ab-angle->ABCF B

Percentage Accurate: 54.0% → 65.8%

Time: 42.6s

Alternatives: 21

Speedup: 5.5×

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

Specification

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 54.0% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 65.8% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(a - b\right) \cdot \left(b + a\right)\\ t_1 := \pi \cdot \frac{angle}{180}\\ t_2 := \sin t_1\\ t_3 := 2 \cdot \sin \left(angle \cdot \frac{{\left(\sqrt{\pi}\right)}^{2}}{-180}\right)\\ \mathbf{if}\;\frac{angle}{180} \leq -5 \cdot 10^{+69}:\\ \;\;\;\;t_3 \cdot t_0\\ \mathbf{elif}\;\frac{angle}{180} \leq 2 \cdot 10^{-54}:\\ \;\;\;\;-0.011111111111111112 \cdot \left(\left(a - b\right) \cdot \left(\left(b + a\right) \cdot \left(\pi \cdot angle\right)\right)\right)\\ \mathbf{elif}\;\frac{angle}{180} \leq 7 \cdot 10^{+49}:\\ \;\;\;\;\left(\left(2 \cdot \left(\left(b - a\right) \cdot \left(b + a\right)\right)\right) \cdot t_2\right) \cdot \cos \left(\left(\pi \cdot angle\right) \cdot 0.005555555555555556\right)\\ \mathbf{elif}\;\frac{angle}{180} \leq 5 \cdot 10^{+64}:\\ \;\;\;\;\left({a}^{2} - {b}^{2}\right) \cdot \left(\cos t_1 \cdot \left(2 \cdot t_2\right)\right)\\ \mathbf{elif}\;\frac{angle}{180} \leq 4 \cdot 10^{+222}:\\ \;\;\;\;t_0 \cdot \left(2 \cdot \sin \left(angle \cdot \frac{\sqrt[3]{{\pi}^{3}}}{-180}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;t_3 \cdot \left(t_0 \cdot \cos \left(angle \cdot \frac{\pi}{-180}\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (a b angle)
 :precision binary64
 (let* ((t_0 (* (- a b) (+ b a)))
        (t_1 (* PI (/ angle 180.0)))
        (t_2 (sin t_1))
        (t_3 (* 2.0 (sin (* angle (/ (pow (sqrt PI) 2.0) -180.0))))))
   (if (<= (/ angle 180.0) -5e+69)
     (* t_3 t_0)
     (if (<= (/ angle 180.0) 2e-54)
       (* -0.011111111111111112 (* (- a b) (* (+ b a) (* PI angle))))
       (if (<= (/ angle 180.0) 7e+49)
         (*
          (* (* 2.0 (* (- b a) (+ b a))) t_2)
          (cos (* (* PI angle) 0.005555555555555556)))
         (if (<= (/ angle 180.0) 5e+64)
           (* (- (pow a 2.0) (pow b 2.0)) (* (cos t_1) (* 2.0 t_2)))
           (if (<= (/ angle 180.0) 4e+222)
             (* t_0 (* 2.0 (sin (* angle (/ (cbrt (pow PI 3.0)) -180.0)))))
             (* t_3 (* t_0 (cos (* angle (/ PI -180.0))))))))))))

C

double code(double a, double b, double angle) {
	double t_0 = (a - b) * (b + a);
	double t_1 = ((double) M_PI) * (angle / 180.0);
	double t_2 = sin(t_1);
	double t_3 = 2.0 * sin((angle * (pow(sqrt(((double) M_PI)), 2.0) / -180.0)));
	double tmp;
	if ((angle / 180.0) <= -5e+69) {
		tmp = t_3 * t_0;
	} else if ((angle / 180.0) <= 2e-54) {
		tmp = -0.011111111111111112 * ((a - b) * ((b + a) * (((double) M_PI) * angle)));
	} else if ((angle / 180.0) <= 7e+49) {
		tmp = ((2.0 * ((b - a) * (b + a))) * t_2) * cos(((((double) M_PI) * angle) * 0.005555555555555556));
	} else if ((angle / 180.0) <= 5e+64) {
		tmp = (pow(a, 2.0) - pow(b, 2.0)) * (cos(t_1) * (2.0 * t_2));
	} else if ((angle / 180.0) <= 4e+222) {
		tmp = t_0 * (2.0 * sin((angle * (cbrt(pow(((double) M_PI), 3.0)) / -180.0))));
	} else {
		tmp = t_3 * (t_0 * cos((angle * (((double) M_PI) / -180.0))));
	}
	return tmp;
}

Java

public static double code(double a, double b, double angle) {
	double t_0 = (a - b) * (b + a);
	double t_1 = Math.PI * (angle / 180.0);
	double t_2 = Math.sin(t_1);
	double t_3 = 2.0 * Math.sin((angle * (Math.pow(Math.sqrt(Math.PI), 2.0) / -180.0)));
	double tmp;
	if ((angle / 180.0) <= -5e+69) {
		tmp = t_3 * t_0;
	} else if ((angle / 180.0) <= 2e-54) {
		tmp = -0.011111111111111112 * ((a - b) * ((b + a) * (Math.PI * angle)));
	} else if ((angle / 180.0) <= 7e+49) {
		tmp = ((2.0 * ((b - a) * (b + a))) * t_2) * Math.cos(((Math.PI * angle) * 0.005555555555555556));
	} else if ((angle / 180.0) <= 5e+64) {
		tmp = (Math.pow(a, 2.0) - Math.pow(b, 2.0)) * (Math.cos(t_1) * (2.0 * t_2));
	} else if ((angle / 180.0) <= 4e+222) {
		tmp = t_0 * (2.0 * Math.sin((angle * (Math.cbrt(Math.pow(Math.PI, 3.0)) / -180.0))));
	} else {
		tmp = t_3 * (t_0 * Math.cos((angle * (Math.PI / -180.0))));
	}
	return tmp;
}

Julia

function code(a, b, angle)
	t_0 = Float64(Float64(a - b) * Float64(b + a))
	t_1 = Float64(pi * Float64(angle / 180.0))
	t_2 = sin(t_1)
	t_3 = Float64(2.0 * sin(Float64(angle * Float64((sqrt(pi) ^ 2.0) / -180.0))))
	tmp = 0.0
	if (Float64(angle / 180.0) <= -5e+69)
		tmp = Float64(t_3 * t_0);
	elseif (Float64(angle / 180.0) <= 2e-54)
		tmp = Float64(-0.011111111111111112 * Float64(Float64(a - b) * Float64(Float64(b + a) * Float64(pi * angle))));
	elseif (Float64(angle / 180.0) <= 7e+49)
		tmp = Float64(Float64(Float64(2.0 * Float64(Float64(b - a) * Float64(b + a))) * t_2) * cos(Float64(Float64(pi * angle) * 0.005555555555555556)));
	elseif (Float64(angle / 180.0) <= 5e+64)
		tmp = Float64(Float64((a ^ 2.0) - (b ^ 2.0)) * Float64(cos(t_1) * Float64(2.0 * t_2)));
	elseif (Float64(angle / 180.0) <= 4e+222)
		tmp = Float64(t_0 * Float64(2.0 * sin(Float64(angle * Float64(cbrt((pi ^ 3.0)) / -180.0)))));
	else
		tmp = Float64(t_3 * Float64(t_0 * cos(Float64(angle * Float64(pi / -180.0)))));
	end
	return tmp
end

Wolfram

code[a_, b_, angle_] := Block[{t$95$0 = N[(N[(a - b), $MachinePrecision] * N[(b + a), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(Pi * N[(angle / 180.0), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[Sin[t$95$1], $MachinePrecision]}, Block[{t$95$3 = N[(2.0 * N[Sin[N[(angle * N[(N[Power[N[Sqrt[Pi], $MachinePrecision], 2.0], $MachinePrecision] / -180.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[(angle / 180.0), $MachinePrecision], -5e+69], N[(t$95$3 * t$95$0), $MachinePrecision], If[LessEqual[N[(angle / 180.0), $MachinePrecision], 2e-54], N[(-0.011111111111111112 * N[(N[(a - b), $MachinePrecision] * N[(N[(b + a), $MachinePrecision] * N[(Pi * angle), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[N[(angle / 180.0), $MachinePrecision], 7e+49], N[(N[(N[(2.0 * N[(N[(b - a), $MachinePrecision] * N[(b + a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * t$95$2), $MachinePrecision] * N[Cos[N[(N[(Pi * angle), $MachinePrecision] * 0.005555555555555556), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[N[(angle / 180.0), $MachinePrecision], 5e+64], N[(N[(N[Power[a, 2.0], $MachinePrecision] - N[Power[b, 2.0], $MachinePrecision]), $MachinePrecision] * N[(N[Cos[t$95$1], $MachinePrecision] * N[(2.0 * t$95$2), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[N[(angle / 180.0), $MachinePrecision], 4e+222], N[(t$95$0 * N[(2.0 * N[Sin[N[(angle * N[(N[Power[N[Power[Pi, 3.0], $MachinePrecision], 1/3], $MachinePrecision] / -180.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(t$95$3 * N[(t$95$0 * N[Cos[N[(angle * N[(Pi / -180.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]]]]]]

Derivation

1. Split input into 6 regimes

2. if (/.f64 angle 180) < -5.00000000000000036e69

   1. Initial program 29.3%

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

   3. Simplified 26.4%

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

      1. unpow2 26.4%

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

      2. unpow2 26.4%

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

      3. difference-of-squares 26.4%

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

   5. Applied egg-rr 26.4%

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

      1. add-cbrt-cube 34.9%

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

      2. pow3 34.9%

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

   7. Applied egg-rr 34.9%

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

   8. Taylor expanded in angle around 0 31.9%

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

      1. rem-cbrt-cube 37.6%

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

      2. add-sqr-sqrt 44.1%

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

      3. pow2 44.1%

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

   10. Applied egg-rr 44.1%

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

   if -5.00000000000000036e69 < (/.f64 angle 180) < 2.0000000000000001e-54

   1. Initial program 73.0%

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

   3. Simplified 73.0%

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

      1. unpow2 73.0%

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

      2. unpow2 73.0%

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

      3. difference-of-squares 76.3%

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

   5. Applied egg-rr 76.3%

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

      1. add-cbrt-cube 75.3%

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

      2. pow3 75.3%

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

   7. Applied egg-rr 75.3%

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

   8. Taylor expanded in angle around 0 74.6%

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

      1. associate-*r* 74.5%

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

      2. associate-*r* 91.1%

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

      3. *-commutative 91.1%

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

      4. *-commutative 91.1%

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

   10. Simplified 91.1%

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

   if 2.0000000000000001e-54 < (/.f64 angle 180) < 6.9999999999999995e49

   1. Initial program 67.0%

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

      1. unpow2 67.0%

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

      2. unpow2 67.0%

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

      3. difference-of-squares 75.4%

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

   3. Applied egg-rr 75.4%

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

   4. Taylor expanded in angle around inf 87.5%

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

   if 6.9999999999999995e49 < (/.f64 angle 180) < 5e64

   1. Initial program 5.2%

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

      1. *-commutative 5.2%

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

      2. associate-*l* 5.2%

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

      3. associate-*l* 5.2%

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

   3. Simplified 5.2%

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

   4. Taylor expanded in b around 0 5.2%

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

      1. neg-mul-1 5.2%

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

      2. +-commutative 5.2%

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

      3. sub-neg 5.2%

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

      4. rem-square-sqrt 1.5%

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

      5. fabs-sqr 1.5%

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

      6. rem-square-sqrt 11.0%

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

      7. fabs-sub 11.0%

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

      8. rem-square-sqrt 9.6%

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

      9. fabs-sqr 9.6%

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

      10. rem-square-sqrt 39.2%

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

   6. Simplified 39.2%

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

   if 5e64 < (/.f64 angle 180) < 4.0000000000000002e222

   1. Initial program 21.4%

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

   3. Simplified 31.0%

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

      1. unpow2 31.0%

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

      2. unpow2 31.0%

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

      3. difference-of-squares 34.2%

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

   5. Applied egg-rr 34.2%

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

      1. add-cbrt-cube 38.6%

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

      2. pow3 38.6%

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

   7. Applied egg-rr 38.6%

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

   8. Taylor expanded in angle around 0 42.8%

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

   if 4.0000000000000002e222 < (/.f64 angle 180)

   1. Initial program 23.5%

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

   3. Simplified 33.0%

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

      1. unpow2 33.0%

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

      2. unpow2 33.0%

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

      3. difference-of-squares 33.0%

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

   5. Applied egg-rr 33.0%

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

      1. add-sqr-sqrt 43.3%

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

      2. pow2 43.3%

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

   7. Applied egg-rr 43.3%

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

4. Final simplification 71.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{angle}{180} \leq -5 \cdot 10^{+69}:\\ \;\;\;\;\left(2 \cdot \sin \left(angle \cdot \frac{{\left(\sqrt{\pi}\right)}^{2}}{-180}\right)\right) \cdot \left(\left(a - b\right) \cdot \left(b + a\right)\right)\\ \mathbf{elif}\;\frac{angle}{180} \leq 2 \cdot 10^{-54}:\\ \;\;\;\;-0.011111111111111112 \cdot \left(\left(a - b\right) \cdot \left(\left(b + a\right) \cdot \left(\pi \cdot angle\right)\right)\right)\\ \mathbf{elif}\;\frac{angle}{180} \leq 7 \cdot 10^{+49}:\\ \;\;\;\;\left(\left(2 \cdot \left(\left(b - a\right) \cdot \left(b + a\right)\right)\right) \cdot \sin \left(\pi \cdot \frac{angle}{180}\right)\right) \cdot \cos \left(\left(\pi \cdot angle\right) \cdot 0.005555555555555556\right)\\ \mathbf{elif}\;\frac{angle}{180} \leq 5 \cdot 10^{+64}:\\ \;\;\;\;\left({a}^{2} - {b}^{2}\right) \cdot \left(\cos \left(\pi \cdot \frac{angle}{180}\right) \cdot \left(2 \cdot \sin \left(\pi \cdot \frac{angle}{180}\right)\right)\right)\\ \mathbf{elif}\;\frac{angle}{180} \leq 4 \cdot 10^{+222}:\\ \;\;\;\;\left(\left(a - b\right) \cdot \left(b + a\right)\right) \cdot \left(2 \cdot \sin \left(angle \cdot \frac{\sqrt[3]{{\pi}^{3}}}{-180}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(2 \cdot \sin \left(angle \cdot \frac{{\left(\sqrt{\pi}\right)}^{2}}{-180}\right)\right) \cdot \left(\left(\left(a - b\right) \cdot \left(b + a\right)\right) \cdot \cos \left(angle \cdot \frac{\pi}{-180}\right)\right)\\ \end{array} \]

Alternative 2: 65.7% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(b - a\right) \cdot \left(b + a\right)\\ t_1 := \pi \cdot \frac{angle}{180}\\ t_2 := \sin t_1\\ t_3 := \left(\pi \cdot angle\right) \cdot 0.005555555555555556\\ t_4 := 2 \cdot t_0\\ t_5 := \cos t_1\\ \mathbf{if}\;\frac{angle}{180} \leq -5 \cdot 10^{+69}:\\ \;\;\;\;\left(2 \cdot \sin \left(angle \cdot \frac{{\left(\sqrt{\pi}\right)}^{2}}{-180}\right)\right) \cdot \left(\left(a - b\right) \cdot \left(b + a\right)\right)\\ \mathbf{elif}\;\frac{angle}{180} \leq 2 \cdot 10^{-54}:\\ \;\;\;\;-0.011111111111111112 \cdot \left(\left(a - b\right) \cdot \left(\left(b + a\right) \cdot \left(\pi \cdot angle\right)\right)\right)\\ \mathbf{elif}\;\frac{angle}{180} \leq 7 \cdot 10^{+49}:\\ \;\;\;\;\left(t_4 \cdot t_2\right) \cdot \cos t_3\\ \mathbf{elif}\;\frac{angle}{180} \leq 5 \cdot 10^{+64}:\\ \;\;\;\;\left({a}^{2} - {b}^{2}\right) \cdot \left(t_5 \cdot \left(2 \cdot t_2\right)\right)\\ \mathbf{elif}\;\frac{angle}{180} \leq 6 \cdot 10^{+162}:\\ \;\;\;\;t_5 \cdot \left(0.011111111111111112 \cdot \left(angle \cdot \left(\pi \cdot t_0\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;t_5 \cdot \left(t_4 \cdot \sin t_3\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (a b angle)
 :precision binary64
 (let* ((t_0 (* (- b a) (+ b a)))
        (t_1 (* PI (/ angle 180.0)))
        (t_2 (sin t_1))
        (t_3 (* (* PI angle) 0.005555555555555556))
        (t_4 (* 2.0 t_0))
        (t_5 (cos t_1)))
   (if (<= (/ angle 180.0) -5e+69)
     (*
      (* 2.0 (sin (* angle (/ (pow (sqrt PI) 2.0) -180.0))))
      (* (- a b) (+ b a)))
     (if (<= (/ angle 180.0) 2e-54)
       (* -0.011111111111111112 (* (- a b) (* (+ b a) (* PI angle))))
       (if (<= (/ angle 180.0) 7e+49)
         (* (* t_4 t_2) (cos t_3))
         (if (<= (/ angle 180.0) 5e+64)
           (* (- (pow a 2.0) (pow b 2.0)) (* t_5 (* 2.0 t_2)))
           (if (<= (/ angle 180.0) 6e+162)
             (* t_5 (* 0.011111111111111112 (* angle (* PI t_0))))
             (* t_5 (* t_4 (sin t_3))))))))))

C

double code(double a, double b, double angle) {
	double t_0 = (b - a) * (b + a);
	double t_1 = ((double) M_PI) * (angle / 180.0);
	double t_2 = sin(t_1);
	double t_3 = (((double) M_PI) * angle) * 0.005555555555555556;
	double t_4 = 2.0 * t_0;
	double t_5 = cos(t_1);
	double tmp;
	if ((angle / 180.0) <= -5e+69) {
		tmp = (2.0 * sin((angle * (pow(sqrt(((double) M_PI)), 2.0) / -180.0)))) * ((a - b) * (b + a));
	} else if ((angle / 180.0) <= 2e-54) {
		tmp = -0.011111111111111112 * ((a - b) * ((b + a) * (((double) M_PI) * angle)));
	} else if ((angle / 180.0) <= 7e+49) {
		tmp = (t_4 * t_2) * cos(t_3);
	} else if ((angle / 180.0) <= 5e+64) {
		tmp = (pow(a, 2.0) - pow(b, 2.0)) * (t_5 * (2.0 * t_2));
	} else if ((angle / 180.0) <= 6e+162) {
		tmp = t_5 * (0.011111111111111112 * (angle * (((double) M_PI) * t_0)));
	} else {
		tmp = t_5 * (t_4 * sin(t_3));
	}
	return tmp;
}

Java

public static double code(double a, double b, double angle) {
	double t_0 = (b - a) * (b + a);
	double t_1 = Math.PI * (angle / 180.0);
	double t_2 = Math.sin(t_1);
	double t_3 = (Math.PI * angle) * 0.005555555555555556;
	double t_4 = 2.0 * t_0;
	double t_5 = Math.cos(t_1);
	double tmp;
	if ((angle / 180.0) <= -5e+69) {
		tmp = (2.0 * Math.sin((angle * (Math.pow(Math.sqrt(Math.PI), 2.0) / -180.0)))) * ((a - b) * (b + a));
	} else if ((angle / 180.0) <= 2e-54) {
		tmp = -0.011111111111111112 * ((a - b) * ((b + a) * (Math.PI * angle)));
	} else if ((angle / 180.0) <= 7e+49) {
		tmp = (t_4 * t_2) * Math.cos(t_3);
	} else if ((angle / 180.0) <= 5e+64) {
		tmp = (Math.pow(a, 2.0) - Math.pow(b, 2.0)) * (t_5 * (2.0 * t_2));
	} else if ((angle / 180.0) <= 6e+162) {
		tmp = t_5 * (0.011111111111111112 * (angle * (Math.PI * t_0)));
	} else {
		tmp = t_5 * (t_4 * Math.sin(t_3));
	}
	return tmp;
}

Python

def code(a, b, angle):
	t_0 = (b - a) * (b + a)
	t_1 = math.pi * (angle / 180.0)
	t_2 = math.sin(t_1)
	t_3 = (math.pi * angle) * 0.005555555555555556
	t_4 = 2.0 * t_0
	t_5 = math.cos(t_1)
	tmp = 0
	if (angle / 180.0) <= -5e+69:
		tmp = (2.0 * math.sin((angle * (math.pow(math.sqrt(math.pi), 2.0) / -180.0)))) * ((a - b) * (b + a))
	elif (angle / 180.0) <= 2e-54:
		tmp = -0.011111111111111112 * ((a - b) * ((b + a) * (math.pi * angle)))
	elif (angle / 180.0) <= 7e+49:
		tmp = (t_4 * t_2) * math.cos(t_3)
	elif (angle / 180.0) <= 5e+64:
		tmp = (math.pow(a, 2.0) - math.pow(b, 2.0)) * (t_5 * (2.0 * t_2))
	elif (angle / 180.0) <= 6e+162:
		tmp = t_5 * (0.011111111111111112 * (angle * (math.pi * t_0)))
	else:
		tmp = t_5 * (t_4 * math.sin(t_3))
	return tmp

Julia

function code(a, b, angle)
	t_0 = Float64(Float64(b - a) * Float64(b + a))
	t_1 = Float64(pi * Float64(angle / 180.0))
	t_2 = sin(t_1)
	t_3 = Float64(Float64(pi * angle) * 0.005555555555555556)
	t_4 = Float64(2.0 * t_0)
	t_5 = cos(t_1)
	tmp = 0.0
	if (Float64(angle / 180.0) <= -5e+69)
		tmp = Float64(Float64(2.0 * sin(Float64(angle * Float64((sqrt(pi) ^ 2.0) / -180.0)))) * Float64(Float64(a - b) * Float64(b + a)));
	elseif (Float64(angle / 180.0) <= 2e-54)
		tmp = Float64(-0.011111111111111112 * Float64(Float64(a - b) * Float64(Float64(b + a) * Float64(pi * angle))));
	elseif (Float64(angle / 180.0) <= 7e+49)
		tmp = Float64(Float64(t_4 * t_2) * cos(t_3));
	elseif (Float64(angle / 180.0) <= 5e+64)
		tmp = Float64(Float64((a ^ 2.0) - (b ^ 2.0)) * Float64(t_5 * Float64(2.0 * t_2)));
	elseif (Float64(angle / 180.0) <= 6e+162)
		tmp = Float64(t_5 * Float64(0.011111111111111112 * Float64(angle * Float64(pi * t_0))));
	else
		tmp = Float64(t_5 * Float64(t_4 * sin(t_3)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, angle)
	t_0 = (b - a) * (b + a);
	t_1 = pi * (angle / 180.0);
	t_2 = sin(t_1);
	t_3 = (pi * angle) * 0.005555555555555556;
	t_4 = 2.0 * t_0;
	t_5 = cos(t_1);
	tmp = 0.0;
	if ((angle / 180.0) <= -5e+69)
		tmp = (2.0 * sin((angle * ((sqrt(pi) ^ 2.0) / -180.0)))) * ((a - b) * (b + a));
	elseif ((angle / 180.0) <= 2e-54)
		tmp = -0.011111111111111112 * ((a - b) * ((b + a) * (pi * angle)));
	elseif ((angle / 180.0) <= 7e+49)
		tmp = (t_4 * t_2) * cos(t_3);
	elseif ((angle / 180.0) <= 5e+64)
		tmp = ((a ^ 2.0) - (b ^ 2.0)) * (t_5 * (2.0 * t_2));
	elseif ((angle / 180.0) <= 6e+162)
		tmp = t_5 * (0.011111111111111112 * (angle * (pi * t_0)));
	else
		tmp = t_5 * (t_4 * sin(t_3));
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, angle_] := Block[{t$95$0 = N[(N[(b - a), $MachinePrecision] * N[(b + a), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(Pi * N[(angle / 180.0), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[Sin[t$95$1], $MachinePrecision]}, Block[{t$95$3 = N[(N[(Pi * angle), $MachinePrecision] * 0.005555555555555556), $MachinePrecision]}, Block[{t$95$4 = N[(2.0 * t$95$0), $MachinePrecision]}, Block[{t$95$5 = N[Cos[t$95$1], $MachinePrecision]}, If[LessEqual[N[(angle / 180.0), $MachinePrecision], -5e+69], N[(N[(2.0 * N[Sin[N[(angle * N[(N[Power[N[Sqrt[Pi], $MachinePrecision], 2.0], $MachinePrecision] / -180.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[(N[(a - b), $MachinePrecision] * N[(b + a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[N[(angle / 180.0), $MachinePrecision], 2e-54], N[(-0.011111111111111112 * N[(N[(a - b), $MachinePrecision] * N[(N[(b + a), $MachinePrecision] * N[(Pi * angle), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[N[(angle / 180.0), $MachinePrecision], 7e+49], N[(N[(t$95$4 * t$95$2), $MachinePrecision] * N[Cos[t$95$3], $MachinePrecision]), $MachinePrecision], If[LessEqual[N[(angle / 180.0), $MachinePrecision], 5e+64], N[(N[(N[Power[a, 2.0], $MachinePrecision] - N[Power[b, 2.0], $MachinePrecision]), $MachinePrecision] * N[(t$95$5 * N[(2.0 * t$95$2), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[N[(angle / 180.0), $MachinePrecision], 6e+162], N[(t$95$5 * N[(0.011111111111111112 * N[(angle * N[(Pi * t$95$0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(t$95$5 * N[(t$95$4 * N[Sin[t$95$3], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]]]]]]]]

Derivation

1. Split input into 6 regimes

2. if (/.f64 angle 180) < -5.00000000000000036e69

   1. Initial program 29.3%

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

   3. Simplified 26.4%

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

      1. unpow2 26.4%

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

      2. unpow2 26.4%

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

      3. difference-of-squares 26.4%

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

   5. Applied egg-rr 26.4%

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

      1. add-cbrt-cube 34.9%

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

      2. pow3 34.9%

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

   7. Applied egg-rr 34.9%

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

   8. Taylor expanded in angle around 0 31.9%

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

      1. rem-cbrt-cube 37.6%

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

      2. add-sqr-sqrt 44.1%

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

      3. pow2 44.1%

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

   10. Applied egg-rr 44.1%

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

   if -5.00000000000000036e69 < (/.f64 angle 180) < 2.0000000000000001e-54

   1. Initial program 73.0%

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

   3. Simplified 73.0%

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

      1. unpow2 73.0%

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

      2. unpow2 73.0%

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

      3. difference-of-squares 76.3%

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

   5. Applied egg-rr 76.3%

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

      1. add-cbrt-cube 75.3%

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

      2. pow3 75.3%

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

   7. Applied egg-rr 75.3%

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

   8. Taylor expanded in angle around 0 74.6%

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

      1. associate-*r* 74.5%

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

      2. associate-*r* 91.1%

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

      3. *-commutative 91.1%

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

      4. *-commutative 91.1%

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

   10. Simplified 91.1%

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

   if 2.0000000000000001e-54 < (/.f64 angle 180) < 6.9999999999999995e49

   1. Initial program 67.0%

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

      1. unpow2 67.0%

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

      2. unpow2 67.0%

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

      3. difference-of-squares 75.4%

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

   3. Applied egg-rr 75.4%

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

   4. Taylor expanded in angle around inf 87.5%

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

   if 6.9999999999999995e49 < (/.f64 angle 180) < 5e64

   1. Initial program 5.2%

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

      1. *-commutative 5.2%

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

      2. associate-*l* 5.2%

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

      3. associate-*l* 5.2%

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

   3. Simplified 5.2%

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

   4. Taylor expanded in b around 0 5.2%

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

      1. neg-mul-1 5.2%

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

      2. +-commutative 5.2%

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

      3. sub-neg 5.2%

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

      4. rem-square-sqrt 1.5%

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

      5. fabs-sqr 1.5%

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

      6. rem-square-sqrt 11.0%

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

      7. fabs-sub 11.0%

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

      8. rem-square-sqrt 9.6%

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

      9. fabs-sqr 9.6%

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

      10. rem-square-sqrt 39.2%

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

   6. Simplified 39.2%

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

   if 5e64 < (/.f64 angle 180) < 5.9999999999999996e162

   1. Initial program 19.6%

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

      1. unpow2 19.6%

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

      2. unpow2 19.6%

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

      3. difference-of-squares 19.6%

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

   3. Applied egg-rr 19.6%

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

   4. Taylor expanded in angle around 0 48.7%

      \[\leadsto \color{blue}{\left(0.011111111111111112 \cdot \left(angle \cdot \left(\pi \cdot \left(\left(a + b\right) \cdot \left(b - a\right)\right)\right)\right)\right)} \cdot \cos \left(\pi \cdot \frac{angle}{180}\right) \]

   if 5.9999999999999996e162 < (/.f64 angle 180)

