ab-angle->ABCF B

Percentage Accurate: 53.7% → 67.5%

Time: 1.2min

Alternatives: 14

Speedup: 32.2×

• 32.8% of points produce a very large (infinite) output. You may want to add a precondition.

Specification

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 53.7% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 67.5% accurate, 0.7× speedup

Math

\[\begin{array}{l} a_m = \left|a\right| \\ b_m = \left|b\right| \\ angle\_m = \left|angle\right| \\ angle\_s = \mathsf{copysign}\left(1, angle\right) \\ \begin{array}{l} t_0 := \pi \cdot \left(angle\_m \cdot 0.011111111111111112\right)\\ t_1 := {b\_m}^{2} - {a\_m}^{2}\\ t_2 := \pi \cdot \left(angle\_m \cdot 0.005555555555555556\right)\\ angle\_s \cdot \begin{array}{l} \mathbf{if}\;\frac{angle\_m}{180} \leq 5 \cdot 10^{+21}:\\ \;\;\;\;\left(b\_m + a\_m\right) \cdot \left(\left(b\_m - a\_m\right) \cdot \sin \left(2 \cdot \left(0.005555555555555556 \cdot \left(angle\_m \cdot \pi\right)\right)\right)\right)\\ \mathbf{elif}\;\frac{angle\_m}{180} \leq 2 \cdot 10^{+105}:\\ \;\;\;\;\sin \left(\mathsf{expm1}\left(\mathsf{log1p}\left(t\_0\right)\right)\right) \cdot {\left(\mathsf{hypot}\left(b\_m, a\_m\right)\right)}^{2}\\ \mathbf{elif}\;\frac{angle\_m}{180} \leq 2 \cdot 10^{+266}:\\ \;\;\;\;\left(\left(b\_m + a\_m\right) \cdot \left(b\_m - a\_m\right)\right) \cdot \left(2 \cdot \left(\sqrt{{\sin t\_2}^{2}} \cdot {\left(\sqrt[3]{\cos t\_2}\right)}^{3}\right)\right)\\ \mathbf{elif}\;\frac{angle\_m}{180} \leq 4 \cdot 10^{+294}:\\ \;\;\;\;\sqrt{{\left(t\_1 \cdot \sin \left(\left(angle\_m \cdot 0.005555555555555556\right) \cdot \left(2 \cdot \pi\right)\right)\right)}^{2}}\\ \mathbf{else}:\\ \;\;\;\;t\_1 \cdot \sqrt{{\sin t\_0}^{2}}\\ \end{array} \end{array} \end{array} \]

FPCore

a_m = (fabs.f64 a)
b_m = (fabs.f64 b)
angle\_m = (fabs.f64 angle)
angle\_s = (copysign.f64 1 angle)
(FPCore (angle_s a_m b_m angle_m)
 :precision binary64
 (let* ((t_0 (* PI (* angle_m 0.011111111111111112)))
        (t_1 (- (pow b_m 2.0) (pow a_m 2.0)))
        (t_2 (* PI (* angle_m 0.005555555555555556))))
   (*
    angle_s
    (if (<= (/ angle_m 180.0) 5e+21)
      (*
       (+ b_m a_m)
       (* (- b_m a_m) (sin (* 2.0 (* 0.005555555555555556 (* angle_m PI))))))
      (if (<= (/ angle_m 180.0) 2e+105)
        (* (sin (expm1 (log1p t_0))) (pow (hypot b_m a_m) 2.0))
        (if (<= (/ angle_m 180.0) 2e+266)
          (*
           (* (+ b_m a_m) (- b_m a_m))
           (* 2.0 (* (sqrt (pow (sin t_2) 2.0)) (pow (cbrt (cos t_2)) 3.0))))
          (if (<= (/ angle_m 180.0) 4e+294)
            (sqrt
             (pow
              (* t_1 (sin (* (* angle_m 0.005555555555555556) (* 2.0 PI))))
              2.0))
            (* t_1 (sqrt (pow (sin t_0) 2.0))))))))))

C

a_m = fabs(a);
b_m = fabs(b);
angle\_m = fabs(angle);
angle\_s = copysign(1.0, angle);
double code(double angle_s, double a_m, double b_m, double angle_m) {
	double t_0 = ((double) M_PI) * (angle_m * 0.011111111111111112);
	double t_1 = pow(b_m, 2.0) - pow(a_m, 2.0);
	double t_2 = ((double) M_PI) * (angle_m * 0.005555555555555556);
	double tmp;
	if ((angle_m / 180.0) <= 5e+21) {
		tmp = (b_m + a_m) * ((b_m - a_m) * sin((2.0 * (0.005555555555555556 * (angle_m * ((double) M_PI))))));
	} else if ((angle_m / 180.0) <= 2e+105) {
		tmp = sin(expm1(log1p(t_0))) * pow(hypot(b_m, a_m), 2.0);
	} else if ((angle_m / 180.0) <= 2e+266) {
		tmp = ((b_m + a_m) * (b_m - a_m)) * (2.0 * (sqrt(pow(sin(t_2), 2.0)) * pow(cbrt(cos(t_2)), 3.0)));
	} else if ((angle_m / 180.0) <= 4e+294) {
		tmp = sqrt(pow((t_1 * sin(((angle_m * 0.005555555555555556) * (2.0 * ((double) M_PI))))), 2.0));
	} else {
		tmp = t_1 * sqrt(pow(sin(t_0), 2.0));
	}
	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 * 0.011111111111111112);
	double t_1 = Math.pow(b_m, 2.0) - Math.pow(a_m, 2.0);
	double t_2 = Math.PI * (angle_m * 0.005555555555555556);
	double tmp;
	if ((angle_m / 180.0) <= 5e+21) {
		tmp = (b_m + a_m) * ((b_m - a_m) * Math.sin((2.0 * (0.005555555555555556 * (angle_m * Math.PI)))));
	} else if ((angle_m / 180.0) <= 2e+105) {
		tmp = Math.sin(Math.expm1(Math.log1p(t_0))) * Math.pow(Math.hypot(b_m, a_m), 2.0);
	} else if ((angle_m / 180.0) <= 2e+266) {
		tmp = ((b_m + a_m) * (b_m - a_m)) * (2.0 * (Math.sqrt(Math.pow(Math.sin(t_2), 2.0)) * Math.pow(Math.cbrt(Math.cos(t_2)), 3.0)));
	} else if ((angle_m / 180.0) <= 4e+294) {
		tmp = Math.sqrt(Math.pow((t_1 * Math.sin(((angle_m * 0.005555555555555556) * (2.0 * Math.PI)))), 2.0));
	} else {
		tmp = t_1 * Math.sqrt(Math.pow(Math.sin(t_0), 2.0));
	}
	return angle_s * tmp;
}

Julia

a_m = abs(a)
b_m = abs(b)
angle\_m = abs(angle)
angle\_s = copysign(1.0, angle)
function code(angle_s, a_m, b_m, angle_m)
	t_0 = Float64(pi * Float64(angle_m * 0.011111111111111112))
	t_1 = Float64((b_m ^ 2.0) - (a_m ^ 2.0))
	t_2 = Float64(pi * Float64(angle_m * 0.005555555555555556))
	tmp = 0.0
	if (Float64(angle_m / 180.0) <= 5e+21)
		tmp = Float64(Float64(b_m + a_m) * Float64(Float64(b_m - a_m) * sin(Float64(2.0 * Float64(0.005555555555555556 * Float64(angle_m * pi))))));
	elseif (Float64(angle_m / 180.0) <= 2e+105)
		tmp = Float64(sin(expm1(log1p(t_0))) * (hypot(b_m, a_m) ^ 2.0));
	elseif (Float64(angle_m / 180.0) <= 2e+266)
		tmp = Float64(Float64(Float64(b_m + a_m) * Float64(b_m - a_m)) * Float64(2.0 * Float64(sqrt((sin(t_2) ^ 2.0)) * (cbrt(cos(t_2)) ^ 3.0))));
	elseif (Float64(angle_m / 180.0) <= 4e+294)
		tmp = sqrt((Float64(t_1 * sin(Float64(Float64(angle_m * 0.005555555555555556) * Float64(2.0 * pi)))) ^ 2.0));
	else
		tmp = Float64(t_1 * sqrt((sin(t_0) ^ 2.0)));
	end
	return Float64(angle_s * tmp)
end

Wolfram

a_m = N[Abs[a], $MachinePrecision]
b_m = N[Abs[b], $MachinePrecision]
angle\_m = N[Abs[angle], $MachinePrecision]
angle\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[angle]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[angle$95$s_, a$95$m_, b$95$m_, angle$95$m_] := Block[{t$95$0 = N[(Pi * N[(angle$95$m * 0.011111111111111112), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[Power[b$95$m, 2.0], $MachinePrecision] - N[Power[a$95$m, 2.0], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(Pi * N[(angle$95$m * 0.005555555555555556), $MachinePrecision]), $MachinePrecision]}, N[(angle$95$s * If[LessEqual[N[(angle$95$m / 180.0), $MachinePrecision], 5e+21], N[(N[(b$95$m + a$95$m), $MachinePrecision] * N[(N[(b$95$m - a$95$m), $MachinePrecision] * N[Sin[N[(2.0 * N[(0.005555555555555556 * N[(angle$95$m * Pi), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[N[(angle$95$m / 180.0), $MachinePrecision], 2e+105], N[(N[Sin[N[(Exp[N[Log[1 + t$95$0], $MachinePrecision]] - 1), $MachinePrecision]], $MachinePrecision] * N[Power[N[Sqrt[b$95$m ^ 2 + a$95$m ^ 2], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision], If[LessEqual[N[(angle$95$m / 180.0), $MachinePrecision], 2e+266], N[(N[(N[(b$95$m + a$95$m), $MachinePrecision] * N[(b$95$m - a$95$m), $MachinePrecision]), $MachinePrecision] * N[(2.0 * N[(N[Sqrt[N[Power[N[Sin[t$95$2], $MachinePrecision], 2.0], $MachinePrecision]], $MachinePrecision] * N[Power[N[Power[N[Cos[t$95$2], $MachinePrecision], 1/3], $MachinePrecision], 3.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[N[(angle$95$m / 180.0), $MachinePrecision], 4e+294], N[Sqrt[N[Power[N[(t$95$1 * N[Sin[N[(N[(angle$95$m * 0.005555555555555556), $MachinePrecision] * N[(2.0 * Pi), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]], $MachinePrecision], N[(t$95$1 * N[Sqrt[N[Power[N[Sin[t$95$0], $MachinePrecision], 2.0], $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]]]]), $MachinePrecision]]]]

Derivation

1. Split input into 5 regimes

2. if (/.f64 angle 180) < 5e21

   1. Initial program 59.0%

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

      1. associate-*l* 59.0%

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

      2. *-commutative 59.0%

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

      3. associate-*l* 59.5%

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

   3. Simplified 59.5%

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

      1. unpow2 59.5%

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

      2. unpow2 59.5%

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

      3. difference-of-squares 62.2%

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

   6. Applied egg-rr 62.2%

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

      1. add-cube-cbrt 62.2%

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

      2. pow3 62.2%

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

      3. div-inv 62.7%

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

      4. metadata-eval 62.7%

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

   8. Applied egg-rr 62.7%

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

      1. pow1 62.7%

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

   10. Applied egg-rr 74.5%

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

   if 5e21 < (/.f64 angle 180) < 1.9999999999999999e105

   1. Initial program 25.4%

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

      1. associate-*l* 25.4%

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

      2. *-commutative 25.4%

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

      3. associate-*l* 25.4%

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

   3. Simplified 25.4%

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

      1. unpow2 25.4%

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

      2. unpow2 25.4%

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

      3. difference-of-squares 25.4%

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

   6. Applied egg-rr 25.4%

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

      1. *-commutative 25.4%

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

      2. difference-of-squares 25.4%

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

      3. unpow2 25.4%

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

      4. unpow2 25.4%

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

      5. sub-neg 25.4%

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

      6. distribute-lft-in 25.4%

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

   8. Applied egg-rr 28.1%

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

      1. distribute-lft-out 28.1%

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

      2. associate-*l* 27.4%

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

      3. rem-square-sqrt 27.4%

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

      4. unpow2 27.4%

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

      5. unpow2 27.4%

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

      6. hypot-undefine 27.4%

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

      7. unpow2 27.4%

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

      8. unpow2 27.4%

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

      9. hypot-undefine 27.4%

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

      10. unpow2 27.4%

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

   10. Simplified 27.4%

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

       1. expm1-log1p-u 38.7%

          \[\leadsto \sin \color{blue}{\left(\mathsf{expm1}\left(\mathsf{log1p}\left(\pi \cdot \left(angle \cdot 0.011111111111111112\right)\right)\right)\right)} \cdot {\left(\mathsf{hypot}\left(b, a\right)\right)}^{2} \]

       2. expm1-undefine 31.7%

          \[\leadsto \sin \color{blue}{\left(e^{\mathsf{log1p}\left(\pi \cdot \left(angle \cdot 0.011111111111111112\right)\right)} - 1\right)} \cdot {\left(\mathsf{hypot}\left(b, a\right)\right)}^{2} \]

   12. Applied egg-rr 31.7%

       \[\leadsto \sin \color{blue}{\left(e^{\mathsf{log1p}\left(\pi \cdot \left(angle \cdot 0.011111111111111112\right)\right)} - 1\right)} \cdot {\left(\mathsf{hypot}\left(b, a\right)\right)}^{2} \]
   13. Step-by-step derivation

       1. expm1-define 38.7%

          \[\leadsto \sin \color{blue}{\left(\mathsf{expm1}\left(\mathsf{log1p}\left(\pi \cdot \left(angle \cdot 0.011111111111111112\right)\right)\right)\right)} \cdot {\left(\mathsf{hypot}\left(b, a\right)\right)}^{2} \]

   14. Simplified 38.7%

       \[\leadsto \sin \color{blue}{\left(\mathsf{expm1}\left(\mathsf{log1p}\left(\pi \cdot \left(angle \cdot 0.011111111111111112\right)\right)\right)\right)} \cdot {\left(\mathsf{hypot}\left(b, a\right)\right)}^{2} \]

   if 1.9999999999999999e105 < (/.f64 angle 180) < 2.0000000000000001e266

   1. Initial program 28.6%

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

      1. associate-*l* 28.6%

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

      2. *-commutative 28.6%

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

      3. associate-*l* 28.6%

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

   3. Simplified 28.6%

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

      1. unpow2 28.6%

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

      2. unpow2 28.6%

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

      3. difference-of-squares 28.6%

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

   6. Applied egg-rr 28.6%

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

      1. add-cube-cbrt 28.6%

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

      2. pow3 28.6%

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

      3. div-inv 29.5%

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

      4. metadata-eval 29.5%

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

   8. Applied egg-rr 29.5%

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

      1. add-sqr-sqrt 20.7%

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

      2. sqrt-unprod 45.1%

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

      3. pow2 45.1%

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

      4. div-inv 45.2%

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

      5. metadata-eval 45.2%

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

   10. Applied egg-rr 45.2%

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

   if 2.0000000000000001e266 < (/.f64 angle 180) < 4.00000000000000027e294

   1. Initial program 28.6%

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

      1. associate-*l* 28.6%

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

      2. *-commutative 28.6%

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

      3. associate-*l* 28.6%

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

   3. Simplified 28.6%

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

      1. add-sqr-sqrt 20.7%

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

      2. sqrt-unprod 21.0%

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

      3. pow2 21.0%

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

      4. 2-sin 21.0%

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

      5. associate-*r* 21.0%

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

      6. div-inv 21.0%

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

      7. metadata-eval 21.0%

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

   6. Applied egg-rr 21.0%

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

   if 4.00000000000000027e294 < (/.f64 angle 180)

   1. Initial program 3.5%

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

      1. associate-*l* 3.5%

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

      2. *-commutative 3.5%

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

      3. associate-*l* 3.5%

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

   3. Simplified 3.5%

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

      1. *-commutative 3.5%

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

      2. sub-neg 3.5%

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

      3. distribute-lft-in 3.5%

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

      4. 2-sin 3.5%

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

      5. associate-*r* 3.5%

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

      6. div-inv 3.5%

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

      7. metadata-eval 3.5%

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

   6. Applied egg-rr 3.5%

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

      1. distribute-lft-out 3.5%

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

      2. sub-neg 3.5%

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

      3. *-commutative 3.5%

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

      4. associate-*r* 3.5%

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

      5. associate-*r* 3.5%

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

      6. *-commutative 3.5%

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

      7. *-commutative 3.5%

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

      8. associate-*r* 3.5%

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

      9. metadata-eval 3.5%

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

   8. Simplified 3.5%

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

      1. add-sqr-sqrt 0.0%

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

      2. sqrt-unprod 61.4%

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

      3. pow2 61.4%

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

      4. *-commutative 61.4%

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

      5. *-commutative 61.4%

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

      6. associate-*l* 60.5%

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

   10. Applied egg-rr 60.5%

       \[\leadsto \left({b}^{2} - {a}^{2}\right) \cdot \color{blue}{\sqrt{{\sin \left(\pi \cdot \left(angle \cdot 0.011111111111111112\right)\right)}^{2}}} \]
3. Recombined 5 regimes into one program.

