdot \pi\right) \cdot \left(\left(\left(b + a\right) \cdot \left(b - a\right)\right) \cdot \cos \left(\frac{\pi}{-180} \cdot angle\right)\right)\right) \]

      3. metadata-eval 28.9%

         \[\leadsto 2 \cdot \left(\sin \left(\left(angle \cdot \color{blue}{0.005555555555555556}\right) \cdot \pi\right) \cdot \left(\left(\left(b + a\right) \cdot \left(b - a\right)\right) \cdot \cos \left(\frac{\pi}{-180} \cdot angle\right)\right)\right) \]

      4. *-commutative 28.9%

         \[\leadsto 2 \cdot \left(\sin \left(\color{blue}{\left(0.005555555555555556 \cdot angle\right)} \cdot \pi\right) \cdot \left(\left(\left(b + a\right) \cdot \left(b - a\right)\right) \cdot \cos \left(\frac{\pi}{-180} \cdot angle\right)\right)\right) \]

      5. associate-*r* 26.4%

         \[\leadsto 2 \cdot \left(\sin \color{blue}{\left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)} \cdot \left(\left(\left(b + a\right) \cdot \left(b - a\right)\right) \cdot \cos \left(\frac{\pi}{-180} \cdot angle\right)\right)\right) \]

      6. add-sqr-sqrt 0.0%

         \[\leadsto 2 \cdot \left(\sin \color{blue}{\left(\sqrt{0.005555555555555556 \cdot \left(angle \cdot \pi\right)} \cdot \sqrt{0.005555555555555556 \cdot \left(angle \cdot \pi\right)}\right)} \cdot \left(\left(\left(b + a\right) \cdot \left(b - a\right)\right) \cdot \cos \left(\frac{\pi}{-180} \cdot angle\right)\right)\right) \]

      7. sqrt-unprod 9.1%

         \[\leadsto 2 \cdot \left(\sin \color{blue}{\left(\sqrt{\left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right) \cdot \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}\right)} \cdot \left(\left(\left(b + a\right) \cdot \left(b - a\right)\right) \cdot \cos \left(\frac{\pi}{-180} \cdot angle\right)\right)\right) \]

      8. associate-*r* 10.6%

         \[\leadsto 2 \cdot \left(\sin \left(\sqrt{\left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right) \cdot \color{blue}{\left(\left(0.005555555555555556 \cdot angle\right) \cdot \pi\right)}}\right) \cdot \left(\left(\left(b + a\right) \cdot \left(b - a\right)\right) \cdot \cos \left(\frac{\pi}{-180} \cdot angle\right)\right)\right) \]

      9. associate-*r* 9.1%

         \[\leadsto 2 \cdot \left(\sin \left(\sqrt{\left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right) \cdot \color{blue}{\left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}}\right) \cdot \left(\left(\left(b + a\right) \cdot \left(b - a\right)\right) \cdot \cos \left(\frac{\pi}{-180} \cdot angle\right)\right)\right) \]

      10. *-commutative 9.1%

          \[\leadsto 2 \cdot \left(\sin \left(\sqrt{\left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right) \cdot \color{blue}{\left(\left(angle \cdot \pi\right) \cdot 0.005555555555555556\right)}}\right) \cdot \left(\left(\left(b + a\right) \cdot \left(b - a\right)\right) \cdot \cos \left(\frac{\pi}{-180} \cdot angle\right)\right)\right) \]

      11. metadata-eval 9.1%

          \[\leadsto 2 \cdot \left(\sin \left(\sqrt{\left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right) \cdot \left(\left(angle \cdot \pi\right) \cdot \color{blue}{\frac{1}{180}}\right)}\right) \cdot \left(\left(\left(b + a\right) \cdot \left(b - a\right)\right) \cdot \cos \left(\frac{\pi}{-180} \cdot angle\right)\right)\right) \]

      12. div-inv 8.7%

          \[\leadsto 2 \cdot \left(\sin \left(\sqrt{\left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right) \cdot \color{blue}{\frac{angle \cdot \pi}{180}}}\right) \cdot \left(\left(\left(b + a\right) \cdot \left(b - a\right)\right) \cdot \cos \left(\frac{\pi}{-180} \cdot angle\right)\right)\right) \]

      13. *-commutative 8.7%

          \[\leadsto 2 \cdot \left(\sin \left(\sqrt{\color{blue}{\left(\left(angle \cdot \pi\right) \cdot 0.005555555555555556\right)} \cdot \frac{angle \cdot \pi}{180}}\right) \cdot \left(\left(\left(b + a\right) \cdot \left(b - a\right)\right) \cdot \cos \left(\frac{\pi}{-180} \cdot angle\right)\right)\right) \]

      14. metadata-eval 8.7%

          \[\leadsto 2 \cdot \left(\sin \left(\sqrt{\left(\left(angle \cdot \pi\right) \cdot \color{blue}{\frac{1}{180}}\right) \cdot \frac{angle \cdot \pi}{180}}\right) \cdot \left(\left(\left(b + a\right) \cdot \left(b - a\right)\right) \cdot \cos \left(\frac{\pi}{-180} \cdot angle\right)\right)\right) \]

      15. div-inv 10.3%

          \[\leadsto 2 \cdot \left(\sin \left(\sqrt{\color{blue}{\frac{angle \cdot \pi}{180}} \cdot \frac{angle \cdot \pi}{180}}\right) \cdot \left(\left(\left(b + a\right) \cdot \left(b - a\right)\right) \cdot \cos \left(\frac{\pi}{-180} \cdot angle\right)\right)\right) \]

      16. frac-times 10.3%

          \[\leadsto 2 \cdot \left(\sin \left(\sqrt{\color{blue}{\frac{\left(angle \cdot \pi\right) \cdot \left(angle \cdot \pi\right)}{180 \cdot 180}}}\right) \cdot \left(\left(\left(b + a\right) \cdot \left(b - a\right)\right) \cdot \cos \left(\frac{\pi}{-180} \cdot angle\right)\right)\right) \]

      17. metadata-eval 10.3%

          \[\leadsto 2 \cdot \left(\sin \left(\sqrt{\frac{\left(angle \cdot \pi\right) \cdot \left(angle \cdot \pi\right)}{\color{blue}{32400}}}\right) \cdot \left(\left(\left(b + a\right) \cdot \left(b - a\right)\right) \cdot \cos \left(\frac{\pi}{-180} \cdot angle\right)\right)\right) \]

      18. metadata-eval 10.3%

          \[\leadsto 2 \cdot \left(\sin \left(\sqrt{\frac{\left(angle \cdot \pi\right) \cdot \left(angle \cdot \pi\right)}{\color{blue}{-180 \cdot -180}}}\right) \cdot \left(\left(\left(b + a\right) \cdot \left(b - a\right)\right) \cdot \cos \left(\frac{\pi}{-180} \cdot angle\right)\right)\right) \]

      19. frac-times 10.3%

          \[\leadsto 2 \cdot \left(\sin \left(\sqrt{\color{blue}{\frac{angle \cdot \pi}{-180} \cdot \frac{angle \cdot \pi}{-180}}}\right) \cdot \left(\left(\left(b + a\right) \cdot \left(b - a\right)\right) \cdot \cos \left(\frac{\pi}{-180} \cdot angle\right)\right)\right) \]

      20. *-commutative 10.3%

          \[\leadsto 2 \cdot \left(\sin \left(\sqrt{\frac{\color{blue}{\pi \cdot angle}}{-180} \cdot \frac{angle \cdot \pi}{-180}}\right) \cdot \left(\left(\left(b + a\right) \cdot \left(b - a\right)\right) \cdot \cos \left(\frac{\pi}{-180} \cdot angle\right)\right)\right) \]

      21. associate-*l/ 9.1%

          \[\leadsto 2 \cdot \left(\sin \left(\sqrt{\color{blue}{\left(\frac{\pi}{-180} \cdot angle\right)} \cdot \frac{angle \cdot \pi}{-180}}\right) \cdot \left(\left(\left(b + a\right) \cdot \left(b - a\right)\right) \cdot \cos \left(\frac{\pi}{-180} \cdot angle\right)\right)\right) \]

   7. Applied egg-rr 34.6%

      \[\leadsto 2 \cdot \left(\sin \color{blue}{\left(\frac{\pi \cdot angle}{-180}\right)} \cdot \left(\left(\left(b + a\right) \cdot \left(b - a\right)\right) \cdot \cos \left(\frac{\pi}{-180} \cdot angle\right)\right)\right) \]

   if -1e20 < (/.f64 angle 180) < 4e11

   1. Initial program 69.7%

      \[\left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin \left(\pi \cdot \frac{angle}{180}\right)\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right) \]

   3. Simplified 69.8%

      \[\leadsto \color{blue}{\sin \left(angle \cdot \frac{\pi}{-180}\right) \cdot \left(-2 \cdot \left(\left({b}^{2} - {a}^{2}\right) \cdot \cos \left(angle \cdot \frac{\pi}{-180}\right)\right)\right)} \]

   4. Taylor expanded in angle around 0 67.7%

      \[\leadsto \color{blue}{0.011111111111111112 \cdot \left(angle \cdot \left(\pi \cdot \left({b}^{2} - {a}^{2}\right)\right)\right)} \]
   5. Step-by-step derivation

      1. unpow2 69.7%

         \[\leadsto 2 \cdot \left(\sin \left(\pi \cdot \frac{angle}{180}\right) \cdot \left(\left(\color{blue}{b \cdot b} - {a}^{2}\right) \cdot \cos \left(\frac{\pi}{-180} \cdot angle\right)\right)\right) \]

      2. unpow2 69.7%

         \[\leadsto 2 \cdot \left(\sin \left(\pi \cdot \frac{angle}{180}\right) \cdot \left(\left(b \cdot b - \color{blue}{a \cdot a}\right) \cdot \cos \left(\frac{\pi}{-180} \cdot angle\right)\right)\right) \]

      3. difference-of-squares 78.7%

         \[\leadsto 2 \cdot \left(\sin \left(\pi \cdot \frac{angle}{180}\right) \cdot \left(\color{blue}{\left(\left(b + a\right) \cdot \left(b - a\right)\right)} \cdot \cos \left(\frac{\pi}{-180} \cdot angle\right)\right)\right) \]

   6. Applied egg-rr 76.7%

      \[\leadsto 0.011111111111111112 \cdot \left(angle \cdot \left(\pi \cdot \color{blue}{\left(\left(b + a\right) \cdot \left(b - a\right)\right)}\right)\right) \]