   1. Initial program 23.8%

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

      1. unpow2 23.8%

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

      2. unpow2 23.8%

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

      3. difference-of-squares 27.5%

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

   3. Applied egg-rr 27.5%

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

   4. Taylor expanded in angle around inf 38.5%

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

4. Final simplification 71.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{angle}{180} \leq -5 \cdot 10^{+69}:\\ \;\;\;\;\left(2 \cdot \sin \left(angle \cdot \frac{{\left(\sqrt{\pi}\right)}^{2}}{-180}\right)\right) \cdot \left(\left(a - b\right) \cdot \left(b + a\right)\right)\\ \mathbf{elif}\;\frac{angle}{180} \leq 2 \cdot 10^{-54}:\\ \;\;\;\;-0.011111111111111112 \cdot \left(\left(a - b\right) \cdot \left(\left(b + a\right) \cdot \left(\pi \cdot angle\right)\right)\right)\\ \mathbf{elif}\;\frac{angle}{180} \leq 7 \cdot 10^{+49}:\\ \;\;\;\;\left(\left(2 \cdot \left(\left(b - a\right) \cdot \left(b + a\right)\right)\right) \cdot \sin \left(\pi \cdot \frac{angle}{180}\right)\right) \cdot \cos \left(\left(\pi \cdot angle\right) \cdot 0.005555555555555556\right)\\ \mathbf{elif}\;\frac{angle}{180} \leq 5 \cdot 10^{+64}:\\ \;\;\;\;\left({a}^{2} - {b}^{2}\right) \cdot \left(\cos \left(\pi \cdot \frac{angle}{180}\right) \cdot \left(2 \cdot \sin \left(\pi \cdot \frac{angle}{180}\right)\right)\right)\\ \mathbf{elif}\;\frac{angle}{180} \leq 6 \cdot 10^{+162}:\\ \;\;\;\;\cos \left(\pi \cdot \frac{angle}{180}\right) \cdot \left(0.011111111111111112 \cdot \left(angle \cdot \left(\pi \cdot \left(\left(b - a\right) \cdot \left(b + a\right)\right)\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\cos \left(\pi \cdot \frac{angle}{180}\right) \cdot \left(\left(2 \cdot \left(\left(b - a\right) \cdot \left(b + a\right)\right)\right) \cdot \sin \left(\left(\pi \cdot angle\right) \cdot 0.005555555555555556\right)\right)\\ \end{array} \]

Alternative 3: 65.9% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(a - b\right) \cdot \left(b + a\right)\\ t_1 := \left(b - a\right) \cdot \left(b + a\right)\\ t_2 := \pi \cdot \frac{angle}{180}\\ \mathbf{if}\;\frac{angle}{180} \leq -5 \cdot 10^{+69}:\\ \;\;\;\;\left(2 \cdot \sin \left(angle \cdot \frac{{\left(\sqrt{\pi}\right)}^{2}}{-180}\right)\right) \cdot t_0\\ \mathbf{elif}\;\frac{angle}{180} \leq 2 \cdot 10^{-54}:\\ \;\;\;\;-0.011111111111111112 \cdot \left(\left(a - b\right) \cdot \left(\left(b + a\right) \cdot \left(\pi \cdot angle\right)\right)\right)\\ \mathbf{elif}\;\frac{angle}{180} \leq 7 \cdot 10^{+49}:\\ \;\;\;\;\left(\left(2 \cdot t_1\right) \cdot \sin t_2\right) \cdot \cos \left(\left(\pi \cdot angle\right) \cdot 0.005555555555555556\right)\\ \mathbf{elif}\;\frac{angle}{180} \leq 5 \cdot 10^{+99}:\\ \;\;\;\;\cos t_2 \cdot \left(0.011111111111111112 \cdot \left(angle \cdot \left(\pi \cdot t_1\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;t_0 \cdot \left(2 \cdot \sin \left(angle \cdot \frac{\sqrt[3]{\pi} \cdot {\left(\sqrt[3]{\pi}\right)}^{2}}{-180}\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (a b angle)
 :precision binary64
 (let* ((t_0 (* (- a b) (+ b a)))
        (t_1 (* (- b a) (+ b a)))
        (t_2 (* PI (/ angle 180.0))))
   (if (<= (/ angle 180.0) -5e+69)
     (* (* 2.0 (sin (* angle (/ (pow (sqrt PI) 2.0) -180.0)))) t_0)
     (if (<= (/ angle 180.0) 2e-54)
       (* -0.011111111111111112 (* (- a b) (* (+ b a) (* PI angle))))
       (if (<= (/ angle 180.0) 7e+49)
         (*
          (* (* 2.0 t_1) (sin t_2))
          (cos (* (* PI angle) 0.005555555555555556)))
         (if (<= (/ angle 180.0) 5e+99)
           (* (cos t_2) (* 0.011111111111111112 (* angle (* PI t_1))))
           (*
            t_0
            (*
             2.0
             (sin
              (* angle (/ (* (cbrt PI) (pow (cbrt PI) 2.0)) -180.0)))))))))))

C

double code(double a, double b, double angle) {
	double t_0 = (a - b) * (b + a);
	double t_1 = (b - a) * (b + a);
	double t_2 = ((double) M_PI) * (angle / 180.0);
	double tmp;
	if ((angle / 180.0) <= -5e+69) {
		tmp = (2.0 * sin((angle * (pow(sqrt(((double) M_PI)), 2.0) / -180.0)))) * t_0;
	} else if ((angle / 180.0) <= 2e-54) {
		tmp = -0.011111111111111112 * ((a - b) * ((b + a) * (((double) M_PI) * angle)));
	} else if ((angle / 180.0) <= 7e+49) {
		tmp = ((2.0 * t_1) * sin(t_2)) * cos(((((double) M_PI) * angle) * 0.005555555555555556));
	} else if ((angle / 180.0) <= 5e+99) {
		tmp = cos(t_2) * (0.011111111111111112 * (angle * (((double) M_PI) * t_1)));
	} else {
		tmp = t_0 * (2.0 * sin((angle * ((cbrt(((double) M_PI)) * pow(cbrt(((double) M_PI)), 2.0)) / -180.0))));
	}
	return tmp;
}

Java

public static double code(double a, double b, double angle) {
	double t_0 = (a - b) * (b + a);
	double t_1 = (b - a) * (b + a);
	double t_2 = Math.PI * (angle / 180.0);
	double tmp;
	if ((angle / 180.0) <= -5e+69) {
		tmp = (2.0 * Math.sin((angle * (Math.pow(Math.sqrt(Math.PI), 2.0) / -180.0)))) * t_0;
	} else if ((angle / 180.0) <= 2e-54) {
		tmp = -0.011111111111111112 * ((a - b) * ((b + a) * (Math.PI * angle)));
	} else if ((angle / 180.0) <= 7e+49) {
		tmp = ((2.0 * t_1) * Math.sin(t_2)) * Math.cos(((Math.PI * angle) * 0.005555555555555556));
	} else if ((angle / 180.0) <= 5e+99) {
		tmp = Math.cos(t_2) * (0.011111111111111112 * (angle * (Math.PI * t_1)));
	} else {
		tmp = t_0 * (2.0 * Math.sin((angle * ((Math.cbrt(Math.PI) * Math.pow(Math.cbrt(Math.PI), 2.0)) / -180.0))));
	}
	return tmp;
}

Julia

function code(a, b, angle)
	t_0 = Float64(Float64(a - b) * Float64(b + a))
	t_1 = Float64(Float64(b - a) * Float64(b + a))
	t_2 = Float64(pi * Float64(angle / 180.0))
	tmp = 0.0
	if (Float64(angle / 180.0) <= -5e+69)
		tmp = Float64(Float64(2.0 * sin(Float64(angle * Float64((sqrt(pi) ^ 2.0) / -180.0)))) * t_0);
	elseif (Float64(angle / 180.0) <= 2e-54)
		tmp = Float64(-0.011111111111111112 * Float64(Float64(a - b) * Float64(Float64(b + a) * Float64(pi * angle))));
	elseif (Float64(angle / 180.0) <= 7e+49)
		tmp = Float64(Float64(Float64(2.0 * t_1) * sin(t_2)) * cos(Float64(Float64(pi * angle) * 0.005555555555555556)));
	elseif (Float64(angle / 180.0) <= 5e+99)
		tmp = Float64(cos(t_2) * Float64(0.011111111111111112 * Float64(angle * Float64(pi * t_1))));
	else
		tmp = Float64(t_0 * Float64(2.0 * sin(Float64(angle * Float64(Float64(cbrt(pi) * (cbrt(pi) ^ 2.0)) / -180.0)))));
	end
	return tmp
end

Wolfram

code[a_, b_, angle_] := Block[{t$95$0 = N[(N[(a - b), $MachinePrecision] * N[(b + a), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[(b - a), $MachinePrecision] * N[(b + a), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(Pi * N[(angle / 180.0), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[(angle / 180.0), $MachinePrecision], -5e+69], N[(N[(2.0 * N[Sin[N[(angle * N[(N[Power[N[Sqrt[Pi], $MachinePrecision], 2.0], $MachinePrecision] / -180.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * t$95$0), $MachinePrecision], If[LessEqual[N[(angle / 180.0), $MachinePrecision], 2e-54], N[(-0.011111111111111112 * N[(N[(a - b), $MachinePrecision] * N[(N[(b + a), $MachinePrecision] * N[(Pi * angle), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[N[(angle / 180.0), $MachinePrecision], 7e+49], N[(N[(N[(2.0 * t$95$1), $MachinePrecision] * N[Sin[t$95$2], $MachinePrecision]), $MachinePrecision] * N[Cos[N[(N[(Pi * angle), $MachinePrecision] * 0.005555555555555556), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[N[(angle / 180.0), $MachinePrecision], 5e+99], N[(N[Cos[t$95$2], $MachinePrecision] * N[(0.011111111111111112 * N[(angle * N[(Pi * t$95$1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(t$95$0 * N[(2.0 * N[Sin[N[(angle * N[(N[(N[Power[Pi, 1/3], $MachinePrecision] * N[Power[N[Power[Pi, 1/3], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] / -180.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]]]]

Derivation

1. Split input into 5 regimes

2. if (/.f64 angle 180) < -5.00000000000000036e69

   1. Initial program 29.3%

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

   3. Simplified 26.4%

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

      1. unpow2 26.4%

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

      2. unpow2 26.4%

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

      3. difference-of-squares 26.4%

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

   5. Applied egg-rr 26.4%

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

      1. add-cbrt-cube 34.9%

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

      2. pow3 34.9%

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

   7. Applied egg-rr 34.9%

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

   8. Taylor expanded in angle around 0 31.9%

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

      1. rem-cbrt-cube 37.6%

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

      2. add-sqr-sqrt 44.1%

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

      3. pow2 44.1%

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

   10. Applied egg-rr 44.1%

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

   if -5.00000000000000036e69 < (/.f64 angle 180) < 2.0000000000000001e-54

   1. Initial program 73.0%

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

   3. Simplified 73.0%

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

      1. unpow2 73.0%

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

      2. unpow2 73.0%

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

      3. difference-of-squares 76.3%

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

   5. Applied egg-rr 76.3%

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

      1. add-cbrt-cube 75.3%

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

      2. pow3 75.3%

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

   7. Applied egg-rr 75.3%

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

   8. Taylor expanded in angle around 0 74.6%

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

      1. associate-*r* 74.5%

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

      2. associate-*r* 91.1%

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

      3. *-commutative 91.1%

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

      4. *-commutative 91.1%

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

   10. Simplified 91.1%

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

   if 2.0000000000000001e-54 < (/.f64 angle 180) < 6.9999999999999995e49

   1. Initial program 67.0%

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

      1. unpow2 67.0%

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

      2. unpow2 67.0%

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

      3. difference-of-squares 75.4%

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

   3. Applied egg-rr 75.4%

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

   4. Taylor expanded in angle around inf 87.5%

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

   if 6.9999999999999995e49 < (/.f64 angle 180) < 5.00000000000000008e99

   1. Initial program 13.7%

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

      1. unpow2 13.7%

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

      2. unpow2 13.7%

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

      3. difference-of-squares 13.9%

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

   3. Applied egg-rr 13.9%

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

   4. Taylor expanded in angle around 0 44.7%

      \[\leadsto \color{blue}{\left(0.011111111111111112 \cdot \left(angle \cdot \left(\pi \cdot \left(\left(a + b\right) \cdot \left(b - a\right)\right)\right)\right)\right)} \cdot \cos \left(\pi \cdot \frac{angle}{180}\right) \]

   if 5.00000000000000008e99 < (/.f64 angle 180)

   1. Initial program 22.9%

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

   3. Simplified 32.3%

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

      1. unpow2 32.3%

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

      2. unpow2 32.3%

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

      3. difference-of-squares 34.9%

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

   5. Applied egg-rr 34.9%

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

      1. add-cbrt-cube 36.1%

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

      2. pow3 36.1%

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

   7. Applied egg-rr 36.1%

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

   8. Taylor expanded in angle around 0 34.4%

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

      1. rem-cbrt-cube 28.8%

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

      2. add-cube-cbrt 38.4%

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

      3. pow2 38.4%

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

   10. Applied egg-rr 38.4%

       \[\leadsto \left(2 \cdot \sin \left(angle \cdot \frac{\color{blue}{{\left(\sqrt[3]{\pi}\right)}^{2} \cdot \sqrt[3]{\pi}}}{-180}\right)\right) \cdot \left(1 \cdot \left(\left(a + b\right) \cdot \left(a - b\right)\right)\right) \]
3. Recombined 5 regimes into one program.

4. Final simplification 71.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{angle}{180} \leq -5 \cdot 10^{+69}:\\ \;\;\;\;\left(2 \cdot \sin \left(angle \cdot \frac{{\left(\sqrt{\pi}\right)}^{2}}{-180}\right)\right) \cdot \left(\left(a - b\right) \cdot \left(b + a\right)\right)\\ \mathbf{elif}\;\frac{angle}{180} \leq 2 \cdot 10^{-54}:\\ \;\;\;\;-0.011111111111111112 \cdot \left(\left(a - b\right) \cdot \left(\left(b + a\right) \cdot \left(\pi \cdot angle\right)\right)\right)\\ \mathbf{elif}\;\frac{angle}{180} \leq 7 \cdot 10^{+49}:\\ \;\;\;\;\left(\left(2 \cdot \left(\left(b - a\right) \cdot \left(b + a\right)\right)\right) \cdot \sin \left(\pi \cdot \frac{angle}{180}\right)\right) \cdot \cos \left(\left(\pi \cdot angle\right) \cdot 0.005555555555555556\right)\\ \mathbf{elif}\;\frac{angle}{180} \leq 5 \cdot 10^{+99}:\\ \;\;\;\;\cos \left(\pi \cdot \frac{angle}{180}\right) \cdot \left(0.011111111111111112 \cdot \left(angle \cdot \left(\pi \cdot \left(\left(b - a\right) \cdot \left(b + a\right)\right)\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\left(a - b\right) \cdot \left(b + a\right)\right) \cdot \left(2 \cdot \sin \left(angle \cdot \frac{\sqrt[3]{\pi} \cdot {\left(\sqrt[3]{\pi}\right)}^{2}}{-180}\right)\right)\\ \end{array} \]

Alternative 4: 59.6% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \pi \cdot \frac{angle}{180}\\ \mathbf{if}\;b \leq 2.8 \cdot 10^{-45}:\\ \;\;\;\;\left(\left(2 \cdot \left(\left(b - a\right) \cdot \left(b + a\right)\right)\right) \cdot \sin t_0\right) \cdot \cos \left(\frac{angle}{180} \cdot {\left(\sqrt{\pi}\right)}^{2}\right)\\ \mathbf{elif}\;b \leq 6.8 \cdot 10^{+289}:\\ \;\;\;\;-0.011111111111111112 \cdot \left(\left(a - b\right) \cdot \left(\left(b + a\right) \cdot \left(\pi \cdot angle\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(2 \cdot \left({b}^{2} \cdot \left|\sin \left(\pi \cdot \left(angle \cdot 0.005555555555555556\right)\right)\right|\right)\right) \cdot \cos t_0\\ \end{array} \end{array} \]

FPCore

(FPCore (a b angle)
 :precision binary64
 (let* ((t_0 (* PI (/ angle 180.0))))
   (if (<= b 2.8e-45)
     (*
      (* (* 2.0 (* (- b a) (+ b a))) (sin t_0))
      (cos (* (/ angle 180.0) (pow (sqrt PI) 2.0))))
     (if (<= b 6.8e+289)
       (* -0.011111111111111112 (* (- a b) (* (+ b a) (* PI angle))))
       (*
        (*
         2.0
         (* (pow b 2.0) (fabs (sin (* PI (* angle 0.005555555555555556))))))
        (cos t_0))))))

C

double code(double a, double b, double angle) {
	double t_0 = ((double) M_PI) * (angle / 180.0);
	double tmp;
	if (b <= 2.8e-45) {
		tmp = ((2.0 * ((b - a) * (b + a))) * sin(t_0)) * cos(((angle / 180.0) * pow(sqrt(((double) M_PI)), 2.0)));
	} else if (b <= 6.8e+289) {
		tmp = -0.011111111111111112 * ((a - b) * ((b + a) * (((double) M_PI) * angle)));
	} else {
		tmp = (2.0 * (pow(b, 2.0) * fabs(sin((((double) M_PI) * (angle * 0.005555555555555556)))))) * cos(t_0);
	}
	return tmp;
}

Java

public static double code(double a, double b, double angle) {
	double t_0 = Math.PI * (angle / 180.0);
	double tmp;
	if (b <= 2.8e-45) {
		tmp = ((2.0 * ((b - a) * (b + a))) * Math.sin(t_0)) * Math.cos(((angle / 180.0) * Math.pow(Math.sqrt(Math.PI), 2.0)));
	} else if (b <= 6.8e+289) {
		tmp = -0.011111111111111112 * ((a - b) * ((b + a) * (Math.PI * angle)));
	} else {
		tmp = (2.0 * (Math.pow(b, 2.0) * Math.abs(Math.sin((Math.PI * (angle * 0.005555555555555556)))))) * Math.cos(t_0);
	}
	return tmp;
}

Python

def code(a, b, angle):
	t_0 = math.pi * (angle / 180.0)
	tmp = 0
	if b <= 2.8e-45:
		tmp = ((2.0 * ((b - a) * (b + a))) * math.sin(t_0)) * math.cos(((angle / 180.0) * math.pow(math.sqrt(math.pi), 2.0)))
	elif b <= 6.8e+289:
		tmp = -0.011111111111111112 * ((a - b) * ((b + a) * (math.pi * angle)))
	else:
		tmp = (2.0 * (math.pow(b, 2.0) * math.fabs(math.sin((math.pi * (angle * 0.005555555555555556)))))) * math.cos(t_0)
	return tmp

Julia

function code(a, b, angle)
	t_0 = Float64(pi * Float64(angle / 180.0))
	tmp = 0.0
	if (b <= 2.8e-45)
		tmp = Float64(Float64(Float64(2.0 * Float64(Float64(b - a) * Float64(b + a))) * sin(t_0)) * cos(Float64(Float64(angle / 180.0) * (sqrt(pi) ^ 2.0))));
	elseif (b <= 6.8e+289)
		tmp = Float64(-0.011111111111111112 * Float64(Float64(a - b) * Float64(Float64(b + a) * Float64(pi * angle))));
	else
		tmp = Float64(Float64(2.0 * Float64((b ^ 2.0) * abs(sin(Float64(pi * Float64(angle * 0.005555555555555556)))))) * cos(t_0));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, angle)
	t_0 = pi * (angle / 180.0);
	tmp = 0.0;
	if (b <= 2.8e-45)
		tmp = ((2.0 * ((b - a) * (b + a))) * sin(t_0)) * cos(((angle / 180.0) * (sqrt(pi) ^ 2.0)));
	elseif (b <= 6.8e+289)
		tmp = -0.011111111111111112 * ((a - b) * ((b + a) * (pi * angle)));
	else
		tmp = (2.0 * ((b ^ 2.0) * abs(sin((pi * (angle * 0.005555555555555556)))))) * cos(t_0);
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, angle_] := Block[{t$95$0 = N[(Pi * N[(angle / 180.0), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[b, 2.8e-45], N[(N[(N[(2.0 * N[(N[(b - a), $MachinePrecision] * N[(b + a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[Sin[t$95$0], $MachinePrecision]), $MachinePrecision] * N[Cos[N[(N[(angle / 180.0), $MachinePrecision] * N[Power[N[Sqrt[Pi], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[b, 6.8e+289], N[(-0.011111111111111112 * N[(N[(a - b), $MachinePrecision] * N[(N[(b + a), $MachinePrecision] * N[(Pi * angle), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(2.0 * N[(N[Power[b, 2.0], $MachinePrecision] * N[Abs[N[Sin[N[(Pi * N[(angle * 0.005555555555555556), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[Cos[t$95$0], $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if b < 2.8000000000000001e-45

   1. Initial program 57.0%

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

      1. unpow2 57.0%

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

      2. unpow2 57.0%

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

      3. difference-of-squares 58.2%

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

   3. Applied egg-rr 58.2%

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

      1. add-sqr-sqrt 59.4%

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

      2. pow2 59.4%

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

   5. Applied egg-rr 58.8%

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

   if 2.8000000000000001e-45 < b < 6.7999999999999997e289

   1. Initial program 45.4%

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

   3. Simplified 47.1%

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

      1. unpow2 47.1%

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

      2. unpow2 47.1%

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

      3. difference-of-squares 52.1%

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

   5. Applied egg-rr 52.1%

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

      1. add-cbrt-cube 61.1%

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

      2. pow3 61.1%

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

   7. Applied egg-rr 61.1%

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

   8. Taylor expanded in angle around 0 56.5%

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

      1. associate-*r* 56.4%

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

      2. associate-*r* 68.2%

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

      3. *-commutative 68.2%

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

      4. *-commutative 68.2%

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

   10. Simplified 68.2%

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

   if 6.7999999999999997e289 < b

   1. Initial program 33.3%

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

   2. Taylor expanded in b around inf 77.8%

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

      1. add-sqr-sqrt 77.8%

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

      2. sqrt-unprod 88.9%

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

      3. pow2 88.9%

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

      4. associate-*r* 88.9%

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

      5. *-commutative 88.9%

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

      6. *-commutative 88.9%

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

   4. Applied egg-rr 88.9%

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

      1. unpow2 88.9%

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

      2. rem-sqrt-square 100.0%

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

   6. Simplified 100.0%

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

4. Final simplification 62.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq 2.8 \cdot 10^{-45}:\\ \;\;\;\;\left(\left(2 \cdot \left(\left(b - a\right) \cdot \left(b + a\right)\right)\right) \cdot \sin \left(\pi \cdot \frac{angle}{180}\right)\right) \cdot \cos \left(\frac{angle}{180} \cdot {\left(\sqrt{\pi}\right)}^{2}\right)\\ \mathbf{elif}\;b \leq 6.8 \cdot 10^{+289}:\\ \;\;\;\;-0.011111111111111112 \cdot \left(\left(a - b\right) \cdot \left(\left(b + a\right) \cdot \left(\pi \cdot angle\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(2 \cdot \left({b}^{2} \cdot \left|\sin \left(\pi \cdot \left(angle \cdot 0.005555555555555556\right)\right)\right|\right)\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right)\\ \end{array} \]

Alternative 5: 65.5% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(a - b\right) \cdot \left(b + a\right)\\ t_1 := \left(b - a\right) \cdot \left(b + a\right)\\ t_2 := \pi \cdot \frac{angle}{180}\\ t_3 := \pi \cdot \left(angle \cdot -0.005555555555555556\right)\\ \mathbf{if}\;\frac{angle}{180} \leq -5 \cdot 10^{+69}:\\ \;\;\;\;\left(2 \cdot \sin \left(angle \cdot \frac{{\left(\sqrt{\pi}\right)}^{2}}{-180}\right)\right) \cdot t_0\\ \mathbf{elif}\;\frac{angle}{180} \leq 2 \cdot 10^{-54}:\\ \;\;\;\;-0.011111111111111112 \cdot \left(\left(a - b\right) \cdot \left(\left(b + a\right) \cdot \left(\pi \cdot angle\right)\right)\right)\\ \mathbf{elif}\;\frac{angle}{180} \leq 7 \cdot 10^{+49}:\\ \;\;\;\;\left(\left(2 \cdot t_1\right) \cdot \sin t_2\right) \cdot \cos \left(\left(\pi \cdot angle\right) \cdot 0.005555555555555556\right)\\ \mathbf{elif}\;\frac{angle}{180} \leq 5 \cdot 10^{+99}:\\ \;\;\;\;\cos t_2 \cdot \left(0.011111111111111112 \cdot \left(angle \cdot \left(\pi \cdot t_1\right)\right)\right)\\ \mathbf{elif}\;\frac{angle}{180} \leq 5 \cdot 10^{+185}:\\ \;\;\;\;\left|\left(t_0 \cdot \cos t_3\right) \cdot \left(2 \cdot \sin t_3\right)\right|\\ \mathbf{else}:\\ \;\;\;\;\sin \left(\left(\pi \cdot angle\right) \cdot 0.011111111111111112\right) \cdot \left({b}^{2} - {a}^{2}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (a b angle)
 :precision binary64
 (let* ((t_0 (* (- a b) (+ b a)))
        (t_1 (* (- b a) (+ b a)))
        (t_2 (* PI (/ angle 180.0)))
        (t_3 (* PI (* angle -0.005555555555555556))))
   (if (<= (/ angle 180.0) -5e+69)
     (* (* 2.0 (sin (* angle (/ (pow (sqrt PI) 2.0) -180.0)))) t_0)
     (if (<= (/ angle 180.0) 2e-54)
       (* -0.011111111111111112 (* (- a b) (* (+ b a) (* PI angle))))
       (if (<= (/ angle 180.0) 7e+49)
         (*
          (* (* 2.0 t_1) (sin t_2))
          (cos (* (* PI angle) 0.005555555555555556)))
         (if (<= (/ angle 180.0) 5e+99)
           (* (cos t_2) (* 0.011111111111111112 (* angle (* PI t_1))))
           (if (<= (/ angle 180.0) 5e+185)
             (fabs (* (* t_0 (cos t_3)) (* 2.0 (sin t_3))))
             (*
              (sin (* (* PI angle) 0.011111111111111112))
              (- (pow b 2.0) (pow a 2.0))))))))))

C

double code(double a, double b, double angle) {
	double t_0 = (a - b) * (b + a);
	double t_1 = (b - a) * (b + a);
	double t_2 = ((double) M_PI) * (angle / 180.0);
	double t_3 = ((double) M_PI) * (angle * -0.005555555555555556);
	double tmp;
	if ((angle / 180.0) <= -5e+69) {
		tmp = (2.0 * sin((angle * (pow(sqrt(((double) M_PI)), 2.0) / -180.0)))) * t_0;
	} else if ((angle / 180.0) <= 2e-54) {
		tmp = -0.011111111111111112 * ((a - b) * ((b + a) * (((double) M_PI) * angle)));
	} else if ((angle / 180.0) <= 7e+49) {
		tmp = ((2.0 * t_1) * sin(t_2)) * cos(((((double) M_PI) * angle) * 0.005555555555555556));
	} else if ((angle / 180.0) <= 5e+99) {
		tmp = cos(t_2) * (0.011111111111111112 * (angle * (((double) M_PI) * t_1)));
	} else if ((angle / 180.0) <= 5e+185) {
		tmp = fabs(((t_0 * cos(t_3)) * (2.0 * sin(t_3))));
	} else {
		tmp = sin(((((double) M_PI) * angle) * 0.011111111111111112)) * (pow(b, 2.0) - pow(a, 2.0));
	}
	return tmp;
}

Java

public static double code(double a, double b, double angle) {
	double t_0 = (a - b) * (b + a);
	double t_1 = (b - a) * (b + a);
	double t_2 = Math.PI * (angle / 180.0);
	double t_3 = Math.PI * (angle * -0.005555555555555556);
	double tmp;
	if ((angle / 180.0) <= -5e+69) {
		tmp = (2.0 * Math.sin((angle * (Math.pow(Math.sqrt(Math.PI), 2.0) / -180.0)))) * t_0;
	} else if ((angle / 180.0) <= 2e-54) {
		tmp = -0.011111111111111112 * ((a - b) * ((b + a) * (Math.PI * angle)));
	} else if ((angle / 180.0) <= 7e+49) {
		tmp = ((2.0 * t_1) * Math.sin(t_2)) * Math.cos(((Math.PI * angle) * 0.005555555555555556));
	} else if ((angle / 180.0) <= 5e+99) {
		tmp = Math.cos(t_2) * (0.011111111111111112 * (angle * (Math.PI * t_1)));
	} else if ((angle / 180.0) <= 5e+185) {
		tmp = Math.abs(((t_0 * Math.cos(t_3)) * (2.0 * Math.sin(t_3))));
	} else {
		tmp = Math.sin(((Math.PI * angle) * 0.011111111111111112)) * (Math.pow(b, 2.0) - Math.pow(a, 2.0));
	}
	return tmp;
}

