ab-angle->ABCF B

Percentage Accurate: 53.1% → 67.5%

Time: 18.6s

Alternatives: 20

Speedup: 32.2×

• 33.4% 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: 53.1% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 67.5% accurate, 0.7× speedup

Math

\[\begin{array}{l} angle\_m = \left|angle\right| \\ angle\_s = \mathsf{copysign}\left(1, angle\right) \\ \begin{array}{l} t_0 := \left(angle\_m \cdot \pi\right) \cdot 0.011111111111111112\\ t_1 := \frac{{\left(\sqrt{\pi}\right)}^{2}}{\frac{180}{angle\_m}}\\ angle\_s \cdot \begin{array}{l} \mathbf{if}\;\frac{angle\_m}{180} \leq 10^{+98}:\\ \;\;\;\;\left(b - a\right) \cdot \left(\left(b + a\right) \cdot \sin t\_0\right)\\ \mathbf{elif}\;\frac{angle\_m}{180} \leq 10^{+135}:\\ \;\;\;\;\frac{2 \cdot \sin \left(e^{0 - \log \left(\frac{180}{angle\_m \cdot \pi}\right)}\right)}{\frac{1}{b \cdot b - a \cdot a}} \cdot \cos \left(\frac{angle\_m}{180} \cdot \pi\right)\\ \mathbf{elif}\;\frac{angle\_m}{180} \leq 10^{+163}:\\ \;\;\;\;\left(b + a\right) \cdot \left(\left(b - a\right) \cdot t\_0\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\left(b + a\right) \cdot \left(2 \cdot \sin t\_1\right)\right) \cdot \left(\left(b - a\right) \cdot \cos t\_1\right)\\ \end{array} \end{array} \end{array} \]

FPCore

angle\_m = (fabs.f64 angle)
angle\_s = (copysign.f64 #s(literal 1 binary64) angle)
(FPCore (angle_s a b angle_m)
 :precision binary64
 (let* ((t_0 (* (* angle_m PI) 0.011111111111111112))
        (t_1 (/ (pow (sqrt PI) 2.0) (/ 180.0 angle_m))))
   (*
    angle_s
    (if (<= (/ angle_m 180.0) 1e+98)
      (* (- b a) (* (+ b a) (sin t_0)))
      (if (<= (/ angle_m 180.0) 1e+135)
        (*
         (/
          (* 2.0 (sin (exp (- 0.0 (log (/ 180.0 (* angle_m PI)))))))
          (/ 1.0 (- (* b b) (* a a))))
         (cos (* (/ angle_m 180.0) PI)))
        (if (<= (/ angle_m 180.0) 1e+163)
          (* (+ b a) (* (- b a) t_0))
          (* (* (+ b a) (* 2.0 (sin t_1))) (* (- b a) (cos t_1)))))))))

C

angle\_m = fabs(angle);
angle\_s = copysign(1.0, angle);
double code(double angle_s, double a, double b, double angle_m) {
	double t_0 = (angle_m * ((double) M_PI)) * 0.011111111111111112;
	double t_1 = pow(sqrt(((double) M_PI)), 2.0) / (180.0 / angle_m);
	double tmp;
	if ((angle_m / 180.0) <= 1e+98) {
		tmp = (b - a) * ((b + a) * sin(t_0));
	} else if ((angle_m / 180.0) <= 1e+135) {
		tmp = ((2.0 * sin(exp((0.0 - log((180.0 / (angle_m * ((double) M_PI)))))))) / (1.0 / ((b * b) - (a * a)))) * cos(((angle_m / 180.0) * ((double) M_PI)));
	} else if ((angle_m / 180.0) <= 1e+163) {
		tmp = (b + a) * ((b - a) * t_0);
	} else {
		tmp = ((b + a) * (2.0 * sin(t_1))) * ((b - a) * cos(t_1));
	}
	return angle_s * tmp;
}

Java

angle\_m = Math.abs(angle);
angle\_s = Math.copySign(1.0, angle);
public static double code(double angle_s, double a, double b, double angle_m) {
	double t_0 = (angle_m * Math.PI) * 0.011111111111111112;
	double t_1 = Math.pow(Math.sqrt(Math.PI), 2.0) / (180.0 / angle_m);
	double tmp;
	if ((angle_m / 180.0) <= 1e+98) {
		tmp = (b - a) * ((b + a) * Math.sin(t_0));
	} else if ((angle_m / 180.0) <= 1e+135) {
		tmp = ((2.0 * Math.sin(Math.exp((0.0 - Math.log((180.0 / (angle_m * Math.PI))))))) / (1.0 / ((b * b) - (a * a)))) * Math.cos(((angle_m / 180.0) * Math.PI));
	} else if ((angle_m / 180.0) <= 1e+163) {
		tmp = (b + a) * ((b - a) * t_0);
	} else {
		tmp = ((b + a) * (2.0 * Math.sin(t_1))) * ((b - a) * Math.cos(t_1));
	}
	return angle_s * tmp;
}

Python

angle\_m = math.fabs(angle)
angle\_s = math.copysign(1.0, angle)
def code(angle_s, a, b, angle_m):
	t_0 = (angle_m * math.pi) * 0.011111111111111112
	t_1 = math.pow(math.sqrt(math.pi), 2.0) / (180.0 / angle_m)
	tmp = 0
	if (angle_m / 180.0) <= 1e+98:
		tmp = (b - a) * ((b + a) * math.sin(t_0))
	elif (angle_m / 180.0) <= 1e+135:
		tmp = ((2.0 * math.sin(math.exp((0.0 - math.log((180.0 / (angle_m * math.pi))))))) / (1.0 / ((b * b) - (a * a)))) * math.cos(((angle_m / 180.0) * math.pi))
	elif (angle_m / 180.0) <= 1e+163:
		tmp = (b + a) * ((b - a) * t_0)
	else:
		tmp = ((b + a) * (2.0 * math.sin(t_1))) * ((b - a) * math.cos(t_1))
	return angle_s * tmp

Julia

angle\_m = abs(angle)
angle\_s = copysign(1.0, angle)
function code(angle_s, a, b, angle_m)
	t_0 = Float64(Float64(angle_m * pi) * 0.011111111111111112)
	t_1 = Float64((sqrt(pi) ^ 2.0) / Float64(180.0 / angle_m))
	tmp = 0.0
	if (Float64(angle_m / 180.0) <= 1e+98)
		tmp = Float64(Float64(b - a) * Float64(Float64(b + a) * sin(t_0)));
	elseif (Float64(angle_m / 180.0) <= 1e+135)
		tmp = Float64(Float64(Float64(2.0 * sin(exp(Float64(0.0 - log(Float64(180.0 / Float64(angle_m * pi))))))) / Float64(1.0 / Float64(Float64(b * b) - Float64(a * a)))) * cos(Float64(Float64(angle_m / 180.0) * pi)));
	elseif (Float64(angle_m / 180.0) <= 1e+163)
		tmp = Float64(Float64(b + a) * Float64(Float64(b - a) * t_0));
	else
		tmp = Float64(Float64(Float64(b + a) * Float64(2.0 * sin(t_1))) * Float64(Float64(b - a) * cos(t_1)));
	end
	return Float64(angle_s * tmp)
end

MATLAB

angle\_m = abs(angle);
angle\_s = sign(angle) * abs(1.0);
function tmp_2 = code(angle_s, a, b, angle_m)
	t_0 = (angle_m * pi) * 0.011111111111111112;
	t_1 = (sqrt(pi) ^ 2.0) / (180.0 / angle_m);
	tmp = 0.0;
	if ((angle_m / 180.0) <= 1e+98)
		tmp = (b - a) * ((b + a) * sin(t_0));
	elseif ((angle_m / 180.0) <= 1e+135)
		tmp = ((2.0 * sin(exp((0.0 - log((180.0 / (angle_m * pi))))))) / (1.0 / ((b * b) - (a * a)))) * cos(((angle_m / 180.0) * pi));
	elseif ((angle_m / 180.0) <= 1e+163)
		tmp = (b + a) * ((b - a) * t_0);
	else
		tmp = ((b + a) * (2.0 * sin(t_1))) * ((b - a) * cos(t_1));
	end
	tmp_2 = angle_s * tmp;
end

Wolfram

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_, b_, angle$95$m_] := Block[{t$95$0 = N[(N[(angle$95$m * Pi), $MachinePrecision] * 0.011111111111111112), $MachinePrecision]}, Block[{t$95$1 = N[(N[Power[N[Sqrt[Pi], $MachinePrecision], 2.0], $MachinePrecision] / N[(180.0 / angle$95$m), $MachinePrecision]), $MachinePrecision]}, N[(angle$95$s * If[LessEqual[N[(angle$95$m / 180.0), $MachinePrecision], 1e+98], N[(N[(b - a), $MachinePrecision] * N[(N[(b + a), $MachinePrecision] * N[Sin[t$95$0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[N[(angle$95$m / 180.0), $MachinePrecision], 1e+135], N[(N[(N[(2.0 * N[Sin[N[Exp[N[(0.0 - N[Log[N[(180.0 / N[(angle$95$m * Pi), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / N[(1.0 / N[(N[(b * b), $MachinePrecision] - N[(a * a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[Cos[N[(N[(angle$95$m / 180.0), $MachinePrecision] * Pi), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[N[(angle$95$m / 180.0), $MachinePrecision], 1e+163], N[(N[(b + a), $MachinePrecision] * N[(N[(b - a), $MachinePrecision] * t$95$0), $MachinePrecision]), $MachinePrecision], N[(N[(N[(b + a), $MachinePrecision] * N[(2.0 * N[Sin[t$95$1], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(N[(b - a), $MachinePrecision] * N[Cos[t$95$1], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]), $MachinePrecision]]]

Derivation

1. Split input into 4 regimes

2. if (/.f64 angle #s(literal 180 binary64)) < 9.99999999999999998e97

   1. Initial program 58.9%

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

      1. associate-*l* N/A

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

      2. *-commutative N/A

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

      3. associate-*l* N/A

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

      4. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\sin \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right), \color{blue}{\left(\cos \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right) \cdot \left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right)\right)}\right) \]

      5. sin-lowering-sin.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sin.f64}\left(\left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right)\right), \left(\color{blue}{\cos \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right)} \cdot \left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right)\right)\right) \]

      6. associate-*r/ N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sin.f64}\left(\left(\frac{\mathsf{PI}\left(\right) \cdot angle}{180}\right)\right), \left(\cos \color{blue}{\left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right)} \cdot \left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right)\right)\right) \]

      7. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sin.f64}\left(\mathsf{/.f64}\left(\left(\mathsf{PI}\left(\right) \cdot angle\right), 180\right)\right), \left(\cos \color{blue}{\left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right)} \cdot \left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right)\right)\right) \]

      8. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sin.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{PI}\left(\right), angle\right), 180\right)\right), \left(\cos \left(\color{blue}{\mathsf{PI}\left(\right)} \cdot \frac{angle}{180}\right) \cdot \left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right)\right)\right) \]

      9. PI-lowering-PI.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sin.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{PI.f64}\left(\right), angle\right), 180\right)\right), \left(\cos \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right) \cdot \left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right)\right)\right) \]

      10. *-commutative N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{sin.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{PI.f64}\left(\right), angle\right), 180\right)\right), \left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \color{blue}{\cos \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right)}\right)\right) \]

      11. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{sin.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{PI.f64}\left(\right), angle\right), 180\right)\right), \mathsf{*.f64}\left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right), \color{blue}{\cos \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right)}\right)\right) \]

   3. Simplified 59.1%

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

      1. associate-*r/ N/A

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

      2. *-commutative N/A

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

      3. associate-*r* N/A

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

      4. associate-*r/ N/A

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

      5. associate-*r* N/A

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

      6. *-commutative N/A

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

      7. *-commutative N/A

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

      8. difference-of-squares N/A

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

   6. Applied egg-rr 75.3%

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

   if 9.99999999999999998e97 < (/.f64 angle #s(literal 180 binary64)) < 9.99999999999999962e134

   1. Initial program 24.1%

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

      1. *-commutative N/A

         \[\leadsto \mathsf{*.f64}\left(\left(\sin \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right) \cdot \left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right)\right), \mathsf{cos.f64}\left(\color{blue}{\mathsf{*.f64}\left(\mathsf{PI.f64}\left(\right), \mathsf{/.f64}\left(angle, 180\right)\right)}\right)\right) \]

      2. pow2 N/A

         \[\leadsto \mathsf{*.f64}\left(\left(\sin \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right) \cdot \left(2 \cdot \left(b \cdot b - {a}^{2}\right)\right)\right), \mathsf{cos.f64}\left(\mathsf{*.f64}\left(\mathsf{PI.f64}\left(\right), \mathsf{/.f64}\left(angle, 180\right)\right)\right)\right) \]

      3. pow2 N/A

         \[\leadsto \mathsf{*.f64}\left(\left(\sin \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right) \cdot \left(2 \cdot \left(b \cdot b - a \cdot a\right)\right)\right), \mathsf{cos.f64}\left(\mathsf{*.f64}\left(\mathsf{PI.f64}\left(\right), \mathsf{/.f64}\left(angle, 180\right)\right)\right)\right) \]

      4. associate-*r* N/A

         \[\leadsto \mathsf{*.f64}\left(\left(\left(\sin \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right) \cdot 2\right) \cdot \left(b \cdot b - a \cdot a\right)\right), \mathsf{cos.f64}\left(\color{blue}{\mathsf{*.f64}\left(\mathsf{PI.f64}\left(\right), \mathsf{/.f64}\left(angle, 180\right)\right)}\right)\right) \]

      5. flip-- N/A

         \[\leadsto \mathsf{*.f64}\left(\left(\left(\sin \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right) \cdot 2\right) \cdot \frac{\left(b \cdot b\right) \cdot \left(b \cdot b\right) - \left(a \cdot a\right) \cdot \left(a \cdot a\right)}{b \cdot b + a \cdot a}\right), \mathsf{cos.f64}\left(\mathsf{*.f64}\left(\mathsf{PI.f64}\left(\right), \color{blue}{\mathsf{/.f64}\left(angle, 180\right)}\right)\right)\right) \]

      6. clear-num N/A

         \[\leadsto \mathsf{*.f64}\left(\left(\left(\sin \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right) \cdot 2\right) \cdot \frac{1}{\frac{b \cdot b + a \cdot a}{\left(b \cdot b\right) \cdot \left(b \cdot b\right) - \left(a \cdot a\right) \cdot \left(a \cdot a\right)}}\right), \mathsf{cos.f64}\left(\mathsf{*.f64}\left(\mathsf{PI.f64}\left(\right), \color{blue}{\mathsf{/.f64}\left(angle, 180\right)}\right)\right)\right) \]

      7. un-div-inv N/A

         \[\leadsto \mathsf{*.f64}\left(\left(\frac{\sin \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right) \cdot 2}{\frac{b \cdot b + a \cdot a}{\left(b \cdot b\right) \cdot \left(b \cdot b\right) - \left(a \cdot a\right) \cdot \left(a \cdot a\right)}}\right), \mathsf{cos.f64}\left(\color{blue}{\mathsf{*.f64}\left(\mathsf{PI.f64}\left(\right), \mathsf{/.f64}\left(angle, 180\right)\right)}\right)\right) \]

      8. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\left(\sin \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right) \cdot 2\right), \left(\frac{b \cdot b + a \cdot a}{\left(b \cdot b\right) \cdot \left(b \cdot b\right) - \left(a \cdot a\right) \cdot \left(a \cdot a\right)}\right)\right), \mathsf{cos.f64}\left(\color{blue}{\mathsf{*.f64}\left(\mathsf{PI.f64}\left(\right), \mathsf{/.f64}\left(angle, 180\right)\right)}\right)\right) \]

   4. Applied egg-rr 31.9%

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

      1. associate-/r/ N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(2, \mathsf{sin.f64}\left(\left(\frac{\mathsf{PI}\left(\right)}{180} \cdot angle\right)\right)\right), \mathsf{/.f64}\left(1, \mathsf{\_.f64}\left(\mathsf{*.f64}\left(b, b\right), \mathsf{*.f64}\left(a, a\right)\right)\right)\right), \mathsf{cos.f64}\left(\mathsf{*.f64}\left(\mathsf{PI.f64}\left(\right), \mathsf{/.f64}\left(angle, 180\right)\right)\right)\right) \]

      2. associate-*l/ N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(2, \mathsf{sin.f64}\left(\left(\frac{\mathsf{PI}\left(\right) \cdot angle}{180}\right)\right)\right), \mathsf{/.f64}\left(1, \mathsf{\_.f64}\left(\mathsf{*.f64}\left(b, b\right), \mathsf{*.f64}\left(a, a\right)\right)\right)\right), \mathsf{cos.f64}\left(\mathsf{*.f64}\left(\mathsf{PI.f64}\left(\right), \mathsf{/.f64}\left(angle, 180\right)\right)\right)\right) \]

      3. clear-num N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(2, \mathsf{sin.f64}\left(\left(\frac{1}{\frac{180}{\mathsf{PI}\left(\right) \cdot angle}}\right)\right)\right), \mathsf{/.f64}\left(1, \mathsf{\_.f64}\left(\mathsf{*.f64}\left(b, b\right), \mathsf{*.f64}\left(a, a\right)\right)\right)\right), \mathsf{cos.f64}\left(\mathsf{*.f64}\left(\mathsf{PI.f64}\left(\right), \mathsf{/.f64}\left(angle, 180\right)\right)\right)\right) \]

      4. inv-pow N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(2, \mathsf{sin.f64}\left(\left({\left(\frac{180}{\mathsf{PI}\left(\right) \cdot angle}\right)}^{-1}\right)\right)\right), \mathsf{/.f64}\left(1, \mathsf{\_.f64}\left(\mathsf{*.f64}\left(b, b\right), \mathsf{*.f64}\left(a, a\right)\right)\right)\right), \mathsf{cos.f64}\left(\mathsf{*.f64}\left(\mathsf{PI.f64}\left(\right), \mathsf{/.f64}\left(angle, 180\right)\right)\right)\right) \]

      5. pow-to-exp N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(2, \mathsf{sin.f64}\left(\left(e^{\log \left(\frac{180}{\mathsf{PI}\left(\right) \cdot angle}\right) \cdot -1}\right)\right)\right), \mathsf{/.f64}\left(1, \mathsf{\_.f64}\left(\mathsf{*.f64}\left(b, b\right), \mathsf{*.f64}\left(a, a\right)\right)\right)\right), \mathsf{cos.f64}\left(\mathsf{*.f64}\left(\mathsf{PI.f64}\left(\right), \mathsf{/.f64}\left(angle, 180\right)\right)\right)\right) \]

      6. exp-lowering-exp.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(2, \mathsf{sin.f64}\left(\mathsf{exp.f64}\left(\left(\log \left(\frac{180}{\mathsf{PI}\left(\right) \cdot angle}\right) \cdot -1\right)\right)\right)\right), \mathsf{/.f64}\left(1, \mathsf{\_.f64}\left(\mathsf{*.f64}\left(b, b\right), \mathsf{*.f64}\left(a, a\right)\right)\right)\right), \mathsf{cos.f64}\left(\mathsf{*.f64}\left(\mathsf{PI.f64}\left(\right), \mathsf{/.f64}\left(angle, 180\right)\right)\right)\right) \]

      7. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(2, \mathsf{sin.f64}\left(\mathsf{exp.f64}\left(\mathsf{*.f64}\left(\log \left(\frac{180}{\mathsf{PI}\left(\right) \cdot angle}\right), -1\right)\right)\right)\right), \mathsf{/.f64}\left(1, \mathsf{\_.f64}\left(\mathsf{*.f64}\left(b, b\right), \mathsf{*.f64}\left(a, a\right)\right)\right)\right), \mathsf{cos.f64}\left(\mathsf{*.f64}\left(\mathsf{PI.f64}\left(\right), \mathsf{/.f64}\left(angle, 180\right)\right)\right)\right) \]

      8. log-lowering-log.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(2, \mathsf{sin.f64}\left(\mathsf{exp.f64}\left(\mathsf{*.f64}\left(\mathsf{log.f64}\left(\left(\frac{180}{\mathsf{PI}\left(\right) \cdot angle}\right)\right), -1\right)\right)\right)\right), \mathsf{/.f64}\left(1, \mathsf{\_.f64}\left(\mathsf{*.f64}\left(b, b\right), \mathsf{*.f64}\left(a, a\right)\right)\right)\right), \mathsf{cos.f64}\left(\mathsf{*.f64}\left(\mathsf{PI.f64}\left(\right), \mathsf{/.f64}\left(angle, 180\right)\right)\right)\right) \]

      9. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(2, \mathsf{sin.f64}\left(\mathsf{exp.f64}\left(\mathsf{*.f64}\left(\mathsf{log.f64}\left(\mathsf{/.f64}\left(180, \left(\mathsf{PI}\left(\right) \cdot angle\right)\right)\right), -1\right)\right)\right)\right), \mathsf{/.f64}\left(1, \mathsf{\_.f64}\left(\mathsf{*.f64}\left(b, b\right), \mathsf{*.f64}\left(a, a\right)\right)\right)\right), \mathsf{cos.f64}\left(\mathsf{*.f64}\left(\mathsf{PI.f64}\left(\right), \mathsf{/.f64}\left(angle, 180\right)\right)\right)\right) \]

      10. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(2, \mathsf{sin.f64}\left(\mathsf{exp.f64}\left(\mathsf{*.f64}\left(\mathsf{log.f64}\left(\mathsf{/.f64}\left(180, \mathsf{*.f64}\left(\mathsf{PI}\left(\right), angle\right)\right)\right), -1\right)\right)\right)\right), \mathsf{/.f64}\left(1, \mathsf{\_.f64}\left(\mathsf{*.f64}\left(b, b\right), \mathsf{*.f64}\left(a, a\right)\right)\right)\right), \mathsf{cos.f64}\left(\mathsf{*.f64}\left(\mathsf{PI.f64}\left(\right), \mathsf{/.f64}\left(angle, 180\right)\right)\right)\right) \]

      11. PI-lowering-PI.f64 34.3%

          \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(2, \mathsf{sin.f64}\left(\mathsf{exp.f64}\left(\mathsf{*.f64}\left(\mathsf{log.f64}\left(\mathsf{/.f64}\left(180, \mathsf{*.f64}\left(\mathsf{PI.f64}\left(\right), angle\right)\right)\right), -1\right)\right)\right)\right), \mathsf{/.f64}\left(1, \mathsf{\_.f64}\left(\mathsf{*.f64}\left(b, b\right), \mathsf{*.f64}\left(a, a\right)\right)\right)\right), \mathsf{cos.f64}\left(\mathsf{*.f64}\left(\mathsf{PI.f64}\left(\right), \mathsf{/.f64}\left(angle, 180\right)\right)\right)\right) \]

   6. Applied egg-rr 34.3%

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

   if 9.99999999999999962e134 < (/.f64 angle #s(literal 180 binary64)) < 9.9999999999999994e162

   1. Initial program 33.8%

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

      1. associate-*l* N/A

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

      2. *-commutative N/A

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

      3. associate-*l* N/A

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

      4. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\sin \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right), \color{blue}{\left(\cos \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right) \cdot \left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right)\right)}\right) \]

      5. sin-lowering-sin.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sin.f64}\left(\left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right)\right), \left(\color{blue}{\cos \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right)} \cdot \left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right)\right)\right) \]

      6. associate-*r/ N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sin.f64}\left(\left(\frac{\mathsf{PI}\left(\right) \cdot angle}{180}\right)\right), \left(\cos \color{blue}{\left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right)} \cdot \left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right)\right)\right) \]