   7. Taylor expanded in angle around 0 76.7%

      \[\leadsto \color{blue}{0.011111111111111112 \cdot \left(angle \cdot \left(\pi \cdot \left(\left(a + b\right) \cdot \left(b - a\right)\right)\right)\right)} \]
   8. Step-by-step derivation

      1. *-commutative 76.7%

         \[\leadsto \color{blue}{\left(angle \cdot \left(\pi \cdot \left(\left(a + b\right) \cdot \left(b - a\right)\right)\right)\right) \cdot 0.011111111111111112} \]

      2. associate-*r* 76.7%

         \[\leadsto \color{blue}{\left(\left(angle \cdot \pi\right) \cdot \left(\left(a + b\right) \cdot \left(b - a\right)\right)\right)} \cdot 0.011111111111111112 \]

      3. +-commutative 76.7%

         \[\leadsto \left(\left(angle \cdot \pi\right) \cdot \left(\color{blue}{\left(b + a\right)} \cdot \left(b - a\right)\right)\right) \cdot 0.011111111111111112 \]

      4. *-commutative 76.7%

         \[\leadsto \left(\left(angle \cdot \pi\right) \cdot \color{blue}{\left(\left(b - a\right) \cdot \left(b + a\right)\right)}\right) \cdot 0.011111111111111112 \]

      5. +-commutative 76.7%

         \[\leadsto \left(\left(angle \cdot \pi\right) \cdot \left(\left(b - a\right) \cdot \color{blue}{\left(a + b\right)}\right)\right) \cdot 0.011111111111111112 \]

   9. Simplified 76.7%

      \[\leadsto \color{blue}{\left(\left(angle \cdot \pi\right) \cdot \left(\left(b - a\right) \cdot \left(a + b\right)\right)\right) \cdot 0.011111111111111112} \]
   10. Step-by-step derivation

       1. associate-*r* 97.4%

          \[\leadsto \color{blue}{\left(\left(\left(angle \cdot \pi\right) \cdot \left(b - a\right)\right) \cdot \left(a + b\right)\right)} \cdot 0.011111111111111112 \]

       2. +-commutative 97.4%

          \[\leadsto \left(\left(\left(angle \cdot \pi\right) \cdot \left(b - a\right)\right) \cdot \color{blue}{\left(b + a\right)}\right) \cdot 0.011111111111111112 \]

       3. distribute-lft-in 92.6%

          \[\leadsto \color{blue}{\left(\left(\left(angle \cdot \pi\right) \cdot \left(b - a\right)\right) \cdot b + \left(\left(angle \cdot \pi\right) \cdot \left(b - a\right)\right) \cdot a\right)} \cdot 0.011111111111111112 \]

       4. *-commutative 92.6%

          \[\leadsto \left(\left(\color{blue}{\left(\pi \cdot angle\right)} \cdot \left(b - a\right)\right) \cdot b + \left(\left(angle \cdot \pi\right) \cdot \left(b - a\right)\right) \cdot a\right) \cdot 0.011111111111111112 \]

       5. associate-*l* 92.5%

          \[\leadsto \left(\color{blue}{\left(\pi \cdot \left(angle \cdot \left(b - a\right)\right)\right)} \cdot b + \left(\left(angle \cdot \pi\right) \cdot \left(b - a\right)\right) \cdot a\right) \cdot 0.011111111111111112 \]

       6. *-commutative 92.5%

          \[\leadsto \left(\left(\pi \cdot \left(angle \cdot \left(b - a\right)\right)\right) \cdot b + \left(\color{blue}{\left(\pi \cdot angle\right)} \cdot \left(b - a\right)\right) \cdot a\right) \cdot 0.011111111111111112 \]

       7. associate-*l* 92.5%

          \[\leadsto \left(\left(\pi \cdot \left(angle \cdot \left(b - a\right)\right)\right) \cdot b + \color{blue}{\left(\pi \cdot \left(angle \cdot \left(b - a\right)\right)\right)} \cdot a\right) \cdot 0.011111111111111112 \]

   11. Applied egg-rr 92.5%

       \[\leadsto \color{blue}{\left(\left(\pi \cdot \left(angle \cdot \left(b - a\right)\right)\right) \cdot b + \left(\pi \cdot \left(angle \cdot \left(b - a\right)\right)\right) \cdot a\right)} \cdot 0.011111111111111112 \]

   if 4e11 < (/.f64 angle 180)

   1. Initial program 24.0%

      \[\left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin \left(\pi \cdot \frac{angle}{180}\right)\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right) \]

   3. Simplified 26.0%

      \[\leadsto \color{blue}{2 \cdot \left(\sin \left(\pi \cdot \frac{angle}{180}\right) \cdot \left(\left({b}^{2} - {a}^{2}\right) \cdot \cos \left(\frac{\pi}{-180} \cdot angle\right)\right)\right)} \]
   4. Step-by-step derivation

      1. unpow2 26.0%

         \[\leadsto 2 \cdot \left(\sin \left(\pi \cdot \frac{angle}{180}\right) \cdot \left(\left(\color{blue}{b \cdot b} - {a}^{2}\right) \cdot \cos \left(\frac{\pi}{-180} \cdot angle\right)\right)\right) \]

      2. unpow2 26.0%

         \[\leadsto 2 \cdot \left(\sin \left(\pi \cdot \frac{angle}{180}\right) \cdot \left(\left(b \cdot b - \color{blue}{a \cdot a}\right) \cdot \cos \left(\frac{\pi}{-180} \cdot angle\right)\right)\right) \]

      3. difference-of-squares 36.3%

         \[\leadsto 2 \cdot \left(\sin \left(\pi \cdot \frac{angle}{180}\right) \cdot \left(\color{blue}{\left(\left(b + a\right) \cdot \left(b - a\right)\right)} \cdot \cos \left(\frac{\pi}{-180} \cdot angle\right)\right)\right) \]

   5. Applied egg-rr 36.3%

      \[\leadsto 2 \cdot \left(\sin \left(\pi \cdot \frac{angle}{180}\right) \cdot \left(\color{blue}{\left(\left(b + a\right) \cdot \left(b - a\right)\right)} \cdot \cos \left(\frac{\pi}{-180} \cdot angle\right)\right)\right) \]
   6. Step-by-step derivation

      1. clear-num 32.0%

         \[\leadsto 2 \cdot \left(\sin \left(\pi \cdot \color{blue}{\frac{1}{\frac{180}{angle}}}\right) \cdot \left(\left(\left(b + a\right) \cdot \left(b - a\right)\right) \cdot \cos \left(\frac{\pi}{-180} \cdot angle\right)\right)\right) \]

      2. un-div-inv 32.1%

         \[\leadsto 2 \cdot \left(\sin \color{blue}{\left(\frac{\pi}{\frac{180}{angle}}\right)} \cdot \left(\left(\left(b + a\right) \cdot \left(b - a\right)\right) \cdot \cos \left(\frac{\pi}{-180} \cdot angle\right)\right)\right) \]

   7. Applied egg-rr 32.1%

      \[\leadsto 2 \cdot \left(\sin \color{blue}{\left(\frac{\pi}{\frac{180}{angle}}\right)} \cdot \left(\left(\left(b + a\right) \cdot \left(b - a\right)\right) \cdot \cos \left(\frac{\pi}{-180} \cdot angle\right)\right)\right) \]

   8. Taylor expanded in angle around 0 41.3%

      \[\leadsto 2 \cdot \left(\sin \left(\frac{\pi}{\frac{180}{angle}}\right) \cdot \left(\left(\left(b + a\right) \cdot \left(b - a\right)\right) \cdot \color{blue}{1}\right)\right) \]
3. Recombined 3 regimes into one program.

4. Final simplification 64.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{angle}{180} \leq -1 \cdot 10^{+20}:\\ \;\;\;\;2 \cdot \left(\sin \left(\frac{angle \cdot \pi}{-180}\right) \cdot \left(\left(\left(b - a\right) \cdot \left(b + a\right)\right) \cdot \cos \left(angle \cdot \frac{\pi}{-180}\right)\right)\right)\\ \mathbf{elif}\;\frac{angle}{180} \leq 400000000000:\\ \;\;\;\;\left(b \cdot \left(\pi \cdot \left(angle \cdot \left(b - a\right)\right)\right) + a \cdot \left(\pi \cdot \left(angle \cdot \left(b - a\right)\right)\right)\right) \cdot 0.011111111111111112\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \left(\left(\left(b - a\right) \cdot \left(b + a\right)\right) \cdot \sin \left(\frac{\pi}{\frac{180}{angle}}\right)\right)\\ \end{array} \]

Alternative 13: 65.1% accurate, 2.7× speedup

Math

\[\begin{array}{l} a = |a|\\ b = |b|\\ \\ \begin{array}{l} t_0 := \pi \cdot \left(angle \cdot \left(b - a\right)\right)\\ t_1 := \left(b - a\right) \cdot \left(b + a\right)\\ \mathbf{if}\;\frac{angle}{180} \leq -1 \cdot 10^{+87}:\\ \;\;\;\;0.011111111111111112 \cdot \left(t_1 \cdot \left|angle \cdot \pi\right|\right)\\ \mathbf{elif}\;\frac{angle}{180} \leq 400000000000:\\ \;\;\;\;\left(b \cdot t_0 + a \cdot t_0\right) \cdot 0.011111111111111112\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \left(t_1 \cdot \sin \left(\frac{\pi}{\frac{180}{angle}}\right)\right)\\ \end{array} \end{array} \]

FPCore

NOTE: a should be positive before calling this function
NOTE: b should be positive before calling this function
(FPCore (a b angle)
 :precision binary64
 (let* ((t_0 (* PI (* angle (- b a)))) (t_1 (* (- b a) (+ b a))))
   (if (<= (/ angle 180.0) -1e+87)
     (* 0.011111111111111112 (* t_1 (fabs (* angle PI))))
     (if (<= (/ angle 180.0) 400000000000.0)
       (* (+ (* b t_0) (* a t_0)) 0.011111111111111112)
       (* 2.0 (* t_1 (sin (/ PI (/ 180.0 angle)))))))))

C

a = abs(a);
b = abs(b);
double code(double a, double b, double angle) {
	double t_0 = ((double) M_PI) * (angle * (b - a));
	double t_1 = (b - a) * (b + a);
	double tmp;
	if ((angle / 180.0) <= -1e+87) {
		tmp = 0.011111111111111112 * (t_1 * fabs((angle * ((double) M_PI))));
	} else if ((angle / 180.0) <= 400000000000.0) {
		tmp = ((b * t_0) + (a * t_0)) * 0.011111111111111112;
	} else {
		tmp = 2.0 * (t_1 * sin((((double) M_PI) / (180.0 / angle))));
	}
	return tmp;
}