Python

def code(a, b, angle):
	t_0 = (a - b) * (b + a)
	t_1 = (b - a) * (b + a)
	t_2 = math.pi * (angle / 180.0)
	t_3 = math.pi * (angle * -0.005555555555555556)
	tmp = 0
	if (angle / 180.0) <= -5e+69:
		tmp = (2.0 * math.sin((angle * (math.pow(math.sqrt(math.pi), 2.0) / -180.0)))) * t_0
	elif (angle / 180.0) <= 2e-54:
		tmp = -0.011111111111111112 * ((a - b) * ((b + a) * (math.pi * angle)))
	elif (angle / 180.0) <= 7e+49:
		tmp = ((2.0 * t_1) * math.sin(t_2)) * math.cos(((math.pi * angle) * 0.005555555555555556))
	elif (angle / 180.0) <= 5e+99:
		tmp = math.cos(t_2) * (0.011111111111111112 * (angle * (math.pi * t_1)))
	elif (angle / 180.0) <= 5e+185:
		tmp = math.fabs(((t_0 * math.cos(t_3)) * (2.0 * math.sin(t_3))))
	else:
		tmp = math.sin(((math.pi * angle) * 0.011111111111111112)) * (math.pow(b, 2.0) - math.pow(a, 2.0))
	return tmp

Julia

function code(a, b, angle)
	t_0 = Float64(Float64(a - b) * Float64(b + a))
	t_1 = Float64(Float64(b - a) * Float64(b + a))
	t_2 = Float64(pi * Float64(angle / 180.0))
	t_3 = Float64(pi * Float64(angle * -0.005555555555555556))
	tmp = 0.0
	if (Float64(angle / 180.0) <= -5e+69)
		tmp = Float64(Float64(2.0 * sin(Float64(angle * Float64((sqrt(pi) ^ 2.0) / -180.0)))) * t_0);
	elseif (Float64(angle / 180.0) <= 2e-54)
		tmp = Float64(-0.011111111111111112 * Float64(Float64(a - b) * Float64(Float64(b + a) * Float64(pi * angle))));
	elseif (Float64(angle / 180.0) <= 7e+49)
		tmp = Float64(Float64(Float64(2.0 * t_1) * sin(t_2)) * cos(Float64(Float64(pi * angle) * 0.005555555555555556)));
	elseif (Float64(angle / 180.0) <= 5e+99)
		tmp = Float64(cos(t_2) * Float64(0.011111111111111112 * Float64(angle * Float64(pi * t_1))));
	elseif (Float64(angle / 180.0) <= 5e+185)
		tmp = abs(Float64(Float64(t_0 * cos(t_3)) * Float64(2.0 * sin(t_3))));
	else
		tmp = Float64(sin(Float64(Float64(pi * angle) * 0.011111111111111112)) * Float64((b ^ 2.0) - (a ^ 2.0)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, angle)
	t_0 = (a - b) * (b + a);
	t_1 = (b - a) * (b + a);
	t_2 = pi * (angle / 180.0);
	t_3 = pi * (angle * -0.005555555555555556);
	tmp = 0.0;
	if ((angle / 180.0) <= -5e+69)
		tmp = (2.0 * sin((angle * ((sqrt(pi) ^ 2.0) / -180.0)))) * t_0;
	elseif ((angle / 180.0) <= 2e-54)
		tmp = -0.011111111111111112 * ((a - b) * ((b + a) * (pi * angle)));
	elseif ((angle / 180.0) <= 7e+49)
		tmp = ((2.0 * t_1) * sin(t_2)) * cos(((pi * angle) * 0.005555555555555556));
	elseif ((angle / 180.0) <= 5e+99)
		tmp = cos(t_2) * (0.011111111111111112 * (angle * (pi * t_1)));
	elseif ((angle / 180.0) <= 5e+185)
		tmp = abs(((t_0 * cos(t_3)) * (2.0 * sin(t_3))));
	else
		tmp = sin(((pi * angle) * 0.011111111111111112)) * ((b ^ 2.0) - (a ^ 2.0));
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, angle_] := Block[{t$95$0 = N[(N[(a - b), $MachinePrecision] * N[(b + a), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[(b - a), $MachinePrecision] * N[(b + a), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(Pi * N[(angle / 180.0), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$3 = N[(Pi * N[(angle * -0.005555555555555556), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[(angle / 180.0), $MachinePrecision], -5e+69], N[(N[(2.0 * N[Sin[N[(angle * N[(N[Power[N[Sqrt[Pi], $MachinePrecision], 2.0], $MachinePrecision] / -180.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * t$95$0), $MachinePrecision], If[LessEqual[N[(angle / 180.0), $MachinePrecision], 2e-54], N[(-0.011111111111111112 * N[(N[(a - b), $MachinePrecision] * N[(N[(b + a), $MachinePrecision] * N[(Pi * angle), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[N[(angle / 180.0), $MachinePrecision], 7e+49], N[(N[(N[(2.0 * t$95$1), $MachinePrecision] * N[Sin[t$95$2], $MachinePrecision]), $MachinePrecision] * N[Cos[N[(N[(Pi * angle), $MachinePrecision] * 0.005555555555555556), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[N[(angle / 180.0), $MachinePrecision], 5e+99], N[(N[Cos[t$95$2], $MachinePrecision] * N[(0.011111111111111112 * N[(angle * N[(Pi * t$95$1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[N[(angle / 180.0), $MachinePrecision], 5e+185], N[Abs[N[(N[(t$95$0 * N[Cos[t$95$3], $MachinePrecision]), $MachinePrecision] * N[(2.0 * N[Sin[t$95$3], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], N[(N[Sin[N[(N[(Pi * angle), $MachinePrecision] * 0.011111111111111112), $MachinePrecision]], $MachinePrecision] * N[(N[Power[b, 2.0], $MachinePrecision] - N[Power[a, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]]]]]]

Derivation

1. Split input into 6 regimes

2. if (/.f64 angle 180) < -5.00000000000000036e69

   1. Initial program 29.3%

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

   3. Simplified 26.4%

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

      1. unpow2 26.4%

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

      2. unpow2 26.4%

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

      3. difference-of-squares 26.4%

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

   5. Applied egg-rr 26.4%

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

      1. add-cbrt-cube 34.9%

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

      2. pow3 34.9%

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

   7. Applied egg-rr 34.9%

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

   8. Taylor expanded in angle around 0 31.9%

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

      1. rem-cbrt-cube 37.6%

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

      2. add-sqr-sqrt 44.1%

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

      3. pow2 44.1%

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

   10. Applied egg-rr 44.1%

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

   if -5.00000000000000036e69 < (/.f64 angle 180) < 2.0000000000000001e-54

   1. Initial program 73.0%

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

   3. Simplified 73.0%

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

      1. unpow2 73.0%

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

      2. unpow2 73.0%

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

      3. difference-of-squares 76.3%

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

   5. Applied egg-rr 76.3%

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

      1. add-cbrt-cube 75.3%

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

      2. pow3 75.3%

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

   7. Applied egg-rr 75.3%

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

   8. Taylor expanded in angle around 0 74.6%

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

      1. associate-*r* 74.5%

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

      2. associate-*r* 91.1%

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

      3. *-commutative 91.1%

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

      4. *-commutative 91.1%

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

   10. Simplified 91.1%

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

   if 2.0000000000000001e-54 < (/.f64 angle 180) < 6.9999999999999995e49

   1. Initial program 67.0%

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

      1. unpow2 67.0%

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

      2. unpow2 67.0%

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

      3. difference-of-squares 75.4%

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

   3. Applied egg-rr 75.4%

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

   4. Taylor expanded in angle around inf 87.5%

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

   if 6.9999999999999995e49 < (/.f64 angle 180) < 5.00000000000000008e99

   1. Initial program 13.7%

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

      1. unpow2 13.7%

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

      2. unpow2 13.7%

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

      3. difference-of-squares 13.9%

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

   3. Applied egg-rr 13.9%

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

   4. Taylor expanded in angle around 0 44.7%

      \[\leadsto \color{blue}{\left(0.011111111111111112 \cdot \left(angle \cdot \left(\pi \cdot \left(\left(a + b\right) \cdot \left(b - a\right)\right)\right)\right)\right)} \cdot \cos \left(\pi \cdot \frac{angle}{180}\right) \]

   if 5.00000000000000008e99 < (/.f64 angle 180) < 4.9999999999999999e185

   1. Initial program 23.0%

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

   3. Simplified 34.8%

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

      1. unpow2 34.8%

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

      2. unpow2 34.8%

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

      3. difference-of-squares 40.1%

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

   5. Applied egg-rr 40.1%

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

      1. add-cbrt-cube 46.6%

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

      2. pow3 46.6%

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

   7. Applied egg-rr 46.6%

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

      1. add-sqr-sqrt 39.3%

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

      2. sqrt-unprod 45.7%

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

      3. pow2 45.7%

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

   9. Applied egg-rr 45.9%

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

       1. unpow2 45.9%

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

       2. rem-sqrt-square 46.5%

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

       3. associate-*r* 46.5%

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

       4. *-commutative 46.5%

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

   11. Simplified 46.1%

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

   if 4.9999999999999999e185 < (/.f64 angle 180)

   1. Initial program 22.7%

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

      1. *-commutative 22.7%

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

      2. associate-*l* 22.7%

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

      3. associate-*l* 22.7%

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

   3. Simplified 22.7%

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

      1. add-cbrt-cube 22.0%

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

      2. pow1/3 17.5%

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

   5. Applied egg-rr 17.5%

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

   6. Taylor expanded in angle around inf 36.6%

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

      1. *-commutative 36.6%

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

   8. Simplified 36.6%

      \[\leadsto \color{blue}{\sin \left(\left(angle \cdot \pi\right) \cdot 0.011111111111111112\right) \cdot \left({b}^{2} - {a}^{2}\right)} \]
3. Recombined 6 regimes into one program.

4. Final simplification 71.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{angle}{180} \leq -5 \cdot 10^{+69}:\\ \;\;\;\;\left(2 \cdot \sin \left(angle \cdot \frac{{\left(\sqrt{\pi}\right)}^{2}}{-180}\right)\right) \cdot \left(\left(a - b\right) \cdot \left(b + a\right)\right)\\ \mathbf{elif}\;\frac{angle}{180} \leq 2 \cdot 10^{-54}:\\ \;\;\;\;-0.011111111111111112 \cdot \left(\left(a - b\right) \cdot \left(\left(b + a\right) \cdot \left(\pi \cdot angle\right)\right)\right)\\ \mathbf{elif}\;\frac{angle}{180} \leq 7 \cdot 10^{+49}:\\ \;\;\;\;\left(\left(2 \cdot \left(\left(b - a\right) \cdot \left(b + a\right)\right)\right) \cdot \sin \left(\pi \cdot \frac{angle}{180}\right)\right) \cdot \cos \left(\left(\pi \cdot angle\right) \cdot 0.005555555555555556\right)\\ \mathbf{elif}\;\frac{angle}{180} \leq 5 \cdot 10^{+99}:\\ \;\;\;\;\cos \left(\pi \cdot \frac{angle}{180}\right) \cdot \left(0.011111111111111112 \cdot \left(angle \cdot \left(\pi \cdot \left(\left(b - a\right) \cdot \left(b + a\right)\right)\right)\right)\right)\\ \mathbf{elif}\;\frac{angle}{180} \leq 5 \cdot 10^{+185}:\\ \;\;\;\;\left|\left(\left(\left(a - b\right) \cdot \left(b + a\right)\right) \cdot \cos \left(\pi \cdot \left(angle \cdot -0.005555555555555556\right)\right)\right) \cdot \left(2 \cdot \sin \left(\pi \cdot \left(angle \cdot -0.005555555555555556\right)\right)\right)\right|\\ \mathbf{else}:\\ \;\;\;\;\sin \left(\left(\pi \cdot angle\right) \cdot 0.011111111111111112\right) \cdot \left({b}^{2} - {a}^{2}\right)\\ \end{array} \]

Alternative 6: 65.4% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(b - a\right) \cdot \left(b + a\right)\\ t_1 := 2 \cdot t_0\\ t_2 := \pi \cdot \frac{angle}{180}\\ t_3 := \cos t_2\\ t_4 := \left(\pi \cdot angle\right) \cdot 0.005555555555555556\\ \mathbf{if}\;\frac{angle}{180} \leq -5 \cdot 10^{+69}:\\ \;\;\;\;\left(2 \cdot \sin \left(angle \cdot \frac{{\left(\sqrt{\pi}\right)}^{2}}{-180}\right)\right) \cdot \left(\left(a - b\right) \cdot \left(b + a\right)\right)\\ \mathbf{elif}\;\frac{angle}{180} \leq 2 \cdot 10^{-54}:\\ \;\;\;\;-0.011111111111111112 \cdot \left(\left(a - b\right) \cdot \left(\left(b + a\right) \cdot \left(\pi \cdot angle\right)\right)\right)\\ \mathbf{elif}\;\frac{angle}{180} \leq 7 \cdot 10^{+49}:\\ \;\;\;\;\left(t_1 \cdot \sin t_2\right) \cdot \cos t_4\\ \mathbf{elif}\;\frac{angle}{180} \leq 5 \cdot 10^{+99}:\\ \;\;\;\;t_3 \cdot \left(0.011111111111111112 \cdot \left(angle \cdot \left(\pi \cdot t_0\right)\right)\right)\\ \mathbf{elif}\;\frac{angle}{180} \leq 5 \cdot 10^{+163}:\\ \;\;\;\;{\left({\left(\sin \left(\left(\pi \cdot angle\right) \cdot 0.011111111111111112\right) \cdot \left(-{a}^{2}\right)\right)}^{3}\right)}^{0.3333333333333333}\\ \mathbf{else}:\\ \;\;\;\;t_3 \cdot \left(t_1 \cdot \sin t_4\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (a b angle)
 :precision binary64
 (let* ((t_0 (* (- b a) (+ b a)))
        (t_1 (* 2.0 t_0))
        (t_2 (* PI (/ angle 180.0)))
        (t_3 (cos t_2))
        (t_4 (* (* PI angle) 0.005555555555555556)))
   (if (<= (/ angle 180.0) -5e+69)
     (*
      (* 2.0 (sin (* angle (/ (pow (sqrt PI) 2.0) -180.0))))
      (* (- a b) (+ b a)))
     (if (<= (/ angle 180.0) 2e-54)
       (* -0.011111111111111112 (* (- a b) (* (+ b a) (* PI angle))))
       (if (<= (/ angle 180.0) 7e+49)
         (* (* t_1 (sin t_2)) (cos t_4))
         (if (<= (/ angle 180.0) 5e+99)
           (* t_3 (* 0.011111111111111112 (* angle (* PI t_0))))
           (if (<= (/ angle 180.0) 5e+163)
             (pow
              (pow
               (* (sin (* (* PI angle) 0.011111111111111112)) (- (pow a 2.0)))
               3.0)
              0.3333333333333333)
             (* t_3 (* t_1 (sin t_4))))))))))

C

double code(double a, double b, double angle) {
	double t_0 = (b - a) * (b + a);
	double t_1 = 2.0 * t_0;
	double t_2 = ((double) M_PI) * (angle / 180.0);
	double t_3 = cos(t_2);
	double t_4 = (((double) M_PI) * angle) * 0.005555555555555556;
	double tmp;
	if ((angle / 180.0) <= -5e+69) {
		tmp = (2.0 * sin((angle * (pow(sqrt(((double) M_PI)), 2.0) / -180.0)))) * ((a - b) * (b + a));
	} else if ((angle / 180.0) <= 2e-54) {
		tmp = -0.011111111111111112 * ((a - b) * ((b + a) * (((double) M_PI) * angle)));
	} else if ((angle / 180.0) <= 7e+49) {
		tmp = (t_1 * sin(t_2)) * cos(t_4);
	} else if ((angle / 180.0) <= 5e+99) {
		tmp = t_3 * (0.011111111111111112 * (angle * (((double) M_PI) * t_0)));
	} else if ((angle / 180.0) <= 5e+163) {
		tmp = pow(pow((sin(((((double) M_PI) * angle) * 0.011111111111111112)) * -pow(a, 2.0)), 3.0), 0.3333333333333333);
	} else {
		tmp = t_3 * (t_1 * sin(t_4));
	}
	return tmp;
}

Java

public static double code(double a, double b, double angle) {
	double t_0 = (b - a) * (b + a);
	double t_1 = 2.0 * t_0;
	double t_2 = Math.PI * (angle / 180.0);
	double t_3 = Math.cos(t_2);
	double t_4 = (Math.PI * angle) * 0.005555555555555556;
	double tmp;
	if ((angle / 180.0) <= -5e+69) {
		tmp = (2.0 * Math.sin((angle * (Math.pow(Math.sqrt(Math.PI), 2.0) / -180.0)))) * ((a - b) * (b + a));
	} else if ((angle / 180.0) <= 2e-54) {
		tmp = -0.011111111111111112 * ((a - b) * ((b + a) * (Math.PI * angle)));
	} else if ((angle / 180.0) <= 7e+49) {
		tmp = (t_1 * Math.sin(t_2)) * Math.cos(t_4);
	} else if ((angle / 180.0) <= 5e+99) {
		tmp = t_3 * (0.011111111111111112 * (angle * (Math.PI * t_0)));
	} else if ((angle / 180.0) <= 5e+163) {
		tmp = Math.pow(Math.pow((Math.sin(((Math.PI * angle) * 0.011111111111111112)) * -Math.pow(a, 2.0)), 3.0), 0.3333333333333333);
	} else {
		tmp = t_3 * (t_1 * Math.sin(t_4));
	}
	return tmp;
}

Python

def code(a, b, angle):
	t_0 = (b - a) * (b + a)
	t_1 = 2.0 * t_0
	t_2 = math.pi * (angle / 180.0)
	t_3 = math.cos(t_2)
	t_4 = (math.pi * angle) * 0.005555555555555556
	tmp = 0
	if (angle / 180.0) <= -5e+69:
		tmp = (2.0 * math.sin((angle * (math.pow(math.sqrt(math.pi), 2.0) / -180.0)))) * ((a - b) * (b + a))
	elif (angle / 180.0) <= 2e-54:
		tmp = -0.011111111111111112 * ((a - b) * ((b + a) * (math.pi * angle)))
	elif (angle / 180.0) <= 7e+49:
		tmp = (t_1 * math.sin(t_2)) * math.cos(t_4)
	elif (angle / 180.0) <= 5e+99:
		tmp = t_3 * (0.011111111111111112 * (angle * (math.pi * t_0)))
	elif (angle / 180.0) <= 5e+163:
		tmp = math.pow(math.pow((math.sin(((math.pi * angle) * 0.011111111111111112)) * -math.pow(a, 2.0)), 3.0), 0.3333333333333333)
	else:
		tmp = t_3 * (t_1 * math.sin(t_4))
	return tmp

Julia

function code(a, b, angle)
	t_0 = Float64(Float64(b - a) * Float64(b + a))
	t_1 = Float64(2.0 * t_0)
	t_2 = Float64(pi * Float64(angle / 180.0))
	t_3 = cos(t_2)
	t_4 = Float64(Float64(pi * angle) * 0.005555555555555556)
	tmp = 0.0
	if (Float64(angle / 180.0) <= -5e+69)
		tmp = Float64(Float64(2.0 * sin(Float64(angle * Float64((sqrt(pi) ^ 2.0) / -180.0)))) * Float64(Float64(a - b) * Float64(b + a)));
	elseif (Float64(angle / 180.0) <= 2e-54)
		tmp = Float64(-0.011111111111111112 * Float64(Float64(a - b) * Float64(Float64(b + a) * Float64(pi * angle))));
	elseif (Float64(angle / 180.0) <= 7e+49)
		tmp = Float64(Float64(t_1 * sin(t_2)) * cos(t_4));
	elseif (Float64(angle / 180.0) <= 5e+99)
		tmp = Float64(t_3 * Float64(0.011111111111111112 * Float64(angle * Float64(pi * t_0))));
	elseif (Float64(angle / 180.0) <= 5e+163)
		tmp = (Float64(sin(Float64(Float64(pi * angle) * 0.011111111111111112)) * Float64(-(a ^ 2.0))) ^ 3.0) ^ 0.3333333333333333;
	else
		tmp = Float64(t_3 * Float64(t_1 * sin(t_4)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, angle)
	t_0 = (b - a) * (b + a);
	t_1 = 2.0 * t_0;
	t_2 = pi * (angle / 180.0);
	t_3 = cos(t_2);
	t_4 = (pi * angle) * 0.005555555555555556;
	tmp = 0.0;
	if ((angle / 180.0) <= -5e+69)
		tmp = (2.0 * sin((angle * ((sqrt(pi) ^ 2.0) / -180.0)))) * ((a - b) * (b + a));
	elseif ((angle / 180.0) <= 2e-54)
		tmp = -0.011111111111111112 * ((a - b) * ((b + a) * (pi * angle)));
	elseif ((angle / 180.0) <= 7e+49)
		tmp = (t_1 * sin(t_2)) * cos(t_4);
	elseif ((angle / 180.0) <= 5e+99)
		tmp = t_3 * (0.011111111111111112 * (angle * (pi * t_0)));
	elseif ((angle / 180.0) <= 5e+163)
		tmp = ((sin(((pi * angle) * 0.011111111111111112)) * -(a ^ 2.0)) ^ 3.0) ^ 0.3333333333333333;
	else
		tmp = t_3 * (t_1 * sin(t_4));
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, angle_] := Block[{t$95$0 = N[(N[(b - a), $MachinePrecision] * N[(b + a), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(2.0 * t$95$0), $MachinePrecision]}, Block[{t$95$2 = N[(Pi * N[(angle / 180.0), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$3 = N[Cos[t$95$2], $MachinePrecision]}, Block[{t$95$4 = N[(N[(Pi * angle), $MachinePrecision] * 0.005555555555555556), $MachinePrecision]}, If[LessEqual[N[(angle / 180.0), $MachinePrecision], -5e+69], N[(N[(2.0 * N[Sin[N[(angle * N[(N[Power[N[Sqrt[Pi], $MachinePrecision], 2.0], $MachinePrecision] / -180.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[(N[(a - b), $MachinePrecision] * N[(b + a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[N[(angle / 180.0), $MachinePrecision], 2e-54], N[(-0.011111111111111112 * N[(N[(a - b), $MachinePrecision] * N[(N[(b + a), $MachinePrecision] * N[(Pi * angle), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[N[(angle / 180.0), $MachinePrecision], 7e+49], N[(N[(t$95$1 * N[Sin[t$95$2], $MachinePrecision]), $MachinePrecision] * N[Cos[t$95$4], $MachinePrecision]), $MachinePrecision], If[LessEqual[N[(angle / 180.0), $MachinePrecision], 5e+99], N[(t$95$3 * N[(0.011111111111111112 * N[(angle * N[(Pi * t$95$0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[N[(angle / 180.0), $MachinePrecision], 5e+163], N[Power[N[Power[N[(N[Sin[N[(N[(Pi * angle), $MachinePrecision] * 0.011111111111111112), $MachinePrecision]], $MachinePrecision] * (-N[Power[a, 2.0], $MachinePrecision])), $MachinePrecision], 3.0], $MachinePrecision], 0.3333333333333333], $MachinePrecision], N[(t$95$3 * N[(t$95$1 * N[Sin[t$95$4], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]]]]]]]

Derivation

1. Split input into 6 regimes

2. if (/.f64 angle 180) < -5.00000000000000036e69

   1. Initial program 29.3%

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

   3. Simplified 26.4%

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

      1. unpow2 26.4%

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

      2. unpow2 26.4%

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

      3. difference-of-squares 26.4%

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

   5. Applied egg-rr 26.4%

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

      1. add-cbrt-cube 34.9%

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

      2. pow3 34.9%

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

   7. Applied egg-rr 34.9%

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

   8. Taylor expanded in angle around 0 31.9%

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

      1. rem-cbrt-cube 37.6%

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

      2. add-sqr-sqrt 44.1%

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

      3. pow2 44.1%

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

   10. Applied egg-rr 44.1%

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

   if -5.00000000000000036e69 < (/.f64 angle 180) < 2.0000000000000001e-54

   1. Initial program 73.0%

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

   3. Simplified 73.0%

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

      1. unpow2 73.0%

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

      2. unpow2 73.0%

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

      3. difference-of-squares 76.3%

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

   5. Applied egg-rr 76.3%

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

      1. add-cbrt-cube 75.3%

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

      2. pow3 75.3%

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

   7. Applied egg-rr 75.3%

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

   8. Taylor expanded in angle around 0 74.6%

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

      1. associate-*r* 74.5%

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

      2. associate-*r* 91.1%

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

      3. *-commutative 91.1%

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

      4. *-commutative 91.1%

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

   10. Simplified 91.1%

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

   if 2.0000000000000001e-54 < (/.f64 angle 180) < 6.9999999999999995e49

   1. Initial program 67.0%

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

      1. unpow2 67.0%

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

      2. unpow2 67.0%

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

      3. difference-of-squares 75.4%

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

   3. Applied egg-rr 75.4%

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

   4. Taylor expanded in angle around inf 87.5%

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

   if 6.9999999999999995e49 < (/.f64 angle 180) < 5.00000000000000008e99

   1. Initial program 13.7%

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

      1. unpow2 13.7%

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

      2. unpow2 13.7%

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

      3. difference-of-squares 13.9%

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

   3. Applied egg-rr 13.9%

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

   4. Taylor expanded in angle around 0 44.7%

      \[\leadsto \color{blue}{\left(0.011111111111111112 \cdot \left(angle \cdot \left(\pi \cdot \left(\left(a + b\right) \cdot \left(b - a\right)\right)\right)\right)\right)} \cdot \cos \left(\pi \cdot \frac{angle}{180}\right) \]

   if 5.00000000000000008e99 < (/.f64 angle 180) < 5e163

   1. Initial program 25.3%

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

      1. *-commutative 25.3%

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

      2. associate-*l* 25.3%

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

      3. associate-*l* 25.3%

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

   3. Simplified 25.3%

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

      1. add-cbrt-cube 25.4%

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

      2. pow1/3 33.7%

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

   5. Applied egg-rr 33.8%

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

   6. Taylor expanded in b around 0 47.4%

      \[\leadsto {\left({\color{blue}{\left(-1 \cdot \left({a}^{2} \cdot \sin \left(0.011111111111111112 \cdot \left(angle \cdot \pi\right)\right)\right)\right)}}^{3}\right)}^{0.3333333333333333} \]
   7. Step-by-step derivation

      1. mul-1-neg 47.4%

         \[\leadsto {\left({\color{blue}{\left(-{a}^{2} \cdot \sin \left(0.011111111111111112 \cdot \left(angle \cdot \pi\right)\right)\right)}}^{3}\right)}^{0.3333333333333333} \]

      2. distribute-rgt-neg-in 47.4%

         \[\leadsto {\left({\color{blue}{\left({a}^{2} \cdot \left(-\sin \left(0.011111111111111112 \cdot \left(angle \cdot \pi\right)\right)\right)\right)}}^{3}\right)}^{0.3333333333333333} \]

      3. *-commutative 47.4%

         \[\leadsto {\left({\left({a}^{2} \cdot \left(-\sin \color{blue}{\left(\left(angle \cdot \pi\right) \cdot 0.011111111111111112\right)}\right)\right)}^{3}\right)}^{0.3333333333333333} \]

   8. Simplified 47.4%

      \[\leadsto {\left({\color{blue}{\left({a}^{2} \cdot \left(-\sin \left(\left(angle \cdot \pi\right) \cdot 0.011111111111111112\right)\right)\right)}}^{3}\right)}^{0.3333333333333333} \]

   if 5e163 < (/.f64 angle 180)

   1. Initial program 21.6%

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

      1. unpow2 21.6%

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

      2. unpow2 21.6%

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

      3. difference-of-squares 25.6%

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

   3. Applied egg-rr 25.6%

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

   4. Taylor expanded in angle around inf 37.5%

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

4. Final simplification 71.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{angle}{180} \leq -5 \cdot 10^{+69}:\\ \;\;\;\;\left(2 \cdot \sin \left(angle \cdot \frac{{\left(\sqrt{\pi}\right)}^{2}}{-180}\right)\right) \cdot \left(\left(a - b\right) \cdot \left(b + a\right)\right)\\ \mathbf{elif}\;\frac{angle}{180} \leq 2 \cdot 10^{-54}:\\ \;\;\;\;-0.011111111111111112 \cdot \left(\left(a - b\right) \cdot \left(\left(b + a\right) \cdot \left(\pi \cdot angle\right)\right)\right)\\ \mathbf{elif}\;\frac{angle}{180} \leq 7 \cdot 10^{+49}:\\ \;\;\;\;\left(\left(2 \cdot \left(\left(b - a\right) \cdot \left(b + a\right)\right)\right) \cdot \sin \left(\pi \cdot \frac{angle}{180}\right)\right) \cdot \cos \left(\left(\pi \cdot angle\right) \cdot 0.005555555555555556\right)\\ \mathbf{elif}\;\frac{angle}{180} \leq 5 \cdot 10^{+99}:\\ \;\;\;\;\cos \left(\pi \cdot \frac{angle}{180}\right) \cdot \left(0.011111111111111112 \cdot \left(angle \cdot \left(\pi \cdot \left(\left(b - a\right) \cdot \left(b + a\right)\right)\right)\right)\right)\\ \mathbf{elif}\;\frac{angle}{180} \leq 5 \cdot 10^{+163}:\\ \;\;\;\;{\left({\left(\sin \left(\left(\pi \cdot angle\right) \cdot 0.011111111111111112\right) \cdot \left(-{a}^{2}\right)\right)}^{3}\right)}^{0.3333333333333333}\\ \mathbf{else}:\\ \;\;\;\;\cos \left(\pi \cdot \frac{angle}{180}\right) \cdot \left(\left(2 \cdot \left(\left(b - a\right) \cdot \left(b + a\right)\right)\right) \cdot \sin \left(\left(\pi \cdot angle\right) \cdot 0.005555555555555556\right)\right)\\ \end{array} \]

