ab-angle->ABCF B

Percentage Accurate: 54.1% → 67.9%

Time: 51.3s

Alternatives: 19

Speedup: 5.5×

• 32.6% of points produce a very large (infinite) output. You may want to add a precondition.

Specification

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 54.1% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 67.9% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(b - a\right) \cdot 2\\ t_1 := angle \cdot \left(0.005555555555555556 \cdot \pi\right)\\ t_2 := \left(b - a\right) \cdot \left(b + a\right)\\ t_3 := \sqrt[3]{\sqrt[3]{t_1}}\\ t_4 := \frac{angle}{180} \cdot \pi\\ t_5 := \cos t_4\\ t_6 := 2 \cdot \sin t_4\\ t_7 := \pi \cdot \left(angle \cdot 0.005555555555555556\right)\\ t_8 := \left(b + a\right) \cdot \sin t_7\\ \mathbf{if}\;\frac{angle}{180} \leq -5 \cdot 10^{+254}:\\ \;\;\;\;t_2 \cdot \left(\left(2 \cdot t_1\right) \cdot t_5\right)\\ \mathbf{elif}\;\frac{angle}{180} \leq -1 \cdot 10^{+172}:\\ \;\;\;\;\left|\left(b - a\right) \cdot \left(\left(b + a\right) \cdot \sin \left(\left(angle \cdot \pi\right) \cdot 0.011111111111111112\right)\right)\right|\\ \mathbf{elif}\;\frac{angle}{180} \leq -2 \cdot 10^{+168}:\\ \;\;\;\;t_2 \cdot \left(t_6 \cdot \cos \left({\left({t_3}^{2}\right)}^{3} \cdot {t_3}^{3}\right)\right)\\ \mathbf{elif}\;\frac{angle}{180} \leq -5 \cdot 10^{+90}:\\ \;\;\;\;t_2 \cdot \left(t_6 \cdot {\left(\sqrt[3]{\cos t_7}\right)}^{3}\right)\\ \mathbf{elif}\;\frac{angle}{180} \leq -4 \cdot 10^{+34}:\\ \;\;\;\;t_2 \cdot \left(t_5 \cdot \left(2 \cdot \sin \left(\sqrt{{t_7}^{2}}\right)\right)\right)\\ \mathbf{elif}\;\frac{angle}{180} \leq 5 \cdot 10^{+61}:\\ \;\;\;\;\left(t_8 \cdot \cos \left(\pi \cdot \sqrt[3]{\frac{angle}{180} \cdot \left(\frac{angle}{180} \cdot \frac{angle}{180}\right)}\right)\right) \cdot t_0\\ \mathbf{else}:\\ \;\;\;\;t_0 \cdot \left(t_8 \cdot \cos \left(\pi \cdot \mathsf{expm1}\left(\mathsf{log1p}\left(\frac{angle}{180}\right)\right)\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (a b angle)
 :precision binary64
 (let* ((t_0 (* (- b a) 2.0))
        (t_1 (* angle (* 0.005555555555555556 PI)))
        (t_2 (* (- b a) (+ b a)))
        (t_3 (cbrt (cbrt t_1)))
        (t_4 (* (/ angle 180.0) PI))
        (t_5 (cos t_4))
        (t_6 (* 2.0 (sin t_4)))
        (t_7 (* PI (* angle 0.005555555555555556)))
        (t_8 (* (+ b a) (sin t_7))))
   (if (<= (/ angle 180.0) -5e+254)
     (* t_2 (* (* 2.0 t_1) t_5))
     (if (<= (/ angle 180.0) -1e+172)
       (fabs
        (* (- b a) (* (+ b a) (sin (* (* angle PI) 0.011111111111111112)))))
       (if (<= (/ angle 180.0) -2e+168)
         (* t_2 (* t_6 (cos (* (pow (pow t_3 2.0) 3.0) (pow t_3 3.0)))))
         (if (<= (/ angle 180.0) -5e+90)
           (* t_2 (* t_6 (pow (cbrt (cos t_7)) 3.0)))
           (if (<= (/ angle 180.0) -4e+34)
             (* t_2 (* t_5 (* 2.0 (sin (sqrt (pow t_7 2.0))))))
             (if (<= (/ angle 180.0) 5e+61)
               (*
                (*
                 t_8
                 (cos
                  (*
                   PI
                   (cbrt
                    (* (/ angle 180.0) (* (/ angle 180.0) (/ angle 180.0)))))))
                t_0)
               (*
                t_0
                (* t_8 (cos (* PI (expm1 (log1p (/ angle 180.0)))))))))))))))

C

double code(double a, double b, double angle) {
	double t_0 = (b - a) * 2.0;
	double t_1 = angle * (0.005555555555555556 * ((double) M_PI));
	double t_2 = (b - a) * (b + a);
	double t_3 = cbrt(cbrt(t_1));
	double t_4 = (angle / 180.0) * ((double) M_PI);
	double t_5 = cos(t_4);
	double t_6 = 2.0 * sin(t_4);
	double t_7 = ((double) M_PI) * (angle * 0.005555555555555556);
	double t_8 = (b + a) * sin(t_7);
	double tmp;
	if ((angle / 180.0) <= -5e+254) {
		tmp = t_2 * ((2.0 * t_1) * t_5);
	} else if ((angle / 180.0) <= -1e+172) {
		tmp = fabs(((b - a) * ((b + a) * sin(((angle * ((double) M_PI)) * 0.011111111111111112)))));
	} else if ((angle / 180.0) <= -2e+168) {
		tmp = t_2 * (t_6 * cos((pow(pow(t_3, 2.0), 3.0) * pow(t_3, 3.0))));
	} else if ((angle / 180.0) <= -5e+90) {
		tmp = t_2 * (t_6 * pow(cbrt(cos(t_7)), 3.0));
	} else if ((angle / 180.0) <= -4e+34) {
		tmp = t_2 * (t_5 * (2.0 * sin(sqrt(pow(t_7, 2.0)))));
	} else if ((angle / 180.0) <= 5e+61) {
		tmp = (t_8 * cos((((double) M_PI) * cbrt(((angle / 180.0) * ((angle / 180.0) * (angle / 180.0))))))) * t_0;
	} else {
		tmp = t_0 * (t_8 * cos((((double) M_PI) * expm1(log1p((angle / 180.0))))));
	}
	return tmp;
}

Java

public static double code(double a, double b, double angle) {
	double t_0 = (b - a) * 2.0;
	double t_1 = angle * (0.005555555555555556 * Math.PI);
	double t_2 = (b - a) * (b + a);
	double t_3 = Math.cbrt(Math.cbrt(t_1));
	double t_4 = (angle / 180.0) * Math.PI;
	double t_5 = Math.cos(t_4);
	double t_6 = 2.0 * Math.sin(t_4);
	double t_7 = Math.PI * (angle * 0.005555555555555556);
	double t_8 = (b + a) * Math.sin(t_7);
	double tmp;
	if ((angle / 180.0) <= -5e+254) {
		tmp = t_2 * ((2.0 * t_1) * t_5);
	} else if ((angle / 180.0) <= -1e+172) {
		tmp = Math.abs(((b - a) * ((b + a) * Math.sin(((angle * Math.PI) * 0.011111111111111112)))));
	} else if ((angle / 180.0) <= -2e+168) {
		tmp = t_2 * (t_6 * Math.cos((Math.pow(Math.pow(t_3, 2.0), 3.0) * Math.pow(t_3, 3.0))));
	} else if ((angle / 180.0) <= -5e+90) {
		tmp = t_2 * (t_6 * Math.pow(Math.cbrt(Math.cos(t_7)), 3.0));
	} else if ((angle / 180.0) <= -4e+34) {
		tmp = t_2 * (t_5 * (2.0 * Math.sin(Math.sqrt(Math.pow(t_7, 2.0)))));
	} else if ((angle / 180.0) <= 5e+61) {
		tmp = (t_8 * Math.cos((Math.PI * Math.cbrt(((angle / 180.0) * ((angle / 180.0) * (angle / 180.0))))))) * t_0;
	} else {
		tmp = t_0 * (t_8 * Math.cos((Math.PI * Math.expm1(Math.log1p((angle / 180.0))))));
	}
	return tmp;
}

Julia

function code(a, b, angle)
	t_0 = Float64(Float64(b - a) * 2.0)
	t_1 = Float64(angle * Float64(0.005555555555555556 * pi))
	t_2 = Float64(Float64(b - a) * Float64(b + a))
	t_3 = cbrt(cbrt(t_1))
	t_4 = Float64(Float64(angle / 180.0) * pi)
	t_5 = cos(t_4)
	t_6 = Float64(2.0 * sin(t_4))
	t_7 = Float64(pi * Float64(angle * 0.005555555555555556))
	t_8 = Float64(Float64(b + a) * sin(t_7))
	tmp = 0.0
	if (Float64(angle / 180.0) <= -5e+254)
		tmp = Float64(t_2 * Float64(Float64(2.0 * t_1) * t_5));
	elseif (Float64(angle / 180.0) <= -1e+172)
		tmp = abs(Float64(Float64(b - a) * Float64(Float64(b + a) * sin(Float64(Float64(angle * pi) * 0.011111111111111112)))));
	elseif (Float64(angle / 180.0) <= -2e+168)
		tmp = Float64(t_2 * Float64(t_6 * cos(Float64(((t_3 ^ 2.0) ^ 3.0) * (t_3 ^ 3.0)))));
	elseif (Float64(angle / 180.0) <= -5e+90)
		tmp = Float64(t_2 * Float64(t_6 * (cbrt(cos(t_7)) ^ 3.0)));
	elseif (Float64(angle / 180.0) <= -4e+34)
		tmp = Float64(t_2 * Float64(t_5 * Float64(2.0 * sin(sqrt((t_7 ^ 2.0))))));
	elseif (Float64(angle / 180.0) <= 5e+61)
		tmp = Float64(Float64(t_8 * cos(Float64(pi * cbrt(Float64(Float64(angle / 180.0) * Float64(Float64(angle / 180.0) * Float64(angle / 180.0))))))) * t_0);
	else
		tmp = Float64(t_0 * Float64(t_8 * cos(Float64(pi * expm1(log1p(Float64(angle / 180.0)))))));
	end
	return tmp
end

Wolfram

code[a_, b_, angle_] := Block[{t$95$0 = N[(N[(b - a), $MachinePrecision] * 2.0), $MachinePrecision]}, Block[{t$95$1 = N[(angle * N[(0.005555555555555556 * Pi), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(N[(b - a), $MachinePrecision] * N[(b + a), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$3 = N[Power[N[Power[t$95$1, 1/3], $MachinePrecision], 1/3], $MachinePrecision]}, Block[{t$95$4 = N[(N[(angle / 180.0), $MachinePrecision] * Pi), $MachinePrecision]}, Block[{t$95$5 = N[Cos[t$95$4], $MachinePrecision]}, Block[{t$95$6 = N[(2.0 * N[Sin[t$95$4], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$7 = N[(Pi * N[(angle * 0.005555555555555556), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$8 = N[(N[(b + a), $MachinePrecision] * N[Sin[t$95$7], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[(angle / 180.0), $MachinePrecision], -5e+254], N[(t$95$2 * N[(N[(2.0 * t$95$1), $MachinePrecision] * t$95$5), $MachinePrecision]), $MachinePrecision], If[LessEqual[N[(angle / 180.0), $MachinePrecision], -1e+172], N[Abs[N[(N[(b - a), $MachinePrecision] * N[(N[(b + a), $MachinePrecision] * N[Sin[N[(N[(angle * Pi), $MachinePrecision] * 0.011111111111111112), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], If[LessEqual[N[(angle / 180.0), $MachinePrecision], -2e+168], N[(t$95$2 * N[(t$95$6 * N[Cos[N[(N[Power[N[Power[t$95$3, 2.0], $MachinePrecision], 3.0], $MachinePrecision] * N[Power[t$95$3, 3.0], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[N[(angle / 180.0), $MachinePrecision], -5e+90], N[(t$95$2 * N[(t$95$6 * N[Power[N[Power[N[Cos[t$95$7], $MachinePrecision], 1/3], $MachinePrecision], 3.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[N[(angle / 180.0), $MachinePrecision], -4e+34], N[(t$95$2 * N[(t$95$5 * N[(2.0 * N[Sin[N[Sqrt[N[Power[t$95$7, 2.0], $MachinePrecision]], $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[N[(angle / 180.0), $MachinePrecision], 5e+61], N[(N[(t$95$8 * N[Cos[N[(Pi * N[Power[N[(N[(angle / 180.0), $MachinePrecision] * N[(N[(angle / 180.0), $MachinePrecision] * N[(angle / 180.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 1/3], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * t$95$0), $MachinePrecision], N[(t$95$0 * N[(t$95$8 * N[Cos[N[(Pi * N[(Exp[N[Log[1 + N[(angle / 180.0), $MachinePrecision]], $MachinePrecision]] - 1), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]]]]]]]]]]]]

Derivation

1. Split input into 7 regimes

2. if (/.f64 angle 180) < -4.99999999999999994e254

   1. Initial program 25.0%

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

      1. *-commutative 25.0%

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

      2. associate-*l* 25.0%

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

      3. associate-*l* 25.0%

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

      4. unpow2 25.0%

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

      5. unpow2 25.0%

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

      6. difference-of-squares 32.7%

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

   3. Simplified 32.7%

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

   4. Taylor expanded in angle around 0 41.5%

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

      1. associate-*r* 41.5%

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

      2. *-commutative 41.5%

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

      3. associate-*l* 41.5%

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

   6. Simplified 41.5%

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

   if -4.99999999999999994e254 < (/.f64 angle 180) < -1.0000000000000001e172

   1. Initial program 17.6%

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

      1. *-commutative 17.6%

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

      2. associate-*l* 17.6%

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

      3. associate-*l* 17.6%

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

      4. unpow2 17.6%

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

      5. unpow2 17.6%

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

      6. difference-of-squares 27.6%

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

   3. Simplified 27.6%

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

      1. add-sqr-sqrt 22.5%

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

      2. sqrt-unprod 53.6%

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

      3. pow2 53.6%

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

   5. Applied egg-rr 53.6%

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

      1. unpow2 53.6%

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

      2. rem-sqrt-square 55.3%

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

      3. *-commutative 55.3%

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

      4. associate-*l* 55.3%

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

      5. +-commutative 55.3%

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

      6. *-commutative 55.3%

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

      7. associate-*r* 55.4%

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

      8. associate-*l* 55.4%

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

      9. *-commutative 55.4%

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

      10. metadata-eval 55.4%

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

   7. Simplified 55.4%

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

   if -1.0000000000000001e172 < (/.f64 angle 180) < -1.9999999999999999e168

   1. Initial program 27.1%

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

      1. *-commutative 27.1%

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

      2. associate-*l* 27.1%

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

      3. associate-*l* 27.1%

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

      4. unpow2 27.1%

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

      5. unpow2 27.1%

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

      6. difference-of-squares 27.1%

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

   3. Simplified 27.1%

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

      1. add-cube-cbrt 27.8%

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

      2. pow3 21.1%

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

      3. div-inv 21.1%

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

      4. metadata-eval 21.1%

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

   5. Applied egg-rr 21.1%

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

      1. add-cube-cbrt 64.2%

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

      2. unpow-prod-down 64.5%

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

   7. Applied egg-rr 64.5%

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

   if -1.9999999999999999e168 < (/.f64 angle 180) < -5.0000000000000004e90

   1. Initial program 46.2%

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

      1. *-commutative 46.2%

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

      2. associate-*l* 46.2%

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

      3. associate-*l* 46.2%

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

      4. unpow2 46.2%

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

      5. unpow2 46.2%

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

      6. difference-of-squares 51.0%

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

   3. Simplified 51.0%

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

      1. add-cube-cbrt 51.0%

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

      2. pow3 51.0%

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

      3. div-inv 51.6%

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

      4. metadata-eval 51.6%

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

   5. Applied egg-rr 51.6%

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

   if -5.0000000000000004e90 < (/.f64 angle 180) < -3.99999999999999978e34

   1. Initial program 11.4%

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

      1. *-commutative 11.4%

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

      2. associate-*l* 11.4%

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

      3. associate-*l* 11.4%

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

      4. unpow2 11.4%

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

      5. unpow2 11.4%

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

      6. difference-of-squares 11.4%

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

   3. Simplified 11.4%

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

      1. add-sqr-sqrt 0.0%

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

      2. sqrt-unprod 46.8%

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

      3. pow2 46.8%

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

      4. div-inv 56.8%

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

      5. metadata-eval 56.8%

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

   5. Applied egg-rr 56.8%

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

   if -3.99999999999999978e34 < (/.f64 angle 180) < 5.00000000000000018e61

   1. Initial program 75.8%

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

      1. *-commutative 75.8%

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

      2. associate-*l* 75.8%

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

      3. associate-*l* 75.8%

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

      4. unpow2 75.8%

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

      5. unpow2 75.8%

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

      6. difference-of-squares 79.1%

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

   3. Simplified 79.1%

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

      1. add-cube-cbrt 79.0%

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

      2. pow3 79.0%

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

      3. div-inv 79.4%

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

      4. metadata-eval 79.4%

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

   5. Applied egg-rr 79.4%

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

   6. Taylor expanded in angle around inf 88.5%

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

      1. associate-*r* 88.5%

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

      2. *-commutative 88.5%

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

      3. *-commutative 88.5%

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

      4. *-commutative 88.5%

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

      5. +-commutative 88.5%

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

      6. *-commutative 88.5%

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

      7. *-commutative 88.5%

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

      8. associate-*r* 89.2%

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

      9. *-commutative 89.2%

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

      10. *-commutative 89.2%

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

      11. associate-*r* 89.6%

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

   8. Simplified 89.6%

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

      1. metadata-eval 89.6%

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

      2. div-inv 89.8%

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

      3. add-cbrt-cube 90.5%

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

   10. Applied egg-rr 90.5%

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

   if 5.00000000000000018e61 < (/.f64 angle 180)

   1. Initial program 30.5%

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

      1. *-commutative 30.5%

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

      2. associate-*l* 30.5%

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

      3. associate-*l* 30.5%

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

      4. unpow2 30.5%

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

      5. unpow2 30.5%

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

      6. difference-of-squares 32.4%

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

   3. Simplified 32.4%

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

      1. add-cube-cbrt 32.4%

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

      2. pow3 32.4%

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

      3. div-inv 32.8%

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

      4. metadata-eval 32.8%

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

   5. Applied egg-rr 32.8%

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

   6. Taylor expanded in angle around inf 26.7%

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

      1. associate-*r* 26.7%

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

      2. *-commutative 26.7%

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

      3. *-commutative 26.7%

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

      4. *-commutative 26.7%

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

      5. +-commutative 26.7%

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

      6. *-commutative 26.7%

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

      7. *-commutative 26.7%

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

      8. associate-*r* 25.2%

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

      9. *-commutative 25.2%

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

      10. *-commutative 25.2%

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

      11. associate-*r* 26.2%

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

   8. Simplified 26.2%

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

      1. metadata-eval 26.2%

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

      2. div-inv 31.1%

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

      3. expm1-log1p-u 45.7%

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

   10. Applied egg-rr 45.7%

       \[\leadsto \left(\left(\left(b + a\right) \cdot \sin \left(\pi \cdot \left(angle \cdot 0.005555555555555556\right)\right)\right) \cdot \cos \left(\pi \cdot \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{angle}{180}\right)\right)}\right)\right) \cdot \left(2 \cdot \left(b - a\right)\right) \]
3. Recombined 7 regimes into one program.

4. Final simplification 69.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{angle}{180} \leq -5 \cdot 10^{+254}:\\ \;\;\;\;\left(\left(b - a\right) \cdot \left(b + a\right)\right) \cdot \left(\left(2 \cdot \left(angle \cdot \left(0.005555555555555556 \cdot \pi\right)\right)\right) \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)\right)\\ \mathbf{elif}\;\frac{angle}{180} \leq -1 \cdot 10^{+172}:\\ \;\;\;\;\left|\left(b - a\right) \cdot \left(\left(b + a\right) \cdot \sin \left(\left(angle \cdot \pi\right) \cdot 0.011111111111111112\right)\right)\right|\\ \mathbf{elif}\;\frac{angle}{180} \leq -2 \cdot 10^{+168}:\\ \;\;\;\;\left(\left(b - a\right) \cdot \left(b + a\right)\right) \cdot \left(\left(2 \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right) \cdot \cos \left({\left({\left(\sqrt[3]{\sqrt[3]{angle \cdot \left(0.005555555555555556 \cdot \pi\right)}}\right)}^{2}\right)}^{3} \cdot {\left(\sqrt[3]{\sqrt[3]{angle \cdot \left(0.005555555555555556 \cdot \pi\right)}}\right)}^{3}\right)\right)\\ \mathbf{elif}\;\frac{angle}{180} \leq -5 \cdot 10^{+90}:\\ \;\;\;\;\left(\left(b - a\right) \cdot \left(b + a\right)\right) \cdot \left(\left(2 \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right) \cdot {\left(\sqrt[3]{\cos \left(\pi \cdot \left(angle \cdot 0.005555555555555556\right)\right)}\right)}^{3}\right)\\ \mathbf{elif}\;\frac{angle}{180} \leq -4 \cdot 10^{+34}:\\ \;\;\;\;\left(\left(b - a\right) \cdot \left(b + a\right)\right) \cdot \left(\cos \left(\frac{angle}{180} \cdot \pi\right) \cdot \left(2 \cdot \sin \left(\sqrt{{\left(\pi \cdot \left(angle \cdot 0.005555555555555556\right)\right)}^{2}}\right)\right)\right)\\ \mathbf{elif}\;\frac{angle}{180} \leq 5 \cdot 10^{+61}:\\ \;\;\;\;\left(\left(\left(b + a\right) \cdot \sin \left(\pi \cdot \left(angle \cdot 0.005555555555555556\right)\right)\right) \cdot \cos \left(\pi \cdot \sqrt[3]{\frac{angle}{180} \cdot \left(\frac{angle}{180} \cdot \frac{angle}{180}\right)}\right)\right) \cdot \left(\left(b - a\right) \cdot 2\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\left(b - a\right) \cdot 2\right) \cdot \left(\left(\left(b + a\right) \cdot \sin \left(\pi \cdot \left(angle \cdot 0.005555555555555556\right)\right)\right) \cdot \cos \left(\pi \cdot \mathsf{expm1}\left(\mathsf{log1p}\left(\frac{angle}{180}\right)\right)\right)\right)\\ \end{array} \]

Alternative 2: 68.1% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{angle}{180} \cdot \pi\\ \mathbf{if}\;\cos t_0 \cdot \left(\sin t_0 \cdot \left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right)\right) \leq -\infty:\\ \;\;\;\;\left(b - a\right) \cdot \left(\left(b + a\right) \cdot \mathsf{fma}\left(-2.2862368541380886 \cdot 10^{-7}, {\left(angle \cdot \pi\right)}^{3}, \left(angle \cdot \pi\right) \cdot 0.011111111111111112\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(b - a\right) \cdot \left(\left(b + a\right) \cdot \sin \left(angle \cdot \left(\pi \cdot 0.011111111111111112\right)\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (a b angle)
 :precision binary64
 (let* ((t_0 (* (/ angle 180.0) PI)))
   (if (<=
        (* (cos t_0) (* (sin t_0) (* 2.0 (- (pow b 2.0) (pow a 2.0)))))
        (- INFINITY))
     (*
      (- b a)
      (*
       (+ b a)
       (fma
        -2.2862368541380886e-7
        (pow (* angle PI) 3.0)
        (* (* angle PI) 0.011111111111111112))))
     (* (- b a) (* (+ b a) (sin (* angle (* PI 0.011111111111111112))))))))

C

double code(double a, double b, double angle) {
	double t_0 = (angle / 180.0) * ((double) M_PI);
	double tmp;
	if ((cos(t_0) * (sin(t_0) * (2.0 * (pow(b, 2.0) - pow(a, 2.0))))) <= -((double) INFINITY)) {
		tmp = (b - a) * ((b + a) * fma(-2.2862368541380886e-7, pow((angle * ((double) M_PI)), 3.0), ((angle * ((double) M_PI)) * 0.011111111111111112)));
	} else {
		tmp = (b - a) * ((b + a) * sin((angle * (((double) M_PI) * 0.011111111111111112))));
	}
	return tmp;
}

Julia

function code(a, b, angle)
	t_0 = Float64(Float64(angle / 180.0) * pi)
	tmp = 0.0
	if (Float64(cos(t_0) * Float64(sin(t_0) * Float64(2.0 * Float64((b ^ 2.0) - (a ^ 2.0))))) <= Float64(-Inf))
		tmp = Float64(Float64(b - a) * Float64(Float64(b + a) * fma(-2.2862368541380886e-7, (Float64(angle * pi) ^ 3.0), Float64(Float64(angle * pi) * 0.011111111111111112))));
	else
		tmp = Float64(Float64(b - a) * Float64(Float64(b + a) * sin(Float64(angle * Float64(pi * 0.011111111111111112)))));
	end
	return tmp
end

Wolfram

code[a_, b_, angle_] := Block[{t$95$0 = N[(N[(angle / 180.0), $MachinePrecision] * Pi), $MachinePrecision]}, If[LessEqual[N[(N[Cos[t$95$0], $MachinePrecision] * N[(N[Sin[t$95$0], $MachinePrecision] * N[(2.0 * N[(N[Power[b, 2.0], $MachinePrecision] - N[Power[a, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], (-Infinity)], N[(N[(b - a), $MachinePrecision] * N[(N[(b + a), $MachinePrecision] * N[(-2.2862368541380886e-7 * N[Power[N[(angle * Pi), $MachinePrecision], 3.0], $MachinePrecision] + N[(N[(angle * Pi), $MachinePrecision] * 0.011111111111111112), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(b - a), $MachinePrecision] * N[(N[(b + a), $MachinePrecision] * N[Sin[N[(angle * N[(Pi * 0.011111111111111112), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 (*.f64 (*.f64 2 (-.f64 (pow.f64 b 2) (pow.f64 a 2))) (sin.f64 (*.f64 (PI.f64) (/.f64 angle 180)))) (cos.f64 (*.f64 (PI.f64) (/.f64 angle 180)))) < -inf.0

   1. Initial program 59.5%

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

      1. *-commutative 59.5%

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

      2. associate-*l* 59.5%

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

      3. associate-*l* 59.5%

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

      4. unpow2 59.5%

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

      5. unpow2 59.5%

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

      6. difference-of-squares 59.5%

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

   3. Simplified 59.5%

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

      1. difference-of-squares 59.5%

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

      2. *-commutative 59.5%

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

      3. sub-neg 59.5%

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

      4. distribute-lft-in 59.5%

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

      5. associate-*l* 59.5%

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

      6. 2-sin 59.5%

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

      7. div-inv 59.5%

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

      8. metadata-eval 59.5%

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

   5. Applied egg-rr 54.0%

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

      1. distribute-lft-out 54.0%

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

      2. sub-neg 54.0%

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

      3. difference-of-squares 54.0%

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

      4. *-commutative 54.0%

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

      5. *-commutative 54.0%

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

      6. associate-*l* 70.8%

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

      7. +-commutative 70.8%

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

      8. *-commutative 70.8%

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

      9. associate-*r* 65.4%

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

      10. associate-*l* 65.4%

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

      11. *-commutative 65.4%

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

      12. metadata-eval 65.4%

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

   7. Simplified 65.4%

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

   8. Taylor expanded in angle around 0 81.7%

      \[\leadsto \left(b - a\right) \cdot \left(\left(a + b\right) \cdot \color{blue}{\left(0.011111111111111112 \cdot \left(angle \cdot \pi\right) + -2.2862368541380886 \cdot 10^{-7} \cdot \left({angle}^{3} \cdot {\pi}^{3}\right)\right)}\right) \]
   9. Step-by-step derivation

      1. +-commutative 81.7%

         \[\leadsto \left(b - a\right) \cdot \left(\left(a + b\right) \cdot \color{blue}{\left(-2.2862368541380886 \cdot 10^{-7} \cdot \left({angle}^{3} \cdot {\pi}^{3}\right) + 0.011111111111111112 \cdot \left(angle \cdot \pi\right)\right)}\right) \]