4. Final simplification 66.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{angle}{180} \leq 5 \cdot 10^{+21}:\\ \;\;\;\;\left(b + a\right) \cdot \left(\left(b - a\right) \cdot \sin \left(2 \cdot \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)\right)\right)\\ \mathbf{elif}\;\frac{angle}{180} \leq 2 \cdot 10^{+105}:\\ \;\;\;\;\sin \left(\mathsf{expm1}\left(\mathsf{log1p}\left(\pi \cdot \left(angle \cdot 0.011111111111111112\right)\right)\right)\right) \cdot {\left(\mathsf{hypot}\left(b, a\right)\right)}^{2}\\ \mathbf{elif}\;\frac{angle}{180} \leq 2 \cdot 10^{+266}:\\ \;\;\;\;\left(\left(b + a\right) \cdot \left(b - a\right)\right) \cdot \left(2 \cdot \left(\sqrt{{\sin \left(\pi \cdot \left(angle \cdot 0.005555555555555556\right)\right)}^{2}} \cdot {\left(\sqrt[3]{\cos \left(\pi \cdot \left(angle \cdot 0.005555555555555556\right)\right)}\right)}^{3}\right)\right)\\ \mathbf{elif}\;\frac{angle}{180} \leq 4 \cdot 10^{+294}:\\ \;\;\;\;\sqrt{{\left(\left({b}^{2} - {a}^{2}\right) \cdot \sin \left(\left(angle \cdot 0.005555555555555556\right) \cdot \left(2 \cdot \pi\right)\right)\right)}^{2}}\\ \mathbf{else}:\\ \;\;\;\;\left({b}^{2} - {a}^{2}\right) \cdot \sqrt{{\sin \left(\pi \cdot \left(angle \cdot 0.011111111111111112\right)\right)}^{2}}\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 65.3% accurate, 0.3× speedup

Math

\[\begin{array}{l} a_m = \left|a\right| \\ b_m = \left|b\right| \\ angle\_m = \left|angle\right| \\ angle\_s = \mathsf{copysign}\left(1, angle\right) \\ \begin{array}{l} t_0 := \frac{angle\_m}{180} \cdot \pi\\ t_1 := \cos t\_0\\ t_2 := \pi \cdot \left(angle\_m \cdot 0.005555555555555556\right)\\ t_3 := \sin t\_2\\ angle\_s \cdot \begin{array}{l} \mathbf{if}\;\left(\left(2 \cdot \left({b\_m}^{2} - {a\_m}^{2}\right)\right) \cdot \sin t\_0\right) \cdot t\_1 \leq 10^{+107}:\\ \;\;\;\;\mathsf{fma}\left(a\_m, \mathsf{fma}\left(a\_m \cdot -2, t\_3 \cdot \cos t\_2, 0\right), 2 \cdot \left({b\_m}^{2} \cdot \left(t\_3 \cdot \cos \left({\left(\sqrt{t\_2}\right)}^{2}\right)\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \left(t\_1 \cdot \left(\mathsf{hypot}\left(b\_m, a\_m\right) \cdot \left(\mathsf{hypot}\left(b\_m, a\_m\right) \cdot t\_3\right)\right)\right)\\ \end{array} \end{array} \end{array} \]

FPCore

a_m = (fabs.f64 a)
b_m = (fabs.f64 b)
angle\_m = (fabs.f64 angle)
angle\_s = (copysign.f64 1 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 0.005555555555555556)))
        (t_3 (sin t_2)))
   (*
    angle_s
    (if (<=
         (* (* (* 2.0 (- (pow b_m 2.0) (pow a_m 2.0))) (sin t_0)) t_1)
         1e+107)
      (fma
       a_m
       (fma (* a_m -2.0) (* t_3 (cos t_2)) 0.0)
       (* 2.0 (* (pow b_m 2.0) (* t_3 (cos (pow (sqrt t_2) 2.0))))))
      (* 2.0 (* t_1 (* (hypot b_m a_m) (* (hypot b_m a_m) t_3))))))))

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 * 0.005555555555555556);
	double t_3 = sin(t_2);
	double tmp;
	if ((((2.0 * (pow(b_m, 2.0) - pow(a_m, 2.0))) * sin(t_0)) * t_1) <= 1e+107) {
		tmp = fma(a_m, fma((a_m * -2.0), (t_3 * cos(t_2)), 0.0), (2.0 * (pow(b_m, 2.0) * (t_3 * cos(pow(sqrt(t_2), 2.0))))));
	} else {
		tmp = 2.0 * (t_1 * (hypot(b_m, a_m) * (hypot(b_m, a_m) * t_3)));
	}
	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 * 0.005555555555555556))
	t_3 = sin(t_2)
	tmp = 0.0
	if (Float64(Float64(Float64(2.0 * Float64((b_m ^ 2.0) - (a_m ^ 2.0))) * sin(t_0)) * t_1) <= 1e+107)
		tmp = fma(a_m, fma(Float64(a_m * -2.0), Float64(t_3 * cos(t_2)), 0.0), Float64(2.0 * Float64((b_m ^ 2.0) * Float64(t_3 * cos((sqrt(t_2) ^ 2.0))))));
	else
		tmp = Float64(2.0 * Float64(t_1 * Float64(hypot(b_m, a_m) * Float64(hypot(b_m, a_m) * t_3))));
	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 * 0.005555555555555556), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$3 = N[Sin[t$95$2], $MachinePrecision]}, N[(angle$95$s * If[LessEqual[N[(N[(N[(2.0 * N[(N[Power[b$95$m, 2.0], $MachinePrecision] - N[Power[a$95$m, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[Sin[t$95$0], $MachinePrecision]), $MachinePrecision] * t$95$1), $MachinePrecision], 1e+107], N[(a$95$m * N[(N[(a$95$m * -2.0), $MachinePrecision] * N[(t$95$3 * N[Cos[t$95$2], $MachinePrecision]), $MachinePrecision] + 0.0), $MachinePrecision] + N[(2.0 * N[(N[Power[b$95$m, 2.0], $MachinePrecision] * N[(t$95$3 * N[Cos[N[Power[N[Sqrt[t$95$2], $MachinePrecision], 2.0], $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(2.0 * N[(t$95$1 * N[(N[Sqrt[b$95$m ^ 2 + a$95$m ^ 2], $MachinePrecision] * N[(N[Sqrt[b$95$m ^ 2 + a$95$m ^ 2], $MachinePrecision] * t$95$3), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]]]]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 (*.f64 (*.f64 2 (-.f64 (pow.f64 b 2) (pow.f64 a 2))) (sin.f64 (*.f64 (PI.f64) (/.f64 angle 180)))) (cos.f64 (*.f64 (PI.f64) (/.f64 angle 180)))) < 9.9999999999999997e106

   1. Initial program 55.0%

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

      1. associate-*l* 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%

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

   7. Taylor expanded in a around 0 60.2%

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

      1. +-commutative 60.2%

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

      2. fma-define 60.2%

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

   9. Simplified 59.4%

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

       1. add-sqr-sqrt 36.3%

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

       2. pow2 36.3%

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

   11. Applied egg-rr 36.3%

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

   if 9.9999999999999997e106 < (*.f64 (*.f64 (*.f64 2 (-.f64 (pow.f64 b 2) (pow.f64 a 2))) (sin.f64 (*.f64 (PI.f64) (/.f64 angle 180)))) (cos.f64 (*.f64 (PI.f64) (/.f64 angle 180))))

   1. Initial program 40.4%

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

      1. associate-*l* 41.6%

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

      2. associate-*l* 41.6%

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

   3. Simplified 41.6%

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

      1. add-sqr-sqrt 29.5%

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

      2. sqrt-unprod 38.6%

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

      3. pow2 38.6%

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

      4. *-commutative 38.6%

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

      5. div-inv 38.5%

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

      6. metadata-eval 38.5%

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

   6. Applied egg-rr 38.5%

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

      1. sqrt-pow1 44.6%

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

      2. metadata-eval 44.6%

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

      3. pow1 44.6%

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

      4. add-sqr-sqrt 27.2%

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

      5. associate-*r* 27.2%

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

   8. Applied egg-rr 47.2%

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

4. Final simplification 39.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right) \cdot \cos \left(\frac{angle}{180} \cdot \pi\right) \leq 10^{+107}:\\ \;\;\;\;\mathsf{fma}\left(a, \mathsf{fma}\left(a \cdot -2, \sin \left(\pi \cdot \left(angle \cdot 0.005555555555555556\right)\right) \cdot \cos \left(\pi \cdot \left(angle \cdot 0.005555555555555556\right)\right), 0\right), 2 \cdot \left({b}^{2} \cdot \left(\sin \left(\pi \cdot \left(angle \cdot 0.005555555555555556\right)\right) \cdot \cos \left({\left(\sqrt{\pi \cdot \left(angle \cdot 0.005555555555555556\right)}\right)}^{2}\right)\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \left(\cos \left(\frac{angle}{180} \cdot \pi\right) \cdot \left(\mathsf{hypot}\left(b, a\right) \cdot \left(\mathsf{hypot}\left(b, a\right) \cdot \sin \left(\pi \cdot \left(angle \cdot 0.005555555555555556\right)\right)\right)\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 67.5% accurate, 0.4× speedup

Math

\[\begin{array}{l} a_m = \left|a\right| \\ b_m = \left|b\right| \\ angle\_m = \left|angle\right| \\ angle\_s = \mathsf{copysign}\left(1, angle\right) \\ \begin{array}{l} t_0 := \pi \cdot \left(angle\_m \cdot 0.005555555555555556\right)\\ t_1 := \sin t\_0\\ angle\_s \cdot \begin{array}{l} \mathbf{if}\;\frac{angle\_m}{180} \leq 5 \cdot 10^{+21}:\\ \;\;\;\;\left(b\_m + a\_m\right) \cdot \left(\left(b\_m - a\_m\right) \cdot \sin \left(2 \cdot \left(0.005555555555555556 \cdot \left(angle\_m \cdot \pi\right)\right)\right)\right)\\ \mathbf{elif}\;\frac{angle\_m}{180} \leq 5 \cdot 10^{+146}:\\ \;\;\;\;\sin \left(\mathsf{expm1}\left(\mathsf{log1p}\left(\pi \cdot \left(angle\_m \cdot 0.011111111111111112\right)\right)\right)\right) \cdot {\left(\mathsf{hypot}\left(b\_m, a\_m\right)\right)}^{2}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(a\_m, \mathsf{fma}\left(a\_m \cdot -2, \cos \left({\left(\sqrt[3]{t\_0}\right)}^{3}\right) \cdot t\_1, 0\right), 2 \cdot \left(\left(t\_1 \cdot \cos \left(\left(angle\_m \cdot 0.005555555555555556\right) \cdot \sqrt[3]{{\pi}^{3}}\right)\right) \cdot {b\_m}^{2}\right)\right)\\ \end{array} \end{array} \end{array} \]

FPCore

a_m = (fabs.f64 a)
b_m = (fabs.f64 b)
angle\_m = (fabs.f64 angle)
angle\_s = (copysign.f64 1 angle)
(FPCore (angle_s a_m b_m angle_m)
 :precision binary64
 (let* ((t_0 (* PI (* angle_m 0.005555555555555556))) (t_1 (sin t_0)))
   (*
    angle_s
    (if (<= (/ angle_m 180.0) 5e+21)
      (*
       (+ b_m a_m)
       (* (- b_m a_m) (sin (* 2.0 (* 0.005555555555555556 (* angle_m PI))))))
      (if (<= (/ angle_m 180.0) 5e+146)
        (*
         (sin (expm1 (log1p (* PI (* angle_m 0.011111111111111112)))))
         (pow (hypot b_m a_m) 2.0))
        (fma
         a_m
         (fma (* a_m -2.0) (* (cos (pow (cbrt t_0) 3.0)) t_1) 0.0)
         (*
          2.0
          (*
           (*
            t_1
            (cos (* (* angle_m 0.005555555555555556) (cbrt (pow PI 3.0)))))
           (pow b_m 2.0)))))))))

C

a_m = fabs(a);
b_m = fabs(b);
angle\_m = fabs(angle);
angle\_s = copysign(1.0, angle);
double code(double angle_s, double a_m, double b_m, double angle_m) {
	double t_0 = ((double) M_PI) * (angle_m * 0.005555555555555556);
	double t_1 = sin(t_0);
	double tmp;
	if ((angle_m / 180.0) <= 5e+21) {
		tmp = (b_m + a_m) * ((b_m - a_m) * sin((2.0 * (0.005555555555555556 * (angle_m * ((double) M_PI))))));
	} else if ((angle_m / 180.0) <= 5e+146) {
		tmp = sin(expm1(log1p((((double) M_PI) * (angle_m * 0.011111111111111112))))) * pow(hypot(b_m, a_m), 2.0);
	} else {
		tmp = fma(a_m, fma((a_m * -2.0), (cos(pow(cbrt(t_0), 3.0)) * t_1), 0.0), (2.0 * ((t_1 * cos(((angle_m * 0.005555555555555556) * cbrt(pow(((double) M_PI), 3.0))))) * pow(b_m, 2.0))));
	}
	return angle_s * tmp;
}

Julia

a_m = abs(a)
b_m = abs(b)
angle\_m = abs(angle)
angle\_s = copysign(1.0, angle)
function code(angle_s, a_m, b_m, angle_m)
	t_0 = Float64(pi * Float64(angle_m * 0.005555555555555556))
	t_1 = sin(t_0)
	tmp = 0.0
	if (Float64(angle_m / 180.0) <= 5e+21)
		tmp = Float64(Float64(b_m + a_m) * Float64(Float64(b_m - a_m) * sin(Float64(2.0 * Float64(0.005555555555555556 * Float64(angle_m * pi))))));
	elseif (Float64(angle_m / 180.0) <= 5e+146)
		tmp = Float64(sin(expm1(log1p(Float64(pi * Float64(angle_m * 0.011111111111111112))))) * (hypot(b_m, a_m) ^ 2.0));
	else
		tmp = fma(a_m, fma(Float64(a_m * -2.0), Float64(cos((cbrt(t_0) ^ 3.0)) * t_1), 0.0), Float64(2.0 * Float64(Float64(t_1 * cos(Float64(Float64(angle_m * 0.005555555555555556) * cbrt((pi ^ 3.0))))) * (b_m ^ 2.0))));
	end
	return Float64(angle_s * tmp)
end

Wolfram

a_m = N[Abs[a], $MachinePrecision]
b_m = N[Abs[b], $MachinePrecision]
angle\_m = N[Abs[angle], $MachinePrecision]
angle\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[angle]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[angle$95$s_, a$95$m_, b$95$m_, angle$95$m_] := Block[{t$95$0 = N[(Pi * N[(angle$95$m * 0.005555555555555556), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[Sin[t$95$0], $MachinePrecision]}, N[(angle$95$s * If[LessEqual[N[(angle$95$m / 180.0), $MachinePrecision], 5e+21], N[(N[(b$95$m + a$95$m), $MachinePrecision] * N[(N[(b$95$m - a$95$m), $MachinePrecision] * N[Sin[N[(2.0 * N[(0.005555555555555556 * N[(angle$95$m * Pi), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[N[(angle$95$m / 180.0), $MachinePrecision], 5e+146], N[(N[Sin[N[(Exp[N[Log[1 + N[(Pi * N[(angle$95$m * 0.011111111111111112), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]] - 1), $MachinePrecision]], $MachinePrecision] * N[Power[N[Sqrt[b$95$m ^ 2 + a$95$m ^ 2], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision], N[(a$95$m * N[(N[(a$95$m * -2.0), $MachinePrecision] * N[(N[Cos[N[Power[N[Power[t$95$0, 1/3], $MachinePrecision], 3.0], $MachinePrecision]], $MachinePrecision] * t$95$1), $MachinePrecision] + 0.0), $MachinePrecision] + N[(2.0 * N[(N[(t$95$1 * N[Cos[N[(N[(angle$95$m * 0.005555555555555556), $MachinePrecision] * N[Power[N[Power[Pi, 3.0], $MachinePrecision], 1/3], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[Power[b$95$m, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if (/.f64 angle 180) < 5e21

   1. Initial program 59.0%

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

      1. associate-*l* 59.0%

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

      2. *-commutative 59.0%

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

      3. associate-*l* 59.5%

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

   3. Simplified 59.5%

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

      1. unpow2 59.5%

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

      2. unpow2 59.5%

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

      3. difference-of-squares 62.2%

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

   6. Applied egg-rr 62.2%

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

      1. add-cube-cbrt 62.2%

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

      2. pow3 62.2%

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

      3. div-inv 62.7%

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

      4. metadata-eval 62.7%

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

   8. Applied egg-rr 62.7%

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

      1. pow1 62.7%

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

   10. Applied egg-rr 74.5%

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

   if 5e21 < (/.f64 angle 180) < 4.9999999999999999e146

   1. Initial program 32.3%

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

      1. associate-*l* 32.3%

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

      2. *-commutative 32.3%

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

      3. associate-*l* 32.3%

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

   3. Simplified 32.3%

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

      1. unpow2 32.3%

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

      2. unpow2 32.3%

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

      3. difference-of-squares 32.3%

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

   6. Applied egg-rr 32.3%

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

      1. *-commutative 32.3%

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

      2. difference-of-squares 32.3%

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

      3. unpow2 32.3%

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

      4. unpow2 32.3%

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

      5. sub-neg 32.3%

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

      6. distribute-lft-in 32.3%

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

   8. Applied egg-rr 26.1%

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

      1. distribute-lft-out 26.1%

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

      2. associate-*l* 25.6%

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

      3. rem-square-sqrt 25.6%

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

      4. unpow2 25.6%

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

      5. unpow2 25.6%

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

      6. hypot-undefine 25.6%

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

      7. unpow2 25.6%

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

      8. unpow2 25.6%

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

      9. hypot-undefine 25.6%

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

      10. unpow2 25.6%

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

   10. Simplified 25.6%

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

       1. expm1-log1p-u 46.1%

          \[\leadsto \sin \color{blue}{\left(\mathsf{expm1}\left(\mathsf{log1p}\left(\pi \cdot \left(angle \cdot 0.011111111111111112\right)\right)\right)\right)} \cdot {\left(\mathsf{hypot}\left(b, a\right)\right)}^{2} \]

       2. expm1-undefine 41.0%

          \[\leadsto \sin \color{blue}{\left(e^{\mathsf{log1p}\left(\pi \cdot \left(angle \cdot 0.011111111111111112\right)\right)} - 1\right)} \cdot {\left(\mathsf{hypot}\left(b, a\right)\right)}^{2} \]

   12. Applied egg-rr 41.0%

       \[\leadsto \sin \color{blue}{\left(e^{\mathsf{log1p}\left(\pi \cdot \left(angle \cdot 0.011111111111111112\right)\right)} - 1\right)} \cdot {\left(\mathsf{hypot}\left(b, a\right)\right)}^{2} \]
   13. Step-by-step derivation

       1. expm1-define 46.1%

          \[\leadsto \sin \color{blue}{\left(\mathsf{expm1}\left(\mathsf{log1p}\left(\pi \cdot \left(angle \cdot 0.011111111111111112\right)\right)\right)\right)} \cdot {\left(\mathsf{hypot}\left(b, a\right)\right)}^{2} \]

   14. Simplified 46.1%

       \[\leadsto \sin \color{blue}{\left(\mathsf{expm1}\left(\mathsf{log1p}\left(\pi \cdot \left(angle \cdot 0.011111111111111112\right)\right)\right)\right)} \cdot {\left(\mathsf{hypot}\left(b, a\right)\right)}^{2} \]

   if 4.9999999999999999e146 < (/.f64 angle 180)

   1. Initial program 22.0%

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

      1. associate-*l* 22.0%

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

      2. *-commutative 22.0%

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

      3. associate-*l* 22.0%

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

   3. Simplified 22.0%

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

      1. unpow2 22.0%

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

      2. unpow2 22.0%

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

      3. difference-of-squares 22.0%

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

   6. Applied egg-rr 22.0%

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

   7. Taylor expanded in a around 0 22.0%

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

      1. +-commutative 22.0%

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

      2. fma-define 22.0%

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

   9. Simplified 26.0%

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

       1. add-cube-cbrt 38.3%

          \[\leadsto \mathsf{fma}\left(a, \mathsf{fma}\left(-2 \cdot a, \cos \color{blue}{\left(\left(\sqrt[3]{\pi \cdot \left(0.005555555555555556 \cdot angle\right)} \cdot \sqrt[3]{\pi \cdot \left(0.005555555555555556 \cdot angle\right)}\right) \cdot \sqrt[3]{\pi \cdot \left(0.005555555555555556 \cdot angle\right)}\right)} \cdot \sin \left(\pi \cdot \left(0.005555555555555556 \cdot angle\right)\right), 0\right), 2 \cdot \left(\left(\cos \left(\pi \cdot \left(0.005555555555555556 \cdot angle\right)\right) \cdot \sin \left(\pi \cdot \left(0.005555555555555556 \cdot angle\right)\right)\right) \cdot {b}^{2}\right)\right) \]

       2. pow3 38.3%

          \[\leadsto \mathsf{fma}\left(a, \mathsf{fma}\left(-2 \cdot a, \cos \color{blue}{\left({\left(\sqrt[3]{\pi \cdot \left(0.005555555555555556 \cdot angle\right)}\right)}^{3}\right)} \cdot \sin \left(\pi \cdot \left(0.005555555555555556 \cdot angle\right)\right), 0\right), 2 \cdot \left(\left(\cos \left(\pi \cdot \left(0.005555555555555556 \cdot angle\right)\right) \cdot \sin \left(\pi \cdot \left(0.005555555555555556 \cdot angle\right)\right)\right) \cdot {b}^{2}\right)\right) \]

