ab-angle->ABCF B

Percentage Accurate: 54.2% → 67.5%

Time: 29.7s

Alternatives: 16

Speedup: 32.2×

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

Specification

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 54.2% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 67.5% accurate, 0.7× speedup

Math

\[\begin{array}{l} a_m = \left|a\right| \\ b_m = \left|b\right| \\ angle\_m = \left|angle\right| \\ angle\_s = \mathsf{copysign}\left(1, angle\right) \\ \begin{array}{l} t_0 := \left(b\_m + a\_m\right) \cdot \left(b\_m - a\_m\right)\\ t_1 := \sin \left(\frac{angle\_m}{180} \cdot \sqrt[3]{{\pi}^{3}}\right)\\ t_2 := \cos \left(\frac{angle\_m}{180} \cdot \pi\right)\\ angle\_s \cdot \begin{array}{l} \mathbf{if}\;\frac{angle\_m}{180} \leq 4 \cdot 10^{+98}:\\ \;\;\;\;{\left(\sqrt[3]{b\_m + a\_m} \cdot \sqrt[3]{\left(b\_m - a\_m\right) \cdot \sin \left(\pi \cdot \left(2 \cdot \left(angle\_m \cdot 0.005555555555555556\right)\right)\right)}\right)}^{3}\\ \mathbf{elif}\;\frac{angle\_m}{180} \leq 2 \cdot 10^{+151}:\\ \;\;\;\;t\_0 \cdot \left(2 \cdot \left(t\_1 \cdot t\_2\right)\right)\\ \mathbf{elif}\;\frac{angle\_m}{180} \leq 10^{+264}:\\ \;\;\;\;t\_0 \cdot \left(2 \cdot \left(t\_1 \cdot \sqrt{{\cos \left(\pi \cdot \left(angle\_m \cdot 0.005555555555555556\right)\right)}^{2}}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;t\_0 \cdot \left(2 \cdot \left(t\_2 \cdot {\left(\sqrt[3]{\sin \left(0.005555555555555556 \cdot \left(angle\_m \cdot \pi\right)\right)}\right)}^{3}\right)\right)\\ \end{array} \end{array} \end{array} \]

FPCore

a_m = (fabs.f64 a)
b_m = (fabs.f64 b)
angle\_m = (fabs.f64 angle)
angle\_s = (copysign.f64 #s(literal 1 binary64) angle)
(FPCore (angle_s a_m b_m angle_m)
 :precision binary64
 (let* ((t_0 (* (+ b_m a_m) (- b_m a_m)))
        (t_1 (sin (* (/ angle_m 180.0) (cbrt (pow PI 3.0)))))
        (t_2 (cos (* (/ angle_m 180.0) PI))))
   (*
    angle_s
    (if (<= (/ angle_m 180.0) 4e+98)
      (pow
       (*
        (cbrt (+ b_m a_m))
        (cbrt
         (*
          (- b_m a_m)
          (sin (* PI (* 2.0 (* angle_m 0.005555555555555556)))))))
       3.0)
      (if (<= (/ angle_m 180.0) 2e+151)
        (* t_0 (* 2.0 (* t_1 t_2)))
        (if (<= (/ angle_m 180.0) 1e+264)
          (*
           t_0
           (*
            2.0
            (*
             t_1
             (sqrt (pow (cos (* PI (* angle_m 0.005555555555555556))) 2.0)))))
          (*
           t_0
           (*
            2.0
            (*
             t_2
             (pow
              (cbrt (sin (* 0.005555555555555556 (* angle_m PI))))
              3.0))))))))))

C

a_m = fabs(a);
b_m = fabs(b);
angle\_m = fabs(angle);
angle\_s = copysign(1.0, angle);
double code(double angle_s, double a_m, double b_m, double angle_m) {
	double t_0 = (b_m + a_m) * (b_m - a_m);
	double t_1 = sin(((angle_m / 180.0) * cbrt(pow(((double) M_PI), 3.0))));
	double t_2 = cos(((angle_m / 180.0) * ((double) M_PI)));
	double tmp;
	if ((angle_m / 180.0) <= 4e+98) {
		tmp = pow((cbrt((b_m + a_m)) * cbrt(((b_m - a_m) * sin((((double) M_PI) * (2.0 * (angle_m * 0.005555555555555556))))))), 3.0);
	} else if ((angle_m / 180.0) <= 2e+151) {
		tmp = t_0 * (2.0 * (t_1 * t_2));
	} else if ((angle_m / 180.0) <= 1e+264) {
		tmp = t_0 * (2.0 * (t_1 * sqrt(pow(cos((((double) M_PI) * (angle_m * 0.005555555555555556))), 2.0))));
	} else {
		tmp = t_0 * (2.0 * (t_2 * pow(cbrt(sin((0.005555555555555556 * (angle_m * ((double) M_PI))))), 3.0)));
	}
	return angle_s * tmp;
}

Java

a_m = Math.abs(a);
b_m = Math.abs(b);
angle\_m = Math.abs(angle);
angle\_s = Math.copySign(1.0, angle);
public static double code(double angle_s, double a_m, double b_m, double angle_m) {
	double t_0 = (b_m + a_m) * (b_m - a_m);
	double t_1 = Math.sin(((angle_m / 180.0) * Math.cbrt(Math.pow(Math.PI, 3.0))));
	double t_2 = Math.cos(((angle_m / 180.0) * Math.PI));
	double tmp;
	if ((angle_m / 180.0) <= 4e+98) {
		tmp = Math.pow((Math.cbrt((b_m + a_m)) * Math.cbrt(((b_m - a_m) * Math.sin((Math.PI * (2.0 * (angle_m * 0.005555555555555556))))))), 3.0);
	} else if ((angle_m / 180.0) <= 2e+151) {
		tmp = t_0 * (2.0 * (t_1 * t_2));
	} else if ((angle_m / 180.0) <= 1e+264) {
		tmp = t_0 * (2.0 * (t_1 * Math.sqrt(Math.pow(Math.cos((Math.PI * (angle_m * 0.005555555555555556))), 2.0))));
	} else {
		tmp = t_0 * (2.0 * (t_2 * Math.pow(Math.cbrt(Math.sin((0.005555555555555556 * (angle_m * Math.PI)))), 3.0)));
	}
	return angle_s * tmp;
}

Julia

a_m = abs(a)
b_m = abs(b)
angle\_m = abs(angle)
angle\_s = copysign(1.0, angle)
function code(angle_s, a_m, b_m, angle_m)
	t_0 = Float64(Float64(b_m + a_m) * Float64(b_m - a_m))
	t_1 = sin(Float64(Float64(angle_m / 180.0) * cbrt((pi ^ 3.0))))
	t_2 = cos(Float64(Float64(angle_m / 180.0) * pi))
	tmp = 0.0
	if (Float64(angle_m / 180.0) <= 4e+98)
		tmp = Float64(cbrt(Float64(b_m + a_m)) * cbrt(Float64(Float64(b_m - a_m) * sin(Float64(pi * Float64(2.0 * Float64(angle_m * 0.005555555555555556))))))) ^ 3.0;
	elseif (Float64(angle_m / 180.0) <= 2e+151)
		tmp = Float64(t_0 * Float64(2.0 * Float64(t_1 * t_2)));
	elseif (Float64(angle_m / 180.0) <= 1e+264)
		tmp = Float64(t_0 * Float64(2.0 * Float64(t_1 * sqrt((cos(Float64(pi * Float64(angle_m * 0.005555555555555556))) ^ 2.0)))));
	else
		tmp = Float64(t_0 * Float64(2.0 * Float64(t_2 * (cbrt(sin(Float64(0.005555555555555556 * Float64(angle_m * pi)))) ^ 3.0))));
	end
	return Float64(angle_s * tmp)
end

Wolfram

a_m = N[Abs[a], $MachinePrecision]
b_m = N[Abs[b], $MachinePrecision]
angle\_m = N[Abs[angle], $MachinePrecision]
angle\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[angle]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[angle$95$s_, a$95$m_, b$95$m_, angle$95$m_] := Block[{t$95$0 = N[(N[(b$95$m + a$95$m), $MachinePrecision] * N[(b$95$m - a$95$m), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[Sin[N[(N[(angle$95$m / 180.0), $MachinePrecision] * N[Power[N[Power[Pi, 3.0], $MachinePrecision], 1/3], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$2 = N[Cos[N[(N[(angle$95$m / 180.0), $MachinePrecision] * Pi), $MachinePrecision]], $MachinePrecision]}, N[(angle$95$s * If[LessEqual[N[(angle$95$m / 180.0), $MachinePrecision], 4e+98], N[Power[N[(N[Power[N[(b$95$m + a$95$m), $MachinePrecision], 1/3], $MachinePrecision] * N[Power[N[(N[(b$95$m - a$95$m), $MachinePrecision] * N[Sin[N[(Pi * N[(2.0 * N[(angle$95$m * 0.005555555555555556), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], 1/3], $MachinePrecision]), $MachinePrecision], 3.0], $MachinePrecision], If[LessEqual[N[(angle$95$m / 180.0), $MachinePrecision], 2e+151], N[(t$95$0 * N[(2.0 * N[(t$95$1 * t$95$2), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[N[(angle$95$m / 180.0), $MachinePrecision], 1e+264], N[(t$95$0 * N[(2.0 * N[(t$95$1 * N[Sqrt[N[Power[N[Cos[N[(Pi * N[(angle$95$m * 0.005555555555555556), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], 2.0], $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(t$95$0 * N[(2.0 * N[(t$95$2 * N[Power[N[Power[N[Sin[N[(0.005555555555555556 * N[(angle$95$m * Pi), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], 1/3], $MachinePrecision], 3.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]), $MachinePrecision]]]]

Derivation

1. Split input into 4 regimes

2. if (/.f64 angle #s(literal 180 binary64)) < 3.99999999999999999e98

   1. Initial program 62.4%

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

      1. associate-*l* 62.4%

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

      2. *-commutative 62.4%

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

      3. associate-*l* 62.9%

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

   3. Simplified 62.9%

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

      1. add-cube-cbrt 62.5%

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

      2. pow3 62.5%

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

      3. 2-sin 62.5%

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

      4. associate-*r* 62.5%

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

      5. div-inv 62.1%

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

      6. metadata-eval 62.1%

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

   6. Applied egg-rr 62.1%

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

      1. unpow2 62.1%

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

      2. unpow2 62.1%

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

      3. difference-of-squares 66.0%

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

      4. metadata-eval 66.0%

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

      5. div-inv 66.8%

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

      6. associate-*l* 66.8%

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

      7. 2-sin 66.8%

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

      8. associate-*l* 76.1%

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

      9. cbrt-prod 76.1%

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

   8. Applied egg-rr 75.3%

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

   if 3.99999999999999999e98 < (/.f64 angle #s(literal 180 binary64)) < 2.00000000000000003e151

   1. Initial program 32.0%

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

      1. associate-*l* 32.0%

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

      2. *-commutative 32.0%

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

      3. associate-*l* 32.0%

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

   3. Simplified 32.0%

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

      1. unpow2 32.0%

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

      2. unpow2 32.0%

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

      3. difference-of-squares 32.0%

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

   6. Applied egg-rr 32.0%

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

      1. add-cbrt-cube 65.2%

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

      2. pow3 65.2%

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

   8. Applied egg-rr 65.2%

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

   if 2.00000000000000003e151 < (/.f64 angle #s(literal 180 binary64)) < 1.00000000000000004e264

   1. Initial program 42.7%

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

      1. associate-*l* 42.7%

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

      2. *-commutative 42.7%

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

      3. associate-*l* 42.7%

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

   3. Simplified 42.7%

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

      1. unpow2 42.7%

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

      2. unpow2 42.7%

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

      3. difference-of-squares 57.0%

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

   6. Applied egg-rr 57.0%

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

      1. add-cbrt-cube 45.6%

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

      2. pow3 45.6%

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

   8. Applied egg-rr 45.6%

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

      1. add-sqr-sqrt 19.1%

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

      2. sqrt-unprod 51.8%

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

      3. pow2 51.8%

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

      4. div-inv 51.8%

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

      5. metadata-eval 51.8%

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

   10. Applied egg-rr 51.8%

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

   if 1.00000000000000004e264 < (/.f64 angle #s(literal 180 binary64))

   1. Initial program 36.9%

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

      1. associate-*l* 36.9%

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

      2. *-commutative 36.9%

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

      3. associate-*l* 36.9%

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

   3. Simplified 36.9%

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

      1. unpow2 36.9%

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

      2. unpow2 36.9%

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

      3. difference-of-squares 36.9%

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

   6. Applied egg-rr 36.9%

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

      1. add-cbrt-cube 48.2%

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

      2. pow3 48.2%

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

   8. Applied egg-rr 48.2%

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

      1. rem-cbrt-cube 36.9%

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

      2. div-inv 36.8%

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

      3. metadata-eval 36.8%

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

      4. add-cube-cbrt 36.8%

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

      5. pow3 36.8%

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

      6. metadata-eval 36.8%

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

      7. div-inv 36.9%

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

      8. rem-cbrt-cube 48.2%

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

      9. rem-cbrt-cube 36.9%

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

      10. associate-*r/ 50.9%

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

      11. div-inv 50.6%

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

      12. metadata-eval 50.6%

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

   10. Applied egg-rr 50.6%

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

4. Final simplification 72.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{angle}{180} \leq 4 \cdot 10^{+98}:\\ \;\;\;\;{\left(\sqrt[3]{b + a} \cdot \sqrt[3]{\left(b - a\right) \cdot \sin \left(\pi \cdot \left(2 \cdot \left(angle \cdot 0.005555555555555556\right)\right)\right)}\right)}^{3}\\ \mathbf{elif}\;\frac{angle}{180} \leq 2 \cdot 10^{+151}:\\ \;\;\;\;\left(\left(b + a\right) \cdot \left(b - a\right)\right) \cdot \left(2 \cdot \left(\sin \left(\frac{angle}{180} \cdot \sqrt[3]{{\pi}^{3}}\right) \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)\right)\right)\\ \mathbf{elif}\;\frac{angle}{180} \leq 10^{+264}:\\ \;\;\;\;\left(\left(b + a\right) \cdot \left(b - a\right)\right) \cdot \left(2 \cdot \left(\sin \left(\frac{angle}{180} \cdot \sqrt[3]{{\pi}^{3}}\right) \cdot \sqrt{{\cos \left(\pi \cdot \left(angle \cdot 0.005555555555555556\right)\right)}^{2}}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\left(b + a\right) \cdot \left(b - a\right)\right) \cdot \left(2 \cdot \left(\cos \left(\frac{angle}{180} \cdot \pi\right) \cdot {\left(\sqrt[3]{\sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}\right)}^{3}\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 67.5% accurate, 0.9× speedup

Math

\[\begin{array}{l} a_m = \left|a\right| \\ b_m = \left|b\right| \\ angle\_m = \left|angle\right| \\ angle\_s = \mathsf{copysign}\left(1, angle\right) \\ \begin{array}{l} t_0 := \left(b\_m + a\_m\right) \cdot \left(b\_m - a\_m\right)\\ t_1 := \sin \left(\frac{angle\_m}{180} \cdot \sqrt[3]{{\pi}^{3}}\right)\\ t_2 := \cos \left(\frac{angle\_m}{180} \cdot \pi\right)\\ angle\_s \cdot \begin{array}{l} \mathbf{if}\;\frac{angle\_m}{180} \leq 4 \cdot 10^{+98}:\\ \;\;\;\;{\left(\sqrt[3]{b\_m + a\_m} \cdot \sqrt[3]{\left(b\_m - a\_m\right) \cdot \sin \left(\pi \cdot \left(2 \cdot \left(angle\_m \cdot 0.005555555555555556\right)\right)\right)}\right)}^{3}\\ \mathbf{elif}\;\frac{angle\_m}{180} \leq 2 \cdot 10^{+155}:\\ \;\;\;\;t\_0 \cdot \left(2 \cdot \left(t\_1 \cdot t\_2\right)\right)\\ \mathbf{elif}\;\frac{angle\_m}{180} \leq 10^{+264}:\\ \;\;\;\;t\_0 \cdot \left(2 \cdot t\_1\right)\\ \mathbf{else}:\\ \;\;\;\;t\_0 \cdot \left(2 \cdot \left(t\_2 \cdot {\left(\sqrt[3]{\sin \left(0.005555555555555556 \cdot \left(angle\_m \cdot \pi\right)\right)}\right)}^{3}\right)\right)\\ \end{array} \end{array} \end{array} \]

FPCore

a_m = (fabs.f64 a)
b_m = (fabs.f64 b)
angle\_m = (fabs.f64 angle)
angle\_s = (copysign.f64 #s(literal 1 binary64) angle)
(FPCore (angle_s a_m b_m angle_m)
 :precision binary64
 (let* ((t_0 (* (+ b_m a_m) (- b_m a_m)))
        (t_1 (sin (* (/ angle_m 180.0) (cbrt (pow PI 3.0)))))
        (t_2 (cos (* (/ angle_m 180.0) PI))))
   (*
    angle_s
    (if (<= (/ angle_m 180.0) 4e+98)
      (pow
       (*
        (cbrt (+ b_m a_m))
        (cbrt
         (*
          (- b_m a_m)
          (sin (* PI (* 2.0 (* angle_m 0.005555555555555556)))))))
       3.0)
      (if (<= (/ angle_m 180.0) 2e+155)
        (* t_0 (* 2.0 (* t_1 t_2)))
        (if (<= (/ angle_m 180.0) 1e+264)
          (* t_0 (* 2.0 t_1))
          (*
           t_0
           (*
            2.0
            (*
             t_2
             (pow
              (cbrt (sin (* 0.005555555555555556 (* angle_m PI))))
              3.0))))))))))

C

a_m = fabs(a);
b_m = fabs(b);
angle\_m = fabs(angle);
angle\_s = copysign(1.0, angle);
double code(double angle_s, double a_m, double b_m, double angle_m) {
	double t_0 = (b_m + a_m) * (b_m - a_m);
	double t_1 = sin(((angle_m / 180.0) * cbrt(pow(((double) M_PI), 3.0))));
	double t_2 = cos(((angle_m / 180.0) * ((double) M_PI)));
	double tmp;
	if ((angle_m / 180.0) <= 4e+98) {
		tmp = pow((cbrt((b_m + a_m)) * cbrt(((b_m - a_m) * sin((((double) M_PI) * (2.0 * (angle_m * 0.005555555555555556))))))), 3.0);
	} else if ((angle_m / 180.0) <= 2e+155) {
		tmp = t_0 * (2.0 * (t_1 * t_2));
	} else if ((angle_m / 180.0) <= 1e+264) {
		tmp = t_0 * (2.0 * t_1);
	} else {
		tmp = t_0 * (2.0 * (t_2 * pow(cbrt(sin((0.005555555555555556 * (angle_m * ((double) M_PI))))), 3.0)));
	}
	return angle_s * tmp;
}

Java

a_m = Math.abs(a);
b_m = Math.abs(b);
angle\_m = Math.abs(angle);
angle\_s = Math.copySign(1.0, angle);
public static double code(double angle_s, double a_m, double b_m, double angle_m) {
	double t_0 = (b_m + a_m) * (b_m - a_m);
	double t_1 = Math.sin(((angle_m / 180.0) * Math.cbrt(Math.pow(Math.PI, 3.0))));
	double t_2 = Math.cos(((angle_m / 180.0) * Math.PI));
	double tmp;
	if ((angle_m / 180.0) <= 4e+98) {
		tmp = Math.pow((Math.cbrt((b_m + a_m)) * Math.cbrt(((b_m - a_m) * Math.sin((Math.PI * (2.0 * (angle_m * 0.005555555555555556))))))), 3.0);
	} else if ((angle_m / 180.0) <= 2e+155) {
		tmp = t_0 * (2.0 * (t_1 * t_2));
	} else if ((angle_m / 180.0) <= 1e+264) {
		tmp = t_0 * (2.0 * t_1);
	} else {
		tmp = t_0 * (2.0 * (t_2 * Math.pow(Math.cbrt(Math.sin((0.005555555555555556 * (angle_m * Math.PI)))), 3.0)));
	}
	return angle_s * tmp;
}

Julia

a_m = abs(a)
b_m = abs(b)
angle\_m = abs(angle)
angle\_s = copysign(1.0, angle)
function code(angle_s, a_m, b_m, angle_m)
	t_0 = Float64(Float64(b_m + a_m) * Float64(b_m - a_m))
	t_1 = sin(Float64(Float64(angle_m / 180.0) * cbrt((pi ^ 3.0))))
	t_2 = cos(Float64(Float64(angle_m / 180.0) * pi))
	tmp = 0.0
	if (Float64(angle_m / 180.0) <= 4e+98)
		tmp = Float64(cbrt(Float64(b_m + a_m)) * cbrt(Float64(Float64(b_m - a_m) * sin(Float64(pi * Float64(2.0 * Float64(angle_m * 0.005555555555555556))))))) ^ 3.0;
	elseif (Float64(angle_m / 180.0) <= 2e+155)
		tmp = Float64(t_0 * Float64(2.0 * Float64(t_1 * t_2)));
	elseif (Float64(angle_m / 180.0) <= 1e+264)
		tmp = Float64(t_0 * Float64(2.0 * t_1));
	else
		tmp = Float64(t_0 * Float64(2.0 * Float64(t_2 * (cbrt(sin(Float64(0.005555555555555556 * Float64(angle_m * pi)))) ^ 3.0))));
	end
	return Float64(angle_s * tmp)
end