      2. *-commutative 81.7%

         \[\leadsto \left(b - a\right) \cdot \left(\left(a + b\right) \cdot \left(-2.2862368541380886 \cdot 10^{-7} \cdot \left({angle}^{3} \cdot {\pi}^{3}\right) + \color{blue}{\left(angle \cdot \pi\right) \cdot 0.011111111111111112}\right)\right) \]

      3. associate-*r* 81.7%

         \[\leadsto \left(b - a\right) \cdot \left(\left(a + b\right) \cdot \left(-2.2862368541380886 \cdot 10^{-7} \cdot \left({angle}^{3} \cdot {\pi}^{3}\right) + \color{blue}{angle \cdot \left(\pi \cdot 0.011111111111111112\right)}\right)\right) \]

      4. fma-def 81.7%

         \[\leadsto \left(b - a\right) \cdot \left(\left(a + b\right) \cdot \color{blue}{\mathsf{fma}\left(-2.2862368541380886 \cdot 10^{-7}, {angle}^{3} \cdot {\pi}^{3}, angle \cdot \left(\pi \cdot 0.011111111111111112\right)\right)}\right) \]

      5. cube-prod 81.7%

         \[\leadsto \left(b - a\right) \cdot \left(\left(a + b\right) \cdot \mathsf{fma}\left(-2.2862368541380886 \cdot 10^{-7}, \color{blue}{{\left(angle \cdot \pi\right)}^{3}}, angle \cdot \left(\pi \cdot 0.011111111111111112\right)\right)\right) \]

      6. associate-*r* 81.7%

         \[\leadsto \left(b - a\right) \cdot \left(\left(a + b\right) \cdot \mathsf{fma}\left(-2.2862368541380886 \cdot 10^{-7}, {\left(angle \cdot \pi\right)}^{3}, \color{blue}{\left(angle \cdot \pi\right) \cdot 0.011111111111111112}\right)\right) \]

      7. *-commutative 81.7%

         \[\leadsto \left(b - a\right) \cdot \left(\left(a + b\right) \cdot \mathsf{fma}\left(-2.2862368541380886 \cdot 10^{-7}, {\left(angle \cdot \pi\right)}^{3}, \color{blue}{0.011111111111111112 \cdot \left(angle \cdot \pi\right)}\right)\right) \]

   10. Simplified 81.7%

       \[\leadsto \left(b - a\right) \cdot \left(\left(a + b\right) \cdot \color{blue}{\mathsf{fma}\left(-2.2862368541380886 \cdot 10^{-7}, {\left(angle \cdot \pi\right)}^{3}, 0.011111111111111112 \cdot \left(angle \cdot \pi\right)\right)}\right) \]

   if -inf.0 < (*.f64 (*.f64 (*.f64 2 (-.f64 (pow.f64 b 2) (pow.f64 a 2))) (sin.f64 (*.f64 (PI.f64) (/.f64 angle 180)))) (cos.f64 (*.f64 (PI.f64) (/.f64 angle 180))))

   1. Initial program 48.6%

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

      1. *-commutative 48.6%

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

      2. associate-*l* 48.6%

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

      3. associate-*l* 48.6%

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

      4. unpow2 48.6%

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

      5. unpow2 48.6%

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

      6. difference-of-squares 53.6%

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

   3. Simplified 53.6%

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

      1. difference-of-squares 48.6%

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

      2. *-commutative 48.6%

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

      3. sub-neg 48.6%

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

      4. distribute-lft-in 48.6%

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

      5. associate-*l* 48.6%

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

      6. 2-sin 48.6%

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

      7. div-inv 48.5%

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

      8. metadata-eval 48.5%

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

   5. Applied egg-rr 48.4%

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

      1. distribute-lft-out 48.4%

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

      2. sub-neg 48.4%

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

      3. difference-of-squares 53.5%

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

      4. *-commutative 53.5%

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

      5. *-commutative 53.5%

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

      6. associate-*l* 55.3%

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

      7. +-commutative 55.3%

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

      8. *-commutative 55.3%

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

      9. associate-*r* 53.7%

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

      10. associate-*l* 53.7%

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

      11. *-commutative 53.7%

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

      12. metadata-eval 53.7%

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

   7. Simplified 53.7%

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

      1. *-un-lft-identity 53.7%

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

      2. associate-*l* 56.0%

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

   9. Applied egg-rr 56.0%

      \[\leadsto \left(b - a\right) \cdot \left(\left(a + b\right) \cdot \color{blue}{\left(1 \cdot \sin \left(angle \cdot \left(\pi \cdot 0.011111111111111112\right)\right)\right)}\right) \]
3. Recombined 2 regimes into one program.

4. Final simplification 61.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\cos \left(\frac{angle}{180} \cdot \pi\right) \cdot \left(\sin \left(\frac{angle}{180} \cdot \pi\right) \cdot \left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right)\right) \leq -\infty:\\ \;\;\;\;\left(b - a\right) \cdot \left(\left(b + a\right) \cdot \mathsf{fma}\left(-2.2862368541380886 \cdot 10^{-7}, {\left(angle \cdot \pi\right)}^{3}, \left(angle \cdot \pi\right) \cdot 0.011111111111111112\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(b - a\right) \cdot \left(\left(b + a\right) \cdot \sin \left(angle \cdot \left(\pi \cdot 0.011111111111111112\right)\right)\right)\\ \end{array} \]

Alternative 3: 68.0% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := angle \cdot \left(0.005555555555555556 \cdot \pi\right)\\ t_1 := \frac{angle}{180} \cdot \pi\\ t_2 := \cos t_1\\ t_3 := \left(b - a\right) \cdot 2\\ t_4 := \left(b - a\right) \cdot \left(b + a\right)\\ t_5 := \pi \cdot \left(angle \cdot 0.005555555555555556\right)\\ t_6 := \left(b + a\right) \cdot \sin t_5\\ \mathbf{if}\;\frac{angle}{180} \leq -5 \cdot 10^{+254}:\\ \;\;\;\;t_4 \cdot \left(\left(2 \cdot t_0\right) \cdot t_2\right)\\ \mathbf{elif}\;\frac{angle}{180} \leq -4 \cdot 10^{+178}:\\ \;\;\;\;\left|\left(b - a\right) \cdot \left(\left(b + a\right) \cdot \sin \left(\left(angle \cdot \pi\right) \cdot 0.011111111111111112\right)\right)\right|\\ \mathbf{elif}\;\frac{angle}{180} \leq -2 \cdot 10^{+168}:\\ \;\;\;\;t_4 \cdot \left(t_2 \cdot \left(2 \cdot \sin \left(\left|t_0\right|\right)\right)\right)\\ \mathbf{elif}\;\frac{angle}{180} \leq -5 \cdot 10^{+90}:\\ \;\;\;\;t_4 \cdot \left(t_2 \cdot \left(2 \cdot \sin t_1\right)\right)\\ \mathbf{elif}\;\frac{angle}{180} \leq -4 \cdot 10^{+34}:\\ \;\;\;\;t_4 \cdot \left(t_2 \cdot \left(2 \cdot \sin \left(\sqrt{{t_5}^{2}}\right)\right)\right)\\ \mathbf{elif}\;\frac{angle}{180} \leq 5 \cdot 10^{+61}:\\ \;\;\;\;\left(t_6 \cdot \cos \left(\pi \cdot \sqrt[3]{\frac{angle}{180} \cdot \left(\frac{angle}{180} \cdot \frac{angle}{180}\right)}\right)\right) \cdot t_3\\ \mathbf{else}:\\ \;\;\;\;t_3 \cdot \left(t_6 \cdot \cos \left(\pi \cdot \mathsf{expm1}\left(\mathsf{log1p}\left(\frac{angle}{180}\right)\right)\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (a b angle)
 :precision binary64
 (let* ((t_0 (* angle (* 0.005555555555555556 PI)))
        (t_1 (* (/ angle 180.0) PI))
        (t_2 (cos t_1))
        (t_3 (* (- b a) 2.0))
        (t_4 (* (- b a) (+ b a)))
        (t_5 (* PI (* angle 0.005555555555555556)))
        (t_6 (* (+ b a) (sin t_5))))
   (if (<= (/ angle 180.0) -5e+254)
     (* t_4 (* (* 2.0 t_0) t_2))
     (if (<= (/ angle 180.0) -4e+178)
       (fabs
        (* (- b a) (* (+ b a) (sin (* (* angle PI) 0.011111111111111112)))))
       (if (<= (/ angle 180.0) -2e+168)
         (* t_4 (* t_2 (* 2.0 (sin (fabs t_0)))))
         (if (<= (/ angle 180.0) -5e+90)
           (* t_4 (* t_2 (* 2.0 (sin t_1))))
           (if (<= (/ angle 180.0) -4e+34)
             (* t_4 (* t_2 (* 2.0 (sin (sqrt (pow t_5 2.0))))))
             (if (<= (/ angle 180.0) 5e+61)
               (*
                (*
                 t_6
                 (cos
                  (*
                   PI
                   (cbrt
                    (* (/ angle 180.0) (* (/ angle 180.0) (/ angle 180.0)))))))
                t_3)
               (*
                t_3
                (* t_6 (cos (* PI (expm1 (log1p (/ angle 180.0)))))))))))))))

C

double code(double a, double b, double angle) {
	double t_0 = angle * (0.005555555555555556 * ((double) M_PI));
	double t_1 = (angle / 180.0) * ((double) M_PI);
	double t_2 = cos(t_1);
	double t_3 = (b - a) * 2.0;
	double t_4 = (b - a) * (b + a);
	double t_5 = ((double) M_PI) * (angle * 0.005555555555555556);
	double t_6 = (b + a) * sin(t_5);
	double tmp;
	if ((angle / 180.0) <= -5e+254) {
		tmp = t_4 * ((2.0 * t_0) * t_2);
	} else if ((angle / 180.0) <= -4e+178) {
		tmp = fabs(((b - a) * ((b + a) * sin(((angle * ((double) M_PI)) * 0.011111111111111112)))));
	} else if ((angle / 180.0) <= -2e+168) {
		tmp = t_4 * (t_2 * (2.0 * sin(fabs(t_0))));
	} else if ((angle / 180.0) <= -5e+90) {
		tmp = t_4 * (t_2 * (2.0 * sin(t_1)));
	} else if ((angle / 180.0) <= -4e+34) {
		tmp = t_4 * (t_2 * (2.0 * sin(sqrt(pow(t_5, 2.0)))));
	} else if ((angle / 180.0) <= 5e+61) {
		tmp = (t_6 * cos((((double) M_PI) * cbrt(((angle / 180.0) * ((angle / 180.0) * (angle / 180.0))))))) * t_3;
	} else {
		tmp = t_3 * (t_6 * cos((((double) M_PI) * expm1(log1p((angle / 180.0))))));
	}
	return tmp;
}

Java

public static double code(double a, double b, double angle) {
	double t_0 = angle * (0.005555555555555556 * Math.PI);
	double t_1 = (angle / 180.0) * Math.PI;
	double t_2 = Math.cos(t_1);
	double t_3 = (b - a) * 2.0;
	double t_4 = (b - a) * (b + a);
	double t_5 = Math.PI * (angle * 0.005555555555555556);
	double t_6 = (b + a) * Math.sin(t_5);
	double tmp;
	if ((angle / 180.0) <= -5e+254) {
		tmp = t_4 * ((2.0 * t_0) * t_2);
	} else if ((angle / 180.0) <= -4e+178) {
		tmp = Math.abs(((b - a) * ((b + a) * Math.sin(((angle * Math.PI) * 0.011111111111111112)))));
	} else if ((angle / 180.0) <= -2e+168) {
		tmp = t_4 * (t_2 * (2.0 * Math.sin(Math.abs(t_0))));
	} else if ((angle / 180.0) <= -5e+90) {
		tmp = t_4 * (t_2 * (2.0 * Math.sin(t_1)));
	} else if ((angle / 180.0) <= -4e+34) {
		tmp = t_4 * (t_2 * (2.0 * Math.sin(Math.sqrt(Math.pow(t_5, 2.0)))));
	} else if ((angle / 180.0) <= 5e+61) {
		tmp = (t_6 * Math.cos((Math.PI * Math.cbrt(((angle / 180.0) * ((angle / 180.0) * (angle / 180.0))))))) * t_3;
	} else {
		tmp = t_3 * (t_6 * Math.cos((Math.PI * Math.expm1(Math.log1p((angle / 180.0))))));
	}
	return tmp;
}

Julia

function code(a, b, angle)
	t_0 = Float64(angle * Float64(0.005555555555555556 * pi))
	t_1 = Float64(Float64(angle / 180.0) * pi)
	t_2 = cos(t_1)
	t_3 = Float64(Float64(b - a) * 2.0)
	t_4 = Float64(Float64(b - a) * Float64(b + a))
	t_5 = Float64(pi * Float64(angle * 0.005555555555555556))
	t_6 = Float64(Float64(b + a) * sin(t_5))
	tmp = 0.0
	if (Float64(angle / 180.0) <= -5e+254)
		tmp = Float64(t_4 * Float64(Float64(2.0 * t_0) * t_2));
	elseif (Float64(angle / 180.0) <= -4e+178)
		tmp = abs(Float64(Float64(b - a) * Float64(Float64(b + a) * sin(Float64(Float64(angle * pi) * 0.011111111111111112)))));
	elseif (Float64(angle / 180.0) <= -2e+168)
		tmp = Float64(t_4 * Float64(t_2 * Float64(2.0 * sin(abs(t_0)))));
	elseif (Float64(angle / 180.0) <= -5e+90)
		tmp = Float64(t_4 * Float64(t_2 * Float64(2.0 * sin(t_1))));
	elseif (Float64(angle / 180.0) <= -4e+34)
		tmp = Float64(t_4 * Float64(t_2 * Float64(2.0 * sin(sqrt((t_5 ^ 2.0))))));
	elseif (Float64(angle / 180.0) <= 5e+61)
		tmp = Float64(Float64(t_6 * cos(Float64(pi * cbrt(Float64(Float64(angle / 180.0) * Float64(Float64(angle / 180.0) * Float64(angle / 180.0))))))) * t_3);
	else
		tmp = Float64(t_3 * Float64(t_6 * cos(Float64(pi * expm1(log1p(Float64(angle / 180.0)))))));
	end
	return tmp
end

Wolfram

code[a_, b_, angle_] := Block[{t$95$0 = N[(angle * N[(0.005555555555555556 * Pi), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[(angle / 180.0), $MachinePrecision] * Pi), $MachinePrecision]}, Block[{t$95$2 = N[Cos[t$95$1], $MachinePrecision]}, Block[{t$95$3 = N[(N[(b - a), $MachinePrecision] * 2.0), $MachinePrecision]}, Block[{t$95$4 = N[(N[(b - a), $MachinePrecision] * N[(b + a), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$5 = N[(Pi * N[(angle * 0.005555555555555556), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$6 = N[(N[(b + a), $MachinePrecision] * N[Sin[t$95$5], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[(angle / 180.0), $MachinePrecision], -5e+254], N[(t$95$4 * N[(N[(2.0 * t$95$0), $MachinePrecision] * t$95$2), $MachinePrecision]), $MachinePrecision], If[LessEqual[N[(angle / 180.0), $MachinePrecision], -4e+178], N[Abs[N[(N[(b - a), $MachinePrecision] * N[(N[(b + a), $MachinePrecision] * N[Sin[N[(N[(angle * Pi), $MachinePrecision] * 0.011111111111111112), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], If[LessEqual[N[(angle / 180.0), $MachinePrecision], -2e+168], N[(t$95$4 * N[(t$95$2 * N[(2.0 * N[Sin[N[Abs[t$95$0], $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[N[(angle / 180.0), $MachinePrecision], -5e+90], N[(t$95$4 * N[(t$95$2 * N[(2.0 * N[Sin[t$95$1], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[N[(angle / 180.0), $MachinePrecision], -4e+34], N[(t$95$4 * N[(t$95$2 * N[(2.0 * N[Sin[N[Sqrt[N[Power[t$95$5, 2.0], $MachinePrecision]], $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[N[(angle / 180.0), $MachinePrecision], 5e+61], N[(N[(t$95$6 * N[Cos[N[(Pi * N[Power[N[(N[(angle / 180.0), $MachinePrecision] * N[(N[(angle / 180.0), $MachinePrecision] * N[(angle / 180.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 1/3], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * t$95$3), $MachinePrecision], N[(t$95$3 * N[(t$95$6 * N[Cos[N[(Pi * N[(Exp[N[Log[1 + N[(angle / 180.0), $MachinePrecision]], $MachinePrecision]] - 1), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]]]]]]]]]]

Derivation

1. Split input into 7 regimes

2. if (/.f64 angle 180) < -4.99999999999999994e254

   1. Initial program 25.0%

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

      1. *-commutative 25.0%

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

      2. associate-*l* 25.0%

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

      3. associate-*l* 25.0%

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

      4. unpow2 25.0%

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

      5. unpow2 25.0%

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

      6. difference-of-squares 32.7%

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

   3. Simplified 32.7%

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

   4. Taylor expanded in angle around 0 41.5%

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

      1. associate-*r* 41.5%

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

      2. *-commutative 41.5%

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

      3. associate-*l* 41.5%

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

   6. Simplified 41.5%

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

   if -4.99999999999999994e254 < (/.f64 angle 180) < -4.0000000000000002e178

   1. Initial program 15.6%

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

      1. *-commutative 15.6%

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

      2. associate-*l* 15.6%

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

      3. associate-*l* 15.6%

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

      4. unpow2 15.6%

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

      5. unpow2 15.6%

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

      6. difference-of-squares 27.2%

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

   3. Simplified 27.2%

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

      1. add-sqr-sqrt 22.1%

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

      2. sqrt-unprod 50.2%

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

      3. pow2 50.2%

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

   5. Applied egg-rr 50.2%

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

      1. unpow2 50.2%

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

      2. rem-sqrt-square 52.1%

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

      3. *-commutative 52.1%

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

      4. associate-*l* 52.1%

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

      5. +-commutative 52.1%

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

      6. *-commutative 52.1%

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

      7. associate-*r* 52.3%

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

      8. associate-*l* 52.3%

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

      9. *-commutative 52.3%

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

      10. metadata-eval 52.3%

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

   7. Simplified 52.3%

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

   if -4.0000000000000002e178 < (/.f64 angle 180) < -1.9999999999999999e168

   1. Initial program 28.4%

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

      1. *-commutative 28.4%

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

      2. associate-*l* 28.4%

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

      3. associate-*l* 28.4%

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

      4. unpow2 28.4%

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

      5. unpow2 28.4%

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

      6. difference-of-squares 28.4%

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

   3. Simplified 28.4%

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

      1. add-sqr-sqrt 0.0%

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

      2. sqrt-unprod 0.0%

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

      3. pow2 0.0%

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

      4. div-inv 0.0%

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

      5. metadata-eval 0.0%

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

   5. Applied egg-rr 0.0%

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

      1. unpow2 0.0%

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

      2. rem-sqrt-square 45.3%

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

      3. *-commutative 45.3%

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

      4. associate-*l* 67.6%

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

   7. Simplified 67.6%

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

   if -1.9999999999999999e168 < (/.f64 angle 180) < -5.0000000000000004e90

   1. Initial program 46.2%

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

      1. *-commutative 46.2%

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

      2. associate-*l* 46.2%

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

      3. associate-*l* 46.2%

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

      4. unpow2 46.2%

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

      5. unpow2 46.2%

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

      6. difference-of-squares 51.0%

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

   3. Simplified 51.0%

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

   if -5.0000000000000004e90 < (/.f64 angle 180) < -3.99999999999999978e34

   1. Initial program 11.4%

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

      1. *-commutative 11.4%

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

      2. associate-*l* 11.4%

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

      3. associate-*l* 11.4%

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

      4. unpow2 11.4%

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

      5. unpow2 11.4%

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

      6. difference-of-squares 11.4%

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

   3. Simplified 11.4%

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

      1. add-sqr-sqrt 0.0%

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

      2. sqrt-unprod 46.8%

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

      3. pow2 46.8%

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

      4. div-inv 56.8%

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

      5. metadata-eval 56.8%

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

   5. Applied egg-rr 56.8%

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

   if -3.99999999999999978e34 < (/.f64 angle 180) < 5.00000000000000018e61

   1. Initial program 75.8%

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

      1. *-commutative 75.8%

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

      2. associate-*l* 75.8%

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

      3. associate-*l* 75.8%

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

      4. unpow2 75.8%

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

      5. unpow2 75.8%

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

      6. difference-of-squares 79.1%

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

   3. Simplified 79.1%

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

      1. add-cube-cbrt 79.0%

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

      2. pow3 79.0%

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

      3. div-inv 79.4%

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

      4. metadata-eval 79.4%

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

   5. Applied egg-rr 79.4%

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

   6. Taylor expanded in angle around inf 88.5%

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

      1. associate-*r* 88.5%

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

      2. *-commutative 88.5%

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

      3. *-commutative 88.5%

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

      4. *-commutative 88.5%

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

      5. +-commutative 88.5%

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

      6. *-commutative 88.5%

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

      7. *-commutative 88.5%

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

      8. associate-*r* 89.2%

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

      9. *-commutative 89.2%

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

      10. *-commutative 89.2%

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

      11. associate-*r* 89.6%

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

   8. Simplified 89.6%

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

      1. metadata-eval 89.6%

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

      2. div-inv 89.8%

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

      3. add-cbrt-cube 90.5%

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

   10. Applied egg-rr 90.5%

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

   if 5.00000000000000018e61 < (/.f64 angle 180)

   1. Initial program 30.5%

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

      1. *-commutative 30.5%

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

      2. associate-*l* 30.5%

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

      3. associate-*l* 30.5%

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

      4. unpow2 30.5%

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

      5. unpow2 30.5%

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

      6. difference-of-squares 32.4%

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

   3. Simplified 32.4%

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

      1. add-cube-cbrt 32.4%

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

      2. pow3 32.4%

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

      3. div-inv 32.8%

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

      4. metadata-eval 32.8%

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

   5. Applied egg-rr 32.8%

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

   6. Taylor expanded in angle around inf 26.7%

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

      1. associate-*r* 26.7%

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

      2. *-commutative 26.7%

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

      3. *-commutative 26.7%

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

      4. *-commutative 26.7%

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

      5. +-commutative 26.7%

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

      6. *-commutative 26.7%

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

      7. *-commutative 26.7%

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

      8. associate-*r* 25.2%

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

      9. *-commutative 25.2%

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

      10. *-commutative 25.2%

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

      11. associate-*r* 26.2%

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

   8. Simplified 26.2%

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

      1. metadata-eval 26.2%

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

      2. div-inv 31.1%

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

      3. expm1-log1p-u 45.7%

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

   10. Applied egg-rr 45.7%

       \[\leadsto \left(\left(\left(b + a\right) \cdot \sin \left(\pi \cdot \left(angle \cdot 0.005555555555555556\right)\right)\right) \cdot \cos \left(\pi \cdot \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{angle}{180}\right)\right)}\right)\right) \cdot \left(2 \cdot \left(b - a\right)\right) \]
3. Recombined 7 regimes into one program.

4. Final simplification 69.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{angle}{180} \leq -5 \cdot 10^{+254}:\\ \;\;\;\;\left(\left(b - a\right) \cdot \left(b + a\right)\right) \cdot \left(\left(2 \cdot \left(angle \cdot \left(0.005555555555555556 \cdot \pi\right)\right)\right) \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)\right)\\ \mathbf{elif}\;\frac{angle}{180} \leq -4 \cdot 10^{+178}:\\ \;\;\;\;\left|\left(b - a\right) \cdot \left(\left(b + a\right) \cdot \sin \left(\left(angle \cdot \pi\right) \cdot 0.011111111111111112\right)\right)\right|\\ \mathbf{elif}\;\frac{angle}{180} \leq -2 \cdot 10^{+168}:\\ \;\;\;\;\left(\left(b - a\right) \cdot \left(b + a\right)\right) \cdot \left(\cos \left(\frac{angle}{180} \cdot \pi\right) \cdot \left(2 \cdot \sin \left(\left|angle \cdot \left(0.005555555555555556 \cdot \pi\right)\right|\right)\right)\right)\\ \mathbf{elif}\;\frac{angle}{180} \leq -5 \cdot 10^{+90}:\\ \;\;\;\;\left(\left(b - a\right) \cdot \left(b + a\right)\right) \cdot \left(\cos \left(\frac{angle}{180} \cdot \pi\right) \cdot \left(2 \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right)\right)\\ \mathbf{elif}\;\frac{angle}{180} \leq -4 \cdot 10^{+34}:\\ \;\;\;\;\left(\left(b - a\right) \cdot \left(b + a\right)\right) \cdot \left(\cos \left(\frac{angle}{180} \cdot \pi\right) \cdot \left(2 \cdot \sin \left(\sqrt{{\left(\pi \cdot \left(angle \cdot 0.005555555555555556\right)\right)}^{2}}\right)\right)\right)\\ \mathbf{elif}\;\frac{angle}{180} \leq 5 \cdot 10^{+61}:\\ \;\;\;\;\left(\left(\left(b + a\right) \cdot \sin \left(\pi \cdot \left(angle \cdot 0.005555555555555556\right)\right)\right) \cdot \cos \left(\pi \cdot \sqrt[3]{\frac{angle}{180} \cdot \left(\frac{angle}{180} \cdot \frac{angle}{180}\right)}\right)\right) \cdot \left(\left(b - a\right) \cdot 2\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\left(b - a\right) \cdot 2\right) \cdot \left(\left(\left(b + a\right) \cdot \sin \left(\pi \cdot \left(angle \cdot 0.005555555555555556\right)\right)\right) \cdot \cos \left(\pi \cdot \mathsf{expm1}\left(\mathsf{log1p}\left(\frac{angle}{180}\right)\right)\right)\right)\\ \end{array} \]

Alternative 4: 67.9% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := angle \cdot \left(0.005555555555555556 \cdot \pi\right)\\ t_1 := \pi \cdot \left(angle \cdot 0.005555555555555556\right)\\ t_2 := \left(b + a\right) \cdot \sin t_1\\ t_3 := \frac{angle}{180} \cdot \pi\\ t_4 := \cos t_3\\ t_5 := \left(b - a\right) \cdot \left(b + a\right)\\ t_6 := \left(b - a\right) \cdot 2\\ \mathbf{if}\;\frac{angle}{180} \leq -5 \cdot 10^{+254}:\\ \;\;\;\;t_5 \cdot \left(\left(2 \cdot t_0\right) \cdot t_4\right)\\ \mathbf{elif}\;\frac{angle}{180} \leq -1.5 \cdot 10^{+172}:\\ \;\;\;\;\left|\left(b - a\right) \cdot \left(\left(b + a\right) \cdot \sin \left(\left(angle \cdot \pi\right) \cdot 0.011111111111111112\right)\right)\right|\\ \mathbf{elif}\;\frac{angle}{180} \leq -2 \cdot 10^{+168}:\\ \;\;\;\;t_5 \cdot \left(t_4 \cdot \left(2 \cdot \sin \left(\left|t_0\right|\right)\right)\right)\\ \mathbf{elif}\;\frac{angle}{180} \leq -5 \cdot 10^{+90}:\\ \;\;\;\;t_5 \cdot \left(\left(2 \cdot \sin t_3\right) \cdot {\left(\sqrt[3]{\cos t_1}\right)}^{3}\right)\\ \mathbf{elif}\;\frac{angle}{180} \leq -4 \cdot 10^{+34}:\\ \;\;\;\;t_5 \cdot \left(t_4 \cdot \left(2 \cdot \sin \left(\sqrt{{t_1}^{2}}\right)\right)\right)\\ \mathbf{elif}\;\frac{angle}{180} \leq 5 \cdot 10^{+61}:\\ \;\;\;\;\left(t_2 \cdot \cos \left(\pi \cdot \sqrt[3]{\frac{angle}{180} \cdot \left(\frac{angle}{180} \cdot \frac{angle}{180}\right)}\right)\right) \cdot t_6\\ \mathbf{else}:\\ \;\;\;\;t_6 \cdot \left(t_2 \cdot \cos \left(\pi \cdot \mathsf{expm1}\left(\mathsf{log1p}\left(\frac{angle}{180}\right)\right)\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (a b angle)
 :precision binary64
 (let* ((t_0 (* angle (* 0.005555555555555556 PI)))
        (t_1 (* PI (* angle 0.005555555555555556)))
        (t_2 (* (+ b a) (sin t_1)))
        (t_3 (* (/ angle 180.0) PI))
        (t_4 (cos t_3))
        (t_5 (* (- b a) (+ b a)))
        (t_6 (* (- b a) 2.0)))
   (if (<= (/ angle 180.0) -5e+254)
     (* t_5 (* (* 2.0 t_0) t_4))
     (if (<= (/ angle 180.0) -1.5e+172)
       (fabs
        (* (- b a) (* (+ b a) (sin (* (* angle PI) 0.011111111111111112)))))
       (if (<= (/ angle 180.0) -2e+168)
         (* t_5 (* t_4 (* 2.0 (sin (fabs t_0)))))
         (if (<= (/ angle 180.0) -5e+90)
           (* t_5 (* (* 2.0 (sin t_3)) (pow (cbrt (cos t_1)) 3.0)))
           (if (<= (/ angle 180.0) -4e+34)
             (* t_5 (* t_4 (* 2.0 (sin (sqrt (pow t_1 2.0))))))
             (if (<= (/ angle 180.0) 5e+61)
               (*
                (*
                 t_2
                 (cos
                  (*
                   PI
                   (cbrt
                    (* (/ angle 180.0) (* (/ angle 180.0) (/ angle 180.0)))))))
                t_6)
               (*
                t_6
                (* t_2 (cos (* PI (expm1 (log1p (/ angle 180.0)))))))))))))))