Alternative 7: 65.7% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(a - b\right) \cdot \left(b + a\right)\\ \mathbf{if}\;\frac{angle}{180} \leq -5 \cdot 10^{+69}:\\ \;\;\;\;\left(2 \cdot \sin \left(angle \cdot \frac{{\left(\sqrt{\pi}\right)}^{2}}{-180}\right)\right) \cdot t_0\\ \mathbf{elif}\;\frac{angle}{180} \leq 2 \cdot 10^{-54}:\\ \;\;\;\;-0.011111111111111112 \cdot \left(\left(a - b\right) \cdot \left(\left(b + a\right) \cdot \left(\pi \cdot angle\right)\right)\right)\\ \mathbf{elif}\;\frac{angle}{180} \leq 7 \cdot 10^{+49}:\\ \;\;\;\;\left(\left(2 \cdot \left(\left(b - a\right) \cdot \left(b + a\right)\right)\right) \cdot \sin \left(\pi \cdot \frac{angle}{180}\right)\right) \cdot \cos \left(\left(\pi \cdot angle\right) \cdot 0.005555555555555556\right)\\ \mathbf{elif}\;\frac{angle}{180} \leq 5 \cdot 10^{+64}:\\ \;\;\;\;\left({a}^{2} - {b}^{2}\right) \cdot \sin \left(\left(\pi \cdot angle\right) \cdot 0.011111111111111112\right)\\ \mathbf{elif}\;\frac{angle}{180} \leq 10^{+288}:\\ \;\;\;\;t_0 \cdot \left(2 \cdot \sin \left(angle \cdot \frac{\sqrt[3]{{\pi}^{3}}}{-180}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(t_0 \cdot \cos \left(angle \cdot \frac{\pi}{-180}\right)\right) \cdot \left(2 \cdot \sin \left(\left(\pi \cdot angle\right) \cdot -0.005555555555555556\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (a b angle)
 :precision binary64
 (let* ((t_0 (* (- a b) (+ b a))))
   (if (<= (/ angle 180.0) -5e+69)
     (* (* 2.0 (sin (* angle (/ (pow (sqrt PI) 2.0) -180.0)))) t_0)
     (if (<= (/ angle 180.0) 2e-54)
       (* -0.011111111111111112 (* (- a b) (* (+ b a) (* PI angle))))
       (if (<= (/ angle 180.0) 7e+49)
         (*
          (* (* 2.0 (* (- b a) (+ b a))) (sin (* PI (/ angle 180.0))))
          (cos (* (* PI angle) 0.005555555555555556)))
         (if (<= (/ angle 180.0) 5e+64)
           (*
            (- (pow a 2.0) (pow b 2.0))
            (sin (* (* PI angle) 0.011111111111111112)))
           (if (<= (/ angle 180.0) 1e+288)
             (* t_0 (* 2.0 (sin (* angle (/ (cbrt (pow PI 3.0)) -180.0)))))
             (*
              (* t_0 (cos (* angle (/ PI -180.0))))
              (* 2.0 (sin (* (* PI angle) -0.005555555555555556)))))))))))

C

double code(double a, double b, double angle) {
	double t_0 = (a - b) * (b + a);
	double tmp;
	if ((angle / 180.0) <= -5e+69) {
		tmp = (2.0 * sin((angle * (pow(sqrt(((double) M_PI)), 2.0) / -180.0)))) * t_0;
	} else if ((angle / 180.0) <= 2e-54) {
		tmp = -0.011111111111111112 * ((a - b) * ((b + a) * (((double) M_PI) * angle)));
	} else if ((angle / 180.0) <= 7e+49) {
		tmp = ((2.0 * ((b - a) * (b + a))) * sin((((double) M_PI) * (angle / 180.0)))) * cos(((((double) M_PI) * angle) * 0.005555555555555556));
	} else if ((angle / 180.0) <= 5e+64) {
		tmp = (pow(a, 2.0) - pow(b, 2.0)) * sin(((((double) M_PI) * angle) * 0.011111111111111112));
	} else if ((angle / 180.0) <= 1e+288) {
		tmp = t_0 * (2.0 * sin((angle * (cbrt(pow(((double) M_PI), 3.0)) / -180.0))));
	} else {
		tmp = (t_0 * cos((angle * (((double) M_PI) / -180.0)))) * (2.0 * sin(((((double) M_PI) * angle) * -0.005555555555555556)));
	}
	return tmp;
}

Java

public static double code(double a, double b, double angle) {
	double t_0 = (a - b) * (b + a);
	double tmp;
	if ((angle / 180.0) <= -5e+69) {
		tmp = (2.0 * Math.sin((angle * (Math.pow(Math.sqrt(Math.PI), 2.0) / -180.0)))) * t_0;
	} else if ((angle / 180.0) <= 2e-54) {
		tmp = -0.011111111111111112 * ((a - b) * ((b + a) * (Math.PI * angle)));
	} else if ((angle / 180.0) <= 7e+49) {
		tmp = ((2.0 * ((b - a) * (b + a))) * Math.sin((Math.PI * (angle / 180.0)))) * Math.cos(((Math.PI * angle) * 0.005555555555555556));
	} else if ((angle / 180.0) <= 5e+64) {
		tmp = (Math.pow(a, 2.0) - Math.pow(b, 2.0)) * Math.sin(((Math.PI * angle) * 0.011111111111111112));
	} else if ((angle / 180.0) <= 1e+288) {
		tmp = t_0 * (2.0 * Math.sin((angle * (Math.cbrt(Math.pow(Math.PI, 3.0)) / -180.0))));
	} else {
		tmp = (t_0 * Math.cos((angle * (Math.PI / -180.0)))) * (2.0 * Math.sin(((Math.PI * angle) * -0.005555555555555556)));
	}
	return tmp;
}

Julia

function code(a, b, angle)
	t_0 = Float64(Float64(a - b) * Float64(b + a))
	tmp = 0.0
	if (Float64(angle / 180.0) <= -5e+69)
		tmp = Float64(Float64(2.0 * sin(Float64(angle * Float64((sqrt(pi) ^ 2.0) / -180.0)))) * t_0);
	elseif (Float64(angle / 180.0) <= 2e-54)
		tmp = Float64(-0.011111111111111112 * Float64(Float64(a - b) * Float64(Float64(b + a) * Float64(pi * angle))));
	elseif (Float64(angle / 180.0) <= 7e+49)
		tmp = Float64(Float64(Float64(2.0 * Float64(Float64(b - a) * Float64(b + a))) * sin(Float64(pi * Float64(angle / 180.0)))) * cos(Float64(Float64(pi * angle) * 0.005555555555555556)));
	elseif (Float64(angle / 180.0) <= 5e+64)
		tmp = Float64(Float64((a ^ 2.0) - (b ^ 2.0)) * sin(Float64(Float64(pi * angle) * 0.011111111111111112)));
	elseif (Float64(angle / 180.0) <= 1e+288)
		tmp = Float64(t_0 * Float64(2.0 * sin(Float64(angle * Float64(cbrt((pi ^ 3.0)) / -180.0)))));
	else
		tmp = Float64(Float64(t_0 * cos(Float64(angle * Float64(pi / -180.0)))) * Float64(2.0 * sin(Float64(Float64(pi * angle) * -0.005555555555555556))));
	end
	return tmp
end

Wolfram

code[a_, b_, angle_] := Block[{t$95$0 = N[(N[(a - b), $MachinePrecision] * N[(b + a), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[(angle / 180.0), $MachinePrecision], -5e+69], N[(N[(2.0 * N[Sin[N[(angle * N[(N[Power[N[Sqrt[Pi], $MachinePrecision], 2.0], $MachinePrecision] / -180.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * t$95$0), $MachinePrecision], If[LessEqual[N[(angle / 180.0), $MachinePrecision], 2e-54], N[(-0.011111111111111112 * N[(N[(a - b), $MachinePrecision] * N[(N[(b + a), $MachinePrecision] * N[(Pi * angle), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[N[(angle / 180.0), $MachinePrecision], 7e+49], N[(N[(N[(2.0 * N[(N[(b - a), $MachinePrecision] * N[(b + a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[Sin[N[(Pi * N[(angle / 180.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[Cos[N[(N[(Pi * angle), $MachinePrecision] * 0.005555555555555556), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[N[(angle / 180.0), $MachinePrecision], 5e+64], N[(N[(N[Power[a, 2.0], $MachinePrecision] - N[Power[b, 2.0], $MachinePrecision]), $MachinePrecision] * N[Sin[N[(N[(Pi * angle), $MachinePrecision] * 0.011111111111111112), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[N[(angle / 180.0), $MachinePrecision], 1e+288], N[(t$95$0 * N[(2.0 * N[Sin[N[(angle * N[(N[Power[N[Power[Pi, 3.0], $MachinePrecision], 1/3], $MachinePrecision] / -180.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(t$95$0 * N[Cos[N[(angle * N[(Pi / -180.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[(2.0 * N[Sin[N[(N[(Pi * angle), $MachinePrecision] * -0.005555555555555556), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]]]

Derivation

1. Split input into 6 regimes

2. if (/.f64 angle 180) < -5.00000000000000036e69

   1. Initial program 29.3%

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

   3. Simplified 26.4%

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

      1. unpow2 26.4%

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

      2. unpow2 26.4%

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

      3. difference-of-squares 26.4%

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

   5. Applied egg-rr 26.4%

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

      1. add-cbrt-cube 34.9%

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

      2. pow3 34.9%

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

   7. Applied egg-rr 34.9%

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

   8. Taylor expanded in angle around 0 31.9%

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

      1. rem-cbrt-cube 37.6%

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

      2. add-sqr-sqrt 44.1%

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

      3. pow2 44.1%

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

   10. Applied egg-rr 44.1%

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

   if -5.00000000000000036e69 < (/.f64 angle 180) < 2.0000000000000001e-54

   1. Initial program 73.0%

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

   3. Simplified 73.0%

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

      1. unpow2 73.0%

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

      2. unpow2 73.0%

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

      3. difference-of-squares 76.3%

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

   5. Applied egg-rr 76.3%

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

      1. add-cbrt-cube 75.3%

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

      2. pow3 75.3%

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

   7. Applied egg-rr 75.3%

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

   8. Taylor expanded in angle around 0 74.6%

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

      1. associate-*r* 74.5%

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

      2. associate-*r* 91.1%

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

      3. *-commutative 91.1%

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

      4. *-commutative 91.1%

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

   10. Simplified 91.1%

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

   if 2.0000000000000001e-54 < (/.f64 angle 180) < 6.9999999999999995e49

   1. Initial program 67.0%

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

      1. unpow2 67.0%

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

      2. unpow2 67.0%

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

      3. difference-of-squares 75.4%

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

   3. Applied egg-rr 75.4%

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

   4. Taylor expanded in angle around inf 87.5%

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

   if 6.9999999999999995e49 < (/.f64 angle 180) < 5e64

   1. Initial program 5.2%

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

   3. Simplified 8.4%

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

   5. Applied egg-rr 36.0%

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

      1. distribute-lft-out 36.0%

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

      2. sub-neg 36.0%

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

      3. *-commutative 36.0%

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

      4. count-2 36.0%

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

      5. *-commutative 36.0%

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

      6. *-commutative 36.0%

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

      7. associate-*r* 36.0%

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

      8. *-commutative 36.0%

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

      9. *-commutative 36.0%

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

      10. associate-*r* 36.1%

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

      11. distribute-rgt-out 36.1%

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

      12. metadata-eval 36.1%

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

   7. Simplified 36.1%

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

   if 5e64 < (/.f64 angle 180) < 1e288

   1. Initial program 21.7%

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

   3. Simplified 29.8%

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

      1. unpow2 29.8%

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

      2. unpow2 29.8%

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

      3. difference-of-squares 32.2%

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

   5. Applied egg-rr 32.2%

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

      1. add-cbrt-cube 32.8%

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

      2. pow3 32.8%

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

   7. Applied egg-rr 32.8%

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

   8. Taylor expanded in angle around 0 41.1%

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

   if 1e288 < (/.f64 angle 180)

   1. Initial program 25.8%

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

   3. Simplified 50.8%

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

      1. unpow2 50.8%

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

      2. unpow2 50.8%

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

      3. difference-of-squares 50.8%

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

   5. Applied egg-rr 50.8%

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

   6. Taylor expanded in angle around inf 55.1%

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

      1. *-commutative 55.1%

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

   8. Simplified 55.1%

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

4. Final simplification 71.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{angle}{180} \leq -5 \cdot 10^{+69}:\\ \;\;\;\;\left(2 \cdot \sin \left(angle \cdot \frac{{\left(\sqrt{\pi}\right)}^{2}}{-180}\right)\right) \cdot \left(\left(a - b\right) \cdot \left(b + a\right)\right)\\ \mathbf{elif}\;\frac{angle}{180} \leq 2 \cdot 10^{-54}:\\ \;\;\;\;-0.011111111111111112 \cdot \left(\left(a - b\right) \cdot \left(\left(b + a\right) \cdot \left(\pi \cdot angle\right)\right)\right)\\ \mathbf{elif}\;\frac{angle}{180} \leq 7 \cdot 10^{+49}:\\ \;\;\;\;\left(\left(2 \cdot \left(\left(b - a\right) \cdot \left(b + a\right)\right)\right) \cdot \sin \left(\pi \cdot \frac{angle}{180}\right)\right) \cdot \cos \left(\left(\pi \cdot angle\right) \cdot 0.005555555555555556\right)\\ \mathbf{elif}\;\frac{angle}{180} \leq 5 \cdot 10^{+64}:\\ \;\;\;\;\left({a}^{2} - {b}^{2}\right) \cdot \sin \left(\left(\pi \cdot angle\right) \cdot 0.011111111111111112\right)\\ \mathbf{elif}\;\frac{angle}{180} \leq 10^{+288}:\\ \;\;\;\;\left(\left(a - b\right) \cdot \left(b + a\right)\right) \cdot \left(2 \cdot \sin \left(angle \cdot \frac{\sqrt[3]{{\pi}^{3}}}{-180}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\left(\left(a - b\right) \cdot \left(b + a\right)\right) \cdot \cos \left(angle \cdot \frac{\pi}{-180}\right)\right) \cdot \left(2 \cdot \sin \left(\left(\pi \cdot angle\right) \cdot -0.005555555555555556\right)\right)\\ \end{array} \]

Alternative 8: 65.0% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(a - b\right) \cdot \left(b + a\right)\\ t_1 := \sin \left(\left(\pi \cdot angle\right) \cdot 0.011111111111111112\right)\\ \mathbf{if}\;\frac{angle}{180} \leq -2 \cdot 10^{+183}:\\ \;\;\;\;t_0 \cdot \left(2 \cdot \sin \left(\left(\pi \cdot angle\right) \cdot -0.005555555555555556\right)\right)\\ \mathbf{elif}\;\frac{angle}{180} \leq -2 \cdot 10^{+76}:\\ \;\;\;\;\left({a}^{2} - {b}^{2}\right) \cdot t_1\\ \mathbf{elif}\;\frac{angle}{180} \leq -1 \cdot 10^{-58}:\\ \;\;\;\;t_1 \cdot \left({b}^{2} - {a}^{2}\right)\\ \mathbf{elif}\;\frac{angle}{180} \leq 2 \cdot 10^{+24}:\\ \;\;\;\;-0.011111111111111112 \cdot \left(\left(a - b\right) \cdot \left(\left(b + a\right) \cdot \left(\pi \cdot angle\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;t_0 \cdot \left(2 \cdot \sin \left(angle \cdot \frac{\sqrt[3]{{\pi}^{3}}}{-180}\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (a b angle)
 :precision binary64
 (let* ((t_0 (* (- a b) (+ b a)))
        (t_1 (sin (* (* PI angle) 0.011111111111111112))))
   (if (<= (/ angle 180.0) -2e+183)
     (* t_0 (* 2.0 (sin (* (* PI angle) -0.005555555555555556))))
     (if (<= (/ angle 180.0) -2e+76)
       (* (- (pow a 2.0) (pow b 2.0)) t_1)
       (if (<= (/ angle 180.0) -1e-58)
         (* t_1 (- (pow b 2.0) (pow a 2.0)))
         (if (<= (/ angle 180.0) 2e+24)
           (* -0.011111111111111112 (* (- a b) (* (+ b a) (* PI angle))))
           (* t_0 (* 2.0 (sin (* angle (/ (cbrt (pow PI 3.0)) -180.0)))))))))))

C

double code(double a, double b, double angle) {
	double t_0 = (a - b) * (b + a);
	double t_1 = sin(((((double) M_PI) * angle) * 0.011111111111111112));
	double tmp;
	if ((angle / 180.0) <= -2e+183) {
		tmp = t_0 * (2.0 * sin(((((double) M_PI) * angle) * -0.005555555555555556)));
	} else if ((angle / 180.0) <= -2e+76) {
		tmp = (pow(a, 2.0) - pow(b, 2.0)) * t_1;
	} else if ((angle / 180.0) <= -1e-58) {
		tmp = t_1 * (pow(b, 2.0) - pow(a, 2.0));
	} else if ((angle / 180.0) <= 2e+24) {
		tmp = -0.011111111111111112 * ((a - b) * ((b + a) * (((double) M_PI) * angle)));
	} else {
		tmp = t_0 * (2.0 * sin((angle * (cbrt(pow(((double) M_PI), 3.0)) / -180.0))));
	}
	return tmp;
}

Java

public static double code(double a, double b, double angle) {
	double t_0 = (a - b) * (b + a);
	double t_1 = Math.sin(((Math.PI * angle) * 0.011111111111111112));
	double tmp;
	if ((angle / 180.0) <= -2e+183) {
		tmp = t_0 * (2.0 * Math.sin(((Math.PI * angle) * -0.005555555555555556)));
	} else if ((angle / 180.0) <= -2e+76) {
		tmp = (Math.pow(a, 2.0) - Math.pow(b, 2.0)) * t_1;
	} else if ((angle / 180.0) <= -1e-58) {
		tmp = t_1 * (Math.pow(b, 2.0) - Math.pow(a, 2.0));
	} else if ((angle / 180.0) <= 2e+24) {
		tmp = -0.011111111111111112 * ((a - b) * ((b + a) * (Math.PI * angle)));
	} else {
		tmp = t_0 * (2.0 * Math.sin((angle * (Math.cbrt(Math.pow(Math.PI, 3.0)) / -180.0))));
	}
	return tmp;
}

Julia

function code(a, b, angle)
	t_0 = Float64(Float64(a - b) * Float64(b + a))
	t_1 = sin(Float64(Float64(pi * angle) * 0.011111111111111112))
	tmp = 0.0
	if (Float64(angle / 180.0) <= -2e+183)
		tmp = Float64(t_0 * Float64(2.0 * sin(Float64(Float64(pi * angle) * -0.005555555555555556))));
	elseif (Float64(angle / 180.0) <= -2e+76)
		tmp = Float64(Float64((a ^ 2.0) - (b ^ 2.0)) * t_1);
	elseif (Float64(angle / 180.0) <= -1e-58)
		tmp = Float64(t_1 * Float64((b ^ 2.0) - (a ^ 2.0)));
	elseif (Float64(angle / 180.0) <= 2e+24)
		tmp = Float64(-0.011111111111111112 * Float64(Float64(a - b) * Float64(Float64(b + a) * Float64(pi * angle))));
	else
		tmp = Float64(t_0 * Float64(2.0 * sin(Float64(angle * Float64(cbrt((pi ^ 3.0)) / -180.0)))));
	end
	return tmp
end

Wolfram

code[a_, b_, angle_] := Block[{t$95$0 = N[(N[(a - b), $MachinePrecision] * N[(b + a), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[Sin[N[(N[(Pi * angle), $MachinePrecision] * 0.011111111111111112), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[N[(angle / 180.0), $MachinePrecision], -2e+183], N[(t$95$0 * N[(2.0 * N[Sin[N[(N[(Pi * angle), $MachinePrecision] * -0.005555555555555556), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[N[(angle / 180.0), $MachinePrecision], -2e+76], N[(N[(N[Power[a, 2.0], $MachinePrecision] - N[Power[b, 2.0], $MachinePrecision]), $MachinePrecision] * t$95$1), $MachinePrecision], If[LessEqual[N[(angle / 180.0), $MachinePrecision], -1e-58], N[(t$95$1 * N[(N[Power[b, 2.0], $MachinePrecision] - N[Power[a, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[N[(angle / 180.0), $MachinePrecision], 2e+24], N[(-0.011111111111111112 * N[(N[(a - b), $MachinePrecision] * N[(N[(b + a), $MachinePrecision] * N[(Pi * angle), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(t$95$0 * N[(2.0 * N[Sin[N[(angle * N[(N[Power[N[Power[Pi, 3.0], $MachinePrecision], 1/3], $MachinePrecision] / -180.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]]]

Derivation

1. Split input into 5 regimes

2. if (/.f64 angle 180) < -1.99999999999999989e183

   1. Initial program 28.7%

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

   3. Simplified 28.2%

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

      1. unpow2 28.2%

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

      2. unpow2 28.2%

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

      3. difference-of-squares 28.2%

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

   5. Applied egg-rr 28.2%

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

      1. add-cbrt-cube 35.4%

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

      2. pow3 35.4%

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

   7. Applied egg-rr 35.4%

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

   8. Taylor expanded in angle around 0 31.7%

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

   9. Taylor expanded in angle around inf 42.1%

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

       1. associate-*r* 42.1%

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

       2. *-commutative 42.1%

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

       3. +-commutative 42.1%

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

   11. Simplified 42.1%

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

   if -1.99999999999999989e183 < (/.f64 angle 180) < -2.0000000000000001e76

   1. Initial program 28.0%

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

   3. Simplified 20.9%

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

   5. Applied egg-rr 31.7%

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

      1. distribute-lft-out 31.7%

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

      2. sub-neg 31.7%

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

      3. *-commutative 31.7%

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

      4. count-2 31.7%

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

      5. *-commutative 31.7%

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

      6. *-commutative 31.7%

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

      7. associate-*r* 43.9%

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

      8. *-commutative 43.9%

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

      9. *-commutative 43.9%

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

      10. associate-*r* 49.1%

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

      11. distribute-rgt-out 49.1%

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

      12. metadata-eval 49.1%

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

   7. Simplified 49.1%

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

   if -2.0000000000000001e76 < (/.f64 angle 180) < -1e-58

   1. Initial program 56.8%

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

      1. *-commutative 56.8%

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

      2. associate-*l* 56.8%

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

      3. associate-*l* 56.9%

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

   3. Simplified 56.9%

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

      1. add-cbrt-cube 40.8%

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

      2. pow1/3 24.5%

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

   5. Applied egg-rr 24.1%

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

   6. Taylor expanded in angle around inf 57.5%

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

      1. *-commutative 57.5%

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

   8. Simplified 57.5%

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

   if -1e-58 < (/.f64 angle 180) < 2e24

   1. Initial program 75.1%

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

   3. Simplified 74.9%

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

      1. unpow2 74.9%

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

      2. unpow2 74.9%

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

      3. difference-of-squares 79.7%

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

   5. Applied egg-rr 79.7%

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

      1. add-cbrt-cube 80.1%

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

      2. pow3 80.1%

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

   7. Applied egg-rr 80.1%

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

   8. Taylor expanded in angle around 0 78.7%

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

      1. associate-*r* 78.6%

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

      2. associate-*r* 94.8%

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

      3. *-commutative 94.8%

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

      4. *-commutative 94.8%

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

   10. Simplified 94.8%

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

   if 2e24 < (/.f64 angle 180)

   1. Initial program 21.7%

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

   3. Simplified 30.1%

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

      1. unpow2 30.1%

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

      2. unpow2 30.1%

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

      3. difference-of-squares 32.0%

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

   5. Applied egg-rr 32.0%

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

      1. add-cbrt-cube 33.9%

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

      2. pow3 33.9%

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

   7. Applied egg-rr 33.9%

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

   8. Taylor expanded in angle around 0 36.7%

      \[\leadsto \left(2 \cdot \sin \left(angle \cdot \frac{\sqrt[3]{{\pi}^{3}}}{-180}\right)\right) \cdot \left(\color{blue}{1} \cdot \left(\left(a + b\right) \cdot \left(a - b\right)\right)\right) \]
3. Recombined 5 regimes into one program.