Wolfram

a_m = N[Abs[a], $MachinePrecision]
b_m = N[Abs[b], $MachinePrecision]
angle\_m = N[Abs[angle], $MachinePrecision]
angle\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[angle]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[angle$95$s_, a$95$m_, b$95$m_, angle$95$m_] := Block[{t$95$0 = N[(N[(b$95$m + a$95$m), $MachinePrecision] * N[(b$95$m - a$95$m), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[Sin[N[(N[(angle$95$m / 180.0), $MachinePrecision] * N[Power[N[Power[Pi, 3.0], $MachinePrecision], 1/3], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$2 = N[Cos[N[(N[(angle$95$m / 180.0), $MachinePrecision] * Pi), $MachinePrecision]], $MachinePrecision]}, N[(angle$95$s * If[LessEqual[N[(angle$95$m / 180.0), $MachinePrecision], 4e+98], N[Power[N[(N[Power[N[(b$95$m + a$95$m), $MachinePrecision], 1/3], $MachinePrecision] * N[Power[N[(N[(b$95$m - a$95$m), $MachinePrecision] * N[Sin[N[(Pi * N[(2.0 * N[(angle$95$m * 0.005555555555555556), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], 1/3], $MachinePrecision]), $MachinePrecision], 3.0], $MachinePrecision], If[LessEqual[N[(angle$95$m / 180.0), $MachinePrecision], 2e+155], N[(t$95$0 * N[(2.0 * N[(t$95$1 * t$95$2), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[N[(angle$95$m / 180.0), $MachinePrecision], 1e+264], N[(t$95$0 * N[(2.0 * t$95$1), $MachinePrecision]), $MachinePrecision], N[(t$95$0 * N[(2.0 * N[(t$95$2 * N[Power[N[Power[N[Sin[N[(0.005555555555555556 * N[(angle$95$m * Pi), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], 1/3], $MachinePrecision], 3.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]), $MachinePrecision]]]]

Derivation

1. Split input into 4 regimes

2. if (/.f64 angle #s(literal 180 binary64)) < 3.99999999999999999e98

   1. Initial program 62.4%

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

      1. associate-*l* 62.4%

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

      2. *-commutative 62.4%

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

      3. associate-*l* 62.9%

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

   3. Simplified 62.9%

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

      1. add-cube-cbrt 62.5%

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

      2. pow3 62.5%

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

      3. 2-sin 62.5%

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

      4. associate-*r* 62.5%

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

      5. div-inv 62.1%

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

      6. metadata-eval 62.1%

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

   6. Applied egg-rr 62.1%

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

      1. unpow2 62.1%

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

      2. unpow2 62.1%

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

      3. difference-of-squares 66.0%

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

      4. metadata-eval 66.0%

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

      5. div-inv 66.8%

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

      6. associate-*l* 66.8%

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

      7. 2-sin 66.8%

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

      8. associate-*l* 76.1%

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

      9. cbrt-prod 76.1%

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

   8. Applied egg-rr 75.3%

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

   if 3.99999999999999999e98 < (/.f64 angle #s(literal 180 binary64)) < 2.00000000000000001e155

   1. Initial program 29.7%

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

      1. associate-*l* 29.7%

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

      2. *-commutative 29.7%

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

      3. associate-*l* 29.7%

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

   3. Simplified 29.7%

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

      1. unpow2 29.7%

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

      2. unpow2 29.7%

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

      3. difference-of-squares 29.7%

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

   6. Applied egg-rr 29.7%

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

      1. add-cbrt-cube 60.5%

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

      2. pow3 60.5%

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

   8. Applied egg-rr 60.5%

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

   if 2.00000000000000001e155 < (/.f64 angle #s(literal 180 binary64)) < 1.00000000000000004e264

   1. Initial program 44.9%

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

      1. associate-*l* 44.9%

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

      2. *-commutative 44.9%

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

      3. associate-*l* 44.9%

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

   3. Simplified 44.9%

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

      1. unpow2 44.9%

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

      2. unpow2 44.9%

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

      3. difference-of-squares 59.9%

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

   6. Applied egg-rr 59.9%

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

      1. add-cbrt-cube 47.8%

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

      2. pow3 47.8%

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

   8. Applied egg-rr 47.8%

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

   9. Taylor expanded in angle around 0 54.1%

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

   if 1.00000000000000004e264 < (/.f64 angle #s(literal 180 binary64))

   1. Initial program 36.9%

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

      1. associate-*l* 36.9%

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

      2. *-commutative 36.9%

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

      3. associate-*l* 36.9%

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

   3. Simplified 36.9%

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

      1. unpow2 36.9%

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

      2. unpow2 36.9%

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

      3. difference-of-squares 36.9%

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

   6. Applied egg-rr 36.9%

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

      1. add-cbrt-cube 48.2%

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

      2. pow3 48.2%

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

   8. Applied egg-rr 48.2%

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

      1. rem-cbrt-cube 36.9%

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

      2. div-inv 36.8%

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

      3. metadata-eval 36.8%

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

      4. add-cube-cbrt 36.8%

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

      5. pow3 36.8%

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

      6. metadata-eval 36.8%

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

      7. div-inv 36.9%

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

      8. rem-cbrt-cube 48.2%

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

      9. rem-cbrt-cube 36.9%

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

      10. associate-*r/ 50.9%

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

      11. div-inv 50.6%

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

      12. metadata-eval 50.6%

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

   10. Applied egg-rr 50.6%

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

4. Final simplification 72.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{angle}{180} \leq 4 \cdot 10^{+98}:\\ \;\;\;\;{\left(\sqrt[3]{b + a} \cdot \sqrt[3]{\left(b - a\right) \cdot \sin \left(\pi \cdot \left(2 \cdot \left(angle \cdot 0.005555555555555556\right)\right)\right)}\right)}^{3}\\ \mathbf{elif}\;\frac{angle}{180} \leq 2 \cdot 10^{+155}:\\ \;\;\;\;\left(\left(b + a\right) \cdot \left(b - a\right)\right) \cdot \left(2 \cdot \left(\sin \left(\frac{angle}{180} \cdot \sqrt[3]{{\pi}^{3}}\right) \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)\right)\right)\\ \mathbf{elif}\;\frac{angle}{180} \leq 10^{+264}:\\ \;\;\;\;\left(\left(b + a\right) \cdot \left(b - a\right)\right) \cdot \left(2 \cdot \sin \left(\frac{angle}{180} \cdot \sqrt[3]{{\pi}^{3}}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\left(b + a\right) \cdot \left(b - a\right)\right) \cdot \left(2 \cdot \left(\cos \left(\frac{angle}{180} \cdot \pi\right) \cdot {\left(\sqrt[3]{\sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}\right)}^{3}\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 67.6% accurate, 0.9× speedup

Math

\[\begin{array}{l} a_m = \left|a\right| \\ b_m = \left|b\right| \\ angle\_m = \left|angle\right| \\ angle\_s = \mathsf{copysign}\left(1, angle\right) \\ \begin{array}{l} t_0 := \left(b\_m + a\_m\right) \cdot \left(b\_m - a\_m\right)\\ t_1 := \pi \cdot \left(angle\_m \cdot 0.005555555555555556\right)\\ t_2 := \frac{angle\_m}{180} \cdot \pi\\ t_3 := \cos t\_2\\ angle\_s \cdot \begin{array}{l} \mathbf{if}\;\frac{angle\_m}{180} \leq 2 \cdot 10^{+86}:\\ \;\;\;\;{\left(\sqrt[3]{b\_m + a\_m} \cdot \sqrt[3]{\left(b\_m - a\_m\right) \cdot \sin \left(\pi \cdot \left(2 \cdot \left(angle\_m \cdot 0.005555555555555556\right)\right)\right)}\right)}^{3}\\ \mathbf{elif}\;\frac{angle\_m}{180} \leq 10^{+222}:\\ \;\;\;\;t\_0 \cdot \left(2 \cdot \left(t\_3 \cdot \sin \left(\sqrt[3]{{\pi}^{3}} \cdot \left(angle\_m \cdot 0.005555555555555556\right)\right)\right)\right)\\ \mathbf{elif}\;\frac{angle\_m}{180} \leq 5 \cdot 10^{+281}:\\ \;\;\;\;t\_0 \cdot \left(2 \cdot \left(\sqrt{{\cos t\_1}^{2}} \cdot \sin t\_1\right)\right)\\ \mathbf{else}:\\ \;\;\;\;t\_0 \cdot \left(2 \cdot \left(t\_3 \cdot \sin t\_2\right)\right)\\ \end{array} \end{array} \end{array} \]

FPCore

a_m = (fabs.f64 a)
b_m = (fabs.f64 b)
angle\_m = (fabs.f64 angle)
angle\_s = (copysign.f64 #s(literal 1 binary64) angle)
(FPCore (angle_s a_m b_m angle_m)
 :precision binary64
 (let* ((t_0 (* (+ b_m a_m) (- b_m a_m)))
        (t_1 (* PI (* angle_m 0.005555555555555556)))
        (t_2 (* (/ angle_m 180.0) PI))
        (t_3 (cos t_2)))
   (*
    angle_s
    (if (<= (/ angle_m 180.0) 2e+86)
      (pow
       (*
        (cbrt (+ b_m a_m))
        (cbrt
         (*
          (- b_m a_m)
          (sin (* PI (* 2.0 (* angle_m 0.005555555555555556)))))))
       3.0)
      (if (<= (/ angle_m 180.0) 1e+222)
        (*
         t_0
         (*
          2.0
          (*
           t_3
           (sin (* (cbrt (pow PI 3.0)) (* angle_m 0.005555555555555556))))))
        (if (<= (/ angle_m 180.0) 5e+281)
          (* t_0 (* 2.0 (* (sqrt (pow (cos t_1) 2.0)) (sin t_1))))
          (* t_0 (* 2.0 (* t_3 (sin t_2))))))))))

C

a_m = fabs(a);
b_m = fabs(b);
angle\_m = fabs(angle);
angle\_s = copysign(1.0, angle);
double code(double angle_s, double a_m, double b_m, double angle_m) {
	double t_0 = (b_m + a_m) * (b_m - a_m);
	double t_1 = ((double) M_PI) * (angle_m * 0.005555555555555556);
	double t_2 = (angle_m / 180.0) * ((double) M_PI);
	double t_3 = cos(t_2);
	double tmp;
	if ((angle_m / 180.0) <= 2e+86) {
		tmp = pow((cbrt((b_m + a_m)) * cbrt(((b_m - a_m) * sin((((double) M_PI) * (2.0 * (angle_m * 0.005555555555555556))))))), 3.0);
	} else if ((angle_m / 180.0) <= 1e+222) {
		tmp = t_0 * (2.0 * (t_3 * sin((cbrt(pow(((double) M_PI), 3.0)) * (angle_m * 0.005555555555555556)))));
	} else if ((angle_m / 180.0) <= 5e+281) {
		tmp = t_0 * (2.0 * (sqrt(pow(cos(t_1), 2.0)) * sin(t_1)));
	} else {
		tmp = t_0 * (2.0 * (t_3 * sin(t_2)));
	}
	return angle_s * tmp;
}

Java

a_m = Math.abs(a);
b_m = Math.abs(b);
angle\_m = Math.abs(angle);
angle\_s = Math.copySign(1.0, angle);
public static double code(double angle_s, double a_m, double b_m, double angle_m) {
	double t_0 = (b_m + a_m) * (b_m - a_m);
	double t_1 = Math.PI * (angle_m * 0.005555555555555556);
	double t_2 = (angle_m / 180.0) * Math.PI;
	double t_3 = Math.cos(t_2);
	double tmp;
	if ((angle_m / 180.0) <= 2e+86) {
		tmp = Math.pow((Math.cbrt((b_m + a_m)) * Math.cbrt(((b_m - a_m) * Math.sin((Math.PI * (2.0 * (angle_m * 0.005555555555555556))))))), 3.0);
	} else if ((angle_m / 180.0) <= 1e+222) {
		tmp = t_0 * (2.0 * (t_3 * Math.sin((Math.cbrt(Math.pow(Math.PI, 3.0)) * (angle_m * 0.005555555555555556)))));
	} else if ((angle_m / 180.0) <= 5e+281) {
		tmp = t_0 * (2.0 * (Math.sqrt(Math.pow(Math.cos(t_1), 2.0)) * Math.sin(t_1)));
	} else {
		tmp = t_0 * (2.0 * (t_3 * Math.sin(t_2)));
	}
	return angle_s * tmp;
}

Julia

a_m = abs(a)
b_m = abs(b)
angle\_m = abs(angle)
angle\_s = copysign(1.0, angle)
function code(angle_s, a_m, b_m, angle_m)
	t_0 = Float64(Float64(b_m + a_m) * Float64(b_m - a_m))
	t_1 = Float64(pi * Float64(angle_m * 0.005555555555555556))
	t_2 = Float64(Float64(angle_m / 180.0) * pi)
	t_3 = cos(t_2)
	tmp = 0.0
	if (Float64(angle_m / 180.0) <= 2e+86)
		tmp = Float64(cbrt(Float64(b_m + a_m)) * cbrt(Float64(Float64(b_m - a_m) * sin(Float64(pi * Float64(2.0 * Float64(angle_m * 0.005555555555555556))))))) ^ 3.0;
	elseif (Float64(angle_m / 180.0) <= 1e+222)
		tmp = Float64(t_0 * Float64(2.0 * Float64(t_3 * sin(Float64(cbrt((pi ^ 3.0)) * Float64(angle_m * 0.005555555555555556))))));
	elseif (Float64(angle_m / 180.0) <= 5e+281)
		tmp = Float64(t_0 * Float64(2.0 * Float64(sqrt((cos(t_1) ^ 2.0)) * sin(t_1))));
	else
		tmp = Float64(t_0 * Float64(2.0 * Float64(t_3 * sin(t_2))));
	end
	return Float64(angle_s * tmp)
end

Wolfram

a_m = N[Abs[a], $MachinePrecision]
b_m = N[Abs[b], $MachinePrecision]
angle\_m = N[Abs[angle], $MachinePrecision]
angle\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[angle]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[angle$95$s_, a$95$m_, b$95$m_, angle$95$m_] := Block[{t$95$0 = N[(N[(b$95$m + a$95$m), $MachinePrecision] * N[(b$95$m - a$95$m), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(Pi * N[(angle$95$m * 0.005555555555555556), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(N[(angle$95$m / 180.0), $MachinePrecision] * Pi), $MachinePrecision]}, Block[{t$95$3 = N[Cos[t$95$2], $MachinePrecision]}, N[(angle$95$s * If[LessEqual[N[(angle$95$m / 180.0), $MachinePrecision], 2e+86], N[Power[N[(N[Power[N[(b$95$m + a$95$m), $MachinePrecision], 1/3], $MachinePrecision] * N[Power[N[(N[(b$95$m - a$95$m), $MachinePrecision] * N[Sin[N[(Pi * N[(2.0 * N[(angle$95$m * 0.005555555555555556), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], 1/3], $MachinePrecision]), $MachinePrecision], 3.0], $MachinePrecision], If[LessEqual[N[(angle$95$m / 180.0), $MachinePrecision], 1e+222], N[(t$95$0 * N[(2.0 * N[(t$95$3 * N[Sin[N[(N[Power[N[Power[Pi, 3.0], $MachinePrecision], 1/3], $MachinePrecision] * N[(angle$95$m * 0.005555555555555556), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[N[(angle$95$m / 180.0), $MachinePrecision], 5e+281], N[(t$95$0 * N[(2.0 * N[(N[Sqrt[N[Power[N[Cos[t$95$1], $MachinePrecision], 2.0], $MachinePrecision]], $MachinePrecision] * N[Sin[t$95$1], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(t$95$0 * N[(2.0 * N[(t$95$3 * N[Sin[t$95$2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]), $MachinePrecision]]]]]

Derivation

1. Split input into 4 regimes

2. if (/.f64 angle #s(literal 180 binary64)) < 2e86

   1. Initial program 63.2%

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

      1. associate-*l* 63.2%

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

      2. *-commutative 63.2%

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

      3. associate-*l* 63.7%

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

   3. Simplified 63.7%

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

      1. add-cube-cbrt 63.3%

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

      2. pow3 63.3%

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

      3. 2-sin 63.3%

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

      4. associate-*r* 63.3%

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

      5. div-inv 62.9%

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

      6. metadata-eval 62.9%

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

   6. Applied egg-rr 62.9%

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

      1. unpow2 62.9%

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

      2. unpow2 62.9%

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

      3. difference-of-squares 66.8%

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

      4. metadata-eval 66.8%

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

      5. div-inv 67.6%

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

      6. associate-*l* 67.6%

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

      7. 2-sin 67.7%

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

      8. associate-*l* 77.1%

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

      9. cbrt-prod 77.1%

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

   8. Applied egg-rr 76.3%

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

   if 2e86 < (/.f64 angle #s(literal 180 binary64)) < 1e222

   1. Initial program 31.9%

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

      1. associate-*l* 31.9%

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

      2. *-commutative 31.9%

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

      3. associate-*l* 31.9%

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

   3. Simplified 31.9%

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

      1. unpow2 31.9%

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

      2. unpow2 31.9%

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

      3. difference-of-squares 42.3%

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

   6. Applied egg-rr 42.3%

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

   7. Taylor expanded in angle around inf 49.5%

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

      1. associate-*r* 41.5%

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

      2. *-commutative 41.5%

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

      3. *-commutative 41.5%

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

      4. *-commutative 41.5%

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

   9. Simplified 41.5%

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

       1. add-cbrt-cube 51.7%

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

       2. pow3 51.7%

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

   11. Applied egg-rr 51.7%

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

   if 1e222 < (/.f64 angle #s(literal 180 binary64)) < 5.00000000000000016e281

   1. Initial program 39.2%

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

      1. associate-*l* 39.2%

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

      2. *-commutative 39.2%

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

      3. associate-*l* 39.2%

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

   3. Simplified 39.2%

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

      1. unpow2 39.2%

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

      2. unpow2 39.2%

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

      3. difference-of-squares 39.2%

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

   6. Applied egg-rr 39.2%

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

   7. Taylor expanded in angle around inf 38.9%

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

      1. associate-*r* 30.1%

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

      2. *-commutative 30.1%

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

      3. *-commutative 30.1%

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

      4. *-commutative 30.1%

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

   9. Simplified 30.1%

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

       1. add-sqr-sqrt 11.1%

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

       2. sqrt-unprod 51.7%

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

       3. pow2 51.7%

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

       4. div-inv 51.7%

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

       5. metadata-eval 51.7%

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

   11. Applied egg-rr 51.7%

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

   if 5.00000000000000016e281 < (/.f64 angle #s(literal 180 binary64))

   1. Initial program 59.1%

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

      1. associate-*l* 59.1%

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

      2. *-commutative 59.1%

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

      3. associate-*l* 59.1%

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

   3. Simplified 59.1%

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

      1. unpow2 59.1%

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

      2. unpow2 59.1%

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

      3. difference-of-squares 59.1%

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

   6. Applied egg-rr 59.1%

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

4. Final simplification 72.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{angle}{180} \leq 2 \cdot 10^{+86}:\\ \;\;\;\;{\left(\sqrt[3]{b + a} \cdot \sqrt[3]{\left(b - a\right) \cdot \sin \left(\pi \cdot \left(2 \cdot \left(angle \cdot 0.005555555555555556\right)\right)\right)}\right)}^{3}\\ \mathbf{elif}\;\frac{angle}{180} \leq 10^{+222}:\\ \;\;\;\;\left(\left(b + a\right) \cdot \left(b - a\right)\right) \cdot \left(2 \cdot \left(\cos \left(\frac{angle}{180} \cdot \pi\right) \cdot \sin \left(\sqrt[3]{{\pi}^{3}} \cdot \left(angle \cdot 0.005555555555555556\right)\right)\right)\right)\\ \mathbf{elif}\;\frac{angle}{180} \leq 5 \cdot 10^{+281}:\\ \;\;\;\;\left(\left(b + a\right) \cdot \left(b - a\right)\right) \cdot \left(2 \cdot \left(\sqrt{{\cos \left(\pi \cdot \left(angle \cdot 0.005555555555555556\right)\right)}^{2}} \cdot \sin \left(\pi \cdot \left(angle \cdot 0.005555555555555556\right)\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\left(b + a\right) \cdot \left(b - a\right)\right) \cdot \left(2 \cdot \left(\cos \left(\frac{angle}{180} \cdot \pi\right) \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 67.5% accurate, 0.9× speedup