C

double code(double a, double b, double angle) {
	double t_0 = angle * (0.005555555555555556 * ((double) M_PI));
	double t_1 = ((double) M_PI) * (angle * 0.005555555555555556);
	double t_2 = (b + a) * sin(t_1);
	double t_3 = (angle / 180.0) * ((double) M_PI);
	double t_4 = cos(t_3);
	double t_5 = (b - a) * (b + a);
	double t_6 = (b - a) * 2.0;
	double tmp;
	if ((angle / 180.0) <= -5e+254) {
		tmp = t_5 * ((2.0 * t_0) * t_4);
	} else if ((angle / 180.0) <= -1.5e+172) {
		tmp = fabs(((b - a) * ((b + a) * sin(((angle * ((double) M_PI)) * 0.011111111111111112)))));
	} else if ((angle / 180.0) <= -2e+168) {
		tmp = t_5 * (t_4 * (2.0 * sin(fabs(t_0))));
	} else if ((angle / 180.0) <= -5e+90) {
		tmp = t_5 * ((2.0 * sin(t_3)) * pow(cbrt(cos(t_1)), 3.0));
	} else if ((angle / 180.0) <= -4e+34) {
		tmp = t_5 * (t_4 * (2.0 * sin(sqrt(pow(t_1, 2.0)))));
	} else if ((angle / 180.0) <= 5e+61) {
		tmp = (t_2 * cos((((double) M_PI) * cbrt(((angle / 180.0) * ((angle / 180.0) * (angle / 180.0))))))) * t_6;
	} else {
		tmp = t_6 * (t_2 * cos((((double) M_PI) * expm1(log1p((angle / 180.0))))));
	}
	return tmp;
}

Java

public static double code(double a, double b, double angle) {
	double t_0 = angle * (0.005555555555555556 * Math.PI);
	double t_1 = Math.PI * (angle * 0.005555555555555556);
	double t_2 = (b + a) * Math.sin(t_1);
	double t_3 = (angle / 180.0) * Math.PI;
	double t_4 = Math.cos(t_3);
	double t_5 = (b - a) * (b + a);
	double t_6 = (b - a) * 2.0;
	double tmp;
	if ((angle / 180.0) <= -5e+254) {
		tmp = t_5 * ((2.0 * t_0) * t_4);
	} else if ((angle / 180.0) <= -1.5e+172) {
		tmp = Math.abs(((b - a) * ((b + a) * Math.sin(((angle * Math.PI) * 0.011111111111111112)))));
	} else if ((angle / 180.0) <= -2e+168) {
		tmp = t_5 * (t_4 * (2.0 * Math.sin(Math.abs(t_0))));
	} else if ((angle / 180.0) <= -5e+90) {
		tmp = t_5 * ((2.0 * Math.sin(t_3)) * Math.pow(Math.cbrt(Math.cos(t_1)), 3.0));
	} else if ((angle / 180.0) <= -4e+34) {
		tmp = t_5 * (t_4 * (2.0 * Math.sin(Math.sqrt(Math.pow(t_1, 2.0)))));
	} else if ((angle / 180.0) <= 5e+61) {
		tmp = (t_2 * Math.cos((Math.PI * Math.cbrt(((angle / 180.0) * ((angle / 180.0) * (angle / 180.0))))))) * t_6;
	} else {
		tmp = t_6 * (t_2 * Math.cos((Math.PI * Math.expm1(Math.log1p((angle / 180.0))))));
	}
	return tmp;
}

Julia

function code(a, b, angle)
	t_0 = Float64(angle * Float64(0.005555555555555556 * pi))
	t_1 = Float64(pi * Float64(angle * 0.005555555555555556))
	t_2 = Float64(Float64(b + a) * sin(t_1))
	t_3 = Float64(Float64(angle / 180.0) * pi)
	t_4 = cos(t_3)
	t_5 = Float64(Float64(b - a) * Float64(b + a))
	t_6 = Float64(Float64(b - a) * 2.0)
	tmp = 0.0
	if (Float64(angle / 180.0) <= -5e+254)
		tmp = Float64(t_5 * Float64(Float64(2.0 * t_0) * t_4));
	elseif (Float64(angle / 180.0) <= -1.5e+172)
		tmp = abs(Float64(Float64(b - a) * Float64(Float64(b + a) * sin(Float64(Float64(angle * pi) * 0.011111111111111112)))));
	elseif (Float64(angle / 180.0) <= -2e+168)
		tmp = Float64(t_5 * Float64(t_4 * Float64(2.0 * sin(abs(t_0)))));
	elseif (Float64(angle / 180.0) <= -5e+90)
		tmp = Float64(t_5 * Float64(Float64(2.0 * sin(t_3)) * (cbrt(cos(t_1)) ^ 3.0)));
	elseif (Float64(angle / 180.0) <= -4e+34)
		tmp = Float64(t_5 * Float64(t_4 * Float64(2.0 * sin(sqrt((t_1 ^ 2.0))))));
	elseif (Float64(angle / 180.0) <= 5e+61)
		tmp = Float64(Float64(t_2 * cos(Float64(pi * cbrt(Float64(Float64(angle / 180.0) * Float64(Float64(angle / 180.0) * Float64(angle / 180.0))))))) * t_6);
	else
		tmp = Float64(t_6 * Float64(t_2 * cos(Float64(pi * expm1(log1p(Float64(angle / 180.0)))))));
	end
	return tmp
end

Wolfram

code[a_, b_, angle_] := Block[{t$95$0 = N[(angle * N[(0.005555555555555556 * Pi), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(Pi * N[(angle * 0.005555555555555556), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(N[(b + a), $MachinePrecision] * N[Sin[t$95$1], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$3 = N[(N[(angle / 180.0), $MachinePrecision] * Pi), $MachinePrecision]}, Block[{t$95$4 = N[Cos[t$95$3], $MachinePrecision]}, Block[{t$95$5 = N[(N[(b - a), $MachinePrecision] * N[(b + a), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$6 = N[(N[(b - a), $MachinePrecision] * 2.0), $MachinePrecision]}, If[LessEqual[N[(angle / 180.0), $MachinePrecision], -5e+254], N[(t$95$5 * N[(N[(2.0 * t$95$0), $MachinePrecision] * t$95$4), $MachinePrecision]), $MachinePrecision], If[LessEqual[N[(angle / 180.0), $MachinePrecision], -1.5e+172], N[Abs[N[(N[(b - a), $MachinePrecision] * N[(N[(b + a), $MachinePrecision] * N[Sin[N[(N[(angle * Pi), $MachinePrecision] * 0.011111111111111112), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], If[LessEqual[N[(angle / 180.0), $MachinePrecision], -2e+168], N[(t$95$5 * N[(t$95$4 * N[(2.0 * N[Sin[N[Abs[t$95$0], $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[N[(angle / 180.0), $MachinePrecision], -5e+90], N[(t$95$5 * N[(N[(2.0 * N[Sin[t$95$3], $MachinePrecision]), $MachinePrecision] * N[Power[N[Power[N[Cos[t$95$1], $MachinePrecision], 1/3], $MachinePrecision], 3.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[N[(angle / 180.0), $MachinePrecision], -4e+34], N[(t$95$5 * N[(t$95$4 * N[(2.0 * N[Sin[N[Sqrt[N[Power[t$95$1, 2.0], $MachinePrecision]], $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[N[(angle / 180.0), $MachinePrecision], 5e+61], N[(N[(t$95$2 * N[Cos[N[(Pi * N[Power[N[(N[(angle / 180.0), $MachinePrecision] * N[(N[(angle / 180.0), $MachinePrecision] * N[(angle / 180.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 1/3], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * t$95$6), $MachinePrecision], N[(t$95$6 * N[(t$95$2 * N[Cos[N[(Pi * N[(Exp[N[Log[1 + N[(angle / 180.0), $MachinePrecision]], $MachinePrecision]] - 1), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]]]]]]]]]]

Derivation

1. Split input into 7 regimes

2. if (/.f64 angle 180) < -4.99999999999999994e254

   1. Initial program 25.0%

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

      1. *-commutative 25.0%

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

      2. associate-*l* 25.0%

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

      3. associate-*l* 25.0%

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

      4. unpow2 25.0%

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

      5. unpow2 25.0%

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

      6. difference-of-squares 32.7%

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

   3. Simplified 32.7%

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

   4. Taylor expanded in angle around 0 41.5%

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

      1. associate-*r* 41.5%

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

      2. *-commutative 41.5%

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

      3. associate-*l* 41.5%

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

   6. Simplified 41.5%

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

   if -4.99999999999999994e254 < (/.f64 angle 180) < -1.5e172

   1. Initial program 18.2%

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

      1. *-commutative 18.2%

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

      2. associate-*l* 18.2%

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

      3. associate-*l* 18.2%

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

      4. unpow2 18.2%

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

      5. unpow2 18.2%

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

      6. difference-of-squares 28.5%

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

   3. Simplified 28.5%

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

      1. add-sqr-sqrt 23.2%

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

      2. sqrt-unprod 52.0%

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

      3. pow2 52.0%

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

   5. Applied egg-rr 52.0%

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

      1. unpow2 52.0%

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

      2. rem-sqrt-square 53.7%

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

      3. *-commutative 53.7%

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

      4. associate-*l* 53.7%

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

      5. +-commutative 53.7%

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

      6. *-commutative 53.7%

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

      7. associate-*r* 53.8%

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

      8. associate-*l* 53.8%

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

      9. *-commutative 53.8%

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

      10. metadata-eval 53.8%

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

   7. Simplified 53.8%

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

   if -1.5e172 < (/.f64 angle 180) < -1.9999999999999999e168

   1. Initial program 22.6%

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

      1. *-commutative 22.6%

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

      2. associate-*l* 22.6%

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

      3. associate-*l* 22.6%

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

      4. unpow2 22.6%

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

      5. unpow2 22.6%

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

      6. difference-of-squares 22.6%

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

   3. Simplified 22.6%

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

      1. add-sqr-sqrt 0.0%

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

      2. sqrt-unprod 0.0%

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

      3. pow2 0.0%

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

      4. div-inv 0.0%

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

      5. metadata-eval 0.0%

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

   5. Applied egg-rr 0.0%

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

      1. unpow2 0.0%

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

      2. rem-sqrt-square 50.9%

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

      3. *-commutative 50.9%

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

      4. associate-*l* 67.6%

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

   7. Simplified 67.6%

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

   if -1.9999999999999999e168 < (/.f64 angle 180) < -5.0000000000000004e90

   1. Initial program 46.2%

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

      1. *-commutative 46.2%

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

      2. associate-*l* 46.2%

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

      3. associate-*l* 46.2%

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

      4. unpow2 46.2%

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

      5. unpow2 46.2%

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

      6. difference-of-squares 51.0%

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

   3. Simplified 51.0%

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

      1. add-cube-cbrt 51.0%

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

      2. pow3 51.0%

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

      3. div-inv 51.6%

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

      4. metadata-eval 51.6%

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

   5. Applied egg-rr 51.6%

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

   if -5.0000000000000004e90 < (/.f64 angle 180) < -3.99999999999999978e34

   1. Initial program 11.4%

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

      1. *-commutative 11.4%

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

      2. associate-*l* 11.4%

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

      3. associate-*l* 11.4%

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

      4. unpow2 11.4%

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

      5. unpow2 11.4%

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

      6. difference-of-squares 11.4%

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

   3. Simplified 11.4%

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

      1. add-sqr-sqrt 0.0%

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

      2. sqrt-unprod 46.8%

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

      3. pow2 46.8%

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

      4. div-inv 56.8%

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

      5. metadata-eval 56.8%

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

   5. Applied egg-rr 56.8%

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

   if -3.99999999999999978e34 < (/.f64 angle 180) < 5.00000000000000018e61

   1. Initial program 75.8%

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

      1. *-commutative 75.8%

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

      2. associate-*l* 75.8%

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

      3. associate-*l* 75.8%

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

      4. unpow2 75.8%

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

      5. unpow2 75.8%

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

      6. difference-of-squares 79.1%

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

   3. Simplified 79.1%

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

      1. add-cube-cbrt 79.0%

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

      2. pow3 79.0%

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

      3. div-inv 79.4%

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

      4. metadata-eval 79.4%

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

   5. Applied egg-rr 79.4%

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

   6. Taylor expanded in angle around inf 88.5%

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

      1. associate-*r* 88.5%

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

      2. *-commutative 88.5%

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

      3. *-commutative 88.5%

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

      4. *-commutative 88.5%

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

      5. +-commutative 88.5%

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

      6. *-commutative 88.5%

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

      7. *-commutative 88.5%

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

      8. associate-*r* 89.2%

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

      9. *-commutative 89.2%

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

      10. *-commutative 89.2%

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

      11. associate-*r* 89.6%

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

   8. Simplified 89.6%

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

      1. metadata-eval 89.6%

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

      2. div-inv 89.8%

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

      3. add-cbrt-cube 90.5%

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

   10. Applied egg-rr 90.5%

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

   if 5.00000000000000018e61 < (/.f64 angle 180)

   1. Initial program 30.5%

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

      1. *-commutative 30.5%

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

      2. associate-*l* 30.5%

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

      3. associate-*l* 30.5%

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

      4. unpow2 30.5%

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

      5. unpow2 30.5%

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

      6. difference-of-squares 32.4%

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

   3. Simplified 32.4%

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

      1. add-cube-cbrt 32.4%

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

      2. pow3 32.4%

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

      3. div-inv 32.8%

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

      4. metadata-eval 32.8%

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

   5. Applied egg-rr 32.8%

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

   6. Taylor expanded in angle around inf 26.7%

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

      1. associate-*r* 26.7%

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

      2. *-commutative 26.7%

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

      3. *-commutative 26.7%

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

      4. *-commutative 26.7%

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

      5. +-commutative 26.7%

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

      6. *-commutative 26.7%

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

      7. *-commutative 26.7%

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

      8. associate-*r* 25.2%

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

      9. *-commutative 25.2%

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

      10. *-commutative 25.2%

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

      11. associate-*r* 26.2%

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

   8. Simplified 26.2%

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

      1. metadata-eval 26.2%

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

      2. div-inv 31.1%

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

      3. expm1-log1p-u 45.7%

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

   10. Applied egg-rr 45.7%

       \[\leadsto \left(\left(\left(b + a\right) \cdot \sin \left(\pi \cdot \left(angle \cdot 0.005555555555555556\right)\right)\right) \cdot \cos \left(\pi \cdot \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{angle}{180}\right)\right)}\right)\right) \cdot \left(2 \cdot \left(b - a\right)\right) \]
3. Recombined 7 regimes into one program.

4. Final simplification 69.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{angle}{180} \leq -5 \cdot 10^{+254}:\\ \;\;\;\;\left(\left(b - a\right) \cdot \left(b + a\right)\right) \cdot \left(\left(2 \cdot \left(angle \cdot \left(0.005555555555555556 \cdot \pi\right)\right)\right) \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)\right)\\ \mathbf{elif}\;\frac{angle}{180} \leq -1.5 \cdot 10^{+172}:\\ \;\;\;\;\left|\left(b - a\right) \cdot \left(\left(b + a\right) \cdot \sin \left(\left(angle \cdot \pi\right) \cdot 0.011111111111111112\right)\right)\right|\\ \mathbf{elif}\;\frac{angle}{180} \leq -2 \cdot 10^{+168}:\\ \;\;\;\;\left(\left(b - a\right) \cdot \left(b + a\right)\right) \cdot \left(\cos \left(\frac{angle}{180} \cdot \pi\right) \cdot \left(2 \cdot \sin \left(\left|angle \cdot \left(0.005555555555555556 \cdot \pi\right)\right|\right)\right)\right)\\ \mathbf{elif}\;\frac{angle}{180} \leq -5 \cdot 10^{+90}:\\ \;\;\;\;\left(\left(b - a\right) \cdot \left(b + a\right)\right) \cdot \left(\left(2 \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right) \cdot {\left(\sqrt[3]{\cos \left(\pi \cdot \left(angle \cdot 0.005555555555555556\right)\right)}\right)}^{3}\right)\\ \mathbf{elif}\;\frac{angle}{180} \leq -4 \cdot 10^{+34}:\\ \;\;\;\;\left(\left(b - a\right) \cdot \left(b + a\right)\right) \cdot \left(\cos \left(\frac{angle}{180} \cdot \pi\right) \cdot \left(2 \cdot \sin \left(\sqrt{{\left(\pi \cdot \left(angle \cdot 0.005555555555555556\right)\right)}^{2}}\right)\right)\right)\\ \mathbf{elif}\;\frac{angle}{180} \leq 5 \cdot 10^{+61}:\\ \;\;\;\;\left(\left(\left(b + a\right) \cdot \sin \left(\pi \cdot \left(angle \cdot 0.005555555555555556\right)\right)\right) \cdot \cos \left(\pi \cdot \sqrt[3]{\frac{angle}{180} \cdot \left(\frac{angle}{180} \cdot \frac{angle}{180}\right)}\right)\right) \cdot \left(\left(b - a\right) \cdot 2\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\left(b - a\right) \cdot 2\right) \cdot \left(\left(\left(b + a\right) \cdot \sin \left(\pi \cdot \left(angle \cdot 0.005555555555555556\right)\right)\right) \cdot \cos \left(\pi \cdot \mathsf{expm1}\left(\mathsf{log1p}\left(\frac{angle}{180}\right)\right)\right)\right)\\ \end{array} \]

Alternative 5: 68.4% accurate, 2.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq 2 \cdot 10^{+144}:\\ \;\;\;\;\left(b - a\right) \cdot \left(\left(b + a\right) \cdot \sin \left(\left(angle \cdot \pi\right) \cdot 0.011111111111111112\right)\right)\\ \mathbf{else}:\\ \;\;\;\;0.011111111111111112 \cdot \left(\left(angle \cdot \left(b - a\right)\right) \cdot \left(\pi \cdot \left(b + a\right)\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (a b angle)
 :precision binary64
 (if (<= b 2e+144)
   (* (- b a) (* (+ b a) (sin (* (* angle PI) 0.011111111111111112))))
   (* 0.011111111111111112 (* (* angle (- b a)) (* PI (+ b a))))))

C

double code(double a, double b, double angle) {
	double tmp;
	if (b <= 2e+144) {
		tmp = (b - a) * ((b + a) * sin(((angle * ((double) M_PI)) * 0.011111111111111112)));
	} else {
		tmp = 0.011111111111111112 * ((angle * (b - a)) * (((double) M_PI) * (b + a)));
	}
	return tmp;
}

Java

public static double code(double a, double b, double angle) {
	double tmp;
	if (b <= 2e+144) {
		tmp = (b - a) * ((b + a) * Math.sin(((angle * Math.PI) * 0.011111111111111112)));
	} else {
		tmp = 0.011111111111111112 * ((angle * (b - a)) * (Math.PI * (b + a)));
	}
	return tmp;
}

Python

def code(a, b, angle):
	tmp = 0
	if b <= 2e+144:
		tmp = (b - a) * ((b + a) * math.sin(((angle * math.pi) * 0.011111111111111112)))
	else:
		tmp = 0.011111111111111112 * ((angle * (b - a)) * (math.pi * (b + a)))
	return tmp

Julia

function code(a, b, angle)
	tmp = 0.0
	if (b <= 2e+144)
		tmp = Float64(Float64(b - a) * Float64(Float64(b + a) * sin(Float64(Float64(angle * pi) * 0.011111111111111112))));
	else
		tmp = Float64(0.011111111111111112 * Float64(Float64(angle * Float64(b - a)) * Float64(pi * Float64(b + a))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, angle)
	tmp = 0.0;
	if (b <= 2e+144)
		tmp = (b - a) * ((b + a) * sin(((angle * pi) * 0.011111111111111112)));
	else
		tmp = 0.011111111111111112 * ((angle * (b - a)) * (pi * (b + a)));
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, angle_] := If[LessEqual[b, 2e+144], N[(N[(b - a), $MachinePrecision] * N[(N[(b + a), $MachinePrecision] * N[Sin[N[(N[(angle * Pi), $MachinePrecision] * 0.011111111111111112), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(0.011111111111111112 * N[(N[(angle * N[(b - a), $MachinePrecision]), $MachinePrecision] * N[(Pi * N[(b + a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b < 2.00000000000000005e144

   1. Initial program 52.8%

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

      1. *-commutative 52.8%

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

      2. associate-*l* 52.8%

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

      3. associate-*l* 52.8%

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

      4. unpow2 52.8%

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

      5. unpow2 52.8%

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

      6. difference-of-squares 55.6%

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

   3. Simplified 55.6%

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

      1. difference-of-squares 52.8%

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

      2. *-commutative 52.8%

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

      3. sub-neg 52.8%

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

      4. distribute-lft-in 52.8%

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

      5. associate-*l* 52.8%

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

      6. 2-sin 52.8%

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

      7. div-inv 51.8%

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

      8. metadata-eval 51.8%

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

   5. Applied egg-rr 50.4%

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

      1. distribute-lft-out 50.4%

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

      2. sub-neg 50.4%

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

      3. difference-of-squares 53.1%

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

      4. *-commutative 53.1%

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

      5. *-commutative 53.1%

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

      6. associate-*l* 56.6%

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

      7. +-commutative 56.6%

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

      8. *-commutative 56.6%

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

      9. associate-*r* 55.5%

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

      10. associate-*l* 55.5%

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

      11. *-commutative 55.5%

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

      12. metadata-eval 55.5%

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

   7. Simplified 55.5%

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

   if 2.00000000000000005e144 < b

   1. Initial program 40.8%

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

      1. *-commutative 40.8%

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

      2. associate-*l* 40.8%

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

      3. associate-*l* 40.8%

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

      4. unpow2 40.8%

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

      5. unpow2 40.8%

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

      6. difference-of-squares 51.0%

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

   3. Simplified 51.0%

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

      1. difference-of-squares 40.8%

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

      2. *-commutative 40.8%

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

      3. sub-neg 40.8%

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

      4. distribute-lft-in 40.8%

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

      5. associate-*l* 40.8%

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

      6. 2-sin 40.8%

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

      7. div-inv 45.8%

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

      8. metadata-eval 45.8%

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

   5. Applied egg-rr 45.8%

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

      1. distribute-lft-out 45.8%

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

      2. sub-neg 45.8%

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

      3. difference-of-squares 56.0%

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

      4. *-commutative 56.0%

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

      5. *-commutative 56.0%

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

      6. associate-*l* 69.8%

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

      7. +-commutative 69.8%

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

      8. *-commutative 69.8%

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

      9. associate-*r* 59.8%

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

      10. associate-*l* 59.8%

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

      11. *-commutative 59.8%

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

      12. metadata-eval 59.8%

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

   7. Simplified 59.8%

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

      1. log1p-expm1-u 59.8%

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

      2. associate-*l* 59.8%

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

   9. Applied egg-rr 59.8%

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

   10. Taylor expanded in angle around 0 56.2%

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

       1. associate-*r* 70.1%

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

       2. *-commutative 70.1%

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

       3. +-commutative 70.1%

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

   12. Simplified 70.1%

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

4. Final simplification 57.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq 2 \cdot 10^{+144}:\\ \;\;\;\;\left(b - a\right) \cdot \left(\left(b + a\right) \cdot \sin \left(\left(angle \cdot \pi\right) \cdot 0.011111111111111112\right)\right)\\ \mathbf{else}:\\ \;\;\;\;0.011111111111111112 \cdot \left(\left(angle \cdot \left(b - a\right)\right) \cdot \left(\pi \cdot \left(b + a\right)\right)\right)\\ \end{array} \]

Alternative 6: 68.5% accurate, 2.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq 2 \cdot 10^{+144}:\\ \;\;\;\;\left(b - a\right) \cdot \left(\left(b + a\right) \cdot \sin \left(angle \cdot \left(\pi \cdot 0.011111111111111112\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;0.011111111111111112 \cdot \left(\left(angle \cdot \left(b - a\right)\right) \cdot \left(\pi \cdot \left(b + a\right)\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (a b angle)
 :precision binary64
 (if (<= b 2e+144)
   (* (- b a) (* (+ b a) (sin (* angle (* PI 0.011111111111111112)))))
   (* 0.011111111111111112 (* (* angle (- b a)) (* PI (+ b a))))))

C

double code(double a, double b, double angle) {
	double tmp;
	if (b <= 2e+144) {
		tmp = (b - a) * ((b + a) * sin((angle * (((double) M_PI) * 0.011111111111111112))));
	} else {
		tmp = 0.011111111111111112 * ((angle * (b - a)) * (((double) M_PI) * (b + a)));
	}
	return tmp;
}

Java

public static double code(double a, double b, double angle) {
	double tmp;
	if (b <= 2e+144) {
		tmp = (b - a) * ((b + a) * Math.sin((angle * (Math.PI * 0.011111111111111112))));
	} else {
		tmp = 0.011111111111111112 * ((angle * (b - a)) * (Math.PI * (b + a)));
	}
	return tmp;
}

Python

def code(a, b, angle):
	tmp = 0
	if b <= 2e+144:
		tmp = (b - a) * ((b + a) * math.sin((angle * (math.pi * 0.011111111111111112))))
	else:
		tmp = 0.011111111111111112 * ((angle * (b - a)) * (math.pi * (b + a)))
	return tmp

Julia

function code(a, b, angle)
	tmp = 0.0
	if (b <= 2e+144)
		tmp = Float64(Float64(b - a) * Float64(Float64(b + a) * sin(Float64(angle * Float64(pi * 0.011111111111111112)))));
	else
		tmp = Float64(0.011111111111111112 * Float64(Float64(angle * Float64(b - a)) * Float64(pi * Float64(b + a))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, angle)
	tmp = 0.0;
	if (b <= 2e+144)
		tmp = (b - a) * ((b + a) * sin((angle * (pi * 0.011111111111111112))));
	else
		tmp = 0.011111111111111112 * ((angle * (b - a)) * (pi * (b + a)));
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, angle_] := If[LessEqual[b, 2e+144], N[(N[(b - a), $MachinePrecision] * N[(N[(b + a), $MachinePrecision] * N[Sin[N[(angle * N[(Pi * 0.011111111111111112), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(0.011111111111111112 * N[(N[(angle * N[(b - a), $MachinePrecision]), $MachinePrecision] * N[(Pi * N[(b + a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b < 2.00000000000000005e144