      7. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sin.f64}\left(\mathsf{/.f64}\left(\left(\mathsf{PI}\left(\right) \cdot angle\right), 180\right)\right), \left(\cos \color{blue}{\left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right)} \cdot \left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right)\right)\right) \]

      8. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sin.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{PI}\left(\right), angle\right), 180\right)\right), \left(\cos \left(\color{blue}{\mathsf{PI}\left(\right)} \cdot \frac{angle}{180}\right) \cdot \left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right)\right)\right) \]

      9. PI-lowering-PI.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sin.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{PI.f64}\left(\right), angle\right), 180\right)\right), \left(\cos \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right) \cdot \left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right)\right)\right) \]

      10. *-commutative N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{sin.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{PI.f64}\left(\right), angle\right), 180\right)\right), \left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \color{blue}{\cos \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right)}\right)\right) \]

      11. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{sin.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{PI.f64}\left(\right), angle\right), 180\right)\right), \mathsf{*.f64}\left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right), \color{blue}{\cos \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right)}\right)\right) \]

   3. Simplified 17.1%

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

   5. Taylor expanded in angle around 0

      \[\leadsto \color{blue}{\frac{1}{90} \cdot \left(angle \cdot \left(\mathsf{PI}\left(\right) \cdot \left({b}^{2} - {a}^{2}\right)\right)\right)} \]
   6. Step-by-step derivation

      1. associate-*r* N/A

         \[\leadsto \frac{1}{90} \cdot \left(\left(angle \cdot \mathsf{PI}\left(\right)\right) \cdot \color{blue}{\left({b}^{2} - {a}^{2}\right)}\right) \]

      2. associate-*r* N/A

         \[\leadsto \left(\frac{1}{90} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right) \cdot \color{blue}{\left({b}^{2} - {a}^{2}\right)} \]

      3. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\left(\frac{1}{90} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right), \color{blue}{\left({b}^{2} - {a}^{2}\right)}\right) \]

      4. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{1}{90}, \left(angle \cdot \mathsf{PI}\left(\right)\right)\right), \left(\color{blue}{{b}^{2}} - {a}^{2}\right)\right) \]

      5. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{1}{90}, \mathsf{*.f64}\left(angle, \mathsf{PI}\left(\right)\right)\right), \left({b}^{\color{blue}{2}} - {a}^{2}\right)\right) \]

      6. PI-lowering-PI.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{1}{90}, \mathsf{*.f64}\left(angle, \mathsf{PI.f64}\left(\right)\right)\right), \left({b}^{2} - {a}^{2}\right)\right) \]

      7. --lowering--.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{1}{90}, \mathsf{*.f64}\left(angle, \mathsf{PI.f64}\left(\right)\right)\right), \mathsf{\_.f64}\left(\left({b}^{2}\right), \color{blue}{\left({a}^{2}\right)}\right)\right) \]

      8. unpow2 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{1}{90}, \mathsf{*.f64}\left(angle, \mathsf{PI.f64}\left(\right)\right)\right), \mathsf{\_.f64}\left(\left(b \cdot b\right), \left({\color{blue}{a}}^{2}\right)\right)\right) \]

      9. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{1}{90}, \mathsf{*.f64}\left(angle, \mathsf{PI.f64}\left(\right)\right)\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(b, b\right), \left({\color{blue}{a}}^{2}\right)\right)\right) \]

      10. unpow2 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{1}{90}, \mathsf{*.f64}\left(angle, \mathsf{PI.f64}\left(\right)\right)\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(b, b\right), \left(a \cdot \color{blue}{a}\right)\right)\right) \]

      11. *-lowering-*.f64 68.3%

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{1}{90}, \mathsf{*.f64}\left(angle, \mathsf{PI.f64}\left(\right)\right)\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(b, b\right), \mathsf{*.f64}\left(a, \color{blue}{a}\right)\right)\right) \]

   7. Simplified 68.3%

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

      1. *-commutative N/A

         \[\leadsto \left(b \cdot b - a \cdot a\right) \cdot \color{blue}{\left(\frac{1}{90} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)} \]

      2. difference-of-squares N/A

         \[\leadsto \left(\left(b + a\right) \cdot \left(b - a\right)\right) \cdot \left(\color{blue}{\frac{1}{90}} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right) \]

      3. associate-*l* N/A

         \[\leadsto \left(b + a\right) \cdot \color{blue}{\left(\left(b - a\right) \cdot \left(\frac{1}{90} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)\right)} \]

      4. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\left(b + a\right), \color{blue}{\left(\left(b - a\right) \cdot \left(\frac{1}{90} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)\right)}\right) \]

      5. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(b, a\right), \left(\color{blue}{\left(b - a\right)} \cdot \left(\frac{1}{90} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)\right)\right) \]

      6. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(b, a\right), \mathsf{*.f64}\left(\left(b - a\right), \color{blue}{\left(\frac{1}{90} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}\right)\right) \]

      7. --lowering--.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(b, a\right), \mathsf{*.f64}\left(\mathsf{\_.f64}\left(b, a\right), \left(\color{blue}{\frac{1}{90}} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)\right)\right) \]

      8. *-commutative N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(b, a\right), \mathsf{*.f64}\left(\mathsf{\_.f64}\left(b, a\right), \left(\frac{1}{90} \cdot \left(\mathsf{PI}\left(\right) \cdot \color{blue}{angle}\right)\right)\right)\right) \]

      9. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(b, a\right), \mathsf{*.f64}\left(\mathsf{\_.f64}\left(b, a\right), \mathsf{*.f64}\left(\frac{1}{90}, \color{blue}{\left(\mathsf{PI}\left(\right) \cdot angle\right)}\right)\right)\right) \]

      10. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(b, a\right), \mathsf{*.f64}\left(\mathsf{\_.f64}\left(b, a\right), \mathsf{*.f64}\left(\frac{1}{90}, \mathsf{*.f64}\left(\mathsf{PI}\left(\right), \color{blue}{angle}\right)\right)\right)\right) \]

      11. PI-lowering-PI.f64 68.3%

          \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(b, a\right), \mathsf{*.f64}\left(\mathsf{\_.f64}\left(b, a\right), \mathsf{*.f64}\left(\frac{1}{90}, \mathsf{*.f64}\left(\mathsf{PI.f64}\left(\right), angle\right)\right)\right)\right) \]

   9. Applied egg-rr 68.3%

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

   if 9.9999999999999994e162 < (/.f64 angle #s(literal 180 binary64))

   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* N/A

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

      2. *-commutative N/A

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

      3. associate-*l* N/A

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

      4. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\sin \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right), \color{blue}{\left(\cos \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right) \cdot \left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right)\right)}\right) \]

      5. sin-lowering-sin.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sin.f64}\left(\left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right)\right), \left(\color{blue}{\cos \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right)} \cdot \left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right)\right)\right) \]

      6. associate-*r/ N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sin.f64}\left(\left(\frac{\mathsf{PI}\left(\right) \cdot angle}{180}\right)\right), \left(\cos \color{blue}{\left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right)} \cdot \left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right)\right)\right) \]

      7. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sin.f64}\left(\mathsf{/.f64}\left(\left(\mathsf{PI}\left(\right) \cdot angle\right), 180\right)\right), \left(\cos \color{blue}{\left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right)} \cdot \left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right)\right)\right) \]

      8. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sin.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{PI}\left(\right), angle\right), 180\right)\right), \left(\cos \left(\color{blue}{\mathsf{PI}\left(\right)} \cdot \frac{angle}{180}\right) \cdot \left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right)\right)\right) \]

      9. PI-lowering-PI.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sin.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{PI.f64}\left(\right), angle\right), 180\right)\right), \left(\cos \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right) \cdot \left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right)\right)\right) \]

      10. *-commutative N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{sin.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{PI.f64}\left(\right), angle\right), 180\right)\right), \left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \color{blue}{\cos \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right)}\right)\right) \]

      11. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{sin.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{PI.f64}\left(\right), angle\right), 180\right)\right), \mathsf{*.f64}\left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right), \color{blue}{\cos \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right)}\right)\right) \]

   3. Simplified 36.8%

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

      1. associate-*r/ N/A

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

      2. associate-*l* N/A

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

      3. associate-*r* N/A

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

      4. difference-of-squares N/A

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

      5. associate-*l* N/A

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

      6. associate-*r* N/A

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

      7. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\left(\left(\sin \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right) \cdot 2\right) \cdot \left(b + a\right)\right), \color{blue}{\left(\left(b - a\right) \cdot \cos \left(\frac{\mathsf{PI}\left(\right) \cdot angle}{180}\right)\right)}\right) \]

   6. Applied egg-rr 32.7%

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

      1. add-sqr-sqrt N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(2, \mathsf{sin.f64}\left(\mathsf{/.f64}\left(\left(\sqrt{\mathsf{PI}\left(\right)} \cdot \sqrt{\mathsf{PI}\left(\right)}\right), \mathsf{/.f64}\left(180, angle\right)\right)\right)\right), \mathsf{+.f64}\left(b, a\right)\right), \mathsf{*.f64}\left(\mathsf{cos.f64}\left(\mathsf{/.f64}\left(\mathsf{PI.f64}\left(\right), \mathsf{/.f64}\left(180, angle\right)\right)\right), \mathsf{\_.f64}\left(b, a\right)\right)\right) \]

      2. pow2 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(2, \mathsf{sin.f64}\left(\mathsf{/.f64}\left(\left({\left(\sqrt{\mathsf{PI}\left(\right)}\right)}^{2}\right), \mathsf{/.f64}\left(180, angle\right)\right)\right)\right), \mathsf{+.f64}\left(b, a\right)\right), \mathsf{*.f64}\left(\mathsf{cos.f64}\left(\mathsf{/.f64}\left(\mathsf{PI.f64}\left(\right), \mathsf{/.f64}\left(180, angle\right)\right)\right), \mathsf{\_.f64}\left(b, a\right)\right)\right) \]

      3. pow-lowering-pow.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(2, \mathsf{sin.f64}\left(\mathsf{/.f64}\left(\mathsf{pow.f64}\left(\left(\sqrt{\mathsf{PI}\left(\right)}\right), 2\right), \mathsf{/.f64}\left(180, angle\right)\right)\right)\right), \mathsf{+.f64}\left(b, a\right)\right), \mathsf{*.f64}\left(\mathsf{cos.f64}\left(\mathsf{/.f64}\left(\mathsf{PI.f64}\left(\right), \mathsf{/.f64}\left(180, angle\right)\right)\right), \mathsf{\_.f64}\left(b, a\right)\right)\right) \]

      4. sqrt-lowering-sqrt.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(2, \mathsf{sin.f64}\left(\mathsf{/.f64}\left(\mathsf{pow.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{PI}\left(\right)\right), 2\right), \mathsf{/.f64}\left(180, angle\right)\right)\right)\right), \mathsf{+.f64}\left(b, a\right)\right), \mathsf{*.f64}\left(\mathsf{cos.f64}\left(\mathsf{/.f64}\left(\mathsf{PI.f64}\left(\right), \mathsf{/.f64}\left(180, angle\right)\right)\right), \mathsf{\_.f64}\left(b, a\right)\right)\right) \]

      5. PI-lowering-PI.f64 34.6%

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(2, \mathsf{sin.f64}\left(\mathsf{/.f64}\left(\mathsf{pow.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{PI.f64}\left(\right)\right), 2\right), \mathsf{/.f64}\left(180, angle\right)\right)\right)\right), \mathsf{+.f64}\left(b, a\right)\right), \mathsf{*.f64}\left(\mathsf{cos.f64}\left(\mathsf{/.f64}\left(\mathsf{PI.f64}\left(\right), \mathsf{/.f64}\left(180, angle\right)\right)\right), \mathsf{\_.f64}\left(b, a\right)\right)\right) \]

   8. Applied egg-rr 34.6%

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

      1. add-sqr-sqrt N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(2, \mathsf{sin.f64}\left(\mathsf{/.f64}\left(\mathsf{pow.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{PI.f64}\left(\right)\right), 2\right), \mathsf{/.f64}\left(180, angle\right)\right)\right)\right), \mathsf{+.f64}\left(b, a\right)\right), \mathsf{*.f64}\left(\mathsf{cos.f64}\left(\mathsf{/.f64}\left(\left(\sqrt{\mathsf{PI}\left(\right)} \cdot \sqrt{\mathsf{PI}\left(\right)}\right), \mathsf{/.f64}\left(180, angle\right)\right)\right), \mathsf{\_.f64}\left(b, a\right)\right)\right) \]

      2. unpow2 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(2, \mathsf{sin.f64}\left(\mathsf{/.f64}\left(\mathsf{pow.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{PI.f64}\left(\right)\right), 2\right), \mathsf{/.f64}\left(180, angle\right)\right)\right)\right), \mathsf{+.f64}\left(b, a\right)\right), \mathsf{*.f64}\left(\mathsf{cos.f64}\left(\mathsf{/.f64}\left(\left({\left(\sqrt{\mathsf{PI}\left(\right)}\right)}^{2}\right), \mathsf{/.f64}\left(180, angle\right)\right)\right), \mathsf{\_.f64}\left(b, a\right)\right)\right) \]

      3. pow-lowering-pow.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(2, \mathsf{sin.f64}\left(\mathsf{/.f64}\left(\mathsf{pow.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{PI.f64}\left(\right)\right), 2\right), \mathsf{/.f64}\left(180, angle\right)\right)\right)\right), \mathsf{+.f64}\left(b, a\right)\right), \mathsf{*.f64}\left(\mathsf{cos.f64}\left(\mathsf{/.f64}\left(\mathsf{pow.f64}\left(\left(\sqrt{\mathsf{PI}\left(\right)}\right), 2\right), \mathsf{/.f64}\left(180, angle\right)\right)\right), \mathsf{\_.f64}\left(b, a\right)\right)\right) \]

      4. sqrt-lowering-sqrt.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(2, \mathsf{sin.f64}\left(\mathsf{/.f64}\left(\mathsf{pow.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{PI.f64}\left(\right)\right), 2\right), \mathsf{/.f64}\left(180, angle\right)\right)\right)\right), \mathsf{+.f64}\left(b, a\right)\right), \mathsf{*.f64}\left(\mathsf{cos.f64}\left(\mathsf{/.f64}\left(\mathsf{pow.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{PI}\left(\right)\right), 2\right), \mathsf{/.f64}\left(180, angle\right)\right)\right), \mathsf{\_.f64}\left(b, a\right)\right)\right) \]

      5. PI-lowering-PI.f64 49.5%

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(2, \mathsf{sin.f64}\left(\mathsf{/.f64}\left(\mathsf{pow.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{PI.f64}\left(\right)\right), 2\right), \mathsf{/.f64}\left(180, angle\right)\right)\right)\right), \mathsf{+.f64}\left(b, a\right)\right), \mathsf{*.f64}\left(\mathsf{cos.f64}\left(\mathsf{/.f64}\left(\mathsf{pow.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{PI.f64}\left(\right)\right), 2\right), \mathsf{/.f64}\left(180, angle\right)\right)\right), \mathsf{\_.f64}\left(b, a\right)\right)\right) \]

   10. Applied egg-rr 49.5%

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

4. Final simplification 71.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{angle}{180} \leq 10^{+98}:\\ \;\;\;\;\left(b - a\right) \cdot \left(\left(b + a\right) \cdot \sin \left(\left(angle \cdot \pi\right) \cdot 0.011111111111111112\right)\right)\\ \mathbf{elif}\;\frac{angle}{180} \leq 10^{+135}:\\ \;\;\;\;\frac{2 \cdot \sin \left(e^{0 - \log \left(\frac{180}{angle \cdot \pi}\right)}\right)}{\frac{1}{b \cdot b - a \cdot a}} \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)\\ \mathbf{elif}\;\frac{angle}{180} \leq 10^{+163}:\\ \;\;\;\;\left(b + a\right) \cdot \left(\left(b - a\right) \cdot \left(\left(angle \cdot \pi\right) \cdot 0.011111111111111112\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\left(b + a\right) \cdot \left(2 \cdot \sin \left(\frac{{\left(\sqrt{\pi}\right)}^{2}}{\frac{180}{angle}}\right)\right)\right) \cdot \left(\left(b - a\right) \cdot \cos \left(\frac{{\left(\sqrt{\pi}\right)}^{2}}{\frac{180}{angle}}\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 67.3% accurate, 0.6× speedup

Math

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

FPCore

angle\_m = (fabs.f64 angle)
angle\_s = (copysign.f64 #s(literal 1 binary64) angle)
(FPCore (angle_s a b angle_m)
 :precision binary64
 (let* ((t_0 (* (/ angle_m 180.0) PI)))
   (*
    angle_s
    (if (<=
         (* (cos t_0) (* (* 2.0 (- (pow b 2.0) (pow a 2.0))) (sin t_0)))
         (- INFINITY))
      (* (+ b a) (* (- b a) (* (* angle_m PI) 0.011111111111111112)))
      (*
       (*
        (+ b a)
        (* 2.0 (sin (/ 1.0 (/ 180.0 (* angle_m (cbrt (* PI (* PI PI)))))))))
       (* (- b a) (cos (/ PI (/ 180.0 angle_m)))))))))

C

angle\_m = fabs(angle);
angle\_s = copysign(1.0, angle);
double code(double angle_s, double a, double b, double angle_m) {
	double t_0 = (angle_m / 180.0) * ((double) M_PI);
	double tmp;
	if ((cos(t_0) * ((2.0 * (pow(b, 2.0) - pow(a, 2.0))) * sin(t_0))) <= -((double) INFINITY)) {
		tmp = (b + a) * ((b - a) * ((angle_m * ((double) M_PI)) * 0.011111111111111112));
	} else {
		tmp = ((b + a) * (2.0 * sin((1.0 / (180.0 / (angle_m * cbrt((((double) M_PI) * (((double) M_PI) * ((double) M_PI)))))))))) * ((b - a) * cos((((double) M_PI) / (180.0 / angle_m))));
	}
	return angle_s * tmp;
}

Java

angle\_m = Math.abs(angle);
angle\_s = Math.copySign(1.0, angle);
public static double code(double angle_s, double a, double b, double angle_m) {
	double t_0 = (angle_m / 180.0) * Math.PI;
	double tmp;
	if ((Math.cos(t_0) * ((2.0 * (Math.pow(b, 2.0) - Math.pow(a, 2.0))) * Math.sin(t_0))) <= -Double.POSITIVE_INFINITY) {
		tmp = (b + a) * ((b - a) * ((angle_m * Math.PI) * 0.011111111111111112));
	} else {
		tmp = ((b + a) * (2.0 * Math.sin((1.0 / (180.0 / (angle_m * Math.cbrt((Math.PI * (Math.PI * Math.PI))))))))) * ((b - a) * Math.cos((Math.PI / (180.0 / angle_m))));
	}
	return angle_s * tmp;
}

Julia

angle\_m = abs(angle)
angle\_s = copysign(1.0, angle)
function code(angle_s, a, b, angle_m)
	t_0 = Float64(Float64(angle_m / 180.0) * pi)
	tmp = 0.0
	if (Float64(cos(t_0) * Float64(Float64(2.0 * Float64((b ^ 2.0) - (a ^ 2.0))) * sin(t_0))) <= Float64(-Inf))
		tmp = Float64(Float64(b + a) * Float64(Float64(b - a) * Float64(Float64(angle_m * pi) * 0.011111111111111112)));
	else
		tmp = Float64(Float64(Float64(b + a) * Float64(2.0 * sin(Float64(1.0 / Float64(180.0 / Float64(angle_m * cbrt(Float64(pi * Float64(pi * pi))))))))) * Float64(Float64(b - a) * cos(Float64(pi / Float64(180.0 / angle_m)))));
	end
	return Float64(angle_s * tmp)
end

Wolfram

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_, b_, angle$95$m_] := Block[{t$95$0 = N[(N[(angle$95$m / 180.0), $MachinePrecision] * Pi), $MachinePrecision]}, N[(angle$95$s * If[LessEqual[N[(N[Cos[t$95$0], $MachinePrecision] * 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]), $MachinePrecision], (-Infinity)], N[(N[(b + a), $MachinePrecision] * N[(N[(b - a), $MachinePrecision] * N[(N[(angle$95$m * Pi), $MachinePrecision] * 0.011111111111111112), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(b + a), $MachinePrecision] * N[(2.0 * N[Sin[N[(1.0 / N[(180.0 / N[(angle$95$m * N[Power[N[(Pi * N[(Pi * Pi), $MachinePrecision]), $MachinePrecision], 1/3], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(N[(b - a), $MachinePrecision] * N[Cos[N[(Pi / N[(180.0 / angle$95$m), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 (*.f64 (*.f64 #s(literal 2 binary64) (-.f64 (pow.f64 b #s(literal 2 binary64)) (pow.f64 a #s(literal 2 binary64)))) (sin.f64 (*.f64 (PI.f64) (/.f64 angle #s(literal 180 binary64))))) (cos.f64 (*.f64 (PI.f64) (/.f64 angle #s(literal 180 binary64))))) < -inf.0

   1. Initial program 53.8%

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

      1. associate-*l* N/A

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

      2. *-commutative N/A

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

      3. associate-*l* N/A

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

      4. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\sin \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right), \color{blue}{\left(\cos \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right) \cdot \left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right)\right)}\right) \]

      5. sin-lowering-sin.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sin.f64}\left(\left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right)\right), \left(\color{blue}{\cos \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right)} \cdot \left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right)\right)\right) \]

      6. associate-*r/ N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sin.f64}\left(\left(\frac{\mathsf{PI}\left(\right) \cdot angle}{180}\right)\right), \left(\cos \color{blue}{\left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right)} \cdot \left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right)\right)\right) \]

      7. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sin.f64}\left(\mathsf{/.f64}\left(\left(\mathsf{PI}\left(\right) \cdot angle\right), 180\right)\right), \left(\cos \color{blue}{\left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right)} \cdot \left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right)\right)\right) \]

      8. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sin.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{PI}\left(\right), angle\right), 180\right)\right), \left(\cos \left(\color{blue}{\mathsf{PI}\left(\right)} \cdot \frac{angle}{180}\right) \cdot \left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right)\right)\right) \]

      9. PI-lowering-PI.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sin.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{PI.f64}\left(\right), angle\right), 180\right)\right), \left(\cos \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right) \cdot \left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right)\right)\right) \]

      10. *-commutative N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{sin.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{PI.f64}\left(\right), angle\right), 180\right)\right), \left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \color{blue}{\cos \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right)}\right)\right) \]

      11. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{sin.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{PI.f64}\left(\right), angle\right), 180\right)\right), \mathsf{*.f64}\left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right), \color{blue}{\cos \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right)}\right)\right) \]

   3. Simplified 53.8%

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

   5. Taylor expanded in angle around 0

      \[\leadsto \color{blue}{\frac{1}{90} \cdot \left(angle \cdot \left(\mathsf{PI}\left(\right) \cdot \left({b}^{2} - {a}^{2}\right)\right)\right)} \]
   6. Step-by-step derivation

      1. associate-*r* N/A

         \[\leadsto \frac{1}{90} \cdot \left(\left(angle \cdot \mathsf{PI}\left(\right)\right) \cdot \color{blue}{\left({b}^{2} - {a}^{2}\right)}\right) \]

      2. associate-*r* N/A

         \[\leadsto \left(\frac{1}{90} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right) \cdot \color{blue}{\left({b}^{2} - {a}^{2}\right)} \]

      3. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\left(\frac{1}{90} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right), \color{blue}{\left({b}^{2} - {a}^{2}\right)}\right) \]