Math

\[\begin{array}{l} a_m = \left|a\right| \\ b_m = \left|b\right| \\ angle\_m = \left|angle\right| \\ angle\_s = \mathsf{copysign}\left(1, angle\right) \\ \begin{array}{l} t_0 := \pi \cdot \left(angle\_m \cdot 0.005555555555555556\right)\\ t_1 := \left(b\_m + a\_m\right) \cdot \left(b\_m - a\_m\right)\\ angle\_s \cdot \begin{array}{l} \mathbf{if}\;\frac{angle\_m}{180} \leq 5 \cdot 10^{+81}:\\ \;\;\;\;{\left(\sqrt[3]{b\_m + a\_m} \cdot \sqrt[3]{\left(b\_m - a\_m\right) \cdot \sin \left(\pi \cdot \left(2 \cdot \left(angle\_m \cdot 0.005555555555555556\right)\right)\right)}\right)}^{3}\\ \mathbf{elif}\;\frac{angle\_m}{180} \leq 2.5 \cdot 10^{+123}:\\ \;\;\;\;\left|\sin \left(\pi \cdot \left(angle\_m \cdot 0.011111111111111112\right)\right) \cdot \left({b\_m}^{2} - {a\_m}^{2}\right)\right|\\ \mathbf{elif}\;\frac{angle\_m}{180} \leq 10^{+140}:\\ \;\;\;\;t\_1 \cdot \left(2 \cdot \left(\cos \left(\frac{angle\_m}{180} \cdot \pi\right) \cdot \sin \left(0.005555555555555556 \cdot \left(angle\_m \cdot \pi\right)\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;t\_1 \cdot \left(2 \cdot \left(\sqrt{{\cos t\_0}^{2}} \cdot \sin t\_0\right)\right)\\ \end{array} \end{array} \end{array} \]

FPCore

a_m = (fabs.f64 a)
b_m = (fabs.f64 b)
angle\_m = (fabs.f64 angle)
angle\_s = (copysign.f64 #s(literal 1 binary64) angle)
(FPCore (angle_s a_m b_m angle_m)
 :precision binary64
 (let* ((t_0 (* PI (* angle_m 0.005555555555555556)))
        (t_1 (* (+ b_m a_m) (- b_m a_m))))
   (*
    angle_s
    (if (<= (/ angle_m 180.0) 5e+81)
      (pow
       (*
        (cbrt (+ b_m a_m))
        (cbrt
         (*
          (- b_m a_m)
          (sin (* PI (* 2.0 (* angle_m 0.005555555555555556)))))))
       3.0)
      (if (<= (/ angle_m 180.0) 2.5e+123)
        (fabs
         (*
          (sin (* PI (* angle_m 0.011111111111111112)))
          (- (pow b_m 2.0) (pow a_m 2.0))))
        (if (<= (/ angle_m 180.0) 1e+140)
          (*
           t_1
           (*
            2.0
            (*
             (cos (* (/ angle_m 180.0) PI))
             (sin (* 0.005555555555555556 (* angle_m PI))))))
          (* t_1 (* 2.0 (* (sqrt (pow (cos t_0) 2.0)) (sin t_0))))))))))

C

a_m = fabs(a);
b_m = fabs(b);
angle\_m = fabs(angle);
angle\_s = copysign(1.0, angle);
double code(double angle_s, double a_m, double b_m, double angle_m) {
	double t_0 = ((double) M_PI) * (angle_m * 0.005555555555555556);
	double t_1 = (b_m + a_m) * (b_m - a_m);
	double tmp;
	if ((angle_m / 180.0) <= 5e+81) {
		tmp = pow((cbrt((b_m + a_m)) * cbrt(((b_m - a_m) * sin((((double) M_PI) * (2.0 * (angle_m * 0.005555555555555556))))))), 3.0);
	} else if ((angle_m / 180.0) <= 2.5e+123) {
		tmp = fabs((sin((((double) M_PI) * (angle_m * 0.011111111111111112))) * (pow(b_m, 2.0) - pow(a_m, 2.0))));
	} else if ((angle_m / 180.0) <= 1e+140) {
		tmp = t_1 * (2.0 * (cos(((angle_m / 180.0) * ((double) M_PI))) * sin((0.005555555555555556 * (angle_m * ((double) M_PI))))));
	} else {
		tmp = t_1 * (2.0 * (sqrt(pow(cos(t_0), 2.0)) * sin(t_0)));
	}
	return angle_s * tmp;
}

Java

a_m = Math.abs(a);
b_m = Math.abs(b);
angle\_m = Math.abs(angle);
angle\_s = Math.copySign(1.0, angle);
public static double code(double angle_s, double a_m, double b_m, double angle_m) {
	double t_0 = Math.PI * (angle_m * 0.005555555555555556);
	double t_1 = (b_m + a_m) * (b_m - a_m);
	double tmp;
	if ((angle_m / 180.0) <= 5e+81) {
		tmp = Math.pow((Math.cbrt((b_m + a_m)) * Math.cbrt(((b_m - a_m) * Math.sin((Math.PI * (2.0 * (angle_m * 0.005555555555555556))))))), 3.0);
	} else if ((angle_m / 180.0) <= 2.5e+123) {
		tmp = Math.abs((Math.sin((Math.PI * (angle_m * 0.011111111111111112))) * (Math.pow(b_m, 2.0) - Math.pow(a_m, 2.0))));
	} else if ((angle_m / 180.0) <= 1e+140) {
		tmp = t_1 * (2.0 * (Math.cos(((angle_m / 180.0) * Math.PI)) * Math.sin((0.005555555555555556 * (angle_m * Math.PI)))));
	} else {
		tmp = t_1 * (2.0 * (Math.sqrt(Math.pow(Math.cos(t_0), 2.0)) * Math.sin(t_0)));
	}
	return angle_s * tmp;
}

Julia

a_m = abs(a)
b_m = abs(b)
angle\_m = abs(angle)
angle\_s = copysign(1.0, angle)
function code(angle_s, a_m, b_m, angle_m)
	t_0 = Float64(pi * Float64(angle_m * 0.005555555555555556))
	t_1 = Float64(Float64(b_m + a_m) * Float64(b_m - a_m))
	tmp = 0.0
	if (Float64(angle_m / 180.0) <= 5e+81)
		tmp = Float64(cbrt(Float64(b_m + a_m)) * cbrt(Float64(Float64(b_m - a_m) * sin(Float64(pi * Float64(2.0 * Float64(angle_m * 0.005555555555555556))))))) ^ 3.0;
	elseif (Float64(angle_m / 180.0) <= 2.5e+123)
		tmp = abs(Float64(sin(Float64(pi * Float64(angle_m * 0.011111111111111112))) * Float64((b_m ^ 2.0) - (a_m ^ 2.0))));
	elseif (Float64(angle_m / 180.0) <= 1e+140)
		tmp = Float64(t_1 * Float64(2.0 * Float64(cos(Float64(Float64(angle_m / 180.0) * pi)) * sin(Float64(0.005555555555555556 * Float64(angle_m * pi))))));
	else
		tmp = Float64(t_1 * Float64(2.0 * Float64(sqrt((cos(t_0) ^ 2.0)) * sin(t_0))));
	end
	return Float64(angle_s * tmp)
end

Wolfram

a_m = N[Abs[a], $MachinePrecision]
b_m = N[Abs[b], $MachinePrecision]
angle\_m = N[Abs[angle], $MachinePrecision]
angle\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[angle]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[angle$95$s_, a$95$m_, b$95$m_, angle$95$m_] := Block[{t$95$0 = N[(Pi * N[(angle$95$m * 0.005555555555555556), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[(b$95$m + a$95$m), $MachinePrecision] * N[(b$95$m - a$95$m), $MachinePrecision]), $MachinePrecision]}, N[(angle$95$s * If[LessEqual[N[(angle$95$m / 180.0), $MachinePrecision], 5e+81], N[Power[N[(N[Power[N[(b$95$m + a$95$m), $MachinePrecision], 1/3], $MachinePrecision] * N[Power[N[(N[(b$95$m - a$95$m), $MachinePrecision] * N[Sin[N[(Pi * N[(2.0 * N[(angle$95$m * 0.005555555555555556), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], 1/3], $MachinePrecision]), $MachinePrecision], 3.0], $MachinePrecision], If[LessEqual[N[(angle$95$m / 180.0), $MachinePrecision], 2.5e+123], N[Abs[N[(N[Sin[N[(Pi * N[(angle$95$m * 0.011111111111111112), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * N[(N[Power[b$95$m, 2.0], $MachinePrecision] - N[Power[a$95$m, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], If[LessEqual[N[(angle$95$m / 180.0), $MachinePrecision], 1e+140], N[(t$95$1 * N[(2.0 * N[(N[Cos[N[(N[(angle$95$m / 180.0), $MachinePrecision] * Pi), $MachinePrecision]], $MachinePrecision] * N[Sin[N[(0.005555555555555556 * N[(angle$95$m * Pi), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(t$95$1 * N[(2.0 * N[(N[Sqrt[N[Power[N[Cos[t$95$0], $MachinePrecision], 2.0], $MachinePrecision]], $MachinePrecision] * N[Sin[t$95$0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]), $MachinePrecision]]]

Derivation

1. Split input into 4 regimes

2. if (/.f64 angle #s(literal 180 binary64)) < 4.9999999999999998e81

   1. Initial program 62.9%

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

      1. associate-*l* 62.9%

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

      2. *-commutative 62.9%

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

      3. associate-*l* 63.3%

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

   3. Simplified 63.3%

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

      1. add-cube-cbrt 62.9%

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

      2. pow3 62.9%

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

      3. 2-sin 62.9%

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

      4. associate-*r* 62.9%

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

      5. div-inv 62.6%

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

      6. metadata-eval 62.6%

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

   6. Applied egg-rr 62.6%

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

      1. unpow2 62.6%

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

      2. unpow2 62.6%

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

      3. difference-of-squares 66.5%

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

      4. metadata-eval 66.5%

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

      5. div-inv 67.3%

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

      6. associate-*l* 67.3%

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

      7. 2-sin 67.3%

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

      8. associate-*l* 76.9%

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

      9. cbrt-prod 76.9%

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

   8. Applied egg-rr 76.0%

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

   if 4.9999999999999998e81 < (/.f64 angle #s(literal 180 binary64)) < 2.49999999999999987e123

   1. Initial program 36.4%

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

      1. associate-*l* 36.4%

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

      2. *-commutative 36.4%

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

      3. associate-*l* 36.4%

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

   3. Simplified 36.4%

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

      1. unpow2 36.4%

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

      2. unpow2 36.4%

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

      3. difference-of-squares 36.6%

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

   6. Applied egg-rr 36.6%

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

   7. Taylor expanded in angle around inf 40.2%

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

      1. associate-*r* 47.7%

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

      2. *-commutative 47.7%

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

      3. *-commutative 47.7%

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

      4. *-commutative 47.7%

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

   9. Simplified 47.7%

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

   11. Applied egg-rr 50.2%

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

       1. unpow2 50.2%

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

       2. rem-sqrt-square 51.7%

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

   13. Simplified 51.7%

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

   if 2.49999999999999987e123 < (/.f64 angle #s(literal 180 binary64)) < 1.00000000000000006e140

   1. Initial program 60.6%

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

      1. associate-*l* 60.6%

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

      2. *-commutative 60.6%

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

      3. associate-*l* 60.6%

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

   3. Simplified 60.6%

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

      1. unpow2 60.6%

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

      2. unpow2 60.6%

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

      3. difference-of-squares 60.6%

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

   6. Applied egg-rr 60.6%

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

   7. Taylor expanded in angle around inf 80.6%

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

   if 1.00000000000000006e140 < (/.f64 angle #s(literal 180 binary64))

   1. Initial program 36.3%

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

      1. associate-*l* 36.3%

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

      2. *-commutative 36.3%

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

      3. associate-*l* 36.3%

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

   3. Simplified 36.3%

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

      1. unpow2 36.3%

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

      2. unpow2 36.3%

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

      3. difference-of-squares 45.7%

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

   6. Applied egg-rr 45.7%

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

   7. Taylor expanded in angle around inf 48.3%

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

      1. associate-*r* 38.7%

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

      2. *-commutative 38.7%

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

      3. *-commutative 38.7%

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

      4. *-commutative 38.7%

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

   9. Simplified 38.7%

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

       1. add-sqr-sqrt 17.4%

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

       2. sqrt-unprod 52.8%

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

       3. pow2 52.8%

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

       4. div-inv 52.8%

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

       5. metadata-eval 52.8%

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

   11. Applied egg-rr 52.8%

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

4. Final simplification 72.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{angle}{180} \leq 5 \cdot 10^{+81}:\\ \;\;\;\;{\left(\sqrt[3]{b + a} \cdot \sqrt[3]{\left(b - a\right) \cdot \sin \left(\pi \cdot \left(2 \cdot \left(angle \cdot 0.005555555555555556\right)\right)\right)}\right)}^{3}\\ \mathbf{elif}\;\frac{angle}{180} \leq 2.5 \cdot 10^{+123}:\\ \;\;\;\;\left|\sin \left(\pi \cdot \left(angle \cdot 0.011111111111111112\right)\right) \cdot \left({b}^{2} - {a}^{2}\right)\right|\\ \mathbf{elif}\;\frac{angle}{180} \leq 10^{+140}:\\ \;\;\;\;\left(\left(b + a\right) \cdot \left(b - a\right)\right) \cdot \left(2 \cdot \left(\cos \left(\frac{angle}{180} \cdot \pi\right) \cdot \sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\left(b + a\right) \cdot \left(b - a\right)\right) \cdot \left(2 \cdot \left(\sqrt{{\cos \left(\pi \cdot \left(angle \cdot 0.005555555555555556\right)\right)}^{2}} \cdot \sin \left(\pi \cdot \left(angle \cdot 0.005555555555555556\right)\right)\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 67.5% accurate, 1.0× speedup

Math

\[\begin{array}{l} a_m = \left|a\right| \\ b_m = \left|b\right| \\ angle\_m = \left|angle\right| \\ angle\_s = \mathsf{copysign}\left(1, angle\right) \\ \begin{array}{l} t_0 := \left(b\_m + a\_m\right) \cdot \left(b\_m - a\_m\right)\\ t_1 := \sin \left(\frac{angle\_m}{180} \cdot \sqrt[3]{{\pi}^{3}}\right)\\ t_2 := \cos \left(\frac{angle\_m}{180} \cdot \pi\right)\\ angle\_s \cdot \begin{array}{l} \mathbf{if}\;\frac{angle\_m}{180} \leq 4 \cdot 10^{+98}:\\ \;\;\;\;{\left(\sqrt[3]{b\_m + a\_m} \cdot \sqrt[3]{\left(b\_m - a\_m\right) \cdot \sin \left(\pi \cdot \left(2 \cdot \left(angle\_m \cdot 0.005555555555555556\right)\right)\right)}\right)}^{3}\\ \mathbf{elif}\;\frac{angle\_m}{180} \leq 2 \cdot 10^{+155}:\\ \;\;\;\;t\_0 \cdot \left(2 \cdot \left(t\_1 \cdot t\_2\right)\right)\\ \mathbf{elif}\;\frac{angle\_m}{180} \leq 10^{+264}:\\ \;\;\;\;t\_0 \cdot \left(2 \cdot t\_1\right)\\ \mathbf{else}:\\ \;\;\;\;t\_0 \cdot \left(2 \cdot \left(t\_2 \cdot \sin \left(0.005555555555555556 \cdot \left(angle\_m \cdot \pi\right)\right)\right)\right)\\ \end{array} \end{array} \end{array} \]

FPCore

a_m = (fabs.f64 a)
b_m = (fabs.f64 b)
angle\_m = (fabs.f64 angle)
angle\_s = (copysign.f64 #s(literal 1 binary64) angle)
(FPCore (angle_s a_m b_m angle_m)
 :precision binary64
 (let* ((t_0 (* (+ b_m a_m) (- b_m a_m)))
        (t_1 (sin (* (/ angle_m 180.0) (cbrt (pow PI 3.0)))))
        (t_2 (cos (* (/ angle_m 180.0) PI))))
   (*
    angle_s
    (if (<= (/ angle_m 180.0) 4e+98)
      (pow
       (*
        (cbrt (+ b_m a_m))
        (cbrt
         (*
          (- b_m a_m)
          (sin (* PI (* 2.0 (* angle_m 0.005555555555555556)))))))
       3.0)
      (if (<= (/ angle_m 180.0) 2e+155)
        (* t_0 (* 2.0 (* t_1 t_2)))
        (if (<= (/ angle_m 180.0) 1e+264)
          (* t_0 (* 2.0 t_1))
          (*
           t_0
           (* 2.0 (* t_2 (sin (* 0.005555555555555556 (* angle_m PI))))))))))))

C

a_m = fabs(a);
b_m = fabs(b);
angle\_m = fabs(angle);
angle\_s = copysign(1.0, angle);
double code(double angle_s, double a_m, double b_m, double angle_m) {
	double t_0 = (b_m + a_m) * (b_m - a_m);
	double t_1 = sin(((angle_m / 180.0) * cbrt(pow(((double) M_PI), 3.0))));
	double t_2 = cos(((angle_m / 180.0) * ((double) M_PI)));
	double tmp;
	if ((angle_m / 180.0) <= 4e+98) {
		tmp = pow((cbrt((b_m + a_m)) * cbrt(((b_m - a_m) * sin((((double) M_PI) * (2.0 * (angle_m * 0.005555555555555556))))))), 3.0);
	} else if ((angle_m / 180.0) <= 2e+155) {
		tmp = t_0 * (2.0 * (t_1 * t_2));
	} else if ((angle_m / 180.0) <= 1e+264) {
		tmp = t_0 * (2.0 * t_1);
	} else {
		tmp = t_0 * (2.0 * (t_2 * sin((0.005555555555555556 * (angle_m * ((double) M_PI))))));
	}
	return angle_s * tmp;
}

Java

a_m = Math.abs(a);
b_m = Math.abs(b);
angle\_m = Math.abs(angle);
angle\_s = Math.copySign(1.0, angle);
public static double code(double angle_s, double a_m, double b_m, double angle_m) {
	double t_0 = (b_m + a_m) * (b_m - a_m);
	double t_1 = Math.sin(((angle_m / 180.0) * Math.cbrt(Math.pow(Math.PI, 3.0))));
	double t_2 = Math.cos(((angle_m / 180.0) * Math.PI));
	double tmp;
	if ((angle_m / 180.0) <= 4e+98) {
		tmp = Math.pow((Math.cbrt((b_m + a_m)) * Math.cbrt(((b_m - a_m) * Math.sin((Math.PI * (2.0 * (angle_m * 0.005555555555555556))))))), 3.0);
	} else if ((angle_m / 180.0) <= 2e+155) {
		tmp = t_0 * (2.0 * (t_1 * t_2));
	} else if ((angle_m / 180.0) <= 1e+264) {
		tmp = t_0 * (2.0 * t_1);
	} else {
		tmp = t_0 * (2.0 * (t_2 * Math.sin((0.005555555555555556 * (angle_m * Math.PI)))));
	}
	return angle_s * tmp;
}

Julia

a_m = abs(a)
b_m = abs(b)
angle\_m = abs(angle)
angle\_s = copysign(1.0, angle)
function code(angle_s, a_m, b_m, angle_m)
	t_0 = Float64(Float64(b_m + a_m) * Float64(b_m - a_m))
	t_1 = sin(Float64(Float64(angle_m / 180.0) * cbrt((pi ^ 3.0))))
	t_2 = cos(Float64(Float64(angle_m / 180.0) * pi))
	tmp = 0.0
	if (Float64(angle_m / 180.0) <= 4e+98)
		tmp = Float64(cbrt(Float64(b_m + a_m)) * cbrt(Float64(Float64(b_m - a_m) * sin(Float64(pi * Float64(2.0 * Float64(angle_m * 0.005555555555555556))))))) ^ 3.0;
	elseif (Float64(angle_m / 180.0) <= 2e+155)
		tmp = Float64(t_0 * Float64(2.0 * Float64(t_1 * t_2)));
	elseif (Float64(angle_m / 180.0) <= 1e+264)
		tmp = Float64(t_0 * Float64(2.0 * t_1));
	else
		tmp = Float64(t_0 * Float64(2.0 * Float64(t_2 * sin(Float64(0.005555555555555556 * Float64(angle_m * pi))))));
	end
	return Float64(angle_s * tmp)
end

Wolfram

a_m = N[Abs[a], $MachinePrecision]
b_m = N[Abs[b], $MachinePrecision]
angle\_m = N[Abs[angle], $MachinePrecision]
angle\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[angle]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[angle$95$s_, a$95$m_, b$95$m_, angle$95$m_] := Block[{t$95$0 = N[(N[(b$95$m + a$95$m), $MachinePrecision] * N[(b$95$m - a$95$m), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[Sin[N[(N[(angle$95$m / 180.0), $MachinePrecision] * N[Power[N[Power[Pi, 3.0], $MachinePrecision], 1/3], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$2 = N[Cos[N[(N[(angle$95$m / 180.0), $MachinePrecision] * Pi), $MachinePrecision]], $MachinePrecision]}, N[(angle$95$s * If[LessEqual[N[(angle$95$m / 180.0), $MachinePrecision], 4e+98], N[Power[N[(N[Power[N[(b$95$m + a$95$m), $MachinePrecision], 1/3], $MachinePrecision] * N[Power[N[(N[(b$95$m - a$95$m), $MachinePrecision] * N[Sin[N[(Pi * N[(2.0 * N[(angle$95$m * 0.005555555555555556), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], 1/3], $MachinePrecision]), $MachinePrecision], 3.0], $MachinePrecision], If[LessEqual[N[(angle$95$m / 180.0), $MachinePrecision], 2e+155], N[(t$95$0 * N[(2.0 * N[(t$95$1 * t$95$2), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[N[(angle$95$m / 180.0), $MachinePrecision], 1e+264], N[(t$95$0 * N[(2.0 * t$95$1), $MachinePrecision]), $MachinePrecision], N[(t$95$0 * N[(2.0 * N[(t$95$2 * N[Sin[N[(0.005555555555555556 * N[(angle$95$m * Pi), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]), $MachinePrecision]]]]