   1. Initial program 52.8%

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

      1. *-commutative 52.8%

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

      2. associate-*l* 52.8%

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

      3. associate-*l* 52.8%

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

      4. unpow2 52.8%

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

      5. unpow2 52.8%

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

      6. difference-of-squares 55.6%

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

   3. Simplified 55.6%

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

      1. difference-of-squares 52.8%

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

      2. *-commutative 52.8%

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

      3. sub-neg 52.8%

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

      4. distribute-lft-in 52.8%

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

      5. associate-*l* 52.8%

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

      6. 2-sin 52.8%

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

      7. div-inv 51.8%

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

      8. metadata-eval 51.8%

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

   5. Applied egg-rr 50.4%

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

      1. distribute-lft-out 50.4%

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

      2. sub-neg 50.4%

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

      3. difference-of-squares 53.1%

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

      4. *-commutative 53.1%

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

      5. *-commutative 53.1%

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

      6. associate-*l* 56.6%

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

      7. +-commutative 56.6%

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

      8. *-commutative 56.6%

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

      9. associate-*r* 55.5%

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

      10. associate-*l* 55.5%

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

      11. *-commutative 55.5%

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

      12. metadata-eval 55.5%

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

   7. Simplified 55.5%

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

      1. *-un-lft-identity 55.5%

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

      2. associate-*l* 57.7%

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

   9. Applied egg-rr 57.7%

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

   if 2.00000000000000005e144 < b

   1. Initial program 40.8%

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

      1. *-commutative 40.8%

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

      2. associate-*l* 40.8%

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

      3. associate-*l* 40.8%

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

      4. unpow2 40.8%

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

      5. unpow2 40.8%

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

      6. difference-of-squares 51.0%

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

   3. Simplified 51.0%

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

      1. difference-of-squares 40.8%

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

      2. *-commutative 40.8%

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

      3. sub-neg 40.8%

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

      4. distribute-lft-in 40.8%

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

      5. associate-*l* 40.8%

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

      6. 2-sin 40.8%

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

      7. div-inv 45.8%

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

      8. metadata-eval 45.8%

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

   5. Applied egg-rr 45.8%

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

      1. distribute-lft-out 45.8%

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

      2. sub-neg 45.8%

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

      3. difference-of-squares 56.0%

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

      4. *-commutative 56.0%

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

      5. *-commutative 56.0%

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

      6. associate-*l* 69.8%

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

      7. +-commutative 69.8%

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

      8. *-commutative 69.8%

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

      9. associate-*r* 59.8%

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

      10. associate-*l* 59.8%

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

      11. *-commutative 59.8%

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

      12. metadata-eval 59.8%

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

   7. Simplified 59.8%

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

      1. log1p-expm1-u 59.8%

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

      2. associate-*l* 59.8%

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

   9. Applied egg-rr 59.8%

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

   10. Taylor expanded in angle around 0 56.2%

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

       1. associate-*r* 70.1%

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

       2. *-commutative 70.1%

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

       3. +-commutative 70.1%

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

   12. Simplified 70.1%

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

4. Final simplification 59.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq 2 \cdot 10^{+144}:\\ \;\;\;\;\left(b - a\right) \cdot \left(\left(b + a\right) \cdot \sin \left(angle \cdot \left(\pi \cdot 0.011111111111111112\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;0.011111111111111112 \cdot \left(\left(angle \cdot \left(b - a\right)\right) \cdot \left(\pi \cdot \left(b + a\right)\right)\right)\\ \end{array} \]

Alternative 7: 54.5% accurate, 2.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq 3.6 \cdot 10^{-118}:\\ \;\;\;\;\left(b - a\right) \cdot \left(a \cdot \sin \left(\pi \cdot \left(angle \cdot 0.011111111111111112\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;0.011111111111111112 \cdot \left(\left(angle \cdot \left(b - a\right)\right) \cdot \left(\pi \cdot \left(b + a\right)\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (a b angle)
 :precision binary64
 (if (<= b 3.6e-118)
   (* (- b a) (* a (sin (* PI (* angle 0.011111111111111112)))))
   (* 0.011111111111111112 (* (* angle (- b a)) (* PI (+ b a))))))

C

double code(double a, double b, double angle) {
	double tmp;
	if (b <= 3.6e-118) {
		tmp = (b - a) * (a * sin((((double) M_PI) * (angle * 0.011111111111111112))));
	} else {
		tmp = 0.011111111111111112 * ((angle * (b - a)) * (((double) M_PI) * (b + a)));
	}
	return tmp;
}

Java

public static double code(double a, double b, double angle) {
	double tmp;
	if (b <= 3.6e-118) {
		tmp = (b - a) * (a * Math.sin((Math.PI * (angle * 0.011111111111111112))));
	} else {
		tmp = 0.011111111111111112 * ((angle * (b - a)) * (Math.PI * (b + a)));
	}
	return tmp;
}

Python

def code(a, b, angle):
	tmp = 0
	if b <= 3.6e-118:
		tmp = (b - a) * (a * math.sin((math.pi * (angle * 0.011111111111111112))))
	else:
		tmp = 0.011111111111111112 * ((angle * (b - a)) * (math.pi * (b + a)))
	return tmp

Julia

function code(a, b, angle)
	tmp = 0.0
	if (b <= 3.6e-118)
		tmp = Float64(Float64(b - a) * Float64(a * sin(Float64(pi * Float64(angle * 0.011111111111111112)))));
	else
		tmp = Float64(0.011111111111111112 * Float64(Float64(angle * Float64(b - a)) * Float64(pi * Float64(b + a))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, angle)
	tmp = 0.0;
	if (b <= 3.6e-118)
		tmp = (b - a) * (a * sin((pi * (angle * 0.011111111111111112))));
	else
		tmp = 0.011111111111111112 * ((angle * (b - a)) * (pi * (b + a)));
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, angle_] := If[LessEqual[b, 3.6e-118], N[(N[(b - a), $MachinePrecision] * N[(a * N[Sin[N[(Pi * N[(angle * 0.011111111111111112), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(0.011111111111111112 * N[(N[(angle * N[(b - a), $MachinePrecision]), $MachinePrecision] * N[(Pi * N[(b + a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b < 3.6000000000000002e-118

   1. Initial program 51.8%

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

      1. *-commutative 51.8%

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

      2. associate-*l* 51.8%

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

      3. associate-*l* 51.8%

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

      4. unpow2 51.8%

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

      5. unpow2 51.8%

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

      6. difference-of-squares 55.8%

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

   3. Simplified 55.8%

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

      1. difference-of-squares 51.8%

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

      2. *-commutative 51.8%

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

      3. sub-neg 51.8%

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

      4. distribute-lft-in 51.8%

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

      5. associate-*l* 51.8%

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

      6. 2-sin 51.8%

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

      7. div-inv 50.5%

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

      8. metadata-eval 50.5%

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

   5. Applied egg-rr 49.1%

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

      1. distribute-lft-out 49.1%

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

      2. sub-neg 49.1%

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

      3. difference-of-squares 53.1%

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

      4. *-commutative 53.1%

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

      5. *-commutative 53.1%

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

      6. associate-*l* 57.4%

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

      7. +-commutative 57.4%

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

      8. *-commutative 57.4%

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

      9. associate-*r* 57.3%

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

      10. associate-*l* 57.3%

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

      11. *-commutative 57.3%

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

      12. metadata-eval 57.3%

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

   7. Simplified 57.3%

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

   8. Taylor expanded in a around inf 38.5%

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

      1. *-commutative 38.5%

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

      2. associate-*r* 36.3%

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

   10. Simplified 36.3%

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

   if 3.6000000000000002e-118 < b

   1. Initial program 49.7%

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

      1. *-commutative 49.7%

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

      2. associate-*l* 49.7%

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

      3. associate-*l* 49.7%

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

      4. unpow2 49.7%

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

      5. unpow2 49.7%

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

      6. difference-of-squares 53.6%

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

   3. Simplified 53.6%

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

      1. difference-of-squares 49.7%

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

      2. *-commutative 49.7%

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

      3. sub-neg 49.7%

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

      4. distribute-lft-in 49.7%

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

      5. associate-*l* 49.7%

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

      6. 2-sin 49.7%

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

      7. div-inv 51.4%

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

      8. metadata-eval 51.4%

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

   5. Applied egg-rr 50.4%

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

      1. distribute-lft-out 50.4%

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

      2. sub-neg 50.4%

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

      3. difference-of-squares 54.2%

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

      4. *-commutative 54.2%

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

      5. *-commutative 54.2%

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

      6. associate-*l* 60.4%

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

      7. +-commutative 60.4%

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

      8. *-commutative 60.4%

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

      9. associate-*r* 54.7%

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

      10. associate-*l* 54.7%

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

      11. *-commutative 54.7%

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

      12. metadata-eval 54.7%

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

   7. Simplified 54.7%

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

      1. log1p-expm1-u 54.7%

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

      2. associate-*l* 57.1%

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

   9. Applied egg-rr 57.1%

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

   10. Taylor expanded in angle around 0 52.4%

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

       1. associate-*r* 58.5%

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

       2. *-commutative 58.5%

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

       3. +-commutative 58.5%

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

   12. Simplified 58.5%

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

4. Final simplification 45.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq 3.6 \cdot 10^{-118}:\\ \;\;\;\;\left(b - a\right) \cdot \left(a \cdot \sin \left(\pi \cdot \left(angle \cdot 0.011111111111111112\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;0.011111111111111112 \cdot \left(\left(angle \cdot \left(b - a\right)\right) \cdot \left(\pi \cdot \left(b + a\right)\right)\right)\\ \end{array} \]

Alternative 8: 54.4% accurate, 2.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq 1.52 \cdot 10^{-118}:\\ \;\;\;\;\left(b - a\right) \cdot \left(a \cdot \sin \left(\left(angle \cdot \pi\right) \cdot 0.011111111111111112\right)\right)\\ \mathbf{else}:\\ \;\;\;\;0.011111111111111112 \cdot \left(\left(angle \cdot \left(b - a\right)\right) \cdot \left(\pi \cdot \left(b + a\right)\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (a b angle)
 :precision binary64
 (if (<= b 1.52e-118)
   (* (- b a) (* a (sin (* (* angle PI) 0.011111111111111112))))
   (* 0.011111111111111112 (* (* angle (- b a)) (* PI (+ b a))))))

C

double code(double a, double b, double angle) {
	double tmp;
	if (b <= 1.52e-118) {
		tmp = (b - a) * (a * sin(((angle * ((double) M_PI)) * 0.011111111111111112)));
	} else {
		tmp = 0.011111111111111112 * ((angle * (b - a)) * (((double) M_PI) * (b + a)));
	}
	return tmp;
}

Java

public static double code(double a, double b, double angle) {
	double tmp;
	if (b <= 1.52e-118) {
		tmp = (b - a) * (a * Math.sin(((angle * Math.PI) * 0.011111111111111112)));
	} else {
		tmp = 0.011111111111111112 * ((angle * (b - a)) * (Math.PI * (b + a)));
	}
	return tmp;
}

Python

def code(a, b, angle):
	tmp = 0
	if b <= 1.52e-118:
		tmp = (b - a) * (a * math.sin(((angle * math.pi) * 0.011111111111111112)))
	else:
		tmp = 0.011111111111111112 * ((angle * (b - a)) * (math.pi * (b + a)))
	return tmp

Julia

function code(a, b, angle)
	tmp = 0.0
	if (b <= 1.52e-118)
		tmp = Float64(Float64(b - a) * Float64(a * sin(Float64(Float64(angle * pi) * 0.011111111111111112))));
	else
		tmp = Float64(0.011111111111111112 * Float64(Float64(angle * Float64(b - a)) * Float64(pi * Float64(b + a))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, angle)
	tmp = 0.0;
	if (b <= 1.52e-118)
		tmp = (b - a) * (a * sin(((angle * pi) * 0.011111111111111112)));
	else
		tmp = 0.011111111111111112 * ((angle * (b - a)) * (pi * (b + a)));
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, angle_] := If[LessEqual[b, 1.52e-118], N[(N[(b - a), $MachinePrecision] * N[(a * N[Sin[N[(N[(angle * Pi), $MachinePrecision] * 0.011111111111111112), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(0.011111111111111112 * N[(N[(angle * N[(b - a), $MachinePrecision]), $MachinePrecision] * N[(Pi * N[(b + a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b < 1.5199999999999999e-118

   1. Initial program 51.8%

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

      1. *-commutative 51.8%

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

      2. associate-*l* 51.8%

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

      3. associate-*l* 51.8%

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

      4. unpow2 51.8%

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

      5. unpow2 51.8%

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

      6. difference-of-squares 55.8%

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

   3. Simplified 55.8%

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

      1. difference-of-squares 51.8%

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

      2. *-commutative 51.8%

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

      3. sub-neg 51.8%

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

      4. distribute-lft-in 51.8%

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

      5. associate-*l* 51.8%

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

      6. 2-sin 51.8%

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

      7. div-inv 50.5%

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

      8. metadata-eval 50.5%

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

   5. Applied egg-rr 49.1%

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

      1. distribute-lft-out 49.1%

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

      2. sub-neg 49.1%

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

      3. difference-of-squares 53.1%

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

      4. *-commutative 53.1%

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

      5. *-commutative 53.1%

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

      6. associate-*l* 57.4%

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

      7. +-commutative 57.4%

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

      8. *-commutative 57.4%

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

      9. associate-*r* 57.3%

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

      10. associate-*l* 57.3%

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

      11. *-commutative 57.3%

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

      12. metadata-eval 57.3%

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

   7. Simplified 57.3%

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

   8. Taylor expanded in a around inf 38.5%

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

   if 1.5199999999999999e-118 < b

   1. Initial program 49.7%

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

      1. *-commutative 49.7%

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

      2. associate-*l* 49.7%

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

      3. associate-*l* 49.7%

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

      4. unpow2 49.7%

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

      5. unpow2 49.7%

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

      6. difference-of-squares 53.6%

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

   3. Simplified 53.6%

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

      1. difference-of-squares 49.7%

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

      2. *-commutative 49.7%

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

      3. sub-neg 49.7%

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

      4. distribute-lft-in 49.7%

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

      5. associate-*l* 49.7%

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

      6. 2-sin 49.7%

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

      7. div-inv 51.4%

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

      8. metadata-eval 51.4%

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

   5. Applied egg-rr 50.4%

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

      1. distribute-lft-out 50.4%

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

      2. sub-neg 50.4%

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

      3. difference-of-squares 54.2%

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

      4. *-commutative 54.2%

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

      5. *-commutative 54.2%

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

      6. associate-*l* 60.4%

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

      7. +-commutative 60.4%

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

      8. *-commutative 60.4%

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

      9. associate-*r* 54.7%

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

      10. associate-*l* 54.7%

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

      11. *-commutative 54.7%

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

      12. metadata-eval 54.7%

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

   7. Simplified 54.7%

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

      1. log1p-expm1-u 54.7%

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

      2. associate-*l* 57.1%

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

   9. Applied egg-rr 57.1%

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

   10. Taylor expanded in angle around 0 52.4%

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

       1. associate-*r* 58.5%

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

       2. *-commutative 58.5%

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

       3. +-commutative 58.5%

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

   12. Simplified 58.5%

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

4. Final simplification 46.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq 1.52 \cdot 10^{-118}:\\ \;\;\;\;\left(b - a\right) \cdot \left(a \cdot \sin \left(\left(angle \cdot \pi\right) \cdot 0.011111111111111112\right)\right)\\ \mathbf{else}:\\ \;\;\;\;0.011111111111111112 \cdot \left(\left(angle \cdot \left(b - a\right)\right) \cdot \left(\pi \cdot \left(b + a\right)\right)\right)\\ \end{array} \]

Alternative 9: 54.5% accurate, 2.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq 5.1 \cdot 10^{-118}:\\ \;\;\;\;\left(b - a\right) \cdot \left(a \cdot \sin \left(angle \cdot \left(\pi \cdot 0.011111111111111112\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;0.011111111111111112 \cdot \left(\left(angle \cdot \left(b - a\right)\right) \cdot \left(\pi \cdot \left(b + a\right)\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (a b angle)
 :precision binary64
 (if (<= b 5.1e-118)
   (* (- b a) (* a (sin (* angle (* PI 0.011111111111111112)))))
   (* 0.011111111111111112 (* (* angle (- b a)) (* PI (+ b a))))))

C

double code(double a, double b, double angle) {
	double tmp;
	if (b <= 5.1e-118) {
		tmp = (b - a) * (a * sin((angle * (((double) M_PI) * 0.011111111111111112))));
	} else {
		tmp = 0.011111111111111112 * ((angle * (b - a)) * (((double) M_PI) * (b + a)));
	}
	return tmp;
}

Java

public static double code(double a, double b, double angle) {
	double tmp;
	if (b <= 5.1e-118) {
		tmp = (b - a) * (a * Math.sin((angle * (Math.PI * 0.011111111111111112))));
	} else {
		tmp = 0.011111111111111112 * ((angle * (b - a)) * (Math.PI * (b + a)));
	}
	return tmp;
}

Python

def code(a, b, angle):
	tmp = 0
	if b <= 5.1e-118:
		tmp = (b - a) * (a * math.sin((angle * (math.pi * 0.011111111111111112))))
	else:
		tmp = 0.011111111111111112 * ((angle * (b - a)) * (math.pi * (b + a)))
	return tmp

Julia

function code(a, b, angle)
	tmp = 0.0
	if (b <= 5.1e-118)
		tmp = Float64(Float64(b - a) * Float64(a * sin(Float64(angle * Float64(pi * 0.011111111111111112)))));
	else
		tmp = Float64(0.011111111111111112 * Float64(Float64(angle * Float64(b - a)) * Float64(pi * Float64(b + a))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, angle)
	tmp = 0.0;
	if (b <= 5.1e-118)
		tmp = (b - a) * (a * sin((angle * (pi * 0.011111111111111112))));
	else
		tmp = 0.011111111111111112 * ((angle * (b - a)) * (pi * (b + a)));
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, angle_] := If[LessEqual[b, 5.1e-118], N[(N[(b - a), $MachinePrecision] * N[(a * N[Sin[N[(angle * N[(Pi * 0.011111111111111112), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(0.011111111111111112 * N[(N[(angle * N[(b - a), $MachinePrecision]), $MachinePrecision] * N[(Pi * N[(b + a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b < 5.09999999999999964e-118

   1. Initial program 51.8%

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

      1. *-commutative 51.8%

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

      2. associate-*l* 51.8%

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

      3. associate-*l* 51.8%

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

      4. unpow2 51.8%

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

      5. unpow2 51.8%

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

      6. difference-of-squares 55.8%

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

   3. Simplified 55.8%

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

      1. difference-of-squares 51.8%

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

      2. *-commutative 51.8%

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

      3. sub-neg 51.8%

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

      4. distribute-lft-in 51.8%

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

      5. associate-*l* 51.8%

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

      6. 2-sin 51.8%

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

      7. div-inv 50.5%

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

      8. metadata-eval 50.5%

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

   5. Applied egg-rr 49.1%

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

      1. distribute-lft-out 49.1%

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

      2. sub-neg 49.1%

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

      3. difference-of-squares 53.1%

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

      4. *-commutative 53.1%

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

      5. *-commutative 53.1%

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

      6. associate-*l* 57.4%

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

      7. +-commutative 57.4%

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

      8. *-commutative 57.4%

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

      9. associate-*r* 57.3%

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

      10. associate-*l* 57.3%

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

      11. *-commutative 57.3%

          \[\leadsto \left(b - a\right) \cdot \left(\left(a + b\right) \cdot \sin \left(\color{blue}{\left(angle \cdot \pi\right)} \cdot \left(0.005555555555555556 \cdot 2\right)\right)\right) \]

      12. metadata-eval 57.3%

          \[\leadsto \left(b - a\right) \cdot \left(\left(a + b\right) \cdot \sin \left(\left(angle \cdot \pi\right) \cdot \color{blue}{0.011111111111111112}\right)\right) \]

   7. Simplified 57.3%

      \[\leadsto \color{blue}{\left(b - a\right) \cdot \left(\left(a + b\right) \cdot \sin \left(\left(angle \cdot \pi\right) \cdot 0.011111111111111112\right)\right)} \]
   8. Step-by-step derivation

      1. log1p-expm1-u 57.3%

         \[\leadsto \left(b - a\right) \cdot \left(\left(a + b\right) \cdot \color{blue}{\mathsf{log1p}\left(\mathsf{expm1}\left(\sin \left(\left(angle \cdot \pi\right) \cdot 0.011111111111111112\right)\right)\right)}\right) \]

      2. associate-*l* 58.7%

         \[\leadsto \left(b - a\right) \cdot \left(\left(a + b\right) \cdot \mathsf{log1p}\left(\mathsf{expm1}\left(\sin \color{blue}{\left(angle \cdot \left(\pi \cdot 0.011111111111111112\right)\right)}\right)\right)\right) \]

   9. Applied egg-rr 58.7%

      \[\leadsto \left(b - a\right) \cdot \left(\left(a + b\right) \cdot \color{blue}{\mathsf{log1p}\left(\mathsf{expm1}\left(\sin \left(angle \cdot \left(\pi \cdot 0.011111111111111112\right)\right)\right)\right)}\right) \]

   10. Taylor expanded in a around inf 38.5%

       \[\leadsto \left(b - a\right) \cdot \color{blue}{\left(\sin \left(0.011111111111111112 \cdot \left(angle \cdot \pi\right)\right) \cdot a\right)} \]
   11. Step-by-step derivation

       1. *-commutative 38.5%

          \[\leadsto \left(b - a\right) \cdot \left(\sin \color{blue}{\left(\left(angle \cdot \pi\right) \cdot 0.011111111111111112\right)} \cdot a\right) \]

       2. associate-*r* 40.0%

          \[\leadsto \left(b - a\right) \cdot \left(\sin \color{blue}{\left(angle \cdot \left(\pi \cdot 0.011111111111111112\right)\right)} \cdot a\right) \]

       3. *-commutative 40.0%

          \[\leadsto \left(b - a\right) \cdot \left(\sin \left(angle \cdot \color{blue}{\left(0.011111111111111112 \cdot \pi\right)}\right) \cdot a\right) \]

   12. Simplified 40.0%

       \[\leadsto \left(b - a\right) \cdot \color{blue}{\left(\sin \left(angle \cdot \left(0.011111111111111112 \cdot \pi\right)\right) \cdot a\right)} \]

   if 5.09999999999999964e-118 < b

   1. Initial program 49.7%

      \[\left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin \left(\pi \cdot \frac{angle}{180}\right)\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right) \]
   2. Step-by-step derivation

      1. *-commutative 49.7%

         \[\leadsto \left(\color{blue}{\left(\left({b}^{2} - {a}^{2}\right) \cdot 2\right)} \cdot \sin \left(\pi \cdot \frac{angle}{180}\right)\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right) \]

      2. associate-*l* 49.7%

         \[\leadsto \color{blue}{\left(\left({b}^{2} - {a}^{2}\right) \cdot \left(2 \cdot \sin \left(\pi \cdot \frac{angle}{180}\right)\right)\right)} \cdot \cos \left(\pi \cdot \frac{angle}{180}\right) \]

      3. associate-*l* 49.7%

         \[\leadsto \color{blue}{\left({b}^{2} - {a}^{2}\right) \cdot \left(\left(2 \cdot \sin \left(\pi \cdot \frac{angle}{180}\right)\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right)\right)} \]

      4. unpow2 49.7%

         \[\leadsto \left(\color{blue}{b \cdot b} - {a}^{2}\right) \cdot \left(\left(2 \cdot \sin \left(\pi \cdot \frac{angle}{180}\right)\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right)\right) \]

      5. unpow2 49.7%

         \[\leadsto \left(b \cdot b - \color{blue}{a \cdot a}\right) \cdot \left(\left(2 \cdot \sin \left(\pi \cdot \frac{angle}{180}\right)\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right)\right) \]

      6. difference-of-squares 53.6%

         \[\leadsto \color{blue}{\left(\left(b + a\right) \cdot \left(b - a\right)\right)} \cdot \left(\left(2 \cdot \sin \left(\pi \cdot \frac{angle}{180}\right)\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right)\right) \]

   3. Simplified 53.6%

      \[\leadsto \color{blue}{\left(\left(b + a\right) \cdot \left(b - a\right)\right) \cdot \left(\left(2 \cdot \sin \left(\pi \cdot \frac{angle}{180}\right)\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right)\right)} \]
   4. Step-by-step derivation

      1. difference-of-squares 49.7%

         \[\leadsto \color{blue}{\left(b \cdot b - a \cdot a\right)} \cdot \left(\left(2 \cdot \sin \left(\pi \cdot \frac{angle}{180}\right)\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right)\right) \]

      2. *-commutative 49.7%

         \[\leadsto \color{blue}{\left(\left(2 \cdot \sin \left(\pi \cdot \frac{angle}{180}\right)\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right)\right) \cdot \left(b \cdot b - a \cdot a\right)} \]

      3. sub-neg 49.7%

         \[\leadsto \left(\left(2 \cdot \sin \left(\pi \cdot \frac{angle}{180}\right)\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right)\right) \cdot \color{blue}{\left(b \cdot b + \left(-a \cdot a\right)\right)} \]

      4. distribute-lft-in 49.7%

         \[\leadsto \color{blue}{\left(\left(2 \cdot \sin \left(\pi \cdot \frac{angle}{180}\right)\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right)\right) \cdot \left(b \cdot b\right) + \left(\left(2 \cdot \sin \left(\pi \cdot \frac{angle}{180}\right)\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right)\right) \cdot \left(-a \cdot a\right)} \]

      5. associate-*l* 49.7%

         \[\leadsto \color{blue}{\left(2 \cdot \left(\sin \left(\pi \cdot \frac{angle}{180}\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right)\right)\right)} \cdot \left(b \cdot b\right) + \left(\left(2 \cdot \sin \left(\pi \cdot \frac{angle}{180}\right)\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right)\right) \cdot \left(-a \cdot a\right) \]

      6. 2-sin 49.7%

         \[\leadsto \color{blue}{\sin \left(2 \cdot \left(\pi \cdot \frac{angle}{180}\right)\right)} \cdot \left(b \cdot b\right) + \left(\left(2 \cdot \sin \left(\pi \cdot \frac{angle}{180}\right)\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right)\right) \cdot \left(-a \cdot a\right) \]

      7. div-inv 51.4%

         \[\leadsto \sin \left(2 \cdot \left(\pi \cdot \color{blue}{\left(angle \cdot \frac{1}{180}\right)}\right)\right) \cdot \left(b \cdot b\right) + \left(\left(2 \cdot \sin \left(\pi \cdot \frac{angle}{180}\right)\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right)\right) \cdot \left(-a \cdot a\right) \]

      8. metadata-eval 51.4%

         \[\leadsto \sin \left(2 \cdot \left(\pi \cdot \left(angle \cdot \color{blue}{0.005555555555555556}\right)\right)\right) \cdot \left(b \cdot b\right) + \left(\left(2 \cdot \sin \left(\pi \cdot \frac{angle}{180}\right)\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right)\right) \cdot \left(-a \cdot a\right) \]

   5. Applied egg-rr 50.4%

      \[\leadsto \color{blue}{\sin \left(2 \cdot \left(\pi \cdot \left(angle \cdot 0.005555555555555556\right)\right)\right) \cdot \left(b \cdot b\right) + \sin \left(2 \cdot \left(\pi \cdot \left(angle \cdot 0.005555555555555556\right)\right)\right) \cdot \left(-a \cdot a\right)} \]
   6. Step-by-step derivation

      1. distribute-lft-out 50.4%

         \[\leadsto \color{blue}{\sin \left(2 \cdot \left(\pi \cdot \left(angle \cdot 0.005555555555555556\right)\right)\right) \cdot \left(b \cdot b + \left(-a \cdot a\right)\right)} \]

      2. sub-neg 50.4%

         \[\leadsto \sin \left(2 \cdot \left(\pi \cdot \left(angle \cdot 0.005555555555555556\right)\right)\right) \cdot \color{blue}{\left(b \cdot b - a \cdot a\right)} \]