      4. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{1}{90}, \left(angle \cdot \mathsf{PI}\left(\right)\right)\right), \left(\color{blue}{{b}^{2}} - {a}^{2}\right)\right) \]

      5. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{1}{90}, \mathsf{*.f64}\left(angle, \mathsf{PI}\left(\right)\right)\right), \left({b}^{\color{blue}{2}} - {a}^{2}\right)\right) \]

      6. PI-lowering-PI.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{1}{90}, \mathsf{*.f64}\left(angle, \mathsf{PI.f64}\left(\right)\right)\right), \left({b}^{2} - {a}^{2}\right)\right) \]

      7. --lowering--.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{1}{90}, \mathsf{*.f64}\left(angle, \mathsf{PI.f64}\left(\right)\right)\right), \mathsf{\_.f64}\left(\left({b}^{2}\right), \color{blue}{\left({a}^{2}\right)}\right)\right) \]

      8. unpow2 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{1}{90}, \mathsf{*.f64}\left(angle, \mathsf{PI.f64}\left(\right)\right)\right), \mathsf{\_.f64}\left(\left(b \cdot b\right), \left({\color{blue}{a}}^{2}\right)\right)\right) \]

      9. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{1}{90}, \mathsf{*.f64}\left(angle, \mathsf{PI.f64}\left(\right)\right)\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(b, b\right), \left({\color{blue}{a}}^{2}\right)\right)\right) \]

      10. unpow2 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{1}{90}, \mathsf{*.f64}\left(angle, \mathsf{PI.f64}\left(\right)\right)\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(b, b\right), \left(a \cdot \color{blue}{a}\right)\right)\right) \]

      11. *-lowering-*.f64 50.4%

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{1}{90}, \mathsf{*.f64}\left(angle, \mathsf{PI.f64}\left(\right)\right)\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(b, b\right), \mathsf{*.f64}\left(a, \color{blue}{a}\right)\right)\right) \]

   7. Simplified 50.4%

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

      1. *-commutative N/A

         \[\leadsto \left(b \cdot b - a \cdot a\right) \cdot \color{blue}{\left(\frac{1}{90} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)} \]

      2. difference-of-squares N/A

         \[\leadsto \left(\left(b + a\right) \cdot \left(b - a\right)\right) \cdot \left(\color{blue}{\frac{1}{90}} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right) \]

      3. associate-*l* N/A

         \[\leadsto \left(b + a\right) \cdot \color{blue}{\left(\left(b - a\right) \cdot \left(\frac{1}{90} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)\right)} \]

      4. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\left(b + a\right), \color{blue}{\left(\left(b - a\right) \cdot \left(\frac{1}{90} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)\right)}\right) \]

      5. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(b, a\right), \left(\color{blue}{\left(b - a\right)} \cdot \left(\frac{1}{90} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)\right)\right) \]

      6. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(b, a\right), \mathsf{*.f64}\left(\left(b - a\right), \color{blue}{\left(\frac{1}{90} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}\right)\right) \]

      7. --lowering--.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(b, a\right), \mathsf{*.f64}\left(\mathsf{\_.f64}\left(b, a\right), \left(\color{blue}{\frac{1}{90}} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)\right)\right) \]

      8. *-commutative N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(b, a\right), \mathsf{*.f64}\left(\mathsf{\_.f64}\left(b, a\right), \left(\frac{1}{90} \cdot \left(\mathsf{PI}\left(\right) \cdot \color{blue}{angle}\right)\right)\right)\right) \]

      9. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(b, a\right), \mathsf{*.f64}\left(\mathsf{\_.f64}\left(b, a\right), \mathsf{*.f64}\left(\frac{1}{90}, \color{blue}{\left(\mathsf{PI}\left(\right) \cdot angle\right)}\right)\right)\right) \]

      10. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(b, a\right), \mathsf{*.f64}\left(\mathsf{\_.f64}\left(b, a\right), \mathsf{*.f64}\left(\frac{1}{90}, \mathsf{*.f64}\left(\mathsf{PI}\left(\right), \color{blue}{angle}\right)\right)\right)\right) \]

      11. PI-lowering-PI.f64 79.1%

          \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(b, a\right), \mathsf{*.f64}\left(\mathsf{\_.f64}\left(b, a\right), \mathsf{*.f64}\left(\frac{1}{90}, \mathsf{*.f64}\left(\mathsf{PI.f64}\left(\right), angle\right)\right)\right)\right) \]

   9. Applied egg-rr 79.1%

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

   if -inf.0 < (*.f64 (*.f64 (*.f64 #s(literal 2 binary64) (-.f64 (pow.f64 b #s(literal 2 binary64)) (pow.f64 a #s(literal 2 binary64)))) (sin.f64 (*.f64 (PI.f64) (/.f64 angle #s(literal 180 binary64))))) (cos.f64 (*.f64 (PI.f64) (/.f64 angle #s(literal 180 binary64)))))

   1. Initial program 54.8%

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

      1. associate-*l* N/A

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

      2. *-commutative N/A

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

      3. associate-*l* N/A

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

      4. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\sin \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right), \color{blue}{\left(\cos \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right) \cdot \left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right)\right)}\right) \]

      5. sin-lowering-sin.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sin.f64}\left(\left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right)\right), \left(\color{blue}{\cos \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right)} \cdot \left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right)\right)\right) \]

      6. associate-*r/ N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sin.f64}\left(\left(\frac{\mathsf{PI}\left(\right) \cdot angle}{180}\right)\right), \left(\cos \color{blue}{\left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right)} \cdot \left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right)\right)\right) \]

      7. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sin.f64}\left(\mathsf{/.f64}\left(\left(\mathsf{PI}\left(\right) \cdot angle\right), 180\right)\right), \left(\cos \color{blue}{\left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right)} \cdot \left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right)\right)\right) \]

      8. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sin.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{PI}\left(\right), angle\right), 180\right)\right), \left(\cos \left(\color{blue}{\mathsf{PI}\left(\right)} \cdot \frac{angle}{180}\right) \cdot \left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right)\right)\right) \]

      9. PI-lowering-PI.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sin.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{PI.f64}\left(\right), angle\right), 180\right)\right), \left(\cos \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right) \cdot \left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right)\right)\right) \]

      10. *-commutative N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{sin.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{PI.f64}\left(\right), angle\right), 180\right)\right), \left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \color{blue}{\cos \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right)}\right)\right) \]

      11. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{sin.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{PI.f64}\left(\right), angle\right), 180\right)\right), \mathsf{*.f64}\left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right), \color{blue}{\cos \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right)}\right)\right) \]

   3. Simplified 54.9%

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

      1. associate-*r/ N/A

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

      2. associate-*l* N/A

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

      3. associate-*r* N/A

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

      4. difference-of-squares N/A

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

      5. associate-*l* N/A

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

      6. associate-*r* N/A

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

      7. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\left(\left(\sin \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right) \cdot 2\right) \cdot \left(b + a\right)\right), \color{blue}{\left(\left(b - a\right) \cdot \cos \left(\frac{\mathsf{PI}\left(\right) \cdot angle}{180}\right)\right)}\right) \]

   6. Applied egg-rr 65.0%

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

      1. associate-/r/ N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(2, \mathsf{sin.f64}\left(\left(\frac{\mathsf{PI}\left(\right)}{180} \cdot angle\right)\right)\right), \mathsf{+.f64}\left(b, a\right)\right), \mathsf{*.f64}\left(\mathsf{cos.f64}\left(\mathsf{/.f64}\left(\mathsf{PI.f64}\left(\right), \mathsf{/.f64}\left(\color{blue}{180}, angle\right)\right)\right), \mathsf{\_.f64}\left(b, a\right)\right)\right) \]

      2. associate-*l/ N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(2, \mathsf{sin.f64}\left(\left(\frac{\mathsf{PI}\left(\right) \cdot angle}{180}\right)\right)\right), \mathsf{+.f64}\left(b, a\right)\right), \mathsf{*.f64}\left(\mathsf{cos.f64}\left(\mathsf{/.f64}\left(\mathsf{PI.f64}\left(\right), \mathsf{/.f64}\left(\color{blue}{180}, angle\right)\right)\right), \mathsf{\_.f64}\left(b, a\right)\right)\right) \]

      3. clear-num N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(2, \mathsf{sin.f64}\left(\left(\frac{1}{\frac{180}{\mathsf{PI}\left(\right) \cdot angle}}\right)\right)\right), \mathsf{+.f64}\left(b, a\right)\right), \mathsf{*.f64}\left(\mathsf{cos.f64}\left(\mathsf{/.f64}\left(\mathsf{PI.f64}\left(\right), \mathsf{/.f64}\left(\color{blue}{180}, angle\right)\right)\right), \mathsf{\_.f64}\left(b, a\right)\right)\right) \]

      4. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(2, \mathsf{sin.f64}\left(\mathsf{/.f64}\left(1, \left(\frac{180}{\mathsf{PI}\left(\right) \cdot angle}\right)\right)\right)\right), \mathsf{+.f64}\left(b, a\right)\right), \mathsf{*.f64}\left(\mathsf{cos.f64}\left(\mathsf{/.f64}\left(\mathsf{PI.f64}\left(\right), \mathsf{/.f64}\left(\color{blue}{180}, angle\right)\right)\right), \mathsf{\_.f64}\left(b, a\right)\right)\right) \]

      5. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(2, \mathsf{sin.f64}\left(\mathsf{/.f64}\left(1, \mathsf{/.f64}\left(180, \left(\mathsf{PI}\left(\right) \cdot angle\right)\right)\right)\right)\right), \mathsf{+.f64}\left(b, a\right)\right), \mathsf{*.f64}\left(\mathsf{cos.f64}\left(\mathsf{/.f64}\left(\mathsf{PI.f64}\left(\right), \mathsf{/.f64}\left(180, angle\right)\right)\right), \mathsf{\_.f64}\left(b, a\right)\right)\right) \]

      6. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(2, \mathsf{sin.f64}\left(\mathsf{/.f64}\left(1, \mathsf{/.f64}\left(180, \mathsf{*.f64}\left(\mathsf{PI}\left(\right), angle\right)\right)\right)\right)\right), \mathsf{+.f64}\left(b, a\right)\right), \mathsf{*.f64}\left(\mathsf{cos.f64}\left(\mathsf{/.f64}\left(\mathsf{PI.f64}\left(\right), \mathsf{/.f64}\left(180, angle\right)\right)\right), \mathsf{\_.f64}\left(b, a\right)\right)\right) \]

      7. PI-lowering-PI.f64 66.3%

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(2, \mathsf{sin.f64}\left(\mathsf{/.f64}\left(1, \mathsf{/.f64}\left(180, \mathsf{*.f64}\left(\mathsf{PI.f64}\left(\right), angle\right)\right)\right)\right)\right), \mathsf{+.f64}\left(b, a\right)\right), \mathsf{*.f64}\left(\mathsf{cos.f64}\left(\mathsf{/.f64}\left(\mathsf{PI.f64}\left(\right), \mathsf{/.f64}\left(180, angle\right)\right)\right), \mathsf{\_.f64}\left(b, a\right)\right)\right) \]

   8. Applied egg-rr 66.3%

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

      1. add-cbrt-cube N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(2, \mathsf{sin.f64}\left(\mathsf{/.f64}\left(1, \mathsf{/.f64}\left(180, \mathsf{*.f64}\left(\left(\sqrt[3]{\left(\mathsf{PI}\left(\right) \cdot \mathsf{PI}\left(\right)\right) \cdot \mathsf{PI}\left(\right)}\right), angle\right)\right)\right)\right)\right), \mathsf{+.f64}\left(b, a\right)\right), \mathsf{*.f64}\left(\mathsf{cos.f64}\left(\mathsf{/.f64}\left(\mathsf{PI.f64}\left(\right), \mathsf{/.f64}\left(180, angle\right)\right)\right), \mathsf{\_.f64}\left(b, a\right)\right)\right) \]

      2. cbrt-lowering-cbrt.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(2, \mathsf{sin.f64}\left(\mathsf{/.f64}\left(1, \mathsf{/.f64}\left(180, \mathsf{*.f64}\left(\mathsf{cbrt.f64}\left(\left(\left(\mathsf{PI}\left(\right) \cdot \mathsf{PI}\left(\right)\right) \cdot \mathsf{PI}\left(\right)\right)\right), angle\right)\right)\right)\right)\right), \mathsf{+.f64}\left(b, a\right)\right), \mathsf{*.f64}\left(\mathsf{cos.f64}\left(\mathsf{/.f64}\left(\mathsf{PI.f64}\left(\right), \mathsf{/.f64}\left(180, angle\right)\right)\right), \mathsf{\_.f64}\left(b, a\right)\right)\right) \]

      3. associate-*l* N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(2, \mathsf{sin.f64}\left(\mathsf{/.f64}\left(1, \mathsf{/.f64}\left(180, \mathsf{*.f64}\left(\mathsf{cbrt.f64}\left(\left(\mathsf{PI}\left(\right) \cdot \left(\mathsf{PI}\left(\right) \cdot \mathsf{PI}\left(\right)\right)\right)\right), angle\right)\right)\right)\right)\right), \mathsf{+.f64}\left(b, a\right)\right), \mathsf{*.f64}\left(\mathsf{cos.f64}\left(\mathsf{/.f64}\left(\mathsf{PI.f64}\left(\right), \mathsf{/.f64}\left(180, angle\right)\right)\right), \mathsf{\_.f64}\left(b, a\right)\right)\right) \]

      4. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(2, \mathsf{sin.f64}\left(\mathsf{/.f64}\left(1, \mathsf{/.f64}\left(180, \mathsf{*.f64}\left(\mathsf{cbrt.f64}\left(\mathsf{*.f64}\left(\mathsf{PI}\left(\right), \left(\mathsf{PI}\left(\right) \cdot \mathsf{PI}\left(\right)\right)\right)\right), angle\right)\right)\right)\right)\right), \mathsf{+.f64}\left(b, a\right)\right), \mathsf{*.f64}\left(\mathsf{cos.f64}\left(\mathsf{/.f64}\left(\mathsf{PI.f64}\left(\right), \mathsf{/.f64}\left(180, angle\right)\right)\right), \mathsf{\_.f64}\left(b, a\right)\right)\right) \]

      5. PI-lowering-PI.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(2, \mathsf{sin.f64}\left(\mathsf{/.f64}\left(1, \mathsf{/.f64}\left(180, \mathsf{*.f64}\left(\mathsf{cbrt.f64}\left(\mathsf{*.f64}\left(\mathsf{PI.f64}\left(\right), \left(\mathsf{PI}\left(\right) \cdot \mathsf{PI}\left(\right)\right)\right)\right), angle\right)\right)\right)\right)\right), \mathsf{+.f64}\left(b, a\right)\right), \mathsf{*.f64}\left(\mathsf{cos.f64}\left(\mathsf{/.f64}\left(\mathsf{PI.f64}\left(\right), \mathsf{/.f64}\left(180, angle\right)\right)\right), \mathsf{\_.f64}\left(b, a\right)\right)\right) \]

      6. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(2, \mathsf{sin.f64}\left(\mathsf{/.f64}\left(1, \mathsf{/.f64}\left(180, \mathsf{*.f64}\left(\mathsf{cbrt.f64}\left(\mathsf{*.f64}\left(\mathsf{PI.f64}\left(\right), \mathsf{*.f64}\left(\mathsf{PI}\left(\right), \mathsf{PI}\left(\right)\right)\right)\right), angle\right)\right)\right)\right)\right), \mathsf{+.f64}\left(b, a\right)\right), \mathsf{*.f64}\left(\mathsf{cos.f64}\left(\mathsf{/.f64}\left(\mathsf{PI.f64}\left(\right), \mathsf{/.f64}\left(180, angle\right)\right)\right), \mathsf{\_.f64}\left(b, a\right)\right)\right) \]

      7. PI-lowering-PI.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(2, \mathsf{sin.f64}\left(\mathsf{/.f64}\left(1, \mathsf{/.f64}\left(180, \mathsf{*.f64}\left(\mathsf{cbrt.f64}\left(\mathsf{*.f64}\left(\mathsf{PI.f64}\left(\right), \mathsf{*.f64}\left(\mathsf{PI.f64}\left(\right), \mathsf{PI}\left(\right)\right)\right)\right), angle\right)\right)\right)\right)\right), \mathsf{+.f64}\left(b, a\right)\right), \mathsf{*.f64}\left(\mathsf{cos.f64}\left(\mathsf{/.f64}\left(\mathsf{PI.f64}\left(\right), \mathsf{/.f64}\left(180, angle\right)\right)\right), \mathsf{\_.f64}\left(b, a\right)\right)\right) \]

      8. PI-lowering-PI.f64 67.9%

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(2, \mathsf{sin.f64}\left(\mathsf{/.f64}\left(1, \mathsf{/.f64}\left(180, \mathsf{*.f64}\left(\mathsf{cbrt.f64}\left(\mathsf{*.f64}\left(\mathsf{PI.f64}\left(\right), \mathsf{*.f64}\left(\mathsf{PI.f64}\left(\right), \mathsf{PI.f64}\left(\right)\right)\right)\right), angle\right)\right)\right)\right)\right), \mathsf{+.f64}\left(b, a\right)\right), \mathsf{*.f64}\left(\mathsf{cos.f64}\left(\mathsf{/.f64}\left(\mathsf{PI.f64}\left(\right), \mathsf{/.f64}\left(180, angle\right)\right)\right), \mathsf{\_.f64}\left(b, a\right)\right)\right) \]

   10. Applied egg-rr 67.9%

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

4. Final simplification 70.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\cos \left(\frac{angle}{180} \cdot \pi\right) \cdot \left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right) \leq -\infty:\\ \;\;\;\;\left(b + a\right) \cdot \left(\left(b - a\right) \cdot \left(\left(angle \cdot \pi\right) \cdot 0.011111111111111112\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\left(b + a\right) \cdot \left(2 \cdot \sin \left(\frac{1}{\frac{180}{angle \cdot \sqrt[3]{\pi \cdot \left(\pi \cdot \pi\right)}}}\right)\right)\right) \cdot \left(\left(b - a\right) \cdot \cos \left(\frac{\pi}{\frac{180}{angle}}\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 67.5% accurate, 1.0× speedup

Math

\[\begin{array}{l} angle\_m = \left|angle\right| \\ angle\_s = \mathsf{copysign}\left(1, angle\right) \\ \begin{array}{l} t_0 := \left(b + a\right) \cdot \left(2 \cdot \sin \left(\frac{{\left(\sqrt{\pi}\right)}^{2}}{\frac{180}{angle\_m}}\right)\right)\\ angle\_s \cdot \begin{array}{l} \mathbf{if}\;\frac{angle\_m}{180} \leq 10^{+98}:\\ \;\;\;\;\left(b - a\right) \cdot \left(\left(b + a\right) \cdot \sin \left(\left(angle\_m \cdot \pi\right) \cdot 0.011111111111111112\right)\right)\\ \mathbf{elif}\;\frac{angle\_m}{180} \leq 2 \cdot 10^{+224}:\\ \;\;\;\;\left(b - a\right) \cdot t\_0\\ \mathbf{else}:\\ \;\;\;\;t\_0 \cdot \left(\left(b - a\right) \cdot \cos \left(\frac{1}{\frac{180}{angle\_m \cdot \pi}}\right)\right)\\ \end{array} \end{array} \end{array} \]

FPCore

angle\_m = (fabs.f64 angle)
angle\_s = (copysign.f64 #s(literal 1 binary64) angle)
(FPCore (angle_s a b angle_m)
 :precision binary64
 (let* ((t_0
         (* (+ b a) (* 2.0 (sin (/ (pow (sqrt PI) 2.0) (/ 180.0 angle_m)))))))
   (*
    angle_s
    (if (<= (/ angle_m 180.0) 1e+98)
      (* (- b a) (* (+ b a) (sin (* (* angle_m PI) 0.011111111111111112))))
      (if (<= (/ angle_m 180.0) 2e+224)
        (* (- b a) t_0)
        (* t_0 (* (- b a) (cos (/ 1.0 (/ 180.0 (* angle_m PI)))))))))))

C

angle\_m = fabs(angle);
angle\_s = copysign(1.0, angle);
double code(double angle_s, double a, double b, double angle_m) {
	double t_0 = (b + a) * (2.0 * sin((pow(sqrt(((double) M_PI)), 2.0) / (180.0 / angle_m))));
	double tmp;
	if ((angle_m / 180.0) <= 1e+98) {
		tmp = (b - a) * ((b + a) * sin(((angle_m * ((double) M_PI)) * 0.011111111111111112)));
	} else if ((angle_m / 180.0) <= 2e+224) {
		tmp = (b - a) * t_0;
	} else {
		tmp = t_0 * ((b - a) * cos((1.0 / (180.0 / (angle_m * ((double) M_PI))))));
	}
	return angle_s * tmp;
}

Java

angle\_m = Math.abs(angle);
angle\_s = Math.copySign(1.0, angle);
public static double code(double angle_s, double a, double b, double angle_m) {
	double t_0 = (b + a) * (2.0 * Math.sin((Math.pow(Math.sqrt(Math.PI), 2.0) / (180.0 / angle_m))));
	double tmp;
	if ((angle_m / 180.0) <= 1e+98) {
		tmp = (b - a) * ((b + a) * Math.sin(((angle_m * Math.PI) * 0.011111111111111112)));
	} else if ((angle_m / 180.0) <= 2e+224) {
		tmp = (b - a) * t_0;
	} else {
		tmp = t_0 * ((b - a) * Math.cos((1.0 / (180.0 / (angle_m * Math.PI)))));
	}
	return angle_s * tmp;
}

Python

angle\_m = math.fabs(angle)
angle\_s = math.copysign(1.0, angle)
def code(angle_s, a, b, angle_m):
	t_0 = (b + a) * (2.0 * math.sin((math.pow(math.sqrt(math.pi), 2.0) / (180.0 / angle_m))))
	tmp = 0
	if (angle_m / 180.0) <= 1e+98:
		tmp = (b - a) * ((b + a) * math.sin(((angle_m * math.pi) * 0.011111111111111112)))
	elif (angle_m / 180.0) <= 2e+224:
		tmp = (b - a) * t_0
	else:
		tmp = t_0 * ((b - a) * math.cos((1.0 / (180.0 / (angle_m * math.pi)))))
	return angle_s * tmp

Julia

angle\_m = abs(angle)
angle\_s = copysign(1.0, angle)
function code(angle_s, a, b, angle_m)
	t_0 = Float64(Float64(b + a) * Float64(2.0 * sin(Float64((sqrt(pi) ^ 2.0) / Float64(180.0 / angle_m)))))
	tmp = 0.0
	if (Float64(angle_m / 180.0) <= 1e+98)
		tmp = Float64(Float64(b - a) * Float64(Float64(b + a) * sin(Float64(Float64(angle_m * pi) * 0.011111111111111112))));
	elseif (Float64(angle_m / 180.0) <= 2e+224)
		tmp = Float64(Float64(b - a) * t_0);
	else
		tmp = Float64(t_0 * Float64(Float64(b - a) * cos(Float64(1.0 / Float64(180.0 / Float64(angle_m * pi))))));
	end
	return Float64(angle_s * tmp)
end