Java

a = Math.abs(a);
b = Math.abs(b);
public static double code(double a, double b, double angle) {
	double t_0 = Math.PI * (angle * (b - a));
	double t_1 = (b - a) * (b + a);
	double tmp;
	if ((angle / 180.0) <= -1e+87) {
		tmp = 0.011111111111111112 * (t_1 * Math.abs((angle * Math.PI)));
	} else if ((angle / 180.0) <= 400000000000.0) {
		tmp = ((b * t_0) + (a * t_0)) * 0.011111111111111112;
	} else {
		tmp = 2.0 * (t_1 * Math.sin((Math.PI / (180.0 / angle))));
	}
	return tmp;
}

Python

a = abs(a)
b = abs(b)
def code(a, b, angle):
	t_0 = math.pi * (angle * (b - a))
	t_1 = (b - a) * (b + a)
	tmp = 0
	if (angle / 180.0) <= -1e+87:
		tmp = 0.011111111111111112 * (t_1 * math.fabs((angle * math.pi)))
	elif (angle / 180.0) <= 400000000000.0:
		tmp = ((b * t_0) + (a * t_0)) * 0.011111111111111112
	else:
		tmp = 2.0 * (t_1 * math.sin((math.pi / (180.0 / angle))))
	return tmp

Julia

a = abs(a)
b = abs(b)
function code(a, b, angle)
	t_0 = Float64(pi * Float64(angle * Float64(b - a)))
	t_1 = Float64(Float64(b - a) * Float64(b + a))
	tmp = 0.0
	if (Float64(angle / 180.0) <= -1e+87)
		tmp = Float64(0.011111111111111112 * Float64(t_1 * abs(Float64(angle * pi))));
	elseif (Float64(angle / 180.0) <= 400000000000.0)
		tmp = Float64(Float64(Float64(b * t_0) + Float64(a * t_0)) * 0.011111111111111112);
	else
		tmp = Float64(2.0 * Float64(t_1 * sin(Float64(pi / Float64(180.0 / angle)))));
	end
	return tmp
end

MATLAB

a = abs(a)
b = abs(b)
function tmp_2 = code(a, b, angle)
	t_0 = pi * (angle * (b - a));
	t_1 = (b - a) * (b + a);
	tmp = 0.0;
	if ((angle / 180.0) <= -1e+87)
		tmp = 0.011111111111111112 * (t_1 * abs((angle * pi)));
	elseif ((angle / 180.0) <= 400000000000.0)
		tmp = ((b * t_0) + (a * t_0)) * 0.011111111111111112;
	else
		tmp = 2.0 * (t_1 * sin((pi / (180.0 / angle))));
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: a should be positive before calling this function
NOTE: b should be positive before calling this function
code[a_, b_, angle_] := Block[{t$95$0 = N[(Pi * N[(angle * N[(b - a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[(b - a), $MachinePrecision] * N[(b + a), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[(angle / 180.0), $MachinePrecision], -1e+87], N[(0.011111111111111112 * N[(t$95$1 * N[Abs[N[(angle * Pi), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[N[(angle / 180.0), $MachinePrecision], 400000000000.0], N[(N[(N[(b * t$95$0), $MachinePrecision] + N[(a * t$95$0), $MachinePrecision]), $MachinePrecision] * 0.011111111111111112), $MachinePrecision], N[(2.0 * N[(t$95$1 * N[Sin[N[(Pi / N[(180.0 / angle), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

2. if (/.f64 angle 180) < -9.9999999999999996e86

   1. Initial program 24.6%

      \[\left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin \left(\pi \cdot \frac{angle}{180}\right)\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right) \]

   3. Simplified 24.3%

      \[\leadsto \color{blue}{\sin \left(angle \cdot \frac{\pi}{-180}\right) \cdot \left(-2 \cdot \left(\left({b}^{2} - {a}^{2}\right) \cdot \cos \left(angle \cdot \frac{\pi}{-180}\right)\right)\right)} \]

   4. Taylor expanded in angle around 0 13.3%

      \[\leadsto \color{blue}{0.011111111111111112 \cdot \left(angle \cdot \left(\pi \cdot \left({b}^{2} - {a}^{2}\right)\right)\right)} \]
   5. Step-by-step derivation

      1. unpow2 24.8%

         \[\leadsto 2 \cdot \left(\sin \left(\pi \cdot \frac{angle}{180}\right) \cdot \left(\left(\color{blue}{b \cdot b} - {a}^{2}\right) \cdot \cos \left(\frac{\pi}{-180} \cdot angle\right)\right)\right) \]

      2. unpow2 24.8%

         \[\leadsto 2 \cdot \left(\sin \left(\pi \cdot \frac{angle}{180}\right) \cdot \left(\left(b \cdot b - \color{blue}{a \cdot a}\right) \cdot \cos \left(\frac{\pi}{-180} \cdot angle\right)\right)\right) \]

      3. difference-of-squares 30.7%

         \[\leadsto 2 \cdot \left(\sin \left(\pi \cdot \frac{angle}{180}\right) \cdot \left(\color{blue}{\left(\left(b + a\right) \cdot \left(b - a\right)\right)} \cdot \cos \left(\frac{\pi}{-180} \cdot angle\right)\right)\right) \]

   6. Applied egg-rr 19.2%

      \[\leadsto 0.011111111111111112 \cdot \left(angle \cdot \left(\pi \cdot \color{blue}{\left(\left(b + a\right) \cdot \left(b - a\right)\right)}\right)\right) \]

   7. Taylor expanded in angle around 0 19.2%

      \[\leadsto \color{blue}{0.011111111111111112 \cdot \left(angle \cdot \left(\pi \cdot \left(\left(a + b\right) \cdot \left(b - a\right)\right)\right)\right)} \]
   8. Step-by-step derivation

      1. *-commutative 19.2%

         \[\leadsto \color{blue}{\left(angle \cdot \left(\pi \cdot \left(\left(a + b\right) \cdot \left(b - a\right)\right)\right)\right) \cdot 0.011111111111111112} \]

      2. associate-*r* 19.2%

         \[\leadsto \color{blue}{\left(\left(angle \cdot \pi\right) \cdot \left(\left(a + b\right) \cdot \left(b - a\right)\right)\right)} \cdot 0.011111111111111112 \]

      3. +-commutative 19.2%

         \[\leadsto \left(\left(angle \cdot \pi\right) \cdot \left(\color{blue}{\left(b + a\right)} \cdot \left(b - a\right)\right)\right) \cdot 0.011111111111111112 \]

      4. *-commutative 19.2%

         \[\leadsto \left(\left(angle \cdot \pi\right) \cdot \color{blue}{\left(\left(b - a\right) \cdot \left(b + a\right)\right)}\right) \cdot 0.011111111111111112 \]

      5. +-commutative 19.2%

         \[\leadsto \left(\left(angle \cdot \pi\right) \cdot \left(\left(b - a\right) \cdot \color{blue}{\left(a + b\right)}\right)\right) \cdot 0.011111111111111112 \]

   9. Simplified 19.2%

      \[\leadsto \color{blue}{\left(\left(angle \cdot \pi\right) \cdot \left(\left(b - a\right) \cdot \left(a + b\right)\right)\right) \cdot 0.011111111111111112} \]
   10. Step-by-step derivation

       1. add-sqr-sqrt 0.0%

          \[\leadsto \left(\color{blue}{\left(\sqrt{angle \cdot \pi} \cdot \sqrt{angle \cdot \pi}\right)} \cdot \left(\left(b - a\right) \cdot \left(a + b\right)\right)\right) \cdot 0.011111111111111112 \]

       2. sqrt-unprod 29.3%

          \[\leadsto \left(\color{blue}{\sqrt{\left(angle \cdot \pi\right) \cdot \left(angle \cdot \pi\right)}} \cdot \left(\left(b - a\right) \cdot \left(a + b\right)\right)\right) \cdot 0.011111111111111112 \]

       3. pow1 29.3%

          \[\leadsto \left(\sqrt{\color{blue}{{\left(angle \cdot \pi\right)}^{1}} \cdot \left(angle \cdot \pi\right)} \cdot \left(\left(b - a\right) \cdot \left(a + b\right)\right)\right) \cdot 0.011111111111111112 \]

       4. pow1 29.3%

          \[\leadsto \left(\sqrt{{\left(angle \cdot \pi\right)}^{1} \cdot \color{blue}{{\left(angle \cdot \pi\right)}^{1}}} \cdot \left(\left(b - a\right) \cdot \left(a + b\right)\right)\right) \cdot 0.011111111111111112 \]

       5. pow-sqr 29.3%

          \[\leadsto \left(\sqrt{\color{blue}{{\left(angle \cdot \pi\right)}^{\left(2 \cdot 1\right)}}} \cdot \left(\left(b - a\right) \cdot \left(a + b\right)\right)\right) \cdot 0.011111111111111112 \]

       6. *-commutative 29.3%

          \[\leadsto \left(\sqrt{{\color{blue}{\left(\pi \cdot angle\right)}}^{\left(2 \cdot 1\right)}} \cdot \left(\left(b - a\right) \cdot \left(a + b\right)\right)\right) \cdot 0.011111111111111112 \]

       7. metadata-eval 29.3%

          \[\leadsto \left(\sqrt{{\left(\pi \cdot angle\right)}^{\color{blue}{2}}} \cdot \left(\left(b - a\right) \cdot \left(a + b\right)\right)\right) \cdot 0.011111111111111112 \]

   11. Applied egg-rr 29.3%

       \[\leadsto \left(\color{blue}{\sqrt{{\left(\pi \cdot angle\right)}^{2}}} \cdot \left(\left(b - a\right) \cdot \left(a + b\right)\right)\right) \cdot 0.011111111111111112 \]
   12. Step-by-step derivation

       1. unpow2 29.3%

          \[\leadsto \left(\sqrt{\color{blue}{\left(\pi \cdot angle\right) \cdot \left(\pi \cdot angle\right)}} \cdot \left(\left(b - a\right) \cdot \left(a + b\right)\right)\right) \cdot 0.011111111111111112 \]

       2. rem-sqrt-square 31.4%

          \[\leadsto \left(\color{blue}{\left|\pi \cdot angle\right|} \cdot \left(\left(b - a\right) \cdot \left(a + b\right)\right)\right) \cdot 0.011111111111111112 \]

       3. *-commutative 31.4%

          \[\leadsto \left(\left|\color{blue}{angle \cdot \pi}\right| \cdot \left(\left(b - a\right) \cdot \left(a + b\right)\right)\right) \cdot 0.011111111111111112 \]