4. Final simplification 70.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{angle}{180} \leq -2 \cdot 10^{+183}:\\ \;\;\;\;\left(\left(a - b\right) \cdot \left(b + a\right)\right) \cdot \left(2 \cdot \sin \left(\left(\pi \cdot angle\right) \cdot -0.005555555555555556\right)\right)\\ \mathbf{elif}\;\frac{angle}{180} \leq -2 \cdot 10^{+76}:\\ \;\;\;\;\left({a}^{2} - {b}^{2}\right) \cdot \sin \left(\left(\pi \cdot angle\right) \cdot 0.011111111111111112\right)\\ \mathbf{elif}\;\frac{angle}{180} \leq -1 \cdot 10^{-58}:\\ \;\;\;\;\sin \left(\left(\pi \cdot angle\right) \cdot 0.011111111111111112\right) \cdot \left({b}^{2} - {a}^{2}\right)\\ \mathbf{elif}\;\frac{angle}{180} \leq 2 \cdot 10^{+24}:\\ \;\;\;\;-0.011111111111111112 \cdot \left(\left(a - b\right) \cdot \left(\left(b + a\right) \cdot \left(\pi \cdot angle\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\left(a - b\right) \cdot \left(b + a\right)\right) \cdot \left(2 \cdot \sin \left(angle \cdot \frac{\sqrt[3]{{\pi}^{3}}}{-180}\right)\right)\\ \end{array} \]

Alternative 9: 65.9% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(a - b\right) \cdot \left(b + a\right)\\ \mathbf{if}\;\frac{angle}{180} \leq -5 \cdot 10^{+69}:\\ \;\;\;\;\left(2 \cdot \sin \left(angle \cdot \frac{{\left(\sqrt{\pi}\right)}^{2}}{-180}\right)\right) \cdot t_0\\ \mathbf{elif}\;\frac{angle}{180} \leq 2 \cdot 10^{-54}:\\ \;\;\;\;-0.011111111111111112 \cdot \left(\left(a - b\right) \cdot \left(\left(b + a\right) \cdot \left(\pi \cdot angle\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(t_0 \cdot \cos \left(angle \cdot \frac{\pi}{-180}\right)\right) \cdot \left(2 \cdot \sin \left(\left(\pi \cdot angle\right) \cdot -0.005555555555555556\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (a b angle)
 :precision binary64
 (let* ((t_0 (* (- a b) (+ b a))))
   (if (<= (/ angle 180.0) -5e+69)
     (* (* 2.0 (sin (* angle (/ (pow (sqrt PI) 2.0) -180.0)))) t_0)
     (if (<= (/ angle 180.0) 2e-54)
       (* -0.011111111111111112 (* (- a b) (* (+ b a) (* PI angle))))
       (*
        (* t_0 (cos (* angle (/ PI -180.0))))
        (* 2.0 (sin (* (* PI angle) -0.005555555555555556))))))))

C

double code(double a, double b, double angle) {
	double t_0 = (a - b) * (b + a);
	double tmp;
	if ((angle / 180.0) <= -5e+69) {
		tmp = (2.0 * sin((angle * (pow(sqrt(((double) M_PI)), 2.0) / -180.0)))) * t_0;
	} else if ((angle / 180.0) <= 2e-54) {
		tmp = -0.011111111111111112 * ((a - b) * ((b + a) * (((double) M_PI) * angle)));
	} else {
		tmp = (t_0 * cos((angle * (((double) M_PI) / -180.0)))) * (2.0 * sin(((((double) M_PI) * angle) * -0.005555555555555556)));
	}
	return tmp;
}

Java

public static double code(double a, double b, double angle) {
	double t_0 = (a - b) * (b + a);
	double tmp;
	if ((angle / 180.0) <= -5e+69) {
		tmp = (2.0 * Math.sin((angle * (Math.pow(Math.sqrt(Math.PI), 2.0) / -180.0)))) * t_0;
	} else if ((angle / 180.0) <= 2e-54) {
		tmp = -0.011111111111111112 * ((a - b) * ((b + a) * (Math.PI * angle)));
	} else {
		tmp = (t_0 * Math.cos((angle * (Math.PI / -180.0)))) * (2.0 * Math.sin(((Math.PI * angle) * -0.005555555555555556)));
	}
	return tmp;
}

Python

def code(a, b, angle):
	t_0 = (a - b) * (b + a)
	tmp = 0
	if (angle / 180.0) <= -5e+69:
		tmp = (2.0 * math.sin((angle * (math.pow(math.sqrt(math.pi), 2.0) / -180.0)))) * t_0
	elif (angle / 180.0) <= 2e-54:
		tmp = -0.011111111111111112 * ((a - b) * ((b + a) * (math.pi * angle)))
	else:
		tmp = (t_0 * math.cos((angle * (math.pi / -180.0)))) * (2.0 * math.sin(((math.pi * angle) * -0.005555555555555556)))
	return tmp

Julia

function code(a, b, angle)
	t_0 = Float64(Float64(a - b) * Float64(b + a))
	tmp = 0.0
	if (Float64(angle / 180.0) <= -5e+69)
		tmp = Float64(Float64(2.0 * sin(Float64(angle * Float64((sqrt(pi) ^ 2.0) / -180.0)))) * t_0);
	elseif (Float64(angle / 180.0) <= 2e-54)
		tmp = Float64(-0.011111111111111112 * Float64(Float64(a - b) * Float64(Float64(b + a) * Float64(pi * angle))));
	else
		tmp = Float64(Float64(t_0 * cos(Float64(angle * Float64(pi / -180.0)))) * Float64(2.0 * sin(Float64(Float64(pi * angle) * -0.005555555555555556))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, angle)
	t_0 = (a - b) * (b + a);
	tmp = 0.0;
	if ((angle / 180.0) <= -5e+69)
		tmp = (2.0 * sin((angle * ((sqrt(pi) ^ 2.0) / -180.0)))) * t_0;
	elseif ((angle / 180.0) <= 2e-54)
		tmp = -0.011111111111111112 * ((a - b) * ((b + a) * (pi * angle)));
	else
		tmp = (t_0 * cos((angle * (pi / -180.0)))) * (2.0 * sin(((pi * angle) * -0.005555555555555556)));
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, angle_] := Block[{t$95$0 = N[(N[(a - b), $MachinePrecision] * N[(b + a), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[(angle / 180.0), $MachinePrecision], -5e+69], N[(N[(2.0 * N[Sin[N[(angle * N[(N[Power[N[Sqrt[Pi], $MachinePrecision], 2.0], $MachinePrecision] / -180.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * t$95$0), $MachinePrecision], If[LessEqual[N[(angle / 180.0), $MachinePrecision], 2e-54], N[(-0.011111111111111112 * N[(N[(a - b), $MachinePrecision] * N[(N[(b + a), $MachinePrecision] * N[(Pi * angle), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(t$95$0 * N[Cos[N[(angle * N[(Pi / -180.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[(2.0 * N[Sin[N[(N[(Pi * angle), $MachinePrecision] * -0.005555555555555556), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if (/.f64 angle 180) < -5.00000000000000036e69

   1. Initial program 29.3%

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

   3. Simplified 26.4%

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

      1. unpow2 26.4%

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

      2. unpow2 26.4%

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

      3. difference-of-squares 26.4%

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

   5. Applied egg-rr 26.4%

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

      1. add-cbrt-cube 34.9%

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

      2. pow3 34.9%

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

   7. Applied egg-rr 34.9%

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

   8. Taylor expanded in angle around 0 31.9%

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

      1. rem-cbrt-cube 37.6%

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

      2. add-sqr-sqrt 44.1%

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

      3. pow2 44.1%

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

   10. Applied egg-rr 44.1%

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

   if -5.00000000000000036e69 < (/.f64 angle 180) < 2.0000000000000001e-54

   1. Initial program 73.0%

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

   3. Simplified 73.0%

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

      1. unpow2 73.0%

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

      2. unpow2 73.0%

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

      3. difference-of-squares 76.3%

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

   5. Applied egg-rr 76.3%

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

      1. add-cbrt-cube 75.3%

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

      2. pow3 75.3%

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

   7. Applied egg-rr 75.3%

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

   8. Taylor expanded in angle around 0 74.6%

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

      1. associate-*r* 74.5%

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

      2. associate-*r* 91.1%

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

      3. *-commutative 91.1%

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

      4. *-commutative 91.1%

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

   10. Simplified 91.1%

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

   if 2.0000000000000001e-54 < (/.f64 angle 180)

   1. Initial program 35.7%

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

   3. Simplified 41.6%

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

      1. unpow2 41.6%

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

      2. unpow2 41.6%

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

      3. difference-of-squares 45.6%

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

   5. Applied egg-rr 45.6%

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

   6. Taylor expanded in angle around inf 47.3%

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

      1. *-commutative 47.3%

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

   8. Simplified 47.3%

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

4. Final simplification 68.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{angle}{180} \leq -5 \cdot 10^{+69}:\\ \;\;\;\;\left(2 \cdot \sin \left(angle \cdot \frac{{\left(\sqrt{\pi}\right)}^{2}}{-180}\right)\right) \cdot \left(\left(a - b\right) \cdot \left(b + a\right)\right)\\ \mathbf{elif}\;\frac{angle}{180} \leq 2 \cdot 10^{-54}:\\ \;\;\;\;-0.011111111111111112 \cdot \left(\left(a - b\right) \cdot \left(\left(b + a\right) \cdot \left(\pi \cdot angle\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\left(\left(a - b\right) \cdot \left(b + a\right)\right) \cdot \cos \left(angle \cdot \frac{\pi}{-180}\right)\right) \cdot \left(2 \cdot \sin \left(\left(\pi \cdot angle\right) \cdot -0.005555555555555556\right)\right)\\ \end{array} \]

Alternative 10: 66.0% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{angle}{180} \leq -5 \cdot 10^{+69}:\\ \;\;\;\;\left(2 \cdot \sin \left(angle \cdot \frac{{\left(\sqrt{\pi}\right)}^{2}}{-180}\right)\right) \cdot \left(\left(a - b\right) \cdot \left(b + a\right)\right)\\ \mathbf{elif}\;\frac{angle}{180} \leq 2 \cdot 10^{-54}:\\ \;\;\;\;-0.011111111111111112 \cdot \left(\left(a - b\right) \cdot \left(\left(b + a\right) \cdot \left(\pi \cdot angle\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\cos \left(\pi \cdot \frac{angle}{180}\right) \cdot \left(\left(2 \cdot \left(\left(b - a\right) \cdot \left(b + a\right)\right)\right) \cdot \sin \left(\left(\pi \cdot angle\right) \cdot 0.005555555555555556\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (a b angle)
 :precision binary64
 (if (<= (/ angle 180.0) -5e+69)
   (*
    (* 2.0 (sin (* angle (/ (pow (sqrt PI) 2.0) -180.0))))
    (* (- a b) (+ b a)))
   (if (<= (/ angle 180.0) 2e-54)
     (* -0.011111111111111112 (* (- a b) (* (+ b a) (* PI angle))))
     (*
      (cos (* PI (/ angle 180.0)))
      (*
       (* 2.0 (* (- b a) (+ b a)))
       (sin (* (* PI angle) 0.005555555555555556)))))))

C

double code(double a, double b, double angle) {
	double tmp;
	if ((angle / 180.0) <= -5e+69) {
		tmp = (2.0 * sin((angle * (pow(sqrt(((double) M_PI)), 2.0) / -180.0)))) * ((a - b) * (b + a));
	} else if ((angle / 180.0) <= 2e-54) {
		tmp = -0.011111111111111112 * ((a - b) * ((b + a) * (((double) M_PI) * angle)));
	} else {
		tmp = cos((((double) M_PI) * (angle / 180.0))) * ((2.0 * ((b - a) * (b + a))) * sin(((((double) M_PI) * angle) * 0.005555555555555556)));
	}
	return tmp;
}

Java

public static double code(double a, double b, double angle) {
	double tmp;
	if ((angle / 180.0) <= -5e+69) {
		tmp = (2.0 * Math.sin((angle * (Math.pow(Math.sqrt(Math.PI), 2.0) / -180.0)))) * ((a - b) * (b + a));
	} else if ((angle / 180.0) <= 2e-54) {
		tmp = -0.011111111111111112 * ((a - b) * ((b + a) * (Math.PI * angle)));
	} else {
		tmp = Math.cos((Math.PI * (angle / 180.0))) * ((2.0 * ((b - a) * (b + a))) * Math.sin(((Math.PI * angle) * 0.005555555555555556)));
	}
	return tmp;
}

Python

def code(a, b, angle):
	tmp = 0
	if (angle / 180.0) <= -5e+69:
		tmp = (2.0 * math.sin((angle * (math.pow(math.sqrt(math.pi), 2.0) / -180.0)))) * ((a - b) * (b + a))
	elif (angle / 180.0) <= 2e-54:
		tmp = -0.011111111111111112 * ((a - b) * ((b + a) * (math.pi * angle)))
	else:
		tmp = math.cos((math.pi * (angle / 180.0))) * ((2.0 * ((b - a) * (b + a))) * math.sin(((math.pi * angle) * 0.005555555555555556)))
	return tmp

Julia

function code(a, b, angle)
	tmp = 0.0
	if (Float64(angle / 180.0) <= -5e+69)
		tmp = Float64(Float64(2.0 * sin(Float64(angle * Float64((sqrt(pi) ^ 2.0) / -180.0)))) * Float64(Float64(a - b) * Float64(b + a)));
	elseif (Float64(angle / 180.0) <= 2e-54)
		tmp = Float64(-0.011111111111111112 * Float64(Float64(a - b) * Float64(Float64(b + a) * Float64(pi * angle))));
	else
		tmp = Float64(cos(Float64(pi * Float64(angle / 180.0))) * Float64(Float64(2.0 * Float64(Float64(b - a) * Float64(b + a))) * sin(Float64(Float64(pi * angle) * 0.005555555555555556))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, angle)
	tmp = 0.0;
	if ((angle / 180.0) <= -5e+69)
		tmp = (2.0 * sin((angle * ((sqrt(pi) ^ 2.0) / -180.0)))) * ((a - b) * (b + a));
	elseif ((angle / 180.0) <= 2e-54)
		tmp = -0.011111111111111112 * ((a - b) * ((b + a) * (pi * angle)));
	else
		tmp = cos((pi * (angle / 180.0))) * ((2.0 * ((b - a) * (b + a))) * sin(((pi * angle) * 0.005555555555555556)));
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, angle_] := If[LessEqual[N[(angle / 180.0), $MachinePrecision], -5e+69], N[(N[(2.0 * N[Sin[N[(angle * N[(N[Power[N[Sqrt[Pi], $MachinePrecision], 2.0], $MachinePrecision] / -180.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[(N[(a - b), $MachinePrecision] * N[(b + a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[N[(angle / 180.0), $MachinePrecision], 2e-54], N[(-0.011111111111111112 * N[(N[(a - b), $MachinePrecision] * N[(N[(b + a), $MachinePrecision] * N[(Pi * angle), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[Cos[N[(Pi * N[(angle / 180.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * N[(N[(2.0 * N[(N[(b - a), $MachinePrecision] * N[(b + a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[Sin[N[(N[(Pi * angle), $MachinePrecision] * 0.005555555555555556), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if (/.f64 angle 180) < -5.00000000000000036e69

   1. Initial program 29.3%

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

   3. Simplified 26.4%

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

      1. unpow2 26.4%

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

      2. unpow2 26.4%

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

      3. difference-of-squares 26.4%

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

   5. Applied egg-rr 26.4%

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

      1. add-cbrt-cube 34.9%

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

      2. pow3 34.9%

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

   7. Applied egg-rr 34.9%

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

   8. Taylor expanded in angle around 0 31.9%

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

      1. rem-cbrt-cube 37.6%

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

      2. add-sqr-sqrt 44.1%

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

      3. pow2 44.1%

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

   10. Applied egg-rr 44.1%

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

   if -5.00000000000000036e69 < (/.f64 angle 180) < 2.0000000000000001e-54

   1. Initial program 73.0%

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

   3. Simplified 73.0%

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

      1. unpow2 73.0%

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

      2. unpow2 73.0%

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

      3. difference-of-squares 76.3%

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

   5. Applied egg-rr 76.3%

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

      1. add-cbrt-cube 75.3%

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

      2. pow3 75.3%

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

   7. Applied egg-rr 75.3%

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

   8. Taylor expanded in angle around 0 74.6%

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

      1. associate-*r* 74.5%

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

      2. associate-*r* 91.1%

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

      3. *-commutative 91.1%

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

      4. *-commutative 91.1%

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

   10. Simplified 91.1%

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

   if 2.0000000000000001e-54 < (/.f64 angle 180)

   1. Initial program 35.7%

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

      1. unpow2 35.7%

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

      2. unpow2 35.7%

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

      3. difference-of-squares 39.8%

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

   3. Applied egg-rr 39.8%

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

   4. Taylor expanded in angle around inf 48.8%

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

4. Final simplification 69.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{angle}{180} \leq -5 \cdot 10^{+69}:\\ \;\;\;\;\left(2 \cdot \sin \left(angle \cdot \frac{{\left(\sqrt{\pi}\right)}^{2}}{-180}\right)\right) \cdot \left(\left(a - b\right) \cdot \left(b + a\right)\right)\\ \mathbf{elif}\;\frac{angle}{180} \leq 2 \cdot 10^{-54}:\\ \;\;\;\;-0.011111111111111112 \cdot \left(\left(a - b\right) \cdot \left(\left(b + a\right) \cdot \left(\pi \cdot angle\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\cos \left(\pi \cdot \frac{angle}{180}\right) \cdot \left(\left(2 \cdot \left(\left(b - a\right) \cdot \left(b + a\right)\right)\right) \cdot \sin \left(\left(\pi \cdot angle\right) \cdot 0.005555555555555556\right)\right)\\ \end{array} \]

Alternative 11: 64.9% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(a - b\right) \cdot \left(b + a\right)\\ \mathbf{if}\;\frac{angle}{180} \leq -5 \cdot 10^{+69}:\\ \;\;\;\;\left(2 \cdot \sin \left(angle \cdot \frac{{\left(\sqrt{\pi}\right)}^{2}}{-180}\right)\right) \cdot t_0\\ \mathbf{elif}\;\frac{angle}{180} \leq 2 \cdot 10^{+24}:\\ \;\;\;\;-0.011111111111111112 \cdot \left(\left(a - b\right) \cdot \left(\left(b + a\right) \cdot \left(\pi \cdot angle\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;t_0 \cdot \left(2 \cdot \sin \left(angle \cdot \frac{\sqrt[3]{{\pi}^{3}}}{-180}\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (a b angle)
 :precision binary64
 (let* ((t_0 (* (- a b) (+ b a))))
   (if (<= (/ angle 180.0) -5e+69)
     (* (* 2.0 (sin (* angle (/ (pow (sqrt PI) 2.0) -180.0)))) t_0)
     (if (<= (/ angle 180.0) 2e+24)
       (* -0.011111111111111112 (* (- a b) (* (+ b a) (* PI angle))))
       (* t_0 (* 2.0 (sin (* angle (/ (cbrt (pow PI 3.0)) -180.0)))))))))

C

double code(double a, double b, double angle) {
	double t_0 = (a - b) * (b + a);
	double tmp;
	if ((angle / 180.0) <= -5e+69) {
		tmp = (2.0 * sin((angle * (pow(sqrt(((double) M_PI)), 2.0) / -180.0)))) * t_0;
	} else if ((angle / 180.0) <= 2e+24) {
		tmp = -0.011111111111111112 * ((a - b) * ((b + a) * (((double) M_PI) * angle)));
	} else {
		tmp = t_0 * (2.0 * sin((angle * (cbrt(pow(((double) M_PI), 3.0)) / -180.0))));
	}
	return tmp;
}

Java

public static double code(double a, double b, double angle) {
	double t_0 = (a - b) * (b + a);
	double tmp;
	if ((angle / 180.0) <= -5e+69) {
		tmp = (2.0 * Math.sin((angle * (Math.pow(Math.sqrt(Math.PI), 2.0) / -180.0)))) * t_0;
	} else if ((angle / 180.0) <= 2e+24) {
		tmp = -0.011111111111111112 * ((a - b) * ((b + a) * (Math.PI * angle)));
	} else {
		tmp = t_0 * (2.0 * Math.sin((angle * (Math.cbrt(Math.pow(Math.PI, 3.0)) / -180.0))));
	}
	return tmp;
}

Julia

function code(a, b, angle)
	t_0 = Float64(Float64(a - b) * Float64(b + a))
	tmp = 0.0
	if (Float64(angle / 180.0) <= -5e+69)
		tmp = Float64(Float64(2.0 * sin(Float64(angle * Float64((sqrt(pi) ^ 2.0) / -180.0)))) * t_0);
	elseif (Float64(angle / 180.0) <= 2e+24)
		tmp = Float64(-0.011111111111111112 * Float64(Float64(a - b) * Float64(Float64(b + a) * Float64(pi * angle))));
	else
		tmp = Float64(t_0 * Float64(2.0 * sin(Float64(angle * Float64(cbrt((pi ^ 3.0)) / -180.0)))));
	end
	return tmp
end

Wolfram

code[a_, b_, angle_] := Block[{t$95$0 = N[(N[(a - b), $MachinePrecision] * N[(b + a), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[(angle / 180.0), $MachinePrecision], -5e+69], N[(N[(2.0 * N[Sin[N[(angle * N[(N[Power[N[Sqrt[Pi], $MachinePrecision], 2.0], $MachinePrecision] / -180.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * t$95$0), $MachinePrecision], If[LessEqual[N[(angle / 180.0), $MachinePrecision], 2e+24], N[(-0.011111111111111112 * N[(N[(a - b), $MachinePrecision] * N[(N[(b + a), $MachinePrecision] * N[(Pi * angle), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(t$95$0 * N[(2.0 * N[Sin[N[(angle * N[(N[Power[N[Power[Pi, 3.0], $MachinePrecision], 1/3], $MachinePrecision] / -180.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if (/.f64 angle 180) < -5.00000000000000036e69

   1. Initial program 29.3%

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

   3. Simplified 26.4%

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

      1. unpow2 26.4%

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

      2. unpow2 26.4%

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

      3. difference-of-squares 26.4%

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

   5. Applied egg-rr 26.4%

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

      1. add-cbrt-cube 34.9%

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

      2. pow3 34.9%

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

   7. Applied egg-rr 34.9%

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

   8. Taylor expanded in angle around 0 31.9%

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

      1. rem-cbrt-cube 37.6%

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

      2. add-sqr-sqrt 44.1%

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

      3. pow2 44.1%

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

   10. Applied egg-rr 44.1%

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

   if -5.00000000000000036e69 < (/.f64 angle 180) < 2e24

   1. Initial program 73.1%

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

   3. Simplified 73.0%

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

      1. unpow2 73.0%

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

      2. unpow2 73.0%

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

      3. difference-of-squares 77.1%

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

   5. Applied egg-rr 77.1%

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

      1. add-cbrt-cube 77.0%

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

      2. pow3 77.0%

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

   7. Applied egg-rr 77.0%

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

   8. Taylor expanded in angle around 0 74.8%

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

      1. associate-*r* 74.7%

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

      2. associate-*r* 89.0%

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

      3. *-commutative 89.0%

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

      4. *-commutative 89.0%

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

   10. Simplified 89.0%

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

   if 2e24 < (/.f64 angle 180)

   1. Initial program 21.7%

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

   3. Simplified 30.1%

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

      1. unpow2 30.1%

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

      2. unpow2 30.1%

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

      3. difference-of-squares 32.0%

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

   5. Applied egg-rr 32.0%

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

      1. add-cbrt-cube 33.9%

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

      2. pow3 33.9%

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

   7. Applied egg-rr 33.9%

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

   8. Taylor expanded in angle around 0 36.7%

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

4. Final simplification 68.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{angle}{180} \leq -5 \cdot 10^{+69}:\\ \;\;\;\;\left(2 \cdot \sin \left(angle \cdot \frac{{\left(\sqrt{\pi}\right)}^{2}}{-180}\right)\right) \cdot \left(\left(a - b\right) \cdot \left(b + a\right)\right)\\ \mathbf{elif}\;\frac{angle}{180} \leq 2 \cdot 10^{+24}:\\ \;\;\;\;-0.011111111111111112 \cdot \left(\left(a - b\right) \cdot \left(\left(b + a\right) \cdot \left(\pi \cdot angle\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\left(a - b\right) \cdot \left(b + a\right)\right) \cdot \left(2 \cdot \sin \left(angle \cdot \frac{\sqrt[3]{{\pi}^{3}}}{-180}\right)\right)\\ \end{array} \]

Alternative 12: 64.3% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \sin \left(\left(\pi \cdot angle\right) \cdot 0.011111111111111112\right)\\ \mathbf{if}\;\frac{angle}{180} \leq -2 \cdot 10^{+183}:\\ \;\;\;\;\left(\left(a - b\right) \cdot \left(b + a\right)\right) \cdot \left(2 \cdot \sin \left(\left(\pi \cdot angle\right) \cdot -0.005555555555555556\right)\right)\\ \mathbf{elif}\;\frac{angle}{180} \leq -2 \cdot 10^{+76}:\\ \;\;\;\;\left({a}^{2} - {b}^{2}\right) \cdot t_0\\ \mathbf{elif}\;\frac{angle}{180} \leq -1 \cdot 10^{-58}:\\ \;\;\;\;t_0 \cdot \left({b}^{2} - {a}^{2}\right)\\ \mathbf{elif}\;\frac{angle}{180} \leq 2 \cdot 10^{+16}:\\ \;\;\;\;-0.011111111111111112 \cdot \left(\left(a - b\right) \cdot \left(\left(b + a\right) \cdot \left(\pi \cdot angle\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\cos \left(\pi \cdot \frac{angle}{180}\right) \cdot \left(0.011111111111111112 \cdot \left(angle \cdot \left(\pi \cdot \left(\left(b - a\right) \cdot \left(b + a\right)\right)\right)\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (a b angle)
 :precision binary64
 (let* ((t_0 (sin (* (* PI angle) 0.011111111111111112))))
   (if (<= (/ angle 180.0) -2e+183)
     (*
      (* (- a b) (+ b a))
      (* 2.0 (sin (* (* PI angle) -0.005555555555555556))))
     (if (<= (/ angle 180.0) -2e+76)
       (* (- (pow a 2.0) (pow b 2.0)) t_0)
       (if (<= (/ angle 180.0) -1e-58)
         (* t_0 (- (pow b 2.0) (pow a 2.0)))
         (if (<= (/ angle 180.0) 2e+16)
           (* -0.011111111111111112 (* (- a b) (* (+ b a) (* PI angle))))
           (*
            (cos (* PI (/ angle 180.0)))
            (*
             0.011111111111111112
             (* angle (* PI (* (- b a) (+ b a))))))))))))

C

double code(double a, double b, double angle) {
	double t_0 = sin(((((double) M_PI) * angle) * 0.011111111111111112));
	double tmp;
	if ((angle / 180.0) <= -2e+183) {
		tmp = ((a - b) * (b + a)) * (2.0 * sin(((((double) M_PI) * angle) * -0.005555555555555556)));
	} else if ((angle / 180.0) <= -2e+76) {
		tmp = (pow(a, 2.0) - pow(b, 2.0)) * t_0;
	} else if ((angle / 180.0) <= -1e-58) {
		tmp = t_0 * (pow(b, 2.0) - pow(a, 2.0));
	} else if ((angle / 180.0) <= 2e+16) {
		tmp = -0.011111111111111112 * ((a - b) * ((b + a) * (((double) M_PI) * angle)));
	} else {
		tmp = cos((((double) M_PI) * (angle / 180.0))) * (0.011111111111111112 * (angle * (((double) M_PI) * ((b - a) * (b + a)))));
	}
	return tmp;
}

Java

public static double code(double a, double b, double angle) {
	double t_0 = Math.sin(((Math.PI * angle) * 0.011111111111111112));
	double tmp;
	if ((angle / 180.0) <= -2e+183) {
		tmp = ((a - b) * (b + a)) * (2.0 * Math.sin(((Math.PI * angle) * -0.005555555555555556)));
	} else if ((angle / 180.0) <= -2e+76) {
		tmp = (Math.pow(a, 2.0) - Math.pow(b, 2.0)) * t_0;
	} else if ((angle / 180.0) <= -1e-58) {
		tmp = t_0 * (Math.pow(b, 2.0) - Math.pow(a, 2.0));
	} else if ((angle / 180.0) <= 2e+16) {
		tmp = -0.011111111111111112 * ((a - b) * ((b + a) * (Math.PI * angle)));
	} else {
		tmp = Math.cos((Math.PI * (angle / 180.0))) * (0.011111111111111112 * (angle * (Math.PI * ((b - a) * (b + a)))));
	}
	return tmp;
}

Python

def code(a, b, angle):
	t_0 = math.sin(((math.pi * angle) * 0.011111111111111112))
	tmp = 0
	if (angle / 180.0) <= -2e+183:
		tmp = ((a - b) * (b + a)) * (2.0 * math.sin(((math.pi * angle) * -0.005555555555555556)))
	elif (angle / 180.0) <= -2e+76:
		tmp = (math.pow(a, 2.0) - math.pow(b, 2.0)) * t_0
	elif (angle / 180.0) <= -1e-58:
		tmp = t_0 * (math.pow(b, 2.0) - math.pow(a, 2.0))
	elif (angle / 180.0) <= 2e+16:
		tmp = -0.011111111111111112 * ((a - b) * ((b + a) * (math.pi * angle)))
	else:
		tmp = math.cos((math.pi * (angle / 180.0))) * (0.011111111111111112 * (angle * (math.pi * ((b - a) * (b + a)))))
	return tmp

Julia

function code(a, b, angle)
	t_0 = sin(Float64(Float64(pi * angle) * 0.011111111111111112))
	tmp = 0.0
	if (Float64(angle / 180.0) <= -2e+183)
		tmp = Float64(Float64(Float64(a - b) * Float64(b + a)) * Float64(2.0 * sin(Float64(Float64(pi * angle) * -0.005555555555555556))));
	elseif (Float64(angle / 180.0) <= -2e+76)
		tmp = Float64(Float64((a ^ 2.0) - (b ^ 2.0)) * t_0);
	elseif (Float64(angle / 180.0) <= -1e-58)
		tmp = Float64(t_0 * Float64((b ^ 2.0) - (a ^ 2.0)));
	elseif (Float64(angle / 180.0) <= 2e+16)
		tmp = Float64(-0.011111111111111112 * Float64(Float64(a - b) * Float64(Float64(b + a) * Float64(pi * angle))));
	else
		tmp = Float64(cos(Float64(pi * Float64(angle / 180.0))) * Float64(0.011111111111111112 * Float64(angle * Float64(pi * Float64(Float64(b - a) * Float64(b + a))))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, angle)
	t_0 = sin(((pi * angle) * 0.011111111111111112));
	tmp = 0.0;
	if ((angle / 180.0) <= -2e+183)
		tmp = ((a - b) * (b + a)) * (2.0 * sin(((pi * angle) * -0.005555555555555556)));
	elseif ((angle / 180.0) <= -2e+76)
		tmp = ((a ^ 2.0) - (b ^ 2.0)) * t_0;
	elseif ((angle / 180.0) <= -1e-58)
		tmp = t_0 * ((b ^ 2.0) - (a ^ 2.0));
	elseif ((angle / 180.0) <= 2e+16)
		tmp = -0.011111111111111112 * ((a - b) * ((b + a) * (pi * angle)));
	else
		tmp = cos((pi * (angle / 180.0))) * (0.011111111111111112 * (angle * (pi * ((b - a) * (b + a)))));
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, angle_] := Block[{t$95$0 = N[Sin[N[(N[(Pi * angle), $MachinePrecision] * 0.011111111111111112), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[N[(angle / 180.0), $MachinePrecision], -2e+183], N[(N[(N[(a - b), $MachinePrecision] * N[(b + a), $MachinePrecision]), $MachinePrecision] * N[(2.0 * N[Sin[N[(N[(Pi * angle), $MachinePrecision] * -0.005555555555555556), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[N[(angle / 180.0), $MachinePrecision], -2e+76], N[(N[(N[Power[a, 2.0], $MachinePrecision] - N[Power[b, 2.0], $MachinePrecision]), $MachinePrecision] * t$95$0), $MachinePrecision], If[LessEqual[N[(angle / 180.0), $MachinePrecision], -1e-58], N[(t$95$0 * N[(N[Power[b, 2.0], $MachinePrecision] - N[Power[a, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[N[(angle / 180.0), $MachinePrecision], 2e+16], N[(-0.011111111111111112 * N[(N[(a - b), $MachinePrecision] * N[(N[(b + a), $MachinePrecision] * N[(Pi * angle), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[Cos[N[(Pi * N[(angle / 180.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * N[(0.011111111111111112 * N[(angle * N[(Pi * N[(N[(b - a), $MachinePrecision] * N[(b + a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]]

Derivation

1. Split input into 5 regimes

2. if (/.f64 angle 180) < -1.99999999999999989e183

   1. Initial program 28.7%

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

   3. Simplified 28.2%

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

      1. unpow2 28.2%

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

      2. unpow2 28.2%

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

      3. difference-of-squares 28.2%

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

   5. Applied egg-rr 28.2%

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

      1. add-cbrt-cube 35.4%

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

      2. pow3 35.4%

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

   7. Applied egg-rr 35.4%

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

   8. Taylor expanded in angle around 0 31.7%

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

   9. Taylor expanded in angle around inf 42.1%

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

       1. associate-*r* 42.1%

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

       2. *-commutative 42.1%

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

       3. +-commutative 42.1%

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

   11. Simplified 42.1%

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

   if -1.99999999999999989e183 < (/.f64 angle 180) < -2.0000000000000001e76

   1. Initial program 28.0%

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

   3. Simplified 20.9%

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

   5. Applied egg-rr 31.7%

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

      1. distribute-lft-out 31.7%

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

      2. sub-neg 31.7%

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

      3. *-commutative 31.7%

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

      4. count-2 31.7%

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

      5. *-commutative 31.7%

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

      6. *-commutative 31.7%

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

      7. associate-*r* 43.9%

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

      8. *-commutative 43.9%

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

      9. *-commutative 43.9%

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

      10. associate-*r* 49.1%

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

      11. distribute-rgt-out 49.1%

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

      12. metadata-eval 49.1%

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

   7. Simplified 49.1%

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

   if -2.0000000000000001e76 < (/.f64 angle 180) < -1e-58

   1. Initial program 56.8%

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

      1. *-commutative 56.8%

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

      2. associate-*l* 56.8%

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

      3. associate-*l* 56.9%

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

   3. Simplified 56.9%

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

      1. add-cbrt-cube 40.8%

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

      2. pow1/3 24.5%

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

   5. Applied egg-rr 24.1%

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

   6. Taylor expanded in angle around inf 57.5%

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

      1. *-commutative 57.5%

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

   8. Simplified 57.5%

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

   if -1e-58 < (/.f64 angle 180) < 2e16

   1. Initial program 75.4%

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

   3. Simplified 75.3%

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

      1. unpow2 75.3%

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

      2. unpow2 75.3%

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

      3. difference-of-squares 80.1%

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

   5. Applied egg-rr 80.1%

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

      1. add-cbrt-cube 80.6%

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

      2. pow3 80.6%

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

   7. Applied egg-rr 80.6%

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

   8. Taylor expanded in angle around 0 79.1%

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

      1. associate-*r* 79.0%

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

      2. associate-*r* 95.4%

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

      3. *-commutative 95.4%

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

      4. *-commutative 95.4%

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

   10. Simplified 95.4%

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

   if 2e16 < (/.f64 angle 180)

   1. Initial program 22.7%

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

      1. unpow2 22.7%

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

      2. unpow2 22.7%

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

      3. difference-of-squares 24.5%

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

   3. Applied egg-rr 24.5%

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

   4. Taylor expanded in angle around 0 36.2%

      \[\leadsto \color{blue}{\left(0.011111111111111112 \cdot \left(angle \cdot \left(\pi \cdot \left(\left(a + b\right) \cdot \left(b - a\right)\right)\right)\right)\right)} \cdot \cos \left(\pi \cdot \frac{angle}{180}\right) \]
3. Recombined 5 regimes into one program.