Derivation

1. Split input into 4 regimes

2. if (/.f64 angle #s(literal 180 binary64)) < 3.99999999999999999e98

   1. Initial program 62.4%

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

      1. associate-*l* 62.4%

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

      2. *-commutative 62.4%

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

      3. associate-*l* 62.9%

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

   3. Simplified 62.9%

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

      1. add-cube-cbrt 62.5%

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

      2. pow3 62.5%

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

      3. 2-sin 62.5%

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

      4. associate-*r* 62.5%

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

      5. div-inv 62.1%

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

      6. metadata-eval 62.1%

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

   6. Applied egg-rr 62.1%

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

      1. unpow2 62.1%

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

      2. unpow2 62.1%

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

      3. difference-of-squares 66.0%

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

      4. metadata-eval 66.0%

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

      5. div-inv 66.8%

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

      6. associate-*l* 66.8%

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

      7. 2-sin 66.8%

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

      8. associate-*l* 76.1%

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

      9. cbrt-prod 76.1%

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

   8. Applied egg-rr 75.3%

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

   if 3.99999999999999999e98 < (/.f64 angle #s(literal 180 binary64)) < 2.00000000000000001e155

   1. Initial program 29.7%

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

      1. associate-*l* 29.7%

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

      2. *-commutative 29.7%

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

      3. associate-*l* 29.7%

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

   3. Simplified 29.7%

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

      1. unpow2 29.7%

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

      2. unpow2 29.7%

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

      3. difference-of-squares 29.7%

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

   6. Applied egg-rr 29.7%

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

      1. add-cbrt-cube 60.5%

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

      2. pow3 60.5%

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

   8. Applied egg-rr 60.5%

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

   if 2.00000000000000001e155 < (/.f64 angle #s(literal 180 binary64)) < 1.00000000000000004e264

   1. Initial program 44.9%

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

      1. associate-*l* 44.9%

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

      2. *-commutative 44.9%

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

      3. associate-*l* 44.9%

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

   3. Simplified 44.9%

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

      1. unpow2 44.9%

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

      2. unpow2 44.9%

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

      3. difference-of-squares 59.9%

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

   6. Applied egg-rr 59.9%

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

      1. add-cbrt-cube 47.8%

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

      2. pow3 47.8%

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

   8. Applied egg-rr 47.8%

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

   9. Taylor expanded in angle around 0 54.1%

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

   if 1.00000000000000004e264 < (/.f64 angle #s(literal 180 binary64))

   1. Initial program 36.9%

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

      1. associate-*l* 36.9%

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

      2. *-commutative 36.9%

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

      3. associate-*l* 36.9%

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

   3. Simplified 36.9%

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

      1. unpow2 36.9%

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

      2. unpow2 36.9%

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

      3. difference-of-squares 36.9%

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

   6. Applied egg-rr 36.9%

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

   7. Taylor expanded in angle around inf 50.6%

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

4. Final simplification 72.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{angle}{180} \leq 4 \cdot 10^{+98}:\\ \;\;\;\;{\left(\sqrt[3]{b + a} \cdot \sqrt[3]{\left(b - a\right) \cdot \sin \left(\pi \cdot \left(2 \cdot \left(angle \cdot 0.005555555555555556\right)\right)\right)}\right)}^{3}\\ \mathbf{elif}\;\frac{angle}{180} \leq 2 \cdot 10^{+155}:\\ \;\;\;\;\left(\left(b + a\right) \cdot \left(b - a\right)\right) \cdot \left(2 \cdot \left(\sin \left(\frac{angle}{180} \cdot \sqrt[3]{{\pi}^{3}}\right) \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)\right)\right)\\ \mathbf{elif}\;\frac{angle}{180} \leq 10^{+264}:\\ \;\;\;\;\left(\left(b + a\right) \cdot \left(b - a\right)\right) \cdot \left(2 \cdot \sin \left(\frac{angle}{180} \cdot \sqrt[3]{{\pi}^{3}}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\left(b + a\right) \cdot \left(b - a\right)\right) \cdot \left(2 \cdot \left(\cos \left(\frac{angle}{180} \cdot \pi\right) \cdot \sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 67.0% accurate, 1.0× speedup

Math

\[\begin{array}{l} a_m = \left|a\right| \\ b_m = \left|b\right| \\ angle\_m = \left|angle\right| \\ angle\_s = \mathsf{copysign}\left(1, angle\right) \\ \begin{array}{l} t_0 := \frac{angle\_m}{180} \cdot \pi\\ t_1 := \cos t\_0\\ t_2 := \pi \cdot \left(angle\_m \cdot 0.011111111111111112\right)\\ t_3 := \left(b\_m + a\_m\right) \cdot \left(b\_m - a\_m\right)\\ angle\_s \cdot \begin{array}{l} \mathbf{if}\;\frac{angle\_m}{180} \leq 4 \cdot 10^{-56}:\\ \;\;\;\;{\left(\sqrt[3]{b\_m + a\_m} \cdot \sqrt[3]{\left(b\_m - a\_m\right) \cdot t\_2}\right)}^{3}\\ \mathbf{elif}\;\frac{angle\_m}{180} \leq 2 \cdot 10^{+86}:\\ \;\;\;\;t\_3 \cdot \left(2 \cdot \left(t\_1 \cdot \sin t\_0\right)\right)\\ \mathbf{elif}\;\frac{angle\_m}{180} \leq 2.5 \cdot 10^{+123}:\\ \;\;\;\;\left|\sin t\_2 \cdot \left({b\_m}^{2} - {a\_m}^{2}\right)\right|\\ \mathbf{elif}\;\frac{angle\_m}{180} \leq 5 \cdot 10^{+140}:\\ \;\;\;\;t\_3 \cdot \left(2 \cdot \left(t\_1 \cdot \sin \left(0.005555555555555556 \cdot \left(angle\_m \cdot \pi\right)\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;t\_3 \cdot \left(2 \cdot \sin \left(\pi \cdot \left(angle\_m \cdot 0.005555555555555556\right)\right)\right)\\ \end{array} \end{array} \end{array} \]

FPCore

a_m = (fabs.f64 a)
b_m = (fabs.f64 b)
angle\_m = (fabs.f64 angle)
angle\_s = (copysign.f64 #s(literal 1 binary64) angle)
(FPCore (angle_s a_m b_m angle_m)
 :precision binary64
 (let* ((t_0 (* (/ angle_m 180.0) PI))
        (t_1 (cos t_0))
        (t_2 (* PI (* angle_m 0.011111111111111112)))
        (t_3 (* (+ b_m a_m) (- b_m a_m))))
   (*
    angle_s
    (if (<= (/ angle_m 180.0) 4e-56)
      (pow (* (cbrt (+ b_m a_m)) (cbrt (* (- b_m a_m) t_2))) 3.0)
      (if (<= (/ angle_m 180.0) 2e+86)
        (* t_3 (* 2.0 (* t_1 (sin t_0))))
        (if (<= (/ angle_m 180.0) 2.5e+123)
          (fabs (* (sin t_2) (- (pow b_m 2.0) (pow a_m 2.0))))
          (if (<= (/ angle_m 180.0) 5e+140)
            (*
             t_3
             (* 2.0 (* t_1 (sin (* 0.005555555555555556 (* angle_m PI))))))
            (*
             t_3
             (* 2.0 (sin (* PI (* angle_m 0.005555555555555556))))))))))))

C

a_m = fabs(a);
b_m = fabs(b);
angle\_m = fabs(angle);
angle\_s = copysign(1.0, angle);
double code(double angle_s, double a_m, double b_m, double angle_m) {
	double t_0 = (angle_m / 180.0) * ((double) M_PI);
	double t_1 = cos(t_0);
	double t_2 = ((double) M_PI) * (angle_m * 0.011111111111111112);
	double t_3 = (b_m + a_m) * (b_m - a_m);
	double tmp;
	if ((angle_m / 180.0) <= 4e-56) {
		tmp = pow((cbrt((b_m + a_m)) * cbrt(((b_m - a_m) * t_2))), 3.0);
	} else if ((angle_m / 180.0) <= 2e+86) {
		tmp = t_3 * (2.0 * (t_1 * sin(t_0)));
	} else if ((angle_m / 180.0) <= 2.5e+123) {
		tmp = fabs((sin(t_2) * (pow(b_m, 2.0) - pow(a_m, 2.0))));
	} else if ((angle_m / 180.0) <= 5e+140) {
		tmp = t_3 * (2.0 * (t_1 * sin((0.005555555555555556 * (angle_m * ((double) M_PI))))));
	} else {
		tmp = t_3 * (2.0 * sin((((double) M_PI) * (angle_m * 0.005555555555555556))));
	}
	return angle_s * tmp;
}

Java

a_m = Math.abs(a);
b_m = Math.abs(b);
angle\_m = Math.abs(angle);
angle\_s = Math.copySign(1.0, angle);
public static double code(double angle_s, double a_m, double b_m, double angle_m) {
	double t_0 = (angle_m / 180.0) * Math.PI;
	double t_1 = Math.cos(t_0);
	double t_2 = Math.PI * (angle_m * 0.011111111111111112);
	double t_3 = (b_m + a_m) * (b_m - a_m);
	double tmp;
	if ((angle_m / 180.0) <= 4e-56) {
		tmp = Math.pow((Math.cbrt((b_m + a_m)) * Math.cbrt(((b_m - a_m) * t_2))), 3.0);
	} else if ((angle_m / 180.0) <= 2e+86) {
		tmp = t_3 * (2.0 * (t_1 * Math.sin(t_0)));
	} else if ((angle_m / 180.0) <= 2.5e+123) {
		tmp = Math.abs((Math.sin(t_2) * (Math.pow(b_m, 2.0) - Math.pow(a_m, 2.0))));
	} else if ((angle_m / 180.0) <= 5e+140) {
		tmp = t_3 * (2.0 * (t_1 * Math.sin((0.005555555555555556 * (angle_m * Math.PI)))));
	} else {
		tmp = t_3 * (2.0 * Math.sin((Math.PI * (angle_m * 0.005555555555555556))));
	}
	return angle_s * tmp;
}

Julia

a_m = abs(a)
b_m = abs(b)
angle\_m = abs(angle)
angle\_s = copysign(1.0, angle)
function code(angle_s, a_m, b_m, angle_m)
	t_0 = Float64(Float64(angle_m / 180.0) * pi)
	t_1 = cos(t_0)
	t_2 = Float64(pi * Float64(angle_m * 0.011111111111111112))
	t_3 = Float64(Float64(b_m + a_m) * Float64(b_m - a_m))
	tmp = 0.0
	if (Float64(angle_m / 180.0) <= 4e-56)
		tmp = Float64(cbrt(Float64(b_m + a_m)) * cbrt(Float64(Float64(b_m - a_m) * t_2))) ^ 3.0;
	elseif (Float64(angle_m / 180.0) <= 2e+86)
		tmp = Float64(t_3 * Float64(2.0 * Float64(t_1 * sin(t_0))));
	elseif (Float64(angle_m / 180.0) <= 2.5e+123)
		tmp = abs(Float64(sin(t_2) * Float64((b_m ^ 2.0) - (a_m ^ 2.0))));
	elseif (Float64(angle_m / 180.0) <= 5e+140)
		tmp = Float64(t_3 * Float64(2.0 * Float64(t_1 * sin(Float64(0.005555555555555556 * Float64(angle_m * pi))))));
	else
		tmp = Float64(t_3 * Float64(2.0 * sin(Float64(pi * Float64(angle_m * 0.005555555555555556)))));
	end
	return Float64(angle_s * tmp)
end

Wolfram

a_m = N[Abs[a], $MachinePrecision]
b_m = N[Abs[b], $MachinePrecision]
angle\_m = N[Abs[angle], $MachinePrecision]
angle\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[angle]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[angle$95$s_, a$95$m_, b$95$m_, angle$95$m_] := Block[{t$95$0 = N[(N[(angle$95$m / 180.0), $MachinePrecision] * Pi), $MachinePrecision]}, Block[{t$95$1 = N[Cos[t$95$0], $MachinePrecision]}, Block[{t$95$2 = N[(Pi * N[(angle$95$m * 0.011111111111111112), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$3 = N[(N[(b$95$m + a$95$m), $MachinePrecision] * N[(b$95$m - a$95$m), $MachinePrecision]), $MachinePrecision]}, N[(angle$95$s * If[LessEqual[N[(angle$95$m / 180.0), $MachinePrecision], 4e-56], N[Power[N[(N[Power[N[(b$95$m + a$95$m), $MachinePrecision], 1/3], $MachinePrecision] * N[Power[N[(N[(b$95$m - a$95$m), $MachinePrecision] * t$95$2), $MachinePrecision], 1/3], $MachinePrecision]), $MachinePrecision], 3.0], $MachinePrecision], If[LessEqual[N[(angle$95$m / 180.0), $MachinePrecision], 2e+86], N[(t$95$3 * N[(2.0 * N[(t$95$1 * N[Sin[t$95$0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[N[(angle$95$m / 180.0), $MachinePrecision], 2.5e+123], N[Abs[N[(N[Sin[t$95$2], $MachinePrecision] * N[(N[Power[b$95$m, 2.0], $MachinePrecision] - N[Power[a$95$m, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], If[LessEqual[N[(angle$95$m / 180.0), $MachinePrecision], 5e+140], N[(t$95$3 * N[(2.0 * N[(t$95$1 * N[Sin[N[(0.005555555555555556 * N[(angle$95$m * Pi), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(t$95$3 * N[(2.0 * N[Sin[N[(Pi * N[(angle$95$m * 0.005555555555555556), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]), $MachinePrecision]]]]]

Derivation

1. Split input into 5 regimes

2. if (/.f64 angle #s(literal 180 binary64)) < 4.0000000000000002e-56

   1. Initial program 62.1%

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

      1. associate-*l* 62.1%

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

      2. *-commutative 62.1%

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

      3. associate-*l* 62.6%

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

   3. Simplified 62.6%

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

      1. add-cube-cbrt 62.2%

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

      2. pow3 62.2%

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

      3. 2-sin 62.2%

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

      4. associate-*r* 62.2%

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

      5. div-inv 61.6%

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

      6. metadata-eval 61.6%

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

   6. Applied egg-rr 61.6%

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

      1. unpow2 61.6%

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

      2. unpow2 61.6%

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

      3. difference-of-squares 66.0%

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

      4. metadata-eval 66.0%

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

      5. div-inv 67.2%

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

      6. associate-*l* 67.2%

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

      7. 2-sin 67.2%

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

      8. associate-*l* 77.8%

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

      9. cbrt-prod 77.8%

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

   8. Applied egg-rr 76.6%

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

   9. Taylor expanded in angle around 0 75.9%

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

       1. associate-*r* 75.9%

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

       2. *-commutative 75.9%

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

   11. Simplified 75.9%

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

   if 4.0000000000000002e-56 < (/.f64 angle #s(literal 180 binary64)) < 2e86

   1. Initial program 71.9%

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

      1. associate-*l* 71.9%

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

      2. *-commutative 71.9%

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

      3. associate-*l* 71.9%

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

   3. Simplified 71.9%

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

      1. unpow2 71.9%

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

      2. unpow2 71.9%

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

      3. difference-of-squares 71.9%

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

   6. Applied egg-rr 71.9%

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

   if 2e86 < (/.f64 angle #s(literal 180 binary64)) < 2.49999999999999987e123

   1. Initial program 18.2%

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

      1. associate-*l* 18.2%

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

      2. *-commutative 18.2%

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

      3. associate-*l* 18.2%

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

   3. Simplified 18.2%

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

      1. unpow2 18.2%

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

      2. unpow2 18.2%

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

      3. difference-of-squares 18.5%

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

   6. Applied egg-rr 18.5%

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

   7. Taylor expanded in angle around inf 23.2%

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

      1. associate-*r* 32.8%

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

      2. *-commutative 32.8%

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

      3. *-commutative 32.8%

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

      4. *-commutative 32.8%

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

   9. Simplified 32.8%

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

   11. Applied egg-rr 36.0%

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

       1. unpow2 36.0%

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

       2. rem-sqrt-square 37.9%

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

   13. Simplified 37.9%

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

   if 2.49999999999999987e123 < (/.f64 angle #s(literal 180 binary64)) < 5.00000000000000008e140

   1. Initial program 60.6%

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

      1. associate-*l* 60.6%

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

      2. *-commutative 60.6%

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

      3. associate-*l* 60.6%

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

   3. Simplified 60.6%

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

      1. unpow2 60.6%

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

      2. unpow2 60.6%

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

      3. difference-of-squares 60.6%

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

   6. Applied egg-rr 60.6%

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

   7. Taylor expanded in angle around inf 80.6%

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

   if 5.00000000000000008e140 < (/.f64 angle #s(literal 180 binary64))

   1. Initial program 36.3%

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

      1. associate-*l* 36.3%

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

      2. *-commutative 36.3%

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

      3. associate-*l* 36.3%

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

   3. Simplified 36.3%

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

      1. unpow2 36.3%

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

      2. unpow2 36.3%

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

      3. difference-of-squares 45.7%

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

   6. Applied egg-rr 45.7%

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

   7. Taylor expanded in angle around inf 48.3%

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

      1. associate-*r* 38.7%

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

      2. *-commutative 38.7%

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

      3. *-commutative 38.7%

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

      4. *-commutative 38.7%

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

   9. Simplified 38.7%

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

   10. Taylor expanded in angle around 0 52.4%

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

4. Final simplification 71.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{angle}{180} \leq 4 \cdot 10^{-56}:\\ \;\;\;\;{\left(\sqrt[3]{b + a} \cdot \sqrt[3]{\left(b - a\right) \cdot \left(\pi \cdot \left(angle \cdot 0.011111111111111112\right)\right)}\right)}^{3}\\ \mathbf{elif}\;\frac{angle}{180} \leq 2 \cdot 10^{+86}:\\ \;\;\;\;\left(\left(b + a\right) \cdot \left(b - a\right)\right) \cdot \left(2 \cdot \left(\cos \left(\frac{angle}{180} \cdot \pi\right) \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right)\right)\\ \mathbf{elif}\;\frac{angle}{180} \leq 2.5 \cdot 10^{+123}:\\ \;\;\;\;\left|\sin \left(\pi \cdot \left(angle \cdot 0.011111111111111112\right)\right) \cdot \left({b}^{2} - {a}^{2}\right)\right|\\ \mathbf{elif}\;\frac{angle}{180} \leq 5 \cdot 10^{+140}:\\ \;\;\;\;\left(\left(b + a\right) \cdot \left(b - a\right)\right) \cdot \left(2 \cdot \left(\cos \left(\frac{angle}{180} \cdot \pi\right) \cdot \sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\left(b + a\right) \cdot \left(b - a\right)\right) \cdot \left(2 \cdot \sin \left(\pi \cdot \left(angle \cdot 0.005555555555555556\right)\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 67.4% accurate, 1.0× speedup

Math

\[\begin{array}{l} a_m = \left|a\right| \\ b_m = \left|b\right| \\ angle\_m = \left|angle\right| \\ angle\_s = \mathsf{copysign}\left(1, angle\right) \\ \begin{array}{l} t_0 := \left(b\_m + a\_m\right) \cdot \left(b\_m - a\_m\right)\\ angle\_s \cdot \begin{array}{l} \mathbf{if}\;\frac{angle\_m}{180} \leq 5 \cdot 10^{+81}:\\ \;\;\;\;{\left(\sqrt[3]{b\_m + a\_m} \cdot \sqrt[3]{\left(b\_m - a\_m\right) \cdot \sin \left(\pi \cdot \left(2 \cdot \left(angle\_m \cdot 0.005555555555555556\right)\right)\right)}\right)}^{3}\\ \mathbf{elif}\;\frac{angle\_m}{180} \leq 2.5 \cdot 10^{+123}:\\ \;\;\;\;\left|\sin \left(\pi \cdot \left(angle\_m \cdot 0.011111111111111112\right)\right) \cdot \left({b\_m}^{2} - {a\_m}^{2}\right)\right|\\ \mathbf{elif}\;\frac{angle\_m}{180} \leq 5 \cdot 10^{+140}:\\ \;\;\;\;t\_0 \cdot \left(2 \cdot \left(\cos \left(\frac{angle\_m}{180} \cdot \pi\right) \cdot \sin \left(0.005555555555555556 \cdot \left(angle\_m \cdot \pi\right)\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;t\_0 \cdot \left(2 \cdot \sin \left(\pi \cdot \left(angle\_m \cdot 0.005555555555555556\right)\right)\right)\\ \end{array} \end{array} \end{array} \]

FPCore

a_m = (fabs.f64 a)
b_m = (fabs.f64 b)
angle\_m = (fabs.f64 angle)
angle\_s = (copysign.f64 #s(literal 1 binary64) angle)
(FPCore (angle_s a_m b_m angle_m)
 :precision binary64
 (let* ((t_0 (* (+ b_m a_m) (- b_m a_m))))
   (*
    angle_s
    (if (<= (/ angle_m 180.0) 5e+81)
      (pow
       (*
        (cbrt (+ b_m a_m))
        (cbrt
         (*
          (- b_m a_m)
          (sin (* PI (* 2.0 (* angle_m 0.005555555555555556)))))))
       3.0)
      (if (<= (/ angle_m 180.0) 2.5e+123)
        (fabs
         (*
          (sin (* PI (* angle_m 0.011111111111111112)))
          (- (pow b_m 2.0) (pow a_m 2.0))))
        (if (<= (/ angle_m 180.0) 5e+140)
          (*
           t_0
           (*
            2.0
            (*
             (cos (* (/ angle_m 180.0) PI))
             (sin (* 0.005555555555555556 (* angle_m PI))))))
          (* t_0 (* 2.0 (sin (* PI (* angle_m 0.005555555555555556)))))))))))