   13. Simplified 31.4%

       \[\leadsto \left(\color{blue}{\left|angle \cdot \pi\right|} \cdot \left(\left(b - a\right) \cdot \left(a + b\right)\right)\right) \cdot 0.011111111111111112 \]

   if -9.9999999999999996e86 < (/.f64 angle 180) < 4e11

   1. Initial program 65.6%

      \[\left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin \left(\pi \cdot \frac{angle}{180}\right)\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right) \]

   3. Simplified 64.6%

      \[\leadsto \color{blue}{\sin \left(angle \cdot \frac{\pi}{-180}\right) \cdot \left(-2 \cdot \left(\left({b}^{2} - {a}^{2}\right) \cdot \cos \left(angle \cdot \frac{\pi}{-180}\right)\right)\right)} \]

   4. Taylor expanded in angle around 0 63.8%

      \[\leadsto \color{blue}{0.011111111111111112 \cdot \left(angle \cdot \left(\pi \cdot \left({b}^{2} - {a}^{2}\right)\right)\right)} \]
   5. Step-by-step derivation

      1. unpow2 64.9%

         \[\leadsto 2 \cdot \left(\sin \left(\pi \cdot \frac{angle}{180}\right) \cdot \left(\left(\color{blue}{b \cdot b} - {a}^{2}\right) \cdot \cos \left(\frac{\pi}{-180} \cdot angle\right)\right)\right) \]

      2. unpow2 64.9%

         \[\leadsto 2 \cdot \left(\sin \left(\pi \cdot \frac{angle}{180}\right) \cdot \left(\left(b \cdot b - \color{blue}{a \cdot a}\right) \cdot \cos \left(\frac{\pi}{-180} \cdot angle\right)\right)\right) \]

      3. difference-of-squares 73.0%

         \[\leadsto 2 \cdot \left(\sin \left(\pi \cdot \frac{angle}{180}\right) \cdot \left(\color{blue}{\left(\left(b + a\right) \cdot \left(b - a\right)\right)} \cdot \cos \left(\frac{\pi}{-180} \cdot angle\right)\right)\right) \]

   6. Applied egg-rr 71.9%

      \[\leadsto 0.011111111111111112 \cdot \left(angle \cdot \left(\pi \cdot \color{blue}{\left(\left(b + a\right) \cdot \left(b - a\right)\right)}\right)\right) \]

   7. Taylor expanded in angle around 0 71.9%

      \[\leadsto \color{blue}{0.011111111111111112 \cdot \left(angle \cdot \left(\pi \cdot \left(\left(a + b\right) \cdot \left(b - a\right)\right)\right)\right)} \]
   8. Step-by-step derivation

      1. *-commutative 71.9%

         \[\leadsto \color{blue}{\left(angle \cdot \left(\pi \cdot \left(\left(a + b\right) \cdot \left(b - a\right)\right)\right)\right) \cdot 0.011111111111111112} \]

      2. associate-*r* 71.9%

         \[\leadsto \color{blue}{\left(\left(angle \cdot \pi\right) \cdot \left(\left(a + b\right) \cdot \left(b - a\right)\right)\right)} \cdot 0.011111111111111112 \]

      3. +-commutative 71.9%

         \[\leadsto \left(\left(angle \cdot \pi\right) \cdot \left(\color{blue}{\left(b + a\right)} \cdot \left(b - a\right)\right)\right) \cdot 0.011111111111111112 \]

      4. *-commutative 71.9%

         \[\leadsto \left(\left(angle \cdot \pi\right) \cdot \color{blue}{\left(\left(b - a\right) \cdot \left(b + a\right)\right)}\right) \cdot 0.011111111111111112 \]

      5. +-commutative 71.9%

         \[\leadsto \left(\left(angle \cdot \pi\right) \cdot \left(\left(b - a\right) \cdot \color{blue}{\left(a + b\right)}\right)\right) \cdot 0.011111111111111112 \]

   9. Simplified 71.9%

      \[\leadsto \color{blue}{\left(\left(angle \cdot \pi\right) \cdot \left(\left(b - a\right) \cdot \left(a + b\right)\right)\right) \cdot 0.011111111111111112} \]
   10. Step-by-step derivation

       1. associate-*r* 90.6%

          \[\leadsto \color{blue}{\left(\left(\left(angle \cdot \pi\right) \cdot \left(b - a\right)\right) \cdot \left(a + b\right)\right)} \cdot 0.011111111111111112 \]

       2. +-commutative 90.6%

          \[\leadsto \left(\left(\left(angle \cdot \pi\right) \cdot \left(b - a\right)\right) \cdot \color{blue}{\left(b + a\right)}\right) \cdot 0.011111111111111112 \]

       3. distribute-lft-in 86.3%

          \[\leadsto \color{blue}{\left(\left(\left(angle \cdot \pi\right) \cdot \left(b - a\right)\right) \cdot b + \left(\left(angle \cdot \pi\right) \cdot \left(b - a\right)\right) \cdot a\right)} \cdot 0.011111111111111112 \]

       4. *-commutative 86.3%

          \[\leadsto \left(\left(\color{blue}{\left(\pi \cdot angle\right)} \cdot \left(b - a\right)\right) \cdot b + \left(\left(angle \cdot \pi\right) \cdot \left(b - a\right)\right) \cdot a\right) \cdot 0.011111111111111112 \]

       5. associate-*l* 86.2%

          \[\leadsto \left(\color{blue}{\left(\pi \cdot \left(angle \cdot \left(b - a\right)\right)\right)} \cdot b + \left(\left(angle \cdot \pi\right) \cdot \left(b - a\right)\right) \cdot a\right) \cdot 0.011111111111111112 \]

       6. *-commutative 86.2%

          \[\leadsto \left(\left(\pi \cdot \left(angle \cdot \left(b - a\right)\right)\right) \cdot b + \left(\color{blue}{\left(\pi \cdot angle\right)} \cdot \left(b - a\right)\right) \cdot a\right) \cdot 0.011111111111111112 \]

       7. associate-*l* 86.2%

          \[\leadsto \left(\left(\pi \cdot \left(angle \cdot \left(b - a\right)\right)\right) \cdot b + \color{blue}{\left(\pi \cdot \left(angle \cdot \left(b - a\right)\right)\right)} \cdot a\right) \cdot 0.011111111111111112 \]

   11. Applied egg-rr 86.2%

       \[\leadsto \color{blue}{\left(\left(\pi \cdot \left(angle \cdot \left(b - a\right)\right)\right) \cdot b + \left(\pi \cdot \left(angle \cdot \left(b - a\right)\right)\right) \cdot a\right)} \cdot 0.011111111111111112 \]

   if 4e11 < (/.f64 angle 180)

   1. Initial program 24.0%

      \[\left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin \left(\pi \cdot \frac{angle}{180}\right)\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right) \]

   3. Simplified 26.0%

      \[\leadsto \color{blue}{2 \cdot \left(\sin \left(\pi \cdot \frac{angle}{180}\right) \cdot \left(\left({b}^{2} - {a}^{2}\right) \cdot \cos \left(\frac{\pi}{-180} \cdot angle\right)\right)\right)} \]
   4. Step-by-step derivation

      1. unpow2 26.0%

         \[\leadsto 2 \cdot \left(\sin \left(\pi \cdot \frac{angle}{180}\right) \cdot \left(\left(\color{blue}{b \cdot b} - {a}^{2}\right) \cdot \cos \left(\frac{\pi}{-180} \cdot angle\right)\right)\right) \]

      2. unpow2 26.0%

         \[\leadsto 2 \cdot \left(\sin \left(\pi \cdot \frac{angle}{180}\right) \cdot \left(\left(b \cdot b - \color{blue}{a \cdot a}\right) \cdot \cos \left(\frac{\pi}{-180} \cdot angle\right)\right)\right) \]

      3. difference-of-squares 36.3%

         \[\leadsto 2 \cdot \left(\sin \left(\pi \cdot \frac{angle}{180}\right) \cdot \left(\color{blue}{\left(\left(b + a\right) \cdot \left(b - a\right)\right)} \cdot \cos \left(\frac{\pi}{-180} \cdot angle\right)\right)\right) \]

   5. Applied egg-rr 36.3%

      \[\leadsto 2 \cdot \left(\sin \left(\pi \cdot \frac{angle}{180}\right) \cdot \left(\color{blue}{\left(\left(b + a\right) \cdot \left(b - a\right)\right)} \cdot \cos \left(\frac{\pi}{-180} \cdot angle\right)\right)\right) \]
   6. Step-by-step derivation

      1. clear-num 32.0%

         \[\leadsto 2 \cdot \left(\sin \left(\pi \cdot \color{blue}{\frac{1}{\frac{180}{angle}}}\right) \cdot \left(\left(\left(b + a\right) \cdot \left(b - a\right)\right) \cdot \cos \left(\frac{\pi}{-180} \cdot angle\right)\right)\right) \]

      2. un-div-inv 32.1%

         \[\leadsto 2 \cdot \left(\sin \color{blue}{\left(\frac{\pi}{\frac{180}{angle}}\right)} \cdot \left(\left(\left(b + a\right) \cdot \left(b - a\right)\right) \cdot \cos \left(\frac{\pi}{-180} \cdot angle\right)\right)\right) \]

   7. Applied egg-rr 32.1%

      \[\leadsto 2 \cdot \left(\sin \color{blue}{\left(\frac{\pi}{\frac{180}{angle}}\right)} \cdot \left(\left(\left(b + a\right) \cdot \left(b - a\right)\right) \cdot \cos \left(\frac{\pi}{-180} \cdot angle\right)\right)\right) \]

   8. Taylor expanded in angle around 0 41.3%

      \[\leadsto 2 \cdot \left(\sin \left(\frac{\pi}{\frac{180}{angle}}\right) \cdot \left(\left(\left(b + a\right) \cdot \left(b - a\right)\right) \cdot \color{blue}{1}\right)\right) \]
3. Recombined 3 regimes into one program.