4. Final simplification 69.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{angle}{180} \leq -2 \cdot 10^{+183}:\\ \;\;\;\;\left(\left(a - b\right) \cdot \left(b + a\right)\right) \cdot \left(2 \cdot \sin \left(\left(\pi \cdot angle\right) \cdot -0.005555555555555556\right)\right)\\ \mathbf{elif}\;\frac{angle}{180} \leq -2 \cdot 10^{+76}:\\ \;\;\;\;\left({a}^{2} - {b}^{2}\right) \cdot \sin \left(\left(\pi \cdot angle\right) \cdot 0.011111111111111112\right)\\ \mathbf{elif}\;\frac{angle}{180} \leq -1 \cdot 10^{-58}:\\ \;\;\;\;\sin \left(\left(\pi \cdot angle\right) \cdot 0.011111111111111112\right) \cdot \left({b}^{2} - {a}^{2}\right)\\ \mathbf{elif}\;\frac{angle}{180} \leq 2 \cdot 10^{+16}:\\ \;\;\;\;-0.011111111111111112 \cdot \left(\left(a - b\right) \cdot \left(\left(b + a\right) \cdot \left(\pi \cdot angle\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\cos \left(\pi \cdot \frac{angle}{180}\right) \cdot \left(0.011111111111111112 \cdot \left(angle \cdot \left(\pi \cdot \left(\left(b - a\right) \cdot \left(b + a\right)\right)\right)\right)\right)\\ \end{array} \]

Alternative 13: 64.7% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;angle \leq -3.3 \cdot 10^{-56}:\\ \;\;\;\;\left(\left(a - b\right) \cdot \left(b + a\right)\right) \cdot \left(2 \cdot \sin \left(angle \cdot \frac{\pi}{-180}\right)\right)\\ \mathbf{elif}\;angle \leq 1.02 \cdot 10^{+20}:\\ \;\;\;\;-0.011111111111111112 \cdot \left(\left(a - b\right) \cdot \left(\left(b + a\right) \cdot \left(\pi \cdot angle\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\cos \left(\pi \cdot \frac{angle}{180}\right) \cdot \left(0.011111111111111112 \cdot \left(angle \cdot \left(\pi \cdot \left(\left(b - a\right) \cdot \left(b + a\right)\right)\right)\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (a b angle)
 :precision binary64
 (if (<= angle -3.3e-56)
   (* (* (- a b) (+ b a)) (* 2.0 (sin (* angle (/ PI -180.0)))))
   (if (<= angle 1.02e+20)
     (* -0.011111111111111112 (* (- a b) (* (+ b a) (* PI angle))))
     (*
      (cos (* PI (/ angle 180.0)))
      (* 0.011111111111111112 (* angle (* PI (* (- b a) (+ b a)))))))))

C

double code(double a, double b, double angle) {
	double tmp;
	if (angle <= -3.3e-56) {
		tmp = ((a - b) * (b + a)) * (2.0 * sin((angle * (((double) M_PI) / -180.0))));
	} else if (angle <= 1.02e+20) {
		tmp = -0.011111111111111112 * ((a - b) * ((b + a) * (((double) M_PI) * angle)));
	} else {
		tmp = cos((((double) M_PI) * (angle / 180.0))) * (0.011111111111111112 * (angle * (((double) M_PI) * ((b - a) * (b + a)))));
	}
	return tmp;
}

Java

public static double code(double a, double b, double angle) {
	double tmp;
	if (angle <= -3.3e-56) {
		tmp = ((a - b) * (b + a)) * (2.0 * Math.sin((angle * (Math.PI / -180.0))));
	} else if (angle <= 1.02e+20) {
		tmp = -0.011111111111111112 * ((a - b) * ((b + a) * (Math.PI * angle)));
	} else {
		tmp = Math.cos((Math.PI * (angle / 180.0))) * (0.011111111111111112 * (angle * (Math.PI * ((b - a) * (b + a)))));
	}
	return tmp;
}

Python

def code(a, b, angle):
	tmp = 0
	if angle <= -3.3e-56:
		tmp = ((a - b) * (b + a)) * (2.0 * math.sin((angle * (math.pi / -180.0))))
	elif angle <= 1.02e+20:
		tmp = -0.011111111111111112 * ((a - b) * ((b + a) * (math.pi * angle)))
	else:
		tmp = math.cos((math.pi * (angle / 180.0))) * (0.011111111111111112 * (angle * (math.pi * ((b - a) * (b + a)))))
	return tmp

Julia

function code(a, b, angle)
	tmp = 0.0
	if (angle <= -3.3e-56)
		tmp = Float64(Float64(Float64(a - b) * Float64(b + a)) * Float64(2.0 * sin(Float64(angle * Float64(pi / -180.0)))));
	elseif (angle <= 1.02e+20)
		tmp = Float64(-0.011111111111111112 * Float64(Float64(a - b) * Float64(Float64(b + a) * Float64(pi * angle))));
	else
		tmp = Float64(cos(Float64(pi * Float64(angle / 180.0))) * Float64(0.011111111111111112 * Float64(angle * Float64(pi * Float64(Float64(b - a) * Float64(b + a))))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, angle)
	tmp = 0.0;
	if (angle <= -3.3e-56)
		tmp = ((a - b) * (b + a)) * (2.0 * sin((angle * (pi / -180.0))));
	elseif (angle <= 1.02e+20)
		tmp = -0.011111111111111112 * ((a - b) * ((b + a) * (pi * angle)));
	else
		tmp = cos((pi * (angle / 180.0))) * (0.011111111111111112 * (angle * (pi * ((b - a) * (b + a)))));
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, angle_] := If[LessEqual[angle, -3.3e-56], N[(N[(N[(a - b), $MachinePrecision] * N[(b + a), $MachinePrecision]), $MachinePrecision] * N[(2.0 * N[Sin[N[(angle * N[(Pi / -180.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[angle, 1.02e+20], N[(-0.011111111111111112 * N[(N[(a - b), $MachinePrecision] * N[(N[(b + a), $MachinePrecision] * N[(Pi * angle), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[Cos[N[(Pi * N[(angle / 180.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * N[(0.011111111111111112 * N[(angle * N[(Pi * N[(N[(b - a), $MachinePrecision] * N[(b + a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if angle < -3.29999999999999984e-56

   1. Initial program 36.2%

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

   3. Simplified 34.1%

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

      1. unpow2 34.1%

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

      2. unpow2 34.1%

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

      3. difference-of-squares 34.1%

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

   5. Applied egg-rr 34.1%

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

   6. Taylor expanded in angle around 0 41.5%

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

   if -3.29999999999999984e-56 < angle < 1.02e20

   1. Initial program 75.4%

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

   3. Simplified 75.3%

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

      1. unpow2 75.3%

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

      2. unpow2 75.3%

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

      3. difference-of-squares 80.1%

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

   5. Applied egg-rr 80.1%

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

      1. add-cbrt-cube 80.6%

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

      2. pow3 80.6%

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

   7. Applied egg-rr 80.6%

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

   8. Taylor expanded in angle around 0 79.1%

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

      1. associate-*r* 79.0%

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

      2. associate-*r* 95.4%

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

      3. *-commutative 95.4%

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

      4. *-commutative 95.4%

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

   10. Simplified 95.4%

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

   if 1.02e20 < angle

   1. Initial program 22.7%

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

      1. unpow2 22.7%

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

      2. unpow2 22.7%

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

      3. difference-of-squares 24.5%

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

   3. Applied egg-rr 24.5%

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

   4. Taylor expanded in angle around 0 36.2%

      \[\leadsto \color{blue}{\left(0.011111111111111112 \cdot \left(angle \cdot \left(\pi \cdot \left(\left(a + b\right) \cdot \left(b - a\right)\right)\right)\right)\right)} \cdot \cos \left(\pi \cdot \frac{angle}{180}\right) \]
3. Recombined 3 regimes into one program.

4. Final simplification 67.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;angle \leq -3.3 \cdot 10^{-56}:\\ \;\;\;\;\left(\left(a - b\right) \cdot \left(b + a\right)\right) \cdot \left(2 \cdot \sin \left(angle \cdot \frac{\pi}{-180}\right)\right)\\ \mathbf{elif}\;angle \leq 1.02 \cdot 10^{+20}:\\ \;\;\;\;-0.011111111111111112 \cdot \left(\left(a - b\right) \cdot \left(\left(b + a\right) \cdot \left(\pi \cdot angle\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\cos \left(\pi \cdot \frac{angle}{180}\right) \cdot \left(0.011111111111111112 \cdot \left(angle \cdot \left(\pi \cdot \left(\left(b - a\right) \cdot \left(b + a\right)\right)\right)\right)\right)\\ \end{array} \]

Alternative 14: 64.1% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(a - b\right) \cdot \left(b + a\right)\\ \mathbf{if}\;angle \leq -2.7 \cdot 10^{-56}:\\ \;\;\;\;t_0 \cdot \left(2 \cdot \sin \left(angle \cdot \frac{\pi}{-180}\right)\right)\\ \mathbf{elif}\;angle \leq 1.3 \cdot 10^{+23}:\\ \;\;\;\;-0.011111111111111112 \cdot \left(\left(a - b\right) \cdot \left(\left(b + a\right) \cdot \left(\pi \cdot angle\right)\right)\right)\\ \mathbf{elif}\;angle \leq 3.5 \cdot 10^{+169}:\\ \;\;\;\;t_0 \cdot \left(2 \cdot \left|\sin \left(\pi \cdot \left(angle \cdot -0.005555555555555556\right)\right)\right|\right)\\ \mathbf{else}:\\ \;\;\;\;-0.011111111111111112 \cdot \left(\left(a - b\right) \cdot \left(a \cdot \left(\pi \cdot angle\right)\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (a b angle)
 :precision binary64
 (let* ((t_0 (* (- a b) (+ b a))))
   (if (<= angle -2.7e-56)
     (* t_0 (* 2.0 (sin (* angle (/ PI -180.0)))))
     (if (<= angle 1.3e+23)
       (* -0.011111111111111112 (* (- a b) (* (+ b a) (* PI angle))))
       (if (<= angle 3.5e+169)
         (* t_0 (* 2.0 (fabs (sin (* PI (* angle -0.005555555555555556))))))
         (* -0.011111111111111112 (* (- a b) (* a (* PI angle)))))))))

C

double code(double a, double b, double angle) {
	double t_0 = (a - b) * (b + a);
	double tmp;
	if (angle <= -2.7e-56) {
		tmp = t_0 * (2.0 * sin((angle * (((double) M_PI) / -180.0))));
	} else if (angle <= 1.3e+23) {
		tmp = -0.011111111111111112 * ((a - b) * ((b + a) * (((double) M_PI) * angle)));
	} else if (angle <= 3.5e+169) {
		tmp = t_0 * (2.0 * fabs(sin((((double) M_PI) * (angle * -0.005555555555555556)))));
	} else {
		tmp = -0.011111111111111112 * ((a - b) * (a * (((double) M_PI) * angle)));
	}
	return tmp;
}

Java

public static double code(double a, double b, double angle) {
	double t_0 = (a - b) * (b + a);
	double tmp;
	if (angle <= -2.7e-56) {
		tmp = t_0 * (2.0 * Math.sin((angle * (Math.PI / -180.0))));
	} else if (angle <= 1.3e+23) {
		tmp = -0.011111111111111112 * ((a - b) * ((b + a) * (Math.PI * angle)));
	} else if (angle <= 3.5e+169) {
		tmp = t_0 * (2.0 * Math.abs(Math.sin((Math.PI * (angle * -0.005555555555555556)))));
	} else {
		tmp = -0.011111111111111112 * ((a - b) * (a * (Math.PI * angle)));
	}
	return tmp;
}

Python

def code(a, b, angle):
	t_0 = (a - b) * (b + a)
	tmp = 0
	if angle <= -2.7e-56:
		tmp = t_0 * (2.0 * math.sin((angle * (math.pi / -180.0))))
	elif angle <= 1.3e+23:
		tmp = -0.011111111111111112 * ((a - b) * ((b + a) * (math.pi * angle)))
	elif angle <= 3.5e+169:
		tmp = t_0 * (2.0 * math.fabs(math.sin((math.pi * (angle * -0.005555555555555556)))))
	else:
		tmp = -0.011111111111111112 * ((a - b) * (a * (math.pi * angle)))
	return tmp

Julia

function code(a, b, angle)
	t_0 = Float64(Float64(a - b) * Float64(b + a))
	tmp = 0.0
	if (angle <= -2.7e-56)
		tmp = Float64(t_0 * Float64(2.0 * sin(Float64(angle * Float64(pi / -180.0)))));
	elseif (angle <= 1.3e+23)
		tmp = Float64(-0.011111111111111112 * Float64(Float64(a - b) * Float64(Float64(b + a) * Float64(pi * angle))));
	elseif (angle <= 3.5e+169)
		tmp = Float64(t_0 * Float64(2.0 * abs(sin(Float64(pi * Float64(angle * -0.005555555555555556))))));
	else
		tmp = Float64(-0.011111111111111112 * Float64(Float64(a - b) * Float64(a * Float64(pi * angle))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, angle)
	t_0 = (a - b) * (b + a);
	tmp = 0.0;
	if (angle <= -2.7e-56)
		tmp = t_0 * (2.0 * sin((angle * (pi / -180.0))));
	elseif (angle <= 1.3e+23)
		tmp = -0.011111111111111112 * ((a - b) * ((b + a) * (pi * angle)));
	elseif (angle <= 3.5e+169)
		tmp = t_0 * (2.0 * abs(sin((pi * (angle * -0.005555555555555556)))));
	else
		tmp = -0.011111111111111112 * ((a - b) * (a * (pi * angle)));
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, angle_] := Block[{t$95$0 = N[(N[(a - b), $MachinePrecision] * N[(b + a), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[angle, -2.7e-56], N[(t$95$0 * N[(2.0 * N[Sin[N[(angle * N[(Pi / -180.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[angle, 1.3e+23], N[(-0.011111111111111112 * N[(N[(a - b), $MachinePrecision] * N[(N[(b + a), $MachinePrecision] * N[(Pi * angle), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[angle, 3.5e+169], N[(t$95$0 * N[(2.0 * N[Abs[N[Sin[N[(Pi * N[(angle * -0.005555555555555556), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(-0.011111111111111112 * N[(N[(a - b), $MachinePrecision] * N[(a * N[(Pi * angle), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 4 regimes

2. if angle < -2.69999999999999995e-56

   1. Initial program 36.2%

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

   3. Simplified 34.1%

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

      1. unpow2 34.1%

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

      2. unpow2 34.1%

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

      3. difference-of-squares 34.1%

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

   5. Applied egg-rr 34.1%

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

   6. Taylor expanded in angle around 0 41.5%

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

   if -2.69999999999999995e-56 < angle < 1.29999999999999996e23

   1. Initial program 75.4%

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

   3. Simplified 75.3%

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

      1. unpow2 75.3%

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

      2. unpow2 75.3%

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

      3. difference-of-squares 80.1%

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

   5. Applied egg-rr 80.1%

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

      1. add-cbrt-cube 80.6%

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

      2. pow3 80.6%

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

   7. Applied egg-rr 80.6%

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

   8. Taylor expanded in angle around 0 79.1%

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

      1. associate-*r* 79.0%

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

      2. associate-*r* 95.4%

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

      3. *-commutative 95.4%

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

      4. *-commutative 95.4%

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

   10. Simplified 95.4%

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

   if 1.29999999999999996e23 < angle < 3.50000000000000019e169

   1. Initial program 25.9%

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

   3. Simplified 32.2%

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

      1. unpow2 32.2%

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

      2. unpow2 32.2%

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

      3. difference-of-squares 32.3%

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

   5. Applied egg-rr 32.3%

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

      1. add-cbrt-cube 37.3%

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

      2. pow3 37.3%

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

   7. Applied egg-rr 37.3%

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

   8. Taylor expanded in angle around 0 38.3%

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

      1. rem-cbrt-cube 32.0%

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

      2. add-sqr-sqrt 16.8%

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

      3. sqrt-unprod 37.9%

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

      4. pow2 37.9%

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

      5. rem-cbrt-cube 37.8%

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

      6. rem-cbrt-cube 37.9%

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

      7. div-inv 37.9%

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

      8. metadata-eval 37.9%

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

   10. Applied egg-rr 37.9%

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

       1. unpow2 37.9%

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

       2. rem-sqrt-square 37.9%

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

       3. associate-*r* 37.9%

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

       4. *-commutative 37.9%

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

       5. associate-*r* 37.9%

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

       6. *-commutative 37.9%

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

       7. *-commutative 37.9%

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

   12. Simplified 37.9%

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

   if 3.50000000000000019e169 < angle

   1. Initial program 18.3%

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

   3. Simplified 28.9%

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

      1. unpow2 28.9%

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

      2. unpow2 28.9%

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

      3. difference-of-squares 33.1%

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

   5. Applied egg-rr 33.1%

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

      1. add-cbrt-cube 30.9%

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

      2. pow3 30.9%

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

   7. Applied egg-rr 30.9%

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

   8. Taylor expanded in angle around 0 31.3%

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

      1. associate-*r* 31.3%

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

      2. associate-*r* 31.3%

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

      3. *-commutative 31.3%

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

      4. *-commutative 31.3%

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

   10. Simplified 31.3%

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

   11. Taylor expanded in a around inf 27.1%

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

4. Final simplification 67.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;angle \leq -2.7 \cdot 10^{-56}:\\ \;\;\;\;\left(\left(a - b\right) \cdot \left(b + a\right)\right) \cdot \left(2 \cdot \sin \left(angle \cdot \frac{\pi}{-180}\right)\right)\\ \mathbf{elif}\;angle \leq 1.3 \cdot 10^{+23}:\\ \;\;\;\;-0.011111111111111112 \cdot \left(\left(a - b\right) \cdot \left(\left(b + a\right) \cdot \left(\pi \cdot angle\right)\right)\right)\\ \mathbf{elif}\;angle \leq 3.5 \cdot 10^{+169}:\\ \;\;\;\;\left(\left(a - b\right) \cdot \left(b + a\right)\right) \cdot \left(2 \cdot \left|\sin \left(\pi \cdot \left(angle \cdot -0.005555555555555556\right)\right)\right|\right)\\ \mathbf{else}:\\ \;\;\;\;-0.011111111111111112 \cdot \left(\left(a - b\right) \cdot \left(a \cdot \left(\pi \cdot angle\right)\right)\right)\\ \end{array} \]

Alternative 15: 61.6% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;{b}^{2} \leq 5 \cdot 10^{-67}:\\ \;\;\;\;\left(\left(a - b\right) \cdot \left(b + a\right)\right) \cdot \left(2 \cdot \sin \left(\left(\pi \cdot angle\right) \cdot -0.005555555555555556\right)\right)\\ \mathbf{else}:\\ \;\;\;\;-0.011111111111111112 \cdot \left(\left(a - b\right) \cdot \left(\left(b + a\right) \cdot \left(\pi \cdot angle\right)\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (a b angle)
 :precision binary64
 (if (<= (pow b 2.0) 5e-67)
   (* (* (- a b) (+ b a)) (* 2.0 (sin (* (* PI angle) -0.005555555555555556))))
   (* -0.011111111111111112 (* (- a b) (* (+ b a) (* PI angle))))))

C

double code(double a, double b, double angle) {
	double tmp;
	if (pow(b, 2.0) <= 5e-67) {
		tmp = ((a - b) * (b + a)) * (2.0 * sin(((((double) M_PI) * angle) * -0.005555555555555556)));
	} else {
		tmp = -0.011111111111111112 * ((a - b) * ((b + a) * (((double) M_PI) * angle)));
	}
	return tmp;
}

Java

public static double code(double a, double b, double angle) {
	double tmp;
	if (Math.pow(b, 2.0) <= 5e-67) {
		tmp = ((a - b) * (b + a)) * (2.0 * Math.sin(((Math.PI * angle) * -0.005555555555555556)));
	} else {
		tmp = -0.011111111111111112 * ((a - b) * ((b + a) * (Math.PI * angle)));
	}
	return tmp;
}

Python

def code(a, b, angle):
	tmp = 0
	if math.pow(b, 2.0) <= 5e-67:
		tmp = ((a - b) * (b + a)) * (2.0 * math.sin(((math.pi * angle) * -0.005555555555555556)))
	else:
		tmp = -0.011111111111111112 * ((a - b) * ((b + a) * (math.pi * angle)))
	return tmp

Julia

function code(a, b, angle)
	tmp = 0.0
	if ((b ^ 2.0) <= 5e-67)
		tmp = Float64(Float64(Float64(a - b) * Float64(b + a)) * Float64(2.0 * sin(Float64(Float64(pi * angle) * -0.005555555555555556))));
	else
		tmp = Float64(-0.011111111111111112 * Float64(Float64(a - b) * Float64(Float64(b + a) * Float64(pi * angle))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, angle)
	tmp = 0.0;
	if ((b ^ 2.0) <= 5e-67)
		tmp = ((a - b) * (b + a)) * (2.0 * sin(((pi * angle) * -0.005555555555555556)));
	else
		tmp = -0.011111111111111112 * ((a - b) * ((b + a) * (pi * angle)));
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, angle_] := If[LessEqual[N[Power[b, 2.0], $MachinePrecision], 5e-67], N[(N[(N[(a - b), $MachinePrecision] * N[(b + a), $MachinePrecision]), $MachinePrecision] * N[(2.0 * N[Sin[N[(N[(Pi * angle), $MachinePrecision] * -0.005555555555555556), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(-0.011111111111111112 * N[(N[(a - b), $MachinePrecision] * N[(N[(b + a), $MachinePrecision] * N[(Pi * angle), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (pow.f64 b 2) < 4.9999999999999999e-67

   1. Initial program 62.6%

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

   3. Simplified 62.9%

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

      1. unpow2 62.9%

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

      2. unpow2 62.9%

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

      3. difference-of-squares 62.9%

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

   5. Applied egg-rr 62.9%

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

      1. add-cbrt-cube 64.5%

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

      2. pow3 64.5%

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

   7. Applied egg-rr 64.5%

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

   8. Taylor expanded in angle around 0 62.4%

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

   9. Taylor expanded in angle around inf 64.7%

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

       1. associate-*r* 64.7%

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

       2. *-commutative 64.7%

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

       3. +-commutative 64.7%

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

   11. Simplified 64.7%

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

   if 4.9999999999999999e-67 < (pow.f64 b 2)

   1. Initial program 45.7%

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

   3. Simplified 47.6%

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

      1. unpow2 47.6%

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

      2. unpow2 47.6%

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

      3. difference-of-squares 52.8%

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

   5. Applied egg-rr 52.8%

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

      1. add-cbrt-cube 55.2%

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

      2. pow3 55.2%

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

   7. Applied egg-rr 55.2%

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

   8. Taylor expanded in angle around 0 50.2%

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

      1. associate-*r* 50.1%

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

      2. associate-*r* 64.0%

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

      3. *-commutative 64.0%

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

      4. *-commutative 64.0%

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

   10. Simplified 64.0%

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

4. Final simplification 64.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;{b}^{2} \leq 5 \cdot 10^{-67}:\\ \;\;\;\;\left(\left(a - b\right) \cdot \left(b + a\right)\right) \cdot \left(2 \cdot \sin \left(\left(\pi \cdot angle\right) \cdot -0.005555555555555556\right)\right)\\ \mathbf{else}:\\ \;\;\;\;-0.011111111111111112 \cdot \left(\left(a - b\right) \cdot \left(\left(b + a\right) \cdot \left(\pi \cdot angle\right)\right)\right)\\ \end{array} \]

Alternative 16: 62.2% accurate, 2.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := -0.011111111111111112 \cdot \left(\left(a - b\right) \cdot \left(\left(b + a\right) \cdot \left(\pi \cdot angle\right)\right)\right)\\ \mathbf{if}\;angle \leq -6 \cdot 10^{+31}:\\ \;\;\;\;\left|t_0\right|\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \end{array} \]

FPCore

(FPCore (a b angle)
 :precision binary64
 (let* ((t_0 (* -0.011111111111111112 (* (- a b) (* (+ b a) (* PI angle))))))
   (if (<= angle -6e+31) (fabs t_0) t_0)))

C

double code(double a, double b, double angle) {
	double t_0 = -0.011111111111111112 * ((a - b) * ((b + a) * (((double) M_PI) * angle)));
	double tmp;
	if (angle <= -6e+31) {
		tmp = fabs(t_0);
	} else {
		tmp = t_0;
	}
	return tmp;
}

Java

public static double code(double a, double b, double angle) {
	double t_0 = -0.011111111111111112 * ((a - b) * ((b + a) * (Math.PI * angle)));
	double tmp;
	if (angle <= -6e+31) {
		tmp = Math.abs(t_0);
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(a, b, angle):
	t_0 = -0.011111111111111112 * ((a - b) * ((b + a) * (math.pi * angle)))
	tmp = 0
	if angle <= -6e+31:
		tmp = math.fabs(t_0)
	else:
		tmp = t_0
	return tmp

Julia

function code(a, b, angle)
	t_0 = Float64(-0.011111111111111112 * Float64(Float64(a - b) * Float64(Float64(b + a) * Float64(pi * angle))))
	tmp = 0.0
	if (angle <= -6e+31)
		tmp = abs(t_0);
	else
		tmp = t_0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, angle)
	t_0 = -0.011111111111111112 * ((a - b) * ((b + a) * (pi * angle)));
	tmp = 0.0;
	if (angle <= -6e+31)
		tmp = abs(t_0);
	else
		tmp = t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, angle_] := Block[{t$95$0 = N[(-0.011111111111111112 * N[(N[(a - b), $MachinePrecision] * N[(N[(b + a), $MachinePrecision] * N[(Pi * angle), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[angle, -6e+31], N[Abs[t$95$0], $MachinePrecision], t$95$0]]