MATLAB

angle\_m = abs(angle);
angle\_s = sign(angle) * abs(1.0);
function tmp_2 = code(angle_s, a, b, angle_m)
	t_0 = (b + a) * (2.0 * sin(((sqrt(pi) ^ 2.0) / (180.0 / angle_m))));
	tmp = 0.0;
	if ((angle_m / 180.0) <= 1e+98)
		tmp = (b - a) * ((b + a) * sin(((angle_m * pi) * 0.011111111111111112)));
	elseif ((angle_m / 180.0) <= 2e+224)
		tmp = (b - a) * t_0;
	else
		tmp = t_0 * ((b - a) * cos((1.0 / (180.0 / (angle_m * pi)))));
	end
	tmp_2 = angle_s * tmp;
end

Wolfram

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_, b_, angle$95$m_] := Block[{t$95$0 = N[(N[(b + a), $MachinePrecision] * N[(2.0 * N[Sin[N[(N[Power[N[Sqrt[Pi], $MachinePrecision], 2.0], $MachinePrecision] / N[(180.0 / angle$95$m), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, N[(angle$95$s * If[LessEqual[N[(angle$95$m / 180.0), $MachinePrecision], 1e+98], N[(N[(b - a), $MachinePrecision] * N[(N[(b + a), $MachinePrecision] * N[Sin[N[(N[(angle$95$m * Pi), $MachinePrecision] * 0.011111111111111112), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[N[(angle$95$m / 180.0), $MachinePrecision], 2e+224], N[(N[(b - a), $MachinePrecision] * t$95$0), $MachinePrecision], N[(t$95$0 * N[(N[(b - a), $MachinePrecision] * N[Cos[N[(1.0 / N[(180.0 / N[(angle$95$m * Pi), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]), $MachinePrecision]]

Derivation

1. Split input into 3 regimes

2. if (/.f64 angle #s(literal 180 binary64)) < 9.99999999999999998e97

   1. Initial program 58.9%

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

      1. associate-*l* N/A

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

      2. *-commutative N/A

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

      3. associate-*l* N/A

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

      4. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\sin \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right), \color{blue}{\left(\cos \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right) \cdot \left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right)\right)}\right) \]

      5. sin-lowering-sin.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sin.f64}\left(\left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right)\right), \left(\color{blue}{\cos \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right)} \cdot \left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right)\right)\right) \]

      6. associate-*r/ N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sin.f64}\left(\left(\frac{\mathsf{PI}\left(\right) \cdot angle}{180}\right)\right), \left(\cos \color{blue}{\left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right)} \cdot \left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right)\right)\right) \]

      7. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sin.f64}\left(\mathsf{/.f64}\left(\left(\mathsf{PI}\left(\right) \cdot angle\right), 180\right)\right), \left(\cos \color{blue}{\left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right)} \cdot \left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right)\right)\right) \]

      8. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sin.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{PI}\left(\right), angle\right), 180\right)\right), \left(\cos \left(\color{blue}{\mathsf{PI}\left(\right)} \cdot \frac{angle}{180}\right) \cdot \left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right)\right)\right) \]

      9. PI-lowering-PI.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sin.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{PI.f64}\left(\right), angle\right), 180\right)\right), \left(\cos \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right) \cdot \left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right)\right)\right) \]

      10. *-commutative N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{sin.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{PI.f64}\left(\right), angle\right), 180\right)\right), \left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \color{blue}{\cos \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right)}\right)\right) \]

      11. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{sin.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{PI.f64}\left(\right), angle\right), 180\right)\right), \mathsf{*.f64}\left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right), \color{blue}{\cos \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right)}\right)\right) \]

   3. Simplified 59.1%

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

      1. associate-*r/ N/A

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

      2. *-commutative N/A

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

      3. associate-*r* N/A

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

      4. associate-*r/ N/A

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

      5. associate-*r* N/A

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

      6. *-commutative N/A

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

      7. *-commutative N/A

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

      8. difference-of-squares N/A

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

   6. Applied egg-rr 75.3%

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

   if 9.99999999999999998e97 < (/.f64 angle #s(literal 180 binary64)) < 1.99999999999999994e224

   1. Initial program 25.9%

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

      1. associate-*l* N/A

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

      2. *-commutative N/A

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

      3. associate-*l* N/A

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

      4. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\sin \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right), \color{blue}{\left(\cos \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right) \cdot \left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right)\right)}\right) \]

      5. sin-lowering-sin.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sin.f64}\left(\left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right)\right), \left(\color{blue}{\cos \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right)} \cdot \left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right)\right)\right) \]

      6. associate-*r/ N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sin.f64}\left(\left(\frac{\mathsf{PI}\left(\right) \cdot angle}{180}\right)\right), \left(\cos \color{blue}{\left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right)} \cdot \left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right)\right)\right) \]

      7. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sin.f64}\left(\mathsf{/.f64}\left(\left(\mathsf{PI}\left(\right) \cdot angle\right), 180\right)\right), \left(\cos \color{blue}{\left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right)} \cdot \left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right)\right)\right) \]

      8. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sin.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{PI}\left(\right), angle\right), 180\right)\right), \left(\cos \left(\color{blue}{\mathsf{PI}\left(\right)} \cdot \frac{angle}{180}\right) \cdot \left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right)\right)\right) \]

      9. PI-lowering-PI.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sin.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{PI.f64}\left(\right), angle\right), 180\right)\right), \left(\cos \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right) \cdot \left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right)\right)\right) \]

      10. *-commutative N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{sin.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{PI.f64}\left(\right), angle\right), 180\right)\right), \left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \color{blue}{\cos \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right)}\right)\right) \]

      11. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{sin.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{PI.f64}\left(\right), angle\right), 180\right)\right), \mathsf{*.f64}\left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right), \color{blue}{\cos \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right)}\right)\right) \]

   3. Simplified 16.4%

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

      1. associate-*r/ N/A

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

      2. associate-*l* N/A

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

      3. associate-*r* N/A

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

      4. difference-of-squares N/A

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

      5. associate-*l* N/A

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

      6. associate-*r* N/A

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

      7. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\left(\left(\sin \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right) \cdot 2\right) \cdot \left(b + a\right)\right), \color{blue}{\left(\left(b - a\right) \cdot \cos \left(\frac{\mathsf{PI}\left(\right) \cdot angle}{180}\right)\right)}\right) \]

   6. Applied egg-rr 25.5%

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

      1. add-sqr-sqrt N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(2, \mathsf{sin.f64}\left(\mathsf{/.f64}\left(\left(\sqrt{\mathsf{PI}\left(\right)} \cdot \sqrt{\mathsf{PI}\left(\right)}\right), \mathsf{/.f64}\left(180, angle\right)\right)\right)\right), \mathsf{+.f64}\left(b, a\right)\right), \mathsf{*.f64}\left(\mathsf{cos.f64}\left(\mathsf{/.f64}\left(\mathsf{PI.f64}\left(\right), \mathsf{/.f64}\left(180, angle\right)\right)\right), \mathsf{\_.f64}\left(b, a\right)\right)\right) \]

      2. pow2 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(2, \mathsf{sin.f64}\left(\mathsf{/.f64}\left(\left({\left(\sqrt{\mathsf{PI}\left(\right)}\right)}^{2}\right), \mathsf{/.f64}\left(180, angle\right)\right)\right)\right), \mathsf{+.f64}\left(b, a\right)\right), \mathsf{*.f64}\left(\mathsf{cos.f64}\left(\mathsf{/.f64}\left(\mathsf{PI.f64}\left(\right), \mathsf{/.f64}\left(180, angle\right)\right)\right), \mathsf{\_.f64}\left(b, a\right)\right)\right) \]

      3. pow-lowering-pow.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(2, \mathsf{sin.f64}\left(\mathsf{/.f64}\left(\mathsf{pow.f64}\left(\left(\sqrt{\mathsf{PI}\left(\right)}\right), 2\right), \mathsf{/.f64}\left(180, angle\right)\right)\right)\right), \mathsf{+.f64}\left(b, a\right)\right), \mathsf{*.f64}\left(\mathsf{cos.f64}\left(\mathsf{/.f64}\left(\mathsf{PI.f64}\left(\right), \mathsf{/.f64}\left(180, angle\right)\right)\right), \mathsf{\_.f64}\left(b, a\right)\right)\right) \]

      4. sqrt-lowering-sqrt.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(2, \mathsf{sin.f64}\left(\mathsf{/.f64}\left(\mathsf{pow.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{PI}\left(\right)\right), 2\right), \mathsf{/.f64}\left(180, angle\right)\right)\right)\right), \mathsf{+.f64}\left(b, a\right)\right), \mathsf{*.f64}\left(\mathsf{cos.f64}\left(\mathsf{/.f64}\left(\mathsf{PI.f64}\left(\right), \mathsf{/.f64}\left(180, angle\right)\right)\right), \mathsf{\_.f64}\left(b, a\right)\right)\right) \]

      5. PI-lowering-PI.f64 21.9%

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(2, \mathsf{sin.f64}\left(\mathsf{/.f64}\left(\mathsf{pow.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{PI.f64}\left(\right)\right), 2\right), \mathsf{/.f64}\left(180, angle\right)\right)\right)\right), \mathsf{+.f64}\left(b, a\right)\right), \mathsf{*.f64}\left(\mathsf{cos.f64}\left(\mathsf{/.f64}\left(\mathsf{PI.f64}\left(\right), \mathsf{/.f64}\left(180, angle\right)\right)\right), \mathsf{\_.f64}\left(b, a\right)\right)\right) \]

   8. Applied egg-rr 21.9%

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

   9. Taylor expanded in angle around 0

      \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(2, \mathsf{sin.f64}\left(\mathsf{/.f64}\left(\mathsf{pow.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{PI.f64}\left(\right)\right), 2\right), \mathsf{/.f64}\left(180, angle\right)\right)\right)\right), \mathsf{+.f64}\left(b, a\right)\right), \mathsf{*.f64}\left(\color{blue}{1}, \mathsf{\_.f64}\left(b, a\right)\right)\right) \]
   10. Step-by-step derivation

   11. Simplified 34.7%

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

   if 1.99999999999999994e224 < (/.f64 angle #s(literal 180 binary64))

   1. Initial program 34.4%

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

      1. associate-*l* N/A

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

      2. *-commutative N/A

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

      3. associate-*l* N/A

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

      4. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\sin \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right), \color{blue}{\left(\cos \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right) \cdot \left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right)\right)}\right) \]

      5. sin-lowering-sin.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sin.f64}\left(\left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right)\right), \left(\color{blue}{\cos \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right)} \cdot \left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right)\right)\right) \]

      6. associate-*r/ N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sin.f64}\left(\left(\frac{\mathsf{PI}\left(\right) \cdot angle}{180}\right)\right), \left(\cos \color{blue}{\left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right)} \cdot \left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right)\right)\right) \]

      7. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sin.f64}\left(\mathsf{/.f64}\left(\left(\mathsf{PI}\left(\right) \cdot angle\right), 180\right)\right), \left(\cos \color{blue}{\left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right)} \cdot \left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right)\right)\right) \]

      8. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sin.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{PI}\left(\right), angle\right), 180\right)\right), \left(\cos \left(\color{blue}{\mathsf{PI}\left(\right)} \cdot \frac{angle}{180}\right) \cdot \left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right)\right)\right) \]

      9. PI-lowering-PI.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sin.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{PI.f64}\left(\right), angle\right), 180\right)\right), \left(\cos \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right) \cdot \left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right)\right)\right) \]

      10. *-commutative N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{sin.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{PI.f64}\left(\right), angle\right), 180\right)\right), \left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \color{blue}{\cos \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right)}\right)\right) \]

      11. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{sin.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{PI.f64}\left(\right), angle\right), 180\right)\right), \mathsf{*.f64}\left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right), \color{blue}{\cos \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right)}\right)\right) \]

   3. Simplified 48.4%

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

      1. associate-*r/ N/A

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

      2. associate-*l* N/A

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

      3. associate-*r* N/A

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

      4. difference-of-squares N/A

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

      5. associate-*l* N/A

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

      6. associate-*r* N/A

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

      7. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\left(\left(\sin \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right) \cdot 2\right) \cdot \left(b + a\right)\right), \color{blue}{\left(\left(b - a\right) \cdot \cos \left(\frac{\mathsf{PI}\left(\right) \cdot angle}{180}\right)\right)}\right) \]

   6. Applied egg-rr 40.0%

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

      1. add-sqr-sqrt N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(2, \mathsf{sin.f64}\left(\mathsf{/.f64}\left(\left(\sqrt{\mathsf{PI}\left(\right)} \cdot \sqrt{\mathsf{PI}\left(\right)}\right), \mathsf{/.f64}\left(180, angle\right)\right)\right)\right), \mathsf{+.f64}\left(b, a\right)\right), \mathsf{*.f64}\left(\mathsf{cos.f64}\left(\mathsf{/.f64}\left(\mathsf{PI.f64}\left(\right), \mathsf{/.f64}\left(180, angle\right)\right)\right), \mathsf{\_.f64}\left(b, a\right)\right)\right) \]

      2. pow2 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(2, \mathsf{sin.f64}\left(\mathsf{/.f64}\left(\left({\left(\sqrt{\mathsf{PI}\left(\right)}\right)}^{2}\right), \mathsf{/.f64}\left(180, angle\right)\right)\right)\right), \mathsf{+.f64}\left(b, a\right)\right), \mathsf{*.f64}\left(\mathsf{cos.f64}\left(\mathsf{/.f64}\left(\mathsf{PI.f64}\left(\right), \mathsf{/.f64}\left(180, angle\right)\right)\right), \mathsf{\_.f64}\left(b, a\right)\right)\right) \]

      3. pow-lowering-pow.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(2, \mathsf{sin.f64}\left(\mathsf{/.f64}\left(\mathsf{pow.f64}\left(\left(\sqrt{\mathsf{PI}\left(\right)}\right), 2\right), \mathsf{/.f64}\left(180, angle\right)\right)\right)\right), \mathsf{+.f64}\left(b, a\right)\right), \mathsf{*.f64}\left(\mathsf{cos.f64}\left(\mathsf{/.f64}\left(\mathsf{PI.f64}\left(\right), \mathsf{/.f64}\left(180, angle\right)\right)\right), \mathsf{\_.f64}\left(b, a\right)\right)\right) \]

      4. sqrt-lowering-sqrt.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(2, \mathsf{sin.f64}\left(\mathsf{/.f64}\left(\mathsf{pow.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{PI}\left(\right)\right), 2\right), \mathsf{/.f64}\left(180, angle\right)\right)\right)\right), \mathsf{+.f64}\left(b, a\right)\right), \mathsf{*.f64}\left(\mathsf{cos.f64}\left(\mathsf{/.f64}\left(\mathsf{PI.f64}\left(\right), \mathsf{/.f64}\left(180, angle\right)\right)\right), \mathsf{\_.f64}\left(b, a\right)\right)\right) \]

      5. PI-lowering-PI.f64 47.8%

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(2, \mathsf{sin.f64}\left(\mathsf{/.f64}\left(\mathsf{pow.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{PI.f64}\left(\right)\right), 2\right), \mathsf{/.f64}\left(180, angle\right)\right)\right)\right), \mathsf{+.f64}\left(b, a\right)\right), \mathsf{*.f64}\left(\mathsf{cos.f64}\left(\mathsf{/.f64}\left(\mathsf{PI.f64}\left(\right), \mathsf{/.f64}\left(180, angle\right)\right)\right), \mathsf{\_.f64}\left(b, a\right)\right)\right) \]

   8. Applied egg-rr 47.8%

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

      1. associate-/r/ N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(2, \mathsf{sin.f64}\left(\mathsf{/.f64}\left(\mathsf{pow.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{PI.f64}\left(\right)\right), 2\right), \mathsf{/.f64}\left(180, angle\right)\right)\right)\right), \mathsf{+.f64}\left(b, a\right)\right), \mathsf{*.f64}\left(\mathsf{cos.f64}\left(\left(\frac{\mathsf{PI}\left(\right)}{180} \cdot angle\right)\right), \mathsf{\_.f64}\left(b, a\right)\right)\right) \]

      2. associate-*l/ N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(2, \mathsf{sin.f64}\left(\mathsf{/.f64}\left(\mathsf{pow.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{PI.f64}\left(\right)\right), 2\right), \mathsf{/.f64}\left(180, angle\right)\right)\right)\right), \mathsf{+.f64}\left(b, a\right)\right), \mathsf{*.f64}\left(\mathsf{cos.f64}\left(\left(\frac{\mathsf{PI}\left(\right) \cdot angle}{180}\right)\right), \mathsf{\_.f64}\left(b, a\right)\right)\right) \]

      3. clear-num N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(2, \mathsf{sin.f64}\left(\mathsf{/.f64}\left(\mathsf{pow.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{PI.f64}\left(\right)\right), 2\right), \mathsf{/.f64}\left(180, angle\right)\right)\right)\right), \mathsf{+.f64}\left(b, a\right)\right), \mathsf{*.f64}\left(\mathsf{cos.f64}\left(\left(\frac{1}{\frac{180}{\mathsf{PI}\left(\right) \cdot angle}}\right)\right), \mathsf{\_.f64}\left(b, a\right)\right)\right) \]

      4. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(2, \mathsf{sin.f64}\left(\mathsf{/.f64}\left(\mathsf{pow.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{PI.f64}\left(\right)\right), 2\right), \mathsf{/.f64}\left(180, angle\right)\right)\right)\right), \mathsf{+.f64}\left(b, a\right)\right), \mathsf{*.f64}\left(\mathsf{cos.f64}\left(\mathsf{/.f64}\left(1, \left(\frac{180}{\mathsf{PI}\left(\right) \cdot angle}\right)\right)\right), \mathsf{\_.f64}\left(b, a\right)\right)\right) \]

      5. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(2, \mathsf{sin.f64}\left(\mathsf{/.f64}\left(\mathsf{pow.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{PI.f64}\left(\right)\right), 2\right), \mathsf{/.f64}\left(180, angle\right)\right)\right)\right), \mathsf{+.f64}\left(b, a\right)\right), \mathsf{*.f64}\left(\mathsf{cos.f64}\left(\mathsf{/.f64}\left(1, \mathsf{/.f64}\left(180, \left(\mathsf{PI}\left(\right) \cdot angle\right)\right)\right)\right), \mathsf{\_.f64}\left(b, a\right)\right)\right) \]

      6. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(2, \mathsf{sin.f64}\left(\mathsf{/.f64}\left(\mathsf{pow.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{PI.f64}\left(\right)\right), 2\right), \mathsf{/.f64}\left(180, angle\right)\right)\right)\right), \mathsf{+.f64}\left(b, a\right)\right), \mathsf{*.f64}\left(\mathsf{cos.f64}\left(\mathsf{/.f64}\left(1, \mathsf{/.f64}\left(180, \mathsf{*.f64}\left(\mathsf{PI}\left(\right), angle\right)\right)\right)\right), \mathsf{\_.f64}\left(b, a\right)\right)\right) \]

      7. PI-lowering-PI.f64 48.2%

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(2, \mathsf{sin.f64}\left(\mathsf{/.f64}\left(\mathsf{pow.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{PI.f64}\left(\right)\right), 2\right), \mathsf{/.f64}\left(180, angle\right)\right)\right)\right), \mathsf{+.f64}\left(b, a\right)\right), \mathsf{*.f64}\left(\mathsf{cos.f64}\left(\mathsf{/.f64}\left(1, \mathsf{/.f64}\left(180, \mathsf{*.f64}\left(\mathsf{PI.f64}\left(\right), angle\right)\right)\right)\right), \mathsf{\_.f64}\left(b, a\right)\right)\right) \]

   10. Applied egg-rr 48.2%

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

4. Final simplification 70.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{angle}{180} \leq 10^{+98}:\\ \;\;\;\;\left(b - a\right) \cdot \left(\left(b + a\right) \cdot \sin \left(\left(angle \cdot \pi\right) \cdot 0.011111111111111112\right)\right)\\ \mathbf{elif}\;\frac{angle}{180} \leq 2 \cdot 10^{+224}:\\ \;\;\;\;\left(b - a\right) \cdot \left(\left(b + a\right) \cdot \left(2 \cdot \sin \left(\frac{{\left(\sqrt{\pi}\right)}^{2}}{\frac{180}{angle}}\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\left(b + a\right) \cdot \left(2 \cdot \sin \left(\frac{{\left(\sqrt{\pi}\right)}^{2}}{\frac{180}{angle}}\right)\right)\right) \cdot \left(\left(b - a\right) \cdot \cos \left(\frac{1}{\frac{180}{angle \cdot \pi}}\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 67.4% accurate, 1.3× speedup

Math

\[\begin{array}{l} angle\_m = \left|angle\right| \\ angle\_s = \mathsf{copysign}\left(1, angle\right) \\ angle\_s \cdot \begin{array}{l} \mathbf{if}\;\frac{angle\_m}{180} \leq 10^{+98}:\\ \;\;\;\;\left(b - a\right) \cdot \left(\left(b + a\right) \cdot \sin \left(\left(angle\_m \cdot \pi\right) \cdot 0.011111111111111112\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(b - a\right) \cdot \left(\left(b + a\right) \cdot \left(2 \cdot \sin \left(\frac{{\left(\sqrt{\pi}\right)}^{2}}{\frac{180}{angle\_m}}\right)\right)\right)\\ \end{array} \end{array} \]

FPCore

angle\_m = (fabs.f64 angle)
angle\_s = (copysign.f64 #s(literal 1 binary64) angle)
(FPCore (angle_s a b angle_m)
 :precision binary64
 (*
  angle_s
  (if (<= (/ angle_m 180.0) 1e+98)
    (* (- b a) (* (+ b a) (sin (* (* angle_m PI) 0.011111111111111112))))
    (*
     (- b a)
     (* (+ b a) (* 2.0 (sin (/ (pow (sqrt PI) 2.0) (/ 180.0 angle_m)))))))))

C

angle\_m = fabs(angle);
angle\_s = copysign(1.0, angle);
double code(double angle_s, double a, double b, double angle_m) {
	double tmp;
	if ((angle_m / 180.0) <= 1e+98) {
		tmp = (b - a) * ((b + a) * sin(((angle_m * ((double) M_PI)) * 0.011111111111111112)));
	} else {
		tmp = (b - a) * ((b + a) * (2.0 * sin((pow(sqrt(((double) M_PI)), 2.0) / (180.0 / angle_m)))));
	}
	return angle_s * tmp;
}

Java

angle\_m = Math.abs(angle);
angle\_s = Math.copySign(1.0, angle);
public static double code(double angle_s, double a, double b, double angle_m) {
	double tmp;
	if ((angle_m / 180.0) <= 1e+98) {
		tmp = (b - a) * ((b + a) * Math.sin(((angle_m * Math.PI) * 0.011111111111111112)));
	} else {
		tmp = (b - a) * ((b + a) * (2.0 * Math.sin((Math.pow(Math.sqrt(Math.PI), 2.0) / (180.0 / angle_m)))));
	}
	return angle_s * tmp;
}

Python

angle\_m = math.fabs(angle)
angle\_s = math.copysign(1.0, angle)
def code(angle_s, a, b, angle_m):
	tmp = 0
	if (angle_m / 180.0) <= 1e+98:
		tmp = (b - a) * ((b + a) * math.sin(((angle_m * math.pi) * 0.011111111111111112)))
	else:
		tmp = (b - a) * ((b + a) * (2.0 * math.sin((math.pow(math.sqrt(math.pi), 2.0) / (180.0 / angle_m)))))
	return angle_s * tmp

Julia

angle\_m = abs(angle)
angle\_s = copysign(1.0, angle)
function code(angle_s, a, b, angle_m)
	tmp = 0.0
	if (Float64(angle_m / 180.0) <= 1e+98)
		tmp = Float64(Float64(b - a) * Float64(Float64(b + a) * sin(Float64(Float64(angle_m * pi) * 0.011111111111111112))));
	else
		tmp = Float64(Float64(b - a) * Float64(Float64(b + a) * Float64(2.0 * sin(Float64((sqrt(pi) ^ 2.0) / Float64(180.0 / angle_m))))));
	end
	return Float64(angle_s * tmp)
end