4. Final simplification 63.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{angle}{180} \leq -1 \cdot 10^{+87}:\\ \;\;\;\;0.011111111111111112 \cdot \left(\left(\left(b - a\right) \cdot \left(b + a\right)\right) \cdot \left|angle \cdot \pi\right|\right)\\ \mathbf{elif}\;\frac{angle}{180} \leq 400000000000:\\ \;\;\;\;\left(b \cdot \left(\pi \cdot \left(angle \cdot \left(b - a\right)\right)\right) + a \cdot \left(\pi \cdot \left(angle \cdot \left(b - a\right)\right)\right)\right) \cdot 0.011111111111111112\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \left(\left(\left(b - a\right) \cdot \left(b + a\right)\right) \cdot \sin \left(\frac{\pi}{\frac{180}{angle}}\right)\right)\\ \end{array} \]

Alternative 14: 56.0% accurate, 2.8× speedup

Math

\[\begin{array}{l} a = |a|\\ b = |b|\\ \\ \begin{array}{l} \mathbf{if}\;b \leq 10^{+229}:\\ \;\;\;\;2 \cdot \left(\left(\left(b - a\right) \cdot \left(b + a\right)\right) \cdot \sin \left(\frac{1}{\frac{180}{angle \cdot \pi}}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;0.011111111111111112 \cdot \left(angle \cdot \left(\pi \cdot {b}^{2}\right)\right)\\ \end{array} \end{array} \]

FPCore

NOTE: a should be positive before calling this function
NOTE: b should be positive before calling this function
(FPCore (a b angle)
 :precision binary64
 (if (<= b 1e+229)
   (* 2.0 (* (* (- b a) (+ b a)) (sin (/ 1.0 (/ 180.0 (* angle PI))))))
   (* 0.011111111111111112 (* angle (* PI (pow b 2.0))))))

C

a = abs(a);
b = abs(b);
double code(double a, double b, double angle) {
	double tmp;
	if (b <= 1e+229) {
		tmp = 2.0 * (((b - a) * (b + a)) * sin((1.0 / (180.0 / (angle * ((double) M_PI))))));
	} else {
		tmp = 0.011111111111111112 * (angle * (((double) M_PI) * pow(b, 2.0)));
	}
	return tmp;
}

Java

a = Math.abs(a);
b = Math.abs(b);
public static double code(double a, double b, double angle) {
	double tmp;
	if (b <= 1e+229) {
		tmp = 2.0 * (((b - a) * (b + a)) * Math.sin((1.0 / (180.0 / (angle * Math.PI)))));
	} else {
		tmp = 0.011111111111111112 * (angle * (Math.PI * Math.pow(b, 2.0)));
	}
	return tmp;
}

Python

a = abs(a)
b = abs(b)
def code(a, b, angle):
	tmp = 0
	if b <= 1e+229:
		tmp = 2.0 * (((b - a) * (b + a)) * math.sin((1.0 / (180.0 / (angle * math.pi)))))
	else:
		tmp = 0.011111111111111112 * (angle * (math.pi * math.pow(b, 2.0)))
	return tmp

Julia

a = abs(a)
b = abs(b)
function code(a, b, angle)
	tmp = 0.0
	if (b <= 1e+229)
		tmp = Float64(2.0 * Float64(Float64(Float64(b - a) * Float64(b + a)) * sin(Float64(1.0 / Float64(180.0 / Float64(angle * pi))))));
	else
		tmp = Float64(0.011111111111111112 * Float64(angle * Float64(pi * (b ^ 2.0))));
	end
	return tmp
end

MATLAB

a = abs(a)
b = abs(b)
function tmp_2 = code(a, b, angle)
	tmp = 0.0;
	if (b <= 1e+229)
		tmp = 2.0 * (((b - a) * (b + a)) * sin((1.0 / (180.0 / (angle * pi)))));
	else
		tmp = 0.011111111111111112 * (angle * (pi * (b ^ 2.0)));
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: a should be positive before calling this function
NOTE: b should be positive before calling this function
code[a_, b_, angle_] := If[LessEqual[b, 1e+229], N[(2.0 * N[(N[(N[(b - a), $MachinePrecision] * N[(b + a), $MachinePrecision]), $MachinePrecision] * N[Sin[N[(1.0 / N[(180.0 / N[(angle * Pi), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(0.011111111111111112 * N[(angle * N[(Pi * N[Power[b, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b < 9.9999999999999999e228

   1. Initial program 47.5%

      \[\left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin \left(\pi \cdot \frac{angle}{180}\right)\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right) \]

   3. Simplified 47.7%

      \[\leadsto \color{blue}{2 \cdot \left(\sin \left(\pi \cdot \frac{angle}{180}\right) \cdot \left(\left({b}^{2} - {a}^{2}\right) \cdot \cos \left(\frac{\pi}{-180} \cdot angle\right)\right)\right)} \]
   4. Step-by-step derivation

      1. unpow2 47.7%

         \[\leadsto 2 \cdot \left(\sin \left(\pi \cdot \frac{angle}{180}\right) \cdot \left(\left(\color{blue}{b \cdot b} - {a}^{2}\right) \cdot \cos \left(\frac{\pi}{-180} \cdot angle\right)\right)\right) \]

      2. unpow2 47.7%

         \[\leadsto 2 \cdot \left(\sin \left(\pi \cdot \frac{angle}{180}\right) \cdot \left(\left(b \cdot b - \color{blue}{a \cdot a}\right) \cdot \cos \left(\frac{\pi}{-180} \cdot angle\right)\right)\right) \]

      3. difference-of-squares 53.3%

         \[\leadsto 2 \cdot \left(\sin \left(\pi \cdot \frac{angle}{180}\right) \cdot \left(\color{blue}{\left(\left(b + a\right) \cdot \left(b - a\right)\right)} \cdot \cos \left(\frac{\pi}{-180} \cdot angle\right)\right)\right) \]

   5. Applied egg-rr 53.3%

      \[\leadsto 2 \cdot \left(\sin \left(\pi \cdot \frac{angle}{180}\right) \cdot \left(\color{blue}{\left(\left(b + a\right) \cdot \left(b - a\right)\right)} \cdot \cos \left(\frac{\pi}{-180} \cdot angle\right)\right)\right) \]
   6. Step-by-step derivation

      1. *-commutative 53.3%

         \[\leadsto 2 \cdot \left(\sin \color{blue}{\left(\frac{angle}{180} \cdot \pi\right)} \cdot \left(\left(\left(b + a\right) \cdot \left(b - a\right)\right) \cdot \cos \left(\frac{\pi}{-180} \cdot angle\right)\right)\right) \]

      2. associate-*l/ 51.5%

         \[\leadsto 2 \cdot \left(\sin \color{blue}{\left(\frac{angle \cdot \pi}{180}\right)} \cdot \left(\left(\left(b + a\right) \cdot \left(b - a\right)\right) \cdot \cos \left(\frac{\pi}{-180} \cdot angle\right)\right)\right) \]

      3. clear-num 50.1%

         \[\leadsto 2 \cdot \left(\sin \color{blue}{\left(\frac{1}{\frac{180}{angle \cdot \pi}}\right)} \cdot \left(\left(\left(b + a\right) \cdot \left(b - a\right)\right) \cdot \cos \left(\frac{\pi}{-180} \cdot angle\right)\right)\right) \]

      4. *-commutative 50.1%

         \[\leadsto 2 \cdot \left(\sin \left(\frac{1}{\frac{180}{\color{blue}{\pi \cdot angle}}}\right) \cdot \left(\left(\left(b + a\right) \cdot \left(b - a\right)\right) \cdot \cos \left(\frac{\pi}{-180} \cdot angle\right)\right)\right) \]

   7. Applied egg-rr 50.1%

      \[\leadsto 2 \cdot \left(\sin \color{blue}{\left(\frac{1}{\frac{180}{\pi \cdot angle}}\right)} \cdot \left(\left(\left(b + a\right) \cdot \left(b - a\right)\right) \cdot \cos \left(\frac{\pi}{-180} \cdot angle\right)\right)\right) \]

   8. Taylor expanded in angle around 0 52.7%

      \[\leadsto 2 \cdot \left(\sin \left(\frac{1}{\frac{180}{\pi \cdot angle}}\right) \cdot \left(\left(\left(b + a\right) \cdot \left(b - a\right)\right) \cdot \color{blue}{1}\right)\right) \]

   if 9.9999999999999999e228 < b

   1. Initial program 34.0%

      \[\left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin \left(\pi \cdot \frac{angle}{180}\right)\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right) \]

   3. Simplified 34.0%

      \[\leadsto \color{blue}{\sin \left(angle \cdot \frac{\pi}{-180}\right) \cdot \left(-2 \cdot \left(\left({b}^{2} - {a}^{2}\right) \cdot \cos \left(angle \cdot \frac{\pi}{-180}\right)\right)\right)} \]

   4. Taylor expanded in angle around 0 34.0%

      \[\leadsto \color{blue}{0.011111111111111112 \cdot \left(angle \cdot \left(\pi \cdot \left({b}^{2} - {a}^{2}\right)\right)\right)} \]

   5. Taylor expanded in b around inf 72.5%

      \[\leadsto 0.011111111111111112 \cdot \color{blue}{\left(angle \cdot \left({b}^{2} \cdot \pi\right)\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 54.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq 10^{+229}:\\ \;\;\;\;2 \cdot \left(\left(\left(b - a\right) \cdot \left(b + a\right)\right) \cdot \sin \left(\frac{1}{\frac{180}{angle \cdot \pi}}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;0.011111111111111112 \cdot \left(angle \cdot \left(\pi \cdot {b}^{2}\right)\right)\\ \end{array} \]

Alternative 15: 54.0% accurate, 2.9× speedup

Math

\[\begin{array}{l} a = |a|\\ b = |b|\\ \\ \begin{array}{l} t_0 := \left(b - a\right) \cdot \left(b + a\right)\\ \mathbf{if}\;angle \leq -3.8 \cdot 10^{+85}:\\ \;\;\;\;0.011111111111111112 \cdot \left(t_0 \cdot \left|angle \cdot \pi\right|\right)\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \left(\left(angle \cdot 0.005555555555555556\right) \cdot \left(\pi \cdot t_0\right)\right)\\ \end{array} \end{array} \]

FPCore

NOTE: a should be positive before calling this function
NOTE: b should be positive before calling this function
(FPCore (a b angle)
 :precision binary64
 (let* ((t_0 (* (- b a) (+ b a))))
   (if (<= angle -3.8e+85)
     (* 0.011111111111111112 (* t_0 (fabs (* angle PI))))
     (* 2.0 (* (* angle 0.005555555555555556) (* PI t_0))))))

C

a = abs(a);
b = abs(b);
double code(double a, double b, double angle) {
	double t_0 = (b - a) * (b + a);
	double tmp;
	if (angle <= -3.8e+85) {
		tmp = 0.011111111111111112 * (t_0 * fabs((angle * ((double) M_PI))));
	} else {
		tmp = 2.0 * ((angle * 0.005555555555555556) * (((double) M_PI) * t_0));
	}
	return tmp;
}