   11. Applied egg-rr 38.3%

       \[\leadsto \mathsf{fma}\left(a, \mathsf{fma}\left(-2 \cdot a, \cos \color{blue}{\left({\left(\sqrt[3]{\pi \cdot \left(0.005555555555555556 \cdot angle\right)}\right)}^{3}\right)} \cdot \sin \left(\pi \cdot \left(0.005555555555555556 \cdot angle\right)\right), 0\right), 2 \cdot \left(\left(\cos \left(\pi \cdot \left(0.005555555555555556 \cdot angle\right)\right) \cdot \sin \left(\pi \cdot \left(0.005555555555555556 \cdot angle\right)\right)\right) \cdot {b}^{2}\right)\right) \]
   12. Step-by-step derivation

       1. add-cbrt-cube 33.9%

          \[\leadsto \mathsf{fma}\left(a, \mathsf{fma}\left(-2 \cdot a, \cos \left({\left(\sqrt[3]{\pi \cdot \left(0.005555555555555556 \cdot angle\right)}\right)}^{3}\right) \cdot \sin \left(\pi \cdot \left(0.005555555555555556 \cdot angle\right)\right), 0\right), 2 \cdot \left(\left(\cos \left(\color{blue}{\sqrt[3]{\left(\pi \cdot \pi\right) \cdot \pi}} \cdot \left(0.005555555555555556 \cdot angle\right)\right) \cdot \sin \left(\pi \cdot \left(0.005555555555555556 \cdot angle\right)\right)\right) \cdot {b}^{2}\right)\right) \]

       2. pow3 33.9%

          \[\leadsto \mathsf{fma}\left(a, \mathsf{fma}\left(-2 \cdot a, \cos \left({\left(\sqrt[3]{\pi \cdot \left(0.005555555555555556 \cdot angle\right)}\right)}^{3}\right) \cdot \sin \left(\pi \cdot \left(0.005555555555555556 \cdot angle\right)\right), 0\right), 2 \cdot \left(\left(\cos \left(\sqrt[3]{\color{blue}{{\pi}^{3}}} \cdot \left(0.005555555555555556 \cdot angle\right)\right) \cdot \sin \left(\pi \cdot \left(0.005555555555555556 \cdot angle\right)\right)\right) \cdot {b}^{2}\right)\right) \]

   13. Applied egg-rr 33.9%

       \[\leadsto \mathsf{fma}\left(a, \mathsf{fma}\left(-2 \cdot a, \cos \left({\left(\sqrt[3]{\pi \cdot \left(0.005555555555555556 \cdot angle\right)}\right)}^{3}\right) \cdot \sin \left(\pi \cdot \left(0.005555555555555556 \cdot angle\right)\right), 0\right), 2 \cdot \left(\left(\cos \left(\color{blue}{\sqrt[3]{{\pi}^{3}}} \cdot \left(0.005555555555555556 \cdot angle\right)\right) \cdot \sin \left(\pi \cdot \left(0.005555555555555556 \cdot angle\right)\right)\right) \cdot {b}^{2}\right)\right) \]
3. Recombined 3 regimes into one program.

4. Final simplification 65.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{angle}{180} \leq 5 \cdot 10^{+21}:\\ \;\;\;\;\left(b + a\right) \cdot \left(\left(b - a\right) \cdot \sin \left(2 \cdot \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)\right)\right)\\ \mathbf{elif}\;\frac{angle}{180} \leq 5 \cdot 10^{+146}:\\ \;\;\;\;\sin \left(\mathsf{expm1}\left(\mathsf{log1p}\left(\pi \cdot \left(angle \cdot 0.011111111111111112\right)\right)\right)\right) \cdot {\left(\mathsf{hypot}\left(b, a\right)\right)}^{2}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(a, \mathsf{fma}\left(a \cdot -2, \cos \left({\left(\sqrt[3]{\pi \cdot \left(angle \cdot 0.005555555555555556\right)}\right)}^{3}\right) \cdot \sin \left(\pi \cdot \left(angle \cdot 0.005555555555555556\right)\right), 0\right), 2 \cdot \left(\left(\sin \left(\pi \cdot \left(angle \cdot 0.005555555555555556\right)\right) \cdot \cos \left(\left(angle \cdot 0.005555555555555556\right) \cdot \sqrt[3]{{\pi}^{3}}\right)\right) \cdot {b}^{2}\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 66.9% accurate, 0.7× speedup

Math

\[\begin{array}{l} a_m = \left|a\right| \\ b_m = \left|b\right| \\ angle\_m = \left|angle\right| \\ angle\_s = \mathsf{copysign}\left(1, angle\right) \\ \begin{array}{l} t_0 := \sin \left(\pi \cdot \left(angle\_m \cdot 0.005555555555555556\right)\right)\\ t_1 := \left(b\_m + a\_m\right) \cdot \left(b\_m - a\_m\right)\\ t_2 := \frac{angle\_m}{180} \cdot \pi\\ t_3 := \cos t\_2\\ angle\_s \cdot \begin{array}{l} \mathbf{if}\;\frac{angle\_m}{180} \leq 5 \cdot 10^{+21}:\\ \;\;\;\;\left(b\_m + a\_m\right) \cdot \left(\left(b\_m - a\_m\right) \cdot \sin \left(2 \cdot \left(0.005555555555555556 \cdot \left(angle\_m \cdot \pi\right)\right)\right)\right)\\ \mathbf{elif}\;\frac{angle\_m}{180} \leq 2 \cdot 10^{+105}:\\ \;\;\;\;\sin \left(\mathsf{expm1}\left(\mathsf{log1p}\left(\pi \cdot \left(angle\_m \cdot 0.011111111111111112\right)\right)\right)\right) \cdot {\left(\mathsf{hypot}\left(b\_m, a\_m\right)\right)}^{2}\\ \mathbf{elif}\;\frac{angle\_m}{180} \leq 5 \cdot 10^{+264}:\\ \;\;\;\;2 \cdot \left(t\_3 \cdot \left(\left({b\_m}^{2} - {a\_m}^{2}\right) \cdot \sqrt{{t\_0}^{2}}\right)\right)\\ \mathbf{elif}\;\frac{angle\_m}{180} \leq 5 \cdot 10^{+281}:\\ \;\;\;\;t\_1 \cdot \left(2 \cdot \left(\sin t\_2 \cdot \cos \left(\frac{angle\_m}{180} \cdot {\left(\sqrt{\pi}\right)}^{2}\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \left(t\_3 \cdot \sqrt{{\left(t\_0 \cdot t\_1\right)}^{2}}\right)\\ \end{array} \end{array} \end{array} \]

FPCore

a_m = (fabs.f64 a)
b_m = (fabs.f64 b)
angle\_m = (fabs.f64 angle)
angle\_s = (copysign.f64 1 angle)
(FPCore (angle_s a_m b_m angle_m)
 :precision binary64
 (let* ((t_0 (sin (* PI (* angle_m 0.005555555555555556))))
        (t_1 (* (+ b_m a_m) (- b_m a_m)))
        (t_2 (* (/ angle_m 180.0) PI))
        (t_3 (cos t_2)))
   (*
    angle_s
    (if (<= (/ angle_m 180.0) 5e+21)
      (*
       (+ b_m a_m)
       (* (- b_m a_m) (sin (* 2.0 (* 0.005555555555555556 (* angle_m PI))))))
      (if (<= (/ angle_m 180.0) 2e+105)
        (*
         (sin (expm1 (log1p (* PI (* angle_m 0.011111111111111112)))))
         (pow (hypot b_m a_m) 2.0))
        (if (<= (/ angle_m 180.0) 5e+264)
          (*
           2.0
           (* t_3 (* (- (pow b_m 2.0) (pow a_m 2.0)) (sqrt (pow t_0 2.0)))))
          (if (<= (/ angle_m 180.0) 5e+281)
            (*
             t_1
             (*
              2.0
              (* (sin t_2) (cos (* (/ angle_m 180.0) (pow (sqrt PI) 2.0))))))
            (* 2.0 (* t_3 (sqrt (pow (* t_0 t_1) 2.0)))))))))))

C

a_m = fabs(a);
b_m = fabs(b);
angle\_m = fabs(angle);
angle\_s = copysign(1.0, angle);
double code(double angle_s, double a_m, double b_m, double angle_m) {
	double t_0 = sin((((double) M_PI) * (angle_m * 0.005555555555555556)));
	double t_1 = (b_m + a_m) * (b_m - a_m);
	double t_2 = (angle_m / 180.0) * ((double) M_PI);
	double t_3 = cos(t_2);
	double tmp;
	if ((angle_m / 180.0) <= 5e+21) {
		tmp = (b_m + a_m) * ((b_m - a_m) * sin((2.0 * (0.005555555555555556 * (angle_m * ((double) M_PI))))));
	} else if ((angle_m / 180.0) <= 2e+105) {
		tmp = sin(expm1(log1p((((double) M_PI) * (angle_m * 0.011111111111111112))))) * pow(hypot(b_m, a_m), 2.0);
	} else if ((angle_m / 180.0) <= 5e+264) {
		tmp = 2.0 * (t_3 * ((pow(b_m, 2.0) - pow(a_m, 2.0)) * sqrt(pow(t_0, 2.0))));
	} else if ((angle_m / 180.0) <= 5e+281) {
		tmp = t_1 * (2.0 * (sin(t_2) * cos(((angle_m / 180.0) * pow(sqrt(((double) M_PI)), 2.0)))));
	} else {
		tmp = 2.0 * (t_3 * sqrt(pow((t_0 * t_1), 2.0)));
	}
	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.sin((Math.PI * (angle_m * 0.005555555555555556)));
	double t_1 = (b_m + a_m) * (b_m - a_m);
	double t_2 = (angle_m / 180.0) * Math.PI;
	double t_3 = Math.cos(t_2);
	double tmp;
	if ((angle_m / 180.0) <= 5e+21) {
		tmp = (b_m + a_m) * ((b_m - a_m) * Math.sin((2.0 * (0.005555555555555556 * (angle_m * Math.PI)))));
	} else if ((angle_m / 180.0) <= 2e+105) {
		tmp = Math.sin(Math.expm1(Math.log1p((Math.PI * (angle_m * 0.011111111111111112))))) * Math.pow(Math.hypot(b_m, a_m), 2.0);
	} else if ((angle_m / 180.0) <= 5e+264) {
		tmp = 2.0 * (t_3 * ((Math.pow(b_m, 2.0) - Math.pow(a_m, 2.0)) * Math.sqrt(Math.pow(t_0, 2.0))));
	} else if ((angle_m / 180.0) <= 5e+281) {
		tmp = t_1 * (2.0 * (Math.sin(t_2) * Math.cos(((angle_m / 180.0) * Math.pow(Math.sqrt(Math.PI), 2.0)))));
	} else {
		tmp = 2.0 * (t_3 * Math.sqrt(Math.pow((t_0 * t_1), 2.0)));
	}
	return angle_s * tmp;
}

Python

a_m = math.fabs(a)
b_m = math.fabs(b)
angle\_m = math.fabs(angle)
angle\_s = math.copysign(1.0, angle)
def code(angle_s, a_m, b_m, angle_m):
	t_0 = math.sin((math.pi * (angle_m * 0.005555555555555556)))
	t_1 = (b_m + a_m) * (b_m - a_m)
	t_2 = (angle_m / 180.0) * math.pi
	t_3 = math.cos(t_2)
	tmp = 0
	if (angle_m / 180.0) <= 5e+21:
		tmp = (b_m + a_m) * ((b_m - a_m) * math.sin((2.0 * (0.005555555555555556 * (angle_m * math.pi)))))
	elif (angle_m / 180.0) <= 2e+105:
		tmp = math.sin(math.expm1(math.log1p((math.pi * (angle_m * 0.011111111111111112))))) * math.pow(math.hypot(b_m, a_m), 2.0)
	elif (angle_m / 180.0) <= 5e+264:
		tmp = 2.0 * (t_3 * ((math.pow(b_m, 2.0) - math.pow(a_m, 2.0)) * math.sqrt(math.pow(t_0, 2.0))))
	elif (angle_m / 180.0) <= 5e+281:
		tmp = t_1 * (2.0 * (math.sin(t_2) * math.cos(((angle_m / 180.0) * math.pow(math.sqrt(math.pi), 2.0)))))
	else:
		tmp = 2.0 * (t_3 * math.sqrt(math.pow((t_0 * t_1), 2.0)))
	return angle_s * tmp

Julia

a_m = abs(a)
b_m = abs(b)
angle\_m = abs(angle)
angle\_s = copysign(1.0, angle)
function code(angle_s, a_m, b_m, angle_m)
	t_0 = sin(Float64(pi * Float64(angle_m * 0.005555555555555556)))
	t_1 = Float64(Float64(b_m + a_m) * Float64(b_m - a_m))
	t_2 = Float64(Float64(angle_m / 180.0) * pi)
	t_3 = cos(t_2)
	tmp = 0.0
	if (Float64(angle_m / 180.0) <= 5e+21)
		tmp = Float64(Float64(b_m + a_m) * Float64(Float64(b_m - a_m) * sin(Float64(2.0 * Float64(0.005555555555555556 * Float64(angle_m * pi))))));
	elseif (Float64(angle_m / 180.0) <= 2e+105)
		tmp = Float64(sin(expm1(log1p(Float64(pi * Float64(angle_m * 0.011111111111111112))))) * (hypot(b_m, a_m) ^ 2.0));
	elseif (Float64(angle_m / 180.0) <= 5e+264)
		tmp = Float64(2.0 * Float64(t_3 * Float64(Float64((b_m ^ 2.0) - (a_m ^ 2.0)) * sqrt((t_0 ^ 2.0)))));
	elseif (Float64(angle_m / 180.0) <= 5e+281)
		tmp = Float64(t_1 * Float64(2.0 * Float64(sin(t_2) * cos(Float64(Float64(angle_m / 180.0) * (sqrt(pi) ^ 2.0))))));
	else
		tmp = Float64(2.0 * Float64(t_3 * sqrt((Float64(t_0 * t_1) ^ 2.0))));
	end
	return Float64(angle_s * tmp)
end

Wolfram

a_m = N[Abs[a], $MachinePrecision]
b_m = N[Abs[b], $MachinePrecision]
angle\_m = N[Abs[angle], $MachinePrecision]
angle\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[angle]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[angle$95$s_, a$95$m_, b$95$m_, angle$95$m_] := Block[{t$95$0 = N[Sin[N[(Pi * N[(angle$95$m * 0.005555555555555556), $MachinePrecision]), $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[(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], 5e+21], N[(N[(b$95$m + a$95$m), $MachinePrecision] * N[(N[(b$95$m - a$95$m), $MachinePrecision] * N[Sin[N[(2.0 * N[(0.005555555555555556 * N[(angle$95$m * Pi), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[N[(angle$95$m / 180.0), $MachinePrecision], 2e+105], N[(N[Sin[N[(Exp[N[Log[1 + N[(Pi * N[(angle$95$m * 0.011111111111111112), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]] - 1), $MachinePrecision]], $MachinePrecision] * N[Power[N[Sqrt[b$95$m ^ 2 + a$95$m ^ 2], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision], If[LessEqual[N[(angle$95$m / 180.0), $MachinePrecision], 5e+264], N[(2.0 * N[(t$95$3 * N[(N[(N[Power[b$95$m, 2.0], $MachinePrecision] - N[Power[a$95$m, 2.0], $MachinePrecision]), $MachinePrecision] * N[Sqrt[N[Power[t$95$0, 2.0], $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[N[(angle$95$m / 180.0), $MachinePrecision], 5e+281], N[(t$95$1 * N[(2.0 * N[(N[Sin[t$95$2], $MachinePrecision] * N[Cos[N[(N[(angle$95$m / 180.0), $MachinePrecision] * N[Power[N[Sqrt[Pi], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(2.0 * N[(t$95$3 * N[Sqrt[N[Power[N[(t$95$0 * t$95$1), $MachinePrecision], 2.0], $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]), $MachinePrecision]]]]]

Derivation

1. Split input into 5 regimes

2. if (/.f64 angle 180) < 5e21

   1. Initial program 59.0%

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

      1. associate-*l* 59.0%

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

      2. *-commutative 59.0%

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

      3. associate-*l* 59.5%

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

   3. Simplified 59.5%

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

      1. unpow2 59.5%

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

      2. unpow2 59.5%

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

      3. difference-of-squares 62.2%

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

   6. Applied egg-rr 62.2%

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

      1. add-cube-cbrt 62.2%

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

      2. pow3 62.2%

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

      3. div-inv 62.7%

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

      4. metadata-eval 62.7%

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

   8. Applied egg-rr 62.7%

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

      1. pow1 62.7%

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

   10. Applied egg-rr 74.5%

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

   if 5e21 < (/.f64 angle 180) < 1.9999999999999999e105

   1. Initial program 25.4%

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

      1. associate-*l* 25.4%

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

      2. *-commutative 25.4%

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

      3. associate-*l* 25.4%

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

   3. Simplified 25.4%

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

      1. unpow2 25.4%

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

      2. unpow2 25.4%

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

      3. difference-of-squares 25.4%

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

   6. Applied egg-rr 25.4%

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

      1. *-commutative 25.4%

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

      2. difference-of-squares 25.4%

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

      3. unpow2 25.4%

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

      4. unpow2 25.4%

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

      5. sub-neg 25.4%

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

      6. distribute-lft-in 25.4%

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

   8. Applied egg-rr 28.1%

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

      1. distribute-lft-out 28.1%

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

      2. associate-*l* 27.4%

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

      3. rem-square-sqrt 27.4%

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

      4. unpow2 27.4%

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

      5. unpow2 27.4%

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

      6. hypot-undefine 27.4%

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

      7. unpow2 27.4%

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

      8. unpow2 27.4%

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

      9. hypot-undefine 27.4%

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

      10. unpow2 27.4%

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

   10. Simplified 27.4%

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

       1. expm1-log1p-u 38.7%

          \[\leadsto \sin \color{blue}{\left(\mathsf{expm1}\left(\mathsf{log1p}\left(\pi \cdot \left(angle \cdot 0.011111111111111112\right)\right)\right)\right)} \cdot {\left(\mathsf{hypot}\left(b, a\right)\right)}^{2} \]

       2. expm1-undefine 31.7%

          \[\leadsto \sin \color{blue}{\left(e^{\mathsf{log1p}\left(\pi \cdot \left(angle \cdot 0.011111111111111112\right)\right)} - 1\right)} \cdot {\left(\mathsf{hypot}\left(b, a\right)\right)}^{2} \]

   12. Applied egg-rr 31.7%

       \[\leadsto \sin \color{blue}{\left(e^{\mathsf{log1p}\left(\pi \cdot \left(angle \cdot 0.011111111111111112\right)\right)} - 1\right)} \cdot {\left(\mathsf{hypot}\left(b, a\right)\right)}^{2} \]
   13. Step-by-step derivation

       1. expm1-define 38.7%

          \[\leadsto \sin \color{blue}{\left(\mathsf{expm1}\left(\mathsf{log1p}\left(\pi \cdot \left(angle \cdot 0.011111111111111112\right)\right)\right)\right)} \cdot {\left(\mathsf{hypot}\left(b, a\right)\right)}^{2} \]

   14. Simplified 38.7%

       \[\leadsto \sin \color{blue}{\left(\mathsf{expm1}\left(\mathsf{log1p}\left(\pi \cdot \left(angle \cdot 0.011111111111111112\right)\right)\right)\right)} \cdot {\left(\mathsf{hypot}\left(b, a\right)\right)}^{2} \]

   if 1.9999999999999999e105 < (/.f64 angle 180) < 5.00000000000000033e264

   1. Initial program 29.4%

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

      1. associate-*l* 29.4%

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

      2. associate-*l* 29.4%

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

   3. Simplified 29.4%

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

      1. add-sqr-sqrt 21.3%

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

      2. sqrt-unprod 43.6%

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

      3. pow2 43.6%

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

      4. div-inv 43.8%

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

      5. metadata-eval 43.8%

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

   6. Applied egg-rr 43.3%

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

   if 5.00000000000000033e264 < (/.f64 angle 180) < 5.00000000000000016e281

   1. Initial program 31.0%

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

      1. associate-*l* 31.0%

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

      2. *-commutative 31.0%

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

      3. associate-*l* 31.0%

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

   3. Simplified 31.0%

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

      1. unpow2 31.0%

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

      2. unpow2 31.0%

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

      3. difference-of-squares 31.0%

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

   6. Applied egg-rr 31.0%

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

      1. add-sqr-sqrt 53.8%

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

      2. pow2 53.8%

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

   8. Applied egg-rr 53.8%

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

   if 5.00000000000000016e281 < (/.f64 angle 180)

   1. Initial program 5.5%

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

      1. associate-*l* 5.5%

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

      2. associate-*l* 5.5%

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

   3. Simplified 5.5%

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

      1. add-sqr-sqrt 3.9%

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

      2. sqrt-unprod 42.0%

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

      3. pow2 42.0%

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

      4. *-commutative 42.0%

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

      5. div-inv 42.0%

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

      6. metadata-eval 42.0%

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

   6. Applied egg-rr 42.0%

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

      1. unpow2 5.5%

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

      2. unpow2 5.5%

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

      3. difference-of-squares 5.5%

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

   8. Applied egg-rr 42.0%

      \[\leadsto 2 \cdot \left(\sqrt{{\left(\sin \left(\pi \cdot \left(angle \cdot 0.005555555555555556\right)\right) \cdot \color{blue}{\left(\left(b + a\right) \cdot \left(b - a\right)\right)}\right)}^{2}} \cdot \cos \left(\pi \cdot \frac{angle}{180}\right)\right) \]
3. Recombined 5 regimes into one program.