MATLAB

angle\_m = abs(angle);
angle\_s = sign(angle) * abs(1.0);
function tmp_2 = code(angle_s, a, b, angle_m)
	tmp = 0.0;
	if ((angle_m / 180.0) <= 1e+98)
		tmp = (b - a) * ((b + a) * sin(((angle_m * pi) * 0.011111111111111112)));
	else
		tmp = (b - a) * ((b + a) * (2.0 * sin(((sqrt(pi) ^ 2.0) / (180.0 / angle_m)))));
	end
	tmp_2 = angle_s * tmp;
end

Wolfram

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_, b_, angle$95$m_] := N[(angle$95$s * If[LessEqual[N[(angle$95$m / 180.0), $MachinePrecision], 1e+98], N[(N[(b - a), $MachinePrecision] * N[(N[(b + a), $MachinePrecision] * N[Sin[N[(N[(angle$95$m * Pi), $MachinePrecision] * 0.011111111111111112), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(b - a), $MachinePrecision] * N[(N[(b + a), $MachinePrecision] * N[(2.0 * N[Sin[N[(N[Power[N[Sqrt[Pi], $MachinePrecision], 2.0], $MachinePrecision] / N[(180.0 / angle$95$m), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if (/.f64 angle #s(literal 180 binary64)) < 9.99999999999999998e97

   1. Initial program 58.9%

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

      1. associate-*l* N/A

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

      2. *-commutative N/A

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

      3. associate-*l* N/A

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

      4. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\sin \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right), \color{blue}{\left(\cos \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right) \cdot \left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right)\right)}\right) \]

      5. sin-lowering-sin.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sin.f64}\left(\left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right)\right), \left(\color{blue}{\cos \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right)} \cdot \left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right)\right)\right) \]

      6. associate-*r/ N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sin.f64}\left(\left(\frac{\mathsf{PI}\left(\right) \cdot angle}{180}\right)\right), \left(\cos \color{blue}{\left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right)} \cdot \left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right)\right)\right) \]

      7. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sin.f64}\left(\mathsf{/.f64}\left(\left(\mathsf{PI}\left(\right) \cdot angle\right), 180\right)\right), \left(\cos \color{blue}{\left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right)} \cdot \left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right)\right)\right) \]

      8. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sin.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{PI}\left(\right), angle\right), 180\right)\right), \left(\cos \left(\color{blue}{\mathsf{PI}\left(\right)} \cdot \frac{angle}{180}\right) \cdot \left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right)\right)\right) \]

      9. PI-lowering-PI.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sin.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{PI.f64}\left(\right), angle\right), 180\right)\right), \left(\cos \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right) \cdot \left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right)\right)\right) \]

      10. *-commutative N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{sin.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{PI.f64}\left(\right), angle\right), 180\right)\right), \left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \color{blue}{\cos \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right)}\right)\right) \]

      11. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{sin.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{PI.f64}\left(\right), angle\right), 180\right)\right), \mathsf{*.f64}\left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right), \color{blue}{\cos \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right)}\right)\right) \]

   3. Simplified 59.1%

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

      1. associate-*r/ N/A

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

      2. *-commutative N/A

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

      3. associate-*r* N/A

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

      4. associate-*r/ N/A

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

      5. associate-*r* N/A

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

      6. *-commutative N/A

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

      7. *-commutative N/A

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

      8. difference-of-squares N/A

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

   6. Applied egg-rr 75.3%

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

   if 9.99999999999999998e97 < (/.f64 angle #s(literal 180 binary64))

   1. Initial program 29.1%

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

      1. associate-*l* N/A

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

      2. *-commutative N/A

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

      3. associate-*l* N/A

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

      4. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\sin \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right), \color{blue}{\left(\cos \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right) \cdot \left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right)\right)}\right) \]

      5. sin-lowering-sin.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sin.f64}\left(\left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right)\right), \left(\color{blue}{\cos \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right)} \cdot \left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right)\right)\right) \]

      6. associate-*r/ N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sin.f64}\left(\left(\frac{\mathsf{PI}\left(\right) \cdot angle}{180}\right)\right), \left(\cos \color{blue}{\left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right)} \cdot \left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right)\right)\right) \]

      7. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sin.f64}\left(\mathsf{/.f64}\left(\left(\mathsf{PI}\left(\right) \cdot angle\right), 180\right)\right), \left(\cos \color{blue}{\left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right)} \cdot \left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right)\right)\right) \]

      8. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sin.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{PI}\left(\right), angle\right), 180\right)\right), \left(\cos \left(\color{blue}{\mathsf{PI}\left(\right)} \cdot \frac{angle}{180}\right) \cdot \left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right)\right)\right) \]

      9. PI-lowering-PI.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sin.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{PI.f64}\left(\right), angle\right), 180\right)\right), \left(\cos \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right) \cdot \left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right)\right)\right) \]

      10. *-commutative N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{sin.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{PI.f64}\left(\right), angle\right), 180\right)\right), \left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \color{blue}{\cos \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right)}\right)\right) \]

      11. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{sin.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{PI.f64}\left(\right), angle\right), 180\right)\right), \mathsf{*.f64}\left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right), \color{blue}{\cos \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right)}\right)\right) \]

   3. Simplified 28.5%

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

      1. associate-*r/ N/A

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

      2. associate-*l* N/A

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

      3. associate-*r* N/A

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

      4. difference-of-squares N/A

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

      5. associate-*l* N/A

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

      6. associate-*r* N/A

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

      7. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\left(\left(\sin \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right) \cdot 2\right) \cdot \left(b + a\right)\right), \color{blue}{\left(\left(b - a\right) \cdot \cos \left(\frac{\mathsf{PI}\left(\right) \cdot angle}{180}\right)\right)}\right) \]

   6. Applied egg-rr 31.0%

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

      1. add-sqr-sqrt N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(2, \mathsf{sin.f64}\left(\mathsf{/.f64}\left(\left(\sqrt{\mathsf{PI}\left(\right)} \cdot \sqrt{\mathsf{PI}\left(\right)}\right), \mathsf{/.f64}\left(180, angle\right)\right)\right)\right), \mathsf{+.f64}\left(b, a\right)\right), \mathsf{*.f64}\left(\mathsf{cos.f64}\left(\mathsf{/.f64}\left(\mathsf{PI.f64}\left(\right), \mathsf{/.f64}\left(180, angle\right)\right)\right), \mathsf{\_.f64}\left(b, a\right)\right)\right) \]

      2. pow2 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(2, \mathsf{sin.f64}\left(\mathsf{/.f64}\left(\left({\left(\sqrt{\mathsf{PI}\left(\right)}\right)}^{2}\right), \mathsf{/.f64}\left(180, angle\right)\right)\right)\right), \mathsf{+.f64}\left(b, a\right)\right), \mathsf{*.f64}\left(\mathsf{cos.f64}\left(\mathsf{/.f64}\left(\mathsf{PI.f64}\left(\right), \mathsf{/.f64}\left(180, angle\right)\right)\right), \mathsf{\_.f64}\left(b, a\right)\right)\right) \]

      3. pow-lowering-pow.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(2, \mathsf{sin.f64}\left(\mathsf{/.f64}\left(\mathsf{pow.f64}\left(\left(\sqrt{\mathsf{PI}\left(\right)}\right), 2\right), \mathsf{/.f64}\left(180, angle\right)\right)\right)\right), \mathsf{+.f64}\left(b, a\right)\right), \mathsf{*.f64}\left(\mathsf{cos.f64}\left(\mathsf{/.f64}\left(\mathsf{PI.f64}\left(\right), \mathsf{/.f64}\left(180, angle\right)\right)\right), \mathsf{\_.f64}\left(b, a\right)\right)\right) \]

      4. sqrt-lowering-sqrt.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(2, \mathsf{sin.f64}\left(\mathsf{/.f64}\left(\mathsf{pow.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{PI}\left(\right)\right), 2\right), \mathsf{/.f64}\left(180, angle\right)\right)\right)\right), \mathsf{+.f64}\left(b, a\right)\right), \mathsf{*.f64}\left(\mathsf{cos.f64}\left(\mathsf{/.f64}\left(\mathsf{PI.f64}\left(\right), \mathsf{/.f64}\left(180, angle\right)\right)\right), \mathsf{\_.f64}\left(b, a\right)\right)\right) \]

      5. PI-lowering-PI.f64 31.7%

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(2, \mathsf{sin.f64}\left(\mathsf{/.f64}\left(\mathsf{pow.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{PI.f64}\left(\right)\right), 2\right), \mathsf{/.f64}\left(180, angle\right)\right)\right)\right), \mathsf{+.f64}\left(b, a\right)\right), \mathsf{*.f64}\left(\mathsf{cos.f64}\left(\mathsf{/.f64}\left(\mathsf{PI.f64}\left(\right), \mathsf{/.f64}\left(180, angle\right)\right)\right), \mathsf{\_.f64}\left(b, a\right)\right)\right) \]

   8. Applied egg-rr 31.7%

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

   9. Taylor expanded in angle around 0

      \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(2, \mathsf{sin.f64}\left(\mathsf{/.f64}\left(\mathsf{pow.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{PI.f64}\left(\right)\right), 2\right), \mathsf{/.f64}\left(180, angle\right)\right)\right)\right), \mathsf{+.f64}\left(b, a\right)\right), \mathsf{*.f64}\left(\color{blue}{1}, \mathsf{\_.f64}\left(b, a\right)\right)\right) \]
   10. Step-by-step derivation

   11. Simplified 40.1%

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

4. Final simplification 70.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{angle}{180} \leq 10^{+98}:\\ \;\;\;\;\left(b - a\right) \cdot \left(\left(b + a\right) \cdot \sin \left(\left(angle \cdot \pi\right) \cdot 0.011111111111111112\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(b - a\right) \cdot \left(\left(b + a\right) \cdot \left(2 \cdot \sin \left(\frac{{\left(\sqrt{\pi}\right)}^{2}}{\frac{180}{angle}}\right)\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 67.6% accurate, 1.8× speedup

Math

\[\begin{array}{l} angle\_m = \left|angle\right| \\ angle\_s = \mathsf{copysign}\left(1, angle\right) \\ \begin{array}{l} t_0 := \frac{\pi}{\frac{180}{angle\_m}}\\ angle\_s \cdot \begin{array}{l} \mathbf{if}\;a \leq 4 \cdot 10^{+186}:\\ \;\;\;\;\left(\left(b - a\right) \cdot \cos t\_0\right) \cdot \left(\left(b + a\right) \cdot \left(2 \cdot \sin \left(\frac{1}{\frac{180}{angle\_m \cdot \pi}}\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\left(b + a\right) \cdot \left(2 \cdot \sin t\_0\right)\right) \cdot \left(\left(b - a\right) \cdot \cos \left(\frac{\pi}{-1} \cdot \frac{0.005555555555555556}{\frac{-1}{angle\_m}}\right)\right)\\ \end{array} \end{array} \end{array} \]

FPCore

angle\_m = (fabs.f64 angle)
angle\_s = (copysign.f64 #s(literal 1 binary64) angle)
(FPCore (angle_s a b angle_m)
 :precision binary64
 (let* ((t_0 (/ PI (/ 180.0 angle_m))))
   (*
    angle_s
    (if (<= a 4e+186)
      (*
       (* (- b a) (cos t_0))
       (* (+ b a) (* 2.0 (sin (/ 1.0 (/ 180.0 (* angle_m PI)))))))
      (*
       (* (+ b a) (* 2.0 (sin t_0)))
       (*
        (- b a)
        (cos (* (/ PI -1.0) (/ 0.005555555555555556 (/ -1.0 angle_m))))))))))

C

angle\_m = fabs(angle);
angle\_s = copysign(1.0, angle);
double code(double angle_s, double a, double b, double angle_m) {
	double t_0 = ((double) M_PI) / (180.0 / angle_m);
	double tmp;
	if (a <= 4e+186) {
		tmp = ((b - a) * cos(t_0)) * ((b + a) * (2.0 * sin((1.0 / (180.0 / (angle_m * ((double) M_PI)))))));
	} else {
		tmp = ((b + a) * (2.0 * sin(t_0))) * ((b - a) * cos(((((double) M_PI) / -1.0) * (0.005555555555555556 / (-1.0 / angle_m)))));
	}
	return angle_s * tmp;
}

Java

angle\_m = Math.abs(angle);
angle\_s = Math.copySign(1.0, angle);
public static double code(double angle_s, double a, double b, double angle_m) {
	double t_0 = Math.PI / (180.0 / angle_m);
	double tmp;
	if (a <= 4e+186) {
		tmp = ((b - a) * Math.cos(t_0)) * ((b + a) * (2.0 * Math.sin((1.0 / (180.0 / (angle_m * Math.PI))))));
	} else {
		tmp = ((b + a) * (2.0 * Math.sin(t_0))) * ((b - a) * Math.cos(((Math.PI / -1.0) * (0.005555555555555556 / (-1.0 / angle_m)))));
	}
	return angle_s * tmp;
}

Python

angle\_m = math.fabs(angle)
angle\_s = math.copysign(1.0, angle)
def code(angle_s, a, b, angle_m):
	t_0 = math.pi / (180.0 / angle_m)
	tmp = 0
	if a <= 4e+186:
		tmp = ((b - a) * math.cos(t_0)) * ((b + a) * (2.0 * math.sin((1.0 / (180.0 / (angle_m * math.pi))))))
	else:
		tmp = ((b + a) * (2.0 * math.sin(t_0))) * ((b - a) * math.cos(((math.pi / -1.0) * (0.005555555555555556 / (-1.0 / angle_m)))))
	return angle_s * tmp

Julia

angle\_m = abs(angle)
angle\_s = copysign(1.0, angle)
function code(angle_s, a, b, angle_m)
	t_0 = Float64(pi / Float64(180.0 / angle_m))
	tmp = 0.0
	if (a <= 4e+186)
		tmp = Float64(Float64(Float64(b - a) * cos(t_0)) * Float64(Float64(b + a) * Float64(2.0 * sin(Float64(1.0 / Float64(180.0 / Float64(angle_m * pi)))))));
	else
		tmp = Float64(Float64(Float64(b + a) * Float64(2.0 * sin(t_0))) * Float64(Float64(b - a) * cos(Float64(Float64(pi / -1.0) * Float64(0.005555555555555556 / Float64(-1.0 / angle_m))))));
	end
	return Float64(angle_s * tmp)
end

MATLAB

angle\_m = abs(angle);
angle\_s = sign(angle) * abs(1.0);
function tmp_2 = code(angle_s, a, b, angle_m)
	t_0 = pi / (180.0 / angle_m);
	tmp = 0.0;
	if (a <= 4e+186)
		tmp = ((b - a) * cos(t_0)) * ((b + a) * (2.0 * sin((1.0 / (180.0 / (angle_m * pi))))));
	else
		tmp = ((b + a) * (2.0 * sin(t_0))) * ((b - a) * cos(((pi / -1.0) * (0.005555555555555556 / (-1.0 / angle_m)))));
	end
	tmp_2 = angle_s * tmp;
end

Wolfram

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_, b_, angle$95$m_] := Block[{t$95$0 = N[(Pi / N[(180.0 / angle$95$m), $MachinePrecision]), $MachinePrecision]}, N[(angle$95$s * If[LessEqual[a, 4e+186], N[(N[(N[(b - a), $MachinePrecision] * N[Cos[t$95$0], $MachinePrecision]), $MachinePrecision] * N[(N[(b + a), $MachinePrecision] * N[(2.0 * N[Sin[N[(1.0 / N[(180.0 / N[(angle$95$m * Pi), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(b + a), $MachinePrecision] * N[(2.0 * N[Sin[t$95$0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(N[(b - a), $MachinePrecision] * N[Cos[N[(N[(Pi / -1.0), $MachinePrecision] * N[(0.005555555555555556 / N[(-1.0 / angle$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if a < 3.99999999999999992e186

   1. Initial program 55.6%

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

      1. associate-*l* N/A

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

      2. *-commutative N/A

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

      3. associate-*l* N/A

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

      4. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\sin \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right), \color{blue}{\left(\cos \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right) \cdot \left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right)\right)}\right) \]

      5. sin-lowering-sin.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sin.f64}\left(\left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right)\right), \left(\color{blue}{\cos \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right)} \cdot \left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right)\right)\right) \]

      6. associate-*r/ N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sin.f64}\left(\left(\frac{\mathsf{PI}\left(\right) \cdot angle}{180}\right)\right), \left(\cos \color{blue}{\left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right)} \cdot \left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right)\right)\right) \]

      7. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sin.f64}\left(\mathsf{/.f64}\left(\left(\mathsf{PI}\left(\right) \cdot angle\right), 180\right)\right), \left(\cos \color{blue}{\left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right)} \cdot \left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right)\right)\right) \]

      8. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sin.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{PI}\left(\right), angle\right), 180\right)\right), \left(\cos \left(\color{blue}{\mathsf{PI}\left(\right)} \cdot \frac{angle}{180}\right) \cdot \left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right)\right)\right) \]

      9. PI-lowering-PI.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sin.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{PI.f64}\left(\right), angle\right), 180\right)\right), \left(\cos \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right) \cdot \left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right)\right)\right) \]

      10. *-commutative N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{sin.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{PI.f64}\left(\right), angle\right), 180\right)\right), \left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \color{blue}{\cos \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right)}\right)\right) \]

      11. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{sin.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{PI.f64}\left(\right), angle\right), 180\right)\right), \mathsf{*.f64}\left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right), \color{blue}{\cos \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right)}\right)\right) \]

   3. Simplified 55.7%

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

      1. associate-*r/ N/A

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

      2. associate-*l* N/A

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

      3. associate-*r* N/A

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

      4. difference-of-squares N/A

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

      5. associate-*l* N/A

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

      6. associate-*r* N/A

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

      7. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\left(\left(\sin \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right) \cdot 2\right) \cdot \left(b + a\right)\right), \color{blue}{\left(\left(b - a\right) \cdot \cos \left(\frac{\mathsf{PI}\left(\right) \cdot angle}{180}\right)\right)}\right) \]

   6. Applied egg-rr 67.5%

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

      1. associate-/r/ N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(2, \mathsf{sin.f64}\left(\left(\frac{\mathsf{PI}\left(\right)}{180} \cdot angle\right)\right)\right), \mathsf{+.f64}\left(b, a\right)\right), \mathsf{*.f64}\left(\mathsf{cos.f64}\left(\mathsf{/.f64}\left(\mathsf{PI.f64}\left(\right), \mathsf{/.f64}\left(\color{blue}{180}, angle\right)\right)\right), \mathsf{\_.f64}\left(b, a\right)\right)\right) \]

      2. associate-*l/ N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(2, \mathsf{sin.f64}\left(\left(\frac{\mathsf{PI}\left(\right) \cdot angle}{180}\right)\right)\right), \mathsf{+.f64}\left(b, a\right)\right), \mathsf{*.f64}\left(\mathsf{cos.f64}\left(\mathsf{/.f64}\left(\mathsf{PI.f64}\left(\right), \mathsf{/.f64}\left(\color{blue}{180}, angle\right)\right)\right), \mathsf{\_.f64}\left(b, a\right)\right)\right) \]

      3. clear-num N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(2, \mathsf{sin.f64}\left(\left(\frac{1}{\frac{180}{\mathsf{PI}\left(\right) \cdot angle}}\right)\right)\right), \mathsf{+.f64}\left(b, a\right)\right), \mathsf{*.f64}\left(\mathsf{cos.f64}\left(\mathsf{/.f64}\left(\mathsf{PI.f64}\left(\right), \mathsf{/.f64}\left(\color{blue}{180}, angle\right)\right)\right), \mathsf{\_.f64}\left(b, a\right)\right)\right) \]

      4. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(2, \mathsf{sin.f64}\left(\mathsf{/.f64}\left(1, \left(\frac{180}{\mathsf{PI}\left(\right) \cdot angle}\right)\right)\right)\right), \mathsf{+.f64}\left(b, a\right)\right), \mathsf{*.f64}\left(\mathsf{cos.f64}\left(\mathsf{/.f64}\left(\mathsf{PI.f64}\left(\right), \mathsf{/.f64}\left(\color{blue}{180}, angle\right)\right)\right), \mathsf{\_.f64}\left(b, a\right)\right)\right) \]

      5. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(2, \mathsf{sin.f64}\left(\mathsf{/.f64}\left(1, \mathsf{/.f64}\left(180, \left(\mathsf{PI}\left(\right) \cdot angle\right)\right)\right)\right)\right), \mathsf{+.f64}\left(b, a\right)\right), \mathsf{*.f64}\left(\mathsf{cos.f64}\left(\mathsf{/.f64}\left(\mathsf{PI.f64}\left(\right), \mathsf{/.f64}\left(180, angle\right)\right)\right), \mathsf{\_.f64}\left(b, a\right)\right)\right) \]

      6. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(2, \mathsf{sin.f64}\left(\mathsf{/.f64}\left(1, \mathsf{/.f64}\left(180, \mathsf{*.f64}\left(\mathsf{PI}\left(\right), angle\right)\right)\right)\right)\right), \mathsf{+.f64}\left(b, a\right)\right), \mathsf{*.f64}\left(\mathsf{cos.f64}\left(\mathsf{/.f64}\left(\mathsf{PI.f64}\left(\right), \mathsf{/.f64}\left(180, angle\right)\right)\right), \mathsf{\_.f64}\left(b, a\right)\right)\right) \]

      7. PI-lowering-PI.f64 69.4%

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(2, \mathsf{sin.f64}\left(\mathsf{/.f64}\left(1, \mathsf{/.f64}\left(180, \mathsf{*.f64}\left(\mathsf{PI.f64}\left(\right), angle\right)\right)\right)\right)\right), \mathsf{+.f64}\left(b, a\right)\right), \mathsf{*.f64}\left(\mathsf{cos.f64}\left(\mathsf{/.f64}\left(\mathsf{PI.f64}\left(\right), \mathsf{/.f64}\left(180, angle\right)\right)\right), \mathsf{\_.f64}\left(b, a\right)\right)\right) \]

   8. Applied egg-rr 69.4%

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

   if 3.99999999999999992e186 < a

   1. Initial program 41.6%

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

      1. associate-*l* N/A

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

      2. *-commutative N/A

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

      3. associate-*l* N/A

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

      4. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\sin \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right), \color{blue}{\left(\cos \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right) \cdot \left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right)\right)}\right) \]

      5. sin-lowering-sin.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sin.f64}\left(\left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right)\right), \left(\color{blue}{\cos \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right)} \cdot \left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right)\right)\right) \]

      6. associate-*r/ N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sin.f64}\left(\left(\frac{\mathsf{PI}\left(\right) \cdot angle}{180}\right)\right), \left(\cos \color{blue}{\left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right)} \cdot \left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right)\right)\right) \]

      7. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sin.f64}\left(\mathsf{/.f64}\left(\left(\mathsf{PI}\left(\right) \cdot angle\right), 180\right)\right), \left(\cos \color{blue}{\left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right)} \cdot \left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right)\right)\right) \]

      8. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sin.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{PI}\left(\right), angle\right), 180\right)\right), \left(\cos \left(\color{blue}{\mathsf{PI}\left(\right)} \cdot \frac{angle}{180}\right) \cdot \left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right)\right)\right) \]