      3. difference-of-squares 54.2%

         \[\leadsto \sin \left(2 \cdot \left(\pi \cdot \left(angle \cdot 0.005555555555555556\right)\right)\right) \cdot \color{blue}{\left(\left(b + a\right) \cdot \left(b - a\right)\right)} \]

      4. *-commutative 54.2%

         \[\leadsto \color{blue}{\left(\left(b + a\right) \cdot \left(b - a\right)\right) \cdot \sin \left(2 \cdot \left(\pi \cdot \left(angle \cdot 0.005555555555555556\right)\right)\right)} \]

      5. *-commutative 54.2%

         \[\leadsto \color{blue}{\left(\left(b - a\right) \cdot \left(b + a\right)\right)} \cdot \sin \left(2 \cdot \left(\pi \cdot \left(angle \cdot 0.005555555555555556\right)\right)\right) \]

      6. associate-*l* 60.4%

         \[\leadsto \color{blue}{\left(b - a\right) \cdot \left(\left(b + a\right) \cdot \sin \left(2 \cdot \left(\pi \cdot \left(angle \cdot 0.005555555555555556\right)\right)\right)\right)} \]

      7. +-commutative 60.4%

         \[\leadsto \left(b - a\right) \cdot \left(\color{blue}{\left(a + b\right)} \cdot \sin \left(2 \cdot \left(\pi \cdot \left(angle \cdot 0.005555555555555556\right)\right)\right)\right) \]

      8. *-commutative 60.4%

         \[\leadsto \left(b - a\right) \cdot \left(\left(a + b\right) \cdot \sin \color{blue}{\left(\left(\pi \cdot \left(angle \cdot 0.005555555555555556\right)\right) \cdot 2\right)}\right) \]

      9. associate-*r* 54.7%

         \[\leadsto \left(b - a\right) \cdot \left(\left(a + b\right) \cdot \sin \left(\color{blue}{\left(\left(\pi \cdot angle\right) \cdot 0.005555555555555556\right)} \cdot 2\right)\right) \]

      10. associate-*l* 54.7%

          \[\leadsto \left(b - a\right) \cdot \left(\left(a + b\right) \cdot \sin \color{blue}{\left(\left(\pi \cdot angle\right) \cdot \left(0.005555555555555556 \cdot 2\right)\right)}\right) \]

      11. *-commutative 54.7%

          \[\leadsto \left(b - a\right) \cdot \left(\left(a + b\right) \cdot \sin \left(\color{blue}{\left(angle \cdot \pi\right)} \cdot \left(0.005555555555555556 \cdot 2\right)\right)\right) \]

      12. metadata-eval 54.7%

          \[\leadsto \left(b - a\right) \cdot \left(\left(a + b\right) \cdot \sin \left(\left(angle \cdot \pi\right) \cdot \color{blue}{0.011111111111111112}\right)\right) \]

   7. Simplified 54.7%

      \[\leadsto \color{blue}{\left(b - a\right) \cdot \left(\left(a + b\right) \cdot \sin \left(\left(angle \cdot \pi\right) \cdot 0.011111111111111112\right)\right)} \]
   8. Step-by-step derivation

      1. log1p-expm1-u 54.7%

         \[\leadsto \left(b - a\right) \cdot \left(\left(a + b\right) \cdot \color{blue}{\mathsf{log1p}\left(\mathsf{expm1}\left(\sin \left(\left(angle \cdot \pi\right) \cdot 0.011111111111111112\right)\right)\right)}\right) \]

      2. associate-*l* 57.1%

         \[\leadsto \left(b - a\right) \cdot \left(\left(a + b\right) \cdot \mathsf{log1p}\left(\mathsf{expm1}\left(\sin \color{blue}{\left(angle \cdot \left(\pi \cdot 0.011111111111111112\right)\right)}\right)\right)\right) \]

   9. Applied egg-rr 57.1%

      \[\leadsto \left(b - a\right) \cdot \left(\left(a + b\right) \cdot \color{blue}{\mathsf{log1p}\left(\mathsf{expm1}\left(\sin \left(angle \cdot \left(\pi \cdot 0.011111111111111112\right)\right)\right)\right)}\right) \]

   10. Taylor expanded in angle around 0 52.4%

       \[\leadsto \color{blue}{0.011111111111111112 \cdot \left(angle \cdot \left(\left(b - a\right) \cdot \left(\pi \cdot \left(a + b\right)\right)\right)\right)} \]
   11. Step-by-step derivation

       1. associate-*r* 58.5%

          \[\leadsto 0.011111111111111112 \cdot \color{blue}{\left(\left(angle \cdot \left(b - a\right)\right) \cdot \left(\pi \cdot \left(a + b\right)\right)\right)} \]

       2. *-commutative 58.5%

          \[\leadsto 0.011111111111111112 \cdot \color{blue}{\left(\left(\pi \cdot \left(a + b\right)\right) \cdot \left(angle \cdot \left(b - a\right)\right)\right)} \]

       3. +-commutative 58.5%

          \[\leadsto 0.011111111111111112 \cdot \left(\left(\pi \cdot \color{blue}{\left(b + a\right)}\right) \cdot \left(angle \cdot \left(b - a\right)\right)\right) \]

   12. Simplified 58.5%

       \[\leadsto \color{blue}{0.011111111111111112 \cdot \left(\left(\pi \cdot \left(b + a\right)\right) \cdot \left(angle \cdot \left(b - a\right)\right)\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 47.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq 5.1 \cdot 10^{-118}:\\ \;\;\;\;\left(b - a\right) \cdot \left(a \cdot \sin \left(angle \cdot \left(\pi \cdot 0.011111111111111112\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;0.011111111111111112 \cdot \left(\left(angle \cdot \left(b - a\right)\right) \cdot \left(\pi \cdot \left(b + a\right)\right)\right)\\ \end{array} \]

Alternative 10: 50.5% accurate, 2.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq 1.05 \cdot 10^{-147}:\\ \;\;\;\;\sin \left(angle \cdot \left(\pi \cdot 0.011111111111111112\right)\right) \cdot \left(a \cdot \left(-a\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(b - a\right) \cdot \left(\left(b + a\right) \cdot \left(\left(angle \cdot \pi\right) \cdot 0.011111111111111112\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (a b angle)
 :precision binary64
 (if (<= b 1.05e-147)
   (* (sin (* angle (* PI 0.011111111111111112))) (* a (- a)))
   (* (- b a) (* (+ b a) (* (* angle PI) 0.011111111111111112)))))

C

double code(double a, double b, double angle) {
	double tmp;
	if (b <= 1.05e-147) {
		tmp = sin((angle * (((double) M_PI) * 0.011111111111111112))) * (a * -a);
	} else {
		tmp = (b - a) * ((b + a) * ((angle * ((double) M_PI)) * 0.011111111111111112));
	}
	return tmp;
}

Java

public static double code(double a, double b, double angle) {
	double tmp;
	if (b <= 1.05e-147) {
		tmp = Math.sin((angle * (Math.PI * 0.011111111111111112))) * (a * -a);
	} else {
		tmp = (b - a) * ((b + a) * ((angle * Math.PI) * 0.011111111111111112));
	}
	return tmp;
}

Python

def code(a, b, angle):
	tmp = 0
	if b <= 1.05e-147:
		tmp = math.sin((angle * (math.pi * 0.011111111111111112))) * (a * -a)
	else:
		tmp = (b - a) * ((b + a) * ((angle * math.pi) * 0.011111111111111112))
	return tmp

Julia

function code(a, b, angle)
	tmp = 0.0
	if (b <= 1.05e-147)
		tmp = Float64(sin(Float64(angle * Float64(pi * 0.011111111111111112))) * Float64(a * Float64(-a)));
	else
		tmp = Float64(Float64(b - a) * Float64(Float64(b + a) * Float64(Float64(angle * pi) * 0.011111111111111112)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, angle)
	tmp = 0.0;
	if (b <= 1.05e-147)
		tmp = sin((angle * (pi * 0.011111111111111112))) * (a * -a);
	else
		tmp = (b - a) * ((b + a) * ((angle * pi) * 0.011111111111111112));
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, angle_] := If[LessEqual[b, 1.05e-147], N[(N[Sin[N[(angle * N[(Pi * 0.011111111111111112), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * N[(a * (-a)), $MachinePrecision]), $MachinePrecision], N[(N[(b - a), $MachinePrecision] * N[(N[(b + a), $MachinePrecision] * N[(N[(angle * Pi), $MachinePrecision] * 0.011111111111111112), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b < 1.05e-147

   1. Initial program 51.1%

      \[\left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin \left(\pi \cdot \frac{angle}{180}\right)\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right) \]
   2. Step-by-step derivation

      1. *-commutative 51.1%

         \[\leadsto \left(\color{blue}{\left(\left({b}^{2} - {a}^{2}\right) \cdot 2\right)} \cdot \sin \left(\pi \cdot \frac{angle}{180}\right)\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right) \]

      2. associate-*l* 51.1%

         \[\leadsto \color{blue}{\left(\left({b}^{2} - {a}^{2}\right) \cdot \left(2 \cdot \sin \left(\pi \cdot \frac{angle}{180}\right)\right)\right)} \cdot \cos \left(\pi \cdot \frac{angle}{180}\right) \]

      3. associate-*l* 51.1%

         \[\leadsto \color{blue}{\left({b}^{2} - {a}^{2}\right) \cdot \left(\left(2 \cdot \sin \left(\pi \cdot \frac{angle}{180}\right)\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right)\right)} \]

      4. unpow2 51.1%

         \[\leadsto \left(\color{blue}{b \cdot b} - {a}^{2}\right) \cdot \left(\left(2 \cdot \sin \left(\pi \cdot \frac{angle}{180}\right)\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right)\right) \]

      5. unpow2 51.1%

         \[\leadsto \left(b \cdot b - \color{blue}{a \cdot a}\right) \cdot \left(\left(2 \cdot \sin \left(\pi \cdot \frac{angle}{180}\right)\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right)\right) \]

      6. difference-of-squares 55.3%

         \[\leadsto \color{blue}{\left(\left(b + a\right) \cdot \left(b - a\right)\right)} \cdot \left(\left(2 \cdot \sin \left(\pi \cdot \frac{angle}{180}\right)\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right)\right) \]

   3. Simplified 55.3%

      \[\leadsto \color{blue}{\left(\left(b + a\right) \cdot \left(b - a\right)\right) \cdot \left(\left(2 \cdot \sin \left(\pi \cdot \frac{angle}{180}\right)\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right)\right)} \]
   4. Step-by-step derivation

      1. difference-of-squares 51.1%

         \[\leadsto \color{blue}{\left(b \cdot b - a \cdot a\right)} \cdot \left(\left(2 \cdot \sin \left(\pi \cdot \frac{angle}{180}\right)\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right)\right) \]

      2. *-commutative 51.1%

         \[\leadsto \color{blue}{\left(\left(2 \cdot \sin \left(\pi \cdot \frac{angle}{180}\right)\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right)\right) \cdot \left(b \cdot b - a \cdot a\right)} \]

      3. sub-neg 51.1%

         \[\leadsto \left(\left(2 \cdot \sin \left(\pi \cdot \frac{angle}{180}\right)\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right)\right) \cdot \color{blue}{\left(b \cdot b + \left(-a \cdot a\right)\right)} \]

      4. distribute-lft-in 51.1%

         \[\leadsto \color{blue}{\left(\left(2 \cdot \sin \left(\pi \cdot \frac{angle}{180}\right)\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right)\right) \cdot \left(b \cdot b\right) + \left(\left(2 \cdot \sin \left(\pi \cdot \frac{angle}{180}\right)\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right)\right) \cdot \left(-a \cdot a\right)} \]

      5. associate-*l* 51.1%

         \[\leadsto \color{blue}{\left(2 \cdot \left(\sin \left(\pi \cdot \frac{angle}{180}\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right)\right)\right)} \cdot \left(b \cdot b\right) + \left(\left(2 \cdot \sin \left(\pi \cdot \frac{angle}{180}\right)\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right)\right) \cdot \left(-a \cdot a\right) \]

      6. 2-sin 51.1%

         \[\leadsto \color{blue}{\sin \left(2 \cdot \left(\pi \cdot \frac{angle}{180}\right)\right)} \cdot \left(b \cdot b\right) + \left(\left(2 \cdot \sin \left(\pi \cdot \frac{angle}{180}\right)\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right)\right) \cdot \left(-a \cdot a\right) \]

      7. div-inv 49.6%

         \[\leadsto \sin \left(2 \cdot \left(\pi \cdot \color{blue}{\left(angle \cdot \frac{1}{180}\right)}\right)\right) \cdot \left(b \cdot b\right) + \left(\left(2 \cdot \sin \left(\pi \cdot \frac{angle}{180}\right)\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right)\right) \cdot \left(-a \cdot a\right) \]

      8. metadata-eval 49.6%

         \[\leadsto \sin \left(2 \cdot \left(\pi \cdot \left(angle \cdot \color{blue}{0.005555555555555556}\right)\right)\right) \cdot \left(b \cdot b\right) + \left(\left(2 \cdot \sin \left(\pi \cdot \frac{angle}{180}\right)\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right)\right) \cdot \left(-a \cdot a\right) \]

   5. Applied egg-rr 48.2%

      \[\leadsto \color{blue}{\sin \left(2 \cdot \left(\pi \cdot \left(angle \cdot 0.005555555555555556\right)\right)\right) \cdot \left(b \cdot b\right) + \sin \left(2 \cdot \left(\pi \cdot \left(angle \cdot 0.005555555555555556\right)\right)\right) \cdot \left(-a \cdot a\right)} \]
   6. Step-by-step derivation

      1. distribute-lft-out 48.2%

         \[\leadsto \color{blue}{\sin \left(2 \cdot \left(\pi \cdot \left(angle \cdot 0.005555555555555556\right)\right)\right) \cdot \left(b \cdot b + \left(-a \cdot a\right)\right)} \]

      2. sub-neg 48.2%

         \[\leadsto \sin \left(2 \cdot \left(\pi \cdot \left(angle \cdot 0.005555555555555556\right)\right)\right) \cdot \color{blue}{\left(b \cdot b - a \cdot a\right)} \]

      3. difference-of-squares 52.4%

         \[\leadsto \sin \left(2 \cdot \left(\pi \cdot \left(angle \cdot 0.005555555555555556\right)\right)\right) \cdot \color{blue}{\left(\left(b + a\right) \cdot \left(b - a\right)\right)} \]

      4. *-commutative 52.4%

         \[\leadsto \color{blue}{\left(\left(b + a\right) \cdot \left(b - a\right)\right) \cdot \sin \left(2 \cdot \left(\pi \cdot \left(angle \cdot 0.005555555555555556\right)\right)\right)} \]

      5. *-commutative 52.4%

         \[\leadsto \color{blue}{\left(\left(b - a\right) \cdot \left(b + a\right)\right)} \cdot \sin \left(2 \cdot \left(\pi \cdot \left(angle \cdot 0.005555555555555556\right)\right)\right) \]

      6. associate-*l* 56.9%

         \[\leadsto \color{blue}{\left(b - a\right) \cdot \left(\left(b + a\right) \cdot \sin \left(2 \cdot \left(\pi \cdot \left(angle \cdot 0.005555555555555556\right)\right)\right)\right)} \]

      7. +-commutative 56.9%

         \[\leadsto \left(b - a\right) \cdot \left(\color{blue}{\left(a + b\right)} \cdot \sin \left(2 \cdot \left(\pi \cdot \left(angle \cdot 0.005555555555555556\right)\right)\right)\right) \]

      8. *-commutative 56.9%

         \[\leadsto \left(b - a\right) \cdot \left(\left(a + b\right) \cdot \sin \color{blue}{\left(\left(\pi \cdot \left(angle \cdot 0.005555555555555556\right)\right) \cdot 2\right)}\right) \]

      9. associate-*r* 56.8%

         \[\leadsto \left(b - a\right) \cdot \left(\left(a + b\right) \cdot \sin \left(\color{blue}{\left(\left(\pi \cdot angle\right) \cdot 0.005555555555555556\right)} \cdot 2\right)\right) \]

      10. associate-*l* 56.8%

          \[\leadsto \left(b - a\right) \cdot \left(\left(a + b\right) \cdot \sin \color{blue}{\left(\left(\pi \cdot angle\right) \cdot \left(0.005555555555555556 \cdot 2\right)\right)}\right) \]

      11. *-commutative 56.8%

          \[\leadsto \left(b - a\right) \cdot \left(\left(a + b\right) \cdot \sin \left(\color{blue}{\left(angle \cdot \pi\right)} \cdot \left(0.005555555555555556 \cdot 2\right)\right)\right) \]

      12. metadata-eval 56.8%

          \[\leadsto \left(b - a\right) \cdot \left(\left(a + b\right) \cdot \sin \left(\left(angle \cdot \pi\right) \cdot \color{blue}{0.011111111111111112}\right)\right) \]

   7. Simplified 56.8%

      \[\leadsto \color{blue}{\left(b - a\right) \cdot \left(\left(a + b\right) \cdot \sin \left(\left(angle \cdot \pi\right) \cdot 0.011111111111111112\right)\right)} \]
   8. Step-by-step derivation

      1. log1p-expm1-u 56.8%

         \[\leadsto \left(b - a\right) \cdot \left(\left(a + b\right) \cdot \color{blue}{\mathsf{log1p}\left(\mathsf{expm1}\left(\sin \left(\left(angle \cdot \pi\right) \cdot 0.011111111111111112\right)\right)\right)}\right) \]

      2. associate-*l* 57.6%

         \[\leadsto \left(b - a\right) \cdot \left(\left(a + b\right) \cdot \mathsf{log1p}\left(\mathsf{expm1}\left(\sin \color{blue}{\left(angle \cdot \left(\pi \cdot 0.011111111111111112\right)\right)}\right)\right)\right) \]

   9. Applied egg-rr 57.6%

      \[\leadsto \left(b - a\right) \cdot \left(\left(a + b\right) \cdot \color{blue}{\mathsf{log1p}\left(\mathsf{expm1}\left(\sin \left(angle \cdot \left(\pi \cdot 0.011111111111111112\right)\right)\right)\right)}\right) \]

   10. Taylor expanded in b around 0 33.0%

       \[\leadsto \color{blue}{-1 \cdot \left(\sin \left(0.011111111111111112 \cdot \left(angle \cdot \pi\right)\right) \cdot {a}^{2}\right)} \]
   11. Step-by-step derivation

       1. mul-1-neg 33.0%

          \[\leadsto \color{blue}{-\sin \left(0.011111111111111112 \cdot \left(angle \cdot \pi\right)\right) \cdot {a}^{2}} \]

       2. distribute-rgt-neg-in 33.0%

          \[\leadsto \color{blue}{\sin \left(0.011111111111111112 \cdot \left(angle \cdot \pi\right)\right) \cdot \left(-{a}^{2}\right)} \]

       3. *-commutative 33.0%

          \[\leadsto \sin \color{blue}{\left(\left(angle \cdot \pi\right) \cdot 0.011111111111111112\right)} \cdot \left(-{a}^{2}\right) \]

       4. associate-*r* 33.6%

          \[\leadsto \sin \color{blue}{\left(angle \cdot \left(\pi \cdot 0.011111111111111112\right)\right)} \cdot \left(-{a}^{2}\right) \]

       5. *-commutative 33.6%

          \[\leadsto \sin \left(angle \cdot \color{blue}{\left(0.011111111111111112 \cdot \pi\right)}\right) \cdot \left(-{a}^{2}\right) \]

       6. unpow2 33.6%

          \[\leadsto \sin \left(angle \cdot \left(0.011111111111111112 \cdot \pi\right)\right) \cdot \left(-\color{blue}{a \cdot a}\right) \]

       7. distribute-rgt-neg-in 33.6%

          \[\leadsto \sin \left(angle \cdot \left(0.011111111111111112 \cdot \pi\right)\right) \cdot \color{blue}{\left(a \cdot \left(-a\right)\right)} \]

   12. Simplified 33.6%

       \[\leadsto \color{blue}{\sin \left(angle \cdot \left(0.011111111111111112 \cdot \pi\right)\right) \cdot \left(a \cdot \left(-a\right)\right)} \]

   if 1.05e-147 < b

   1. Initial program 50.8%

      \[\left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin \left(\pi \cdot \frac{angle}{180}\right)\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right) \]
   2. Step-by-step derivation

      1. *-commutative 50.8%

         \[\leadsto \left(\color{blue}{\left(\left({b}^{2} - {a}^{2}\right) \cdot 2\right)} \cdot \sin \left(\pi \cdot \frac{angle}{180}\right)\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right) \]

      2. associate-*l* 50.8%

         \[\leadsto \color{blue}{\left(\left({b}^{2} - {a}^{2}\right) \cdot \left(2 \cdot \sin \left(\pi \cdot \frac{angle}{180}\right)\right)\right)} \cdot \cos \left(\pi \cdot \frac{angle}{180}\right) \]

      3. associate-*l* 50.8%

         \[\leadsto \color{blue}{\left({b}^{2} - {a}^{2}\right) \cdot \left(\left(2 \cdot \sin \left(\pi \cdot \frac{angle}{180}\right)\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right)\right)} \]

      4. unpow2 50.8%

         \[\leadsto \left(\color{blue}{b \cdot b} - {a}^{2}\right) \cdot \left(\left(2 \cdot \sin \left(\pi \cdot \frac{angle}{180}\right)\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right)\right) \]

      5. unpow2 50.8%

         \[\leadsto \left(b \cdot b - \color{blue}{a \cdot a}\right) \cdot \left(\left(2 \cdot \sin \left(\pi \cdot \frac{angle}{180}\right)\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right)\right) \]

      6. difference-of-squares 54.4%

         \[\leadsto \color{blue}{\left(\left(b + a\right) \cdot \left(b - a\right)\right)} \cdot \left(\left(2 \cdot \sin \left(\pi \cdot \frac{angle}{180}\right)\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right)\right) \]

   3. Simplified 54.4%

      \[\leadsto \color{blue}{\left(\left(b + a\right) \cdot \left(b - a\right)\right) \cdot \left(\left(2 \cdot \sin \left(\pi \cdot \frac{angle}{180}\right)\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right)\right)} \]
   4. Step-by-step derivation

      1. difference-of-squares 50.8%

         \[\leadsto \color{blue}{\left(b \cdot b - a \cdot a\right)} \cdot \left(\left(2 \cdot \sin \left(\pi \cdot \frac{angle}{180}\right)\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right)\right) \]

      2. *-commutative 50.8%

         \[\leadsto \color{blue}{\left(\left(2 \cdot \sin \left(\pi \cdot \frac{angle}{180}\right)\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right)\right) \cdot \left(b \cdot b - a \cdot a\right)} \]

      3. sub-neg 50.8%

         \[\leadsto \left(\left(2 \cdot \sin \left(\pi \cdot \frac{angle}{180}\right)\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right)\right) \cdot \color{blue}{\left(b \cdot b + \left(-a \cdot a\right)\right)} \]

      4. distribute-lft-in 50.8%

         \[\leadsto \color{blue}{\left(\left(2 \cdot \sin \left(\pi \cdot \frac{angle}{180}\right)\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right)\right) \cdot \left(b \cdot b\right) + \left(\left(2 \cdot \sin \left(\pi \cdot \frac{angle}{180}\right)\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right)\right) \cdot \left(-a \cdot a\right)} \]

      5. associate-*l* 50.8%

         \[\leadsto \color{blue}{\left(2 \cdot \left(\sin \left(\pi \cdot \frac{angle}{180}\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right)\right)\right)} \cdot \left(b \cdot b\right) + \left(\left(2 \cdot \sin \left(\pi \cdot \frac{angle}{180}\right)\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right)\right) \cdot \left(-a \cdot a\right) \]

      6. 2-sin 50.8%

         \[\leadsto \color{blue}{\sin \left(2 \cdot \left(\pi \cdot \frac{angle}{180}\right)\right)} \cdot \left(b \cdot b\right) + \left(\left(2 \cdot \sin \left(\pi \cdot \frac{angle}{180}\right)\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right)\right) \cdot \left(-a \cdot a\right) \]

      7. div-inv 52.4%

         \[\leadsto \sin \left(2 \cdot \left(\pi \cdot \color{blue}{\left(angle \cdot \frac{1}{180}\right)}\right)\right) \cdot \left(b \cdot b\right) + \left(\left(2 \cdot \sin \left(\pi \cdot \frac{angle}{180}\right)\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right)\right) \cdot \left(-a \cdot a\right) \]

      8. metadata-eval 52.4%

         \[\leadsto \sin \left(2 \cdot \left(\pi \cdot \left(angle \cdot \color{blue}{0.005555555555555556}\right)\right)\right) \cdot \left(b \cdot b\right) + \left(\left(2 \cdot \sin \left(\pi \cdot \frac{angle}{180}\right)\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right)\right) \cdot \left(-a \cdot a\right) \]

   5. Applied egg-rr 51.4%

      \[\leadsto \color{blue}{\sin \left(2 \cdot \left(\pi \cdot \left(angle \cdot 0.005555555555555556\right)\right)\right) \cdot \left(b \cdot b\right) + \sin \left(2 \cdot \left(\pi \cdot \left(angle \cdot 0.005555555555555556\right)\right)\right) \cdot \left(-a \cdot a\right)} \]
   6. Step-by-step derivation

      1. distribute-lft-out 51.4%

         \[\leadsto \color{blue}{\sin \left(2 \cdot \left(\pi \cdot \left(angle \cdot 0.005555555555555556\right)\right)\right) \cdot \left(b \cdot b + \left(-a \cdot a\right)\right)} \]

      2. sub-neg 51.4%

         \[\leadsto \sin \left(2 \cdot \left(\pi \cdot \left(angle \cdot 0.005555555555555556\right)\right)\right) \cdot \color{blue}{\left(b \cdot b - a \cdot a\right)} \]

      3. difference-of-squares 55.0%

         \[\leadsto \sin \left(2 \cdot \left(\pi \cdot \left(angle \cdot 0.005555555555555556\right)\right)\right) \cdot \color{blue}{\left(\left(b + a\right) \cdot \left(b - a\right)\right)} \]

      4. *-commutative 55.0%

         \[\leadsto \color{blue}{\left(\left(b + a\right) \cdot \left(b - a\right)\right) \cdot \sin \left(2 \cdot \left(\pi \cdot \left(angle \cdot 0.005555555555555556\right)\right)\right)} \]

      5. *-commutative 55.0%

         \[\leadsto \color{blue}{\left(\left(b - a\right) \cdot \left(b + a\right)\right)} \cdot \sin \left(2 \cdot \left(\pi \cdot \left(angle \cdot 0.005555555555555556\right)\right)\right) \]

      6. associate-*l* 60.7%

         \[\leadsto \color{blue}{\left(b - a\right) \cdot \left(\left(b + a\right) \cdot \sin \left(2 \cdot \left(\pi \cdot \left(angle \cdot 0.005555555555555556\right)\right)\right)\right)} \]

      7. +-commutative 60.7%

         \[\leadsto \left(b - a\right) \cdot \left(\color{blue}{\left(a + b\right)} \cdot \sin \left(2 \cdot \left(\pi \cdot \left(angle \cdot 0.005555555555555556\right)\right)\right)\right) \]

      8. *-commutative 60.7%

         \[\leadsto \left(b - a\right) \cdot \left(\left(a + b\right) \cdot \sin \color{blue}{\left(\left(\pi \cdot \left(angle \cdot 0.005555555555555556\right)\right) \cdot 2\right)}\right) \]

      9. associate-*r* 55.4%

         \[\leadsto \left(b - a\right) \cdot \left(\left(a + b\right) \cdot \sin \left(\color{blue}{\left(\left(\pi \cdot angle\right) \cdot 0.005555555555555556\right)} \cdot 2\right)\right) \]

      10. associate-*l* 55.4%

          \[\leadsto \left(b - a\right) \cdot \left(\left(a + b\right) \cdot \sin \color{blue}{\left(\left(\pi \cdot angle\right) \cdot \left(0.005555555555555556 \cdot 2\right)\right)}\right) \]

      11. *-commutative 55.4%

          \[\leadsto \left(b - a\right) \cdot \left(\left(a + b\right) \cdot \sin \left(\color{blue}{\left(angle \cdot \pi\right)} \cdot \left(0.005555555555555556 \cdot 2\right)\right)\right) \]

      12. metadata-eval 55.4%

          \[\leadsto \left(b - a\right) \cdot \left(\left(a + b\right) \cdot \sin \left(\left(angle \cdot \pi\right) \cdot \color{blue}{0.011111111111111112}\right)\right) \]

   7. Simplified 55.4%

      \[\leadsto \color{blue}{\left(b - a\right) \cdot \left(\left(a + b\right) \cdot \sin \left(\left(angle \cdot \pi\right) \cdot 0.011111111111111112\right)\right)} \]

   8. Taylor expanded in angle around 0 58.9%

      \[\leadsto \left(b - a\right) \cdot \left(\left(a + b\right) \cdot \color{blue}{\left(0.011111111111111112 \cdot \left(angle \cdot \pi\right)\right)}\right) \]
3. Recombined 2 regimes into one program.