4. Final simplification 66.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{angle}{180} \leq 5 \cdot 10^{+21}:\\ \;\;\;\;\left(b + a\right) \cdot \left(\left(b - a\right) \cdot \sin \left(2 \cdot \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)\right)\right)\\ \mathbf{elif}\;\frac{angle}{180} \leq 2 \cdot 10^{+105}:\\ \;\;\;\;\sin \left(\mathsf{expm1}\left(\mathsf{log1p}\left(\pi \cdot \left(angle \cdot 0.011111111111111112\right)\right)\right)\right) \cdot {\left(\mathsf{hypot}\left(b, a\right)\right)}^{2}\\ \mathbf{elif}\;\frac{angle}{180} \leq 5 \cdot 10^{+264}:\\ \;\;\;\;2 \cdot \left(\cos \left(\frac{angle}{180} \cdot \pi\right) \cdot \left(\left({b}^{2} - {a}^{2}\right) \cdot \sqrt{{\sin \left(\pi \cdot \left(angle \cdot 0.005555555555555556\right)\right)}^{2}}\right)\right)\\ \mathbf{elif}\;\frac{angle}{180} \leq 5 \cdot 10^{+281}:\\ \;\;\;\;\left(\left(b + a\right) \cdot \left(b - a\right)\right) \cdot \left(2 \cdot \left(\sin \left(\frac{angle}{180} \cdot \pi\right) \cdot \cos \left(\frac{angle}{180} \cdot {\left(\sqrt{\pi}\right)}^{2}\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \left(\cos \left(\frac{angle}{180} \cdot \pi\right) \cdot \sqrt{{\left(\sin \left(\pi \cdot \left(angle \cdot 0.005555555555555556\right)\right) \cdot \left(\left(b + a\right) \cdot \left(b - a\right)\right)\right)}^{2}}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 61.1% accurate, 0.8× speedup

Math

\[\begin{array}{l} a_m = \left|a\right| \\ b_m = \left|b\right| \\ angle\_m = \left|angle\right| \\ angle\_s = \mathsf{copysign}\left(1, angle\right) \\ \begin{array}{l} t_0 := {b\_m}^{2} - {a\_m}^{2}\\ angle\_s \cdot \begin{array}{l} \mathbf{if}\;t\_0 \leq -5 \cdot 10^{+273}:\\ \;\;\;\;0.011111111111111112 \cdot \left(a\_m \cdot \left(angle\_m \cdot \left(\pi \cdot \left(b\_m - b\_m\right)\right) - a\_m \cdot \left(angle\_m \cdot \pi\right)\right) + angle\_m \cdot \left(\pi \cdot {b\_m}^{2}\right)\right)\\ \mathbf{elif}\;t\_0 \leq 5 \cdot 10^{+292}:\\ \;\;\;\;\left(\left(b\_m + a\_m\right) \cdot \left(b\_m - a\_m\right)\right) \cdot \sin \left(0.011111111111111112 \cdot \left(angle\_m \cdot \pi\right)\right)\\ \mathbf{else}:\\ \;\;\;\;-0.011111111111111112 \cdot \left({a\_m}^{2} \cdot \left(angle\_m \cdot \pi\right)\right) + b\_m \cdot \left(0.011111111111111112 \cdot \left(angle\_m \cdot \left(b\_m \cdot \pi\right)\right) + 0.011111111111111112 \cdot \left(angle\_m \cdot \left(\pi \cdot \left(a\_m - a\_m\right)\right)\right)\right)\\ \end{array} \end{array} \end{array} \]

FPCore

a_m = (fabs.f64 a)
b_m = (fabs.f64 b)
angle\_m = (fabs.f64 angle)
angle\_s = (copysign.f64 1 angle)
(FPCore (angle_s a_m b_m angle_m)
 :precision binary64
 (let* ((t_0 (- (pow b_m 2.0) (pow a_m 2.0))))
   (*
    angle_s
    (if (<= t_0 -5e+273)
      (*
       0.011111111111111112
       (+
        (* a_m (- (* angle_m (* PI (- b_m b_m))) (* a_m (* angle_m PI))))
        (* angle_m (* PI (pow b_m 2.0)))))
      (if (<= t_0 5e+292)
        (*
         (* (+ b_m a_m) (- b_m a_m))
         (sin (* 0.011111111111111112 (* angle_m PI))))
        (+
         (* -0.011111111111111112 (* (pow a_m 2.0) (* angle_m PI)))
         (*
          b_m
          (+
           (* 0.011111111111111112 (* angle_m (* b_m PI)))
           (* 0.011111111111111112 (* angle_m (* PI (- a_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 = pow(b_m, 2.0) - pow(a_m, 2.0);
	double tmp;
	if (t_0 <= -5e+273) {
		tmp = 0.011111111111111112 * ((a_m * ((angle_m * (((double) M_PI) * (b_m - b_m))) - (a_m * (angle_m * ((double) M_PI))))) + (angle_m * (((double) M_PI) * pow(b_m, 2.0))));
	} else if (t_0 <= 5e+292) {
		tmp = ((b_m + a_m) * (b_m - a_m)) * sin((0.011111111111111112 * (angle_m * ((double) M_PI))));
	} else {
		tmp = (-0.011111111111111112 * (pow(a_m, 2.0) * (angle_m * ((double) M_PI)))) + (b_m * ((0.011111111111111112 * (angle_m * (b_m * ((double) M_PI)))) + (0.011111111111111112 * (angle_m * (((double) M_PI) * (a_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 = Math.pow(b_m, 2.0) - Math.pow(a_m, 2.0);
	double tmp;
	if (t_0 <= -5e+273) {
		tmp = 0.011111111111111112 * ((a_m * ((angle_m * (Math.PI * (b_m - b_m))) - (a_m * (angle_m * Math.PI)))) + (angle_m * (Math.PI * Math.pow(b_m, 2.0))));
	} else if (t_0 <= 5e+292) {
		tmp = ((b_m + a_m) * (b_m - a_m)) * Math.sin((0.011111111111111112 * (angle_m * Math.PI)));
	} else {
		tmp = (-0.011111111111111112 * (Math.pow(a_m, 2.0) * (angle_m * Math.PI))) + (b_m * ((0.011111111111111112 * (angle_m * (b_m * Math.PI))) + (0.011111111111111112 * (angle_m * (Math.PI * (a_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 = math.pow(b_m, 2.0) - math.pow(a_m, 2.0)
	tmp = 0
	if t_0 <= -5e+273:
		tmp = 0.011111111111111112 * ((a_m * ((angle_m * (math.pi * (b_m - b_m))) - (a_m * (angle_m * math.pi)))) + (angle_m * (math.pi * math.pow(b_m, 2.0))))
	elif t_0 <= 5e+292:
		tmp = ((b_m + a_m) * (b_m - a_m)) * math.sin((0.011111111111111112 * (angle_m * math.pi)))
	else:
		tmp = (-0.011111111111111112 * (math.pow(a_m, 2.0) * (angle_m * math.pi))) + (b_m * ((0.011111111111111112 * (angle_m * (b_m * math.pi))) + (0.011111111111111112 * (angle_m * (math.pi * (a_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((b_m ^ 2.0) - (a_m ^ 2.0))
	tmp = 0.0
	if (t_0 <= -5e+273)
		tmp = Float64(0.011111111111111112 * Float64(Float64(a_m * Float64(Float64(angle_m * Float64(pi * Float64(b_m - b_m))) - Float64(a_m * Float64(angle_m * pi)))) + Float64(angle_m * Float64(pi * (b_m ^ 2.0)))));
	elseif (t_0 <= 5e+292)
		tmp = Float64(Float64(Float64(b_m + a_m) * Float64(b_m - a_m)) * sin(Float64(0.011111111111111112 * Float64(angle_m * pi))));
	else
		tmp = Float64(Float64(-0.011111111111111112 * Float64((a_m ^ 2.0) * Float64(angle_m * pi))) + Float64(b_m * Float64(Float64(0.011111111111111112 * Float64(angle_m * Float64(b_m * pi))) + Float64(0.011111111111111112 * Float64(angle_m * Float64(pi * Float64(a_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 = (b_m ^ 2.0) - (a_m ^ 2.0);
	tmp = 0.0;
	if (t_0 <= -5e+273)
		tmp = 0.011111111111111112 * ((a_m * ((angle_m * (pi * (b_m - b_m))) - (a_m * (angle_m * pi)))) + (angle_m * (pi * (b_m ^ 2.0))));
	elseif (t_0 <= 5e+292)
		tmp = ((b_m + a_m) * (b_m - a_m)) * sin((0.011111111111111112 * (angle_m * pi)));
	else
		tmp = (-0.011111111111111112 * ((a_m ^ 2.0) * (angle_m * pi))) + (b_m * ((0.011111111111111112 * (angle_m * (b_m * pi))) + (0.011111111111111112 * (angle_m * (pi * (a_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[(N[Power[b$95$m, 2.0], $MachinePrecision] - N[Power[a$95$m, 2.0], $MachinePrecision]), $MachinePrecision]}, N[(angle$95$s * If[LessEqual[t$95$0, -5e+273], N[(0.011111111111111112 * N[(N[(a$95$m * N[(N[(angle$95$m * N[(Pi * N[(b$95$m - b$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(a$95$m * N[(angle$95$m * Pi), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(angle$95$m * N[(Pi * N[Power[b$95$m, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$0, 5e+292], N[(N[(N[(b$95$m + a$95$m), $MachinePrecision] * N[(b$95$m - a$95$m), $MachinePrecision]), $MachinePrecision] * N[Sin[N[(0.011111111111111112 * N[(angle$95$m * Pi), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(N[(-0.011111111111111112 * N[(N[Power[a$95$m, 2.0], $MachinePrecision] * N[(angle$95$m * Pi), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(b$95$m * N[(N[(0.011111111111111112 * N[(angle$95$m * N[(b$95$m * Pi), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(0.011111111111111112 * N[(angle$95$m * N[(Pi * N[(a$95$m - a$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]), $MachinePrecision]]

Derivation

1. Split input into 3 regimes

2. if (-.f64 (pow.f64 b 2) (pow.f64 a 2)) < -4.99999999999999961e273

   1. Initial program 45.1%

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

      1. associate-*l* 45.1%

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

      2. *-commutative 45.1%

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

      3. associate-*l* 46.8%

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

   3. Simplified 46.8%

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

   5. Taylor expanded in angle around 0 54.5%

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

      1. unpow2 46.8%

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

      2. unpow2 46.8%

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

      3. difference-of-squares 46.8%

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

   7. Applied egg-rr 54.5%

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

   8. Taylor expanded in a around 0 69.3%

      \[\leadsto 0.011111111111111112 \cdot \color{blue}{\left(a \cdot \left(-1 \cdot \left(a \cdot \left(angle \cdot \pi\right)\right) + angle \cdot \left(\pi \cdot \left(b + -1 \cdot b\right)\right)\right) + angle \cdot \left({b}^{2} \cdot \pi\right)\right)} \]

   if -4.99999999999999961e273 < (-.f64 (pow.f64 b 2) (pow.f64 a 2)) < 4.9999999999999996e292

   1. Initial program 56.5%

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

      1. associate-*l* 56.5%

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

      2. *-commutative 56.5%

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

      3. associate-*l* 56.5%

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

   3. Simplified 56.5%

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

      1. *-commutative 56.5%

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

      2. sub-neg 56.5%

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

      3. distribute-lft-in 56.5%

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

      4. 2-sin 56.5%

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

      5. associate-*r* 56.5%

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

      6. div-inv 56.9%

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

      7. metadata-eval 56.9%

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

   6. Applied egg-rr 57.0%

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

      1. distribute-lft-out 57.0%

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

      2. sub-neg 57.0%

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

      3. *-commutative 57.0%

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

      4. associate-*r* 57.0%

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

      5. associate-*r* 56.7%

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

      6. *-commutative 56.7%

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

      7. *-commutative 56.7%

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

      8. associate-*r* 56.7%

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

      9. metadata-eval 56.7%

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

   8. Simplified 56.7%

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

      1. unpow2 56.5%

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

      2. unpow2 56.5%

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

      3. difference-of-squares 56.5%

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

   10. Applied egg-rr 56.7%

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

   if 4.9999999999999996e292 < (-.f64 (pow.f64 b 2) (pow.f64 a 2))

   1. Initial program 42.6%

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

      1. associate-*l* 42.6%

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

      2. *-commutative 42.6%

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

      3. associate-*l* 42.6%

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

   3. Simplified 42.6%

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

   5. Taylor expanded in angle around 0 45.6%

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

      1. unpow2 42.6%

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

      2. unpow2 42.6%

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

      3. difference-of-squares 50.7%

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

   7. Applied egg-rr 53.7%

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

   8. Taylor expanded in b around 0 63.1%

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

4. Final simplification 61.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;{b}^{2} - {a}^{2} \leq -5 \cdot 10^{+273}:\\ \;\;\;\;0.011111111111111112 \cdot \left(a \cdot \left(angle \cdot \left(\pi \cdot \left(b - b\right)\right) - a \cdot \left(angle \cdot \pi\right)\right) + angle \cdot \left(\pi \cdot {b}^{2}\right)\right)\\ \mathbf{elif}\;{b}^{2} - {a}^{2} \leq 5 \cdot 10^{+292}:\\ \;\;\;\;\left(\left(b + a\right) \cdot \left(b - a\right)\right) \cdot \sin \left(0.011111111111111112 \cdot \left(angle \cdot \pi\right)\right)\\ \mathbf{else}:\\ \;\;\;\;-0.011111111111111112 \cdot \left({a}^{2} \cdot \left(angle \cdot \pi\right)\right) + b \cdot \left(0.011111111111111112 \cdot \left(angle \cdot \left(b \cdot \pi\right)\right) + 0.011111111111111112 \cdot \left(angle \cdot \left(\pi \cdot \left(a - a\right)\right)\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 67.9% accurate, 0.8× speedup

Math

\[\begin{array}{l} a_m = \left|a\right| \\ b_m = \left|b\right| \\ angle\_m = \left|angle\right| \\ angle\_s = \mathsf{copysign}\left(1, angle\right) \\ \begin{array}{l} t_0 := \sin \left(\pi \cdot \left(angle\_m \cdot 0.005555555555555556\right)\right)\\ t_1 := \left(b\_m + a\_m\right) \cdot \left(b\_m - a\_m\right)\\ t_2 := \frac{angle\_m}{180} \cdot \pi\\ t_3 := \cos t\_2\\ angle\_s \cdot \begin{array}{l} \mathbf{if}\;\frac{angle\_m}{180} \leq 5 \cdot 10^{+21}:\\ \;\;\;\;\left(b\_m + a\_m\right) \cdot \left(\left(b\_m - a\_m\right) \cdot \sin \left(2 \cdot \left(0.005555555555555556 \cdot \left(angle\_m \cdot \pi\right)\right)\right)\right)\\ \mathbf{elif}\;\frac{angle\_m}{180} \leq 2 \cdot 10^{+105}:\\ \;\;\;\;\sin \left(\mathsf{expm1}\left(\mathsf{log1p}\left(\pi \cdot \left(angle\_m \cdot 0.011111111111111112\right)\right)\right)\right) \cdot {\left(\mathsf{hypot}\left(b\_m, a\_m\right)\right)}^{2}\\ \mathbf{elif}\;\frac{angle\_m}{180} \leq 5 \cdot 10^{+264}:\\ \;\;\;\;t\_1 \cdot \left(2 \cdot \left(t\_3 \cdot \sqrt{{t\_0}^{2}}\right)\right)\\ \mathbf{elif}\;\frac{angle\_m}{180} \leq 5 \cdot 10^{+281}:\\ \;\;\;\;t\_1 \cdot \left(2 \cdot \left(\sin t\_2 \cdot \cos \left(\frac{angle\_m}{180} \cdot {\left(\sqrt{\pi}\right)}^{2}\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \left(t\_3 \cdot \sqrt{{\left(t\_0 \cdot t\_1\right)}^{2}}\right)\\ \end{array} \end{array} \end{array} \]

FPCore

a_m = (fabs.f64 a)
b_m = (fabs.f64 b)
angle\_m = (fabs.f64 angle)
angle\_s = (copysign.f64 1 angle)
(FPCore (angle_s a_m b_m angle_m)
 :precision binary64
 (let* ((t_0 (sin (* PI (* angle_m 0.005555555555555556))))
        (t_1 (* (+ b_m a_m) (- b_m a_m)))
        (t_2 (* (/ angle_m 180.0) PI))
        (t_3 (cos t_2)))
   (*
    angle_s
    (if (<= (/ angle_m 180.0) 5e+21)
      (*
       (+ b_m a_m)
       (* (- b_m a_m) (sin (* 2.0 (* 0.005555555555555556 (* angle_m PI))))))
      (if (<= (/ angle_m 180.0) 2e+105)
        (*
         (sin (expm1 (log1p (* PI (* angle_m 0.011111111111111112)))))
         (pow (hypot b_m a_m) 2.0))
        (if (<= (/ angle_m 180.0) 5e+264)
          (* t_1 (* 2.0 (* t_3 (sqrt (pow t_0 2.0)))))
          (if (<= (/ angle_m 180.0) 5e+281)
            (*
             t_1
             (*
              2.0
              (* (sin t_2) (cos (* (/ angle_m 180.0) (pow (sqrt PI) 2.0))))))
            (* 2.0 (* t_3 (sqrt (pow (* t_0 t_1) 2.0)))))))))))