C

a_m = fabs(a);
b_m = fabs(b);
angle\_m = fabs(angle);
angle\_s = copysign(1.0, angle);
double code(double angle_s, double a_m, double b_m, double angle_m) {
	double t_0 = (b_m + a_m) * (b_m - a_m);
	double tmp;
	if ((angle_m / 180.0) <= 5e+81) {
		tmp = pow((cbrt((b_m + a_m)) * cbrt(((b_m - a_m) * sin((((double) M_PI) * (2.0 * (angle_m * 0.005555555555555556))))))), 3.0);
	} else if ((angle_m / 180.0) <= 2.5e+123) {
		tmp = fabs((sin((((double) M_PI) * (angle_m * 0.011111111111111112))) * (pow(b_m, 2.0) - pow(a_m, 2.0))));
	} else if ((angle_m / 180.0) <= 5e+140) {
		tmp = t_0 * (2.0 * (cos(((angle_m / 180.0) * ((double) M_PI))) * sin((0.005555555555555556 * (angle_m * ((double) M_PI))))));
	} else {
		tmp = t_0 * (2.0 * sin((((double) M_PI) * (angle_m * 0.005555555555555556))));
	}
	return angle_s * tmp;
}

Java

a_m = Math.abs(a);
b_m = Math.abs(b);
angle\_m = Math.abs(angle);
angle\_s = Math.copySign(1.0, angle);
public static double code(double angle_s, double a_m, double b_m, double angle_m) {
	double t_0 = (b_m + a_m) * (b_m - a_m);
	double tmp;
	if ((angle_m / 180.0) <= 5e+81) {
		tmp = Math.pow((Math.cbrt((b_m + a_m)) * Math.cbrt(((b_m - a_m) * Math.sin((Math.PI * (2.0 * (angle_m * 0.005555555555555556))))))), 3.0);
	} else if ((angle_m / 180.0) <= 2.5e+123) {
		tmp = Math.abs((Math.sin((Math.PI * (angle_m * 0.011111111111111112))) * (Math.pow(b_m, 2.0) - Math.pow(a_m, 2.0))));
	} else if ((angle_m / 180.0) <= 5e+140) {
		tmp = t_0 * (2.0 * (Math.cos(((angle_m / 180.0) * Math.PI)) * Math.sin((0.005555555555555556 * (angle_m * Math.PI)))));
	} else {
		tmp = t_0 * (2.0 * Math.sin((Math.PI * (angle_m * 0.005555555555555556))));
	}
	return angle_s * tmp;
}

Julia

a_m = abs(a)
b_m = abs(b)
angle\_m = abs(angle)
angle\_s = copysign(1.0, angle)
function code(angle_s, a_m, b_m, angle_m)
	t_0 = Float64(Float64(b_m + a_m) * Float64(b_m - a_m))
	tmp = 0.0
	if (Float64(angle_m / 180.0) <= 5e+81)
		tmp = Float64(cbrt(Float64(b_m + a_m)) * cbrt(Float64(Float64(b_m - a_m) * sin(Float64(pi * Float64(2.0 * Float64(angle_m * 0.005555555555555556))))))) ^ 3.0;
	elseif (Float64(angle_m / 180.0) <= 2.5e+123)
		tmp = abs(Float64(sin(Float64(pi * Float64(angle_m * 0.011111111111111112))) * Float64((b_m ^ 2.0) - (a_m ^ 2.0))));
	elseif (Float64(angle_m / 180.0) <= 5e+140)
		tmp = Float64(t_0 * Float64(2.0 * Float64(cos(Float64(Float64(angle_m / 180.0) * pi)) * sin(Float64(0.005555555555555556 * Float64(angle_m * pi))))));
	else
		tmp = Float64(t_0 * Float64(2.0 * sin(Float64(pi * Float64(angle_m * 0.005555555555555556)))));
	end
	return Float64(angle_s * tmp)
end

Wolfram

a_m = N[Abs[a], $MachinePrecision]
b_m = N[Abs[b], $MachinePrecision]
angle\_m = N[Abs[angle], $MachinePrecision]
angle\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[angle]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[angle$95$s_, a$95$m_, b$95$m_, angle$95$m_] := Block[{t$95$0 = N[(N[(b$95$m + a$95$m), $MachinePrecision] * N[(b$95$m - a$95$m), $MachinePrecision]), $MachinePrecision]}, N[(angle$95$s * If[LessEqual[N[(angle$95$m / 180.0), $MachinePrecision], 5e+81], N[Power[N[(N[Power[N[(b$95$m + a$95$m), $MachinePrecision], 1/3], $MachinePrecision] * N[Power[N[(N[(b$95$m - a$95$m), $MachinePrecision] * N[Sin[N[(Pi * N[(2.0 * N[(angle$95$m * 0.005555555555555556), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], 1/3], $MachinePrecision]), $MachinePrecision], 3.0], $MachinePrecision], If[LessEqual[N[(angle$95$m / 180.0), $MachinePrecision], 2.5e+123], N[Abs[N[(N[Sin[N[(Pi * N[(angle$95$m * 0.011111111111111112), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * N[(N[Power[b$95$m, 2.0], $MachinePrecision] - N[Power[a$95$m, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], If[LessEqual[N[(angle$95$m / 180.0), $MachinePrecision], 5e+140], N[(t$95$0 * N[(2.0 * N[(N[Cos[N[(N[(angle$95$m / 180.0), $MachinePrecision] * Pi), $MachinePrecision]], $MachinePrecision] * N[Sin[N[(0.005555555555555556 * N[(angle$95$m * Pi), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(t$95$0 * N[(2.0 * N[Sin[N[(Pi * N[(angle$95$m * 0.005555555555555556), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]), $MachinePrecision]]

Derivation

1. Split input into 4 regimes

2. if (/.f64 angle #s(literal 180 binary64)) < 4.9999999999999998e81

   1. Initial program 62.9%

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

      1. associate-*l* 62.9%

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

      2. *-commutative 62.9%

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

      3. associate-*l* 63.3%

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

   3. Simplified 63.3%

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

      1. add-cube-cbrt 62.9%

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

      2. pow3 62.9%

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

      3. 2-sin 62.9%

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

      4. associate-*r* 62.9%

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

      5. div-inv 62.6%

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

      6. metadata-eval 62.6%

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

   6. Applied egg-rr 62.6%

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

      1. unpow2 62.6%

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

      2. unpow2 62.6%

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

      3. difference-of-squares 66.5%

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

      4. metadata-eval 66.5%

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

      5. div-inv 67.3%

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

      6. associate-*l* 67.3%

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

      7. 2-sin 67.3%

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

      8. associate-*l* 76.9%

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

      9. cbrt-prod 76.9%

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

   8. Applied egg-rr 76.0%

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

   if 4.9999999999999998e81 < (/.f64 angle #s(literal 180 binary64)) < 2.49999999999999987e123

   1. Initial program 36.4%

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

      1. associate-*l* 36.4%

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

      2. *-commutative 36.4%

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

      3. associate-*l* 36.4%

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

   3. Simplified 36.4%

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

      1. unpow2 36.4%

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

      2. unpow2 36.4%

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

      3. difference-of-squares 36.6%

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

   6. Applied egg-rr 36.6%

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

   7. Taylor expanded in angle around inf 40.2%

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

      1. associate-*r* 47.7%

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

      2. *-commutative 47.7%

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

      3. *-commutative 47.7%

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

      4. *-commutative 47.7%

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

   9. Simplified 47.7%

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

   11. Applied egg-rr 50.2%

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

       1. unpow2 50.2%

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

       2. rem-sqrt-square 51.7%

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

   13. Simplified 51.7%

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

   if 2.49999999999999987e123 < (/.f64 angle #s(literal 180 binary64)) < 5.00000000000000008e140

   1. Initial program 60.6%

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

      1. associate-*l* 60.6%

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

      2. *-commutative 60.6%

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

      3. associate-*l* 60.6%

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

   3. Simplified 60.6%

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

      1. unpow2 60.6%

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

      2. unpow2 60.6%

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

      3. difference-of-squares 60.6%

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

   6. Applied egg-rr 60.6%

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

   7. Taylor expanded in angle around inf 80.6%

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

   if 5.00000000000000008e140 < (/.f64 angle #s(literal 180 binary64))

   1. Initial program 36.3%

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

      1. associate-*l* 36.3%

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

      2. *-commutative 36.3%

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

      3. associate-*l* 36.3%

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

   3. Simplified 36.3%

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

      1. unpow2 36.3%

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

      2. unpow2 36.3%

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

      3. difference-of-squares 45.7%

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

   6. Applied egg-rr 45.7%

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

   7. Taylor expanded in angle around inf 48.3%

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

      1. associate-*r* 38.7%

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

      2. *-commutative 38.7%

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

      3. *-commutative 38.7%

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

      4. *-commutative 38.7%

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

   9. Simplified 38.7%

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

   10. Taylor expanded in angle around 0 52.4%

       \[\leadsto \left(\left(b + a\right) \cdot \left(b - a\right)\right) \cdot \left(2 \cdot \left(\sin \left(\pi \cdot \left(0.005555555555555556 \cdot angle\right)\right) \cdot \color{blue}{1}\right)\right) \]
3. Recombined 4 regimes into one program.

4. Final simplification 72.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{angle}{180} \leq 5 \cdot 10^{+81}:\\ \;\;\;\;{\left(\sqrt[3]{b + a} \cdot \sqrt[3]{\left(b - a\right) \cdot \sin \left(\pi \cdot \left(2 \cdot \left(angle \cdot 0.005555555555555556\right)\right)\right)}\right)}^{3}\\ \mathbf{elif}\;\frac{angle}{180} \leq 2.5 \cdot 10^{+123}:\\ \;\;\;\;\left|\sin \left(\pi \cdot \left(angle \cdot 0.011111111111111112\right)\right) \cdot \left({b}^{2} - {a}^{2}\right)\right|\\ \mathbf{elif}\;\frac{angle}{180} \leq 5 \cdot 10^{+140}:\\ \;\;\;\;\left(\left(b + a\right) \cdot \left(b - a\right)\right) \cdot \left(2 \cdot \left(\cos \left(\frac{angle}{180} \cdot \pi\right) \cdot \sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\left(b + a\right) \cdot \left(b - a\right)\right) \cdot \left(2 \cdot \sin \left(\pi \cdot \left(angle \cdot 0.005555555555555556\right)\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 67.1% accurate, 1.3× speedup

Math

\[\begin{array}{l} a_m = \left|a\right| \\ b_m = \left|b\right| \\ angle\_m = \left|angle\right| \\ angle\_s = \mathsf{copysign}\left(1, angle\right) \\ \begin{array}{l} t_0 := \frac{angle\_m}{180} \cdot \pi\\ t_1 := \left(b\_m + a\_m\right) \cdot \left(b\_m - a\_m\right)\\ angle\_s \cdot \begin{array}{l} \mathbf{if}\;\frac{angle\_m}{180} \leq 4 \cdot 10^{-56}:\\ \;\;\;\;{\left(\sqrt[3]{b\_m + a\_m} \cdot \sqrt[3]{\left(b\_m - a\_m\right) \cdot \left(\pi \cdot \left(angle\_m \cdot 0.011111111111111112\right)\right)}\right)}^{3}\\ \mathbf{elif}\;\frac{angle\_m}{180} \leq 5 \cdot 10^{+140}:\\ \;\;\;\;t\_1 \cdot \left(2 \cdot \left(\cos t\_0 \cdot \sin t\_0\right)\right)\\ \mathbf{else}:\\ \;\;\;\;t\_1 \cdot \left(2 \cdot \sin \left(\pi \cdot \left(angle\_m \cdot 0.005555555555555556\right)\right)\right)\\ \end{array} \end{array} \end{array} \]

FPCore

a_m = (fabs.f64 a)
b_m = (fabs.f64 b)
angle\_m = (fabs.f64 angle)
angle\_s = (copysign.f64 #s(literal 1 binary64) angle)
(FPCore (angle_s a_m b_m angle_m)
 :precision binary64
 (let* ((t_0 (* (/ angle_m 180.0) PI)) (t_1 (* (+ b_m a_m) (- b_m a_m))))
   (*
    angle_s
    (if (<= (/ angle_m 180.0) 4e-56)
      (pow
       (*
        (cbrt (+ b_m a_m))
        (cbrt (* (- b_m a_m) (* PI (* angle_m 0.011111111111111112)))))
       3.0)
      (if (<= (/ angle_m 180.0) 5e+140)
        (* t_1 (* 2.0 (* (cos t_0) (sin t_0))))
        (* t_1 (* 2.0 (sin (* PI (* angle_m 0.005555555555555556))))))))))

C

a_m = fabs(a);
b_m = fabs(b);
angle\_m = fabs(angle);
angle\_s = copysign(1.0, angle);
double code(double angle_s, double a_m, double b_m, double angle_m) {
	double t_0 = (angle_m / 180.0) * ((double) M_PI);
	double t_1 = (b_m + a_m) * (b_m - a_m);
	double tmp;
	if ((angle_m / 180.0) <= 4e-56) {
		tmp = pow((cbrt((b_m + a_m)) * cbrt(((b_m - a_m) * (((double) M_PI) * (angle_m * 0.011111111111111112))))), 3.0);
	} else if ((angle_m / 180.0) <= 5e+140) {
		tmp = t_1 * (2.0 * (cos(t_0) * sin(t_0)));
	} else {
		tmp = t_1 * (2.0 * sin((((double) M_PI) * (angle_m * 0.005555555555555556))));
	}
	return angle_s * tmp;
}

Java

a_m = Math.abs(a);
b_m = Math.abs(b);
angle\_m = Math.abs(angle);
angle\_s = Math.copySign(1.0, angle);
public static double code(double angle_s, double a_m, double b_m, double angle_m) {
	double t_0 = (angle_m / 180.0) * Math.PI;
	double t_1 = (b_m + a_m) * (b_m - a_m);
	double tmp;
	if ((angle_m / 180.0) <= 4e-56) {
		tmp = Math.pow((Math.cbrt((b_m + a_m)) * Math.cbrt(((b_m - a_m) * (Math.PI * (angle_m * 0.011111111111111112))))), 3.0);
	} else if ((angle_m / 180.0) <= 5e+140) {
		tmp = t_1 * (2.0 * (Math.cos(t_0) * Math.sin(t_0)));
	} else {
		tmp = t_1 * (2.0 * Math.sin((Math.PI * (angle_m * 0.005555555555555556))));
	}
	return angle_s * tmp;
}

Julia

a_m = abs(a)
b_m = abs(b)
angle\_m = abs(angle)
angle\_s = copysign(1.0, angle)
function code(angle_s, a_m, b_m, angle_m)
	t_0 = Float64(Float64(angle_m / 180.0) * pi)
	t_1 = Float64(Float64(b_m + a_m) * Float64(b_m - a_m))
	tmp = 0.0
	if (Float64(angle_m / 180.0) <= 4e-56)
		tmp = Float64(cbrt(Float64(b_m + a_m)) * cbrt(Float64(Float64(b_m - a_m) * Float64(pi * Float64(angle_m * 0.011111111111111112))))) ^ 3.0;
	elseif (Float64(angle_m / 180.0) <= 5e+140)
		tmp = Float64(t_1 * Float64(2.0 * Float64(cos(t_0) * sin(t_0))));
	else
		tmp = Float64(t_1 * Float64(2.0 * sin(Float64(pi * Float64(angle_m * 0.005555555555555556)))));
	end
	return Float64(angle_s * tmp)
end

Wolfram

a_m = N[Abs[a], $MachinePrecision]
b_m = N[Abs[b], $MachinePrecision]
angle\_m = N[Abs[angle], $MachinePrecision]
angle\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[angle]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[angle$95$s_, a$95$m_, b$95$m_, angle$95$m_] := Block[{t$95$0 = N[(N[(angle$95$m / 180.0), $MachinePrecision] * Pi), $MachinePrecision]}, Block[{t$95$1 = N[(N[(b$95$m + a$95$m), $MachinePrecision] * N[(b$95$m - a$95$m), $MachinePrecision]), $MachinePrecision]}, N[(angle$95$s * If[LessEqual[N[(angle$95$m / 180.0), $MachinePrecision], 4e-56], N[Power[N[(N[Power[N[(b$95$m + a$95$m), $MachinePrecision], 1/3], $MachinePrecision] * N[Power[N[(N[(b$95$m - a$95$m), $MachinePrecision] * N[(Pi * N[(angle$95$m * 0.011111111111111112), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 1/3], $MachinePrecision]), $MachinePrecision], 3.0], $MachinePrecision], If[LessEqual[N[(angle$95$m / 180.0), $MachinePrecision], 5e+140], N[(t$95$1 * N[(2.0 * N[(N[Cos[t$95$0], $MachinePrecision] * N[Sin[t$95$0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(t$95$1 * N[(2.0 * N[Sin[N[(Pi * N[(angle$95$m * 0.005555555555555556), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if (/.f64 angle #s(literal 180 binary64)) < 4.0000000000000002e-56

   1. Initial program 62.1%

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

      1. associate-*l* 62.1%

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

      2. *-commutative 62.1%

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

      3. associate-*l* 62.6%

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

   3. Simplified 62.6%

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

      1. add-cube-cbrt 62.2%

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

      2. pow3 62.2%

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

      3. 2-sin 62.2%

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

      4. associate-*r* 62.2%

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

      5. div-inv 61.6%

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

      6. metadata-eval 61.6%

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

   6. Applied egg-rr 61.6%

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

      1. unpow2 61.6%

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

      2. unpow2 61.6%

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

      3. difference-of-squares 66.0%

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

      4. metadata-eval 66.0%

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

      5. div-inv 67.2%

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

      6. associate-*l* 67.2%

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

      7. 2-sin 67.2%

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

      8. associate-*l* 77.8%

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

      9. cbrt-prod 77.8%

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

   8. Applied egg-rr 76.6%

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

   9. Taylor expanded in angle around 0 75.9%

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

       1. associate-*r* 75.9%

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

       2. *-commutative 75.9%

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

   11. Simplified 75.9%

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

   if 4.0000000000000002e-56 < (/.f64 angle #s(literal 180 binary64)) < 5.00000000000000008e140

   1. Initial program 59.9%

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

      1. associate-*l* 59.9%

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

      2. *-commutative 59.9%

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

      3. associate-*l* 59.9%

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

   3. Simplified 59.9%

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

      1. unpow2 59.9%

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

      2. unpow2 59.9%

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

      3. difference-of-squares 59.9%

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

   6. Applied egg-rr 59.9%

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

   if 5.00000000000000008e140 < (/.f64 angle #s(literal 180 binary64))

   1. Initial program 36.3%

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

      1. associate-*l* 36.3%

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

      2. *-commutative 36.3%

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

      3. associate-*l* 36.3%

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

   3. Simplified 36.3%

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

      1. unpow2 36.3%

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

      2. unpow2 36.3%

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

      3. difference-of-squares 45.7%

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

   6. Applied egg-rr 45.7%

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

   7. Taylor expanded in angle around inf 48.3%

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

      1. associate-*r* 38.7%

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

      2. *-commutative 38.7%

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

      3. *-commutative 38.7%

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

      4. *-commutative 38.7%

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

   9. Simplified 38.7%

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

   10. Taylor expanded in angle around 0 52.4%

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

4. Final simplification 70.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{angle}{180} \leq 4 \cdot 10^{-56}:\\ \;\;\;\;{\left(\sqrt[3]{b + a} \cdot \sqrt[3]{\left(b - a\right) \cdot \left(\pi \cdot \left(angle \cdot 0.011111111111111112\right)\right)}\right)}^{3}\\ \mathbf{elif}\;\frac{angle}{180} \leq 5 \cdot 10^{+140}:\\ \;\;\;\;\left(\left(b + a\right) \cdot \left(b - a\right)\right) \cdot \left(2 \cdot \left(\cos \left(\frac{angle}{180} \cdot \pi\right) \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\left(b + a\right) \cdot \left(b - a\right)\right) \cdot \left(2 \cdot \sin \left(\pi \cdot \left(angle \cdot 0.005555555555555556\right)\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 66.0% accurate, 1.8× speedup