4. Final simplification 44.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq 1.05 \cdot 10^{-147}:\\ \;\;\;\;\sin \left(angle \cdot \left(\pi \cdot 0.011111111111111112\right)\right) \cdot \left(a \cdot \left(-a\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(b - a\right) \cdot \left(\left(b + a\right) \cdot \left(\left(angle \cdot \pi\right) \cdot 0.011111111111111112\right)\right)\\ \end{array} \]

Alternative 11: 50.4% accurate, 2.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(angle \cdot \pi\right) \cdot 0.011111111111111112\\ \mathbf{if}\;b \leq 1.04 \cdot 10^{-147}:\\ \;\;\;\;\sin t_0 \cdot \left(a \cdot \left(-a\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(b - a\right) \cdot \left(\left(b + a\right) \cdot t_0\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (a b angle)
 :precision binary64
 (let* ((t_0 (* (* angle PI) 0.011111111111111112)))
   (if (<= b 1.04e-147)
     (* (sin t_0) (* a (- a)))
     (* (- b a) (* (+ b a) t_0)))))

C

double code(double a, double b, double angle) {
	double t_0 = (angle * ((double) M_PI)) * 0.011111111111111112;
	double tmp;
	if (b <= 1.04e-147) {
		tmp = sin(t_0) * (a * -a);
	} else {
		tmp = (b - a) * ((b + a) * t_0);
	}
	return tmp;
}

Java

public static double code(double a, double b, double angle) {
	double t_0 = (angle * Math.PI) * 0.011111111111111112;
	double tmp;
	if (b <= 1.04e-147) {
		tmp = Math.sin(t_0) * (a * -a);
	} else {
		tmp = (b - a) * ((b + a) * t_0);
	}
	return tmp;
}

Python

def code(a, b, angle):
	t_0 = (angle * math.pi) * 0.011111111111111112
	tmp = 0
	if b <= 1.04e-147:
		tmp = math.sin(t_0) * (a * -a)
	else:
		tmp = (b - a) * ((b + a) * t_0)
	return tmp

Julia

function code(a, b, angle)
	t_0 = Float64(Float64(angle * pi) * 0.011111111111111112)
	tmp = 0.0
	if (b <= 1.04e-147)
		tmp = Float64(sin(t_0) * Float64(a * Float64(-a)));
	else
		tmp = Float64(Float64(b - a) * Float64(Float64(b + a) * t_0));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, angle)
	t_0 = (angle * pi) * 0.011111111111111112;
	tmp = 0.0;
	if (b <= 1.04e-147)
		tmp = sin(t_0) * (a * -a);
	else
		tmp = (b - a) * ((b + a) * t_0);
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, angle_] := Block[{t$95$0 = N[(N[(angle * Pi), $MachinePrecision] * 0.011111111111111112), $MachinePrecision]}, If[LessEqual[b, 1.04e-147], N[(N[Sin[t$95$0], $MachinePrecision] * N[(a * (-a)), $MachinePrecision]), $MachinePrecision], N[(N[(b - a), $MachinePrecision] * N[(N[(b + a), $MachinePrecision] * t$95$0), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if b < 1.04000000000000003e-147

   1. Initial program 51.1%

      \[\left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin \left(\pi \cdot \frac{angle}{180}\right)\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right) \]
   2. Step-by-step derivation

      1. *-commutative 51.1%

         \[\leadsto \left(\color{blue}{\left(\left({b}^{2} - {a}^{2}\right) \cdot 2\right)} \cdot \sin \left(\pi \cdot \frac{angle}{180}\right)\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right) \]

      2. associate-*l* 51.1%

         \[\leadsto \color{blue}{\left(\left({b}^{2} - {a}^{2}\right) \cdot \left(2 \cdot \sin \left(\pi \cdot \frac{angle}{180}\right)\right)\right)} \cdot \cos \left(\pi \cdot \frac{angle}{180}\right) \]

      3. associate-*l* 51.1%

         \[\leadsto \color{blue}{\left({b}^{2} - {a}^{2}\right) \cdot \left(\left(2 \cdot \sin \left(\pi \cdot \frac{angle}{180}\right)\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right)\right)} \]

      4. unpow2 51.1%

         \[\leadsto \left(\color{blue}{b \cdot b} - {a}^{2}\right) \cdot \left(\left(2 \cdot \sin \left(\pi \cdot \frac{angle}{180}\right)\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right)\right) \]

      5. unpow2 51.1%

         \[\leadsto \left(b \cdot b - \color{blue}{a \cdot a}\right) \cdot \left(\left(2 \cdot \sin \left(\pi \cdot \frac{angle}{180}\right)\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right)\right) \]

      6. difference-of-squares 55.3%

         \[\leadsto \color{blue}{\left(\left(b + a\right) \cdot \left(b - a\right)\right)} \cdot \left(\left(2 \cdot \sin \left(\pi \cdot \frac{angle}{180}\right)\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right)\right) \]

   3. Simplified 55.3%

      \[\leadsto \color{blue}{\left(\left(b + a\right) \cdot \left(b - a\right)\right) \cdot \left(\left(2 \cdot \sin \left(\pi \cdot \frac{angle}{180}\right)\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right)\right)} \]
   4. Step-by-step derivation

      1. difference-of-squares 51.1%

         \[\leadsto \color{blue}{\left(b \cdot b - a \cdot a\right)} \cdot \left(\left(2 \cdot \sin \left(\pi \cdot \frac{angle}{180}\right)\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right)\right) \]

      2. *-commutative 51.1%

         \[\leadsto \color{blue}{\left(\left(2 \cdot \sin \left(\pi \cdot \frac{angle}{180}\right)\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right)\right) \cdot \left(b \cdot b - a \cdot a\right)} \]

      3. prod-diff 42.4%

         \[\leadsto \left(\left(2 \cdot \sin \left(\pi \cdot \frac{angle}{180}\right)\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right)\right) \cdot \color{blue}{\left(\mathsf{fma}\left(b, b, -a \cdot a\right) + \mathsf{fma}\left(-a, a, a \cdot a\right)\right)} \]

      4. fma-neg 42.4%

         \[\leadsto \left(\left(2 \cdot \sin \left(\pi \cdot \frac{angle}{180}\right)\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right)\right) \cdot \left(\color{blue}{\left(b \cdot b - a \cdot a\right)} + \mathsf{fma}\left(-a, a, a \cdot a\right)\right) \]

      5. distribute-lft-in 42.4%

         \[\leadsto \color{blue}{\left(\left(2 \cdot \sin \left(\pi \cdot \frac{angle}{180}\right)\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right)\right) \cdot \left(b \cdot b - a \cdot a\right) + \left(\left(2 \cdot \sin \left(\pi \cdot \frac{angle}{180}\right)\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right)\right) \cdot \mathsf{fma}\left(-a, a, a \cdot a\right)} \]

   5. Applied egg-rr 42.4%

      \[\leadsto \color{blue}{\left(\left(b + a\right) \cdot \left(b - a\right)\right) \cdot \sin \left(2 \cdot \left(\pi \cdot \left(angle \cdot 0.005555555555555556\right)\right)\right) + \sin \left(2 \cdot \left(\pi \cdot \left(angle \cdot 0.005555555555555556\right)\right)\right) \cdot \mathsf{fma}\left(-a, a, a \cdot a\right)} \]
   6. Step-by-step derivation

      1. *-commutative 42.4%

         \[\leadsto \left(\left(b + a\right) \cdot \left(b - a\right)\right) \cdot \sin \left(2 \cdot \left(\pi \cdot \left(angle \cdot 0.005555555555555556\right)\right)\right) + \color{blue}{\mathsf{fma}\left(-a, a, a \cdot a\right) \cdot \sin \left(2 \cdot \left(\pi \cdot \left(angle \cdot 0.005555555555555556\right)\right)\right)} \]

      2. distribute-rgt-out 42.4%

         \[\leadsto \color{blue}{\sin \left(2 \cdot \left(\pi \cdot \left(angle \cdot 0.005555555555555556\right)\right)\right) \cdot \left(\left(b + a\right) \cdot \left(b - a\right) + \mathsf{fma}\left(-a, a, a \cdot a\right)\right)} \]

      3. *-commutative 42.4%

         \[\leadsto \color{blue}{\left(\left(b + a\right) \cdot \left(b - a\right) + \mathsf{fma}\left(-a, a, a \cdot a\right)\right) \cdot \sin \left(2 \cdot \left(\pi \cdot \left(angle \cdot 0.005555555555555556\right)\right)\right)} \]

   7. Simplified 40.8%

      \[\leadsto \color{blue}{\mathsf{fma}\left(b, b, 0 \cdot \left(a \cdot a\right) - a \cdot a\right) \cdot \sin \left(\left(angle \cdot \pi\right) \cdot 0.011111111111111112\right)} \]

   8. Taylor expanded in b around 0 33.0%

      \[\leadsto \color{blue}{\left(-1 \cdot {a}^{2}\right)} \cdot \sin \left(\left(angle \cdot \pi\right) \cdot 0.011111111111111112\right) \]
   9. Step-by-step derivation

      1. mul-1-neg 33.0%

         \[\leadsto \color{blue}{\left(-{a}^{2}\right)} \cdot \sin \left(\left(angle \cdot \pi\right) \cdot 0.011111111111111112\right) \]

      2. unpow2 33.0%

         \[\leadsto \left(-\color{blue}{a \cdot a}\right) \cdot \sin \left(\left(angle \cdot \pi\right) \cdot 0.011111111111111112\right) \]

      3. distribute-rgt-neg-in 33.0%

         \[\leadsto \color{blue}{\left(a \cdot \left(-a\right)\right)} \cdot \sin \left(\left(angle \cdot \pi\right) \cdot 0.011111111111111112\right) \]

   10. Simplified 33.0%

       \[\leadsto \color{blue}{\left(a \cdot \left(-a\right)\right)} \cdot \sin \left(\left(angle \cdot \pi\right) \cdot 0.011111111111111112\right) \]

   if 1.04000000000000003e-147 < b

   1. Initial program 50.8%

      \[\left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin \left(\pi \cdot \frac{angle}{180}\right)\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right) \]
   2. Step-by-step derivation

      1. *-commutative 50.8%

         \[\leadsto \left(\color{blue}{\left(\left({b}^{2} - {a}^{2}\right) \cdot 2\right)} \cdot \sin \left(\pi \cdot \frac{angle}{180}\right)\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right) \]

      2. associate-*l* 50.8%

         \[\leadsto \color{blue}{\left(\left({b}^{2} - {a}^{2}\right) \cdot \left(2 \cdot \sin \left(\pi \cdot \frac{angle}{180}\right)\right)\right)} \cdot \cos \left(\pi \cdot \frac{angle}{180}\right) \]

      3. associate-*l* 50.8%

         \[\leadsto \color{blue}{\left({b}^{2} - {a}^{2}\right) \cdot \left(\left(2 \cdot \sin \left(\pi \cdot \frac{angle}{180}\right)\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right)\right)} \]

      4. unpow2 50.8%

         \[\leadsto \left(\color{blue}{b \cdot b} - {a}^{2}\right) \cdot \left(\left(2 \cdot \sin \left(\pi \cdot \frac{angle}{180}\right)\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right)\right) \]

      5. unpow2 50.8%

         \[\leadsto \left(b \cdot b - \color{blue}{a \cdot a}\right) \cdot \left(\left(2 \cdot \sin \left(\pi \cdot \frac{angle}{180}\right)\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right)\right) \]

      6. difference-of-squares 54.4%

         \[\leadsto \color{blue}{\left(\left(b + a\right) \cdot \left(b - a\right)\right)} \cdot \left(\left(2 \cdot \sin \left(\pi \cdot \frac{angle}{180}\right)\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right)\right) \]

   3. Simplified 54.4%

      \[\leadsto \color{blue}{\left(\left(b + a\right) \cdot \left(b - a\right)\right) \cdot \left(\left(2 \cdot \sin \left(\pi \cdot \frac{angle}{180}\right)\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right)\right)} \]
   4. Step-by-step derivation

      1. difference-of-squares 50.8%

         \[\leadsto \color{blue}{\left(b \cdot b - a \cdot a\right)} \cdot \left(\left(2 \cdot \sin \left(\pi \cdot \frac{angle}{180}\right)\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right)\right) \]

      2. *-commutative 50.8%

         \[\leadsto \color{blue}{\left(\left(2 \cdot \sin \left(\pi \cdot \frac{angle}{180}\right)\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right)\right) \cdot \left(b \cdot b - a \cdot a\right)} \]

      3. sub-neg 50.8%

         \[\leadsto \left(\left(2 \cdot \sin \left(\pi \cdot \frac{angle}{180}\right)\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right)\right) \cdot \color{blue}{\left(b \cdot b + \left(-a \cdot a\right)\right)} \]

      4. distribute-lft-in 50.8%

         \[\leadsto \color{blue}{\left(\left(2 \cdot \sin \left(\pi \cdot \frac{angle}{180}\right)\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right)\right) \cdot \left(b \cdot b\right) + \left(\left(2 \cdot \sin \left(\pi \cdot \frac{angle}{180}\right)\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right)\right) \cdot \left(-a \cdot a\right)} \]

      5. associate-*l* 50.8%

         \[\leadsto \color{blue}{\left(2 \cdot \left(\sin \left(\pi \cdot \frac{angle}{180}\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right)\right)\right)} \cdot \left(b \cdot b\right) + \left(\left(2 \cdot \sin \left(\pi \cdot \frac{angle}{180}\right)\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right)\right) \cdot \left(-a \cdot a\right) \]

      6. 2-sin 50.8%

         \[\leadsto \color{blue}{\sin \left(2 \cdot \left(\pi \cdot \frac{angle}{180}\right)\right)} \cdot \left(b \cdot b\right) + \left(\left(2 \cdot \sin \left(\pi \cdot \frac{angle}{180}\right)\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right)\right) \cdot \left(-a \cdot a\right) \]

      7. div-inv 52.4%

         \[\leadsto \sin \left(2 \cdot \left(\pi \cdot \color{blue}{\left(angle \cdot \frac{1}{180}\right)}\right)\right) \cdot \left(b \cdot b\right) + \left(\left(2 \cdot \sin \left(\pi \cdot \frac{angle}{180}\right)\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right)\right) \cdot \left(-a \cdot a\right) \]

      8. metadata-eval 52.4%

         \[\leadsto \sin \left(2 \cdot \left(\pi \cdot \left(angle \cdot \color{blue}{0.005555555555555556}\right)\right)\right) \cdot \left(b \cdot b\right) + \left(\left(2 \cdot \sin \left(\pi \cdot \frac{angle}{180}\right)\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right)\right) \cdot \left(-a \cdot a\right) \]

   5. Applied egg-rr 51.4%

      \[\leadsto \color{blue}{\sin \left(2 \cdot \left(\pi \cdot \left(angle \cdot 0.005555555555555556\right)\right)\right) \cdot \left(b \cdot b\right) + \sin \left(2 \cdot \left(\pi \cdot \left(angle \cdot 0.005555555555555556\right)\right)\right) \cdot \left(-a \cdot a\right)} \]
   6. Step-by-step derivation

      1. distribute-lft-out 51.4%

         \[\leadsto \color{blue}{\sin \left(2 \cdot \left(\pi \cdot \left(angle \cdot 0.005555555555555556\right)\right)\right) \cdot \left(b \cdot b + \left(-a \cdot a\right)\right)} \]

      2. sub-neg 51.4%

         \[\leadsto \sin \left(2 \cdot \left(\pi \cdot \left(angle \cdot 0.005555555555555556\right)\right)\right) \cdot \color{blue}{\left(b \cdot b - a \cdot a\right)} \]

      3. difference-of-squares 55.0%

         \[\leadsto \sin \left(2 \cdot \left(\pi \cdot \left(angle \cdot 0.005555555555555556\right)\right)\right) \cdot \color{blue}{\left(\left(b + a\right) \cdot \left(b - a\right)\right)} \]

      4. *-commutative 55.0%

         \[\leadsto \color{blue}{\left(\left(b + a\right) \cdot \left(b - a\right)\right) \cdot \sin \left(2 \cdot \left(\pi \cdot \left(angle \cdot 0.005555555555555556\right)\right)\right)} \]

      5. *-commutative 55.0%

         \[\leadsto \color{blue}{\left(\left(b - a\right) \cdot \left(b + a\right)\right)} \cdot \sin \left(2 \cdot \left(\pi \cdot \left(angle \cdot 0.005555555555555556\right)\right)\right) \]

      6. associate-*l* 60.7%

         \[\leadsto \color{blue}{\left(b - a\right) \cdot \left(\left(b + a\right) \cdot \sin \left(2 \cdot \left(\pi \cdot \left(angle \cdot 0.005555555555555556\right)\right)\right)\right)} \]

      7. +-commutative 60.7%

         \[\leadsto \left(b - a\right) \cdot \left(\color{blue}{\left(a + b\right)} \cdot \sin \left(2 \cdot \left(\pi \cdot \left(angle \cdot 0.005555555555555556\right)\right)\right)\right) \]

      8. *-commutative 60.7%

         \[\leadsto \left(b - a\right) \cdot \left(\left(a + b\right) \cdot \sin \color{blue}{\left(\left(\pi \cdot \left(angle \cdot 0.005555555555555556\right)\right) \cdot 2\right)}\right) \]

      9. associate-*r* 55.4%

         \[\leadsto \left(b - a\right) \cdot \left(\left(a + b\right) \cdot \sin \left(\color{blue}{\left(\left(\pi \cdot angle\right) \cdot 0.005555555555555556\right)} \cdot 2\right)\right) \]

      10. associate-*l* 55.4%

          \[\leadsto \left(b - a\right) \cdot \left(\left(a + b\right) \cdot \sin \color{blue}{\left(\left(\pi \cdot angle\right) \cdot \left(0.005555555555555556 \cdot 2\right)\right)}\right) \]

      11. *-commutative 55.4%

          \[\leadsto \left(b - a\right) \cdot \left(\left(a + b\right) \cdot \sin \left(\color{blue}{\left(angle \cdot \pi\right)} \cdot \left(0.005555555555555556 \cdot 2\right)\right)\right) \]

      12. metadata-eval 55.4%

          \[\leadsto \left(b - a\right) \cdot \left(\left(a + b\right) \cdot \sin \left(\left(angle \cdot \pi\right) \cdot \color{blue}{0.011111111111111112}\right)\right) \]

   7. Simplified 55.4%

      \[\leadsto \color{blue}{\left(b - a\right) \cdot \left(\left(a + b\right) \cdot \sin \left(\left(angle \cdot \pi\right) \cdot 0.011111111111111112\right)\right)} \]

   8. Taylor expanded in angle around 0 58.9%

      \[\leadsto \left(b - a\right) \cdot \left(\left(a + b\right) \cdot \color{blue}{\left(0.011111111111111112 \cdot \left(angle \cdot \pi\right)\right)}\right) \]
3. Recombined 2 regimes into one program.

4. Final simplification 44.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq 1.04 \cdot 10^{-147}:\\ \;\;\;\;\sin \left(\left(angle \cdot \pi\right) \cdot 0.011111111111111112\right) \cdot \left(a \cdot \left(-a\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(b - a\right) \cdot \left(\left(b + a\right) \cdot \left(\left(angle \cdot \pi\right) \cdot 0.011111111111111112\right)\right)\\ \end{array} \]

Alternative 12: 42.4% accurate, 5.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;a \leq 1.32 \cdot 10^{-40}:\\ \;\;\;\;0.011111111111111112 \cdot \left(angle \cdot \left(\pi \cdot \left(b \cdot b\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;0.011111111111111112 \cdot \left(angle \cdot \left(\left(b - a\right) \cdot \left(a \cdot \pi\right)\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (a b angle)
 :precision binary64
 (if (<= a 1.32e-40)
   (* 0.011111111111111112 (* angle (* PI (* b b))))
   (* 0.011111111111111112 (* angle (* (- b a) (* a PI))))))

C

double code(double a, double b, double angle) {
	double tmp;
	if (a <= 1.32e-40) {
		tmp = 0.011111111111111112 * (angle * (((double) M_PI) * (b * b)));
	} else {
		tmp = 0.011111111111111112 * (angle * ((b - a) * (a * ((double) M_PI))));
	}
	return tmp;
}

Java

public static double code(double a, double b, double angle) {
	double tmp;
	if (a <= 1.32e-40) {
		tmp = 0.011111111111111112 * (angle * (Math.PI * (b * b)));
	} else {
		tmp = 0.011111111111111112 * (angle * ((b - a) * (a * Math.PI)));
	}
	return tmp;
}

Python

def code(a, b, angle):
	tmp = 0
	if a <= 1.32e-40:
		tmp = 0.011111111111111112 * (angle * (math.pi * (b * b)))
	else:
		tmp = 0.011111111111111112 * (angle * ((b - a) * (a * math.pi)))
	return tmp

Julia

function code(a, b, angle)
	tmp = 0.0
	if (a <= 1.32e-40)
		tmp = Float64(0.011111111111111112 * Float64(angle * Float64(pi * Float64(b * b))));
	else
		tmp = Float64(0.011111111111111112 * Float64(angle * Float64(Float64(b - a) * Float64(a * pi))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, angle)
	tmp = 0.0;
	if (a <= 1.32e-40)
		tmp = 0.011111111111111112 * (angle * (pi * (b * b)));
	else
		tmp = 0.011111111111111112 * (angle * ((b - a) * (a * pi)));
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, angle_] := If[LessEqual[a, 1.32e-40], N[(0.011111111111111112 * N[(angle * N[(Pi * N[(b * b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(0.011111111111111112 * N[(angle * N[(N[(b - a), $MachinePrecision] * N[(a * Pi), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if a < 1.32000000000000009e-40

   1. Initial program 52.9%

      \[\left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin \left(\pi \cdot \frac{angle}{180}\right)\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right) \]
   2. Step-by-step derivation

      1. *-commutative 52.9%

         \[\leadsto \left(\color{blue}{\left(\left({b}^{2} - {a}^{2}\right) \cdot 2\right)} \cdot \sin \left(\pi \cdot \frac{angle}{180}\right)\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right) \]

      2. associate-*l* 52.9%

         \[\leadsto \color{blue}{\left(\left({b}^{2} - {a}^{2}\right) \cdot \left(2 \cdot \sin \left(\pi \cdot \frac{angle}{180}\right)\right)\right)} \cdot \cos \left(\pi \cdot \frac{angle}{180}\right) \]

      3. associate-*l* 52.9%

         \[\leadsto \color{blue}{\left({b}^{2} - {a}^{2}\right) \cdot \left(\left(2 \cdot \sin \left(\pi \cdot \frac{angle}{180}\right)\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right)\right)} \]

      4. unpow2 52.9%

         \[\leadsto \left(\color{blue}{b \cdot b} - {a}^{2}\right) \cdot \left(\left(2 \cdot \sin \left(\pi \cdot \frac{angle}{180}\right)\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right)\right) \]

      5. unpow2 52.9%

         \[\leadsto \left(b \cdot b - \color{blue}{a \cdot a}\right) \cdot \left(\left(2 \cdot \sin \left(\pi \cdot \frac{angle}{180}\right)\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right)\right) \]

      6. difference-of-squares 55.2%

         \[\leadsto \color{blue}{\left(\left(b + a\right) \cdot \left(b - a\right)\right)} \cdot \left(\left(2 \cdot \sin \left(\pi \cdot \frac{angle}{180}\right)\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right)\right) \]

   3. Simplified 55.2%

      \[\leadsto \color{blue}{\left(\left(b + a\right) \cdot \left(b - a\right)\right) \cdot \left(\left(2 \cdot \sin \left(\pi \cdot \frac{angle}{180}\right)\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right)\right)} \]

   4. Taylor expanded in angle around 0 51.6%

      \[\leadsto \color{blue}{0.011111111111111112 \cdot \left(angle \cdot \left(\left(b - a\right) \cdot \left(\pi \cdot \left(a + b\right)\right)\right)\right)} \]

   5. Taylor expanded in b around inf 37.8%

      \[\leadsto 0.011111111111111112 \cdot \left(angle \cdot \color{blue}{\left({b}^{2} \cdot \pi\right)}\right) \]
   6. Step-by-step derivation

      1. *-commutative 37.8%

         \[\leadsto 0.011111111111111112 \cdot \left(angle \cdot \color{blue}{\left(\pi \cdot {b}^{2}\right)}\right) \]

      2. unpow2 37.8%

         \[\leadsto 0.011111111111111112 \cdot \left(angle \cdot \left(\pi \cdot \color{blue}{\left(b \cdot b\right)}\right)\right) \]

   7. Simplified 37.8%

      \[\leadsto 0.011111111111111112 \cdot \left(angle \cdot \color{blue}{\left(\pi \cdot \left(b \cdot b\right)\right)}\right) \]

   if 1.32000000000000009e-40 < a

   1. Initial program 45.9%

      \[\left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin \left(\pi \cdot \frac{angle}{180}\right)\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right) \]
   2. Step-by-step derivation

      1. *-commutative 45.9%

         \[\leadsto \left(\color{blue}{\left(\left({b}^{2} - {a}^{2}\right) \cdot 2\right)} \cdot \sin \left(\pi \cdot \frac{angle}{180}\right)\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right) \]

      2. associate-*l* 45.9%

         \[\leadsto \color{blue}{\left(\left({b}^{2} - {a}^{2}\right) \cdot \left(2 \cdot \sin \left(\pi \cdot \frac{angle}{180}\right)\right)\right)} \cdot \cos \left(\pi \cdot \frac{angle}{180}\right) \]

      3. associate-*l* 45.9%

         \[\leadsto \color{blue}{\left({b}^{2} - {a}^{2}\right) \cdot \left(\left(2 \cdot \sin \left(\pi \cdot \frac{angle}{180}\right)\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right)\right)} \]

      4. unpow2 45.9%

         \[\leadsto \left(\color{blue}{b \cdot b} - {a}^{2}\right) \cdot \left(\left(2 \cdot \sin \left(\pi \cdot \frac{angle}{180}\right)\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right)\right) \]

      5. unpow2 45.9%

         \[\leadsto \left(b \cdot b - \color{blue}{a \cdot a}\right) \cdot \left(\left(2 \cdot \sin \left(\pi \cdot \frac{angle}{180}\right)\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right)\right) \]

      6. difference-of-squares 54.2%

         \[\leadsto \color{blue}{\left(\left(b + a\right) \cdot \left(b - a\right)\right)} \cdot \left(\left(2 \cdot \sin \left(\pi \cdot \frac{angle}{180}\right)\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right)\right) \]

   3. Simplified 54.2%

      \[\leadsto \color{blue}{\left(\left(b + a\right) \cdot \left(b - a\right)\right) \cdot \left(\left(2 \cdot \sin \left(\pi \cdot \frac{angle}{180}\right)\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right)\right)} \]

   4. Taylor expanded in angle around 0 49.8%

      \[\leadsto \color{blue}{0.011111111111111112 \cdot \left(angle \cdot \left(\left(b - a\right) \cdot \left(\pi \cdot \left(a + b\right)\right)\right)\right)} \]

   5. Taylor expanded in a around inf 41.5%

      \[\leadsto 0.011111111111111112 \cdot \left(angle \cdot \left(\left(b - a\right) \cdot \color{blue}{\left(a \cdot \pi\right)}\right)\right) \]
   6. Step-by-step derivation

      1. *-commutative 41.5%

         \[\leadsto 0.011111111111111112 \cdot \left(angle \cdot \left(\left(b - a\right) \cdot \color{blue}{\left(\pi \cdot a\right)}\right)\right) \]

   7. Simplified 41.5%

      \[\leadsto 0.011111111111111112 \cdot \left(angle \cdot \left(\left(b - a\right) \cdot \color{blue}{\left(\pi \cdot a\right)}\right)\right) \]
3. Recombined 2 regimes into one program.