C

a_m = fabs(a);
b_m = fabs(b);
angle\_m = fabs(angle);
angle\_s = copysign(1.0, angle);
double code(double angle_s, double a_m, double b_m, double angle_m) {
	double t_0 = sin((((double) M_PI) * (angle_m * 0.005555555555555556)));
	double t_1 = (b_m + a_m) * (b_m - a_m);
	double t_2 = (angle_m / 180.0) * ((double) M_PI);
	double t_3 = cos(t_2);
	double tmp;
	if ((angle_m / 180.0) <= 5e+21) {
		tmp = (b_m + a_m) * ((b_m - a_m) * sin((2.0 * (0.005555555555555556 * (angle_m * ((double) M_PI))))));
	} else if ((angle_m / 180.0) <= 2e+105) {
		tmp = sin(expm1(log1p((((double) M_PI) * (angle_m * 0.011111111111111112))))) * pow(hypot(b_m, a_m), 2.0);
	} else if ((angle_m / 180.0) <= 5e+264) {
		tmp = t_1 * (2.0 * (t_3 * sqrt(pow(t_0, 2.0))));
	} else if ((angle_m / 180.0) <= 5e+281) {
		tmp = t_1 * (2.0 * (sin(t_2) * cos(((angle_m / 180.0) * pow(sqrt(((double) M_PI)), 2.0)))));
	} else {
		tmp = 2.0 * (t_3 * sqrt(pow((t_0 * t_1), 2.0)));
	}
	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.sin((Math.PI * (angle_m * 0.005555555555555556)));
	double t_1 = (b_m + a_m) * (b_m - a_m);
	double t_2 = (angle_m / 180.0) * Math.PI;
	double t_3 = Math.cos(t_2);
	double tmp;
	if ((angle_m / 180.0) <= 5e+21) {
		tmp = (b_m + a_m) * ((b_m - a_m) * Math.sin((2.0 * (0.005555555555555556 * (angle_m * Math.PI)))));
	} else if ((angle_m / 180.0) <= 2e+105) {
		tmp = Math.sin(Math.expm1(Math.log1p((Math.PI * (angle_m * 0.011111111111111112))))) * Math.pow(Math.hypot(b_m, a_m), 2.0);
	} else if ((angle_m / 180.0) <= 5e+264) {
		tmp = t_1 * (2.0 * (t_3 * Math.sqrt(Math.pow(t_0, 2.0))));
	} else if ((angle_m / 180.0) <= 5e+281) {
		tmp = t_1 * (2.0 * (Math.sin(t_2) * Math.cos(((angle_m / 180.0) * Math.pow(Math.sqrt(Math.PI), 2.0)))));
	} else {
		tmp = 2.0 * (t_3 * Math.sqrt(Math.pow((t_0 * t_1), 2.0)));
	}
	return angle_s * tmp;
}

Python

a_m = math.fabs(a)
b_m = math.fabs(b)
angle\_m = math.fabs(angle)
angle\_s = math.copysign(1.0, angle)
def code(angle_s, a_m, b_m, angle_m):
	t_0 = math.sin((math.pi * (angle_m * 0.005555555555555556)))
	t_1 = (b_m + a_m) * (b_m - a_m)
	t_2 = (angle_m / 180.0) * math.pi
	t_3 = math.cos(t_2)
	tmp = 0
	if (angle_m / 180.0) <= 5e+21:
		tmp = (b_m + a_m) * ((b_m - a_m) * math.sin((2.0 * (0.005555555555555556 * (angle_m * math.pi)))))
	elif (angle_m / 180.0) <= 2e+105:
		tmp = math.sin(math.expm1(math.log1p((math.pi * (angle_m * 0.011111111111111112))))) * math.pow(math.hypot(b_m, a_m), 2.0)
	elif (angle_m / 180.0) <= 5e+264:
		tmp = t_1 * (2.0 * (t_3 * math.sqrt(math.pow(t_0, 2.0))))
	elif (angle_m / 180.0) <= 5e+281:
		tmp = t_1 * (2.0 * (math.sin(t_2) * math.cos(((angle_m / 180.0) * math.pow(math.sqrt(math.pi), 2.0)))))
	else:
		tmp = 2.0 * (t_3 * math.sqrt(math.pow((t_0 * t_1), 2.0)))
	return angle_s * tmp

Julia

a_m = abs(a)
b_m = abs(b)
angle\_m = abs(angle)
angle\_s = copysign(1.0, angle)
function code(angle_s, a_m, b_m, angle_m)
	t_0 = sin(Float64(pi * Float64(angle_m * 0.005555555555555556)))
	t_1 = Float64(Float64(b_m + a_m) * Float64(b_m - a_m))
	t_2 = Float64(Float64(angle_m / 180.0) * pi)
	t_3 = cos(t_2)
	tmp = 0.0
	if (Float64(angle_m / 180.0) <= 5e+21)
		tmp = Float64(Float64(b_m + a_m) * Float64(Float64(b_m - a_m) * sin(Float64(2.0 * Float64(0.005555555555555556 * Float64(angle_m * pi))))));
	elseif (Float64(angle_m / 180.0) <= 2e+105)
		tmp = Float64(sin(expm1(log1p(Float64(pi * Float64(angle_m * 0.011111111111111112))))) * (hypot(b_m, a_m) ^ 2.0));
	elseif (Float64(angle_m / 180.0) <= 5e+264)
		tmp = Float64(t_1 * Float64(2.0 * Float64(t_3 * sqrt((t_0 ^ 2.0)))));
	elseif (Float64(angle_m / 180.0) <= 5e+281)
		tmp = Float64(t_1 * Float64(2.0 * Float64(sin(t_2) * cos(Float64(Float64(angle_m / 180.0) * (sqrt(pi) ^ 2.0))))));
	else
		tmp = Float64(2.0 * Float64(t_3 * sqrt((Float64(t_0 * t_1) ^ 2.0))));
	end
	return Float64(angle_s * tmp)
end

Wolfram

a_m = N[Abs[a], $MachinePrecision]
b_m = N[Abs[b], $MachinePrecision]
angle\_m = N[Abs[angle], $MachinePrecision]
angle\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[angle]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[angle$95$s_, a$95$m_, b$95$m_, angle$95$m_] := Block[{t$95$0 = N[Sin[N[(Pi * N[(angle$95$m * 0.005555555555555556), $MachinePrecision]), $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[(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], 5e+21], N[(N[(b$95$m + a$95$m), $MachinePrecision] * N[(N[(b$95$m - a$95$m), $MachinePrecision] * N[Sin[N[(2.0 * N[(0.005555555555555556 * N[(angle$95$m * Pi), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[N[(angle$95$m / 180.0), $MachinePrecision], 2e+105], N[(N[Sin[N[(Exp[N[Log[1 + N[(Pi * N[(angle$95$m * 0.011111111111111112), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]] - 1), $MachinePrecision]], $MachinePrecision] * N[Power[N[Sqrt[b$95$m ^ 2 + a$95$m ^ 2], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision], If[LessEqual[N[(angle$95$m / 180.0), $MachinePrecision], 5e+264], N[(t$95$1 * N[(2.0 * N[(t$95$3 * N[Sqrt[N[Power[t$95$0, 2.0], $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[N[(angle$95$m / 180.0), $MachinePrecision], 5e+281], N[(t$95$1 * N[(2.0 * N[(N[Sin[t$95$2], $MachinePrecision] * N[Cos[N[(N[(angle$95$m / 180.0), $MachinePrecision] * N[Power[N[Sqrt[Pi], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(2.0 * N[(t$95$3 * N[Sqrt[N[Power[N[(t$95$0 * t$95$1), $MachinePrecision], 2.0], $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]), $MachinePrecision]]]]]

Derivation

1. Split input into 5 regimes

2. if (/.f64 angle 180) < 5e21

   1. Initial program 59.0%

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

      1. associate-*l* 59.0%

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

      2. *-commutative 59.0%

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

      3. associate-*l* 59.5%

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

   3. Simplified 59.5%

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

      1. unpow2 59.5%

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

      2. unpow2 59.5%

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

      3. difference-of-squares 62.2%

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

   6. Applied egg-rr 62.2%

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

      1. add-cube-cbrt 62.2%

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

      2. pow3 62.2%

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

      3. div-inv 62.7%

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

      4. metadata-eval 62.7%

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

   8. Applied egg-rr 62.7%

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

      1. pow1 62.7%

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

   10. Applied egg-rr 74.5%

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

   if 5e21 < (/.f64 angle 180) < 1.9999999999999999e105

   1. Initial program 25.4%

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

      1. associate-*l* 25.4%

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

      2. *-commutative 25.4%

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

      3. associate-*l* 25.4%

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

   3. Simplified 25.4%

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

      1. unpow2 25.4%

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

      2. unpow2 25.4%

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

      3. difference-of-squares 25.4%

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

   6. Applied egg-rr 25.4%

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

      1. *-commutative 25.4%

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

      2. difference-of-squares 25.4%

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

      3. unpow2 25.4%

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

      4. unpow2 25.4%

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

      5. sub-neg 25.4%

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

      6. distribute-lft-in 25.4%

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

   8. Applied egg-rr 28.1%

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

      1. distribute-lft-out 28.1%

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

      2. associate-*l* 27.4%

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

      3. rem-square-sqrt 27.4%

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

      4. unpow2 27.4%

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

      5. unpow2 27.4%

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

      6. hypot-undefine 27.4%

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

      7. unpow2 27.4%

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

      8. unpow2 27.4%

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

      9. hypot-undefine 27.4%

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

      10. unpow2 27.4%

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

   10. Simplified 27.4%

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

       1. expm1-log1p-u 38.7%

          \[\leadsto \sin \color{blue}{\left(\mathsf{expm1}\left(\mathsf{log1p}\left(\pi \cdot \left(angle \cdot 0.011111111111111112\right)\right)\right)\right)} \cdot {\left(\mathsf{hypot}\left(b, a\right)\right)}^{2} \]

       2. expm1-undefine 31.7%

          \[\leadsto \sin \color{blue}{\left(e^{\mathsf{log1p}\left(\pi \cdot \left(angle \cdot 0.011111111111111112\right)\right)} - 1\right)} \cdot {\left(\mathsf{hypot}\left(b, a\right)\right)}^{2} \]

   12. Applied egg-rr 31.7%

       \[\leadsto \sin \color{blue}{\left(e^{\mathsf{log1p}\left(\pi \cdot \left(angle \cdot 0.011111111111111112\right)\right)} - 1\right)} \cdot {\left(\mathsf{hypot}\left(b, a\right)\right)}^{2} \]
   13. Step-by-step derivation

       1. expm1-define 38.7%

          \[\leadsto \sin \color{blue}{\left(\mathsf{expm1}\left(\mathsf{log1p}\left(\pi \cdot \left(angle \cdot 0.011111111111111112\right)\right)\right)\right)} \cdot {\left(\mathsf{hypot}\left(b, a\right)\right)}^{2} \]

   14. Simplified 38.7%

       \[\leadsto \sin \color{blue}{\left(\mathsf{expm1}\left(\mathsf{log1p}\left(\pi \cdot \left(angle \cdot 0.011111111111111112\right)\right)\right)\right)} \cdot {\left(\mathsf{hypot}\left(b, a\right)\right)}^{2} \]

   if 1.9999999999999999e105 < (/.f64 angle 180) < 5.00000000000000033e264

   1. Initial program 29.4%

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

      1. associate-*l* 29.4%

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

      2. *-commutative 29.4%

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

      3. associate-*l* 29.4%

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

   3. Simplified 29.4%

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

      1. unpow2 29.4%

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

      2. unpow2 29.4%

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

      3. difference-of-squares 29.4%

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

   6. Applied egg-rr 29.4%

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

      1. add-sqr-sqrt 21.3%

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

      2. sqrt-unprod 43.6%

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

      3. pow2 43.6%

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

      4. div-inv 43.8%

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

      5. metadata-eval 43.8%

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

   8. Applied egg-rr 43.3%

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

   if 5.00000000000000033e264 < (/.f64 angle 180) < 5.00000000000000016e281

   1. Initial program 31.0%

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

      1. associate-*l* 31.0%

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

      2. *-commutative 31.0%

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

      3. associate-*l* 31.0%

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

   3. Simplified 31.0%

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

      1. unpow2 31.0%

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

      2. unpow2 31.0%

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

      3. difference-of-squares 31.0%

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

   6. Applied egg-rr 31.0%

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

      1. add-sqr-sqrt 53.8%

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

      2. pow2 53.8%

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

   8. Applied egg-rr 53.8%

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

   if 5.00000000000000016e281 < (/.f64 angle 180)

   1. Initial program 5.5%

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

      1. associate-*l* 5.5%

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

      2. associate-*l* 5.5%

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

   3. Simplified 5.5%

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

      1. add-sqr-sqrt 3.9%

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

      2. sqrt-unprod 42.0%

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

      3. pow2 42.0%

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

      4. *-commutative 42.0%

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

      5. div-inv 42.0%

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

      6. metadata-eval 42.0%

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

   6. Applied egg-rr 42.0%

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

      1. unpow2 5.5%

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

      2. unpow2 5.5%

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

      3. difference-of-squares 5.5%

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

   8. Applied egg-rr 42.0%

      \[\leadsto 2 \cdot \left(\sqrt{{\left(\sin \left(\pi \cdot \left(angle \cdot 0.005555555555555556\right)\right) \cdot \color{blue}{\left(\left(b + a\right) \cdot \left(b - a\right)\right)}\right)}^{2}} \cdot \cos \left(\pi \cdot \frac{angle}{180}\right)\right) \]
3. Recombined 5 regimes into one program.

4. Final simplification 66.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{angle}{180} \leq 5 \cdot 10^{+21}:\\ \;\;\;\;\left(b + a\right) \cdot \left(\left(b - a\right) \cdot \sin \left(2 \cdot \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)\right)\right)\\ \mathbf{elif}\;\frac{angle}{180} \leq 2 \cdot 10^{+105}:\\ \;\;\;\;\sin \left(\mathsf{expm1}\left(\mathsf{log1p}\left(\pi \cdot \left(angle \cdot 0.011111111111111112\right)\right)\right)\right) \cdot {\left(\mathsf{hypot}\left(b, a\right)\right)}^{2}\\ \mathbf{elif}\;\frac{angle}{180} \leq 5 \cdot 10^{+264}:\\ \;\;\;\;\left(\left(b + a\right) \cdot \left(b - a\right)\right) \cdot \left(2 \cdot \left(\cos \left(\frac{angle}{180} \cdot \pi\right) \cdot \sqrt{{\sin \left(\pi \cdot \left(angle \cdot 0.005555555555555556\right)\right)}^{2}}\right)\right)\\ \mathbf{elif}\;\frac{angle}{180} \leq 5 \cdot 10^{+281}:\\ \;\;\;\;\left(\left(b + a\right) \cdot \left(b - a\right)\right) \cdot \left(2 \cdot \left(\sin \left(\frac{angle}{180} \cdot \pi\right) \cdot \cos \left(\frac{angle}{180} \cdot {\left(\sqrt{\pi}\right)}^{2}\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \left(\cos \left(\frac{angle}{180} \cdot \pi\right) \cdot \sqrt{{\left(\sin \left(\pi \cdot \left(angle \cdot 0.005555555555555556\right)\right) \cdot \left(\left(b + a\right) \cdot \left(b - a\right)\right)\right)}^{2}}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 67.2% accurate, 1.0× speedup

Math

\[\begin{array}{l} a_m = \left|a\right| \\ b_m = \left|b\right| \\ angle\_m = \left|angle\right| \\ angle\_s = \mathsf{copysign}\left(1, angle\right) \\ angle\_s \cdot \begin{array}{l} \mathbf{if}\;{b\_m}^{2} - {a\_m}^{2} \leq -5 \cdot 10^{+273}:\\ \;\;\;\;0.011111111111111112 \cdot \mathsf{fma}\left(angle\_m, \pi \cdot {b\_m}^{2}, a\_m \cdot \left(angle\_m \cdot 0 - \pi \cdot \left(angle\_m \cdot a\_m\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(b\_m + a\_m\right) \cdot \left(\left(b\_m - a\_m\right) \cdot \sin \left(2 \cdot \left(0.005555555555555556 \cdot \left(angle\_m \cdot \pi\right)\right)\right)\right)\\ \end{array} \end{array} \]

FPCore

a_m = (fabs.f64 a)
b_m = (fabs.f64 b)
angle\_m = (fabs.f64 angle)
angle\_s = (copysign.f64 1 angle)
(FPCore (angle_s a_m b_m angle_m)
 :precision binary64
 (*
  angle_s
  (if (<= (- (pow b_m 2.0) (pow a_m 2.0)) -5e+273)
    (*
     0.011111111111111112
     (fma
      angle_m
      (* PI (pow b_m 2.0))
      (* a_m (- (* angle_m 0.0) (* PI (* angle_m a_m))))))
    (*
     (+ b_m a_m)
     (* (- b_m a_m) (sin (* 2.0 (* 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 tmp;
	if ((pow(b_m, 2.0) - pow(a_m, 2.0)) <= -5e+273) {
		tmp = 0.011111111111111112 * fma(angle_m, (((double) M_PI) * pow(b_m, 2.0)), (a_m * ((angle_m * 0.0) - (((double) M_PI) * (angle_m * a_m)))));
	} else {
		tmp = (b_m + a_m) * ((b_m - a_m) * sin((2.0 * (0.005555555555555556 * (angle_m * ((double) M_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 (Float64((b_m ^ 2.0) - (a_m ^ 2.0)) <= -5e+273)
		tmp = Float64(0.011111111111111112 * fma(angle_m, Float64(pi * (b_m ^ 2.0)), Float64(a_m * Float64(Float64(angle_m * 0.0) - Float64(pi * Float64(angle_m * a_m))))));
	else
		tmp = Float64(Float64(b_m + a_m) * Float64(Float64(b_m - a_m) * sin(Float64(2.0 * 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_] := N[(angle$95$s * If[LessEqual[N[(N[Power[b$95$m, 2.0], $MachinePrecision] - N[Power[a$95$m, 2.0], $MachinePrecision]), $MachinePrecision], -5e+273], N[(0.011111111111111112 * N[(angle$95$m * N[(Pi * N[Power[b$95$m, 2.0], $MachinePrecision]), $MachinePrecision] + N[(a$95$m * N[(N[(angle$95$m * 0.0), $MachinePrecision] - N[(Pi * N[(angle$95$m * a$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(b$95$m + a$95$m), $MachinePrecision] * N[(N[(b$95$m - a$95$m), $MachinePrecision] * N[Sin[N[(2.0 * N[(0.005555555555555556 * N[(angle$95$m * Pi), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if (-.f64 (pow.f64 b 2) (pow.f64 a 2)) < -4.99999999999999961e273