Java

a = Math.abs(a);
b = Math.abs(b);
public static double code(double a, double b, double angle) {
	double t_0 = (b - a) * (b + a);
	double tmp;
	if (angle <= -3.8e+85) {
		tmp = 0.011111111111111112 * (t_0 * Math.abs((angle * Math.PI)));
	} else {
		tmp = 2.0 * ((angle * 0.005555555555555556) * (Math.PI * t_0));
	}
	return tmp;
}

Python

a = abs(a)
b = abs(b)
def code(a, b, angle):
	t_0 = (b - a) * (b + a)
	tmp = 0
	if angle <= -3.8e+85:
		tmp = 0.011111111111111112 * (t_0 * math.fabs((angle * math.pi)))
	else:
		tmp = 2.0 * ((angle * 0.005555555555555556) * (math.pi * t_0))
	return tmp

Julia

a = abs(a)
b = abs(b)
function code(a, b, angle)
	t_0 = Float64(Float64(b - a) * Float64(b + a))
	tmp = 0.0
	if (angle <= -3.8e+85)
		tmp = Float64(0.011111111111111112 * Float64(t_0 * abs(Float64(angle * pi))));
	else
		tmp = Float64(2.0 * Float64(Float64(angle * 0.005555555555555556) * Float64(pi * t_0)));
	end
	return tmp
end

MATLAB

a = abs(a)
b = abs(b)
function tmp_2 = code(a, b, angle)
	t_0 = (b - a) * (b + a);
	tmp = 0.0;
	if (angle <= -3.8e+85)
		tmp = 0.011111111111111112 * (t_0 * abs((angle * pi)));
	else
		tmp = 2.0 * ((angle * 0.005555555555555556) * (pi * t_0));
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: a should be positive before calling this function
NOTE: b should be positive before calling this function
code[a_, b_, angle_] := Block[{t$95$0 = N[(N[(b - a), $MachinePrecision] * N[(b + a), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[angle, -3.8e+85], N[(0.011111111111111112 * N[(t$95$0 * N[Abs[N[(angle * Pi), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(2.0 * N[(N[(angle * 0.005555555555555556), $MachinePrecision] * N[(Pi * t$95$0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if angle < -3.79999999999999992e85

   1. Initial program 24.6%

      \[\left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin \left(\pi \cdot \frac{angle}{180}\right)\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right) \]

   3. Simplified 24.3%

      \[\leadsto \color{blue}{\sin \left(angle \cdot \frac{\pi}{-180}\right) \cdot \left(-2 \cdot \left(\left({b}^{2} - {a}^{2}\right) \cdot \cos \left(angle \cdot \frac{\pi}{-180}\right)\right)\right)} \]

   4. Taylor expanded in angle around 0 13.3%

      \[\leadsto \color{blue}{0.011111111111111112 \cdot \left(angle \cdot \left(\pi \cdot \left({b}^{2} - {a}^{2}\right)\right)\right)} \]
   5. Step-by-step derivation

      1. unpow2 24.8%

         \[\leadsto 2 \cdot \left(\sin \left(\pi \cdot \frac{angle}{180}\right) \cdot \left(\left(\color{blue}{b \cdot b} - {a}^{2}\right) \cdot \cos \left(\frac{\pi}{-180} \cdot angle\right)\right)\right) \]

      2. unpow2 24.8%

         \[\leadsto 2 \cdot \left(\sin \left(\pi \cdot \frac{angle}{180}\right) \cdot \left(\left(b \cdot b - \color{blue}{a \cdot a}\right) \cdot \cos \left(\frac{\pi}{-180} \cdot angle\right)\right)\right) \]

      3. difference-of-squares 30.7%

         \[\leadsto 2 \cdot \left(\sin \left(\pi \cdot \frac{angle}{180}\right) \cdot \left(\color{blue}{\left(\left(b + a\right) \cdot \left(b - a\right)\right)} \cdot \cos \left(\frac{\pi}{-180} \cdot angle\right)\right)\right) \]

   6. Applied egg-rr 19.2%

      \[\leadsto 0.011111111111111112 \cdot \left(angle \cdot \left(\pi \cdot \color{blue}{\left(\left(b + a\right) \cdot \left(b - a\right)\right)}\right)\right) \]

   7. Taylor expanded in angle around 0 19.2%

      \[\leadsto \color{blue}{0.011111111111111112 \cdot \left(angle \cdot \left(\pi \cdot \left(\left(a + b\right) \cdot \left(b - a\right)\right)\right)\right)} \]
   8. Step-by-step derivation

      1. *-commutative 19.2%

         \[\leadsto \color{blue}{\left(angle \cdot \left(\pi \cdot \left(\left(a + b\right) \cdot \left(b - a\right)\right)\right)\right) \cdot 0.011111111111111112} \]

      2. associate-*r* 19.2%

         \[\leadsto \color{blue}{\left(\left(angle \cdot \pi\right) \cdot \left(\left(a + b\right) \cdot \left(b - a\right)\right)\right)} \cdot 0.011111111111111112 \]

      3. +-commutative 19.2%

         \[\leadsto \left(\left(angle \cdot \pi\right) \cdot \left(\color{blue}{\left(b + a\right)} \cdot \left(b - a\right)\right)\right) \cdot 0.011111111111111112 \]

      4. *-commutative 19.2%

         \[\leadsto \left(\left(angle \cdot \pi\right) \cdot \color{blue}{\left(\left(b - a\right) \cdot \left(b + a\right)\right)}\right) \cdot 0.011111111111111112 \]

      5. +-commutative 19.2%

         \[\leadsto \left(\left(angle \cdot \pi\right) \cdot \left(\left(b - a\right) \cdot \color{blue}{\left(a + b\right)}\right)\right) \cdot 0.011111111111111112 \]

   9. Simplified 19.2%

      \[\leadsto \color{blue}{\left(\left(angle \cdot \pi\right) \cdot \left(\left(b - a\right) \cdot \left(a + b\right)\right)\right) \cdot 0.011111111111111112} \]
   10. Step-by-step derivation

       1. add-sqr-sqrt 0.0%

          \[\leadsto \left(\color{blue}{\left(\sqrt{angle \cdot \pi} \cdot \sqrt{angle \cdot \pi}\right)} \cdot \left(\left(b - a\right) \cdot \left(a + b\right)\right)\right) \cdot 0.011111111111111112 \]

       2. sqrt-unprod 29.3%

          \[\leadsto \left(\color{blue}{\sqrt{\left(angle \cdot \pi\right) \cdot \left(angle \cdot \pi\right)}} \cdot \left(\left(b - a\right) \cdot \left(a + b\right)\right)\right) \cdot 0.011111111111111112 \]

       3. pow1 29.3%

          \[\leadsto \left(\sqrt{\color{blue}{{\left(angle \cdot \pi\right)}^{1}} \cdot \left(angle \cdot \pi\right)} \cdot \left(\left(b - a\right) \cdot \left(a + b\right)\right)\right) \cdot 0.011111111111111112 \]

       4. pow1 29.3%

          \[\leadsto \left(\sqrt{{\left(angle \cdot \pi\right)}^{1} \cdot \color{blue}{{\left(angle \cdot \pi\right)}^{1}}} \cdot \left(\left(b - a\right) \cdot \left(a + b\right)\right)\right) \cdot 0.011111111111111112 \]

       5. pow-sqr 29.3%

          \[\leadsto \left(\sqrt{\color{blue}{{\left(angle \cdot \pi\right)}^{\left(2 \cdot 1\right)}}} \cdot \left(\left(b - a\right) \cdot \left(a + b\right)\right)\right) \cdot 0.011111111111111112 \]

       6. *-commutative 29.3%

          \[\leadsto \left(\sqrt{{\color{blue}{\left(\pi \cdot angle\right)}}^{\left(2 \cdot 1\right)}} \cdot \left(\left(b - a\right) \cdot \left(a + b\right)\right)\right) \cdot 0.011111111111111112 \]

       7. metadata-eval 29.3%

          \[\leadsto \left(\sqrt{{\left(\pi \cdot angle\right)}^{\color{blue}{2}}} \cdot \left(\left(b - a\right) \cdot \left(a + b\right)\right)\right) \cdot 0.011111111111111112 \]

   11. Applied egg-rr 29.3%

       \[\leadsto \left(\color{blue}{\sqrt{{\left(\pi \cdot angle\right)}^{2}}} \cdot \left(\left(b - a\right) \cdot \left(a + b\right)\right)\right) \cdot 0.011111111111111112 \]
   12. Step-by-step derivation

       1. unpow2 29.3%

          \[\leadsto \left(\sqrt{\color{blue}{\left(\pi \cdot angle\right) \cdot \left(\pi \cdot angle\right)}} \cdot \left(\left(b - a\right) \cdot \left(a + b\right)\right)\right) \cdot 0.011111111111111112 \]

       2. rem-sqrt-square 31.4%

          \[\leadsto \left(\color{blue}{\left|\pi \cdot angle\right|} \cdot \left(\left(b - a\right) \cdot \left(a + b\right)\right)\right) \cdot 0.011111111111111112 \]

       3. *-commutative 31.4%

          \[\leadsto \left(\left|\color{blue}{angle \cdot \pi}\right| \cdot \left(\left(b - a\right) \cdot \left(a + b\right)\right)\right) \cdot 0.011111111111111112 \]

   13. Simplified 31.4%

       \[\leadsto \left(\color{blue}{\left|angle \cdot \pi\right|} \cdot \left(\left(b - a\right) \cdot \left(a + b\right)\right)\right) \cdot 0.011111111111111112 \]

   if -3.79999999999999992e85 < angle

   1. Initial program 51.8%

      \[\left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin \left(\pi \cdot \frac{angle}{180}\right)\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right) \]

   3. Simplified 52.0%

      \[\leadsto \color{blue}{2 \cdot \left(\sin \left(\pi \cdot \frac{angle}{180}\right) \cdot \left(\left({b}^{2} - {a}^{2}\right) \cdot \cos \left(\frac{\pi}{-180} \cdot angle\right)\right)\right)} \]
   4. Step-by-step derivation

      1. unpow2 52.0%

         \[\leadsto 2 \cdot \left(\sin \left(\pi \cdot \frac{angle}{180}\right) \cdot \left(\left(\color{blue}{b \cdot b} - {a}^{2}\right) \cdot \cos \left(\frac{\pi}{-180} \cdot angle\right)\right)\right) \]