Math

\[\begin{array}{l} a_m = \left|a\right| \\ b_m = \left|b\right| \\ angle\_m = \left|angle\right| \\ angle\_s = \mathsf{copysign}\left(1, angle\right) \\ \begin{array}{l} t_0 := \left(b\_m + a\_m\right) \cdot \left(b\_m - a\_m\right)\\ t_1 := angle\_m \cdot \left(\left(b\_m + a\_m\right) \cdot \pi\right)\\ t_2 := \frac{angle\_m}{180} \cdot \pi\\ angle\_s \cdot \begin{array}{l} \mathbf{if}\;\frac{angle\_m}{180} \leq 5 \cdot 10^{-29}:\\ \;\;\;\;0.011111111111111112 \cdot \left(b\_m \cdot t\_1 - a\_m \cdot t\_1\right)\\ \mathbf{elif}\;\frac{angle\_m}{180} \leq 5 \cdot 10^{+140}:\\ \;\;\;\;t\_0 \cdot \left(2 \cdot \left(\cos t\_2 \cdot \sin t\_2\right)\right)\\ \mathbf{else}:\\ \;\;\;\;t\_0 \cdot \left(2 \cdot \sin \left(\pi \cdot \left(angle\_m \cdot 0.005555555555555556\right)\right)\right)\\ \end{array} \end{array} \end{array} \]

FPCore

a_m = (fabs.f64 a)
b_m = (fabs.f64 b)
angle\_m = (fabs.f64 angle)
angle\_s = (copysign.f64 #s(literal 1 binary64) angle)
(FPCore (angle_s a_m b_m angle_m)
 :precision binary64
 (let* ((t_0 (* (+ b_m a_m) (- b_m a_m)))
        (t_1 (* angle_m (* (+ b_m a_m) PI)))
        (t_2 (* (/ angle_m 180.0) PI)))
   (*
    angle_s
    (if (<= (/ angle_m 180.0) 5e-29)
      (* 0.011111111111111112 (- (* b_m t_1) (* a_m t_1)))
      (if (<= (/ angle_m 180.0) 5e+140)
        (* t_0 (* 2.0 (* (cos t_2) (sin t_2))))
        (* t_0 (* 2.0 (sin (* PI (* angle_m 0.005555555555555556))))))))))

C

a_m = fabs(a);
b_m = fabs(b);
angle\_m = fabs(angle);
angle\_s = copysign(1.0, angle);
double code(double angle_s, double a_m, double b_m, double angle_m) {
	double t_0 = (b_m + a_m) * (b_m - a_m);
	double t_1 = angle_m * ((b_m + a_m) * ((double) M_PI));
	double t_2 = (angle_m / 180.0) * ((double) M_PI);
	double tmp;
	if ((angle_m / 180.0) <= 5e-29) {
		tmp = 0.011111111111111112 * ((b_m * t_1) - (a_m * t_1));
	} else if ((angle_m / 180.0) <= 5e+140) {
		tmp = t_0 * (2.0 * (cos(t_2) * sin(t_2)));
	} else {
		tmp = t_0 * (2.0 * sin((((double) M_PI) * (angle_m * 0.005555555555555556))));
	}
	return angle_s * tmp;
}

Java

a_m = Math.abs(a);
b_m = Math.abs(b);
angle\_m = Math.abs(angle);
angle\_s = Math.copySign(1.0, angle);
public static double code(double angle_s, double a_m, double b_m, double angle_m) {
	double t_0 = (b_m + a_m) * (b_m - a_m);
	double t_1 = angle_m * ((b_m + a_m) * Math.PI);
	double t_2 = (angle_m / 180.0) * Math.PI;
	double tmp;
	if ((angle_m / 180.0) <= 5e-29) {
		tmp = 0.011111111111111112 * ((b_m * t_1) - (a_m * t_1));
	} else if ((angle_m / 180.0) <= 5e+140) {
		tmp = t_0 * (2.0 * (Math.cos(t_2) * Math.sin(t_2)));
	} else {
		tmp = t_0 * (2.0 * Math.sin((Math.PI * (angle_m * 0.005555555555555556))));
	}
	return angle_s * tmp;
}

Python

a_m = math.fabs(a)
b_m = math.fabs(b)
angle\_m = math.fabs(angle)
angle\_s = math.copysign(1.0, angle)
def code(angle_s, a_m, b_m, angle_m):
	t_0 = (b_m + a_m) * (b_m - a_m)
	t_1 = angle_m * ((b_m + a_m) * math.pi)
	t_2 = (angle_m / 180.0) * math.pi
	tmp = 0
	if (angle_m / 180.0) <= 5e-29:
		tmp = 0.011111111111111112 * ((b_m * t_1) - (a_m * t_1))
	elif (angle_m / 180.0) <= 5e+140:
		tmp = t_0 * (2.0 * (math.cos(t_2) * math.sin(t_2)))
	else:
		tmp = t_0 * (2.0 * math.sin((math.pi * (angle_m * 0.005555555555555556))))
	return angle_s * tmp

Julia

a_m = abs(a)
b_m = abs(b)
angle\_m = abs(angle)
angle\_s = copysign(1.0, angle)
function code(angle_s, a_m, b_m, angle_m)
	t_0 = Float64(Float64(b_m + a_m) * Float64(b_m - a_m))
	t_1 = Float64(angle_m * Float64(Float64(b_m + a_m) * pi))
	t_2 = Float64(Float64(angle_m / 180.0) * pi)
	tmp = 0.0
	if (Float64(angle_m / 180.0) <= 5e-29)
		tmp = Float64(0.011111111111111112 * Float64(Float64(b_m * t_1) - Float64(a_m * t_1)));
	elseif (Float64(angle_m / 180.0) <= 5e+140)
		tmp = Float64(t_0 * Float64(2.0 * Float64(cos(t_2) * sin(t_2))));
	else
		tmp = Float64(t_0 * Float64(2.0 * sin(Float64(pi * Float64(angle_m * 0.005555555555555556)))));
	end
	return Float64(angle_s * tmp)
end

MATLAB

a_m = abs(a);
b_m = abs(b);
angle\_m = abs(angle);
angle\_s = sign(angle) * abs(1.0);
function tmp_2 = code(angle_s, a_m, b_m, angle_m)
	t_0 = (b_m + a_m) * (b_m - a_m);
	t_1 = angle_m * ((b_m + a_m) * pi);
	t_2 = (angle_m / 180.0) * pi;
	tmp = 0.0;
	if ((angle_m / 180.0) <= 5e-29)
		tmp = 0.011111111111111112 * ((b_m * t_1) - (a_m * t_1));
	elseif ((angle_m / 180.0) <= 5e+140)
		tmp = t_0 * (2.0 * (cos(t_2) * sin(t_2)));
	else
		tmp = t_0 * (2.0 * sin((pi * (angle_m * 0.005555555555555556))));
	end
	tmp_2 = angle_s * tmp;
end

Wolfram

a_m = N[Abs[a], $MachinePrecision]
b_m = N[Abs[b], $MachinePrecision]
angle\_m = N[Abs[angle], $MachinePrecision]
angle\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[angle]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[angle$95$s_, a$95$m_, b$95$m_, angle$95$m_] := Block[{t$95$0 = N[(N[(b$95$m + a$95$m), $MachinePrecision] * N[(b$95$m - a$95$m), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(angle$95$m * N[(N[(b$95$m + a$95$m), $MachinePrecision] * Pi), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(N[(angle$95$m / 180.0), $MachinePrecision] * Pi), $MachinePrecision]}, N[(angle$95$s * If[LessEqual[N[(angle$95$m / 180.0), $MachinePrecision], 5e-29], N[(0.011111111111111112 * N[(N[(b$95$m * t$95$1), $MachinePrecision] - N[(a$95$m * t$95$1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[N[(angle$95$m / 180.0), $MachinePrecision], 5e+140], N[(t$95$0 * N[(2.0 * N[(N[Cos[t$95$2], $MachinePrecision] * N[Sin[t$95$2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(t$95$0 * N[(2.0 * N[Sin[N[(Pi * N[(angle$95$m * 0.005555555555555556), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if (/.f64 angle #s(literal 180 binary64)) < 4.99999999999999986e-29

   1. Initial program 62.9%

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

      1. associate-*l* 62.9%

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

      2. *-commutative 62.9%

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

      3. associate-*l* 63.4%

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

   3. Simplified 63.4%

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

   5. Taylor expanded in angle around 0 62.1%

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

      1. unpow2 62.1%

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

      2. unpow2 62.1%

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

      3. difference-of-squares 67.5%

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

   7. Applied egg-rr 67.5%

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

   8. Taylor expanded in angle around 0 67.5%

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

      1. associate-*r* 67.4%

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

   10. Simplified 67.4%

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

       1. associate-*r* 76.6%

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

       2. sub-neg 76.6%

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

       3. distribute-lft-in 73.4%

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

       4. +-commutative 73.4%

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

       5. +-commutative 73.4%

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

   12. Applied egg-rr 73.4%

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

   if 4.99999999999999986e-29 < (/.f64 angle #s(literal 180 binary64)) < 5.00000000000000008e140

   1. Initial program 54.9%

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

      1. associate-*l* 55.0%

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

      2. *-commutative 55.0%

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

      3. associate-*l* 55.0%

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

   3. Simplified 55.0%

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

      1. unpow2 55.0%

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

      2. unpow2 55.0%

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

      3. difference-of-squares 55.0%

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

   6. Applied egg-rr 55.0%

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

   if 5.00000000000000008e140 < (/.f64 angle #s(literal 180 binary64))

   1. Initial program 36.3%

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

      1. associate-*l* 36.3%

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

      2. *-commutative 36.3%

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

      3. associate-*l* 36.3%

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

   3. Simplified 36.3%

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

      1. unpow2 36.3%

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

      2. unpow2 36.3%

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

      3. difference-of-squares 45.7%

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

   6. Applied egg-rr 45.7%

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

   7. Taylor expanded in angle around inf 48.3%

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

      1. associate-*r* 38.7%

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

      2. *-commutative 38.7%

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

      3. *-commutative 38.7%

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

      4. *-commutative 38.7%

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

   9. Simplified 38.7%

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

   10. Taylor expanded in angle around 0 52.4%

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

4. Final simplification 68.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{angle}{180} \leq 5 \cdot 10^{-29}:\\ \;\;\;\;0.011111111111111112 \cdot \left(b \cdot \left(angle \cdot \left(\left(b + a\right) \cdot \pi\right)\right) - a \cdot \left(angle \cdot \left(\left(b + a\right) \cdot \pi\right)\right)\right)\\ \mathbf{elif}\;\frac{angle}{180} \leq 5 \cdot 10^{+140}:\\ \;\;\;\;\left(\left(b + a\right) \cdot \left(b - a\right)\right) \cdot \left(2 \cdot \left(\cos \left(\frac{angle}{180} \cdot \pi\right) \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\left(b + a\right) \cdot \left(b - a\right)\right) \cdot \left(2 \cdot \sin \left(\pi \cdot \left(angle \cdot 0.005555555555555556\right)\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 66.0% accurate, 1.8× speedup

Math

\[\begin{array}{l} a_m = \left|a\right| \\ b_m = \left|b\right| \\ angle\_m = \left|angle\right| \\ angle\_s = \mathsf{copysign}\left(1, angle\right) \\ \begin{array}{l} t_0 := \left(b\_m + a\_m\right) \cdot \left(b\_m - a\_m\right)\\ t_1 := angle\_m \cdot \left(\left(b\_m + a\_m\right) \cdot \pi\right)\\ t_2 := \frac{angle\_m}{180} \cdot \pi\\ angle\_s \cdot \begin{array}{l} \mathbf{if}\;\frac{angle\_m}{180} \leq 5 \cdot 10^{-29}:\\ \;\;\;\;0.011111111111111112 \cdot \left(b\_m \cdot t\_1 - a\_m \cdot t\_1\right)\\ \mathbf{elif}\;\frac{angle\_m}{180} \leq 5 \cdot 10^{+140}:\\ \;\;\;\;2 \cdot \left(\cos t\_2 \cdot \left(t\_0 \cdot \sin t\_2\right)\right)\\ \mathbf{else}:\\ \;\;\;\;t\_0 \cdot \left(2 \cdot \sin \left(\pi \cdot \left(angle\_m \cdot 0.005555555555555556\right)\right)\right)\\ \end{array} \end{array} \end{array} \]

FPCore

a_m = (fabs.f64 a)
b_m = (fabs.f64 b)
angle\_m = (fabs.f64 angle)
angle\_s = (copysign.f64 #s(literal 1 binary64) angle)
(FPCore (angle_s a_m b_m angle_m)
 :precision binary64
 (let* ((t_0 (* (+ b_m a_m) (- b_m a_m)))
        (t_1 (* angle_m (* (+ b_m a_m) PI)))
        (t_2 (* (/ angle_m 180.0) PI)))
   (*
    angle_s
    (if (<= (/ angle_m 180.0) 5e-29)
      (* 0.011111111111111112 (- (* b_m t_1) (* a_m t_1)))
      (if (<= (/ angle_m 180.0) 5e+140)
        (* 2.0 (* (cos t_2) (* t_0 (sin t_2))))
        (* t_0 (* 2.0 (sin (* PI (* angle_m 0.005555555555555556))))))))))

C

a_m = fabs(a);
b_m = fabs(b);
angle\_m = fabs(angle);
angle\_s = copysign(1.0, angle);
double code(double angle_s, double a_m, double b_m, double angle_m) {
	double t_0 = (b_m + a_m) * (b_m - a_m);
	double t_1 = angle_m * ((b_m + a_m) * ((double) M_PI));
	double t_2 = (angle_m / 180.0) * ((double) M_PI);
	double tmp;
	if ((angle_m / 180.0) <= 5e-29) {
		tmp = 0.011111111111111112 * ((b_m * t_1) - (a_m * t_1));
	} else if ((angle_m / 180.0) <= 5e+140) {
		tmp = 2.0 * (cos(t_2) * (t_0 * sin(t_2)));
	} else {
		tmp = t_0 * (2.0 * sin((((double) M_PI) * (angle_m * 0.005555555555555556))));
	}
	return angle_s * tmp;
}

Java

a_m = Math.abs(a);
b_m = Math.abs(b);
angle\_m = Math.abs(angle);
angle\_s = Math.copySign(1.0, angle);
public static double code(double angle_s, double a_m, double b_m, double angle_m) {
	double t_0 = (b_m + a_m) * (b_m - a_m);
	double t_1 = angle_m * ((b_m + a_m) * Math.PI);
	double t_2 = (angle_m / 180.0) * Math.PI;
	double tmp;
	if ((angle_m / 180.0) <= 5e-29) {
		tmp = 0.011111111111111112 * ((b_m * t_1) - (a_m * t_1));
	} else if ((angle_m / 180.0) <= 5e+140) {
		tmp = 2.0 * (Math.cos(t_2) * (t_0 * Math.sin(t_2)));
	} else {
		tmp = t_0 * (2.0 * Math.sin((Math.PI * (angle_m * 0.005555555555555556))));
	}
	return angle_s * tmp;
}

Python

a_m = math.fabs(a)
b_m = math.fabs(b)
angle\_m = math.fabs(angle)
angle\_s = math.copysign(1.0, angle)
def code(angle_s, a_m, b_m, angle_m):
	t_0 = (b_m + a_m) * (b_m - a_m)
	t_1 = angle_m * ((b_m + a_m) * math.pi)
	t_2 = (angle_m / 180.0) * math.pi
	tmp = 0
	if (angle_m / 180.0) <= 5e-29:
		tmp = 0.011111111111111112 * ((b_m * t_1) - (a_m * t_1))
	elif (angle_m / 180.0) <= 5e+140:
		tmp = 2.0 * (math.cos(t_2) * (t_0 * math.sin(t_2)))
	else:
		tmp = t_0 * (2.0 * math.sin((math.pi * (angle_m * 0.005555555555555556))))
	return angle_s * tmp

Julia

a_m = abs(a)
b_m = abs(b)
angle\_m = abs(angle)
angle\_s = copysign(1.0, angle)
function code(angle_s, a_m, b_m, angle_m)
	t_0 = Float64(Float64(b_m + a_m) * Float64(b_m - a_m))
	t_1 = Float64(angle_m * Float64(Float64(b_m + a_m) * pi))
	t_2 = Float64(Float64(angle_m / 180.0) * pi)
	tmp = 0.0
	if (Float64(angle_m / 180.0) <= 5e-29)
		tmp = Float64(0.011111111111111112 * Float64(Float64(b_m * t_1) - Float64(a_m * t_1)));
	elseif (Float64(angle_m / 180.0) <= 5e+140)
		tmp = Float64(2.0 * Float64(cos(t_2) * Float64(t_0 * sin(t_2))));
	else
		tmp = Float64(t_0 * Float64(2.0 * sin(Float64(pi * Float64(angle_m * 0.005555555555555556)))));
	end
	return Float64(angle_s * tmp)
end

MATLAB

a_m = abs(a);
b_m = abs(b);
angle\_m = abs(angle);
angle\_s = sign(angle) * abs(1.0);
function tmp_2 = code(angle_s, a_m, b_m, angle_m)
	t_0 = (b_m + a_m) * (b_m - a_m);
	t_1 = angle_m * ((b_m + a_m) * pi);
	t_2 = (angle_m / 180.0) * pi;
	tmp = 0.0;
	if ((angle_m / 180.0) <= 5e-29)
		tmp = 0.011111111111111112 * ((b_m * t_1) - (a_m * t_1));
	elseif ((angle_m / 180.0) <= 5e+140)
		tmp = 2.0 * (cos(t_2) * (t_0 * sin(t_2)));
	else
		tmp = t_0 * (2.0 * sin((pi * (angle_m * 0.005555555555555556))));
	end
	tmp_2 = angle_s * tmp;
end

Wolfram

a_m = N[Abs[a], $MachinePrecision]
b_m = N[Abs[b], $MachinePrecision]
angle\_m = N[Abs[angle], $MachinePrecision]
angle\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[angle]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[angle$95$s_, a$95$m_, b$95$m_, angle$95$m_] := Block[{t$95$0 = N[(N[(b$95$m + a$95$m), $MachinePrecision] * N[(b$95$m - a$95$m), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(angle$95$m * N[(N[(b$95$m + a$95$m), $MachinePrecision] * Pi), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(N[(angle$95$m / 180.0), $MachinePrecision] * Pi), $MachinePrecision]}, N[(angle$95$s * If[LessEqual[N[(angle$95$m / 180.0), $MachinePrecision], 5e-29], N[(0.011111111111111112 * N[(N[(b$95$m * t$95$1), $MachinePrecision] - N[(a$95$m * t$95$1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[N[(angle$95$m / 180.0), $MachinePrecision], 5e+140], N[(2.0 * N[(N[Cos[t$95$2], $MachinePrecision] * N[(t$95$0 * N[Sin[t$95$2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(t$95$0 * N[(2.0 * N[Sin[N[(Pi * N[(angle$95$m * 0.005555555555555556), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if (/.f64 angle #s(literal 180 binary64)) < 4.99999999999999986e-29

   1. Initial program 62.9%

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

      1. associate-*l* 62.9%

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

      2. *-commutative 62.9%

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

      3. associate-*l* 63.4%

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

   3. Simplified 63.4%

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

   5. Taylor expanded in angle around 0 62.1%

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

      1. unpow2 62.1%

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

      2. unpow2 62.1%

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

      3. difference-of-squares 67.5%

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

   7. Applied egg-rr 67.5%

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

   8. Taylor expanded in angle around 0 67.5%

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

      1. associate-*r* 67.4%

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

   10. Simplified 67.4%

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

       1. associate-*r* 76.6%

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

       2. sub-neg 76.6%

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

       3. distribute-lft-in 73.4%

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

       4. +-commutative 73.4%

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

       5. +-commutative 73.4%

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

   12. Applied egg-rr 73.4%

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

   if 4.99999999999999986e-29 < (/.f64 angle #s(literal 180 binary64)) < 5.00000000000000008e140

   1. Initial program 54.9%

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

      1. associate-*l* 54.9%

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

      2. associate-*l* 54.9%

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

   3. Simplified 54.9%

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

      1. unpow2 54.9%

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

      2. unpow2 54.9%

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

      3. difference-of-squares 55.0%

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

   6. Applied egg-rr 55.0%

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

   if 5.00000000000000008e140 < (/.f64 angle #s(literal 180 binary64))

   1. Initial program 36.3%

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

      1. associate-*l* 36.3%

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

      2. *-commutative 36.3%

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

      3. associate-*l* 36.3%

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

   3. Simplified 36.3%

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

      1. unpow2 36.3%

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

      2. unpow2 36.3%

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

      3. difference-of-squares 45.7%

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

   6. Applied egg-rr 45.7%

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

   7. Taylor expanded in angle around inf 48.3%

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

      1. associate-*r* 38.7%

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

      2. *-commutative 38.7%

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

      3. *-commutative 38.7%

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

      4. *-commutative 38.7%

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

   9. Simplified 38.7%

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

   10. Taylor expanded in angle around 0 52.4%

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

4. Final simplification 68.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{angle}{180} \leq 5 \cdot 10^{-29}:\\ \;\;\;\;0.011111111111111112 \cdot \left(b \cdot \left(angle \cdot \left(\left(b + a\right) \cdot \pi\right)\right) - a \cdot \left(angle \cdot \left(\left(b + a\right) \cdot \pi\right)\right)\right)\\ \mathbf{elif}\;\frac{angle}{180} \leq 5 \cdot 10^{+140}:\\ \;\;\;\;2 \cdot \left(\cos \left(\frac{angle}{180} \cdot \pi\right) \cdot \left(\left(\left(b + a\right) \cdot \left(b - a\right)\right) \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\left(b + a\right) \cdot \left(b - a\right)\right) \cdot \left(2 \cdot \sin \left(\pi \cdot \left(angle \cdot 0.005555555555555556\right)\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 66.1% accurate, 1.8× speedup