      9. PI-lowering-PI.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sin.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{PI.f64}\left(\right), angle\right), 180\right)\right), \left(\cos \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right) \cdot \left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right)\right)\right) \]

      10. *-commutative N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{sin.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{PI.f64}\left(\right), angle\right), 180\right)\right), \left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \color{blue}{\cos \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right)}\right)\right) \]

      11. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{sin.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{PI.f64}\left(\right), angle\right), 180\right)\right), \mathsf{*.f64}\left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right), \color{blue}{\cos \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right)}\right)\right) \]

   3. Simplified 41.6%

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

      1. associate-*r/ N/A

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

      2. associate-*l* N/A

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

      3. associate-*r* N/A

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

      4. difference-of-squares N/A

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

      5. associate-*l* N/A

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

      6. associate-*r* N/A

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

      7. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\left(\left(\sin \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right) \cdot 2\right) \cdot \left(b + a\right)\right), \color{blue}{\left(\left(b - a\right) \cdot \cos \left(\frac{\mathsf{PI}\left(\right) \cdot angle}{180}\right)\right)}\right) \]

   6. Applied egg-rr 83.2%

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

      1. clear-num N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(2, \mathsf{sin.f64}\left(\mathsf{/.f64}\left(\mathsf{PI.f64}\left(\right), \mathsf{/.f64}\left(180, angle\right)\right)\right)\right), \mathsf{+.f64}\left(b, a\right)\right), \mathsf{*.f64}\left(\mathsf{cos.f64}\left(\left(\frac{\mathsf{PI}\left(\right)}{\frac{1}{\frac{angle}{180}}}\right)\right), \mathsf{\_.f64}\left(b, a\right)\right)\right) \]

      2. add-sqr-sqrt N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(2, \mathsf{sin.f64}\left(\mathsf{/.f64}\left(\mathsf{PI.f64}\left(\right), \mathsf{/.f64}\left(180, angle\right)\right)\right)\right), \mathsf{+.f64}\left(b, a\right)\right), \mathsf{*.f64}\left(\mathsf{cos.f64}\left(\left(\frac{\sqrt{\mathsf{PI}\left(\right)} \cdot \sqrt{\mathsf{PI}\left(\right)}}{\frac{1}{\frac{angle}{180}}}\right)\right), \mathsf{\_.f64}\left(b, a\right)\right)\right) \]

      3. frac-2neg N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(2, \mathsf{sin.f64}\left(\mathsf{/.f64}\left(\mathsf{PI.f64}\left(\right), \mathsf{/.f64}\left(180, angle\right)\right)\right)\right), \mathsf{+.f64}\left(b, a\right)\right), \mathsf{*.f64}\left(\mathsf{cos.f64}\left(\left(\frac{\sqrt{\mathsf{PI}\left(\right)} \cdot \sqrt{\mathsf{PI}\left(\right)}}{\frac{1}{\frac{\mathsf{neg}\left(angle\right)}{\mathsf{neg}\left(180\right)}}}\right)\right), \mathsf{\_.f64}\left(b, a\right)\right)\right) \]

      4. associate-/r/ N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(2, \mathsf{sin.f64}\left(\mathsf{/.f64}\left(\mathsf{PI.f64}\left(\right), \mathsf{/.f64}\left(180, angle\right)\right)\right)\right), \mathsf{+.f64}\left(b, a\right)\right), \mathsf{*.f64}\left(\mathsf{cos.f64}\left(\left(\frac{\sqrt{\mathsf{PI}\left(\right)} \cdot \sqrt{\mathsf{PI}\left(\right)}}{\frac{1}{\mathsf{neg}\left(angle\right)} \cdot \left(\mathsf{neg}\left(180\right)\right)}\right)\right), \mathsf{\_.f64}\left(b, a\right)\right)\right) \]

      5. times-frac N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(2, \mathsf{sin.f64}\left(\mathsf{/.f64}\left(\mathsf{PI.f64}\left(\right), \mathsf{/.f64}\left(180, angle\right)\right)\right)\right), \mathsf{+.f64}\left(b, a\right)\right), \mathsf{*.f64}\left(\mathsf{cos.f64}\left(\left(\frac{\sqrt{\mathsf{PI}\left(\right)}}{\frac{1}{\mathsf{neg}\left(angle\right)}} \cdot \frac{\sqrt{\mathsf{PI}\left(\right)}}{\mathsf{neg}\left(180\right)}\right)\right), \mathsf{\_.f64}\left(b, a\right)\right)\right) \]

      6. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(2, \mathsf{sin.f64}\left(\mathsf{/.f64}\left(\mathsf{PI.f64}\left(\right), \mathsf{/.f64}\left(180, angle\right)\right)\right)\right), \mathsf{+.f64}\left(b, a\right)\right), \mathsf{*.f64}\left(\mathsf{cos.f64}\left(\mathsf{*.f64}\left(\left(\frac{\sqrt{\mathsf{PI}\left(\right)}}{\frac{1}{\mathsf{neg}\left(angle\right)}}\right), \left(\frac{\sqrt{\mathsf{PI}\left(\right)}}{\mathsf{neg}\left(180\right)}\right)\right)\right), \mathsf{\_.f64}\left(b, a\right)\right)\right) \]

      7. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(2, \mathsf{sin.f64}\left(\mathsf{/.f64}\left(\mathsf{PI.f64}\left(\right), \mathsf{/.f64}\left(180, angle\right)\right)\right)\right), \mathsf{+.f64}\left(b, a\right)\right), \mathsf{*.f64}\left(\mathsf{cos.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(\left(\sqrt{\mathsf{PI}\left(\right)}\right), \left(\frac{1}{\mathsf{neg}\left(angle\right)}\right)\right), \left(\frac{\sqrt{\mathsf{PI}\left(\right)}}{\mathsf{neg}\left(180\right)}\right)\right)\right), \mathsf{\_.f64}\left(b, a\right)\right)\right) \]

      8. sqrt-lowering-sqrt.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(2, \mathsf{sin.f64}\left(\mathsf{/.f64}\left(\mathsf{PI.f64}\left(\right), \mathsf{/.f64}\left(180, angle\right)\right)\right)\right), \mathsf{+.f64}\left(b, a\right)\right), \mathsf{*.f64}\left(\mathsf{cos.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{PI}\left(\right)\right), \left(\frac{1}{\mathsf{neg}\left(angle\right)}\right)\right), \left(\frac{\sqrt{\mathsf{PI}\left(\right)}}{\mathsf{neg}\left(180\right)}\right)\right)\right), \mathsf{\_.f64}\left(b, a\right)\right)\right) \]

      9. PI-lowering-PI.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(2, \mathsf{sin.f64}\left(\mathsf{/.f64}\left(\mathsf{PI.f64}\left(\right), \mathsf{/.f64}\left(180, angle\right)\right)\right)\right), \mathsf{+.f64}\left(b, a\right)\right), \mathsf{*.f64}\left(\mathsf{cos.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{PI.f64}\left(\right)\right), \left(\frac{1}{\mathsf{neg}\left(angle\right)}\right)\right), \left(\frac{\sqrt{\mathsf{PI}\left(\right)}}{\mathsf{neg}\left(180\right)}\right)\right)\right), \mathsf{\_.f64}\left(b, a\right)\right)\right) \]

      10. frac-2neg N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(2, \mathsf{sin.f64}\left(\mathsf{/.f64}\left(\mathsf{PI.f64}\left(\right), \mathsf{/.f64}\left(180, angle\right)\right)\right)\right), \mathsf{+.f64}\left(b, a\right)\right), \mathsf{*.f64}\left(\mathsf{cos.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{PI.f64}\left(\right)\right), \left(\frac{\mathsf{neg}\left(1\right)}{\mathsf{neg}\left(\left(\mathsf{neg}\left(angle\right)\right)\right)}\right)\right), \left(\frac{\sqrt{\mathsf{PI}\left(\right)}}{\mathsf{neg}\left(180\right)}\right)\right)\right), \mathsf{\_.f64}\left(b, a\right)\right)\right) \]

      11. metadata-eval N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(2, \mathsf{sin.f64}\left(\mathsf{/.f64}\left(\mathsf{PI.f64}\left(\right), \mathsf{/.f64}\left(180, angle\right)\right)\right)\right), \mathsf{+.f64}\left(b, a\right)\right), \mathsf{*.f64}\left(\mathsf{cos.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{PI.f64}\left(\right)\right), \left(\frac{-1}{\mathsf{neg}\left(\left(\mathsf{neg}\left(angle\right)\right)\right)}\right)\right), \left(\frac{\sqrt{\mathsf{PI}\left(\right)}}{\mathsf{neg}\left(180\right)}\right)\right)\right), \mathsf{\_.f64}\left(b, a\right)\right)\right) \]

      12. remove-double-neg N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(2, \mathsf{sin.f64}\left(\mathsf{/.f64}\left(\mathsf{PI.f64}\left(\right), \mathsf{/.f64}\left(180, angle\right)\right)\right)\right), \mathsf{+.f64}\left(b, a\right)\right), \mathsf{*.f64}\left(\mathsf{cos.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{PI.f64}\left(\right)\right), \left(\frac{-1}{angle}\right)\right), \left(\frac{\sqrt{\mathsf{PI}\left(\right)}}{\mathsf{neg}\left(180\right)}\right)\right)\right), \mathsf{\_.f64}\left(b, a\right)\right)\right) \]

      13. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(2, \mathsf{sin.f64}\left(\mathsf{/.f64}\left(\mathsf{PI.f64}\left(\right), \mathsf{/.f64}\left(180, angle\right)\right)\right)\right), \mathsf{+.f64}\left(b, a\right)\right), \mathsf{*.f64}\left(\mathsf{cos.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{PI.f64}\left(\right)\right), \mathsf{/.f64}\left(-1, angle\right)\right), \left(\frac{\sqrt{\mathsf{PI}\left(\right)}}{\mathsf{neg}\left(180\right)}\right)\right)\right), \mathsf{\_.f64}\left(b, a\right)\right)\right) \]

      14. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(2, \mathsf{sin.f64}\left(\mathsf{/.f64}\left(\mathsf{PI.f64}\left(\right), \mathsf{/.f64}\left(180, angle\right)\right)\right)\right), \mathsf{+.f64}\left(b, a\right)\right), \mathsf{*.f64}\left(\mathsf{cos.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{PI.f64}\left(\right)\right), \mathsf{/.f64}\left(-1, angle\right)\right), \mathsf{/.f64}\left(\left(\sqrt{\mathsf{PI}\left(\right)}\right), \left(\mathsf{neg}\left(180\right)\right)\right)\right)\right), \mathsf{\_.f64}\left(b, a\right)\right)\right) \]

      15. sqrt-lowering-sqrt.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(2, \mathsf{sin.f64}\left(\mathsf{/.f64}\left(\mathsf{PI.f64}\left(\right), \mathsf{/.f64}\left(180, angle\right)\right)\right)\right), \mathsf{+.f64}\left(b, a\right)\right), \mathsf{*.f64}\left(\mathsf{cos.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{PI.f64}\left(\right)\right), \mathsf{/.f64}\left(-1, angle\right)\right), \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{PI}\left(\right)\right), \left(\mathsf{neg}\left(180\right)\right)\right)\right)\right), \mathsf{\_.f64}\left(b, a\right)\right)\right) \]

      16. PI-lowering-PI.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(2, \mathsf{sin.f64}\left(\mathsf{/.f64}\left(\mathsf{PI.f64}\left(\right), \mathsf{/.f64}\left(180, angle\right)\right)\right)\right), \mathsf{+.f64}\left(b, a\right)\right), \mathsf{*.f64}\left(\mathsf{cos.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{PI.f64}\left(\right)\right), \mathsf{/.f64}\left(-1, angle\right)\right), \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{PI.f64}\left(\right)\right), \left(\mathsf{neg}\left(180\right)\right)\right)\right)\right), \mathsf{\_.f64}\left(b, a\right)\right)\right) \]

      17. metadata-eval 83.2%

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(2, \mathsf{sin.f64}\left(\mathsf{/.f64}\left(\mathsf{PI.f64}\left(\right), \mathsf{/.f64}\left(180, angle\right)\right)\right)\right), \mathsf{+.f64}\left(b, a\right)\right), \mathsf{*.f64}\left(\mathsf{cos.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{PI.f64}\left(\right)\right), \mathsf{/.f64}\left(-1, angle\right)\right), \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{PI.f64}\left(\right)\right), -180\right)\right)\right), \mathsf{\_.f64}\left(b, a\right)\right)\right) \]

   8. Applied egg-rr 83.2%

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

      1. frac-2neg N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(2, \mathsf{sin.f64}\left(\mathsf{/.f64}\left(\mathsf{PI.f64}\left(\right), \mathsf{/.f64}\left(180, angle\right)\right)\right)\right), \mathsf{+.f64}\left(b, a\right)\right), \mathsf{*.f64}\left(\mathsf{cos.f64}\left(\left(\frac{\mathsf{neg}\left(\sqrt{\mathsf{PI}\left(\right)}\right)}{\mathsf{neg}\left(\frac{-1}{angle}\right)} \cdot \frac{\sqrt{\mathsf{PI}\left(\right)}}{-180}\right)\right), \mathsf{\_.f64}\left(b, a\right)\right)\right) \]

      2. frac-2neg N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(2, \mathsf{sin.f64}\left(\mathsf{/.f64}\left(\mathsf{PI.f64}\left(\right), \mathsf{/.f64}\left(180, angle\right)\right)\right)\right), \mathsf{+.f64}\left(b, a\right)\right), \mathsf{*.f64}\left(\mathsf{cos.f64}\left(\left(\frac{\mathsf{neg}\left(\sqrt{\mathsf{PI}\left(\right)}\right)}{\mathsf{neg}\left(\frac{-1}{angle}\right)} \cdot \frac{\mathsf{neg}\left(\sqrt{\mathsf{PI}\left(\right)}\right)}{\mathsf{neg}\left(-180\right)}\right)\right), \mathsf{\_.f64}\left(b, a\right)\right)\right) \]

      3. metadata-eval N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(2, \mathsf{sin.f64}\left(\mathsf{/.f64}\left(\mathsf{PI.f64}\left(\right), \mathsf{/.f64}\left(180, angle\right)\right)\right)\right), \mathsf{+.f64}\left(b, a\right)\right), \mathsf{*.f64}\left(\mathsf{cos.f64}\left(\left(\frac{\mathsf{neg}\left(\sqrt{\mathsf{PI}\left(\right)}\right)}{\mathsf{neg}\left(\frac{-1}{angle}\right)} \cdot \frac{\mathsf{neg}\left(\sqrt{\mathsf{PI}\left(\right)}\right)}{180}\right)\right), \mathsf{\_.f64}\left(b, a\right)\right)\right) \]

      4. frac-times N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(2, \mathsf{sin.f64}\left(\mathsf{/.f64}\left(\mathsf{PI.f64}\left(\right), \mathsf{/.f64}\left(180, angle\right)\right)\right)\right), \mathsf{+.f64}\left(b, a\right)\right), \mathsf{*.f64}\left(\mathsf{cos.f64}\left(\left(\frac{\left(\mathsf{neg}\left(\sqrt{\mathsf{PI}\left(\right)}\right)\right) \cdot \left(\mathsf{neg}\left(\sqrt{\mathsf{PI}\left(\right)}\right)\right)}{\left(\mathsf{neg}\left(\frac{-1}{angle}\right)\right) \cdot 180}\right)\right), \mathsf{\_.f64}\left(b, a\right)\right)\right) \]

      5. sqr-neg N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(2, \mathsf{sin.f64}\left(\mathsf{/.f64}\left(\mathsf{PI.f64}\left(\right), \mathsf{/.f64}\left(180, angle\right)\right)\right)\right), \mathsf{+.f64}\left(b, a\right)\right), \mathsf{*.f64}\left(\mathsf{cos.f64}\left(\left(\frac{\sqrt{\mathsf{PI}\left(\right)} \cdot \sqrt{\mathsf{PI}\left(\right)}}{\left(\mathsf{neg}\left(\frac{-1}{angle}\right)\right) \cdot 180}\right)\right), \mathsf{\_.f64}\left(b, a\right)\right)\right) \]

      6. add-sqr-sqrt N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(2, \mathsf{sin.f64}\left(\mathsf{/.f64}\left(\mathsf{PI.f64}\left(\right), \mathsf{/.f64}\left(180, angle\right)\right)\right)\right), \mathsf{+.f64}\left(b, a\right)\right), \mathsf{*.f64}\left(\mathsf{cos.f64}\left(\left(\frac{\mathsf{PI}\left(\right)}{\left(\mathsf{neg}\left(\frac{-1}{angle}\right)\right) \cdot 180}\right)\right), \mathsf{\_.f64}\left(b, a\right)\right)\right) \]

      7. *-commutative N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(2, \mathsf{sin.f64}\left(\mathsf{/.f64}\left(\mathsf{PI.f64}\left(\right), \mathsf{/.f64}\left(180, angle\right)\right)\right)\right), \mathsf{+.f64}\left(b, a\right)\right), \mathsf{*.f64}\left(\mathsf{cos.f64}\left(\left(\frac{\mathsf{PI}\left(\right)}{180 \cdot \left(\mathsf{neg}\left(\frac{-1}{angle}\right)\right)}\right)\right), \mathsf{\_.f64}\left(b, a\right)\right)\right) \]

      8. associate-/r* N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(2, \mathsf{sin.f64}\left(\mathsf{/.f64}\left(\mathsf{PI.f64}\left(\right), \mathsf{/.f64}\left(180, angle\right)\right)\right)\right), \mathsf{+.f64}\left(b, a\right)\right), \mathsf{*.f64}\left(\mathsf{cos.f64}\left(\left(\frac{\frac{\mathsf{PI}\left(\right)}{180}}{\mathsf{neg}\left(\frac{-1}{angle}\right)}\right)\right), \mathsf{\_.f64}\left(b, a\right)\right)\right) \]

      9. div-inv N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(2, \mathsf{sin.f64}\left(\mathsf{/.f64}\left(\mathsf{PI.f64}\left(\right), \mathsf{/.f64}\left(180, angle\right)\right)\right)\right), \mathsf{+.f64}\left(b, a\right)\right), \mathsf{*.f64}\left(\mathsf{cos.f64}\left(\left(\frac{\mathsf{PI}\left(\right) \cdot \frac{1}{180}}{\mathsf{neg}\left(\frac{-1}{angle}\right)}\right)\right), \mathsf{\_.f64}\left(b, a\right)\right)\right) \]

      10. neg-mul-1 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(2, \mathsf{sin.f64}\left(\mathsf{/.f64}\left(\mathsf{PI.f64}\left(\right), \mathsf{/.f64}\left(180, angle\right)\right)\right)\right), \mathsf{+.f64}\left(b, a\right)\right), \mathsf{*.f64}\left(\mathsf{cos.f64}\left(\left(\frac{\mathsf{PI}\left(\right) \cdot \frac{1}{180}}{-1 \cdot \frac{-1}{angle}}\right)\right), \mathsf{\_.f64}\left(b, a\right)\right)\right) \]

      11. times-frac N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(2, \mathsf{sin.f64}\left(\mathsf{/.f64}\left(\mathsf{PI.f64}\left(\right), \mathsf{/.f64}\left(180, angle\right)\right)\right)\right), \mathsf{+.f64}\left(b, a\right)\right), \mathsf{*.f64}\left(\mathsf{cos.f64}\left(\left(\frac{\mathsf{PI}\left(\right)}{-1} \cdot \frac{\frac{1}{180}}{\frac{-1}{angle}}\right)\right), \mathsf{\_.f64}\left(b, a\right)\right)\right) \]

      12. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(2, \mathsf{sin.f64}\left(\mathsf{/.f64}\left(\mathsf{PI.f64}\left(\right), \mathsf{/.f64}\left(180, angle\right)\right)\right)\right), \mathsf{+.f64}\left(b, a\right)\right), \mathsf{*.f64}\left(\mathsf{cos.f64}\left(\mathsf{*.f64}\left(\left(\frac{\mathsf{PI}\left(\right)}{-1}\right), \left(\frac{\frac{1}{180}}{\frac{-1}{angle}}\right)\right)\right), \mathsf{\_.f64}\left(b, a\right)\right)\right) \]

      13. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(2, \mathsf{sin.f64}\left(\mathsf{/.f64}\left(\mathsf{PI.f64}\left(\right), \mathsf{/.f64}\left(180, angle\right)\right)\right)\right), \mathsf{+.f64}\left(b, a\right)\right), \mathsf{*.f64}\left(\mathsf{cos.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{PI}\left(\right), -1\right), \left(\frac{\frac{1}{180}}{\frac{-1}{angle}}\right)\right)\right), \mathsf{\_.f64}\left(b, a\right)\right)\right) \]

      14. PI-lowering-PI.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(2, \mathsf{sin.f64}\left(\mathsf{/.f64}\left(\mathsf{PI.f64}\left(\right), \mathsf{/.f64}\left(180, angle\right)\right)\right)\right), \mathsf{+.f64}\left(b, a\right)\right), \mathsf{*.f64}\left(\mathsf{cos.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{PI.f64}\left(\right), -1\right), \left(\frac{\frac{1}{180}}{\frac{-1}{angle}}\right)\right)\right), \mathsf{\_.f64}\left(b, a\right)\right)\right) \]

      15. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(2, \mathsf{sin.f64}\left(\mathsf{/.f64}\left(\mathsf{PI.f64}\left(\right), \mathsf{/.f64}\left(180, angle\right)\right)\right)\right), \mathsf{+.f64}\left(b, a\right)\right), \mathsf{*.f64}\left(\mathsf{cos.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{PI.f64}\left(\right), -1\right), \mathsf{/.f64}\left(\left(\frac{1}{180}\right), \left(\frac{-1}{angle}\right)\right)\right)\right), \mathsf{\_.f64}\left(b, a\right)\right)\right) \]

      16. metadata-eval N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(2, \mathsf{sin.f64}\left(\mathsf{/.f64}\left(\mathsf{PI.f64}\left(\right), \mathsf{/.f64}\left(180, angle\right)\right)\right)\right), \mathsf{+.f64}\left(b, a\right)\right), \mathsf{*.f64}\left(\mathsf{cos.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{PI.f64}\left(\right), -1\right), \mathsf{/.f64}\left(\frac{1}{180}, \left(\frac{-1}{angle}\right)\right)\right)\right), \mathsf{\_.f64}\left(b, a\right)\right)\right) \]

      17. /-lowering-/.f64 83.2%

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(2, \mathsf{sin.f64}\left(\mathsf{/.f64}\left(\mathsf{PI.f64}\left(\right), \mathsf{/.f64}\left(180, angle\right)\right)\right)\right), \mathsf{+.f64}\left(b, a\right)\right), \mathsf{*.f64}\left(\mathsf{cos.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{PI.f64}\left(\right), -1\right), \mathsf{/.f64}\left(\frac{1}{180}, \mathsf{/.f64}\left(-1, angle\right)\right)\right)\right), \mathsf{\_.f64}\left(b, a\right)\right)\right) \]

   10. Applied egg-rr 83.2%

       \[\leadsto \left(\left(2 \cdot \sin \left(\frac{\pi}{\frac{180}{angle}}\right)\right) \cdot \left(b + a\right)\right) \cdot \left(\cos \color{blue}{\left(\frac{\pi}{-1} \cdot \frac{0.005555555555555556}{\frac{-1}{angle}}\right)} \cdot \left(b - a\right)\right) \]
3. Recombined 2 regimes into one program.