      2. unpow2 52.0%

         \[\leadsto 2 \cdot \left(\sin \left(\pi \cdot \frac{angle}{180}\right) \cdot \left(\left(b \cdot b - \color{blue}{a \cdot a}\right) \cdot \cos \left(\frac{\pi}{-180} \cdot angle\right)\right)\right) \]

      3. difference-of-squares 60.9%

         \[\leadsto 2 \cdot \left(\sin \left(\pi \cdot \frac{angle}{180}\right) \cdot \left(\color{blue}{\left(\left(b + a\right) \cdot \left(b - a\right)\right)} \cdot \cos \left(\frac{\pi}{-180} \cdot angle\right)\right)\right) \]

   5. Applied egg-rr 60.9%

      \[\leadsto 2 \cdot \left(\sin \left(\pi \cdot \frac{angle}{180}\right) \cdot \left(\color{blue}{\left(\left(b + a\right) \cdot \left(b - a\right)\right)} \cdot \cos \left(\frac{\pi}{-180} \cdot angle\right)\right)\right) \]

   6. Taylor expanded in angle around 0 56.9%

      \[\leadsto 2 \cdot \color{blue}{\left(0.005555555555555556 \cdot \left(angle \cdot \left(\pi \cdot \left(\left(a + b\right) \cdot \left(b - a\right)\right)\right)\right)\right)} \]
   7. Step-by-step derivation

      1. associate-*r* 56.9%

         \[\leadsto 2 \cdot \color{blue}{\left(\left(0.005555555555555556 \cdot angle\right) \cdot \left(\pi \cdot \left(\left(a + b\right) \cdot \left(b - a\right)\right)\right)\right)} \]

   8. Simplified 56.9%

      \[\leadsto 2 \cdot \color{blue}{\left(\left(0.005555555555555556 \cdot angle\right) \cdot \left(\pi \cdot \left(\left(a + b\right) \cdot \left(b - a\right)\right)\right)\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 51.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;angle \leq -3.8 \cdot 10^{+85}:\\ \;\;\;\;0.011111111111111112 \cdot \left(\left(\left(b - a\right) \cdot \left(b + a\right)\right) \cdot \left|angle \cdot \pi\right|\right)\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \left(\left(angle \cdot 0.005555555555555556\right) \cdot \left(\pi \cdot \left(\left(b - a\right) \cdot \left(b + a\right)\right)\right)\right)\\ \end{array} \]

Alternative 16: 56.0% accurate, 2.9× speedup

Math

\[\begin{array}{l} a = |a|\\ b = |b|\\ \\ 2 \cdot \left(\left(\left(b - a\right) \cdot \left(b + a\right)\right) \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right) \end{array} \]

FPCore

NOTE: a should be positive before calling this function
NOTE: b should be positive before calling this function
(FPCore (a b angle)
 :precision binary64
 (* 2.0 (* (* (- b a) (+ b a)) (sin (* (/ angle 180.0) PI)))))

C

a = abs(a);
b = abs(b);
double code(double a, double b, double angle) {
	return 2.0 * (((b - a) * (b + a)) * sin(((angle / 180.0) * ((double) M_PI))));
}

Java

a = Math.abs(a);
b = Math.abs(b);
public static double code(double a, double b, double angle) {
	return 2.0 * (((b - a) * (b + a)) * Math.sin(((angle / 180.0) * Math.PI)));
}

Python

a = abs(a)
b = abs(b)
def code(a, b, angle):
	return 2.0 * (((b - a) * (b + a)) * math.sin(((angle / 180.0) * math.pi)))

Julia

a = abs(a)
b = abs(b)
function code(a, b, angle)
	return Float64(2.0 * Float64(Float64(Float64(b - a) * Float64(b + a)) * sin(Float64(Float64(angle / 180.0) * pi))))
end

MATLAB

a = abs(a)
b = abs(b)
function tmp = code(a, b, angle)
	tmp = 2.0 * (((b - a) * (b + a)) * sin(((angle / 180.0) * pi)));
end

Wolfram

NOTE: a should be positive before calling this function
NOTE: b should be positive before calling this function
code[a_, b_, angle_] := N[(2.0 * N[(N[(N[(b - a), $MachinePrecision] * N[(b + a), $MachinePrecision]), $MachinePrecision] * N[Sin[N[(N[(angle / 180.0), $MachinePrecision] * Pi), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 46.4%

   \[\left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin \left(\pi \cdot \frac{angle}{180}\right)\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right) \]

3. Simplified 46.6%

   \[\leadsto \color{blue}{2 \cdot \left(\sin \left(\pi \cdot \frac{angle}{180}\right) \cdot \left(\left({b}^{2} - {a}^{2}\right) \cdot \cos \left(\frac{\pi}{-180} \cdot angle\right)\right)\right)} \]
4. Step-by-step derivation

   1. unpow2 46.6%

      \[\leadsto 2 \cdot \left(\sin \left(\pi \cdot \frac{angle}{180}\right) \cdot \left(\left(\color{blue}{b \cdot b} - {a}^{2}\right) \cdot \cos \left(\frac{\pi}{-180} \cdot angle\right)\right)\right) \]

   2. unpow2 46.6%

      \[\leadsto 2 \cdot \left(\sin \left(\pi \cdot \frac{angle}{180}\right) \cdot \left(\left(b \cdot b - \color{blue}{a \cdot a}\right) \cdot \cos \left(\frac{\pi}{-180} \cdot angle\right)\right)\right) \]

   3. difference-of-squares 54.9%

      \[\leadsto 2 \cdot \left(\sin \left(\pi \cdot \frac{angle}{180}\right) \cdot \left(\color{blue}{\left(\left(b + a\right) \cdot \left(b - a\right)\right)} \cdot \cos \left(\frac{\pi}{-180} \cdot angle\right)\right)\right) \]

5. Applied egg-rr 54.9%

   \[\leadsto 2 \cdot \left(\sin \left(\pi \cdot \frac{angle}{180}\right) \cdot \left(\color{blue}{\left(\left(b + a\right) \cdot \left(b - a\right)\right)} \cdot \cos \left(\frac{\pi}{-180} \cdot angle\right)\right)\right) \]

6. Taylor expanded in angle around 0 53.0%

   \[\leadsto 2 \cdot \left(\sin \left(\pi \cdot \frac{angle}{180}\right) \cdot \left(\left(\left(b + a\right) \cdot \left(b - a\right)\right) \cdot \color{blue}{1}\right)\right) \]

7. Final simplification 53.0%

   \[\leadsto 2 \cdot \left(\left(\left(b - a\right) \cdot \left(b + a\right)\right) \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right) \]

Alternative 17: 54.3% accurate, 5.4× speedup

Math

\[\begin{array}{l} a = |a|\\ b = |b|\\ \\ 2 \cdot \left(\left(angle \cdot 0.005555555555555556\right) \cdot \left(\pi \cdot \left(\left(b - a\right) \cdot \left(b + a\right)\right)\right)\right) \end{array} \]

FPCore

NOTE: a should be positive before calling this function
NOTE: b should be positive before calling this function
(FPCore (a b angle)
 :precision binary64
 (* 2.0 (* (* angle 0.005555555555555556) (* PI (* (- b a) (+ b a))))))

C

a = abs(a);
b = abs(b);
double code(double a, double b, double angle) {
	return 2.0 * ((angle * 0.005555555555555556) * (((double) M_PI) * ((b - a) * (b + a))));
}

Java

a = Math.abs(a);
b = Math.abs(b);
public static double code(double a, double b, double angle) {
	return 2.0 * ((angle * 0.005555555555555556) * (Math.PI * ((b - a) * (b + a))));
}

Python

a = abs(a)
b = abs(b)
def code(a, b, angle):
	return 2.0 * ((angle * 0.005555555555555556) * (math.pi * ((b - a) * (b + a))))

Julia

a = abs(a)
b = abs(b)
function code(a, b, angle)
	return Float64(2.0 * Float64(Float64(angle * 0.005555555555555556) * Float64(pi * Float64(Float64(b - a) * Float64(b + a)))))
end

MATLAB

a = abs(a)
b = abs(b)
function tmp = code(a, b, angle)
	tmp = 2.0 * ((angle * 0.005555555555555556) * (pi * ((b - a) * (b + a))));
end

Wolfram

NOTE: a should be positive before calling this function
NOTE: b should be positive before calling this function
code[a_, b_, angle_] := N[(2.0 * N[(N[(angle * 0.005555555555555556), $MachinePrecision] * N[(Pi * N[(N[(b - a), $MachinePrecision] * N[(b + a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 46.4%

   \[\left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin \left(\pi \cdot \frac{angle}{180}\right)\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right) \]

3. Simplified 46.6%

   \[\leadsto \color{blue}{2 \cdot \left(\sin \left(\pi \cdot \frac{angle}{180}\right) \cdot \left(\left({b}^{2} - {a}^{2}\right) \cdot \cos \left(\frac{\pi}{-180} \cdot angle\right)\right)\right)} \]
4. Step-by-step derivation

   1. unpow2 46.6%

      \[\leadsto 2 \cdot \left(\sin \left(\pi \cdot \frac{angle}{180}\right) \cdot \left(\left(\color{blue}{b \cdot b} - {a}^{2}\right) \cdot \cos \left(\frac{\pi}{-180} \cdot angle\right)\right)\right) \]

   2. unpow2 46.6%

      \[\leadsto 2 \cdot \left(\sin \left(\pi \cdot \frac{angle}{180}\right) \cdot \left(\left(b \cdot b - \color{blue}{a \cdot a}\right) \cdot \cos \left(\frac{\pi}{-180} \cdot angle\right)\right)\right) \]

   3. difference-of-squares 54.9%

      \[\leadsto 2 \cdot \left(\sin \left(\pi \cdot \frac{angle}{180}\right) \cdot \left(\color{blue}{\left(\left(b + a\right) \cdot \left(b - a\right)\right)} \cdot \cos \left(\frac{\pi}{-180} \cdot angle\right)\right)\right) \]

5. Applied egg-rr 54.9%

   \[\leadsto 2 \cdot \left(\sin \left(\pi \cdot \frac{angle}{180}\right) \cdot \left(\color{blue}{\left(\left(b + a\right) \cdot \left(b - a\right)\right)} \cdot \cos \left(\frac{\pi}{-180} \cdot angle\right)\right)\right) \]

6. Taylor expanded in angle around 0 49.4%

   \[\leadsto 2 \cdot \color{blue}{\left(0.005555555555555556 \cdot \left(angle \cdot \left(\pi \cdot \left(\left(a + b\right) \cdot \left(b - a\right)\right)\right)\right)\right)} \]
7. Step-by-step derivation