Math

\[\begin{array}{l} a_m = \left|a\right| \\ b_m = \left|b\right| \\ angle\_m = \left|angle\right| \\ angle\_s = \mathsf{copysign}\left(1, angle\right) \\ \begin{array}{l} t_0 := angle\_m \cdot \left(\left(b\_m + a\_m\right) \cdot \pi\right)\\ t_1 := \left(b\_m + a\_m\right) \cdot \left(b\_m - a\_m\right)\\ t_2 := 0.005555555555555556 \cdot \left(angle\_m \cdot \pi\right)\\ angle\_s \cdot \begin{array}{l} \mathbf{if}\;\frac{angle\_m}{180} \leq 5 \cdot 10^{-29}:\\ \;\;\;\;0.011111111111111112 \cdot \left(b\_m \cdot t\_0 - a\_m \cdot t\_0\right)\\ \mathbf{elif}\;\frac{angle\_m}{180} \leq 2 \cdot 10^{+28}:\\ \;\;\;\;2 \cdot \left(\cos t\_2 \cdot \left(t\_1 \cdot \sin t\_2\right)\right)\\ \mathbf{else}:\\ \;\;\;\;t\_1 \cdot \left(2 \cdot \sin \left(\pi \cdot \left(angle\_m \cdot 0.005555555555555556\right)\right)\right)\\ \end{array} \end{array} \end{array} \]

FPCore

a_m = (fabs.f64 a)
b_m = (fabs.f64 b)
angle\_m = (fabs.f64 angle)
angle\_s = (copysign.f64 #s(literal 1 binary64) angle)
(FPCore (angle_s a_m b_m angle_m)
 :precision binary64
 (let* ((t_0 (* angle_m (* (+ b_m a_m) PI)))
        (t_1 (* (+ b_m a_m) (- b_m a_m)))
        (t_2 (* 0.005555555555555556 (* angle_m PI))))
   (*
    angle_s
    (if (<= (/ angle_m 180.0) 5e-29)
      (* 0.011111111111111112 (- (* b_m t_0) (* a_m t_0)))
      (if (<= (/ angle_m 180.0) 2e+28)
        (* 2.0 (* (cos t_2) (* t_1 (sin t_2))))
        (* t_1 (* 2.0 (sin (* PI (* angle_m 0.005555555555555556))))))))))

C

a_m = fabs(a);
b_m = fabs(b);
angle\_m = fabs(angle);
angle\_s = copysign(1.0, angle);
double code(double angle_s, double a_m, double b_m, double angle_m) {
	double t_0 = angle_m * ((b_m + a_m) * ((double) M_PI));
	double t_1 = (b_m + a_m) * (b_m - a_m);
	double t_2 = 0.005555555555555556 * (angle_m * ((double) M_PI));
	double tmp;
	if ((angle_m / 180.0) <= 5e-29) {
		tmp = 0.011111111111111112 * ((b_m * t_0) - (a_m * t_0));
	} else if ((angle_m / 180.0) <= 2e+28) {
		tmp = 2.0 * (cos(t_2) * (t_1 * sin(t_2)));
	} else {
		tmp = t_1 * (2.0 * sin((((double) M_PI) * (angle_m * 0.005555555555555556))));
	}
	return angle_s * tmp;
}

Java

a_m = Math.abs(a);
b_m = Math.abs(b);
angle\_m = Math.abs(angle);
angle\_s = Math.copySign(1.0, angle);
public static double code(double angle_s, double a_m, double b_m, double angle_m) {
	double t_0 = angle_m * ((b_m + a_m) * Math.PI);
	double t_1 = (b_m + a_m) * (b_m - a_m);
	double t_2 = 0.005555555555555556 * (angle_m * Math.PI);
	double tmp;
	if ((angle_m / 180.0) <= 5e-29) {
		tmp = 0.011111111111111112 * ((b_m * t_0) - (a_m * t_0));
	} else if ((angle_m / 180.0) <= 2e+28) {
		tmp = 2.0 * (Math.cos(t_2) * (t_1 * Math.sin(t_2)));
	} else {
		tmp = t_1 * (2.0 * Math.sin((Math.PI * (angle_m * 0.005555555555555556))));
	}
	return angle_s * tmp;
}

Python

a_m = math.fabs(a)
b_m = math.fabs(b)
angle\_m = math.fabs(angle)
angle\_s = math.copysign(1.0, angle)
def code(angle_s, a_m, b_m, angle_m):
	t_0 = angle_m * ((b_m + a_m) * math.pi)
	t_1 = (b_m + a_m) * (b_m - a_m)
	t_2 = 0.005555555555555556 * (angle_m * math.pi)
	tmp = 0
	if (angle_m / 180.0) <= 5e-29:
		tmp = 0.011111111111111112 * ((b_m * t_0) - (a_m * t_0))
	elif (angle_m / 180.0) <= 2e+28:
		tmp = 2.0 * (math.cos(t_2) * (t_1 * math.sin(t_2)))
	else:
		tmp = t_1 * (2.0 * math.sin((math.pi * (angle_m * 0.005555555555555556))))
	return angle_s * tmp

Julia

a_m = abs(a)
b_m = abs(b)
angle\_m = abs(angle)
angle\_s = copysign(1.0, angle)
function code(angle_s, a_m, b_m, angle_m)
	t_0 = Float64(angle_m * Float64(Float64(b_m + a_m) * pi))
	t_1 = Float64(Float64(b_m + a_m) * Float64(b_m - a_m))
	t_2 = Float64(0.005555555555555556 * Float64(angle_m * pi))
	tmp = 0.0
	if (Float64(angle_m / 180.0) <= 5e-29)
		tmp = Float64(0.011111111111111112 * Float64(Float64(b_m * t_0) - Float64(a_m * t_0)));
	elseif (Float64(angle_m / 180.0) <= 2e+28)
		tmp = Float64(2.0 * Float64(cos(t_2) * Float64(t_1 * sin(t_2))));
	else
		tmp = Float64(t_1 * Float64(2.0 * sin(Float64(pi * Float64(angle_m * 0.005555555555555556)))));
	end
	return Float64(angle_s * tmp)
end

MATLAB

a_m = abs(a);
b_m = abs(b);
angle\_m = abs(angle);
angle\_s = sign(angle) * abs(1.0);
function tmp_2 = code(angle_s, a_m, b_m, angle_m)
	t_0 = angle_m * ((b_m + a_m) * pi);
	t_1 = (b_m + a_m) * (b_m - a_m);
	t_2 = 0.005555555555555556 * (angle_m * pi);
	tmp = 0.0;
	if ((angle_m / 180.0) <= 5e-29)
		tmp = 0.011111111111111112 * ((b_m * t_0) - (a_m * t_0));
	elseif ((angle_m / 180.0) <= 2e+28)
		tmp = 2.0 * (cos(t_2) * (t_1 * sin(t_2)));
	else
		tmp = t_1 * (2.0 * sin((pi * (angle_m * 0.005555555555555556))));
	end
	tmp_2 = angle_s * tmp;
end

Wolfram

a_m = N[Abs[a], $MachinePrecision]
b_m = N[Abs[b], $MachinePrecision]
angle\_m = N[Abs[angle], $MachinePrecision]
angle\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[angle]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[angle$95$s_, a$95$m_, b$95$m_, angle$95$m_] := Block[{t$95$0 = N[(angle$95$m * N[(N[(b$95$m + a$95$m), $MachinePrecision] * Pi), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[(b$95$m + a$95$m), $MachinePrecision] * N[(b$95$m - a$95$m), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(0.005555555555555556 * N[(angle$95$m * Pi), $MachinePrecision]), $MachinePrecision]}, N[(angle$95$s * If[LessEqual[N[(angle$95$m / 180.0), $MachinePrecision], 5e-29], N[(0.011111111111111112 * N[(N[(b$95$m * t$95$0), $MachinePrecision] - N[(a$95$m * t$95$0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[N[(angle$95$m / 180.0), $MachinePrecision], 2e+28], N[(2.0 * N[(N[Cos[t$95$2], $MachinePrecision] * N[(t$95$1 * N[Sin[t$95$2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(t$95$1 * N[(2.0 * N[Sin[N[(Pi * N[(angle$95$m * 0.005555555555555556), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if (/.f64 angle #s(literal 180 binary64)) < 4.99999999999999986e-29

   1. Initial program 62.9%

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

      1. associate-*l* 62.9%

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

      2. *-commutative 62.9%

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

      3. associate-*l* 63.4%

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

   3. Simplified 63.4%

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

   5. Taylor expanded in angle around 0 62.1%

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

      1. unpow2 62.1%

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

      2. unpow2 62.1%

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

      3. difference-of-squares 67.5%

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

   7. Applied egg-rr 67.5%

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

   8. Taylor expanded in angle around 0 67.5%

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

      1. associate-*r* 67.4%

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

   10. Simplified 67.4%

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

       1. associate-*r* 76.6%

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

       2. sub-neg 76.6%

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

       3. distribute-lft-in 73.4%

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

       4. +-commutative 73.4%

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

       5. +-commutative 73.4%

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

   12. Applied egg-rr 73.4%

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

   if 4.99999999999999986e-29 < (/.f64 angle #s(literal 180 binary64)) < 1.99999999999999992e28

   1. Initial program 92.0%

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

      1. associate-*l* 92.2%

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

      2. *-commutative 92.2%

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

      3. associate-*l* 92.2%

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

   3. Simplified 92.2%

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

      1. unpow2 92.2%

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

      2. unpow2 92.2%

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

      3. difference-of-squares 92.2%

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

   6. Applied egg-rr 92.2%

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

   7. Taylor expanded in angle around inf 92.0%

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

   if 1.99999999999999992e28 < (/.f64 angle #s(literal 180 binary64))

   1. Initial program 39.0%

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

      1. associate-*l* 39.0%

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

      2. *-commutative 39.0%

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

      3. associate-*l* 39.0%

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

   3. Simplified 39.0%

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

      1. unpow2 39.0%

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

      2. unpow2 39.0%

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

      3. difference-of-squares 44.4%

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

   6. Applied egg-rr 44.4%

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

   7. Taylor expanded in angle around inf 46.3%

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

      1. associate-*r* 42.0%

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

      2. *-commutative 42.0%

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

      3. *-commutative 42.0%

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

      4. *-commutative 42.0%

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

   9. Simplified 42.0%

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

   10. Taylor expanded in angle around 0 42.2%

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

4. Final simplification 67.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{angle}{180} \leq 5 \cdot 10^{-29}:\\ \;\;\;\;0.011111111111111112 \cdot \left(b \cdot \left(angle \cdot \left(\left(b + a\right) \cdot \pi\right)\right) - a \cdot \left(angle \cdot \left(\left(b + a\right) \cdot \pi\right)\right)\right)\\ \mathbf{elif}\;\frac{angle}{180} \leq 2 \cdot 10^{+28}:\\ \;\;\;\;2 \cdot \left(\cos \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right) \cdot \left(\left(\left(b + a\right) \cdot \left(b - a\right)\right) \cdot \sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\left(b + a\right) \cdot \left(b - a\right)\right) \cdot \left(2 \cdot \sin \left(\pi \cdot \left(angle \cdot 0.005555555555555556\right)\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 12: 64.6% accurate, 3.4× speedup

Math

\[\begin{array}{l} a_m = \left|a\right| \\ b_m = \left|b\right| \\ angle\_m = \left|angle\right| \\ angle\_s = \mathsf{copysign}\left(1, angle\right) \\ \begin{array}{l} t_0 := angle\_m \cdot \left(\left(b\_m + a\_m\right) \cdot \pi\right)\\ angle\_s \cdot \begin{array}{l} \mathbf{if}\;\frac{angle\_m}{180} \leq 5 \cdot 10^{-29}:\\ \;\;\;\;0.011111111111111112 \cdot \left(b\_m \cdot t\_0 - a\_m \cdot t\_0\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\left(b\_m + a\_m\right) \cdot \left(b\_m - a\_m\right)\right) \cdot \left(2 \cdot \sin \left(\pi \cdot \left(angle\_m \cdot 0.005555555555555556\right)\right)\right)\\ \end{array} \end{array} \end{array} \]

FPCore

a_m = (fabs.f64 a)
b_m = (fabs.f64 b)
angle\_m = (fabs.f64 angle)
angle\_s = (copysign.f64 #s(literal 1 binary64) angle)
(FPCore (angle_s a_m b_m angle_m)
 :precision binary64
 (let* ((t_0 (* angle_m (* (+ b_m a_m) PI))))
   (*
    angle_s
    (if (<= (/ angle_m 180.0) 5e-29)
      (* 0.011111111111111112 (- (* b_m t_0) (* a_m t_0)))
      (*
       (* (+ b_m a_m) (- b_m a_m))
       (* 2.0 (sin (* PI (* angle_m 0.005555555555555556)))))))))

C

a_m = fabs(a);
b_m = fabs(b);
angle\_m = fabs(angle);
angle\_s = copysign(1.0, angle);
double code(double angle_s, double a_m, double b_m, double angle_m) {
	double t_0 = angle_m * ((b_m + a_m) * ((double) M_PI));
	double tmp;
	if ((angle_m / 180.0) <= 5e-29) {
		tmp = 0.011111111111111112 * ((b_m * t_0) - (a_m * t_0));
	} else {
		tmp = ((b_m + a_m) * (b_m - a_m)) * (2.0 * sin((((double) M_PI) * (angle_m * 0.005555555555555556))));
	}
	return angle_s * tmp;
}

Java

a_m = Math.abs(a);
b_m = Math.abs(b);
angle\_m = Math.abs(angle);
angle\_s = Math.copySign(1.0, angle);
public static double code(double angle_s, double a_m, double b_m, double angle_m) {
	double t_0 = angle_m * ((b_m + a_m) * Math.PI);
	double tmp;
	if ((angle_m / 180.0) <= 5e-29) {
		tmp = 0.011111111111111112 * ((b_m * t_0) - (a_m * t_0));
	} else {
		tmp = ((b_m + a_m) * (b_m - a_m)) * (2.0 * Math.sin((Math.PI * (angle_m * 0.005555555555555556))));
	}
	return angle_s * tmp;
}

Python

a_m = math.fabs(a)
b_m = math.fabs(b)
angle\_m = math.fabs(angle)
angle\_s = math.copysign(1.0, angle)
def code(angle_s, a_m, b_m, angle_m):
	t_0 = angle_m * ((b_m + a_m) * math.pi)
	tmp = 0
	if (angle_m / 180.0) <= 5e-29:
		tmp = 0.011111111111111112 * ((b_m * t_0) - (a_m * t_0))
	else:
		tmp = ((b_m + a_m) * (b_m - a_m)) * (2.0 * math.sin((math.pi * (angle_m * 0.005555555555555556))))
	return angle_s * tmp

Julia

a_m = abs(a)
b_m = abs(b)
angle\_m = abs(angle)
angle\_s = copysign(1.0, angle)
function code(angle_s, a_m, b_m, angle_m)
	t_0 = Float64(angle_m * Float64(Float64(b_m + a_m) * pi))
	tmp = 0.0
	if (Float64(angle_m / 180.0) <= 5e-29)
		tmp = Float64(0.011111111111111112 * Float64(Float64(b_m * t_0) - Float64(a_m * t_0)));
	else
		tmp = Float64(Float64(Float64(b_m + a_m) * Float64(b_m - a_m)) * Float64(2.0 * sin(Float64(pi * Float64(angle_m * 0.005555555555555556)))));
	end
	return Float64(angle_s * tmp)
end

MATLAB

a_m = abs(a);
b_m = abs(b);
angle\_m = abs(angle);
angle\_s = sign(angle) * abs(1.0);
function tmp_2 = code(angle_s, a_m, b_m, angle_m)
	t_0 = angle_m * ((b_m + a_m) * pi);
	tmp = 0.0;
	if ((angle_m / 180.0) <= 5e-29)
		tmp = 0.011111111111111112 * ((b_m * t_0) - (a_m * t_0));
	else
		tmp = ((b_m + a_m) * (b_m - a_m)) * (2.0 * sin((pi * (angle_m * 0.005555555555555556))));
	end
	tmp_2 = angle_s * tmp;
end

Wolfram

a_m = N[Abs[a], $MachinePrecision]
b_m = N[Abs[b], $MachinePrecision]
angle\_m = N[Abs[angle], $MachinePrecision]
angle\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[angle]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[angle$95$s_, a$95$m_, b$95$m_, angle$95$m_] := Block[{t$95$0 = N[(angle$95$m * N[(N[(b$95$m + a$95$m), $MachinePrecision] * Pi), $MachinePrecision]), $MachinePrecision]}, N[(angle$95$s * If[LessEqual[N[(angle$95$m / 180.0), $MachinePrecision], 5e-29], N[(0.011111111111111112 * N[(N[(b$95$m * t$95$0), $MachinePrecision] - N[(a$95$m * t$95$0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(b$95$m + a$95$m), $MachinePrecision] * N[(b$95$m - a$95$m), $MachinePrecision]), $MachinePrecision] * N[(2.0 * N[Sin[N[(Pi * N[(angle$95$m * 0.005555555555555556), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 angle #s(literal 180 binary64)) < 4.99999999999999986e-29

   1. Initial program 62.9%

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

      1. associate-*l* 62.9%

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

      2. *-commutative 62.9%

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

      3. associate-*l* 63.4%

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

   3. Simplified 63.4%

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

   5. Taylor expanded in angle around 0 62.1%

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

      1. unpow2 62.1%

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

      2. unpow2 62.1%

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

      3. difference-of-squares 67.5%

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

   7. Applied egg-rr 67.5%

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

   8. Taylor expanded in angle around 0 67.5%

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

      1. associate-*r* 67.4%

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

   10. Simplified 67.4%

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

       1. associate-*r* 76.6%

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

       2. sub-neg 76.6%

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

       3. distribute-lft-in 73.4%

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

       4. +-commutative 73.4%

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

       5. +-commutative 73.4%

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

   12. Applied egg-rr 73.4%

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

   if 4.99999999999999986e-29 < (/.f64 angle #s(literal 180 binary64))

   1. Initial program 45.6%

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

      1. associate-*l* 45.6%

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

      2. *-commutative 45.6%

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

      3. associate-*l* 45.6%

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

   3. Simplified 45.6%

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

      1. unpow2 45.6%

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

      2. unpow2 45.6%

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

      3. difference-of-squares 50.3%

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

   6. Applied egg-rr 50.3%

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

   7. Taylor expanded in angle around inf 52.1%

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

      1. associate-*r* 48.3%

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

      2. *-commutative 48.3%

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

      3. *-commutative 48.3%

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

      4. *-commutative 48.3%

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

   9. Simplified 48.3%

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

   10. Taylor expanded in angle around 0 43.8%

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

4. Final simplification 66.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{angle}{180} \leq 5 \cdot 10^{-29}:\\ \;\;\;\;0.011111111111111112 \cdot \left(b \cdot \left(angle \cdot \left(\left(b + a\right) \cdot \pi\right)\right) - a \cdot \left(angle \cdot \left(\left(b + a\right) \cdot \pi\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\left(b + a\right) \cdot \left(b - a\right)\right) \cdot \left(2 \cdot \sin \left(\pi \cdot \left(angle \cdot 0.005555555555555556\right)\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 13: 62.5% accurate, 16.1× speedup

Math

\[\begin{array}{l} a_m = \left|a\right| \\ b_m = \left|b\right| \\ angle\_m = \left|angle\right| \\ angle\_s = \mathsf{copysign}\left(1, angle\right) \\ \begin{array}{l} t_0 := angle\_m \cdot \left(\left(b\_m + a\_m\right) \cdot \pi\right)\\ angle\_s \cdot \begin{array}{l} \mathbf{if}\;angle\_m \leq 2.7 \cdot 10^{-51}:\\ \;\;\;\;0.011111111111111112 \cdot \left(b\_m \cdot t\_0 - a\_m \cdot t\_0\right)\\ \mathbf{else}:\\ \;\;\;\;0.011111111111111112 \cdot \left(angle\_m \cdot \left(\pi \cdot \left(\left(b\_m + a\_m\right) \cdot \left(b\_m - a\_m\right)\right)\right)\right)\\ \end{array} \end{array} \end{array} \]

FPCore

a_m = (fabs.f64 a)
b_m = (fabs.f64 b)
angle\_m = (fabs.f64 angle)
angle\_s = (copysign.f64 #s(literal 1 binary64) angle)
(FPCore (angle_s a_m b_m angle_m)
 :precision binary64
 (let* ((t_0 (* angle_m (* (+ b_m a_m) PI))))
   (*
    angle_s
    (if (<= angle_m 2.7e-51)
      (* 0.011111111111111112 (- (* b_m t_0) (* a_m t_0)))
      (*
       0.011111111111111112
       (* angle_m (* PI (* (+ b_m a_m) (- b_m a_m)))))))))

C

a_m = fabs(a);
b_m = fabs(b);
angle\_m = fabs(angle);
angle\_s = copysign(1.0, angle);
double code(double angle_s, double a_m, double b_m, double angle_m) {
	double t_0 = angle_m * ((b_m + a_m) * ((double) M_PI));
	double tmp;
	if (angle_m <= 2.7e-51) {
		tmp = 0.011111111111111112 * ((b_m * t_0) - (a_m * t_0));
	} else {
		tmp = 0.011111111111111112 * (angle_m * (((double) M_PI) * ((b_m + a_m) * (b_m - a_m))));
	}
	return angle_s * tmp;
}