   1. Initial program 45.1%

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

      1. associate-*l* 45.1%

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

      2. *-commutative 45.1%

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

      3. associate-*l* 46.8%

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

   3. Simplified 46.8%

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

   5. Taylor expanded in angle around 0 54.5%

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

      1. unpow2 46.8%

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

      2. unpow2 46.8%

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

      3. difference-of-squares 46.8%

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

   7. Applied egg-rr 54.5%

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

      1. associate-*r* 54.5%

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

      2. sub-neg 54.5%

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

      3. distribute-lft-in 50.7%

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

   9. Applied egg-rr 50.7%

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

   10. Taylor expanded in a around 0 69.3%

       \[\leadsto 0.011111111111111112 \cdot \color{blue}{\left(a \cdot \left(-1 \cdot \left(a \cdot \left(angle \cdot \pi\right)\right) + angle \cdot \left(-1 \cdot \left(b \cdot \pi\right) + b \cdot \pi\right)\right) + angle \cdot \left({b}^{2} \cdot \pi\right)\right)} \]
   11. Step-by-step derivation

       1. +-commutative 69.3%

          \[\leadsto 0.011111111111111112 \cdot \color{blue}{\left(angle \cdot \left({b}^{2} \cdot \pi\right) + a \cdot \left(-1 \cdot \left(a \cdot \left(angle \cdot \pi\right)\right) + angle \cdot \left(-1 \cdot \left(b \cdot \pi\right) + b \cdot \pi\right)\right)\right)} \]

       2. fma-define 74.7%

          \[\leadsto 0.011111111111111112 \cdot \color{blue}{\mathsf{fma}\left(angle, {b}^{2} \cdot \pi, a \cdot \left(-1 \cdot \left(a \cdot \left(angle \cdot \pi\right)\right) + angle \cdot \left(-1 \cdot \left(b \cdot \pi\right) + b \cdot \pi\right)\right)\right)} \]

       3. *-commutative 74.7%

          \[\leadsto 0.011111111111111112 \cdot \mathsf{fma}\left(angle, \color{blue}{\pi \cdot {b}^{2}}, a \cdot \left(-1 \cdot \left(a \cdot \left(angle \cdot \pi\right)\right) + angle \cdot \left(-1 \cdot \left(b \cdot \pi\right) + b \cdot \pi\right)\right)\right) \]

       4. +-commutative 74.7%

          \[\leadsto 0.011111111111111112 \cdot \mathsf{fma}\left(angle, \pi \cdot {b}^{2}, a \cdot \color{blue}{\left(angle \cdot \left(-1 \cdot \left(b \cdot \pi\right) + b \cdot \pi\right) + -1 \cdot \left(a \cdot \left(angle \cdot \pi\right)\right)\right)}\right) \]

       5. fma-define 74.7%

          \[\leadsto 0.011111111111111112 \cdot \mathsf{fma}\left(angle, \pi \cdot {b}^{2}, a \cdot \color{blue}{\mathsf{fma}\left(angle, -1 \cdot \left(b \cdot \pi\right) + b \cdot \pi, -1 \cdot \left(a \cdot \left(angle \cdot \pi\right)\right)\right)}\right) \]

       6. distribute-lft1-in 74.7%

          \[\leadsto 0.011111111111111112 \cdot \mathsf{fma}\left(angle, \pi \cdot {b}^{2}, a \cdot \mathsf{fma}\left(angle, \color{blue}{\left(-1 + 1\right) \cdot \left(b \cdot \pi\right)}, -1 \cdot \left(a \cdot \left(angle \cdot \pi\right)\right)\right)\right) \]

       7. metadata-eval 74.7%

          \[\leadsto 0.011111111111111112 \cdot \mathsf{fma}\left(angle, \pi \cdot {b}^{2}, a \cdot \mathsf{fma}\left(angle, \color{blue}{0} \cdot \left(b \cdot \pi\right), -1 \cdot \left(a \cdot \left(angle \cdot \pi\right)\right)\right)\right) \]

       8. mul0-lft 74.7%

          \[\leadsto 0.011111111111111112 \cdot \mathsf{fma}\left(angle, \pi \cdot {b}^{2}, a \cdot \mathsf{fma}\left(angle, \color{blue}{0}, -1 \cdot \left(a \cdot \left(angle \cdot \pi\right)\right)\right)\right) \]

       9. mul-1-neg 74.7%

          \[\leadsto 0.011111111111111112 \cdot \mathsf{fma}\left(angle, \pi \cdot {b}^{2}, a \cdot \mathsf{fma}\left(angle, 0, \color{blue}{-a \cdot \left(angle \cdot \pi\right)}\right)\right) \]

       10. fma-neg 74.7%

           \[\leadsto 0.011111111111111112 \cdot \mathsf{fma}\left(angle, \pi \cdot {b}^{2}, a \cdot \color{blue}{\left(angle \cdot 0 - a \cdot \left(angle \cdot \pi\right)\right)}\right) \]

       11. associate-*r* 74.7%

           \[\leadsto 0.011111111111111112 \cdot \mathsf{fma}\left(angle, \pi \cdot {b}^{2}, a \cdot \left(angle \cdot 0 - \color{blue}{\left(a \cdot angle\right) \cdot \pi}\right)\right) \]

   12. Simplified 74.7%

       \[\leadsto 0.011111111111111112 \cdot \color{blue}{\mathsf{fma}\left(angle, \pi \cdot {b}^{2}, a \cdot \left(angle \cdot 0 - \left(a \cdot angle\right) \cdot \pi\right)\right)} \]

   if -4.99999999999999961e273 < (-.f64 (pow.f64 b 2) (pow.f64 a 2))

   1. Initial program 52.0%

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

      1. associate-*l* 52.0%

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

      2. *-commutative 52.0%

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

      3. associate-*l* 52.0%

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

   3. Simplified 52.0%

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

      1. unpow2 52.0%

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

      2. unpow2 52.0%

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

      3. difference-of-squares 54.6%

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

   6. Applied egg-rr 54.6%

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

      1. add-cube-cbrt 54.6%

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

      2. pow3 54.6%

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

      3. div-inv 55.2%

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

      4. metadata-eval 55.2%

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

   8. Applied egg-rr 55.2%

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

      1. pow1 55.2%

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

   10. Applied egg-rr 61.2%

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

4. Final simplification 64.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;{b}^{2} - {a}^{2} \leq -5 \cdot 10^{+273}:\\ \;\;\;\;0.011111111111111112 \cdot \mathsf{fma}\left(angle, \pi \cdot {b}^{2}, a \cdot \left(angle \cdot 0 - \pi \cdot \left(angle \cdot a\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(b + a\right) \cdot \left(\left(b - a\right) \cdot \sin \left(2 \cdot \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 60.4% accurate, 3.1× speedup

Math

\[\begin{array}{l} a_m = \left|a\right| \\ b_m = \left|b\right| \\ angle\_m = \left|angle\right| \\ angle\_s = \mathsf{copysign}\left(1, angle\right) \\ \begin{array}{l} t_0 := \left(b\_m + a\_m\right) \cdot \left(b\_m - a\_m\right)\\ angle\_s \cdot \begin{array}{l} \mathbf{if}\;\frac{angle\_m}{180} \leq 2 \cdot 10^{-272}:\\ \;\;\;\;0.011111111111111112 \cdot \left(a\_m \cdot \left(angle\_m \cdot \left(\pi \cdot \left(b\_m - b\_m\right)\right) - a\_m \cdot \left(angle\_m \cdot \pi\right)\right) + angle\_m \cdot \left(\pi \cdot {b\_m}^{2}\right)\right)\\ \mathbf{elif}\;\frac{angle\_m}{180} \leq 5 \cdot 10^{-129}:\\ \;\;\;\;0.011111111111111112 \cdot \left(b\_m \cdot \left(\pi \cdot \left(angle\_m \cdot b\_m\right)\right) - {a\_m}^{2} \cdot \left(angle\_m \cdot \pi\right)\right)\\ \mathbf{elif}\;\frac{angle\_m}{180} \leq 10^{+73}:\\ \;\;\;\;t\_0 \cdot \sin \left(0.011111111111111112 \cdot \left(angle\_m \cdot \pi\right)\right)\\ \mathbf{else}:\\ \;\;\;\;t\_0 \cdot \left(2 \cdot \sin \left(\frac{angle\_m}{180} \cdot \pi\right)\right)\\ \end{array} \end{array} \end{array} \]

FPCore

a_m = (fabs.f64 a)
b_m = (fabs.f64 b)
angle\_m = (fabs.f64 angle)
angle\_s = (copysign.f64 1 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) 2e-272)
      (*
       0.011111111111111112
       (+
        (* a_m (- (* angle_m (* PI (- b_m b_m))) (* a_m (* angle_m PI))))
        (* angle_m (* PI (pow b_m 2.0)))))
      (if (<= (/ angle_m 180.0) 5e-129)
        (*
         0.011111111111111112
         (- (* b_m (* PI (* angle_m b_m))) (* (pow a_m 2.0) (* angle_m PI))))
        (if (<= (/ angle_m 180.0) 1e+73)
          (* t_0 (sin (* 0.011111111111111112 (* angle_m PI))))
          (* t_0 (* 2.0 (sin (* (/ angle_m 180.0) 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 tmp;
	if ((angle_m / 180.0) <= 2e-272) {
		tmp = 0.011111111111111112 * ((a_m * ((angle_m * (((double) M_PI) * (b_m - b_m))) - (a_m * (angle_m * ((double) M_PI))))) + (angle_m * (((double) M_PI) * pow(b_m, 2.0))));
	} else if ((angle_m / 180.0) <= 5e-129) {
		tmp = 0.011111111111111112 * ((b_m * (((double) M_PI) * (angle_m * b_m))) - (pow(a_m, 2.0) * (angle_m * ((double) M_PI))));
	} else if ((angle_m / 180.0) <= 1e+73) {
		tmp = t_0 * sin((0.011111111111111112 * (angle_m * ((double) M_PI))));
	} else {
		tmp = t_0 * (2.0 * sin(((angle_m / 180.0) * ((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 tmp;
	if ((angle_m / 180.0) <= 2e-272) {
		tmp = 0.011111111111111112 * ((a_m * ((angle_m * (Math.PI * (b_m - b_m))) - (a_m * (angle_m * Math.PI)))) + (angle_m * (Math.PI * Math.pow(b_m, 2.0))));
	} else if ((angle_m / 180.0) <= 5e-129) {
		tmp = 0.011111111111111112 * ((b_m * (Math.PI * (angle_m * b_m))) - (Math.pow(a_m, 2.0) * (angle_m * Math.PI)));
	} else if ((angle_m / 180.0) <= 1e+73) {
		tmp = t_0 * Math.sin((0.011111111111111112 * (angle_m * Math.PI)));
	} else {
		tmp = t_0 * (2.0 * Math.sin(((angle_m / 180.0) * Math.PI)));
	}
	return angle_s * tmp;
}

Python

a_m = math.fabs(a)
b_m = math.fabs(b)
angle\_m = math.fabs(angle)
angle\_s = math.copysign(1.0, angle)
def code(angle_s, a_m, b_m, angle_m):
	t_0 = (b_m + a_m) * (b_m - a_m)
	tmp = 0
	if (angle_m / 180.0) <= 2e-272:
		tmp = 0.011111111111111112 * ((a_m * ((angle_m * (math.pi * (b_m - b_m))) - (a_m * (angle_m * math.pi)))) + (angle_m * (math.pi * math.pow(b_m, 2.0))))
	elif (angle_m / 180.0) <= 5e-129:
		tmp = 0.011111111111111112 * ((b_m * (math.pi * (angle_m * b_m))) - (math.pow(a_m, 2.0) * (angle_m * math.pi)))
	elif (angle_m / 180.0) <= 1e+73:
		tmp = t_0 * math.sin((0.011111111111111112 * (angle_m * math.pi)))
	else:
		tmp = t_0 * (2.0 * math.sin(((angle_m / 180.0) * 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))
	tmp = 0.0
	if (Float64(angle_m / 180.0) <= 2e-272)
		tmp = Float64(0.011111111111111112 * Float64(Float64(a_m * Float64(Float64(angle_m * Float64(pi * Float64(b_m - b_m))) - Float64(a_m * Float64(angle_m * pi)))) + Float64(angle_m * Float64(pi * (b_m ^ 2.0)))));
	elseif (Float64(angle_m / 180.0) <= 5e-129)
		tmp = Float64(0.011111111111111112 * Float64(Float64(b_m * Float64(pi * Float64(angle_m * b_m))) - Float64((a_m ^ 2.0) * Float64(angle_m * pi))));
	elseif (Float64(angle_m / 180.0) <= 1e+73)
		tmp = Float64(t_0 * sin(Float64(0.011111111111111112 * Float64(angle_m * pi))));
	else
		tmp = Float64(t_0 * Float64(2.0 * sin(Float64(Float64(angle_m / 180.0) * pi))));
	end
	return Float64(angle_s * tmp)
end

MATLAB

a_m = abs(a);
b_m = abs(b);
angle\_m = abs(angle);
angle\_s = sign(angle) * abs(1.0);
function tmp_2 = code(angle_s, a_m, b_m, angle_m)
	t_0 = (b_m + a_m) * (b_m - a_m);
	tmp = 0.0;
	if ((angle_m / 180.0) <= 2e-272)
		tmp = 0.011111111111111112 * ((a_m * ((angle_m * (pi * (b_m - b_m))) - (a_m * (angle_m * pi)))) + (angle_m * (pi * (b_m ^ 2.0))));
	elseif ((angle_m / 180.0) <= 5e-129)
		tmp = 0.011111111111111112 * ((b_m * (pi * (angle_m * b_m))) - ((a_m ^ 2.0) * (angle_m * pi)));
	elseif ((angle_m / 180.0) <= 1e+73)
		tmp = t_0 * sin((0.011111111111111112 * (angle_m * pi)));
	else
		tmp = t_0 * (2.0 * sin(((angle_m / 180.0) * pi)));
	end
	tmp_2 = angle_s * tmp;
end

Wolfram

a_m = N[Abs[a], $MachinePrecision]
b_m = N[Abs[b], $MachinePrecision]
angle\_m = N[Abs[angle], $MachinePrecision]
angle\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[angle]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[angle$95$s_, a$95$m_, b$95$m_, angle$95$m_] := Block[{t$95$0 = N[(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], 2e-272], N[(0.011111111111111112 * N[(N[(a$95$m * N[(N[(angle$95$m * N[(Pi * N[(b$95$m - b$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(a$95$m * N[(angle$95$m * Pi), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(angle$95$m * N[(Pi * N[Power[b$95$m, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[N[(angle$95$m / 180.0), $MachinePrecision], 5e-129], N[(0.011111111111111112 * N[(N[(b$95$m * N[(Pi * N[(angle$95$m * b$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(N[Power[a$95$m, 2.0], $MachinePrecision] * N[(angle$95$m * Pi), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[N[(angle$95$m / 180.0), $MachinePrecision], 1e+73], N[(t$95$0 * N[Sin[N[(0.011111111111111112 * N[(angle$95$m * Pi), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(t$95$0 * N[(2.0 * N[Sin[N[(N[(angle$95$m / 180.0), $MachinePrecision] * Pi), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]), $MachinePrecision]]

Derivation

1. Split input into 4 regimes

2. if (/.f64 angle 180) < 1.99999999999999986e-272

   1. Initial program 47.7%

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

      1. associate-*l* 47.7%

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

      2. *-commutative 47.7%

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

      3. associate-*l* 48.4%

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

   3. Simplified 48.4%

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

   5. Taylor expanded in angle around 0 50.5%

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

      1. unpow2 48.4%

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

      2. unpow2 48.4%

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

      3. difference-of-squares 51.6%

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

   7. Applied egg-rr 53.7%

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

   8. Taylor expanded in a around 0 54.2%

      \[\leadsto 0.011111111111111112 \cdot \color{blue}{\left(a \cdot \left(-1 \cdot \left(a \cdot \left(angle \cdot \pi\right)\right) + angle \cdot \left(\pi \cdot \left(b + -1 \cdot b\right)\right)\right) + angle \cdot \left({b}^{2} \cdot \pi\right)\right)} \]

   if 1.99999999999999986e-272 < (/.f64 angle 180) < 5.00000000000000027e-129

   1. Initial program 71.1%

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

      1. associate-*l* 71.1%

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

      2. *-commutative 71.1%

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

      3. associate-*l* 71.1%

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

   3. Simplified 71.1%

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

   5. Taylor expanded in angle around 0 71.0%

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

      1. unpow2 71.1%

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

      2. unpow2 71.1%

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

      3. difference-of-squares 75.1%

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

   7. Applied egg-rr 75.0%

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

   8. Taylor expanded in b around 0 83.9%

      \[\leadsto 0.011111111111111112 \cdot \color{blue}{\left(-1 \cdot \left({a}^{2} \cdot \left(angle \cdot \pi\right)\right) + b \cdot \left(angle \cdot \left(b \cdot \pi\right) + angle \cdot \left(\pi \cdot \left(a + -1 \cdot a\right)\right)\right)\right)} \]
   9. Step-by-step derivation

      1. +-commutative 83.9%

         \[\leadsto 0.011111111111111112 \cdot \color{blue}{\left(b \cdot \left(angle \cdot \left(b \cdot \pi\right) + angle \cdot \left(\pi \cdot \left(a + -1 \cdot a\right)\right)\right) + -1 \cdot \left({a}^{2} \cdot \left(angle \cdot \pi\right)\right)\right)} \]

      2. mul-1-neg 83.9%

         \[\leadsto 0.011111111111111112 \cdot \left(b \cdot \left(angle \cdot \left(b \cdot \pi\right) + angle \cdot \left(\pi \cdot \left(a + -1 \cdot a\right)\right)\right) + \color{blue}{\left(-{a}^{2} \cdot \left(angle \cdot \pi\right)\right)}\right) \]

      3. unsub-neg 83.9%

         \[\leadsto 0.011111111111111112 \cdot \color{blue}{\left(b \cdot \left(angle \cdot \left(b \cdot \pi\right) + angle \cdot \left(\pi \cdot \left(a + -1 \cdot a\right)\right)\right) - {a}^{2} \cdot \left(angle \cdot \pi\right)\right)} \]

      4. distribute-lft-out 83.9%

         \[\leadsto 0.011111111111111112 \cdot \left(b \cdot \color{blue}{\left(angle \cdot \left(b \cdot \pi + \pi \cdot \left(a + -1 \cdot a\right)\right)\right)} - {a}^{2} \cdot \left(angle \cdot \pi\right)\right) \]

      5. *-commutative 83.9%

         \[\leadsto 0.011111111111111112 \cdot \left(b \cdot \left(angle \cdot \left(b \cdot \pi + \color{blue}{\left(a + -1 \cdot a\right) \cdot \pi}\right)\right) - {a}^{2} \cdot \left(angle \cdot \pi\right)\right) \]

      6. distribute-rgt1-in 83.9%

         \[\leadsto 0.011111111111111112 \cdot \left(b \cdot \left(angle \cdot \left(b \cdot \pi + \color{blue}{\left(\left(-1 + 1\right) \cdot a\right)} \cdot \pi\right)\right) - {a}^{2} \cdot \left(angle \cdot \pi\right)\right) \]