4. Final simplification 38.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \leq 1.32 \cdot 10^{-40}:\\ \;\;\;\;0.011111111111111112 \cdot \left(angle \cdot \left(\pi \cdot \left(b \cdot b\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;0.011111111111111112 \cdot \left(angle \cdot \left(\left(b - a\right) \cdot \left(a \cdot \pi\right)\right)\right)\\ \end{array} \]

Alternative 13: 43.9% accurate, 5.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;a \leq 8 \cdot 10^{-40}:\\ \;\;\;\;0.011111111111111112 \cdot \left(angle \cdot \left(\left(b - a\right) \cdot \left(b \cdot \pi\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;0.011111111111111112 \cdot \left(angle \cdot \left(\left(b - a\right) \cdot \left(a \cdot \pi\right)\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (a b angle)
 :precision binary64
 (if (<= a 8e-40)
   (* 0.011111111111111112 (* angle (* (- b a) (* b PI))))
   (* 0.011111111111111112 (* angle (* (- b a) (* a PI))))))

C

double code(double a, double b, double angle) {
	double tmp;
	if (a <= 8e-40) {
		tmp = 0.011111111111111112 * (angle * ((b - a) * (b * ((double) M_PI))));
	} else {
		tmp = 0.011111111111111112 * (angle * ((b - a) * (a * ((double) M_PI))));
	}
	return tmp;
}

Java

public static double code(double a, double b, double angle) {
	double tmp;
	if (a <= 8e-40) {
		tmp = 0.011111111111111112 * (angle * ((b - a) * (b * Math.PI)));
	} else {
		tmp = 0.011111111111111112 * (angle * ((b - a) * (a * Math.PI)));
	}
	return tmp;
}

Python

def code(a, b, angle):
	tmp = 0
	if a <= 8e-40:
		tmp = 0.011111111111111112 * (angle * ((b - a) * (b * math.pi)))
	else:
		tmp = 0.011111111111111112 * (angle * ((b - a) * (a * math.pi)))
	return tmp

Julia

function code(a, b, angle)
	tmp = 0.0
	if (a <= 8e-40)
		tmp = Float64(0.011111111111111112 * Float64(angle * Float64(Float64(b - a) * Float64(b * pi))));
	else
		tmp = Float64(0.011111111111111112 * Float64(angle * Float64(Float64(b - a) * Float64(a * pi))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, angle)
	tmp = 0.0;
	if (a <= 8e-40)
		tmp = 0.011111111111111112 * (angle * ((b - a) * (b * pi)));
	else
		tmp = 0.011111111111111112 * (angle * ((b - a) * (a * pi)));
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, angle_] := If[LessEqual[a, 8e-40], N[(0.011111111111111112 * N[(angle * N[(N[(b - a), $MachinePrecision] * N[(b * Pi), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(0.011111111111111112 * N[(angle * N[(N[(b - a), $MachinePrecision] * N[(a * Pi), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if a < 7.9999999999999994e-40

   1. Initial program 52.9%

      \[\left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin \left(\pi \cdot \frac{angle}{180}\right)\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right) \]
   2. Step-by-step derivation

      1. *-commutative 52.9%

         \[\leadsto \left(\color{blue}{\left(\left({b}^{2} - {a}^{2}\right) \cdot 2\right)} \cdot \sin \left(\pi \cdot \frac{angle}{180}\right)\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right) \]

      2. associate-*l* 52.9%

         \[\leadsto \color{blue}{\left(\left({b}^{2} - {a}^{2}\right) \cdot \left(2 \cdot \sin \left(\pi \cdot \frac{angle}{180}\right)\right)\right)} \cdot \cos \left(\pi \cdot \frac{angle}{180}\right) \]

      3. associate-*l* 52.9%

         \[\leadsto \color{blue}{\left({b}^{2} - {a}^{2}\right) \cdot \left(\left(2 \cdot \sin \left(\pi \cdot \frac{angle}{180}\right)\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right)\right)} \]

      4. unpow2 52.9%

         \[\leadsto \left(\color{blue}{b \cdot b} - {a}^{2}\right) \cdot \left(\left(2 \cdot \sin \left(\pi \cdot \frac{angle}{180}\right)\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right)\right) \]

      5. unpow2 52.9%

         \[\leadsto \left(b \cdot b - \color{blue}{a \cdot a}\right) \cdot \left(\left(2 \cdot \sin \left(\pi \cdot \frac{angle}{180}\right)\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right)\right) \]

      6. difference-of-squares 55.2%

         \[\leadsto \color{blue}{\left(\left(b + a\right) \cdot \left(b - a\right)\right)} \cdot \left(\left(2 \cdot \sin \left(\pi \cdot \frac{angle}{180}\right)\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right)\right) \]

   3. Simplified 55.2%

      \[\leadsto \color{blue}{\left(\left(b + a\right) \cdot \left(b - a\right)\right) \cdot \left(\left(2 \cdot \sin \left(\pi \cdot \frac{angle}{180}\right)\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right)\right)} \]

   4. Taylor expanded in angle around 0 51.6%

      \[\leadsto \color{blue}{0.011111111111111112 \cdot \left(angle \cdot \left(\left(b - a\right) \cdot \left(\pi \cdot \left(a + b\right)\right)\right)\right)} \]

   5. Taylor expanded in a around 0 40.0%

      \[\leadsto 0.011111111111111112 \cdot \left(angle \cdot \left(\left(b - a\right) \cdot \color{blue}{\left(b \cdot \pi\right)}\right)\right) \]
   6. Step-by-step derivation

      1. *-commutative 40.0%

         \[\leadsto 0.011111111111111112 \cdot \left(angle \cdot \left(\left(b - a\right) \cdot \color{blue}{\left(\pi \cdot b\right)}\right)\right) \]

   7. Simplified 40.0%

      \[\leadsto 0.011111111111111112 \cdot \left(angle \cdot \left(\left(b - a\right) \cdot \color{blue}{\left(\pi \cdot b\right)}\right)\right) \]

   if 7.9999999999999994e-40 < a

   1. Initial program 45.9%

      \[\left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin \left(\pi \cdot \frac{angle}{180}\right)\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right) \]
   2. Step-by-step derivation

      1. *-commutative 45.9%

         \[\leadsto \left(\color{blue}{\left(\left({b}^{2} - {a}^{2}\right) \cdot 2\right)} \cdot \sin \left(\pi \cdot \frac{angle}{180}\right)\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right) \]

      2. associate-*l* 45.9%

         \[\leadsto \color{blue}{\left(\left({b}^{2} - {a}^{2}\right) \cdot \left(2 \cdot \sin \left(\pi \cdot \frac{angle}{180}\right)\right)\right)} \cdot \cos \left(\pi \cdot \frac{angle}{180}\right) \]

      3. associate-*l* 45.9%

         \[\leadsto \color{blue}{\left({b}^{2} - {a}^{2}\right) \cdot \left(\left(2 \cdot \sin \left(\pi \cdot \frac{angle}{180}\right)\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right)\right)} \]

      4. unpow2 45.9%

         \[\leadsto \left(\color{blue}{b \cdot b} - {a}^{2}\right) \cdot \left(\left(2 \cdot \sin \left(\pi \cdot \frac{angle}{180}\right)\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right)\right) \]

      5. unpow2 45.9%

         \[\leadsto \left(b \cdot b - \color{blue}{a \cdot a}\right) \cdot \left(\left(2 \cdot \sin \left(\pi \cdot \frac{angle}{180}\right)\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right)\right) \]

      6. difference-of-squares 54.2%

         \[\leadsto \color{blue}{\left(\left(b + a\right) \cdot \left(b - a\right)\right)} \cdot \left(\left(2 \cdot \sin \left(\pi \cdot \frac{angle}{180}\right)\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right)\right) \]

   3. Simplified 54.2%

      \[\leadsto \color{blue}{\left(\left(b + a\right) \cdot \left(b - a\right)\right) \cdot \left(\left(2 \cdot \sin \left(\pi \cdot \frac{angle}{180}\right)\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right)\right)} \]

   4. Taylor expanded in angle around 0 49.8%

      \[\leadsto \color{blue}{0.011111111111111112 \cdot \left(angle \cdot \left(\left(b - a\right) \cdot \left(\pi \cdot \left(a + b\right)\right)\right)\right)} \]

   5. Taylor expanded in a around inf 41.5%

      \[\leadsto 0.011111111111111112 \cdot \left(angle \cdot \left(\left(b - a\right) \cdot \color{blue}{\left(a \cdot \pi\right)}\right)\right) \]
   6. Step-by-step derivation

      1. *-commutative 41.5%

         \[\leadsto 0.011111111111111112 \cdot \left(angle \cdot \left(\left(b - a\right) \cdot \color{blue}{\left(\pi \cdot a\right)}\right)\right) \]

   7. Simplified 41.5%

      \[\leadsto 0.011111111111111112 \cdot \left(angle \cdot \left(\left(b - a\right) \cdot \color{blue}{\left(\pi \cdot a\right)}\right)\right) \]
3. Recombined 2 regimes into one program.

4. Final simplification 40.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \leq 8 \cdot 10^{-40}:\\ \;\;\;\;0.011111111111111112 \cdot \left(angle \cdot \left(\left(b - a\right) \cdot \left(b \cdot \pi\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;0.011111111111111112 \cdot \left(angle \cdot \left(\left(b - a\right) \cdot \left(a \cdot \pi\right)\right)\right)\\ \end{array} \]

Alternative 14: 55.0% accurate, 5.5× speedup

Math

\[\begin{array}{l} \\ 0.011111111111111112 \cdot \left(angle \cdot \left(\left(b - a\right) \cdot \left(\pi \cdot \left(b + a\right)\right)\right)\right) \end{array} \]

FPCore

(FPCore (a b angle)
 :precision binary64
 (* 0.011111111111111112 (* angle (* (- b a) (* PI (+ b a))))))

C

double code(double a, double b, double angle) {
	return 0.011111111111111112 * (angle * ((b - a) * (((double) M_PI) * (b + a))));
}

Java

public static double code(double a, double b, double angle) {
	return 0.011111111111111112 * (angle * ((b - a) * (Math.PI * (b + a))));
}

Python

def code(a, b, angle):
	return 0.011111111111111112 * (angle * ((b - a) * (math.pi * (b + a))))

Julia

function code(a, b, angle)
	return Float64(0.011111111111111112 * Float64(angle * Float64(Float64(b - a) * Float64(pi * Float64(b + a)))))
end

MATLAB

function tmp = code(a, b, angle)
	tmp = 0.011111111111111112 * (angle * ((b - a) * (pi * (b + a))));
end

Wolfram

code[a_, b_, angle_] := N[(0.011111111111111112 * N[(angle * N[(N[(b - a), $MachinePrecision] * N[(Pi * N[(b + a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 51.0%

   \[\left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin \left(\pi \cdot \frac{angle}{180}\right)\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right) \]
2. Step-by-step derivation

   1. *-commutative 51.0%

      \[\leadsto \left(\color{blue}{\left(\left({b}^{2} - {a}^{2}\right) \cdot 2\right)} \cdot \sin \left(\pi \cdot \frac{angle}{180}\right)\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right) \]

   2. associate-*l* 51.0%

      \[\leadsto \color{blue}{\left(\left({b}^{2} - {a}^{2}\right) \cdot \left(2 \cdot \sin \left(\pi \cdot \frac{angle}{180}\right)\right)\right)} \cdot \cos \left(\pi \cdot \frac{angle}{180}\right) \]

   3. associate-*l* 51.0%

      \[\leadsto \color{blue}{\left({b}^{2} - {a}^{2}\right) \cdot \left(\left(2 \cdot \sin \left(\pi \cdot \frac{angle}{180}\right)\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right)\right)} \]

   4. unpow2 51.0%

      \[\leadsto \left(\color{blue}{b \cdot b} - {a}^{2}\right) \cdot \left(\left(2 \cdot \sin \left(\pi \cdot \frac{angle}{180}\right)\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right)\right) \]

   5. unpow2 51.0%

      \[\leadsto \left(b \cdot b - \color{blue}{a \cdot a}\right) \cdot \left(\left(2 \cdot \sin \left(\pi \cdot \frac{angle}{180}\right)\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right)\right) \]

   6. difference-of-squares 54.9%

      \[\leadsto \color{blue}{\left(\left(b + a\right) \cdot \left(b - a\right)\right)} \cdot \left(\left(2 \cdot \sin \left(\pi \cdot \frac{angle}{180}\right)\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right)\right) \]

3. Simplified 54.9%

   \[\leadsto \color{blue}{\left(\left(b + a\right) \cdot \left(b - a\right)\right) \cdot \left(\left(2 \cdot \sin \left(\pi \cdot \frac{angle}{180}\right)\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right)\right)} \]

4. Taylor expanded in angle around 0 51.1%

   \[\leadsto \color{blue}{0.011111111111111112 \cdot \left(angle \cdot \left(\left(b - a\right) \cdot \left(\pi \cdot \left(a + b\right)\right)\right)\right)} \]

5. Final simplification 51.1%

   \[\leadsto 0.011111111111111112 \cdot \left(angle \cdot \left(\left(b - a\right) \cdot \left(\pi \cdot \left(b + a\right)\right)\right)\right) \]

Alternative 15: 63.1% accurate, 5.5× speedup

Math

\[\begin{array}{l} \\ 0.011111111111111112 \cdot \left(\left(angle \cdot \left(b - a\right)\right) \cdot \left(\pi \cdot \left(b + a\right)\right)\right) \end{array} \]

FPCore

(FPCore (a b angle)
 :precision binary64
 (* 0.011111111111111112 (* (* angle (- b a)) (* PI (+ b a)))))

C

double code(double a, double b, double angle) {
	return 0.011111111111111112 * ((angle * (b - a)) * (((double) M_PI) * (b + a)));
}

Java

public static double code(double a, double b, double angle) {
	return 0.011111111111111112 * ((angle * (b - a)) * (Math.PI * (b + a)));
}

Python

def code(a, b, angle):
	return 0.011111111111111112 * ((angle * (b - a)) * (math.pi * (b + a)))

Julia

function code(a, b, angle)
	return Float64(0.011111111111111112 * Float64(Float64(angle * Float64(b - a)) * Float64(pi * Float64(b + a))))
end

MATLAB

function tmp = code(a, b, angle)
	tmp = 0.011111111111111112 * ((angle * (b - a)) * (pi * (b + a)));
end

Wolfram

code[a_, b_, angle_] := N[(0.011111111111111112 * N[(N[(angle * N[(b - a), $MachinePrecision]), $MachinePrecision] * N[(Pi * N[(b + a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 51.0%

   \[\left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin \left(\pi \cdot \frac{angle}{180}\right)\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right) \]
2. Step-by-step derivation

   1. *-commutative 51.0%

      \[\leadsto \left(\color{blue}{\left(\left({b}^{2} - {a}^{2}\right) \cdot 2\right)} \cdot \sin \left(\pi \cdot \frac{angle}{180}\right)\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right) \]

   2. associate-*l* 51.0%

      \[\leadsto \color{blue}{\left(\left({b}^{2} - {a}^{2}\right) \cdot \left(2 \cdot \sin \left(\pi \cdot \frac{angle}{180}\right)\right)\right)} \cdot \cos \left(\pi \cdot \frac{angle}{180}\right) \]

   3. associate-*l* 51.0%

      \[\leadsto \color{blue}{\left({b}^{2} - {a}^{2}\right) \cdot \left(\left(2 \cdot \sin \left(\pi \cdot \frac{angle}{180}\right)\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right)\right)} \]

   4. unpow2 51.0%

      \[\leadsto \left(\color{blue}{b \cdot b} - {a}^{2}\right) \cdot \left(\left(2 \cdot \sin \left(\pi \cdot \frac{angle}{180}\right)\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right)\right) \]

   5. unpow2 51.0%

      \[\leadsto \left(b \cdot b - \color{blue}{a \cdot a}\right) \cdot \left(\left(2 \cdot \sin \left(\pi \cdot \frac{angle}{180}\right)\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right)\right) \]

   6. difference-of-squares 54.9%

      \[\leadsto \color{blue}{\left(\left(b + a\right) \cdot \left(b - a\right)\right)} \cdot \left(\left(2 \cdot \sin \left(\pi \cdot \frac{angle}{180}\right)\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right)\right) \]

3. Simplified 54.9%

   \[\leadsto \color{blue}{\left(\left(b + a\right) \cdot \left(b - a\right)\right) \cdot \left(\left(2 \cdot \sin \left(\pi \cdot \frac{angle}{180}\right)\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right)\right)} \]
4. Step-by-step derivation

   1. difference-of-squares 51.0%

      \[\leadsto \color{blue}{\left(b \cdot b - a \cdot a\right)} \cdot \left(\left(2 \cdot \sin \left(\pi \cdot \frac{angle}{180}\right)\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right)\right) \]

   2. *-commutative 51.0%

      \[\leadsto \color{blue}{\left(\left(2 \cdot \sin \left(\pi \cdot \frac{angle}{180}\right)\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right)\right) \cdot \left(b \cdot b - a \cdot a\right)} \]

   3. sub-neg 51.0%

      \[\leadsto \left(\left(2 \cdot \sin \left(\pi \cdot \frac{angle}{180}\right)\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right)\right) \cdot \color{blue}{\left(b \cdot b + \left(-a \cdot a\right)\right)} \]

   4. distribute-lft-in 51.0%

      \[\leadsto \color{blue}{\left(\left(2 \cdot \sin \left(\pi \cdot \frac{angle}{180}\right)\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right)\right) \cdot \left(b \cdot b\right) + \left(\left(2 \cdot \sin \left(\pi \cdot \frac{angle}{180}\right)\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right)\right) \cdot \left(-a \cdot a\right)} \]

   5. associate-*l* 51.0%

      \[\leadsto \color{blue}{\left(2 \cdot \left(\sin \left(\pi \cdot \frac{angle}{180}\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right)\right)\right)} \cdot \left(b \cdot b\right) + \left(\left(2 \cdot \sin \left(\pi \cdot \frac{angle}{180}\right)\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right)\right) \cdot \left(-a \cdot a\right) \]

   6. 2-sin 51.0%

      \[\leadsto \color{blue}{\sin \left(2 \cdot \left(\pi \cdot \frac{angle}{180}\right)\right)} \cdot \left(b \cdot b\right) + \left(\left(2 \cdot \sin \left(\pi \cdot \frac{angle}{180}\right)\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right)\right) \cdot \left(-a \cdot a\right) \]

   7. div-inv 50.9%

      \[\leadsto \sin \left(2 \cdot \left(\pi \cdot \color{blue}{\left(angle \cdot \frac{1}{180}\right)}\right)\right) \cdot \left(b \cdot b\right) + \left(\left(2 \cdot \sin \left(\pi \cdot \frac{angle}{180}\right)\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right)\right) \cdot \left(-a \cdot a\right) \]

   8. metadata-eval 50.9%

      \[\leadsto \sin \left(2 \cdot \left(\pi \cdot \left(angle \cdot \color{blue}{0.005555555555555556}\right)\right)\right) \cdot \left(b \cdot b\right) + \left(\left(2 \cdot \sin \left(\pi \cdot \frac{angle}{180}\right)\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right)\right) \cdot \left(-a \cdot a\right) \]

5. Applied egg-rr 49.6%

   \[\leadsto \color{blue}{\sin \left(2 \cdot \left(\pi \cdot \left(angle \cdot 0.005555555555555556\right)\right)\right) \cdot \left(b \cdot b\right) + \sin \left(2 \cdot \left(\pi \cdot \left(angle \cdot 0.005555555555555556\right)\right)\right) \cdot \left(-a \cdot a\right)} \]
6. Step-by-step derivation

   1. distribute-lft-out 49.6%

      \[\leadsto \color{blue}{\sin \left(2 \cdot \left(\pi \cdot \left(angle \cdot 0.005555555555555556\right)\right)\right) \cdot \left(b \cdot b + \left(-a \cdot a\right)\right)} \]

   2. sub-neg 49.6%

      \[\leadsto \sin \left(2 \cdot \left(\pi \cdot \left(angle \cdot 0.005555555555555556\right)\right)\right) \cdot \color{blue}{\left(b \cdot b - a \cdot a\right)} \]

   3. difference-of-squares 53.6%

      \[\leadsto \sin \left(2 \cdot \left(\pi \cdot \left(angle \cdot 0.005555555555555556\right)\right)\right) \cdot \color{blue}{\left(\left(b + a\right) \cdot \left(b - a\right)\right)} \]

   4. *-commutative 53.6%

      \[\leadsto \color{blue}{\left(\left(b + a\right) \cdot \left(b - a\right)\right) \cdot \sin \left(2 \cdot \left(\pi \cdot \left(angle \cdot 0.005555555555555556\right)\right)\right)} \]

   5. *-commutative 53.6%

      \[\leadsto \color{blue}{\left(\left(b - a\right) \cdot \left(b + a\right)\right)} \cdot \sin \left(2 \cdot \left(\pi \cdot \left(angle \cdot 0.005555555555555556\right)\right)\right) \]

   6. associate-*l* 58.6%

      \[\leadsto \color{blue}{\left(b - a\right) \cdot \left(\left(b + a\right) \cdot \sin \left(2 \cdot \left(\pi \cdot \left(angle \cdot 0.005555555555555556\right)\right)\right)\right)} \]

   7. +-commutative 58.6%

      \[\leadsto \left(b - a\right) \cdot \left(\color{blue}{\left(a + b\right)} \cdot \sin \left(2 \cdot \left(\pi \cdot \left(angle \cdot 0.005555555555555556\right)\right)\right)\right) \]

   8. *-commutative 58.6%

      \[\leadsto \left(b - a\right) \cdot \left(\left(a + b\right) \cdot \sin \color{blue}{\left(\left(\pi \cdot \left(angle \cdot 0.005555555555555556\right)\right) \cdot 2\right)}\right) \]

   9. associate-*r* 56.2%

      \[\leadsto \left(b - a\right) \cdot \left(\left(a + b\right) \cdot \sin \left(\color{blue}{\left(\left(\pi \cdot angle\right) \cdot 0.005555555555555556\right)} \cdot 2\right)\right) \]

   10. associate-*l* 56.2%

       \[\leadsto \left(b - a\right) \cdot \left(\left(a + b\right) \cdot \sin \color{blue}{\left(\left(\pi \cdot angle\right) \cdot \left(0.005555555555555556 \cdot 2\right)\right)}\right) \]

   11. *-commutative 56.2%

       \[\leadsto \left(b - a\right) \cdot \left(\left(a + b\right) \cdot \sin \left(\color{blue}{\left(angle \cdot \pi\right)} \cdot \left(0.005555555555555556 \cdot 2\right)\right)\right) \]

   12. metadata-eval 56.2%

       \[\leadsto \left(b - a\right) \cdot \left(\left(a + b\right) \cdot \sin \left(\left(angle \cdot \pi\right) \cdot \color{blue}{0.011111111111111112}\right)\right) \]

7. Simplified 56.2%

   \[\leadsto \color{blue}{\left(b - a\right) \cdot \left(\left(a + b\right) \cdot \sin \left(\left(angle \cdot \pi\right) \cdot 0.011111111111111112\right)\right)} \]
8. Step-by-step derivation

   1. log1p-expm1-u 56.2%

      \[\leadsto \left(b - a\right) \cdot \left(\left(a + b\right) \cdot \color{blue}{\mathsf{log1p}\left(\mathsf{expm1}\left(\sin \left(\left(angle \cdot \pi\right) \cdot 0.011111111111111112\right)\right)\right)}\right) \]

   2. associate-*l* 58.0%

      \[\leadsto \left(b - a\right) \cdot \left(\left(a + b\right) \cdot \mathsf{log1p}\left(\mathsf{expm1}\left(\sin \color{blue}{\left(angle \cdot \left(\pi \cdot 0.011111111111111112\right)\right)}\right)\right)\right) \]

9. Applied egg-rr 58.0%

   \[\leadsto \left(b - a\right) \cdot \left(\left(a + b\right) \cdot \color{blue}{\mathsf{log1p}\left(\mathsf{expm1}\left(\sin \left(angle \cdot \left(\pi \cdot 0.011111111111111112\right)\right)\right)\right)}\right) \]

10. Taylor expanded in angle around 0 51.1%

    \[\leadsto \color{blue}{0.011111111111111112 \cdot \left(angle \cdot \left(\left(b - a\right) \cdot \left(\pi \cdot \left(a + b\right)\right)\right)\right)} \]
11. Step-by-step derivation

    1. associate-*r* 55.4%

       \[\leadsto 0.011111111111111112 \cdot \color{blue}{\left(\left(angle \cdot \left(b - a\right)\right) \cdot \left(\pi \cdot \left(a + b\right)\right)\right)} \]

    2. *-commutative 55.4%

       \[\leadsto 0.011111111111111112 \cdot \color{blue}{\left(\left(\pi \cdot \left(a + b\right)\right) \cdot \left(angle \cdot \left(b - a\right)\right)\right)} \]

    3. +-commutative 55.4%

       \[\leadsto 0.011111111111111112 \cdot \left(\left(\pi \cdot \color{blue}{\left(b + a\right)}\right) \cdot \left(angle \cdot \left(b - a\right)\right)\right) \]

12. Simplified 55.4%

    \[\leadsto \color{blue}{0.011111111111111112 \cdot \left(\left(\pi \cdot \left(b + a\right)\right) \cdot \left(angle \cdot \left(b - a\right)\right)\right)} \]

13. Final simplification 55.4%

    \[\leadsto 0.011111111111111112 \cdot \left(\left(angle \cdot \left(b - a\right)\right) \cdot \left(\pi \cdot \left(b + a\right)\right)\right) \]

Alternative 16: 63.2% accurate, 5.5× speedup

Math

\[\begin{array}{l} \\ \left(b - a\right) \cdot \left(\left(b + a\right) \cdot \left(\left(angle \cdot \pi\right) \cdot 0.011111111111111112\right)\right) \end{array} \]

FPCore

(FPCore (a b angle)
 :precision binary64
 (* (- b a) (* (+ b a) (* (* angle PI) 0.011111111111111112))))

C

double code(double a, double b, double angle) {
	return (b - a) * ((b + a) * ((angle * ((double) M_PI)) * 0.011111111111111112));
}

Java

public static double code(double a, double b, double angle) {
	return (b - a) * ((b + a) * ((angle * Math.PI) * 0.011111111111111112));
}

Python

def code(a, b, angle):
	return (b - a) * ((b + a) * ((angle * math.pi) * 0.011111111111111112))

Julia

function code(a, b, angle)
	return Float64(Float64(b - a) * Float64(Float64(b + a) * Float64(Float64(angle * pi) * 0.011111111111111112)))
end

MATLAB

function tmp = code(a, b, angle)
	tmp = (b - a) * ((b + a) * ((angle * pi) * 0.011111111111111112));
end

Wolfram

code[a_, b_, angle_] := N[(N[(b - a), $MachinePrecision] * N[(N[(b + a), $MachinePrecision] * N[(N[(angle * Pi), $MachinePrecision] * 0.011111111111111112), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 51.0%

   \[\left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin \left(\pi \cdot \frac{angle}{180}\right)\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right) \]
2. Step-by-step derivation