Derivation

1. Split input into 2 regimes

2. if angle < -5.99999999999999978e31

   1. Initial program 29.1%

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

   3. Simplified 26.6%

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

      1. unpow2 26.6%

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

      2. unpow2 26.6%

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

      3. difference-of-squares 26.6%

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

   5. Applied egg-rr 26.6%

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

   6. Taylor expanded in angle around 0 22.6%

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

      1. add-sqr-sqrt 14.0%

         \[\leadsto \color{blue}{\sqrt{-0.011111111111111112 \cdot \left(angle \cdot \left(\pi \cdot \left(\left(a + b\right) \cdot \left(a - b\right)\right)\right)\right)} \cdot \sqrt{-0.011111111111111112 \cdot \left(angle \cdot \left(\pi \cdot \left(\left(a + b\right) \cdot \left(a - b\right)\right)\right)\right)}} \]

      2. sqrt-unprod 29.3%

         \[\leadsto \color{blue}{\sqrt{\left(-0.011111111111111112 \cdot \left(angle \cdot \left(\pi \cdot \left(\left(a + b\right) \cdot \left(a - b\right)\right)\right)\right)\right) \cdot \left(-0.011111111111111112 \cdot \left(angle \cdot \left(\pi \cdot \left(\left(a + b\right) \cdot \left(a - b\right)\right)\right)\right)\right)}} \]

      3. pow2 29.3%

         \[\leadsto \sqrt{\color{blue}{{\left(-0.011111111111111112 \cdot \left(angle \cdot \left(\pi \cdot \left(\left(a + b\right) \cdot \left(a - b\right)\right)\right)\right)\right)}^{2}}} \]

      4. *-commutative 29.3%

         \[\leadsto \sqrt{{\color{blue}{\left(\left(angle \cdot \left(\pi \cdot \left(\left(a + b\right) \cdot \left(a - b\right)\right)\right)\right) \cdot -0.011111111111111112\right)}}^{2}} \]

      5. associate-*r* 29.3%

         \[\leadsto \sqrt{{\left(\color{blue}{\left(\left(angle \cdot \pi\right) \cdot \left(\left(a + b\right) \cdot \left(a - b\right)\right)\right)} \cdot -0.011111111111111112\right)}^{2}} \]

   8. Applied egg-rr 29.3%

      \[\leadsto \color{blue}{\sqrt{{\left(\left(\left(angle \cdot \pi\right) \cdot \left(\left(a + b\right) \cdot \left(a - b\right)\right)\right) \cdot -0.011111111111111112\right)}^{2}}} \]
   9. Step-by-step derivation

      1. unpow2 29.3%

         \[\leadsto \sqrt{\color{blue}{\left(\left(\left(angle \cdot \pi\right) \cdot \left(\left(a + b\right) \cdot \left(a - b\right)\right)\right) \cdot -0.011111111111111112\right) \cdot \left(\left(\left(angle \cdot \pi\right) \cdot \left(\left(a + b\right) \cdot \left(a - b\right)\right)\right) \cdot -0.011111111111111112\right)}} \]

      2. rem-sqrt-square 29.5%

         \[\leadsto \color{blue}{\left|\left(\left(angle \cdot \pi\right) \cdot \left(\left(a + b\right) \cdot \left(a - b\right)\right)\right) \cdot -0.011111111111111112\right|} \]

      3. *-commutative 29.5%

         \[\leadsto \left|\color{blue}{-0.011111111111111112 \cdot \left(\left(angle \cdot \pi\right) \cdot \left(\left(a + b\right) \cdot \left(a - b\right)\right)\right)}\right| \]

      4. associate-*r* 26.4%

         \[\leadsto \left|-0.011111111111111112 \cdot \color{blue}{\left(\left(\left(angle \cdot \pi\right) \cdot \left(a + b\right)\right) \cdot \left(a - b\right)\right)}\right| \]

      5. *-commutative 26.4%

         \[\leadsto \left|-0.011111111111111112 \cdot \color{blue}{\left(\left(a - b\right) \cdot \left(\left(angle \cdot \pi\right) \cdot \left(a + b\right)\right)\right)}\right| \]

      6. *-commutative 26.4%

         \[\leadsto \left|-0.011111111111111112 \cdot \left(\left(a - b\right) \cdot \color{blue}{\left(\left(a + b\right) \cdot \left(angle \cdot \pi\right)\right)}\right)\right| \]

   10. Simplified 26.4%

       \[\leadsto \color{blue}{\left|-0.011111111111111112 \cdot \left(\left(a - b\right) \cdot \left(\left(a + b\right) \cdot \left(angle \cdot \pi\right)\right)\right)\right|} \]

   if -5.99999999999999978e31 < angle

   1. Initial program 61.1%

      \[\left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin \left(\pi \cdot \frac{angle}{180}\right)\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right) \]

   3. Simplified 63.4%

      \[\leadsto \color{blue}{\left(2 \cdot \sin \left(angle \cdot \frac{\pi}{-180}\right)\right) \cdot \left(\cos \left(angle \cdot \frac{\pi}{-180}\right) \cdot \left({a}^{2} - {b}^{2}\right)\right)} \]
   4. Step-by-step derivation

      1. unpow2 63.4%

         \[\leadsto \left(2 \cdot \sin \left(angle \cdot \frac{\pi}{-180}\right)\right) \cdot \left(\cos \left(angle \cdot \frac{\pi}{-180}\right) \cdot \left(\color{blue}{a \cdot a} - {b}^{2}\right)\right) \]

      2. unpow2 63.4%

         \[\leadsto \left(2 \cdot \sin \left(angle \cdot \frac{\pi}{-180}\right)\right) \cdot \left(\cos \left(angle \cdot \frac{\pi}{-180}\right) \cdot \left(a \cdot a - \color{blue}{b \cdot b}\right)\right) \]

      3. difference-of-squares 67.1%

         \[\leadsto \left(2 \cdot \sin \left(angle \cdot \frac{\pi}{-180}\right)\right) \cdot \left(\cos \left(angle \cdot \frac{\pi}{-180}\right) \cdot \color{blue}{\left(\left(a + b\right) \cdot \left(a - b\right)\right)}\right) \]

   5. Applied egg-rr 67.1%

      \[\leadsto \left(2 \cdot \sin \left(angle \cdot \frac{\pi}{-180}\right)\right) \cdot \left(\cos \left(angle \cdot \frac{\pi}{-180}\right) \cdot \color{blue}{\left(\left(a + b\right) \cdot \left(a - b\right)\right)}\right) \]
   6. Step-by-step derivation

      1. add-cbrt-cube 68.0%

         \[\leadsto \left(2 \cdot \sin \left(angle \cdot \frac{\color{blue}{\sqrt[3]{\left(\pi \cdot \pi\right) \cdot \pi}}}{-180}\right)\right) \cdot \left(\cos \left(angle \cdot \frac{\pi}{-180}\right) \cdot \left(\left(a + b\right) \cdot \left(a - b\right)\right)\right) \]

      2. pow3 68.0%

         \[\leadsto \left(2 \cdot \sin \left(angle \cdot \frac{\sqrt[3]{\color{blue}{{\pi}^{3}}}}{-180}\right)\right) \cdot \left(\cos \left(angle \cdot \frac{\pi}{-180}\right) \cdot \left(\left(a + b\right) \cdot \left(a - b\right)\right)\right) \]

   7. Applied egg-rr 68.0%

      \[\leadsto \left(2 \cdot \sin \left(angle \cdot \frac{\color{blue}{\sqrt[3]{{\pi}^{3}}}}{-180}\right)\right) \cdot \left(\cos \left(angle \cdot \frac{\pi}{-180}\right) \cdot \left(\left(a + b\right) \cdot \left(a - b\right)\right)\right) \]

   8. Taylor expanded in angle around 0 65.1%

      \[\leadsto \color{blue}{-0.011111111111111112 \cdot \left(angle \cdot \left(\pi \cdot \left(\left(a + b\right) \cdot \left(a - b\right)\right)\right)\right)} \]
   9. Step-by-step derivation

      1. associate-*r* 65.0%

         \[\leadsto -0.011111111111111112 \cdot \color{blue}{\left(\left(angle \cdot \pi\right) \cdot \left(\left(a + b\right) \cdot \left(a - b\right)\right)\right)} \]

      2. associate-*r* 75.1%

         \[\leadsto -0.011111111111111112 \cdot \color{blue}{\left(\left(\left(angle \cdot \pi\right) \cdot \left(a + b\right)\right) \cdot \left(a - b\right)\right)} \]

      3. *-commutative 75.1%

         \[\leadsto -0.011111111111111112 \cdot \color{blue}{\left(\left(a - b\right) \cdot \left(\left(angle \cdot \pi\right) \cdot \left(a + b\right)\right)\right)} \]

      4. *-commutative 75.1%

         \[\leadsto -0.011111111111111112 \cdot \left(\left(a - b\right) \cdot \color{blue}{\left(\left(a + b\right) \cdot \left(angle \cdot \pi\right)\right)}\right) \]

   10. Simplified 75.1%

       \[\leadsto \color{blue}{-0.011111111111111112 \cdot \left(\left(a - b\right) \cdot \left(\left(a + b\right) \cdot \left(angle \cdot \pi\right)\right)\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 63.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;angle \leq -6 \cdot 10^{+31}:\\ \;\;\;\;\left|-0.011111111111111112 \cdot \left(\left(a - b\right) \cdot \left(\left(b + a\right) \cdot \left(\pi \cdot angle\right)\right)\right)\right|\\ \mathbf{else}:\\ \;\;\;\;-0.011111111111111112 \cdot \left(\left(a - b\right) \cdot \left(\left(b + a\right) \cdot \left(\pi \cdot angle\right)\right)\right)\\ \end{array} \]

Alternative 17: 55.4% accurate, 5.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq 1.05 \cdot 10^{+129}:\\ \;\;\;\;-0.011111111111111112 \cdot \left(angle \cdot \left(\pi \cdot \left(\left(a - b\right) \cdot \left(b + a\right)\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;-0.011111111111111112 \cdot \left(\left(a - b\right) \cdot \left(angle \cdot \left(b \cdot \pi\right)\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (a b angle)
 :precision binary64
 (if (<= b 1.05e+129)
   (* -0.011111111111111112 (* angle (* PI (* (- a b) (+ b a)))))
   (* -0.011111111111111112 (* (- a b) (* angle (* b PI))))))

C

double code(double a, double b, double angle) {
	double tmp;
	if (b <= 1.05e+129) {
		tmp = -0.011111111111111112 * (angle * (((double) M_PI) * ((a - b) * (b + a))));
	} else {
		tmp = -0.011111111111111112 * ((a - b) * (angle * (b * ((double) M_PI))));
	}
	return tmp;
}

Java

public static double code(double a, double b, double angle) {
	double tmp;
	if (b <= 1.05e+129) {
		tmp = -0.011111111111111112 * (angle * (Math.PI * ((a - b) * (b + a))));
	} else {
		tmp = -0.011111111111111112 * ((a - b) * (angle * (b * Math.PI)));
	}
	return tmp;
}

Python

def code(a, b, angle):
	tmp = 0
	if b <= 1.05e+129:
		tmp = -0.011111111111111112 * (angle * (math.pi * ((a - b) * (b + a))))
	else:
		tmp = -0.011111111111111112 * ((a - b) * (angle * (b * math.pi)))
	return tmp

Julia

function code(a, b, angle)
	tmp = 0.0
	if (b <= 1.05e+129)
		tmp = Float64(-0.011111111111111112 * Float64(angle * Float64(pi * Float64(Float64(a - b) * Float64(b + a)))));
	else
		tmp = Float64(-0.011111111111111112 * Float64(Float64(a - b) * Float64(angle * Float64(b * pi))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, angle)
	tmp = 0.0;
	if (b <= 1.05e+129)
		tmp = -0.011111111111111112 * (angle * (pi * ((a - b) * (b + a))));
	else
		tmp = -0.011111111111111112 * ((a - b) * (angle * (b * pi)));
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, angle_] := If[LessEqual[b, 1.05e+129], N[(-0.011111111111111112 * N[(angle * N[(Pi * N[(N[(a - b), $MachinePrecision] * N[(b + a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(-0.011111111111111112 * N[(N[(a - b), $MachinePrecision] * N[(angle * N[(b * Pi), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b < 1.04999999999999998e129

   1. Initial program 57.9%

      \[\left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin \left(\pi \cdot \frac{angle}{180}\right)\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right) \]

   3. Simplified 58.3%

      \[\leadsto \color{blue}{\left(2 \cdot \sin \left(angle \cdot \frac{\pi}{-180}\right)\right) \cdot \left(\cos \left(angle \cdot \frac{\pi}{-180}\right) \cdot \left({a}^{2} - {b}^{2}\right)\right)} \]
   4. Step-by-step derivation

      1. unpow2 58.3%

         \[\leadsto \left(2 \cdot \sin \left(angle \cdot \frac{\pi}{-180}\right)\right) \cdot \left(\cos \left(angle \cdot \frac{\pi}{-180}\right) \cdot \left(\color{blue}{a \cdot a} - {b}^{2}\right)\right) \]

      2. unpow2 58.3%

         \[\leadsto \left(2 \cdot \sin \left(angle \cdot \frac{\pi}{-180}\right)\right) \cdot \left(\cos \left(angle \cdot \frac{\pi}{-180}\right) \cdot \left(a \cdot a - \color{blue}{b \cdot b}\right)\right) \]

      3. difference-of-squares 59.3%

         \[\leadsto \left(2 \cdot \sin \left(angle \cdot \frac{\pi}{-180}\right)\right) \cdot \left(\cos \left(angle \cdot \frac{\pi}{-180}\right) \cdot \color{blue}{\left(\left(a + b\right) \cdot \left(a - b\right)\right)}\right) \]

   5. Applied egg-rr 59.3%

      \[\leadsto \left(2 \cdot \sin \left(angle \cdot \frac{\pi}{-180}\right)\right) \cdot \left(\cos \left(angle \cdot \frac{\pi}{-180}\right) \cdot \color{blue}{\left(\left(a + b\right) \cdot \left(a - b\right)\right)}\right) \]

   6. Taylor expanded in angle around 0 55.2%

      \[\leadsto \color{blue}{-0.011111111111111112 \cdot \left(angle \cdot \left(\pi \cdot \left(\left(a + b\right) \cdot \left(a - b\right)\right)\right)\right)} \]

   if 1.04999999999999998e129 < b

   1. Initial program 28.7%

      \[\left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin \left(\pi \cdot \frac{angle}{180}\right)\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right) \]

   3. Simplified 33.7%

      \[\leadsto \color{blue}{\left(2 \cdot \sin \left(angle \cdot \frac{\pi}{-180}\right)\right) \cdot \left(\cos \left(angle \cdot \frac{\pi}{-180}\right) \cdot \left({a}^{2} - {b}^{2}\right)\right)} \]
   4. Step-by-step derivation

      1. unpow2 33.7%

         \[\leadsto \left(2 \cdot \sin \left(angle \cdot \frac{\pi}{-180}\right)\right) \cdot \left(\cos \left(angle \cdot \frac{\pi}{-180}\right) \cdot \left(\color{blue}{a \cdot a} - {b}^{2}\right)\right) \]

      2. unpow2 33.7%

         \[\leadsto \left(2 \cdot \sin \left(angle \cdot \frac{\pi}{-180}\right)\right) \cdot \left(\cos \left(angle \cdot \frac{\pi}{-180}\right) \cdot \left(a \cdot a - \color{blue}{b \cdot b}\right)\right) \]

      3. difference-of-squares 46.6%

         \[\leadsto \left(2 \cdot \sin \left(angle \cdot \frac{\pi}{-180}\right)\right) \cdot \left(\cos \left(angle \cdot \frac{\pi}{-180}\right) \cdot \color{blue}{\left(\left(a + b\right) \cdot \left(a - b\right)\right)}\right) \]

   5. Applied egg-rr 46.6%

      \[\leadsto \left(2 \cdot \sin \left(angle \cdot \frac{\pi}{-180}\right)\right) \cdot \left(\cos \left(angle \cdot \frac{\pi}{-180}\right) \cdot \color{blue}{\left(\left(a + b\right) \cdot \left(a - b\right)\right)}\right) \]
   6. Step-by-step derivation

      1. add-cbrt-cube 59.5%

         \[\leadsto \left(2 \cdot \sin \left(angle \cdot \frac{\color{blue}{\sqrt[3]{\left(\pi \cdot \pi\right) \cdot \pi}}}{-180}\right)\right) \cdot \left(\cos \left(angle \cdot \frac{\pi}{-180}\right) \cdot \left(\left(a + b\right) \cdot \left(a - b\right)\right)\right) \]

      2. pow3 59.5%

         \[\leadsto \left(2 \cdot \sin \left(angle \cdot \frac{\sqrt[3]{\color{blue}{{\pi}^{3}}}}{-180}\right)\right) \cdot \left(\cos \left(angle \cdot \frac{\pi}{-180}\right) \cdot \left(\left(a + b\right) \cdot \left(a - b\right)\right)\right) \]

   7. Applied egg-rr 59.5%

      \[\leadsto \left(2 \cdot \sin \left(angle \cdot \frac{\color{blue}{\sqrt[3]{{\pi}^{3}}}}{-180}\right)\right) \cdot \left(\cos \left(angle \cdot \frac{\pi}{-180}\right) \cdot \left(\left(a + b\right) \cdot \left(a - b\right)\right)\right) \]

   8. Taylor expanded in angle around 0 52.5%

      \[\leadsto \color{blue}{-0.011111111111111112 \cdot \left(angle \cdot \left(\pi \cdot \left(\left(a + b\right) \cdot \left(a - b\right)\right)\right)\right)} \]
   9. Step-by-step derivation

      1. associate-*r* 52.5%

         \[\leadsto -0.011111111111111112 \cdot \color{blue}{\left(\left(angle \cdot \pi\right) \cdot \left(\left(a + b\right) \cdot \left(a - b\right)\right)\right)} \]

      2. associate-*r* 64.2%

         \[\leadsto -0.011111111111111112 \cdot \color{blue}{\left(\left(\left(angle \cdot \pi\right) \cdot \left(a + b\right)\right) \cdot \left(a - b\right)\right)} \]

      3. *-commutative 64.2%

         \[\leadsto -0.011111111111111112 \cdot \color{blue}{\left(\left(a - b\right) \cdot \left(\left(angle \cdot \pi\right) \cdot \left(a + b\right)\right)\right)} \]

      4. *-commutative 64.2%

         \[\leadsto -0.011111111111111112 \cdot \left(\left(a - b\right) \cdot \color{blue}{\left(\left(a + b\right) \cdot \left(angle \cdot \pi\right)\right)}\right) \]

   10. Simplified 64.2%

       \[\leadsto \color{blue}{-0.011111111111111112 \cdot \left(\left(a - b\right) \cdot \left(\left(a + b\right) \cdot \left(angle \cdot \pi\right)\right)\right)} \]

   11. Taylor expanded in a around 0 59.2%

       \[\leadsto -0.011111111111111112 \cdot \left(\left(a - b\right) \cdot \color{blue}{\left(angle \cdot \left(b \cdot \pi\right)\right)}\right) \]
3. Recombined 2 regimes into one program.

4. Final simplification 55.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq 1.05 \cdot 10^{+129}:\\ \;\;\;\;-0.011111111111111112 \cdot \left(angle \cdot \left(\pi \cdot \left(\left(a - b\right) \cdot \left(b + a\right)\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;-0.011111111111111112 \cdot \left(\left(a - b\right) \cdot \left(angle \cdot \left(b \cdot \pi\right)\right)\right)\\ \end{array} \]

Alternative 18: 46.4% accurate, 5.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;a \leq 6.5 \cdot 10^{+37}:\\ \;\;\;\;-0.011111111111111112 \cdot \left(\left(a - b\right) \cdot \left(angle \cdot \left(b \cdot \pi\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;-0.011111111111111112 \cdot \left(\left(a - b\right) \cdot \left(a \cdot \left(\pi \cdot angle\right)\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (a b angle)
 :precision binary64
 (if (<= a 6.5e+37)
   (* -0.011111111111111112 (* (- a b) (* angle (* b PI))))
   (* -0.011111111111111112 (* (- a b) (* a (* PI angle))))))

C

double code(double a, double b, double angle) {
	double tmp;
	if (a <= 6.5e+37) {
		tmp = -0.011111111111111112 * ((a - b) * (angle * (b * ((double) M_PI))));
	} else {
		tmp = -0.011111111111111112 * ((a - b) * (a * (((double) M_PI) * angle)));
	}
	return tmp;
}

Java

public static double code(double a, double b, double angle) {
	double tmp;
	if (a <= 6.5e+37) {
		tmp = -0.011111111111111112 * ((a - b) * (angle * (b * Math.PI)));
	} else {
		tmp = -0.011111111111111112 * ((a - b) * (a * (Math.PI * angle)));
	}
	return tmp;
}

Python

def code(a, b, angle):
	tmp = 0
	if a <= 6.5e+37:
		tmp = -0.011111111111111112 * ((a - b) * (angle * (b * math.pi)))
	else:
		tmp = -0.011111111111111112 * ((a - b) * (a * (math.pi * angle)))
	return tmp

Julia

function code(a, b, angle)
	tmp = 0.0
	if (a <= 6.5e+37)
		tmp = Float64(-0.011111111111111112 * Float64(Float64(a - b) * Float64(angle * Float64(b * pi))));
	else
		tmp = Float64(-0.011111111111111112 * Float64(Float64(a - b) * Float64(a * Float64(pi * angle))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, angle)
	tmp = 0.0;
	if (a <= 6.5e+37)
		tmp = -0.011111111111111112 * ((a - b) * (angle * (b * pi)));
	else
		tmp = -0.011111111111111112 * ((a - b) * (a * (pi * angle)));
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, angle_] := If[LessEqual[a, 6.5e+37], N[(-0.011111111111111112 * N[(N[(a - b), $MachinePrecision] * N[(angle * N[(b * Pi), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(-0.011111111111111112 * N[(N[(a - b), $MachinePrecision] * N[(a * N[(Pi * angle), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if a < 6.4999999999999998e37

   1. Initial program 53.8%

      \[\left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin \left(\pi \cdot \frac{angle}{180}\right)\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right) \]

   3. Simplified 55.4%

      \[\leadsto \color{blue}{\left(2 \cdot \sin \left(angle \cdot \frac{\pi}{-180}\right)\right) \cdot \left(\cos \left(angle \cdot \frac{\pi}{-180}\right) \cdot \left({a}^{2} - {b}^{2}\right)\right)} \]
   4. Step-by-step derivation

      1. unpow2 55.4%

         \[\leadsto \left(2 \cdot \sin \left(angle \cdot \frac{\pi}{-180}\right)\right) \cdot \left(\cos \left(angle \cdot \frac{\pi}{-180}\right) \cdot \left(\color{blue}{a \cdot a} - {b}^{2}\right)\right) \]

      2. unpow2 55.4%

         \[\leadsto \left(2 \cdot \sin \left(angle \cdot \frac{\pi}{-180}\right)\right) \cdot \left(\cos \left(angle \cdot \frac{\pi}{-180}\right) \cdot \left(a \cdot a - \color{blue}{b \cdot b}\right)\right) \]

      3. difference-of-squares 56.1%

         \[\leadsto \left(2 \cdot \sin \left(angle \cdot \frac{\pi}{-180}\right)\right) \cdot \left(\cos \left(angle \cdot \frac{\pi}{-180}\right) \cdot \color{blue}{\left(\left(a + b\right) \cdot \left(a - b\right)\right)}\right) \]

   5. Applied egg-rr 56.1%

      \[\leadsto \left(2 \cdot \sin \left(angle \cdot \frac{\pi}{-180}\right)\right) \cdot \left(\cos \left(angle \cdot \frac{\pi}{-180}\right) \cdot \color{blue}{\left(\left(a + b\right) \cdot \left(a - b\right)\right)}\right) \]
   6. Step-by-step derivation

      1. add-cbrt-cube 57.4%

         \[\leadsto \left(2 \cdot \sin \left(angle \cdot \frac{\color{blue}{\sqrt[3]{\left(\pi \cdot \pi\right) \cdot \pi}}}{-180}\right)\right) \cdot \left(\cos \left(angle \cdot \frac{\pi}{-180}\right) \cdot \left(\left(a + b\right) \cdot \left(a - b\right)\right)\right) \]

      2. pow3 57.4%

         \[\leadsto \left(2 \cdot \sin \left(angle \cdot \frac{\sqrt[3]{\color{blue}{{\pi}^{3}}}}{-180}\right)\right) \cdot \left(\cos \left(angle \cdot \frac{\pi}{-180}\right) \cdot \left(\left(a + b\right) \cdot \left(a - b\right)\right)\right) \]

   7. Applied egg-rr 57.4%

      \[\leadsto \left(2 \cdot \sin \left(angle \cdot \frac{\color{blue}{\sqrt[3]{{\pi}^{3}}}}{-180}\right)\right) \cdot \left(\cos \left(angle \cdot \frac{\pi}{-180}\right) \cdot \left(\left(a + b\right) \cdot \left(a - b\right)\right)\right) \]

   8. Taylor expanded in angle around 0 53.9%

      \[\leadsto \color{blue}{-0.011111111111111112 \cdot \left(angle \cdot \left(\pi \cdot \left(\left(a + b\right) \cdot \left(a - b\right)\right)\right)\right)} \]
   9. Step-by-step derivation

      1. associate-*r* 53.9%

         \[\leadsto -0.011111111111111112 \cdot \color{blue}{\left(\left(angle \cdot \pi\right) \cdot \left(\left(a + b\right) \cdot \left(a - b\right)\right)\right)} \]

      2. associate-*r* 60.9%

         \[\leadsto -0.011111111111111112 \cdot \color{blue}{\left(\left(\left(angle \cdot \pi\right) \cdot \left(a + b\right)\right) \cdot \left(a - b\right)\right)} \]

      3. *-commutative 60.9%

         \[\leadsto -0.011111111111111112 \cdot \color{blue}{\left(\left(a - b\right) \cdot \left(\left(angle \cdot \pi\right) \cdot \left(a + b\right)\right)\right)} \]

      4. *-commutative 60.9%

         \[\leadsto -0.011111111111111112 \cdot \left(\left(a - b\right) \cdot \color{blue}{\left(\left(a + b\right) \cdot \left(angle \cdot \pi\right)\right)}\right) \]

   10. Simplified 60.9%

       \[\leadsto \color{blue}{-0.011111111111111112 \cdot \left(\left(a - b\right) \cdot \left(\left(a + b\right) \cdot \left(angle \cdot \pi\right)\right)\right)} \]

   11. Taylor expanded in a around 0 43.9%

       \[\leadsto -0.011111111111111112 \cdot \left(\left(a - b\right) \cdot \color{blue}{\left(angle \cdot \left(b \cdot \pi\right)\right)}\right) \]

   if 6.4999999999999998e37 < a

   1. Initial program 51.5%

      \[\left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin \left(\pi \cdot \frac{angle}{180}\right)\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right) \]

   3. Simplified 51.1%

      \[\leadsto \color{blue}{\left(2 \cdot \sin \left(angle \cdot \frac{\pi}{-180}\right)\right) \cdot \left(\cos \left(angle \cdot \frac{\pi}{-180}\right) \cdot \left({a}^{2} - {b}^{2}\right)\right)} \]
   4. Step-by-step derivation

      1. unpow2 51.1%

         \[\leadsto \left(2 \cdot \sin \left(angle \cdot \frac{\pi}{-180}\right)\right) \cdot \left(\cos \left(angle \cdot \frac{\pi}{-180}\right) \cdot \left(\color{blue}{a \cdot a} - {b}^{2}\right)\right) \]

      2. unpow2 51.1%

         \[\leadsto \left(2 \cdot \sin \left(angle \cdot \frac{\pi}{-180}\right)\right) \cdot \left(\cos \left(angle \cdot \frac{\pi}{-180}\right) \cdot \left(a \cdot a - \color{blue}{b \cdot b}\right)\right) \]

      3. difference-of-squares 61.8%

         \[\leadsto \left(2 \cdot \sin \left(angle \cdot \frac{\pi}{-180}\right)\right) \cdot \left(\cos \left(angle \cdot \frac{\pi}{-180}\right) \cdot \color{blue}{\left(\left(a + b\right) \cdot \left(a - b\right)\right)}\right) \]

   5. Applied egg-rr 61.8%

      \[\leadsto \left(2 \cdot \sin \left(angle \cdot \frac{\pi}{-180}\right)\right) \cdot \left(\cos \left(angle \cdot \frac{\pi}{-180}\right) \cdot \color{blue}{\left(\left(a + b\right) \cdot \left(a - b\right)\right)}\right) \]
   6. Step-by-step derivation

      1. add-cbrt-cube 66.1%

         \[\leadsto \left(2 \cdot \sin \left(angle \cdot \frac{\color{blue}{\sqrt[3]{\left(\pi \cdot \pi\right) \cdot \pi}}}{-180}\right)\right) \cdot \left(\cos \left(angle \cdot \frac{\pi}{-180}\right) \cdot \left(\left(a + b\right) \cdot \left(a - b\right)\right)\right) \]

      2. pow3 66.1%

         \[\leadsto \left(2 \cdot \sin \left(angle \cdot \frac{\sqrt[3]{\color{blue}{{\pi}^{3}}}}{-180}\right)\right) \cdot \left(\cos \left(angle \cdot \frac{\pi}{-180}\right) \cdot \left(\left(a + b\right) \cdot \left(a - b\right)\right)\right) \]

   7. Applied egg-rr 66.1%

      \[\leadsto \left(2 \cdot \sin \left(angle \cdot \frac{\color{blue}{\sqrt[3]{{\pi}^{3}}}}{-180}\right)\right) \cdot \left(\cos \left(angle \cdot \frac{\pi}{-180}\right) \cdot \left(\left(a + b\right) \cdot \left(a - b\right)\right)\right) \]

   8. Taylor expanded in angle around 0 57.7%

      \[\leadsto \color{blue}{-0.011111111111111112 \cdot \left(angle \cdot \left(\pi \cdot \left(\left(a + b\right) \cdot \left(a - b\right)\right)\right)\right)} \]
   9. Step-by-step derivation

      1. associate-*r* 57.6%

         \[\leadsto -0.011111111111111112 \cdot \color{blue}{\left(\left(angle \cdot \pi\right) \cdot \left(\left(a + b\right) \cdot \left(a - b\right)\right)\right)} \]

      2. associate-*r* 64.3%

         \[\leadsto -0.011111111111111112 \cdot \color{blue}{\left(\left(\left(angle \cdot \pi\right) \cdot \left(a + b\right)\right) \cdot \left(a - b\right)\right)} \]

      3. *-commutative 64.3%

         \[\leadsto -0.011111111111111112 \cdot \color{blue}{\left(\left(a - b\right) \cdot \left(\left(angle \cdot \pi\right) \cdot \left(a + b\right)\right)\right)} \]

      4. *-commutative 64.3%

         \[\leadsto -0.011111111111111112 \cdot \left(\left(a - b\right) \cdot \color{blue}{\left(\left(a + b\right) \cdot \left(angle \cdot \pi\right)\right)}\right) \]

   10. Simplified 64.3%

       \[\leadsto \color{blue}{-0.011111111111111112 \cdot \left(\left(a - b\right) \cdot \left(\left(a + b\right) \cdot \left(angle \cdot \pi\right)\right)\right)} \]

   11. Taylor expanded in a around inf 58.0%

       \[\leadsto -0.011111111111111112 \cdot \left(\left(a - b\right) \cdot \color{blue}{\left(a \cdot \left(angle \cdot \pi\right)\right)}\right) \]
3. Recombined 2 regimes into one program.