      7. metadata-eval 83.9%

         \[\leadsto 0.011111111111111112 \cdot \left(b \cdot \left(angle \cdot \left(b \cdot \pi + \left(\color{blue}{0} \cdot a\right) \cdot \pi\right)\right) - {a}^{2} \cdot \left(angle \cdot \pi\right)\right) \]

      8. mul0-lft 83.9%

         \[\leadsto 0.011111111111111112 \cdot \left(b \cdot \left(angle \cdot \left(b \cdot \pi + \color{blue}{0} \cdot \pi\right)\right) - {a}^{2} \cdot \left(angle \cdot \pi\right)\right) \]

      9. distribute-rgt-out 83.9%

         \[\leadsto 0.011111111111111112 \cdot \left(b \cdot \left(angle \cdot \color{blue}{\left(\pi \cdot \left(b + 0\right)\right)}\right) - {a}^{2} \cdot \left(angle \cdot \pi\right)\right) \]

      10. +-rgt-identity 83.9%

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

      11. *-commutative 83.9%

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

      12. associate-*r* 84.0%

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

   10. Simplified 84.0%

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

   if 5.00000000000000027e-129 < (/.f64 angle 180) < 9.99999999999999983e72

   1. Initial program 77.6%

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

      1. associate-*l* 77.6%

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

      2. *-commutative 77.6%

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

      3. associate-*l* 77.6%

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

   3. Simplified 77.6%

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

      1. *-commutative 77.6%

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

      2. sub-neg 77.6%

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

      3. distribute-lft-in 77.6%

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

      4. 2-sin 77.6%

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

      5. associate-*r* 77.6%

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

      6. div-inv 77.6%

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

      7. metadata-eval 77.6%

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

   6. Applied egg-rr 77.7%

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

      1. distribute-lft-out 77.7%

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

      2. sub-neg 77.7%

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

      3. *-commutative 77.7%

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

      4. associate-*r* 77.7%

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

      5. associate-*r* 77.2%

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

      6. *-commutative 77.2%

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

      7. *-commutative 77.2%

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

      8. associate-*r* 77.2%

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

      9. metadata-eval 77.2%

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

   8. Simplified 77.2%

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

      1. unpow2 77.6%

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

      2. unpow2 77.6%

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

      3. difference-of-squares 77.8%

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

   10. Applied egg-rr 77.4%

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

   if 9.99999999999999983e72 < (/.f64 angle 180)

   1. Initial program 25.1%

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

      1. associate-*l* 25.1%

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

      2. *-commutative 25.1%

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

      3. associate-*l* 25.1%

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

   3. Simplified 25.1%

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

      1. unpow2 25.1%

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

      2. unpow2 25.1%

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

      3. difference-of-squares 25.1%

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

   6. Applied egg-rr 25.1%

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

   7. Taylor expanded in angle around 0 30.4%

      \[\leadsto \left(\left(b + a\right) \cdot \left(b - a\right)\right) \cdot \left(2 \cdot \left(\sin \left(\pi \cdot \frac{angle}{180}\right) \cdot \color{blue}{1}\right)\right) \]
3. Recombined 4 regimes into one program.

4. Final simplification 56.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{angle}{180} \leq 2 \cdot 10^{-272}:\\ \;\;\;\;0.011111111111111112 \cdot \left(a \cdot \left(angle \cdot \left(\pi \cdot \left(b - b\right)\right) - a \cdot \left(angle \cdot \pi\right)\right) + angle \cdot \left(\pi \cdot {b}^{2}\right)\right)\\ \mathbf{elif}\;\frac{angle}{180} \leq 5 \cdot 10^{-129}:\\ \;\;\;\;0.011111111111111112 \cdot \left(b \cdot \left(\pi \cdot \left(angle \cdot b\right)\right) - {a}^{2} \cdot \left(angle \cdot \pi\right)\right)\\ \mathbf{elif}\;\frac{angle}{180} \leq 10^{+73}:\\ \;\;\;\;\left(\left(b + a\right) \cdot \left(b - a\right)\right) \cdot \sin \left(0.011111111111111112 \cdot \left(angle \cdot \pi\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\left(b + a\right) \cdot \left(b - a\right)\right) \cdot \left(2 \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 60.4% accurate, 3.2× speedup

Math

\[\begin{array}{l} a_m = \left|a\right| \\ b_m = \left|b\right| \\ angle\_m = \left|angle\right| \\ angle\_s = \mathsf{copysign}\left(1, angle\right) \\ \begin{array}{l} t_0 := \left(b\_m + a\_m\right) \cdot \left(b\_m - a\_m\right)\\ angle\_s \cdot \begin{array}{l} \mathbf{if}\;\frac{angle\_m}{180} \leq 5 \cdot 10^{-129}:\\ \;\;\;\;0.011111111111111112 \cdot \left(b\_m \cdot \left(\pi \cdot \left(angle\_m \cdot b\_m\right)\right) - {a\_m}^{2} \cdot \left(angle\_m \cdot \pi\right)\right)\\ \mathbf{elif}\;\frac{angle\_m}{180} \leq 10^{+73}:\\ \;\;\;\;t\_0 \cdot \sin \left(0.011111111111111112 \cdot \left(angle\_m \cdot \pi\right)\right)\\ \mathbf{else}:\\ \;\;\;\;t\_0 \cdot \left(2 \cdot \sin \left(\frac{angle\_m}{180} \cdot \pi\right)\right)\\ \end{array} \end{array} \end{array} \]

FPCore

a_m = (fabs.f64 a)
b_m = (fabs.f64 b)
angle\_m = (fabs.f64 angle)
angle\_s = (copysign.f64 1 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-129)
      (*
       0.011111111111111112
       (- (* b_m (* PI (* angle_m b_m))) (* (pow a_m 2.0) (* angle_m PI))))
      (if (<= (/ angle_m 180.0) 1e+73)
        (* t_0 (sin (* 0.011111111111111112 (* angle_m PI))))
        (* t_0 (* 2.0 (sin (* (/ angle_m 180.0) 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 tmp;
	if ((angle_m / 180.0) <= 5e-129) {
		tmp = 0.011111111111111112 * ((b_m * (((double) M_PI) * (angle_m * b_m))) - (pow(a_m, 2.0) * (angle_m * ((double) M_PI))));
	} else if ((angle_m / 180.0) <= 1e+73) {
		tmp = t_0 * sin((0.011111111111111112 * (angle_m * ((double) M_PI))));
	} else {
		tmp = t_0 * (2.0 * sin(((angle_m / 180.0) * ((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 tmp;
	if ((angle_m / 180.0) <= 5e-129) {
		tmp = 0.011111111111111112 * ((b_m * (Math.PI * (angle_m * b_m))) - (Math.pow(a_m, 2.0) * (angle_m * Math.PI)));
	} else if ((angle_m / 180.0) <= 1e+73) {
		tmp = t_0 * Math.sin((0.011111111111111112 * (angle_m * Math.PI)));
	} else {
		tmp = t_0 * (2.0 * Math.sin(((angle_m / 180.0) * Math.PI)));
	}
	return angle_s * tmp;
}

Python

a_m = math.fabs(a)
b_m = math.fabs(b)
angle\_m = math.fabs(angle)
angle\_s = math.copysign(1.0, angle)
def code(angle_s, a_m, b_m, angle_m):
	t_0 = (b_m + a_m) * (b_m - a_m)
	tmp = 0
	if (angle_m / 180.0) <= 5e-129:
		tmp = 0.011111111111111112 * ((b_m * (math.pi * (angle_m * b_m))) - (math.pow(a_m, 2.0) * (angle_m * math.pi)))
	elif (angle_m / 180.0) <= 1e+73:
		tmp = t_0 * math.sin((0.011111111111111112 * (angle_m * math.pi)))
	else:
		tmp = t_0 * (2.0 * math.sin(((angle_m / 180.0) * 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))
	tmp = 0.0
	if (Float64(angle_m / 180.0) <= 5e-129)
		tmp = Float64(0.011111111111111112 * Float64(Float64(b_m * Float64(pi * Float64(angle_m * b_m))) - Float64((a_m ^ 2.0) * Float64(angle_m * pi))));
	elseif (Float64(angle_m / 180.0) <= 1e+73)
		tmp = Float64(t_0 * sin(Float64(0.011111111111111112 * Float64(angle_m * pi))));
	else
		tmp = Float64(t_0 * Float64(2.0 * sin(Float64(Float64(angle_m / 180.0) * pi))));
	end
	return Float64(angle_s * tmp)
end

MATLAB

a_m = abs(a);
b_m = abs(b);
angle\_m = abs(angle);
angle\_s = sign(angle) * abs(1.0);
function tmp_2 = code(angle_s, a_m, b_m, angle_m)
	t_0 = (b_m + a_m) * (b_m - a_m);
	tmp = 0.0;
	if ((angle_m / 180.0) <= 5e-129)
		tmp = 0.011111111111111112 * ((b_m * (pi * (angle_m * b_m))) - ((a_m ^ 2.0) * (angle_m * pi)));
	elseif ((angle_m / 180.0) <= 1e+73)
		tmp = t_0 * sin((0.011111111111111112 * (angle_m * pi)));
	else
		tmp = t_0 * (2.0 * sin(((angle_m / 180.0) * pi)));
	end
	tmp_2 = angle_s * tmp;
end

Wolfram

a_m = N[Abs[a], $MachinePrecision]
b_m = N[Abs[b], $MachinePrecision]
angle\_m = N[Abs[angle], $MachinePrecision]
angle\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[angle]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[angle$95$s_, a$95$m_, b$95$m_, angle$95$m_] := Block[{t$95$0 = N[(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-129], N[(0.011111111111111112 * N[(N[(b$95$m * N[(Pi * N[(angle$95$m * b$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(N[Power[a$95$m, 2.0], $MachinePrecision] * N[(angle$95$m * Pi), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[N[(angle$95$m / 180.0), $MachinePrecision], 1e+73], N[(t$95$0 * N[Sin[N[(0.011111111111111112 * N[(angle$95$m * Pi), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(t$95$0 * N[(2.0 * N[Sin[N[(N[(angle$95$m / 180.0), $MachinePrecision] * Pi), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]), $MachinePrecision]]

Derivation

1. Split input into 3 regimes

2. if (/.f64 angle 180) < 5.00000000000000027e-129

   1. Initial program 52.2%

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

      1. associate-*l* 52.2%

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

      2. *-commutative 52.2%

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

      3. associate-*l* 52.8%

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

   3. Simplified 52.8%

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

   5. Taylor expanded in angle around 0 54.5%

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

      1. unpow2 52.8%

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

      2. unpow2 52.8%

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

      3. difference-of-squares 56.2%

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

   7. Applied egg-rr 57.8%

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

   8. Taylor expanded in b around 0 59.3%

      \[\leadsto 0.011111111111111112 \cdot \color{blue}{\left(-1 \cdot \left({a}^{2} \cdot \left(angle \cdot \pi\right)\right) + b \cdot \left(angle \cdot \left(b \cdot \pi\right) + angle \cdot \left(\pi \cdot \left(a + -1 \cdot a\right)\right)\right)\right)} \]
   9. Step-by-step derivation

      1. +-commutative 59.3%

         \[\leadsto 0.011111111111111112 \cdot \color{blue}{\left(b \cdot \left(angle \cdot \left(b \cdot \pi\right) + angle \cdot \left(\pi \cdot \left(a + -1 \cdot a\right)\right)\right) + -1 \cdot \left({a}^{2} \cdot \left(angle \cdot \pi\right)\right)\right)} \]

      2. mul-1-neg 59.3%

         \[\leadsto 0.011111111111111112 \cdot \left(b \cdot \left(angle \cdot \left(b \cdot \pi\right) + angle \cdot \left(\pi \cdot \left(a + -1 \cdot a\right)\right)\right) + \color{blue}{\left(-{a}^{2} \cdot \left(angle \cdot \pi\right)\right)}\right) \]

      3. unsub-neg 59.3%

         \[\leadsto 0.011111111111111112 \cdot \color{blue}{\left(b \cdot \left(angle \cdot \left(b \cdot \pi\right) + angle \cdot \left(\pi \cdot \left(a + -1 \cdot a\right)\right)\right) - {a}^{2} \cdot \left(angle \cdot \pi\right)\right)} \]

      4. distribute-lft-out 59.3%

         \[\leadsto 0.011111111111111112 \cdot \left(b \cdot \color{blue}{\left(angle \cdot \left(b \cdot \pi + \pi \cdot \left(a + -1 \cdot a\right)\right)\right)} - {a}^{2} \cdot \left(angle \cdot \pi\right)\right) \]

      5. *-commutative 59.3%

         \[\leadsto 0.011111111111111112 \cdot \left(b \cdot \left(angle \cdot \left(b \cdot \pi + \color{blue}{\left(a + -1 \cdot a\right) \cdot \pi}\right)\right) - {a}^{2} \cdot \left(angle \cdot \pi\right)\right) \]

      6. distribute-rgt1-in 59.3%

         \[\leadsto 0.011111111111111112 \cdot \left(b \cdot \left(angle \cdot \left(b \cdot \pi + \color{blue}{\left(\left(-1 + 1\right) \cdot a\right)} \cdot \pi\right)\right) - {a}^{2} \cdot \left(angle \cdot \pi\right)\right) \]

      7. metadata-eval 59.3%

         \[\leadsto 0.011111111111111112 \cdot \left(b \cdot \left(angle \cdot \left(b \cdot \pi + \left(\color{blue}{0} \cdot a\right) \cdot \pi\right)\right) - {a}^{2} \cdot \left(angle \cdot \pi\right)\right) \]

      8. mul0-lft 59.3%

         \[\leadsto 0.011111111111111112 \cdot \left(b \cdot \left(angle \cdot \left(b \cdot \pi + \color{blue}{0} \cdot \pi\right)\right) - {a}^{2} \cdot \left(angle \cdot \pi\right)\right) \]

      9. distribute-rgt-out 59.3%

         \[\leadsto 0.011111111111111112 \cdot \left(b \cdot \left(angle \cdot \color{blue}{\left(\pi \cdot \left(b + 0\right)\right)}\right) - {a}^{2} \cdot \left(angle \cdot \pi\right)\right) \]

      10. +-rgt-identity 59.3%

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

      11. *-commutative 59.3%

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

      12. associate-*r* 59.4%

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

   10. Simplified 59.4%

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

   if 5.00000000000000027e-129 < (/.f64 angle 180) < 9.99999999999999983e72

   1. Initial program 77.6%

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

      1. associate-*l* 77.6%

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

      2. *-commutative 77.6%

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

      3. associate-*l* 77.6%

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

   3. Simplified 77.6%

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

      1. *-commutative 77.6%

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

      2. sub-neg 77.6%

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

      3. distribute-lft-in 77.6%

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

      4. 2-sin 77.6%

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

      5. associate-*r* 77.6%

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

      6. div-inv 77.6%

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

      7. metadata-eval 77.6%

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

   6. Applied egg-rr 77.7%

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

      1. distribute-lft-out 77.7%

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

      2. sub-neg 77.7%

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

      3. *-commutative 77.7%

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

      4. associate-*r* 77.7%

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

      5. associate-*r* 77.2%

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

      6. *-commutative 77.2%

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

      7. *-commutative 77.2%

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

      8. associate-*r* 77.2%

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

      9. metadata-eval 77.2%

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

   8. Simplified 77.2%

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

      1. unpow2 77.6%

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

      2. unpow2 77.6%

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

      3. difference-of-squares 77.8%

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

   10. Applied egg-rr 77.4%

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

   if 9.99999999999999983e72 < (/.f64 angle 180)

   1. Initial program 25.1%

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

      1. associate-*l* 25.1%

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

      2. *-commutative 25.1%

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

      3. associate-*l* 25.1%

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

   3. Simplified 25.1%

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

      1. unpow2 25.1%

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

      2. unpow2 25.1%

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

      3. difference-of-squares 25.1%

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

   6. Applied egg-rr 25.1%

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

   7. Taylor expanded in angle around 0 30.4%

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

4. Final simplification 56.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{angle}{180} \leq 5 \cdot 10^{-129}:\\ \;\;\;\;0.011111111111111112 \cdot \left(b \cdot \left(\pi \cdot \left(angle \cdot b\right)\right) - {a}^{2} \cdot \left(angle \cdot \pi\right)\right)\\ \mathbf{elif}\;\frac{angle}{180} \leq 10^{+73}:\\ \;\;\;\;\left(\left(b + a\right) \cdot \left(b - a\right)\right) \cdot \sin \left(0.011111111111111112 \cdot \left(angle \cdot \pi\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\left(b + a\right) \cdot \left(b - a\right)\right) \cdot \left(2 \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 59.6% accurate, 3.4× speedup

Math

\[\begin{array}{l} a_m = \left|a\right| \\ b_m = \left|b\right| \\ angle\_m = \left|angle\right| \\ angle\_s = \mathsf{copysign}\left(1, angle\right) \\ \begin{array}{l} t_0 := \left(b\_m + a\_m\right) \cdot \left(b\_m - a\_m\right)\\ angle\_s \cdot \begin{array}{l} \mathbf{if}\;angle\_m \leq 1.35 \cdot 10^{-122}:\\ \;\;\;\;0.011111111111111112 \cdot \left(b\_m \cdot \left(\pi \cdot \left(angle\_m \cdot b\_m\right)\right) - {a\_m}^{2} \cdot \left(angle\_m \cdot \pi\right)\right)\\ \mathbf{elif}\;angle\_m \leq 5.2 \cdot 10^{+163}:\\ \;\;\;\;t\_0 \cdot \sin \left(0.011111111111111112 \cdot \left(angle\_m \cdot \pi\right)\right)\\ \mathbf{else}:\\ \;\;\;\;0.011111111111111112 \cdot \left(angle\_m \cdot \left(\pi \cdot t\_0\right)\right)\\ \end{array} \end{array} \end{array} \]