   1. *-commutative 51.0%

      \[\leadsto \left(\color{blue}{\left(\left({b}^{2} - {a}^{2}\right) \cdot 2\right)} \cdot \sin \left(\pi \cdot \frac{angle}{180}\right)\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right) \]

   2. associate-*l* 51.0%

      \[\leadsto \color{blue}{\left(\left({b}^{2} - {a}^{2}\right) \cdot \left(2 \cdot \sin \left(\pi \cdot \frac{angle}{180}\right)\right)\right)} \cdot \cos \left(\pi \cdot \frac{angle}{180}\right) \]

   3. associate-*l* 51.0%

      \[\leadsto \color{blue}{\left({b}^{2} - {a}^{2}\right) \cdot \left(\left(2 \cdot \sin \left(\pi \cdot \frac{angle}{180}\right)\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right)\right)} \]

   4. unpow2 51.0%

      \[\leadsto \left(\color{blue}{b \cdot b} - {a}^{2}\right) \cdot \left(\left(2 \cdot \sin \left(\pi \cdot \frac{angle}{180}\right)\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right)\right) \]

   5. unpow2 51.0%

      \[\leadsto \left(b \cdot b - \color{blue}{a \cdot a}\right) \cdot \left(\left(2 \cdot \sin \left(\pi \cdot \frac{angle}{180}\right)\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right)\right) \]

   6. difference-of-squares 54.9%

      \[\leadsto \color{blue}{\left(\left(b + a\right) \cdot \left(b - a\right)\right)} \cdot \left(\left(2 \cdot \sin \left(\pi \cdot \frac{angle}{180}\right)\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right)\right) \]

3. Simplified 54.9%

   \[\leadsto \color{blue}{\left(\left(b + a\right) \cdot \left(b - a\right)\right) \cdot \left(\left(2 \cdot \sin \left(\pi \cdot \frac{angle}{180}\right)\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right)\right)} \]
4. Step-by-step derivation

   1. difference-of-squares 51.0%

      \[\leadsto \color{blue}{\left(b \cdot b - a \cdot a\right)} \cdot \left(\left(2 \cdot \sin \left(\pi \cdot \frac{angle}{180}\right)\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right)\right) \]

   2. *-commutative 51.0%

      \[\leadsto \color{blue}{\left(\left(2 \cdot \sin \left(\pi \cdot \frac{angle}{180}\right)\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right)\right) \cdot \left(b \cdot b - a \cdot a\right)} \]

   3. sub-neg 51.0%

      \[\leadsto \left(\left(2 \cdot \sin \left(\pi \cdot \frac{angle}{180}\right)\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right)\right) \cdot \color{blue}{\left(b \cdot b + \left(-a \cdot a\right)\right)} \]

   4. distribute-lft-in 51.0%

      \[\leadsto \color{blue}{\left(\left(2 \cdot \sin \left(\pi \cdot \frac{angle}{180}\right)\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right)\right) \cdot \left(b \cdot b\right) + \left(\left(2 \cdot \sin \left(\pi \cdot \frac{angle}{180}\right)\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right)\right) \cdot \left(-a \cdot a\right)} \]

   5. associate-*l* 51.0%

      \[\leadsto \color{blue}{\left(2 \cdot \left(\sin \left(\pi \cdot \frac{angle}{180}\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right)\right)\right)} \cdot \left(b \cdot b\right) + \left(\left(2 \cdot \sin \left(\pi \cdot \frac{angle}{180}\right)\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right)\right) \cdot \left(-a \cdot a\right) \]

   6. 2-sin 51.0%

      \[\leadsto \color{blue}{\sin \left(2 \cdot \left(\pi \cdot \frac{angle}{180}\right)\right)} \cdot \left(b \cdot b\right) + \left(\left(2 \cdot \sin \left(\pi \cdot \frac{angle}{180}\right)\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right)\right) \cdot \left(-a \cdot a\right) \]

   7. div-inv 50.9%

      \[\leadsto \sin \left(2 \cdot \left(\pi \cdot \color{blue}{\left(angle \cdot \frac{1}{180}\right)}\right)\right) \cdot \left(b \cdot b\right) + \left(\left(2 \cdot \sin \left(\pi \cdot \frac{angle}{180}\right)\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right)\right) \cdot \left(-a \cdot a\right) \]

   8. metadata-eval 50.9%

      \[\leadsto \sin \left(2 \cdot \left(\pi \cdot \left(angle \cdot \color{blue}{0.005555555555555556}\right)\right)\right) \cdot \left(b \cdot b\right) + \left(\left(2 \cdot \sin \left(\pi \cdot \frac{angle}{180}\right)\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right)\right) \cdot \left(-a \cdot a\right) \]

5. Applied egg-rr 49.6%

   \[\leadsto \color{blue}{\sin \left(2 \cdot \left(\pi \cdot \left(angle \cdot 0.005555555555555556\right)\right)\right) \cdot \left(b \cdot b\right) + \sin \left(2 \cdot \left(\pi \cdot \left(angle \cdot 0.005555555555555556\right)\right)\right) \cdot \left(-a \cdot a\right)} \]
6. Step-by-step derivation

   1. distribute-lft-out 49.6%

      \[\leadsto \color{blue}{\sin \left(2 \cdot \left(\pi \cdot \left(angle \cdot 0.005555555555555556\right)\right)\right) \cdot \left(b \cdot b + \left(-a \cdot a\right)\right)} \]

   2. sub-neg 49.6%

      \[\leadsto \sin \left(2 \cdot \left(\pi \cdot \left(angle \cdot 0.005555555555555556\right)\right)\right) \cdot \color{blue}{\left(b \cdot b - a \cdot a\right)} \]

   3. difference-of-squares 53.6%

      \[\leadsto \sin \left(2 \cdot \left(\pi \cdot \left(angle \cdot 0.005555555555555556\right)\right)\right) \cdot \color{blue}{\left(\left(b + a\right) \cdot \left(b - a\right)\right)} \]

   4. *-commutative 53.6%

      \[\leadsto \color{blue}{\left(\left(b + a\right) \cdot \left(b - a\right)\right) \cdot \sin \left(2 \cdot \left(\pi \cdot \left(angle \cdot 0.005555555555555556\right)\right)\right)} \]

   5. *-commutative 53.6%

      \[\leadsto \color{blue}{\left(\left(b - a\right) \cdot \left(b + a\right)\right)} \cdot \sin \left(2 \cdot \left(\pi \cdot \left(angle \cdot 0.005555555555555556\right)\right)\right) \]

   6. associate-*l* 58.6%

      \[\leadsto \color{blue}{\left(b - a\right) \cdot \left(\left(b + a\right) \cdot \sin \left(2 \cdot \left(\pi \cdot \left(angle \cdot 0.005555555555555556\right)\right)\right)\right)} \]

   7. +-commutative 58.6%

      \[\leadsto \left(b - a\right) \cdot \left(\color{blue}{\left(a + b\right)} \cdot \sin \left(2 \cdot \left(\pi \cdot \left(angle \cdot 0.005555555555555556\right)\right)\right)\right) \]

   8. *-commutative 58.6%

      \[\leadsto \left(b - a\right) \cdot \left(\left(a + b\right) \cdot \sin \color{blue}{\left(\left(\pi \cdot \left(angle \cdot 0.005555555555555556\right)\right) \cdot 2\right)}\right) \]

   9. associate-*r* 56.2%

      \[\leadsto \left(b - a\right) \cdot \left(\left(a + b\right) \cdot \sin \left(\color{blue}{\left(\left(\pi \cdot angle\right) \cdot 0.005555555555555556\right)} \cdot 2\right)\right) \]

   10. associate-*l* 56.2%

       \[\leadsto \left(b - a\right) \cdot \left(\left(a + b\right) \cdot \sin \color{blue}{\left(\left(\pi \cdot angle\right) \cdot \left(0.005555555555555556 \cdot 2\right)\right)}\right) \]

   11. *-commutative 56.2%

       \[\leadsto \left(b - a\right) \cdot \left(\left(a + b\right) \cdot \sin \left(\color{blue}{\left(angle \cdot \pi\right)} \cdot \left(0.005555555555555556 \cdot 2\right)\right)\right) \]

   12. metadata-eval 56.2%

       \[\leadsto \left(b - a\right) \cdot \left(\left(a + b\right) \cdot \sin \left(\left(angle \cdot \pi\right) \cdot \color{blue}{0.011111111111111112}\right)\right) \]

7. Simplified 56.2%

   \[\leadsto \color{blue}{\left(b - a\right) \cdot \left(\left(a + b\right) \cdot \sin \left(\left(angle \cdot \pi\right) \cdot 0.011111111111111112\right)\right)} \]

8. Taylor expanded in angle around 0 55.4%

   \[\leadsto \left(b - a\right) \cdot \left(\left(a + b\right) \cdot \color{blue}{\left(0.011111111111111112 \cdot \left(angle \cdot \pi\right)\right)}\right) \]

9. Final simplification 55.4%

   \[\leadsto \left(b - a\right) \cdot \left(\left(b + a\right) \cdot \left(\left(angle \cdot \pi\right) \cdot 0.011111111111111112\right)\right) \]

Alternative 17: 41.4% accurate, 5.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;a \leq 90000000000:\\ \;\;\;\;0.011111111111111112 \cdot \left(angle \cdot \left(\pi \cdot \left(b \cdot b\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(angle \cdot \left(a \cdot a\right)\right) \cdot \left(\pi \cdot -0.011111111111111112\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (a b angle)
 :precision binary64
 (if (<= a 90000000000.0)
   (* 0.011111111111111112 (* angle (* PI (* b b))))
   (* (* angle (* a a)) (* PI -0.011111111111111112))))

C

double code(double a, double b, double angle) {
	double tmp;
	if (a <= 90000000000.0) {
		tmp = 0.011111111111111112 * (angle * (((double) M_PI) * (b * b)));
	} else {
		tmp = (angle * (a * a)) * (((double) M_PI) * -0.011111111111111112);
	}
	return tmp;
}

Java

public static double code(double a, double b, double angle) {
	double tmp;
	if (a <= 90000000000.0) {
		tmp = 0.011111111111111112 * (angle * (Math.PI * (b * b)));
	} else {
		tmp = (angle * (a * a)) * (Math.PI * -0.011111111111111112);
	}
	return tmp;
}

Python

def code(a, b, angle):
	tmp = 0
	if a <= 90000000000.0:
		tmp = 0.011111111111111112 * (angle * (math.pi * (b * b)))
	else:
		tmp = (angle * (a * a)) * (math.pi * -0.011111111111111112)
	return tmp

Julia

function code(a, b, angle)
	tmp = 0.0
	if (a <= 90000000000.0)
		tmp = Float64(0.011111111111111112 * Float64(angle * Float64(pi * Float64(b * b))));
	else
		tmp = Float64(Float64(angle * Float64(a * a)) * Float64(pi * -0.011111111111111112));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, angle)
	tmp = 0.0;
	if (a <= 90000000000.0)
		tmp = 0.011111111111111112 * (angle * (pi * (b * b)));
	else
		tmp = (angle * (a * a)) * (pi * -0.011111111111111112);
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, angle_] := If[LessEqual[a, 90000000000.0], N[(0.011111111111111112 * N[(angle * N[(Pi * N[(b * b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(angle * N[(a * a), $MachinePrecision]), $MachinePrecision] * N[(Pi * -0.011111111111111112), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if a < 9e10

   1. Initial program 53.2%

      \[\left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin \left(\pi \cdot \frac{angle}{180}\right)\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right) \]
   2. Step-by-step derivation

      1. *-commutative 53.2%

         \[\leadsto \left(\color{blue}{\left(\left({b}^{2} - {a}^{2}\right) \cdot 2\right)} \cdot \sin \left(\pi \cdot \frac{angle}{180}\right)\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right) \]

      2. associate-*l* 53.2%

         \[\leadsto \color{blue}{\left(\left({b}^{2} - {a}^{2}\right) \cdot \left(2 \cdot \sin \left(\pi \cdot \frac{angle}{180}\right)\right)\right)} \cdot \cos \left(\pi \cdot \frac{angle}{180}\right) \]

      3. associate-*l* 53.2%

         \[\leadsto \color{blue}{\left({b}^{2} - {a}^{2}\right) \cdot \left(\left(2 \cdot \sin \left(\pi \cdot \frac{angle}{180}\right)\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right)\right)} \]

      4. unpow2 53.2%

         \[\leadsto \left(\color{blue}{b \cdot b} - {a}^{2}\right) \cdot \left(\left(2 \cdot \sin \left(\pi \cdot \frac{angle}{180}\right)\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right)\right) \]

      5. unpow2 53.2%

         \[\leadsto \left(b \cdot b - \color{blue}{a \cdot a}\right) \cdot \left(\left(2 \cdot \sin \left(\pi \cdot \frac{angle}{180}\right)\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right)\right) \]

      6. difference-of-squares 55.3%

         \[\leadsto \color{blue}{\left(\left(b + a\right) \cdot \left(b - a\right)\right)} \cdot \left(\left(2 \cdot \sin \left(\pi \cdot \frac{angle}{180}\right)\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right)\right) \]

   3. Simplified 55.3%

      \[\leadsto \color{blue}{\left(\left(b + a\right) \cdot \left(b - a\right)\right) \cdot \left(\left(2 \cdot \sin \left(\pi \cdot \frac{angle}{180}\right)\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right)\right)} \]

   4. Taylor expanded in angle around 0 51.8%

      \[\leadsto \color{blue}{0.011111111111111112 \cdot \left(angle \cdot \left(\left(b - a\right) \cdot \left(\pi \cdot \left(a + b\right)\right)\right)\right)} \]

   5. Taylor expanded in b around inf 37.2%

      \[\leadsto 0.011111111111111112 \cdot \left(angle \cdot \color{blue}{\left({b}^{2} \cdot \pi\right)}\right) \]
   6. Step-by-step derivation

      1. *-commutative 37.2%

         \[\leadsto 0.011111111111111112 \cdot \left(angle \cdot \color{blue}{\left(\pi \cdot {b}^{2}\right)}\right) \]

      2. unpow2 37.2%

         \[\leadsto 0.011111111111111112 \cdot \left(angle \cdot \left(\pi \cdot \color{blue}{\left(b \cdot b\right)}\right)\right) \]

   7. Simplified 37.2%

      \[\leadsto 0.011111111111111112 \cdot \left(angle \cdot \color{blue}{\left(\pi \cdot \left(b \cdot b\right)\right)}\right) \]

   if 9e10 < a

   1. Initial program 43.0%

      \[\left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin \left(\pi \cdot \frac{angle}{180}\right)\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right) \]
   2. Step-by-step derivation

      1. *-commutative 43.0%

         \[\leadsto \left(\color{blue}{\left(\left({b}^{2} - {a}^{2}\right) \cdot 2\right)} \cdot \sin \left(\pi \cdot \frac{angle}{180}\right)\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right) \]

      2. associate-*l* 43.0%

         \[\leadsto \color{blue}{\left(\left({b}^{2} - {a}^{2}\right) \cdot \left(2 \cdot \sin \left(\pi \cdot \frac{angle}{180}\right)\right)\right)} \cdot \cos \left(\pi \cdot \frac{angle}{180}\right) \]

      3. associate-*l* 43.0%

         \[\leadsto \color{blue}{\left({b}^{2} - {a}^{2}\right) \cdot \left(\left(2 \cdot \sin \left(\pi \cdot \frac{angle}{180}\right)\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right)\right)} \]

      4. unpow2 43.0%

         \[\leadsto \left(\color{blue}{b \cdot b} - {a}^{2}\right) \cdot \left(\left(2 \cdot \sin \left(\pi \cdot \frac{angle}{180}\right)\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right)\right) \]

      5. unpow2 43.0%

         \[\leadsto \left(b \cdot b - \color{blue}{a \cdot a}\right) \cdot \left(\left(2 \cdot \sin \left(\pi \cdot \frac{angle}{180}\right)\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right)\right) \]

      6. difference-of-squares 53.5%

         \[\leadsto \color{blue}{\left(\left(b + a\right) \cdot \left(b - a\right)\right)} \cdot \left(\left(2 \cdot \sin \left(\pi \cdot \frac{angle}{180}\right)\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right)\right) \]

   3. Simplified 53.5%

      \[\leadsto \color{blue}{\left(\left(b + a\right) \cdot \left(b - a\right)\right) \cdot \left(\left(2 \cdot \sin \left(\pi \cdot \frac{angle}{180}\right)\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right)\right)} \]

   4. Taylor expanded in angle around 0 48.4%

      \[\leadsto \color{blue}{0.011111111111111112 \cdot \left(angle \cdot \left(\left(b - a\right) \cdot \left(\pi \cdot \left(a + b\right)\right)\right)\right)} \]

   5. Taylor expanded in b around 0 41.4%

      \[\leadsto \color{blue}{-0.011111111111111112 \cdot \left(angle \cdot \left({a}^{2} \cdot \pi\right)\right)} \]
   6. Step-by-step derivation

      1. *-commutative 41.4%

         \[\leadsto \color{blue}{\left(angle \cdot \left({a}^{2} \cdot \pi\right)\right) \cdot -0.011111111111111112} \]

      2. associate-*r* 41.5%

         \[\leadsto \color{blue}{\left(\left(angle \cdot {a}^{2}\right) \cdot \pi\right)} \cdot -0.011111111111111112 \]

      3. associate-*l* 41.5%

         \[\leadsto \color{blue}{\left(angle \cdot {a}^{2}\right) \cdot \left(\pi \cdot -0.011111111111111112\right)} \]

      4. unpow2 41.5%

         \[\leadsto \left(angle \cdot \color{blue}{\left(a \cdot a\right)}\right) \cdot \left(\pi \cdot -0.011111111111111112\right) \]

   7. Simplified 41.5%

      \[\leadsto \color{blue}{\left(angle \cdot \left(a \cdot a\right)\right) \cdot \left(\pi \cdot -0.011111111111111112\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 38.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \leq 90000000000:\\ \;\;\;\;0.011111111111111112 \cdot \left(angle \cdot \left(\pi \cdot \left(b \cdot b\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(angle \cdot \left(a \cdot a\right)\right) \cdot \left(\pi \cdot -0.011111111111111112\right)\\ \end{array} \]

Alternative 18: 35.2% accurate, 5.7× speedup

Math

\[\begin{array}{l} \\ 0.011111111111111112 \cdot \left(angle \cdot \left(\pi \cdot \left(b \cdot b\right)\right)\right) \end{array} \]

FPCore

(FPCore (a b angle)
 :precision binary64
 (* 0.011111111111111112 (* angle (* PI (* b b)))))

C

double code(double a, double b, double angle) {
	return 0.011111111111111112 * (angle * (((double) M_PI) * (b * b)));
}

Java

public static double code(double a, double b, double angle) {
	return 0.011111111111111112 * (angle * (Math.PI * (b * b)));
}

Python

def code(a, b, angle):
	return 0.011111111111111112 * (angle * (math.pi * (b * b)))

Julia

function code(a, b, angle)
	return Float64(0.011111111111111112 * Float64(angle * Float64(pi * Float64(b * b))))
end

MATLAB

function tmp = code(a, b, angle)
	tmp = 0.011111111111111112 * (angle * (pi * (b * b)));
end

Wolfram

code[a_, b_, angle_] := N[(0.011111111111111112 * N[(angle * N[(Pi * N[(b * b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 51.0%

   \[\left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin \left(\pi \cdot \frac{angle}{180}\right)\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right) \]
2. Step-by-step derivation

   1. *-commutative 51.0%

      \[\leadsto \left(\color{blue}{\left(\left({b}^{2} - {a}^{2}\right) \cdot 2\right)} \cdot \sin \left(\pi \cdot \frac{angle}{180}\right)\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right) \]

   2. associate-*l* 51.0%

      \[\leadsto \color{blue}{\left(\left({b}^{2} - {a}^{2}\right) \cdot \left(2 \cdot \sin \left(\pi \cdot \frac{angle}{180}\right)\right)\right)} \cdot \cos \left(\pi \cdot \frac{angle}{180}\right) \]

   3. associate-*l* 51.0%

      \[\leadsto \color{blue}{\left({b}^{2} - {a}^{2}\right) \cdot \left(\left(2 \cdot \sin \left(\pi \cdot \frac{angle}{180}\right)\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right)\right)} \]

   4. unpow2 51.0%

      \[\leadsto \left(\color{blue}{b \cdot b} - {a}^{2}\right) \cdot \left(\left(2 \cdot \sin \left(\pi \cdot \frac{angle}{180}\right)\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right)\right) \]

   5. unpow2 51.0%

      \[\leadsto \left(b \cdot b - \color{blue}{a \cdot a}\right) \cdot \left(\left(2 \cdot \sin \left(\pi \cdot \frac{angle}{180}\right)\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right)\right) \]

   6. difference-of-squares 54.9%

      \[\leadsto \color{blue}{\left(\left(b + a\right) \cdot \left(b - a\right)\right)} \cdot \left(\left(2 \cdot \sin \left(\pi \cdot \frac{angle}{180}\right)\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right)\right) \]

3. Simplified 54.9%

   \[\leadsto \color{blue}{\left(\left(b + a\right) \cdot \left(b - a\right)\right) \cdot \left(\left(2 \cdot \sin \left(\pi \cdot \frac{angle}{180}\right)\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right)\right)} \]

4. Taylor expanded in angle around 0 51.1%

   \[\leadsto \color{blue}{0.011111111111111112 \cdot \left(angle \cdot \left(\left(b - a\right) \cdot \left(\pi \cdot \left(a + b\right)\right)\right)\right)} \]

5. Taylor expanded in b around inf 33.2%

   \[\leadsto 0.011111111111111112 \cdot \left(angle \cdot \color{blue}{\left({b}^{2} \cdot \pi\right)}\right) \]
6. Step-by-step derivation

   1. *-commutative 33.2%

      \[\leadsto 0.011111111111111112 \cdot \left(angle \cdot \color{blue}{\left(\pi \cdot {b}^{2}\right)}\right) \]

   2. unpow2 33.2%

      \[\leadsto 0.011111111111111112 \cdot \left(angle \cdot \left(\pi \cdot \color{blue}{\left(b \cdot b\right)}\right)\right) \]

7. Simplified 33.2%

   \[\leadsto 0.011111111111111112 \cdot \left(angle \cdot \color{blue}{\left(\pi \cdot \left(b \cdot b\right)\right)}\right) \]

8. Final simplification 33.2%

   \[\leadsto 0.011111111111111112 \cdot \left(angle \cdot \left(\pi \cdot \left(b \cdot b\right)\right)\right) \]

Alternative 19: 35.2% accurate, 5.7× speedup

Math

\[\begin{array}{l} \\ 0.011111111111111112 \cdot \left(\left(angle \cdot \pi\right) \cdot \left(b \cdot b\right)\right) \end{array} \]

FPCore

(FPCore (a b angle)
 :precision binary64
 (* 0.011111111111111112 (* (* angle PI) (* b b))))

C

double code(double a, double b, double angle) {
	return 0.011111111111111112 * ((angle * ((double) M_PI)) * (b * b));
}

Java

public static double code(double a, double b, double angle) {
	return 0.011111111111111112 * ((angle * Math.PI) * (b * b));
}

Python

def code(a, b, angle):
	return 0.011111111111111112 * ((angle * math.pi) * (b * b))

Julia

function code(a, b, angle)
	return Float64(0.011111111111111112 * Float64(Float64(angle * pi) * Float64(b * b)))
end

MATLAB

function tmp = code(a, b, angle)
	tmp = 0.011111111111111112 * ((angle * pi) * (b * b));
end

Wolfram

code[a_, b_, angle_] := N[(0.011111111111111112 * N[(N[(angle * Pi), $MachinePrecision] * N[(b * b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 51.0%

   \[\left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin \left(\pi \cdot \frac{angle}{180}\right)\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right) \]
2. Step-by-step derivation

   1. *-commutative 51.0%

      \[\leadsto \left(\color{blue}{\left(\left({b}^{2} - {a}^{2}\right) \cdot 2\right)} \cdot \sin \left(\pi \cdot \frac{angle}{180}\right)\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right) \]

   2. associate-*l* 51.0%

      \[\leadsto \color{blue}{\left(\left({b}^{2} - {a}^{2}\right) \cdot \left(2 \cdot \sin \left(\pi \cdot \frac{angle}{180}\right)\right)\right)} \cdot \cos \left(\pi \cdot \frac{angle}{180}\right) \]

   3. associate-*l* 51.0%

      \[\leadsto \color{blue}{\left({b}^{2} - {a}^{2}\right) \cdot \left(\left(2 \cdot \sin \left(\pi \cdot \frac{angle}{180}\right)\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right)\right)} \]

   4. unpow2 51.0%

      \[\leadsto \left(\color{blue}{b \cdot b} - {a}^{2}\right) \cdot \left(\left(2 \cdot \sin \left(\pi \cdot \frac{angle}{180}\right)\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right)\right) \]

   5. unpow2 51.0%

      \[\leadsto \left(b \cdot b - \color{blue}{a \cdot a}\right) \cdot \left(\left(2 \cdot \sin \left(\pi \cdot \frac{angle}{180}\right)\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right)\right) \]

   6. difference-of-squares 54.9%

      \[\leadsto \color{blue}{\left(\left(b + a\right) \cdot \left(b - a\right)\right)} \cdot \left(\left(2 \cdot \sin \left(\pi \cdot \frac{angle}{180}\right)\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right)\right) \]

3. Simplified 54.9%

   \[\leadsto \color{blue}{\left(\left(b + a\right) \cdot \left(b - a\right)\right) \cdot \left(\left(2 \cdot \sin \left(\pi \cdot \frac{angle}{180}\right)\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right)\right)} \]

4. Taylor expanded in angle around 0 51.1%

   \[\leadsto \color{blue}{0.011111111111111112 \cdot \left(angle \cdot \left(\left(b - a\right) \cdot \left(\pi \cdot \left(a + b\right)\right)\right)\right)} \]

5. Taylor expanded in b around inf 33.2%

   \[\leadsto 0.011111111111111112 \cdot \color{blue}{\left(angle \cdot \left({b}^{2} \cdot \pi\right)\right)} \]
6. Step-by-step derivation

   1. associate-*r* 33.2%

      \[\leadsto 0.011111111111111112 \cdot \color{blue}{\left(\left(angle \cdot {b}^{2}\right) \cdot \pi\right)} \]

   2. unpow2 33.2%

      \[\leadsto 0.011111111111111112 \cdot \left(\left(angle \cdot \color{blue}{\left(b \cdot b\right)}\right) \cdot \pi\right) \]

7. Simplified 33.2%

   \[\leadsto 0.011111111111111112 \cdot \color{blue}{\left(\left(angle \cdot \left(b \cdot b\right)\right) \cdot \pi\right)} \]

8. Taylor expanded in angle around 0 33.2%

   \[\leadsto 0.011111111111111112 \cdot \color{blue}{\left(angle \cdot \left({b}^{2} \cdot \pi\right)\right)} \]
9. Step-by-step derivation

   1. *-commutative 33.2%

      \[\leadsto 0.011111111111111112 \cdot \left(angle \cdot \color{blue}{\left(\pi \cdot {b}^{2}\right)}\right) \]

   2. associate-*r* 33.2%

      \[\leadsto 0.011111111111111112 \cdot \color{blue}{\left(\left(angle \cdot \pi\right) \cdot {b}^{2}\right)} \]

   3. unpow2 33.2%

      \[\leadsto 0.011111111111111112 \cdot \left(\left(angle \cdot \pi\right) \cdot \color{blue}{\left(b \cdot b\right)}\right) \]

10. Simplified 33.2%

    \[\leadsto 0.011111111111111112 \cdot \color{blue}{\left(\left(angle \cdot \pi\right) \cdot \left(b \cdot b\right)\right)} \]

11. Final simplification 33.2%

    \[\leadsto 0.011111111111111112 \cdot \left(\left(angle \cdot \pi\right) \cdot \left(b \cdot b\right)\right) \]

Reproduce

herbie shell --seed 2023274 
(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)))))