4. Final simplification 47.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \leq 6.5 \cdot 10^{+37}:\\ \;\;\;\;-0.011111111111111112 \cdot \left(\left(a - b\right) \cdot \left(angle \cdot \left(b \cdot \pi\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;-0.011111111111111112 \cdot \left(\left(a - b\right) \cdot \left(a \cdot \left(\pi \cdot angle\right)\right)\right)\\ \end{array} \]

Alternative 19: 46.4% accurate, 5.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;a \leq 2 \cdot 10^{+47}:\\ \;\;\;\;-0.011111111111111112 \cdot \left(\left(a - b\right) \cdot \left(angle \cdot \left(b \cdot \pi\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;-0.011111111111111112 \cdot \left(\left(a - b\right) \cdot \left(angle \cdot \left(a \cdot \pi\right)\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (a b angle)
 :precision binary64
 (if (<= a 2e+47)
   (* -0.011111111111111112 (* (- a b) (* angle (* b PI))))
   (* -0.011111111111111112 (* (- a b) (* angle (* a PI))))))

C

double code(double a, double b, double angle) {
	double tmp;
	if (a <= 2e+47) {
		tmp = -0.011111111111111112 * ((a - b) * (angle * (b * ((double) M_PI))));
	} else {
		tmp = -0.011111111111111112 * ((a - b) * (angle * (a * ((double) M_PI))));
	}
	return tmp;
}

Java

public static double code(double a, double b, double angle) {
	double tmp;
	if (a <= 2e+47) {
		tmp = -0.011111111111111112 * ((a - b) * (angle * (b * Math.PI)));
	} else {
		tmp = -0.011111111111111112 * ((a - b) * (angle * (a * Math.PI)));
	}
	return tmp;
}

Python

def code(a, b, angle):
	tmp = 0
	if a <= 2e+47:
		tmp = -0.011111111111111112 * ((a - b) * (angle * (b * math.pi)))
	else:
		tmp = -0.011111111111111112 * ((a - b) * (angle * (a * math.pi)))
	return tmp

Julia

function code(a, b, angle)
	tmp = 0.0
	if (a <= 2e+47)
		tmp = Float64(-0.011111111111111112 * Float64(Float64(a - b) * Float64(angle * Float64(b * pi))));
	else
		tmp = Float64(-0.011111111111111112 * Float64(Float64(a - b) * Float64(angle * Float64(a * pi))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, angle)
	tmp = 0.0;
	if (a <= 2e+47)
		tmp = -0.011111111111111112 * ((a - b) * (angle * (b * pi)));
	else
		tmp = -0.011111111111111112 * ((a - b) * (angle * (a * pi)));
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, angle_] := If[LessEqual[a, 2e+47], N[(-0.011111111111111112 * N[(N[(a - b), $MachinePrecision] * N[(angle * N[(b * Pi), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(-0.011111111111111112 * N[(N[(a - b), $MachinePrecision] * N[(angle * N[(a * Pi), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if a < 2.0000000000000001e47

   1. Initial program 53.6%

      \[\left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin \left(\pi \cdot \frac{angle}{180}\right)\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right) \]

   3. Simplified 55.1%

      \[\leadsto \color{blue}{\left(2 \cdot \sin \left(angle \cdot \frac{\pi}{-180}\right)\right) \cdot \left(\cos \left(angle \cdot \frac{\pi}{-180}\right) \cdot \left({a}^{2} - {b}^{2}\right)\right)} \]
   4. Step-by-step derivation

      1. unpow2 55.1%

         \[\leadsto \left(2 \cdot \sin \left(angle \cdot \frac{\pi}{-180}\right)\right) \cdot \left(\cos \left(angle \cdot \frac{\pi}{-180}\right) \cdot \left(\color{blue}{a \cdot a} - {b}^{2}\right)\right) \]

      2. unpow2 55.1%

         \[\leadsto \left(2 \cdot \sin \left(angle \cdot \frac{\pi}{-180}\right)\right) \cdot \left(\cos \left(angle \cdot \frac{\pi}{-180}\right) \cdot \left(a \cdot a - \color{blue}{b \cdot b}\right)\right) \]

      3. difference-of-squares 55.8%

         \[\leadsto \left(2 \cdot \sin \left(angle \cdot \frac{\pi}{-180}\right)\right) \cdot \left(\cos \left(angle \cdot \frac{\pi}{-180}\right) \cdot \color{blue}{\left(\left(a + b\right) \cdot \left(a - b\right)\right)}\right) \]

   5. Applied egg-rr 55.8%

      \[\leadsto \left(2 \cdot \sin \left(angle \cdot \frac{\pi}{-180}\right)\right) \cdot \left(\cos \left(angle \cdot \frac{\pi}{-180}\right) \cdot \color{blue}{\left(\left(a + b\right) \cdot \left(a - b\right)\right)}\right) \]
   6. Step-by-step derivation

      1. add-cbrt-cube 57.2%

         \[\leadsto \left(2 \cdot \sin \left(angle \cdot \frac{\color{blue}{\sqrt[3]{\left(\pi \cdot \pi\right) \cdot \pi}}}{-180}\right)\right) \cdot \left(\cos \left(angle \cdot \frac{\pi}{-180}\right) \cdot \left(\left(a + b\right) \cdot \left(a - b\right)\right)\right) \]

      2. pow3 57.2%

         \[\leadsto \left(2 \cdot \sin \left(angle \cdot \frac{\sqrt[3]{\color{blue}{{\pi}^{3}}}}{-180}\right)\right) \cdot \left(\cos \left(angle \cdot \frac{\pi}{-180}\right) \cdot \left(\left(a + b\right) \cdot \left(a - b\right)\right)\right) \]

   7. Applied egg-rr 57.2%

      \[\leadsto \left(2 \cdot \sin \left(angle \cdot \frac{\color{blue}{\sqrt[3]{{\pi}^{3}}}}{-180}\right)\right) \cdot \left(\cos \left(angle \cdot \frac{\pi}{-180}\right) \cdot \left(\left(a + b\right) \cdot \left(a - b\right)\right)\right) \]

   8. Taylor expanded in angle around 0 53.7%

      \[\leadsto \color{blue}{-0.011111111111111112 \cdot \left(angle \cdot \left(\pi \cdot \left(\left(a + b\right) \cdot \left(a - b\right)\right)\right)\right)} \]
   9. Step-by-step derivation

      1. associate-*r* 53.6%

         \[\leadsto -0.011111111111111112 \cdot \color{blue}{\left(\left(angle \cdot \pi\right) \cdot \left(\left(a + b\right) \cdot \left(a - b\right)\right)\right)} \]

      2. associate-*r* 60.6%

         \[\leadsto -0.011111111111111112 \cdot \color{blue}{\left(\left(\left(angle \cdot \pi\right) \cdot \left(a + b\right)\right) \cdot \left(a - b\right)\right)} \]

      3. *-commutative 60.6%

         \[\leadsto -0.011111111111111112 \cdot \color{blue}{\left(\left(a - b\right) \cdot \left(\left(angle \cdot \pi\right) \cdot \left(a + b\right)\right)\right)} \]

      4. *-commutative 60.6%

         \[\leadsto -0.011111111111111112 \cdot \left(\left(a - b\right) \cdot \color{blue}{\left(\left(a + b\right) \cdot \left(angle \cdot \pi\right)\right)}\right) \]

   10. Simplified 60.6%

       \[\leadsto \color{blue}{-0.011111111111111112 \cdot \left(\left(a - b\right) \cdot \left(\left(a + b\right) \cdot \left(angle \cdot \pi\right)\right)\right)} \]

   11. Taylor expanded in a around 0 43.8%

       \[\leadsto -0.011111111111111112 \cdot \left(\left(a - b\right) \cdot \color{blue}{\left(angle \cdot \left(b \cdot \pi\right)\right)}\right) \]

   if 2.0000000000000001e47 < a

   1. Initial program 52.4%

      \[\left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin \left(\pi \cdot \frac{angle}{180}\right)\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right) \]

   3. Simplified 52.0%

      \[\leadsto \color{blue}{\left(2 \cdot \sin \left(angle \cdot \frac{\pi}{-180}\right)\right) \cdot \left(\cos \left(angle \cdot \frac{\pi}{-180}\right) \cdot \left({a}^{2} - {b}^{2}\right)\right)} \]
   4. Step-by-step derivation

      1. unpow2 52.0%

         \[\leadsto \left(2 \cdot \sin \left(angle \cdot \frac{\pi}{-180}\right)\right) \cdot \left(\cos \left(angle \cdot \frac{\pi}{-180}\right) \cdot \left(\color{blue}{a \cdot a} - {b}^{2}\right)\right) \]

      2. unpow2 52.0%

         \[\leadsto \left(2 \cdot \sin \left(angle \cdot \frac{\pi}{-180}\right)\right) \cdot \left(\cos \left(angle \cdot \frac{\pi}{-180}\right) \cdot \left(a \cdot a - \color{blue}{b \cdot b}\right)\right) \]

      3. difference-of-squares 62.9%

         \[\leadsto \left(2 \cdot \sin \left(angle \cdot \frac{\pi}{-180}\right)\right) \cdot \left(\cos \left(angle \cdot \frac{\pi}{-180}\right) \cdot \color{blue}{\left(\left(a + b\right) \cdot \left(a - b\right)\right)}\right) \]

   5. Applied egg-rr 62.9%

      \[\leadsto \left(2 \cdot \sin \left(angle \cdot \frac{\pi}{-180}\right)\right) \cdot \left(\cos \left(angle \cdot \frac{\pi}{-180}\right) \cdot \color{blue}{\left(\left(a + b\right) \cdot \left(a - b\right)\right)}\right) \]
   6. Step-by-step derivation

      1. add-cbrt-cube 67.0%

         \[\leadsto \left(2 \cdot \sin \left(angle \cdot \frac{\color{blue}{\sqrt[3]{\left(\pi \cdot \pi\right) \cdot \pi}}}{-180}\right)\right) \cdot \left(\cos \left(angle \cdot \frac{\pi}{-180}\right) \cdot \left(\left(a + b\right) \cdot \left(a - b\right)\right)\right) \]

      2. pow3 67.0%

         \[\leadsto \left(2 \cdot \sin \left(angle \cdot \frac{\sqrt[3]{\color{blue}{{\pi}^{3}}}}{-180}\right)\right) \cdot \left(\cos \left(angle \cdot \frac{\pi}{-180}\right) \cdot \left(\left(a + b\right) \cdot \left(a - b\right)\right)\right) \]

   7. Applied egg-rr 67.0%

      \[\leadsto \left(2 \cdot \sin \left(angle \cdot \frac{\color{blue}{\sqrt[3]{{\pi}^{3}}}}{-180}\right)\right) \cdot \left(\cos \left(angle \cdot \frac{\pi}{-180}\right) \cdot \left(\left(a + b\right) \cdot \left(a - b\right)\right)\right) \]

   8. Taylor expanded in angle around 0 58.7%

      \[\leadsto \color{blue}{-0.011111111111111112 \cdot \left(angle \cdot \left(\pi \cdot \left(\left(a + b\right) \cdot \left(a - b\right)\right)\right)\right)} \]
   9. Step-by-step derivation

      1. associate-*r* 58.7%

         \[\leadsto -0.011111111111111112 \cdot \color{blue}{\left(\left(angle \cdot \pi\right) \cdot \left(\left(a + b\right) \cdot \left(a - b\right)\right)\right)} \]

      2. associate-*r* 65.5%

         \[\leadsto -0.011111111111111112 \cdot \color{blue}{\left(\left(\left(angle \cdot \pi\right) \cdot \left(a + b\right)\right) \cdot \left(a - b\right)\right)} \]

      3. *-commutative 65.5%

         \[\leadsto -0.011111111111111112 \cdot \color{blue}{\left(\left(a - b\right) \cdot \left(\left(angle \cdot \pi\right) \cdot \left(a + b\right)\right)\right)} \]

      4. *-commutative 65.5%

         \[\leadsto -0.011111111111111112 \cdot \left(\left(a - b\right) \cdot \color{blue}{\left(\left(a + b\right) \cdot \left(angle \cdot \pi\right)\right)}\right) \]

   10. Simplified 65.5%

       \[\leadsto \color{blue}{-0.011111111111111112 \cdot \left(\left(a - b\right) \cdot \left(\left(a + b\right) \cdot \left(angle \cdot \pi\right)\right)\right)} \]

   11. Taylor expanded in a around inf 59.0%

       \[\leadsto -0.011111111111111112 \cdot \left(\left(a - b\right) \cdot \color{blue}{\left(a \cdot \left(angle \cdot \pi\right)\right)}\right) \]
   12. Step-by-step derivation

       1. *-commutative 59.0%

          \[\leadsto -0.011111111111111112 \cdot \left(\left(a - b\right) \cdot \color{blue}{\left(\left(angle \cdot \pi\right) \cdot a\right)}\right) \]

       2. associate-*l* 59.1%

          \[\leadsto -0.011111111111111112 \cdot \left(\left(a - b\right) \cdot \color{blue}{\left(angle \cdot \left(\pi \cdot a\right)\right)}\right) \]

   13. Simplified 59.1%

       \[\leadsto -0.011111111111111112 \cdot \left(\left(a - b\right) \cdot \color{blue}{\left(angle \cdot \left(\pi \cdot a\right)\right)}\right) \]
3. Recombined 2 regimes into one program.

4. Final simplification 47.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \leq 2 \cdot 10^{+47}:\\ \;\;\;\;-0.011111111111111112 \cdot \left(\left(a - b\right) \cdot \left(angle \cdot \left(b \cdot \pi\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;-0.011111111111111112 \cdot \left(\left(a - b\right) \cdot \left(angle \cdot \left(a \cdot \pi\right)\right)\right)\\ \end{array} \]

Alternative 20: 61.7% accurate, 5.5× speedup

Math

\[\begin{array}{l} \\ -0.011111111111111112 \cdot \left(\left(a - b\right) \cdot \left(\left(b + a\right) \cdot \left(\pi \cdot angle\right)\right)\right) \end{array} \]

FPCore

(FPCore (a b angle)
 :precision binary64
 (* -0.011111111111111112 (* (- a b) (* (+ b a) (* PI angle)))))

C

double code(double a, double b, double angle) {
	return -0.011111111111111112 * ((a - b) * ((b + a) * (((double) M_PI) * angle)));
}

Java

public static double code(double a, double b, double angle) {
	return -0.011111111111111112 * ((a - b) * ((b + a) * (Math.PI * angle)));
}

Python

def code(a, b, angle):
	return -0.011111111111111112 * ((a - b) * ((b + a) * (math.pi * angle)))

Julia

function code(a, b, angle)
	return Float64(-0.011111111111111112 * Float64(Float64(a - b) * Float64(Float64(b + a) * Float64(pi * angle))))
end

MATLAB

function tmp = code(a, b, angle)
	tmp = -0.011111111111111112 * ((a - b) * ((b + a) * (pi * angle)));
end

Wolfram

code[a_, b_, angle_] := N[(-0.011111111111111112 * N[(N[(a - b), $MachinePrecision] * N[(N[(b + a), $MachinePrecision] * N[(Pi * angle), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 53.3%

   \[\left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin \left(\pi \cdot \frac{angle}{180}\right)\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right) \]

3. Simplified 54.5%

   \[\leadsto \color{blue}{\left(2 \cdot \sin \left(angle \cdot \frac{\pi}{-180}\right)\right) \cdot \left(\cos \left(angle \cdot \frac{\pi}{-180}\right) \cdot \left({a}^{2} - {b}^{2}\right)\right)} \]
4. Step-by-step derivation

   1. unpow2 54.5%

      \[\leadsto \left(2 \cdot \sin \left(angle \cdot \frac{\pi}{-180}\right)\right) \cdot \left(\cos \left(angle \cdot \frac{\pi}{-180}\right) \cdot \left(\color{blue}{a \cdot a} - {b}^{2}\right)\right) \]

   2. unpow2 54.5%

      \[\leadsto \left(2 \cdot \sin \left(angle \cdot \frac{\pi}{-180}\right)\right) \cdot \left(\cos \left(angle \cdot \frac{\pi}{-180}\right) \cdot \left(a \cdot a - \color{blue}{b \cdot b}\right)\right) \]

   3. difference-of-squares 57.3%

      \[\leadsto \left(2 \cdot \sin \left(angle \cdot \frac{\pi}{-180}\right)\right) \cdot \left(\cos \left(angle \cdot \frac{\pi}{-180}\right) \cdot \color{blue}{\left(\left(a + b\right) \cdot \left(a - b\right)\right)}\right) \]

5. Applied egg-rr 57.3%

   \[\leadsto \left(2 \cdot \sin \left(angle \cdot \frac{\pi}{-180}\right)\right) \cdot \left(\cos \left(angle \cdot \frac{\pi}{-180}\right) \cdot \color{blue}{\left(\left(a + b\right) \cdot \left(a - b\right)\right)}\right) \]
6. Step-by-step derivation

   1. add-cbrt-cube 59.3%

      \[\leadsto \left(2 \cdot \sin \left(angle \cdot \frac{\color{blue}{\sqrt[3]{\left(\pi \cdot \pi\right) \cdot \pi}}}{-180}\right)\right) \cdot \left(\cos \left(angle \cdot \frac{\pi}{-180}\right) \cdot \left(\left(a + b\right) \cdot \left(a - b\right)\right)\right) \]

   2. pow3 59.3%

      \[\leadsto \left(2 \cdot \sin \left(angle \cdot \frac{\sqrt[3]{\color{blue}{{\pi}^{3}}}}{-180}\right)\right) \cdot \left(\cos \left(angle \cdot \frac{\pi}{-180}\right) \cdot \left(\left(a + b\right) \cdot \left(a - b\right)\right)\right) \]

7. Applied egg-rr 59.3%

   \[\leadsto \left(2 \cdot \sin \left(angle \cdot \frac{\color{blue}{\sqrt[3]{{\pi}^{3}}}}{-180}\right)\right) \cdot \left(\cos \left(angle \cdot \frac{\pi}{-180}\right) \cdot \left(\left(a + b\right) \cdot \left(a - b\right)\right)\right) \]

8. Taylor expanded in angle around 0 54.8%

   \[\leadsto \color{blue}{-0.011111111111111112 \cdot \left(angle \cdot \left(\pi \cdot \left(\left(a + b\right) \cdot \left(a - b\right)\right)\right)\right)} \]
9. Step-by-step derivation

   1. associate-*r* 54.7%

      \[\leadsto -0.011111111111111112 \cdot \color{blue}{\left(\left(angle \cdot \pi\right) \cdot \left(\left(a + b\right) \cdot \left(a - b\right)\right)\right)} \]

   2. associate-*r* 61.6%

      \[\leadsto -0.011111111111111112 \cdot \color{blue}{\left(\left(\left(angle \cdot \pi\right) \cdot \left(a + b\right)\right) \cdot \left(a - b\right)\right)} \]

   3. *-commutative 61.6%

      \[\leadsto -0.011111111111111112 \cdot \color{blue}{\left(\left(a - b\right) \cdot \left(\left(angle \cdot \pi\right) \cdot \left(a + b\right)\right)\right)} \]

   4. *-commutative 61.6%

      \[\leadsto -0.011111111111111112 \cdot \left(\left(a - b\right) \cdot \color{blue}{\left(\left(a + b\right) \cdot \left(angle \cdot \pi\right)\right)}\right) \]

10. Simplified 61.6%

    \[\leadsto \color{blue}{-0.011111111111111112 \cdot \left(\left(a - b\right) \cdot \left(\left(a + b\right) \cdot \left(angle \cdot \pi\right)\right)\right)} \]

11. Final simplification 61.6%

    \[\leadsto -0.011111111111111112 \cdot \left(\left(a - b\right) \cdot \left(\left(b + a\right) \cdot \left(\pi \cdot angle\right)\right)\right) \]

Alternative 21: 40.1% accurate, 5.6× speedup

Math

\[\begin{array}{l} \\ -0.011111111111111112 \cdot \left(\left(a - b\right) \cdot \left(a \cdot \left(\pi \cdot angle\right)\right)\right) \end{array} \]

FPCore

(FPCore (a b angle)
 :precision binary64
 (* -0.011111111111111112 (* (- a b) (* a (* PI angle)))))

C

double code(double a, double b, double angle) {
	return -0.011111111111111112 * ((a - b) * (a * (((double) M_PI) * angle)));
}

Java

public static double code(double a, double b, double angle) {
	return -0.011111111111111112 * ((a - b) * (a * (Math.PI * angle)));
}

Python

def code(a, b, angle):
	return -0.011111111111111112 * ((a - b) * (a * (math.pi * angle)))

Julia

function code(a, b, angle)
	return Float64(-0.011111111111111112 * Float64(Float64(a - b) * Float64(a * Float64(pi * angle))))
end

MATLAB

function tmp = code(a, b, angle)
	tmp = -0.011111111111111112 * ((a - b) * (a * (pi * angle)));
end

Wolfram

code[a_, b_, angle_] := N[(-0.011111111111111112 * N[(N[(a - b), $MachinePrecision] * N[(a * N[(Pi * angle), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 53.3%

   \[\left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin \left(\pi \cdot \frac{angle}{180}\right)\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right) \]

3. Simplified 54.5%

   \[\leadsto \color{blue}{\left(2 \cdot \sin \left(angle \cdot \frac{\pi}{-180}\right)\right) \cdot \left(\cos \left(angle \cdot \frac{\pi}{-180}\right) \cdot \left({a}^{2} - {b}^{2}\right)\right)} \]
4. Step-by-step derivation

   1. unpow2 54.5%

      \[\leadsto \left(2 \cdot \sin \left(angle \cdot \frac{\pi}{-180}\right)\right) \cdot \left(\cos \left(angle \cdot \frac{\pi}{-180}\right) \cdot \left(\color{blue}{a \cdot a} - {b}^{2}\right)\right) \]

   2. unpow2 54.5%

      \[\leadsto \left(2 \cdot \sin \left(angle \cdot \frac{\pi}{-180}\right)\right) \cdot \left(\cos \left(angle \cdot \frac{\pi}{-180}\right) \cdot \left(a \cdot a - \color{blue}{b \cdot b}\right)\right) \]

   3. difference-of-squares 57.3%

      \[\leadsto \left(2 \cdot \sin \left(angle \cdot \frac{\pi}{-180}\right)\right) \cdot \left(\cos \left(angle \cdot \frac{\pi}{-180}\right) \cdot \color{blue}{\left(\left(a + b\right) \cdot \left(a - b\right)\right)}\right) \]

5. Applied egg-rr 57.3%

   \[\leadsto \left(2 \cdot \sin \left(angle \cdot \frac{\pi}{-180}\right)\right) \cdot \left(\cos \left(angle \cdot \frac{\pi}{-180}\right) \cdot \color{blue}{\left(\left(a + b\right) \cdot \left(a - b\right)\right)}\right) \]
6. Step-by-step derivation

   1. add-cbrt-cube 59.3%

      \[\leadsto \left(2 \cdot \sin \left(angle \cdot \frac{\color{blue}{\sqrt[3]{\left(\pi \cdot \pi\right) \cdot \pi}}}{-180}\right)\right) \cdot \left(\cos \left(angle \cdot \frac{\pi}{-180}\right) \cdot \left(\left(a + b\right) \cdot \left(a - b\right)\right)\right) \]

   2. pow3 59.3%

      \[\leadsto \left(2 \cdot \sin \left(angle \cdot \frac{\sqrt[3]{\color{blue}{{\pi}^{3}}}}{-180}\right)\right) \cdot \left(\cos \left(angle \cdot \frac{\pi}{-180}\right) \cdot \left(\left(a + b\right) \cdot \left(a - b\right)\right)\right) \]

7. Applied egg-rr 59.3%

   \[\leadsto \left(2 \cdot \sin \left(angle \cdot \frac{\color{blue}{\sqrt[3]{{\pi}^{3}}}}{-180}\right)\right) \cdot \left(\cos \left(angle \cdot \frac{\pi}{-180}\right) \cdot \left(\left(a + b\right) \cdot \left(a - b\right)\right)\right) \]

8. Taylor expanded in angle around 0 54.8%

   \[\leadsto \color{blue}{-0.011111111111111112 \cdot \left(angle \cdot \left(\pi \cdot \left(\left(a + b\right) \cdot \left(a - b\right)\right)\right)\right)} \]
9. Step-by-step derivation

   1. associate-*r* 54.7%

      \[\leadsto -0.011111111111111112 \cdot \color{blue}{\left(\left(angle \cdot \pi\right) \cdot \left(\left(a + b\right) \cdot \left(a - b\right)\right)\right)} \]

   2. associate-*r* 61.6%

      \[\leadsto -0.011111111111111112 \cdot \color{blue}{\left(\left(\left(angle \cdot \pi\right) \cdot \left(a + b\right)\right) \cdot \left(a - b\right)\right)} \]

   3. *-commutative 61.6%

      \[\leadsto -0.011111111111111112 \cdot \color{blue}{\left(\left(a - b\right) \cdot \left(\left(angle \cdot \pi\right) \cdot \left(a + b\right)\right)\right)} \]

   4. *-commutative 61.6%

      \[\leadsto -0.011111111111111112 \cdot \left(\left(a - b\right) \cdot \color{blue}{\left(\left(a + b\right) \cdot \left(angle \cdot \pi\right)\right)}\right) \]

10. Simplified 61.6%

    \[\leadsto \color{blue}{-0.011111111111111112 \cdot \left(\left(a - b\right) \cdot \left(\left(a + b\right) \cdot \left(angle \cdot \pi\right)\right)\right)} \]

11. Taylor expanded in a around inf 40.5%

    \[\leadsto -0.011111111111111112 \cdot \left(\left(a - b\right) \cdot \color{blue}{\left(a \cdot \left(angle \cdot \pi\right)\right)}\right) \]

12. Final simplification 40.5%

    \[\leadsto -0.011111111111111112 \cdot \left(\left(a - b\right) \cdot \left(a \cdot \left(\pi \cdot angle\right)\right)\right) \]

Reproduce

herbie shell --seed 2023299 
(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)))))