4. Final simplification 70.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \leq 4 \cdot 10^{+186}:\\ \;\;\;\;\left(\left(b - a\right) \cdot \cos \left(\frac{\pi}{\frac{180}{angle}}\right)\right) \cdot \left(\left(b + a\right) \cdot \left(2 \cdot \sin \left(\frac{1}{\frac{180}{angle \cdot \pi}}\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\left(b + a\right) \cdot \left(2 \cdot \sin \left(\frac{\pi}{\frac{180}{angle}}\right)\right)\right) \cdot \left(\left(b - a\right) \cdot \cos \left(\frac{\pi}{-1} \cdot \frac{0.005555555555555556}{\frac{-1}{angle}}\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 67.6% accurate, 1.8× speedup

Math

\[\begin{array}{l} angle\_m = \left|angle\right| \\ angle\_s = \mathsf{copysign}\left(1, angle\right) \\ \begin{array}{l} t_0 := \frac{\pi}{\frac{180}{angle\_m}}\\ angle\_s \cdot \begin{array}{l} \mathbf{if}\;a \leq 2.5 \cdot 10^{+192}:\\ \;\;\;\;\left(\left(b - a\right) \cdot \cos t\_0\right) \cdot \left(\left(b + a\right) \cdot \left(2 \cdot \sin \left(\frac{1}{\frac{180}{angle\_m \cdot \pi}}\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\cos \left(\frac{angle\_m}{180} \cdot \pi\right) \cdot \left(\left(\left(b + a\right) \cdot 2\right) \cdot \frac{\sin t\_0}{\frac{1}{b - a}}\right)\\ \end{array} \end{array} \end{array} \]

FPCore

angle\_m = (fabs.f64 angle)
angle\_s = (copysign.f64 #s(literal 1 binary64) angle)
(FPCore (angle_s a b angle_m)
 :precision binary64
 (let* ((t_0 (/ PI (/ 180.0 angle_m))))
   (*
    angle_s
    (if (<= a 2.5e+192)
      (*
       (* (- b a) (cos t_0))
       (* (+ b a) (* 2.0 (sin (/ 1.0 (/ 180.0 (* angle_m PI)))))))
      (*
       (cos (* (/ angle_m 180.0) PI))
       (* (* (+ b a) 2.0) (/ (sin t_0) (/ 1.0 (- b a)))))))))

C

angle\_m = fabs(angle);
angle\_s = copysign(1.0, angle);
double code(double angle_s, double a, double b, double angle_m) {
	double t_0 = ((double) M_PI) / (180.0 / angle_m);
	double tmp;
	if (a <= 2.5e+192) {
		tmp = ((b - a) * cos(t_0)) * ((b + a) * (2.0 * sin((1.0 / (180.0 / (angle_m * ((double) M_PI)))))));
	} else {
		tmp = cos(((angle_m / 180.0) * ((double) M_PI))) * (((b + a) * 2.0) * (sin(t_0) / (1.0 / (b - a))));
	}
	return angle_s * tmp;
}

Java

angle\_m = Math.abs(angle);
angle\_s = Math.copySign(1.0, angle);
public static double code(double angle_s, double a, double b, double angle_m) {
	double t_0 = Math.PI / (180.0 / angle_m);
	double tmp;
	if (a <= 2.5e+192) {
		tmp = ((b - a) * Math.cos(t_0)) * ((b + a) * (2.0 * Math.sin((1.0 / (180.0 / (angle_m * Math.PI))))));
	} else {
		tmp = Math.cos(((angle_m / 180.0) * Math.PI)) * (((b + a) * 2.0) * (Math.sin(t_0) / (1.0 / (b - a))));
	}
	return angle_s * tmp;
}

Python

angle\_m = math.fabs(angle)
angle\_s = math.copysign(1.0, angle)
def code(angle_s, a, b, angle_m):
	t_0 = math.pi / (180.0 / angle_m)
	tmp = 0
	if a <= 2.5e+192:
		tmp = ((b - a) * math.cos(t_0)) * ((b + a) * (2.0 * math.sin((1.0 / (180.0 / (angle_m * math.pi))))))
	else:
		tmp = math.cos(((angle_m / 180.0) * math.pi)) * (((b + a) * 2.0) * (math.sin(t_0) / (1.0 / (b - a))))
	return angle_s * tmp

Julia

angle\_m = abs(angle)
angle\_s = copysign(1.0, angle)
function code(angle_s, a, b, angle_m)
	t_0 = Float64(pi / Float64(180.0 / angle_m))
	tmp = 0.0
	if (a <= 2.5e+192)
		tmp = Float64(Float64(Float64(b - a) * cos(t_0)) * Float64(Float64(b + a) * Float64(2.0 * sin(Float64(1.0 / Float64(180.0 / Float64(angle_m * pi)))))));
	else
		tmp = Float64(cos(Float64(Float64(angle_m / 180.0) * pi)) * Float64(Float64(Float64(b + a) * 2.0) * Float64(sin(t_0) / Float64(1.0 / Float64(b - a)))));
	end
	return Float64(angle_s * tmp)
end

MATLAB

angle\_m = abs(angle);
angle\_s = sign(angle) * abs(1.0);
function tmp_2 = code(angle_s, a, b, angle_m)
	t_0 = pi / (180.0 / angle_m);
	tmp = 0.0;
	if (a <= 2.5e+192)
		tmp = ((b - a) * cos(t_0)) * ((b + a) * (2.0 * sin((1.0 / (180.0 / (angle_m * pi))))));
	else
		tmp = cos(((angle_m / 180.0) * pi)) * (((b + a) * 2.0) * (sin(t_0) / (1.0 / (b - a))));
	end
	tmp_2 = angle_s * tmp;
end

Wolfram

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_, b_, angle$95$m_] := Block[{t$95$0 = N[(Pi / N[(180.0 / angle$95$m), $MachinePrecision]), $MachinePrecision]}, N[(angle$95$s * If[LessEqual[a, 2.5e+192], N[(N[(N[(b - a), $MachinePrecision] * N[Cos[t$95$0], $MachinePrecision]), $MachinePrecision] * N[(N[(b + a), $MachinePrecision] * N[(2.0 * N[Sin[N[(1.0 / N[(180.0 / N[(angle$95$m * Pi), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[Cos[N[(N[(angle$95$m / 180.0), $MachinePrecision] * Pi), $MachinePrecision]], $MachinePrecision] * N[(N[(N[(b + a), $MachinePrecision] * 2.0), $MachinePrecision] * N[(N[Sin[t$95$0], $MachinePrecision] / N[(1.0 / N[(b - a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if a < 2.50000000000000017e192

   1. Initial program 55.4%

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

      1. associate-*l* N/A

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

      2. *-commutative N/A

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

      3. associate-*l* N/A

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

      4. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\sin \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right), \color{blue}{\left(\cos \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right) \cdot \left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right)\right)}\right) \]

      5. sin-lowering-sin.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sin.f64}\left(\left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right)\right), \left(\color{blue}{\cos \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right)} \cdot \left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right)\right)\right) \]

      6. associate-*r/ N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sin.f64}\left(\left(\frac{\mathsf{PI}\left(\right) \cdot angle}{180}\right)\right), \left(\cos \color{blue}{\left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right)} \cdot \left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right)\right)\right) \]

      7. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sin.f64}\left(\mathsf{/.f64}\left(\left(\mathsf{PI}\left(\right) \cdot angle\right), 180\right)\right), \left(\cos \color{blue}{\left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right)} \cdot \left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right)\right)\right) \]

      8. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sin.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{PI}\left(\right), angle\right), 180\right)\right), \left(\cos \left(\color{blue}{\mathsf{PI}\left(\right)} \cdot \frac{angle}{180}\right) \cdot \left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right)\right)\right) \]

      9. PI-lowering-PI.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sin.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{PI.f64}\left(\right), angle\right), 180\right)\right), \left(\cos \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right) \cdot \left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right)\right)\right) \]

      10. *-commutative N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{sin.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{PI.f64}\left(\right), angle\right), 180\right)\right), \left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \color{blue}{\cos \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right)}\right)\right) \]

      11. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{sin.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{PI.f64}\left(\right), angle\right), 180\right)\right), \mathsf{*.f64}\left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right), \color{blue}{\cos \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right)}\right)\right) \]

   3. Simplified 55.4%

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

      1. associate-*r/ N/A

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

      2. associate-*l* N/A

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

      3. associate-*r* N/A

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

      4. difference-of-squares N/A

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

      5. associate-*l* N/A

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

      6. associate-*r* N/A

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

      7. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\left(\left(\sin \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right) \cdot 2\right) \cdot \left(b + a\right)\right), \color{blue}{\left(\left(b - a\right) \cdot \cos \left(\frac{\mathsf{PI}\left(\right) \cdot angle}{180}\right)\right)}\right) \]

   6. Applied egg-rr 67.9%

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

      1. associate-/r/ N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(2, \mathsf{sin.f64}\left(\left(\frac{\mathsf{PI}\left(\right)}{180} \cdot angle\right)\right)\right), \mathsf{+.f64}\left(b, a\right)\right), \mathsf{*.f64}\left(\mathsf{cos.f64}\left(\mathsf{/.f64}\left(\mathsf{PI.f64}\left(\right), \mathsf{/.f64}\left(\color{blue}{180}, angle\right)\right)\right), \mathsf{\_.f64}\left(b, a\right)\right)\right) \]

      2. associate-*l/ N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(2, \mathsf{sin.f64}\left(\left(\frac{\mathsf{PI}\left(\right) \cdot angle}{180}\right)\right)\right), \mathsf{+.f64}\left(b, a\right)\right), \mathsf{*.f64}\left(\mathsf{cos.f64}\left(\mathsf{/.f64}\left(\mathsf{PI.f64}\left(\right), \mathsf{/.f64}\left(\color{blue}{180}, angle\right)\right)\right), \mathsf{\_.f64}\left(b, a\right)\right)\right) \]

      3. clear-num N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(2, \mathsf{sin.f64}\left(\left(\frac{1}{\frac{180}{\mathsf{PI}\left(\right) \cdot angle}}\right)\right)\right), \mathsf{+.f64}\left(b, a\right)\right), \mathsf{*.f64}\left(\mathsf{cos.f64}\left(\mathsf{/.f64}\left(\mathsf{PI.f64}\left(\right), \mathsf{/.f64}\left(\color{blue}{180}, angle\right)\right)\right), \mathsf{\_.f64}\left(b, a\right)\right)\right) \]

      4. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(2, \mathsf{sin.f64}\left(\mathsf{/.f64}\left(1, \left(\frac{180}{\mathsf{PI}\left(\right) \cdot angle}\right)\right)\right)\right), \mathsf{+.f64}\left(b, a\right)\right), \mathsf{*.f64}\left(\mathsf{cos.f64}\left(\mathsf{/.f64}\left(\mathsf{PI.f64}\left(\right), \mathsf{/.f64}\left(\color{blue}{180}, angle\right)\right)\right), \mathsf{\_.f64}\left(b, a\right)\right)\right) \]

      5. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(2, \mathsf{sin.f64}\left(\mathsf{/.f64}\left(1, \mathsf{/.f64}\left(180, \left(\mathsf{PI}\left(\right) \cdot angle\right)\right)\right)\right)\right), \mathsf{+.f64}\left(b, a\right)\right), \mathsf{*.f64}\left(\mathsf{cos.f64}\left(\mathsf{/.f64}\left(\mathsf{PI.f64}\left(\right), \mathsf{/.f64}\left(180, angle\right)\right)\right), \mathsf{\_.f64}\left(b, a\right)\right)\right) \]

      6. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(2, \mathsf{sin.f64}\left(\mathsf{/.f64}\left(1, \mathsf{/.f64}\left(180, \mathsf{*.f64}\left(\mathsf{PI}\left(\right), angle\right)\right)\right)\right)\right), \mathsf{+.f64}\left(b, a\right)\right), \mathsf{*.f64}\left(\mathsf{cos.f64}\left(\mathsf{/.f64}\left(\mathsf{PI.f64}\left(\right), \mathsf{/.f64}\left(180, angle\right)\right)\right), \mathsf{\_.f64}\left(b, a\right)\right)\right) \]

      7. PI-lowering-PI.f64 69.7%

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(2, \mathsf{sin.f64}\left(\mathsf{/.f64}\left(1, \mathsf{/.f64}\left(180, \mathsf{*.f64}\left(\mathsf{PI.f64}\left(\right), angle\right)\right)\right)\right)\right), \mathsf{+.f64}\left(b, a\right)\right), \mathsf{*.f64}\left(\mathsf{cos.f64}\left(\mathsf{/.f64}\left(\mathsf{PI.f64}\left(\right), \mathsf{/.f64}\left(180, angle\right)\right)\right), \mathsf{\_.f64}\left(b, a\right)\right)\right) \]

   8. Applied egg-rr 69.7%

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

   if 2.50000000000000017e192 < a

   1. Initial program 42.7%

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

      1. *-commutative N/A

         \[\leadsto \mathsf{*.f64}\left(\left(\sin \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right) \cdot \left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right)\right), \mathsf{cos.f64}\left(\color{blue}{\mathsf{*.f64}\left(\mathsf{PI.f64}\left(\right), \mathsf{/.f64}\left(angle, 180\right)\right)}\right)\right) \]

      2. pow2 N/A

         \[\leadsto \mathsf{*.f64}\left(\left(\sin \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right) \cdot \left(2 \cdot \left(b \cdot b - {a}^{2}\right)\right)\right), \mathsf{cos.f64}\left(\mathsf{*.f64}\left(\mathsf{PI.f64}\left(\right), \mathsf{/.f64}\left(angle, 180\right)\right)\right)\right) \]

      3. pow2 N/A

         \[\leadsto \mathsf{*.f64}\left(\left(\sin \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right) \cdot \left(2 \cdot \left(b \cdot b - a \cdot a\right)\right)\right), \mathsf{cos.f64}\left(\mathsf{*.f64}\left(\mathsf{PI.f64}\left(\right), \mathsf{/.f64}\left(angle, 180\right)\right)\right)\right) \]

      4. associate-*r* N/A

         \[\leadsto \mathsf{*.f64}\left(\left(\left(\sin \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right) \cdot 2\right) \cdot \left(b \cdot b - a \cdot a\right)\right), \mathsf{cos.f64}\left(\color{blue}{\mathsf{*.f64}\left(\mathsf{PI.f64}\left(\right), \mathsf{/.f64}\left(angle, 180\right)\right)}\right)\right) \]

      5. flip-- N/A

         \[\leadsto \mathsf{*.f64}\left(\left(\left(\sin \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right) \cdot 2\right) \cdot \frac{\left(b \cdot b\right) \cdot \left(b \cdot b\right) - \left(a \cdot a\right) \cdot \left(a \cdot a\right)}{b \cdot b + a \cdot a}\right), \mathsf{cos.f64}\left(\mathsf{*.f64}\left(\mathsf{PI.f64}\left(\right), \color{blue}{\mathsf{/.f64}\left(angle, 180\right)}\right)\right)\right) \]

      6. clear-num N/A

         \[\leadsto \mathsf{*.f64}\left(\left(\left(\sin \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right) \cdot 2\right) \cdot \frac{1}{\frac{b \cdot b + a \cdot a}{\left(b \cdot b\right) \cdot \left(b \cdot b\right) - \left(a \cdot a\right) \cdot \left(a \cdot a\right)}}\right), \mathsf{cos.f64}\left(\mathsf{*.f64}\left(\mathsf{PI.f64}\left(\right), \color{blue}{\mathsf{/.f64}\left(angle, 180\right)}\right)\right)\right) \]

      7. un-div-inv N/A

         \[\leadsto \mathsf{*.f64}\left(\left(\frac{\sin \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right) \cdot 2}{\frac{b \cdot b + a \cdot a}{\left(b \cdot b\right) \cdot \left(b \cdot b\right) - \left(a \cdot a\right) \cdot \left(a \cdot a\right)}}\right), \mathsf{cos.f64}\left(\color{blue}{\mathsf{*.f64}\left(\mathsf{PI.f64}\left(\right), \mathsf{/.f64}\left(angle, 180\right)\right)}\right)\right) \]

      8. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\left(\sin \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right) \cdot 2\right), \left(\frac{b \cdot b + a \cdot a}{\left(b \cdot b\right) \cdot \left(b \cdot b\right) - \left(a \cdot a\right) \cdot \left(a \cdot a\right)}\right)\right), \mathsf{cos.f64}\left(\color{blue}{\mathsf{*.f64}\left(\mathsf{PI.f64}\left(\right), \mathsf{/.f64}\left(angle, 180\right)\right)}\right)\right) \]

   4. Applied egg-rr 42.7%

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

      1. difference-of-squares N/A

         \[\leadsto \mathsf{*.f64}\left(\left(\frac{2 \cdot \sin \left(\frac{\mathsf{PI}\left(\right)}{\frac{180}{angle}}\right)}{\frac{1}{\left(b + a\right) \cdot \left(b - a\right)}}\right), \mathsf{cos.f64}\left(\mathsf{*.f64}\left(\mathsf{PI.f64}\left(\right), \mathsf{/.f64}\left(angle, \color{blue}{180}\right)\right)\right)\right) \]

      2. associate-/r* N/A

         \[\leadsto \mathsf{*.f64}\left(\left(\frac{2 \cdot \sin \left(\frac{\mathsf{PI}\left(\right)}{\frac{180}{angle}}\right)}{\frac{\frac{1}{b + a}}{b - a}}\right), \mathsf{cos.f64}\left(\mathsf{*.f64}\left(\mathsf{PI.f64}\left(\right), \color{blue}{\mathsf{/.f64}\left(angle, 180\right)}\right)\right)\right) \]

      3. div-inv N/A

         \[\leadsto \mathsf{*.f64}\left(\left(\frac{2 \cdot \sin \left(\frac{\mathsf{PI}\left(\right)}{\frac{180}{angle}}\right)}{\frac{1}{b + a} \cdot \frac{1}{b - a}}\right), \mathsf{cos.f64}\left(\mathsf{*.f64}\left(\mathsf{PI.f64}\left(\right), \color{blue}{\mathsf{/.f64}\left(angle, 180\right)}\right)\right)\right) \]

      4. times-frac N/A

         \[\leadsto \mathsf{*.f64}\left(\left(\frac{2}{\frac{1}{b + a}} \cdot \frac{\sin \left(\frac{\mathsf{PI}\left(\right)}{\frac{180}{angle}}\right)}{\frac{1}{b - a}}\right), \mathsf{cos.f64}\left(\color{blue}{\mathsf{*.f64}\left(\mathsf{PI.f64}\left(\right), \mathsf{/.f64}\left(angle, 180\right)\right)}\right)\right) \]

      5. un-div-inv N/A

         \[\leadsto \mathsf{*.f64}\left(\left(\left(2 \cdot \frac{1}{\frac{1}{b + a}}\right) \cdot \frac{\sin \left(\frac{\mathsf{PI}\left(\right)}{\frac{180}{angle}}\right)}{\frac{1}{b - a}}\right), \mathsf{cos.f64}\left(\mathsf{*.f64}\left(\color{blue}{\mathsf{PI.f64}\left(\right)}, \mathsf{/.f64}\left(angle, 180\right)\right)\right)\right) \]

      6. flip-+ N/A

         \[\leadsto \mathsf{*.f64}\left(\left(\left(2 \cdot \frac{1}{\frac{1}{\frac{b \cdot b - a \cdot a}{b - a}}}\right) \cdot \frac{\sin \left(\frac{\mathsf{PI}\left(\right)}{\frac{180}{angle}}\right)}{\frac{1}{b - a}}\right), \mathsf{cos.f64}\left(\mathsf{*.f64}\left(\mathsf{PI.f64}\left(\right), \mathsf{/.f64}\left(angle, 180\right)\right)\right)\right) \]

      7. clear-num N/A

         \[\leadsto \mathsf{*.f64}\left(\left(\left(2 \cdot \frac{1}{\frac{b - a}{b \cdot b - a \cdot a}}\right) \cdot \frac{\sin \left(\frac{\mathsf{PI}\left(\right)}{\frac{180}{angle}}\right)}{\frac{1}{b - a}}\right), \mathsf{cos.f64}\left(\mathsf{*.f64}\left(\mathsf{PI.f64}\left(\right), \mathsf{/.f64}\left(angle, 180\right)\right)\right)\right) \]

      8. clear-num N/A

         \[\leadsto \mathsf{*.f64}\left(\left(\left(2 \cdot \frac{b \cdot b - a \cdot a}{b - a}\right) \cdot \frac{\sin \left(\frac{\mathsf{PI}\left(\right)}{\frac{180}{angle}}\right)}{\frac{1}{b - a}}\right), \mathsf{cos.f64}\left(\mathsf{*.f64}\left(\mathsf{PI.f64}\left(\right), \mathsf{/.f64}\left(angle, 180\right)\right)\right)\right) \]

      9. flip-+ N/A

         \[\leadsto \mathsf{*.f64}\left(\left(\left(2 \cdot \left(b + a\right)\right) \cdot \frac{\sin \left(\frac{\mathsf{PI}\left(\right)}{\frac{180}{angle}}\right)}{\frac{1}{b - a}}\right), \mathsf{cos.f64}\left(\mathsf{*.f64}\left(\mathsf{PI.f64}\left(\right), \mathsf{/.f64}\left(angle, 180\right)\right)\right)\right) \]

      10. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\left(2 \cdot \left(b + a\right)\right), \left(\frac{\sin \left(\frac{\mathsf{PI}\left(\right)}{\frac{180}{angle}}\right)}{\frac{1}{b - a}}\right)\right), \mathsf{cos.f64}\left(\color{blue}{\mathsf{*.f64}\left(\mathsf{PI.f64}\left(\right), \mathsf{/.f64}\left(angle, 180\right)\right)}\right)\right) \]

      11. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(2, \left(b + a\right)\right), \left(\frac{\sin \left(\frac{\mathsf{PI}\left(\right)}{\frac{180}{angle}}\right)}{\frac{1}{b - a}}\right)\right), \mathsf{cos.f64}\left(\mathsf{*.f64}\left(\color{blue}{\mathsf{PI.f64}\left(\right)}, \mathsf{/.f64}\left(angle, 180\right)\right)\right)\right) \]

      12. +-lowering-+.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(2, \mathsf{+.f64}\left(b, a\right)\right), \left(\frac{\sin \left(\frac{\mathsf{PI}\left(\right)}{\frac{180}{angle}}\right)}{\frac{1}{b - a}}\right)\right), \mathsf{cos.f64}\left(\mathsf{*.f64}\left(\mathsf{PI.f64}\left(\right), \mathsf{/.f64}\left(angle, 180\right)\right)\right)\right) \]

      13. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(2, \mathsf{+.f64}\left(b, a\right)\right), \mathsf{/.f64}\left(\sin \left(\frac{\mathsf{PI}\left(\right)}{\frac{180}{angle}}\right), \left(\frac{1}{b - a}\right)\right)\right), \mathsf{cos.f64}\left(\mathsf{*.f64}\left(\mathsf{PI.f64}\left(\right), \color{blue}{\mathsf{/.f64}\left(angle, 180\right)}\right)\right)\right) \]

   6. Applied egg-rr 79.7%

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

4. Final simplification 70.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \leq 2.5 \cdot 10^{+192}:\\ \;\;\;\;\left(\left(b - a\right) \cdot \cos \left(\frac{\pi}{\frac{180}{angle}}\right)\right) \cdot \left(\left(b + a\right) \cdot \left(2 \cdot \sin \left(\frac{1}{\frac{180}{angle \cdot \pi}}\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\cos \left(\frac{angle}{180} \cdot \pi\right) \cdot \left(\left(\left(b + a\right) \cdot 2\right) \cdot \frac{\sin \left(\frac{\pi}{\frac{180}{angle}}\right)}{\frac{1}{b - a}}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 67.7% accurate, 1.8× speedup