FPCore

a_m = (fabs.f64 a)
b_m = (fabs.f64 b)
angle\_m = (fabs.f64 angle)
angle\_s = (copysign.f64 1 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 1.35e-122)
      (*
       0.011111111111111112
       (- (* b_m (* PI (* angle_m b_m))) (* (pow a_m 2.0) (* angle_m PI))))
      (if (<= angle_m 5.2e+163)
        (* t_0 (sin (* 0.011111111111111112 (* angle_m PI))))
        (* 0.011111111111111112 (* angle_m (* PI 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 = (b_m + a_m) * (b_m - a_m);
	double tmp;
	if (angle_m <= 1.35e-122) {
		tmp = 0.011111111111111112 * ((b_m * (((double) M_PI) * (angle_m * b_m))) - (pow(a_m, 2.0) * (angle_m * ((double) M_PI))));
	} else if (angle_m <= 5.2e+163) {
		tmp = t_0 * sin((0.011111111111111112 * (angle_m * ((double) M_PI))));
	} else {
		tmp = 0.011111111111111112 * (angle_m * (((double) M_PI) * 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 = (b_m + a_m) * (b_m - a_m);
	double tmp;
	if (angle_m <= 1.35e-122) {
		tmp = 0.011111111111111112 * ((b_m * (Math.PI * (angle_m * b_m))) - (Math.pow(a_m, 2.0) * (angle_m * Math.PI)));
	} else if (angle_m <= 5.2e+163) {
		tmp = t_0 * Math.sin((0.011111111111111112 * (angle_m * Math.PI)));
	} else {
		tmp = 0.011111111111111112 * (angle_m * (Math.PI * t_0));
	}
	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)
	tmp = 0
	if angle_m <= 1.35e-122:
		tmp = 0.011111111111111112 * ((b_m * (math.pi * (angle_m * b_m))) - (math.pow(a_m, 2.0) * (angle_m * math.pi)))
	elif angle_m <= 5.2e+163:
		tmp = t_0 * math.sin((0.011111111111111112 * (angle_m * math.pi)))
	else:
		tmp = 0.011111111111111112 * (angle_m * (math.pi * 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(Float64(b_m + a_m) * Float64(b_m - a_m))
	tmp = 0.0
	if (angle_m <= 1.35e-122)
		tmp = Float64(0.011111111111111112 * Float64(Float64(b_m * Float64(pi * Float64(angle_m * b_m))) - Float64((a_m ^ 2.0) * Float64(angle_m * pi))));
	elseif (angle_m <= 5.2e+163)
		tmp = Float64(t_0 * sin(Float64(0.011111111111111112 * Float64(angle_m * pi))));
	else
		tmp = Float64(0.011111111111111112 * Float64(angle_m * Float64(pi * t_0)));
	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);
	tmp = 0.0;
	if (angle_m <= 1.35e-122)
		tmp = 0.011111111111111112 * ((b_m * (pi * (angle_m * b_m))) - ((a_m ^ 2.0) * (angle_m * pi)));
	elseif (angle_m <= 5.2e+163)
		tmp = t_0 * sin((0.011111111111111112 * (angle_m * pi)));
	else
		tmp = 0.011111111111111112 * (angle_m * (pi * t_0));
	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]}, N[(angle$95$s * If[LessEqual[angle$95$m, 1.35e-122], N[(0.011111111111111112 * N[(N[(b$95$m * N[(Pi * N[(angle$95$m * b$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(N[Power[a$95$m, 2.0], $MachinePrecision] * N[(angle$95$m * Pi), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[angle$95$m, 5.2e+163], N[(t$95$0 * N[Sin[N[(0.011111111111111112 * N[(angle$95$m * Pi), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(0.011111111111111112 * N[(angle$95$m * N[(Pi * t$95$0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]), $MachinePrecision]]

Derivation

1. Split input into 3 regimes

2. if angle < 1.35000000000000005e-122

   1. Initial program 52.2%

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

      1. associate-*l* 52.2%

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

      2. *-commutative 52.2%

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

      3. associate-*l* 52.8%

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

   3. Simplified 52.8%

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

   5. Taylor expanded in angle around 0 54.5%

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

      1. unpow2 52.8%

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

      2. unpow2 52.8%

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

      3. difference-of-squares 56.2%

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

   7. Applied egg-rr 57.8%

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

   8. Taylor expanded in b around 0 59.3%

      \[\leadsto 0.011111111111111112 \cdot \color{blue}{\left(-1 \cdot \left({a}^{2} \cdot \left(angle \cdot \pi\right)\right) + b \cdot \left(angle \cdot \left(b \cdot \pi\right) + angle \cdot \left(\pi \cdot \left(a + -1 \cdot a\right)\right)\right)\right)} \]
   9. Step-by-step derivation

      1. +-commutative 59.3%

         \[\leadsto 0.011111111111111112 \cdot \color{blue}{\left(b \cdot \left(angle \cdot \left(b \cdot \pi\right) + angle \cdot \left(\pi \cdot \left(a + -1 \cdot a\right)\right)\right) + -1 \cdot \left({a}^{2} \cdot \left(angle \cdot \pi\right)\right)\right)} \]

      2. mul-1-neg 59.3%

         \[\leadsto 0.011111111111111112 \cdot \left(b \cdot \left(angle \cdot \left(b \cdot \pi\right) + angle \cdot \left(\pi \cdot \left(a + -1 \cdot a\right)\right)\right) + \color{blue}{\left(-{a}^{2} \cdot \left(angle \cdot \pi\right)\right)}\right) \]

      3. unsub-neg 59.3%

         \[\leadsto 0.011111111111111112 \cdot \color{blue}{\left(b \cdot \left(angle \cdot \left(b \cdot \pi\right) + angle \cdot \left(\pi \cdot \left(a + -1 \cdot a\right)\right)\right) - {a}^{2} \cdot \left(angle \cdot \pi\right)\right)} \]

      4. distribute-lft-out 59.3%

         \[\leadsto 0.011111111111111112 \cdot \left(b \cdot \color{blue}{\left(angle \cdot \left(b \cdot \pi + \pi \cdot \left(a + -1 \cdot a\right)\right)\right)} - {a}^{2} \cdot \left(angle \cdot \pi\right)\right) \]

      5. *-commutative 59.3%

         \[\leadsto 0.011111111111111112 \cdot \left(b \cdot \left(angle \cdot \left(b \cdot \pi + \color{blue}{\left(a + -1 \cdot a\right) \cdot \pi}\right)\right) - {a}^{2} \cdot \left(angle \cdot \pi\right)\right) \]

      6. distribute-rgt1-in 59.3%

         \[\leadsto 0.011111111111111112 \cdot \left(b \cdot \left(angle \cdot \left(b \cdot \pi + \color{blue}{\left(\left(-1 + 1\right) \cdot a\right)} \cdot \pi\right)\right) - {a}^{2} \cdot \left(angle \cdot \pi\right)\right) \]

      7. metadata-eval 59.3%

         \[\leadsto 0.011111111111111112 \cdot \left(b \cdot \left(angle \cdot \left(b \cdot \pi + \left(\color{blue}{0} \cdot a\right) \cdot \pi\right)\right) - {a}^{2} \cdot \left(angle \cdot \pi\right)\right) \]

      8. mul0-lft 59.3%

         \[\leadsto 0.011111111111111112 \cdot \left(b \cdot \left(angle \cdot \left(b \cdot \pi + \color{blue}{0} \cdot \pi\right)\right) - {a}^{2} \cdot \left(angle \cdot \pi\right)\right) \]

      9. distribute-rgt-out 59.3%

         \[\leadsto 0.011111111111111112 \cdot \left(b \cdot \left(angle \cdot \color{blue}{\left(\pi \cdot \left(b + 0\right)\right)}\right) - {a}^{2} \cdot \left(angle \cdot \pi\right)\right) \]

      10. +-rgt-identity 59.3%

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

      11. *-commutative 59.3%

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

      12. associate-*r* 59.4%

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

   10. Simplified 59.4%

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

   if 1.35000000000000005e-122 < angle < 5.2000000000000003e163

   1. Initial program 62.6%

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

      1. associate-*l* 62.6%

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

      2. *-commutative 62.6%

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

      3. associate-*l* 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. *-commutative 62.6%

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

      2. sub-neg 62.6%

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

      3. distribute-lft-in 62.6%

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

      4. 2-sin 62.6%

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

      5. associate-*r* 62.6%

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

      6. div-inv 62.9%

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

      7. metadata-eval 62.9%

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

   6. Applied egg-rr 62.9%

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

      1. distribute-lft-out 62.9%

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

      2. sub-neg 62.9%

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

      3. *-commutative 62.9%

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

      4. associate-*r* 62.9%

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

      5. associate-*r* 63.2%

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

      6. *-commutative 63.2%

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

      7. *-commutative 63.2%

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

      8. associate-*r* 63.2%

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

      9. metadata-eval 63.2%

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

   8. Simplified 63.2%

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

      1. unpow2 62.6%

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

      2. unpow2 62.6%

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

      3. difference-of-squares 62.8%

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

   10. Applied egg-rr 63.3%

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

   if 5.2000000000000003e163 < angle

   1. Initial program 19.0%

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

      1. associate-*l* 19.0%

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

      2. *-commutative 19.0%

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

      3. associate-*l* 19.0%

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

   3. Simplified 19.0%

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

   5. Taylor expanded in angle around 0 25.2%

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

      1. unpow2 19.0%

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

      2. unpow2 19.0%

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

      3. difference-of-squares 19.0%

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

   7. Applied egg-rr 25.2%

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

4. Final simplification 55.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;angle \leq 1.35 \cdot 10^{-122}:\\ \;\;\;\;0.011111111111111112 \cdot \left(b \cdot \left(\pi \cdot \left(angle \cdot b\right)\right) - {a}^{2} \cdot \left(angle \cdot \pi\right)\right)\\ \mathbf{elif}\;angle \leq 5.2 \cdot 10^{+163}:\\ \;\;\;\;\left(\left(b + a\right) \cdot \left(b - a\right)\right) \cdot \sin \left(0.011111111111111112 \cdot \left(angle \cdot \pi\right)\right)\\ \mathbf{else}:\\ \;\;\;\;0.011111111111111112 \cdot \left(angle \cdot \left(\pi \cdot \left(\left(b + a\right) \cdot \left(b - a\right)\right)\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 56.7% accurate, 3.6× speedup

Math

\[\begin{array}{l} a_m = \left|a\right| \\ b_m = \left|b\right| \\ angle\_m = \left|angle\right| \\ angle\_s = \mathsf{copysign}\left(1, angle\right) \\ \begin{array}{l} t_0 := \left(b\_m + a\_m\right) \cdot \left(b\_m - a\_m\right)\\ angle\_s \cdot \begin{array}{l} \mathbf{if}\;angle\_m \leq 5.2 \cdot 10^{+163}:\\ \;\;\;\;t\_0 \cdot \sin \left(0.011111111111111112 \cdot \left(angle\_m \cdot \pi\right)\right)\\ \mathbf{else}:\\ \;\;\;\;0.011111111111111112 \cdot \left(angle\_m \cdot \left(\pi \cdot t\_0\right)\right)\\ \end{array} \end{array} \end{array} \]

FPCore

a_m = (fabs.f64 a)
b_m = (fabs.f64 b)
angle\_m = (fabs.f64 angle)
angle\_s = (copysign.f64 1 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 5.2e+163)
      (* t_0 (sin (* 0.011111111111111112 (* angle_m PI))))
      (* 0.011111111111111112 (* angle_m (* PI 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 = (b_m + a_m) * (b_m - a_m);
	double tmp;
	if (angle_m <= 5.2e+163) {
		tmp = t_0 * sin((0.011111111111111112 * (angle_m * ((double) M_PI))));
	} else {
		tmp = 0.011111111111111112 * (angle_m * (((double) M_PI) * 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 = (b_m + a_m) * (b_m - a_m);
	double tmp;
	if (angle_m <= 5.2e+163) {
		tmp = t_0 * Math.sin((0.011111111111111112 * (angle_m * Math.PI)));
	} else {
		tmp = 0.011111111111111112 * (angle_m * (Math.PI * t_0));
	}
	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)
	tmp = 0
	if angle_m <= 5.2e+163:
		tmp = t_0 * math.sin((0.011111111111111112 * (angle_m * math.pi)))
	else:
		tmp = 0.011111111111111112 * (angle_m * (math.pi * 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(Float64(b_m + a_m) * Float64(b_m - a_m))
	tmp = 0.0
	if (angle_m <= 5.2e+163)
		tmp = Float64(t_0 * sin(Float64(0.011111111111111112 * Float64(angle_m * pi))));
	else
		tmp = Float64(0.011111111111111112 * Float64(angle_m * Float64(pi * t_0)));
	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);
	tmp = 0.0;
	if (angle_m <= 5.2e+163)
		tmp = t_0 * sin((0.011111111111111112 * (angle_m * pi)));
	else
		tmp = 0.011111111111111112 * (angle_m * (pi * t_0));
	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]}, N[(angle$95$s * If[LessEqual[angle$95$m, 5.2e+163], N[(t$95$0 * N[Sin[N[(0.011111111111111112 * N[(angle$95$m * Pi), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(0.011111111111111112 * N[(angle$95$m * N[(Pi * t$95$0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if angle < 5.2000000000000003e163

   1. Initial program 55.3%

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

      1. associate-*l* 55.4%

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

      2. *-commutative 55.4%

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

      3. associate-*l* 55.8%

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

   3. Simplified 55.8%

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

      1. *-commutative 55.8%

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

      2. sub-neg 55.8%

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

      3. distribute-lft-in 55.8%

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

      4. 2-sin 55.8%

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

      5. associate-*r* 55.8%

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

      6. div-inv 56.3%

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

      7. metadata-eval 56.3%

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

   6. Applied egg-rr 56.3%

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

      1. distribute-lft-out 56.3%

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

      2. sub-neg 56.3%

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

      3. *-commutative 56.3%

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

      4. associate-*r* 56.3%

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

      5. associate-*r* 56.4%

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

      6. *-commutative 56.4%

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

      7. *-commutative 56.4%

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

      8. associate-*r* 56.4%

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

      9. metadata-eval 56.4%

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

   8. Simplified 56.4%

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

      1. unpow2 55.8%

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

      2. unpow2 55.8%

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

      3. difference-of-squares 58.1%

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

   10. Applied egg-rr 58.3%

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

   if 5.2000000000000003e163 < angle

   1. Initial program 19.0%

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

      1. associate-*l* 19.0%

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

      2. *-commutative 19.0%

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

      3. associate-*l* 19.0%

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

   3. Simplified 19.0%

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

   5. Taylor expanded in angle around 0 25.2%

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

      1. unpow2 19.0%

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

      2. unpow2 19.0%

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

      3. difference-of-squares 19.0%

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

   7. Applied egg-rr 25.2%

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

4. Final simplification 53.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;angle \leq 5.2 \cdot 10^{+163}:\\ \;\;\;\;\left(\left(b + a\right) \cdot \left(b - a\right)\right) \cdot \sin \left(0.011111111111111112 \cdot \left(angle \cdot \pi\right)\right)\\ \mathbf{else}:\\ \;\;\;\;0.011111111111111112 \cdot \left(angle \cdot \left(\pi \cdot \left(\left(b + a\right) \cdot \left(b - a\right)\right)\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 12: 67.6% accurate, 3.6× speedup

Math

\[\begin{array}{l} a_m = \left|a\right| \\ b_m = \left|b\right| \\ angle\_m = \left|angle\right| \\ angle\_s = \mathsf{copysign}\left(1, angle\right) \\ angle\_s \cdot \left(\left(b\_m + a\_m\right) \cdot \left(\left(b\_m - a\_m\right) \cdot \sin \left(2 \cdot \left(0.005555555555555556 \cdot \left(angle\_m \cdot \pi\right)\right)\right)\right)\right) \end{array} \]

FPCore

a_m = (fabs.f64 a)
b_m = (fabs.f64 b)
angle\_m = (fabs.f64 angle)
angle\_s = (copysign.f64 1 angle)
(FPCore (angle_s a_m b_m angle_m)
 :precision binary64
 (*
  angle_s
  (*
   (+ b_m a_m)
   (* (- b_m a_m) (sin (* 2.0 (* 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) {
	return angle_s * ((b_m + a_m) * ((b_m - a_m) * sin((2.0 * (0.005555555555555556 * (angle_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 * ((b_m + a_m) * ((b_m - a_m) * Math.sin((2.0 * (0.005555555555555556 * (angle_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 * ((b_m + a_m) * ((b_m - a_m) * math.sin((2.0 * (0.005555555555555556 * (angle_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(Float64(b_m + a_m) * Float64(Float64(b_m - a_m) * sin(Float64(2.0 * Float64(0.005555555555555556 * Float64(angle_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 * ((b_m + a_m) * ((b_m - a_m) * sin((2.0 * (0.005555555555555556 * (angle_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[(N[(b$95$m + a$95$m), $MachinePrecision] * N[(N[(b$95$m - a$95$m), $MachinePrecision] * N[Sin[N[(2.0 * N[(0.005555555555555556 * N[(angle$95$m * Pi), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 50.5%

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

   1. associate-*l* 50.5%

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

   2. *-commutative 50.5%

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

   3. associate-*l* 50.9%

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

3. Simplified 50.9%

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

   1. unpow2 50.9%

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

   2. unpow2 50.9%

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

   3. difference-of-squares 53.0%

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

6. Applied egg-rr 53.0%

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

   1. add-cube-cbrt 52.9%

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

   2. pow3 52.9%

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

   3. div-inv 53.8%

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

   4. metadata-eval 53.8%

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

8. Applied egg-rr 53.8%

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

   1. pow1 53.8%

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

10. Applied egg-rr 62.1%

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

11. Final simplification 62.1%

    \[\leadsto \left(b + a\right) \cdot \left(\left(b - a\right) \cdot \sin \left(2 \cdot \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)\right)\right) \]
12. Add Preprocessing

Alternative 13: 54.1% 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 1 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 50.5%

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

   1. associate-*l* 50.5%

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

   2. *-commutative 50.5%

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

   3. associate-*l* 50.9%

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

3. Simplified 50.9%

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

5. Taylor expanded in angle around 0 50.0%

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

   1. unpow2 50.9%

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

   2. unpow2 50.9%

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

   3. difference-of-squares 53.0%

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

7. Applied egg-rr 52.1%

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

8. Final simplification 52.1%

   \[\leadsto 0.011111111111111112 \cdot \left(angle \cdot \left(\pi \cdot \left(\left(b + a\right) \cdot \left(b - a\right)\right)\right)\right) \]
9. Add Preprocessing

Alternative 14: 54.1% accurate, 32.2× speedup

Math

\[\begin{array}{l} a_m = \left|a\right| \\ b_m = \left|b\right| \\ angle\_m = \left|angle\right| \\ angle\_s = \mathsf{copysign}\left(1, angle\right) \\ angle\_s \cdot \left(0.011111111111111112 \cdot \left(angle\_m \cdot \left(\left(b\_m - a\_m\right) \cdot \left(\left(b\_m + a\_m\right) \cdot \pi\right)\right)\right)\right) \end{array} \]

FPCore

a_m = (fabs.f64 a)
b_m = (fabs.f64 b)
angle\_m = (fabs.f64 angle)
angle\_s = (copysign.f64 1 angle)
(FPCore (angle_s a_m b_m angle_m)
 :precision binary64
 (*
  angle_s
  (* 0.011111111111111112 (* angle_m (* (- b_m a_m) (* (+ b_m a_m) PI))))))

C

a_m = fabs(a);
b_m = fabs(b);
angle\_m = fabs(angle);
angle\_s = copysign(1.0, angle);
double code(double angle_s, double a_m, double b_m, double angle_m) {
	return angle_s * (0.011111111111111112 * (angle_m * ((b_m - a_m) * ((b_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) * ((b_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) * ((b_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(Float64(b_m + 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) * ((b_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[(N[(b$95$m + a$95$m), $MachinePrecision] * Pi), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 50.5%

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

   1. associate-*l* 50.5%

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

   2. *-commutative 50.5%

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

   3. associate-*l* 50.9%

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

3. Simplified 50.9%

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

5. Taylor expanded in angle around 0 50.0%

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

   1. unpow2 50.9%

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

   2. unpow2 50.9%

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

   3. difference-of-squares 53.0%

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

7. Applied egg-rr 52.1%

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

   1. associate-*r* 52.1%

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

   2. sub-neg 52.1%

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

   3. distribute-lft-in 48.9%

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

9. Applied egg-rr 48.9%

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

    1. distribute-lft-out 52.1%

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

    2. sub-neg 52.1%

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

11. Applied egg-rr 52.1%

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

12. Final simplification 52.1%

    \[\leadsto 0.011111111111111112 \cdot \left(angle \cdot \left(\left(b - a\right) \cdot \left(\left(b + a\right) \cdot \pi\right)\right)\right) \]
13. Add Preprocessing

Reproduce

herbie shell --seed 2024080 
(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)))))