Java

a_m = Math.abs(a);
b_m = Math.abs(b);
angle\_m = Math.abs(angle);
angle\_s = Math.copySign(1.0, angle);
public static double code(double angle_s, double a_m, double b_m, double angle_m) {
	double t_0 = angle_m * ((b_m + a_m) * Math.PI);
	double tmp;
	if (angle_m <= 2.7e-51) {
		tmp = 0.011111111111111112 * ((b_m * t_0) - (a_m * t_0));
	} else {
		tmp = 0.011111111111111112 * (angle_m * (Math.PI * ((b_m + a_m) * (b_m - a_m))));
	}
	return angle_s * tmp;
}

Python

a_m = math.fabs(a)
b_m = math.fabs(b)
angle\_m = math.fabs(angle)
angle\_s = math.copysign(1.0, angle)
def code(angle_s, a_m, b_m, angle_m):
	t_0 = angle_m * ((b_m + a_m) * math.pi)
	tmp = 0
	if angle_m <= 2.7e-51:
		tmp = 0.011111111111111112 * ((b_m * t_0) - (a_m * t_0))
	else:
		tmp = 0.011111111111111112 * (angle_m * (math.pi * ((b_m + a_m) * (b_m - a_m))))
	return angle_s * tmp

Julia

a_m = abs(a)
b_m = abs(b)
angle\_m = abs(angle)
angle\_s = copysign(1.0, angle)
function code(angle_s, a_m, b_m, angle_m)
	t_0 = Float64(angle_m * Float64(Float64(b_m + a_m) * pi))
	tmp = 0.0
	if (angle_m <= 2.7e-51)
		tmp = Float64(0.011111111111111112 * Float64(Float64(b_m * t_0) - Float64(a_m * t_0)));
	else
		tmp = Float64(0.011111111111111112 * Float64(angle_m * Float64(pi * Float64(Float64(b_m + a_m) * Float64(b_m - a_m)))));
	end
	return Float64(angle_s * tmp)
end

MATLAB

a_m = abs(a);
b_m = abs(b);
angle\_m = abs(angle);
angle\_s = sign(angle) * abs(1.0);
function tmp_2 = code(angle_s, a_m, b_m, angle_m)
	t_0 = angle_m * ((b_m + a_m) * pi);
	tmp = 0.0;
	if (angle_m <= 2.7e-51)
		tmp = 0.011111111111111112 * ((b_m * t_0) - (a_m * t_0));
	else
		tmp = 0.011111111111111112 * (angle_m * (pi * ((b_m + a_m) * (b_m - a_m))));
	end
	tmp_2 = angle_s * tmp;
end

Wolfram

a_m = N[Abs[a], $MachinePrecision]
b_m = N[Abs[b], $MachinePrecision]
angle\_m = N[Abs[angle], $MachinePrecision]
angle\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[angle]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[angle$95$s_, a$95$m_, b$95$m_, angle$95$m_] := Block[{t$95$0 = N[(angle$95$m * N[(N[(b$95$m + a$95$m), $MachinePrecision] * Pi), $MachinePrecision]), $MachinePrecision]}, N[(angle$95$s * If[LessEqual[angle$95$m, 2.7e-51], N[(0.011111111111111112 * N[(N[(b$95$m * t$95$0), $MachinePrecision] - N[(a$95$m * t$95$0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(0.011111111111111112 * N[(angle$95$m * N[(Pi * N[(N[(b$95$m + a$95$m), $MachinePrecision] * N[(b$95$m - a$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if angle < 2.6999999999999997e-51

   1. Initial program 62.1%

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

      1. associate-*l* 62.1%

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

      2. *-commutative 62.1%

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

      3. associate-*l* 62.6%

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

   3. Simplified 62.6%

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

   5. Taylor expanded in angle around 0 61.3%

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

      1. unpow2 61.3%

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

      2. unpow2 61.3%

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

      3. difference-of-squares 66.8%

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

   7. Applied egg-rr 66.8%

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

   8. Taylor expanded in angle around 0 66.8%

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

      1. associate-*r* 66.8%

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

   10. Simplified 66.8%

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

       1. associate-*r* 76.1%

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

       2. sub-neg 76.1%

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

       3. distribute-lft-in 72.8%

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

       4. +-commutative 72.8%

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

       5. +-commutative 72.8%

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

   12. Applied egg-rr 72.8%

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

   if 2.6999999999999997e-51 < angle

   1. Initial program 48.8%

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

      1. associate-*l* 48.8%

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

      2. *-commutative 48.8%

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

      3. associate-*l* 48.8%

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

   3. Simplified 48.8%

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

   5. Taylor expanded in angle around 0 38.2%

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

      1. unpow2 38.2%

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

      2. unpow2 38.2%

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

      3. difference-of-squares 39.7%

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

   7. Applied egg-rr 39.7%

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

4. Final simplification 64.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;angle \leq 2.7 \cdot 10^{-51}:\\ \;\;\;\;0.011111111111111112 \cdot \left(b \cdot \left(angle \cdot \left(\left(b + a\right) \cdot \pi\right)\right) - a \cdot \left(angle \cdot \left(\left(b + a\right) \cdot \pi\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;0.011111111111111112 \cdot \left(angle \cdot \left(\pi \cdot \left(\left(b + a\right) \cdot \left(b - a\right)\right)\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 14: 49.7% accurate, 26.2× speedup

Math

\[\begin{array}{l} a_m = \left|a\right| \\ b_m = \left|b\right| \\ angle\_m = \left|angle\right| \\ angle\_s = \mathsf{copysign}\left(1, angle\right) \\ angle\_s \cdot \begin{array}{l} \mathbf{if}\;a\_m \leq 8.5 \cdot 10^{-16}:\\ \;\;\;\;0.011111111111111112 \cdot \left(angle\_m \cdot \left(\left(b\_m - a\_m\right) \cdot \left(b\_m \cdot \pi\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;0.011111111111111112 \cdot \left(angle\_m \cdot \left(\left(b\_m - a\_m\right) \cdot \left(a\_m \cdot \pi\right)\right)\right)\\ \end{array} \end{array} \]

FPCore

a_m = (fabs.f64 a)
b_m = (fabs.f64 b)
angle\_m = (fabs.f64 angle)
angle\_s = (copysign.f64 #s(literal 1 binary64) angle)
(FPCore (angle_s a_m b_m angle_m)
 :precision binary64
 (*
  angle_s
  (if (<= a_m 8.5e-16)
    (* 0.011111111111111112 (* angle_m (* (- b_m a_m) (* b_m PI))))
    (* 0.011111111111111112 (* angle_m (* (- b_m a_m) (* a_m PI)))))))

C

a_m = fabs(a);
b_m = fabs(b);
angle\_m = fabs(angle);
angle\_s = copysign(1.0, angle);
double code(double angle_s, double a_m, double b_m, double angle_m) {
	double tmp;
	if (a_m <= 8.5e-16) {
		tmp = 0.011111111111111112 * (angle_m * ((b_m - a_m) * (b_m * ((double) M_PI))));
	} else {
		tmp = 0.011111111111111112 * (angle_m * ((b_m - a_m) * (a_m * ((double) M_PI))));
	}
	return angle_s * tmp;
}

Java

a_m = Math.abs(a);
b_m = Math.abs(b);
angle\_m = Math.abs(angle);
angle\_s = Math.copySign(1.0, angle);
public static double code(double angle_s, double a_m, double b_m, double angle_m) {
	double tmp;
	if (a_m <= 8.5e-16) {
		tmp = 0.011111111111111112 * (angle_m * ((b_m - a_m) * (b_m * Math.PI)));
	} else {
		tmp = 0.011111111111111112 * (angle_m * ((b_m - a_m) * (a_m * Math.PI)));
	}
	return angle_s * tmp;
}

Python

a_m = math.fabs(a)
b_m = math.fabs(b)
angle\_m = math.fabs(angle)
angle\_s = math.copysign(1.0, angle)
def code(angle_s, a_m, b_m, angle_m):
	tmp = 0
	if a_m <= 8.5e-16:
		tmp = 0.011111111111111112 * (angle_m * ((b_m - a_m) * (b_m * math.pi)))
	else:
		tmp = 0.011111111111111112 * (angle_m * ((b_m - a_m) * (a_m * math.pi)))
	return angle_s * tmp

Julia

a_m = abs(a)
b_m = abs(b)
angle\_m = abs(angle)
angle\_s = copysign(1.0, angle)
function code(angle_s, a_m, b_m, angle_m)
	tmp = 0.0
	if (a_m <= 8.5e-16)
		tmp = Float64(0.011111111111111112 * Float64(angle_m * Float64(Float64(b_m - a_m) * Float64(b_m * pi))));
	else
		tmp = Float64(0.011111111111111112 * Float64(angle_m * Float64(Float64(b_m - a_m) * Float64(a_m * pi))));
	end
	return Float64(angle_s * tmp)
end

MATLAB

a_m = abs(a);
b_m = abs(b);
angle\_m = abs(angle);
angle\_s = sign(angle) * abs(1.0);
function tmp_2 = code(angle_s, a_m, b_m, angle_m)
	tmp = 0.0;
	if (a_m <= 8.5e-16)
		tmp = 0.011111111111111112 * (angle_m * ((b_m - a_m) * (b_m * pi)));
	else
		tmp = 0.011111111111111112 * (angle_m * ((b_m - a_m) * (a_m * pi)));
	end
	tmp_2 = angle_s * tmp;
end

Wolfram

a_m = N[Abs[a], $MachinePrecision]
b_m = N[Abs[b], $MachinePrecision]
angle\_m = N[Abs[angle], $MachinePrecision]
angle\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[angle]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[angle$95$s_, a$95$m_, b$95$m_, angle$95$m_] := N[(angle$95$s * If[LessEqual[a$95$m, 8.5e-16], N[(0.011111111111111112 * N[(angle$95$m * N[(N[(b$95$m - a$95$m), $MachinePrecision] * N[(b$95$m * Pi), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(0.011111111111111112 * N[(angle$95$m * N[(N[(b$95$m - a$95$m), $MachinePrecision] * N[(a$95$m * Pi), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if a < 8.5000000000000001e-16

   1. Initial program 61.7%

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

      1. associate-*l* 61.7%

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

      2. *-commutative 61.7%

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

      3. associate-*l* 62.2%

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

   3. Simplified 62.2%

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

   5. Taylor expanded in angle around 0 58.5%

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

      1. unpow2 58.5%

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

      2. unpow2 58.5%

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

      3. difference-of-squares 60.7%

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

   7. Applied egg-rr 60.7%

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

   8. Taylor expanded in angle around 0 60.7%

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

      1. associate-*r* 60.7%

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

   10. Simplified 60.7%

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

   11. Taylor expanded in a around 0 46.1%

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

       1. *-commutative 46.1%

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

   13. Simplified 46.1%

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

   if 8.5000000000000001e-16 < a

   1. Initial program 48.9%

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

      1. associate-*l* 48.9%

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

      2. *-commutative 48.9%

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

      3. associate-*l* 48.9%

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

   3. Simplified 48.9%

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

   5. Taylor expanded in angle around 0 44.8%

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

      1. unpow2 44.8%

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

      2. unpow2 44.8%

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

      3. difference-of-squares 56.1%

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

   7. Applied egg-rr 56.1%

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

   8. Taylor expanded in angle around 0 56.1%

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

      1. associate-*r* 56.1%

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

   10. Simplified 56.1%

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

   11. Taylor expanded in a around inf 44.7%

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

       1. *-commutative 44.7%

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

   13. Simplified 44.7%

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

4. Final simplification 45.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \leq 8.5 \cdot 10^{-16}:\\ \;\;\;\;0.011111111111111112 \cdot \left(angle \cdot \left(\left(b - a\right) \cdot \left(b \cdot \pi\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;0.011111111111111112 \cdot \left(angle \cdot \left(\left(b - a\right) \cdot \left(a \cdot \pi\right)\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 15: 54.3% accurate, 32.2× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

a_m = N[Abs[a], $MachinePrecision]
b_m = N[Abs[b], $MachinePrecision]
angle\_m = N[Abs[angle], $MachinePrecision]
angle\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[angle]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[angle$95$s_, a$95$m_, b$95$m_, angle$95$m_] := N[(angle$95$s * N[(0.011111111111111112 * N[(angle$95$m * N[(Pi * N[(N[(b$95$m + a$95$m), $MachinePrecision] * N[(b$95$m - a$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 58.6%

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

   1. associate-*l* 58.6%

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

   2. *-commutative 58.6%

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

   3. associate-*l* 58.9%

      \[\leadsto \color{blue}{\left({b}^{2} - {a}^{2}\right) \cdot \left(2 \cdot \left(\sin \left(\pi \cdot \frac{angle}{180}\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right)\right)\right)} \]

3. Simplified 58.9%

   \[\leadsto \color{blue}{\left({b}^{2} - {a}^{2}\right) \cdot \left(2 \cdot \left(\sin \left(\pi \cdot \frac{angle}{180}\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right)\right)\right)} \]
4. Add Preprocessing

5. Taylor expanded in angle around 0 55.2%

   \[\leadsto \color{blue}{0.011111111111111112 \cdot \left(angle \cdot \left(\pi \cdot \left({b}^{2} - {a}^{2}\right)\right)\right)} \]
6. Step-by-step derivation

   1. unpow2 55.2%

      \[\leadsto 0.011111111111111112 \cdot \left(angle \cdot \left(\pi \cdot \left(\color{blue}{b \cdot b} - {a}^{2}\right)\right)\right) \]

   2. unpow2 55.2%

      \[\leadsto 0.011111111111111112 \cdot \left(angle \cdot \left(\pi \cdot \left(b \cdot b - \color{blue}{a \cdot a}\right)\right)\right) \]

   3. difference-of-squares 59.6%

      \[\leadsto 0.011111111111111112 \cdot \left(angle \cdot \left(\pi \cdot \color{blue}{\left(\left(b + a\right) \cdot \left(b - a\right)\right)}\right)\right) \]

7. Applied egg-rr 59.6%

   \[\leadsto 0.011111111111111112 \cdot \left(angle \cdot \left(\pi \cdot \color{blue}{\left(\left(b + a\right) \cdot \left(b - a\right)\right)}\right)\right) \]
8. Add Preprocessing

Alternative 16: 38.2% accurate, 38.1× speedup

Math

\[\begin{array}{l} a_m = \left|a\right| \\ b_m = \left|b\right| \\ angle\_m = \left|angle\right| \\ angle\_s = \mathsf{copysign}\left(1, angle\right) \\ angle\_s \cdot \left(0.011111111111111112 \cdot \left(angle\_m \cdot \left(\left(b\_m - a\_m\right) \cdot \left(a\_m \cdot \pi\right)\right)\right)\right) \end{array} \]

FPCore

a_m = (fabs.f64 a)
b_m = (fabs.f64 b)
angle\_m = (fabs.f64 angle)
angle\_s = (copysign.f64 #s(literal 1 binary64) angle)
(FPCore (angle_s a_m b_m angle_m)
 :precision binary64
 (* angle_s (* 0.011111111111111112 (* angle_m (* (- b_m a_m) (* a_m PI))))))

C

a_m = fabs(a);
b_m = fabs(b);
angle\_m = fabs(angle);
angle\_s = copysign(1.0, angle);
double code(double angle_s, double a_m, double b_m, double angle_m) {
	return angle_s * (0.011111111111111112 * (angle_m * ((b_m - a_m) * (a_m * ((double) M_PI)))));
}

Java

a_m = Math.abs(a);
b_m = Math.abs(b);
angle\_m = Math.abs(angle);
angle\_s = Math.copySign(1.0, angle);
public static double code(double angle_s, double a_m, double b_m, double angle_m) {
	return angle_s * (0.011111111111111112 * (angle_m * ((b_m - a_m) * (a_m * Math.PI))));
}

Python

a_m = math.fabs(a)
b_m = math.fabs(b)
angle\_m = math.fabs(angle)
angle\_s = math.copysign(1.0, angle)
def code(angle_s, a_m, b_m, angle_m):
	return angle_s * (0.011111111111111112 * (angle_m * ((b_m - a_m) * (a_m * math.pi))))

Julia

a_m = abs(a)
b_m = abs(b)
angle\_m = abs(angle)
angle\_s = copysign(1.0, angle)
function code(angle_s, a_m, b_m, angle_m)
	return Float64(angle_s * Float64(0.011111111111111112 * Float64(angle_m * Float64(Float64(b_m - a_m) * Float64(a_m * pi)))))
end

MATLAB

a_m = abs(a);
b_m = abs(b);
angle\_m = abs(angle);
angle\_s = sign(angle) * abs(1.0);
function tmp = code(angle_s, a_m, b_m, angle_m)
	tmp = angle_s * (0.011111111111111112 * (angle_m * ((b_m - a_m) * (a_m * pi))));
end

Wolfram

a_m = N[Abs[a], $MachinePrecision]
b_m = N[Abs[b], $MachinePrecision]
angle\_m = N[Abs[angle], $MachinePrecision]
angle\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[angle]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[angle$95$s_, a$95$m_, b$95$m_, angle$95$m_] := N[(angle$95$s * N[(0.011111111111111112 * N[(angle$95$m * N[(N[(b$95$m - a$95$m), $MachinePrecision] * N[(a$95$m * Pi), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 58.6%

   \[\left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin \left(\pi \cdot \frac{angle}{180}\right)\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right) \]
2. Step-by-step derivation

   1. associate-*l* 58.6%

      \[\leadsto \color{blue}{\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \left(\sin \left(\pi \cdot \frac{angle}{180}\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right)\right)} \]

   2. *-commutative 58.6%

      \[\leadsto \color{blue}{\left(\left({b}^{2} - {a}^{2}\right) \cdot 2\right)} \cdot \left(\sin \left(\pi \cdot \frac{angle}{180}\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right)\right) \]

   3. associate-*l* 58.9%

      \[\leadsto \color{blue}{\left({b}^{2} - {a}^{2}\right) \cdot \left(2 \cdot \left(\sin \left(\pi \cdot \frac{angle}{180}\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right)\right)\right)} \]

3. Simplified 58.9%

   \[\leadsto \color{blue}{\left({b}^{2} - {a}^{2}\right) \cdot \left(2 \cdot \left(\sin \left(\pi \cdot \frac{angle}{180}\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right)\right)\right)} \]
4. Add Preprocessing

5. Taylor expanded in angle around 0 55.2%

   \[\leadsto \color{blue}{0.011111111111111112 \cdot \left(angle \cdot \left(\pi \cdot \left({b}^{2} - {a}^{2}\right)\right)\right)} \]
6. Step-by-step derivation

   1. unpow2 55.2%

      \[\leadsto 0.011111111111111112 \cdot \left(angle \cdot \left(\pi \cdot \left(\color{blue}{b \cdot b} - {a}^{2}\right)\right)\right) \]

   2. unpow2 55.2%

      \[\leadsto 0.011111111111111112 \cdot \left(angle \cdot \left(\pi \cdot \left(b \cdot b - \color{blue}{a \cdot a}\right)\right)\right) \]

   3. difference-of-squares 59.6%

      \[\leadsto 0.011111111111111112 \cdot \left(angle \cdot \left(\pi \cdot \color{blue}{\left(\left(b + a\right) \cdot \left(b - a\right)\right)}\right)\right) \]

7. Applied egg-rr 59.6%

   \[\leadsto 0.011111111111111112 \cdot \left(angle \cdot \left(\pi \cdot \color{blue}{\left(\left(b + a\right) \cdot \left(b - a\right)\right)}\right)\right) \]

8. Taylor expanded in angle around 0 59.6%

   \[\leadsto 0.011111111111111112 \cdot \color{blue}{\left(angle \cdot \left(\pi \cdot \left(\left(a + b\right) \cdot \left(b - a\right)\right)\right)\right)} \]
9. Step-by-step derivation

   1. associate-*r* 59.6%

      \[\leadsto 0.011111111111111112 \cdot \left(angle \cdot \color{blue}{\left(\left(\pi \cdot \left(a + b\right)\right) \cdot \left(b - a\right)\right)}\right) \]

10. Simplified 59.6%

    \[\leadsto 0.011111111111111112 \cdot \color{blue}{\left(angle \cdot \left(\left(\pi \cdot \left(a + b\right)\right) \cdot \left(b - a\right)\right)\right)} \]

11. Taylor expanded in a around inf 41.3%

    \[\leadsto 0.011111111111111112 \cdot \left(angle \cdot \left(\color{blue}{\left(a \cdot \pi\right)} \cdot \left(b - a\right)\right)\right) \]
12. Step-by-step derivation

    1. *-commutative 41.3%

       \[\leadsto 0.011111111111111112 \cdot \left(angle \cdot \left(\color{blue}{\left(\pi \cdot a\right)} \cdot \left(b - a\right)\right)\right) \]

13. Simplified 41.3%

    \[\leadsto 0.011111111111111112 \cdot \left(angle \cdot \left(\color{blue}{\left(\pi \cdot a\right)} \cdot \left(b - a\right)\right)\right) \]

14. Final simplification 41.3%

    \[\leadsto 0.011111111111111112 \cdot \left(angle \cdot \left(\left(b - a\right) \cdot \left(a \cdot \pi\right)\right)\right) \]
15. Add Preprocessing

Reproduce

herbie shell --seed 2024101 
(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)))))