   1. associate-*r* 49.4%

      \[\leadsto 2 \cdot \color{blue}{\left(\left(0.005555555555555556 \cdot angle\right) \cdot \left(\pi \cdot \left(\left(a + b\right) \cdot \left(b - a\right)\right)\right)\right)} \]

8. Simplified 49.4%

   \[\leadsto 2 \cdot \color{blue}{\left(\left(0.005555555555555556 \cdot angle\right) \cdot \left(\pi \cdot \left(\left(a + b\right) \cdot \left(b - a\right)\right)\right)\right)} \]

9. Final simplification 49.4%

   \[\leadsto 2 \cdot \left(\left(angle \cdot 0.005555555555555556\right) \cdot \left(\pi \cdot \left(\left(b - a\right) \cdot \left(b + a\right)\right)\right)\right) \]

Alternative 18: 54.3% accurate, 5.5× speedup

Math

\[\begin{array}{l} a = |a|\\ b = |b|\\ \\ 0.011111111111111112 \cdot \left(angle \cdot \left(\pi \cdot \left(\left(b - a\right) \cdot \left(b + a\right)\right)\right)\right) \end{array} \]

FPCore

NOTE: a should be positive before calling this function
NOTE: b should be positive before calling this function
(FPCore (a b angle)
 :precision binary64
 (* 0.011111111111111112 (* angle (* PI (* (- b a) (+ b a))))))

C

a = abs(a);
b = abs(b);
double code(double a, double b, double angle) {
	return 0.011111111111111112 * (angle * (((double) M_PI) * ((b - a) * (b + a))));
}

Java

a = Math.abs(a);
b = Math.abs(b);
public static double code(double a, double b, double angle) {
	return 0.011111111111111112 * (angle * (Math.PI * ((b - a) * (b + a))));
}

Python

a = abs(a)
b = abs(b)
def code(a, b, angle):
	return 0.011111111111111112 * (angle * (math.pi * ((b - a) * (b + a))))

Julia

a = abs(a)
b = abs(b)
function code(a, b, angle)
	return Float64(0.011111111111111112 * Float64(angle * Float64(pi * Float64(Float64(b - a) * Float64(b + a)))))
end

MATLAB

a = abs(a)
b = abs(b)
function tmp = code(a, b, angle)
	tmp = 0.011111111111111112 * (angle * (pi * ((b - a) * (b + a))));
end

Wolfram

NOTE: a should be positive before calling this function
NOTE: b should be positive before calling this function
code[a_, b_, angle_] := N[(0.011111111111111112 * N[(angle * N[(Pi * N[(N[(b - a), $MachinePrecision] * N[(b + a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 46.4%

   \[\left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin \left(\pi \cdot \frac{angle}{180}\right)\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right) \]

3. Simplified 46.1%

   \[\leadsto \color{blue}{\sin \left(angle \cdot \frac{\pi}{-180}\right) \cdot \left(-2 \cdot \left(\left({b}^{2} - {a}^{2}\right) \cdot \cos \left(angle \cdot \frac{\pi}{-180}\right)\right)\right)} \]

4. Taylor expanded in angle around 0 41.2%

   \[\leadsto \color{blue}{0.011111111111111112 \cdot \left(angle \cdot \left(\pi \cdot \left({b}^{2} - {a}^{2}\right)\right)\right)} \]
5. Step-by-step derivation

   1. unpow2 46.6%

      \[\leadsto 2 \cdot \left(\sin \left(\pi \cdot \frac{angle}{180}\right) \cdot \left(\left(\color{blue}{b \cdot b} - {a}^{2}\right) \cdot \cos \left(\frac{\pi}{-180} \cdot angle\right)\right)\right) \]

   2. unpow2 46.6%

      \[\leadsto 2 \cdot \left(\sin \left(\pi \cdot \frac{angle}{180}\right) \cdot \left(\left(b \cdot b - \color{blue}{a \cdot a}\right) \cdot \cos \left(\frac{\pi}{-180} \cdot angle\right)\right)\right) \]

   3. difference-of-squares 54.9%

      \[\leadsto 2 \cdot \left(\sin \left(\pi \cdot \frac{angle}{180}\right) \cdot \left(\color{blue}{\left(\left(b + a\right) \cdot \left(b - a\right)\right)} \cdot \cos \left(\frac{\pi}{-180} \cdot angle\right)\right)\right) \]

6. Applied egg-rr 49.4%

   \[\leadsto 0.011111111111111112 \cdot \left(angle \cdot \left(\pi \cdot \color{blue}{\left(\left(b + a\right) \cdot \left(b - a\right)\right)}\right)\right) \]

7. Final simplification 49.4%

   \[\leadsto 0.011111111111111112 \cdot \left(angle \cdot \left(\pi \cdot \left(\left(b - a\right) \cdot \left(b + a\right)\right)\right)\right) \]

Alternative 19: 54.3% accurate, 5.5× speedup

Math

\[\begin{array}{l} a = |a|\\ b = |b|\\ \\ 0.011111111111111112 \cdot \left(\left(\left(b - a\right) \cdot \left(b + a\right)\right) \cdot \left(angle \cdot \pi\right)\right) \end{array} \]

FPCore

NOTE: a should be positive before calling this function
NOTE: b should be positive before calling this function
(FPCore (a b angle)
 :precision binary64
 (* 0.011111111111111112 (* (* (- b a) (+ b a)) (* angle PI))))

C

a = abs(a);
b = abs(b);
double code(double a, double b, double angle) {
	return 0.011111111111111112 * (((b - a) * (b + a)) * (angle * ((double) M_PI)));
}

Java

a = Math.abs(a);
b = Math.abs(b);
public static double code(double a, double b, double angle) {
	return 0.011111111111111112 * (((b - a) * (b + a)) * (angle * Math.PI));
}

Python

a = abs(a)
b = abs(b)
def code(a, b, angle):
	return 0.011111111111111112 * (((b - a) * (b + a)) * (angle * math.pi))

Julia

a = abs(a)
b = abs(b)
function code(a, b, angle)
	return Float64(0.011111111111111112 * Float64(Float64(Float64(b - a) * Float64(b + a)) * Float64(angle * pi)))
end

MATLAB

a = abs(a)
b = abs(b)
function tmp = code(a, b, angle)
	tmp = 0.011111111111111112 * (((b - a) * (b + a)) * (angle * pi));
end

Wolfram

NOTE: a should be positive before calling this function
NOTE: b should be positive before calling this function
code[a_, b_, angle_] := N[(0.011111111111111112 * N[(N[(N[(b - a), $MachinePrecision] * N[(b + a), $MachinePrecision]), $MachinePrecision] * N[(angle * Pi), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 46.4%

   \[\left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin \left(\pi \cdot \frac{angle}{180}\right)\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right) \]

3. Simplified 46.1%

   \[\leadsto \color{blue}{\sin \left(angle \cdot \frac{\pi}{-180}\right) \cdot \left(-2 \cdot \left(\left({b}^{2} - {a}^{2}\right) \cdot \cos \left(angle \cdot \frac{\pi}{-180}\right)\right)\right)} \]

4. Taylor expanded in angle around 0 41.2%

   \[\leadsto \color{blue}{0.011111111111111112 \cdot \left(angle \cdot \left(\pi \cdot \left({b}^{2} - {a}^{2}\right)\right)\right)} \]
5. Step-by-step derivation

   1. unpow2 46.6%

      \[\leadsto 2 \cdot \left(\sin \left(\pi \cdot \frac{angle}{180}\right) \cdot \left(\left(\color{blue}{b \cdot b} - {a}^{2}\right) \cdot \cos \left(\frac{\pi}{-180} \cdot angle\right)\right)\right) \]

   2. unpow2 46.6%

      \[\leadsto 2 \cdot \left(\sin \left(\pi \cdot \frac{angle}{180}\right) \cdot \left(\left(b \cdot b - \color{blue}{a \cdot a}\right) \cdot \cos \left(\frac{\pi}{-180} \cdot angle\right)\right)\right) \]

   3. difference-of-squares 54.9%

      \[\leadsto 2 \cdot \left(\sin \left(\pi \cdot \frac{angle}{180}\right) \cdot \left(\color{blue}{\left(\left(b + a\right) \cdot \left(b - a\right)\right)} \cdot \cos \left(\frac{\pi}{-180} \cdot angle\right)\right)\right) \]

6. Applied egg-rr 49.4%

   \[\leadsto 0.011111111111111112 \cdot \left(angle \cdot \left(\pi \cdot \color{blue}{\left(\left(b + a\right) \cdot \left(b - a\right)\right)}\right)\right) \]

7. Taylor expanded in angle around 0 49.4%

   \[\leadsto \color{blue}{0.011111111111111112 \cdot \left(angle \cdot \left(\pi \cdot \left(\left(a + b\right) \cdot \left(b - a\right)\right)\right)\right)} \]
8. Step-by-step derivation

   1. *-commutative 49.4%

      \[\leadsto \color{blue}{\left(angle \cdot \left(\pi \cdot \left(\left(a + b\right) \cdot \left(b - a\right)\right)\right)\right) \cdot 0.011111111111111112} \]

   2. associate-*r* 49.4%

      \[\leadsto \color{blue}{\left(\left(angle \cdot \pi\right) \cdot \left(\left(a + b\right) \cdot \left(b - a\right)\right)\right)} \cdot 0.011111111111111112 \]

   3. +-commutative 49.4%

      \[\leadsto \left(\left(angle \cdot \pi\right) \cdot \left(\color{blue}{\left(b + a\right)} \cdot \left(b - a\right)\right)\right) \cdot 0.011111111111111112 \]

   4. *-commutative 49.4%

      \[\leadsto \left(\left(angle \cdot \pi\right) \cdot \color{blue}{\left(\left(b - a\right) \cdot \left(b + a\right)\right)}\right) \cdot 0.011111111111111112 \]

   5. +-commutative 49.4%

      \[\leadsto \left(\left(angle \cdot \pi\right) \cdot \left(\left(b - a\right) \cdot \color{blue}{\left(a + b\right)}\right)\right) \cdot 0.011111111111111112 \]

9. Simplified 49.4%

   \[\leadsto \color{blue}{\left(\left(angle \cdot \pi\right) \cdot \left(\left(b - a\right) \cdot \left(a + b\right)\right)\right) \cdot 0.011111111111111112} \]

10. Final simplification 49.4%

    \[\leadsto 0.011111111111111112 \cdot \left(\left(\left(b - a\right) \cdot \left(b + a\right)\right) \cdot \left(angle \cdot \pi\right)\right) \]

Reproduce

herbie shell --seed 2023305 
(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)))))