Math

\[\begin{array}{l} angle\_m = \left|angle\right| \\ angle\_s = \mathsf{copysign}\left(1, angle\right) \\ \begin{array}{l} t_0 := \frac{angle\_m}{\frac{180}{\pi}}\\ angle\_s \cdot \begin{array}{l} \mathbf{if}\;a \leq 4.1 \cdot 10^{+186}:\\ \;\;\;\;\left(\left(b - a\right) \cdot \cos \left(\frac{\pi}{\frac{180}{angle\_m}}\right)\right) \cdot \left(\left(b + a\right) \cdot \left(2 \cdot \sin \left(\frac{1}{\frac{180}{angle\_m \cdot \pi}}\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(b - a\right) \cdot \left(\left(2 \cdot \sin t\_0\right) \cdot \left(\left(b + a\right) \cdot \cos t\_0\right)\right)\\ \end{array} \end{array} \end{array} \]

FPCore

angle\_m = (fabs.f64 angle)
angle\_s = (copysign.f64 #s(literal 1 binary64) angle)
(FPCore (angle_s a b angle_m)
 :precision binary64
 (let* ((t_0 (/ angle_m (/ 180.0 PI))))
   (*
    angle_s
    (if (<= a 4.1e+186)
      (*
       (* (- b a) (cos (/ PI (/ 180.0 angle_m))))
       (* (+ b a) (* 2.0 (sin (/ 1.0 (/ 180.0 (* angle_m PI)))))))
      (* (- b a) (* (* 2.0 (sin t_0)) (* (+ b a) (cos t_0))))))))

C

angle\_m = fabs(angle);
angle\_s = copysign(1.0, angle);
double code(double angle_s, double a, double b, double angle_m) {
	double t_0 = angle_m / (180.0 / ((double) M_PI));
	double tmp;
	if (a <= 4.1e+186) {
		tmp = ((b - a) * cos((((double) M_PI) / (180.0 / angle_m)))) * ((b + a) * (2.0 * sin((1.0 / (180.0 / (angle_m * ((double) M_PI)))))));
	} else {
		tmp = (b - a) * ((2.0 * sin(t_0)) * ((b + a) * cos(t_0)));
	}
	return angle_s * tmp;
}

Java

angle\_m = Math.abs(angle);
angle\_s = Math.copySign(1.0, angle);
public static double code(double angle_s, double a, double b, double angle_m) {
	double t_0 = angle_m / (180.0 / Math.PI);
	double tmp;
	if (a <= 4.1e+186) {
		tmp = ((b - a) * Math.cos((Math.PI / (180.0 / angle_m)))) * ((b + a) * (2.0 * Math.sin((1.0 / (180.0 / (angle_m * Math.PI))))));
	} else {
		tmp = (b - a) * ((2.0 * Math.sin(t_0)) * ((b + a) * Math.cos(t_0)));
	}
	return angle_s * tmp;
}

Python

angle\_m = math.fabs(angle)
angle\_s = math.copysign(1.0, angle)
def code(angle_s, a, b, angle_m):
	t_0 = angle_m / (180.0 / math.pi)
	tmp = 0
	if a <= 4.1e+186:
		tmp = ((b - a) * math.cos((math.pi / (180.0 / angle_m)))) * ((b + a) * (2.0 * math.sin((1.0 / (180.0 / (angle_m * math.pi))))))
	else:
		tmp = (b - a) * ((2.0 * math.sin(t_0)) * ((b + a) * math.cos(t_0)))
	return angle_s * tmp

Julia

angle\_m = abs(angle)
angle\_s = copysign(1.0, angle)
function code(angle_s, a, b, angle_m)
	t_0 = Float64(angle_m / Float64(180.0 / pi))
	tmp = 0.0
	if (a <= 4.1e+186)
		tmp = Float64(Float64(Float64(b - a) * cos(Float64(pi / Float64(180.0 / angle_m)))) * Float64(Float64(b + a) * Float64(2.0 * sin(Float64(1.0 / Float64(180.0 / Float64(angle_m * pi)))))));
	else
		tmp = Float64(Float64(b - a) * Float64(Float64(2.0 * sin(t_0)) * Float64(Float64(b + a) * cos(t_0))));
	end
	return Float64(angle_s * tmp)
end

MATLAB

angle\_m = abs(angle);
angle\_s = sign(angle) * abs(1.0);
function tmp_2 = code(angle_s, a, b, angle_m)
	t_0 = angle_m / (180.0 / pi);
	tmp = 0.0;
	if (a <= 4.1e+186)
		tmp = ((b - a) * cos((pi / (180.0 / angle_m)))) * ((b + a) * (2.0 * sin((1.0 / (180.0 / (angle_m * pi))))));
	else
		tmp = (b - a) * ((2.0 * sin(t_0)) * ((b + a) * cos(t_0)));
	end
	tmp_2 = angle_s * tmp;
end

Wolfram

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_, b_, angle$95$m_] := Block[{t$95$0 = N[(angle$95$m / N[(180.0 / Pi), $MachinePrecision]), $MachinePrecision]}, N[(angle$95$s * If[LessEqual[a, 4.1e+186], N[(N[(N[(b - a), $MachinePrecision] * N[Cos[N[(Pi / N[(180.0 / angle$95$m), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[(N[(b + a), $MachinePrecision] * N[(2.0 * N[Sin[N[(1.0 / N[(180.0 / N[(angle$95$m * Pi), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(b - a), $MachinePrecision] * N[(N[(2.0 * N[Sin[t$95$0], $MachinePrecision]), $MachinePrecision] * N[(N[(b + a), $MachinePrecision] * N[Cos[t$95$0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if a < 4.1e186

   1. Initial program 55.6%

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

      1. associate-*l* N/A

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

      2. *-commutative N/A

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

      3. associate-*l* N/A

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

      4. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\sin \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right), \color{blue}{\left(\cos \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right) \cdot \left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right)\right)}\right) \]

      5. sin-lowering-sin.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sin.f64}\left(\left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right)\right), \left(\color{blue}{\cos \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right)} \cdot \left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right)\right)\right) \]

      6. associate-*r/ N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sin.f64}\left(\left(\frac{\mathsf{PI}\left(\right) \cdot angle}{180}\right)\right), \left(\cos \color{blue}{\left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right)} \cdot \left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right)\right)\right) \]

      7. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sin.f64}\left(\mathsf{/.f64}\left(\left(\mathsf{PI}\left(\right) \cdot angle\right), 180\right)\right), \left(\cos \color{blue}{\left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right)} \cdot \left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right)\right)\right) \]

      8. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sin.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{PI}\left(\right), angle\right), 180\right)\right), \left(\cos \left(\color{blue}{\mathsf{PI}\left(\right)} \cdot \frac{angle}{180}\right) \cdot \left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right)\right)\right) \]

      9. PI-lowering-PI.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sin.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{PI.f64}\left(\right), angle\right), 180\right)\right), \left(\cos \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right) \cdot \left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right)\right)\right) \]

      10. *-commutative N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{sin.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{PI.f64}\left(\right), angle\right), 180\right)\right), \left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \color{blue}{\cos \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right)}\right)\right) \]

      11. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{sin.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{PI.f64}\left(\right), angle\right), 180\right)\right), \mathsf{*.f64}\left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right), \color{blue}{\cos \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right)}\right)\right) \]

   3. Simplified 55.7%

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

      1. associate-*r/ N/A

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

      2. associate-*l* N/A

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

      3. associate-*r* N/A

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

      4. difference-of-squares N/A

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

      5. associate-*l* N/A

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

      6. associate-*r* N/A

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

      7. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\left(\left(\sin \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right) \cdot 2\right) \cdot \left(b + a\right)\right), \color{blue}{\left(\left(b - a\right) \cdot \cos \left(\frac{\mathsf{PI}\left(\right) \cdot angle}{180}\right)\right)}\right) \]

   6. Applied egg-rr 67.5%

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

      1. associate-/r/ N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(2, \mathsf{sin.f64}\left(\left(\frac{\mathsf{PI}\left(\right)}{180} \cdot angle\right)\right)\right), \mathsf{+.f64}\left(b, a\right)\right), \mathsf{*.f64}\left(\mathsf{cos.f64}\left(\mathsf{/.f64}\left(\mathsf{PI.f64}\left(\right), \mathsf{/.f64}\left(\color{blue}{180}, angle\right)\right)\right), \mathsf{\_.f64}\left(b, a\right)\right)\right) \]

      2. associate-*l/ N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(2, \mathsf{sin.f64}\left(\left(\frac{\mathsf{PI}\left(\right) \cdot angle}{180}\right)\right)\right), \mathsf{+.f64}\left(b, a\right)\right), \mathsf{*.f64}\left(\mathsf{cos.f64}\left(\mathsf{/.f64}\left(\mathsf{PI.f64}\left(\right), \mathsf{/.f64}\left(\color{blue}{180}, angle\right)\right)\right), \mathsf{\_.f64}\left(b, a\right)\right)\right) \]

      3. clear-num N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(2, \mathsf{sin.f64}\left(\left(\frac{1}{\frac{180}{\mathsf{PI}\left(\right) \cdot angle}}\right)\right)\right), \mathsf{+.f64}\left(b, a\right)\right), \mathsf{*.f64}\left(\mathsf{cos.f64}\left(\mathsf{/.f64}\left(\mathsf{PI.f64}\left(\right), \mathsf{/.f64}\left(\color{blue}{180}, angle\right)\right)\right), \mathsf{\_.f64}\left(b, a\right)\right)\right) \]

      4. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(2, \mathsf{sin.f64}\left(\mathsf{/.f64}\left(1, \left(\frac{180}{\mathsf{PI}\left(\right) \cdot angle}\right)\right)\right)\right), \mathsf{+.f64}\left(b, a\right)\right), \mathsf{*.f64}\left(\mathsf{cos.f64}\left(\mathsf{/.f64}\left(\mathsf{PI.f64}\left(\right), \mathsf{/.f64}\left(\color{blue}{180}, angle\right)\right)\right), \mathsf{\_.f64}\left(b, a\right)\right)\right) \]

      5. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(2, \mathsf{sin.f64}\left(\mathsf{/.f64}\left(1, \mathsf{/.f64}\left(180, \left(\mathsf{PI}\left(\right) \cdot angle\right)\right)\right)\right)\right), \mathsf{+.f64}\left(b, a\right)\right), \mathsf{*.f64}\left(\mathsf{cos.f64}\left(\mathsf{/.f64}\left(\mathsf{PI.f64}\left(\right), \mathsf{/.f64}\left(180, angle\right)\right)\right), \mathsf{\_.f64}\left(b, a\right)\right)\right) \]

      6. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(2, \mathsf{sin.f64}\left(\mathsf{/.f64}\left(1, \mathsf{/.f64}\left(180, \mathsf{*.f64}\left(\mathsf{PI}\left(\right), angle\right)\right)\right)\right)\right), \mathsf{+.f64}\left(b, a\right)\right), \mathsf{*.f64}\left(\mathsf{cos.f64}\left(\mathsf{/.f64}\left(\mathsf{PI.f64}\left(\right), \mathsf{/.f64}\left(180, angle\right)\right)\right), \mathsf{\_.f64}\left(b, a\right)\right)\right) \]

      7. PI-lowering-PI.f64 69.4%

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(2, \mathsf{sin.f64}\left(\mathsf{/.f64}\left(1, \mathsf{/.f64}\left(180, \mathsf{*.f64}\left(\mathsf{PI.f64}\left(\right), angle\right)\right)\right)\right)\right), \mathsf{+.f64}\left(b, a\right)\right), \mathsf{*.f64}\left(\mathsf{cos.f64}\left(\mathsf{/.f64}\left(\mathsf{PI.f64}\left(\right), \mathsf{/.f64}\left(180, angle\right)\right)\right), \mathsf{\_.f64}\left(b, a\right)\right)\right) \]

   8. Applied egg-rr 69.4%

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

   if 4.1e186 < a

   1. Initial program 41.6%

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

      1. associate-*l* N/A

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

      2. *-commutative N/A

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

      3. associate-*l* N/A

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

      4. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\sin \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right), \color{blue}{\left(\cos \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right) \cdot \left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right)\right)}\right) \]

      5. sin-lowering-sin.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sin.f64}\left(\left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right)\right), \left(\color{blue}{\cos \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right)} \cdot \left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right)\right)\right) \]

      6. associate-*r/ N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sin.f64}\left(\left(\frac{\mathsf{PI}\left(\right) \cdot angle}{180}\right)\right), \left(\cos \color{blue}{\left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right)} \cdot \left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right)\right)\right) \]

      7. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sin.f64}\left(\mathsf{/.f64}\left(\left(\mathsf{PI}\left(\right) \cdot angle\right), 180\right)\right), \left(\cos \color{blue}{\left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right)} \cdot \left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right)\right)\right) \]

      8. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sin.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{PI}\left(\right), angle\right), 180\right)\right), \left(\cos \left(\color{blue}{\mathsf{PI}\left(\right)} \cdot \frac{angle}{180}\right) \cdot \left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right)\right)\right) \]

      9. PI-lowering-PI.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sin.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{PI.f64}\left(\right), angle\right), 180\right)\right), \left(\cos \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right) \cdot \left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right)\right)\right) \]

      10. *-commutative N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{sin.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{PI.f64}\left(\right), angle\right), 180\right)\right), \left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \color{blue}{\cos \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right)}\right)\right) \]

      11. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{sin.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{PI.f64}\left(\right), angle\right), 180\right)\right), \mathsf{*.f64}\left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right), \color{blue}{\cos \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right)}\right)\right) \]

   3. Simplified 41.6%

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

      1. associate-*r/ N/A

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

      2. associate-*l* N/A

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

      3. associate-*r* N/A

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

      4. difference-of-squares N/A

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

      5. associate-*l* N/A

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

      6. associate-*r* N/A

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

      7. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\left(\left(\sin \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right) \cdot 2\right) \cdot \left(b + a\right)\right), \color{blue}{\left(\left(b - a\right) \cdot \cos \left(\frac{\mathsf{PI}\left(\right) \cdot angle}{180}\right)\right)}\right) \]

   6. Applied egg-rr 83.2%

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

      1. add-sqr-sqrt N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(2, \mathsf{sin.f64}\left(\mathsf{/.f64}\left(\left(\sqrt{\mathsf{PI}\left(\right)} \cdot \sqrt{\mathsf{PI}\left(\right)}\right), \mathsf{/.f64}\left(180, angle\right)\right)\right)\right), \mathsf{+.f64}\left(b, a\right)\right), \mathsf{*.f64}\left(\mathsf{cos.f64}\left(\mathsf{/.f64}\left(\mathsf{PI.f64}\left(\right), \mathsf{/.f64}\left(180, angle\right)\right)\right), \mathsf{\_.f64}\left(b, a\right)\right)\right) \]

      2. pow2 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(2, \mathsf{sin.f64}\left(\mathsf{/.f64}\left(\left({\left(\sqrt{\mathsf{PI}\left(\right)}\right)}^{2}\right), \mathsf{/.f64}\left(180, angle\right)\right)\right)\right), \mathsf{+.f64}\left(b, a\right)\right), \mathsf{*.f64}\left(\mathsf{cos.f64}\left(\mathsf{/.f64}\left(\mathsf{PI.f64}\left(\right), \mathsf{/.f64}\left(180, angle\right)\right)\right), \mathsf{\_.f64}\left(b, a\right)\right)\right) \]

      3. pow-lowering-pow.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(2, \mathsf{sin.f64}\left(\mathsf{/.f64}\left(\mathsf{pow.f64}\left(\left(\sqrt{\mathsf{PI}\left(\right)}\right), 2\right), \mathsf{/.f64}\left(180, angle\right)\right)\right)\right), \mathsf{+.f64}\left(b, a\right)\right), \mathsf{*.f64}\left(\mathsf{cos.f64}\left(\mathsf{/.f64}\left(\mathsf{PI.f64}\left(\right), \mathsf{/.f64}\left(180, angle\right)\right)\right), \mathsf{\_.f64}\left(b, a\right)\right)\right) \]

      4. sqrt-lowering-sqrt.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(2, \mathsf{sin.f64}\left(\mathsf{/.f64}\left(\mathsf{pow.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{PI}\left(\right)\right), 2\right), \mathsf{/.f64}\left(180, angle\right)\right)\right)\right), \mathsf{+.f64}\left(b, a\right)\right), \mathsf{*.f64}\left(\mathsf{cos.f64}\left(\mathsf{/.f64}\left(\mathsf{PI.f64}\left(\right), \mathsf{/.f64}\left(180, angle\right)\right)\right), \mathsf{\_.f64}\left(b, a\right)\right)\right) \]

      5. PI-lowering-PI.f64 71.8%

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(2, \mathsf{sin.f64}\left(\mathsf{/.f64}\left(\mathsf{pow.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{PI.f64}\left(\right)\right), 2\right), \mathsf{/.f64}\left(180, angle\right)\right)\right)\right), \mathsf{+.f64}\left(b, a\right)\right), \mathsf{*.f64}\left(\mathsf{cos.f64}\left(\mathsf{/.f64}\left(\mathsf{PI.f64}\left(\right), \mathsf{/.f64}\left(180, angle\right)\right)\right), \mathsf{\_.f64}\left(b, a\right)\right)\right) \]

   8. Applied egg-rr 71.8%

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

      1. add-sqr-sqrt N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(2, \mathsf{sin.f64}\left(\mathsf{/.f64}\left(\mathsf{pow.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{PI.f64}\left(\right)\right), 2\right), \mathsf{/.f64}\left(180, angle\right)\right)\right)\right), \mathsf{+.f64}\left(b, a\right)\right), \mathsf{*.f64}\left(\mathsf{cos.f64}\left(\mathsf{/.f64}\left(\left(\sqrt{\mathsf{PI}\left(\right)} \cdot \sqrt{\mathsf{PI}\left(\right)}\right), \mathsf{/.f64}\left(180, angle\right)\right)\right), \mathsf{\_.f64}\left(b, a\right)\right)\right) \]

      2. unpow2 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(2, \mathsf{sin.f64}\left(\mathsf{/.f64}\left(\mathsf{pow.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{PI.f64}\left(\right)\right), 2\right), \mathsf{/.f64}\left(180, angle\right)\right)\right)\right), \mathsf{+.f64}\left(b, a\right)\right), \mathsf{*.f64}\left(\mathsf{cos.f64}\left(\mathsf{/.f64}\left(\left({\left(\sqrt{\mathsf{PI}\left(\right)}\right)}^{2}\right), \mathsf{/.f64}\left(180, angle\right)\right)\right), \mathsf{\_.f64}\left(b, a\right)\right)\right) \]

      3. pow-lowering-pow.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(2, \mathsf{sin.f64}\left(\mathsf{/.f64}\left(\mathsf{pow.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{PI.f64}\left(\right)\right), 2\right), \mathsf{/.f64}\left(180, angle\right)\right)\right)\right), \mathsf{+.f64}\left(b, a\right)\right), \mathsf{*.f64}\left(\mathsf{cos.f64}\left(\mathsf{/.f64}\left(\mathsf{pow.f64}\left(\left(\sqrt{\mathsf{PI}\left(\right)}\right), 2\right), \mathsf{/.f64}\left(180, angle\right)\right)\right), \mathsf{\_.f64}\left(b, a\right)\right)\right) \]

      4. sqrt-lowering-sqrt.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(2, \mathsf{sin.f64}\left(\mathsf{/.f64}\left(\mathsf{pow.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{PI.f64}\left(\right)\right), 2\right), \mathsf{/.f64}\left(180, angle\right)\right)\right)\right), \mathsf{+.f64}\left(b, a\right)\right), \mathsf{*.f64}\left(\mathsf{cos.f64}\left(\mathsf{/.f64}\left(\mathsf{pow.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{PI}\left(\right)\right), 2\right), \mathsf{/.f64}\left(180, angle\right)\right)\right), \mathsf{\_.f64}\left(b, a\right)\right)\right) \]

      5. PI-lowering-PI.f64 88.4%

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(2, \mathsf{sin.f64}\left(\mathsf{/.f64}\left(\mathsf{pow.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{PI.f64}\left(\right)\right), 2\right), \mathsf{/.f64}\left(180, angle\right)\right)\right)\right), \mathsf{+.f64}\left(b, a\right)\right), \mathsf{*.f64}\left(\mathsf{cos.f64}\left(\mathsf{/.f64}\left(\mathsf{pow.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{PI.f64}\left(\right)\right), 2\right), \mathsf{/.f64}\left(180, angle\right)\right)\right), \mathsf{\_.f64}\left(b, a\right)\right)\right) \]

   10. Applied egg-rr 88.4%

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

       1. associate-*r* N/A

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

       2. *-commutative N/A

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

       3. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\left(b - a\right), \color{blue}{\left(\left(\left(2 \cdot \sin \left(\frac{{\left(\sqrt{\mathsf{PI}\left(\right)}\right)}^{2}}{\frac{180}{angle}}\right)\right) \cdot \left(b + a\right)\right) \cdot \cos \left(\frac{{\left(\sqrt{\mathsf{PI}\left(\right)}\right)}^{2}}{\frac{180}{angle}}\right)\right)}\right) \]

       4. --lowering--.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{\_.f64}\left(b, a\right), \left(\color{blue}{\left(\left(2 \cdot \sin \left(\frac{{\left(\sqrt{\mathsf{PI}\left(\right)}\right)}^{2}}{\frac{180}{angle}}\right)\right) \cdot \left(b + a\right)\right)} \cdot \cos \left(\frac{{\left(\sqrt{\mathsf{PI}\left(\right)}\right)}^{2}}{\frac{180}{angle}}\right)\right)\right) \]

       5. associate-*l* N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{\_.f64}\left(b, a\right), \left(\left(2 \cdot \left(\sin \left(\frac{{\left(\sqrt{\mathsf{PI}\left(\right)}\right)}^{2}}{\frac{180}{angle}}\right) \cdot \left(b + a\right)\right)\right) \cdot \cos \color{blue}{\left(\frac{{\left(\sqrt{\mathsf{PI}\left(\right)}\right)}^{2}}{\frac{180}{angle}}\right)}\right)\right) \]

       6. unpow2 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{\_.f64}\left(b, a\right), \left(\left(2 \cdot \left(\sin \left(\frac{\sqrt{\mathsf{PI}\left(\right)} \cdot \sqrt{\mathsf{PI}\left(\right)}}{\frac{180}{angle}}\right) \cdot \left(b + a\right)\right)\right) \cdot \cos \left(\frac{{\left(\sqrt{\mathsf{PI}\left(\right)}\right)}^{2}}{\frac{180}{angle}}\right)\right)\right) \]

       7. add-sqr-sqrt N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{\_.f64}\left(b, a\right), \left(\left(2 \cdot \left(\sin \left(\frac{\mathsf{PI}\left(\right)}{\frac{180}{angle}}\right) \cdot \left(b + a\right)\right)\right) \cdot \cos \left(\frac{{\left(\sqrt{\mathsf{PI}\left(\right)}\right)}^{2}}{\frac{180}{angle}}\right)\right)\right) \]

       8. associate-*l* N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{\_.f64}\left(b, a\right), \left(\left(\left(2 \cdot \sin \left(\frac{\mathsf{PI}\left(\right)}{\frac{180}{angle}}\right)\right) \cdot \left(b + a\right)\right) \cdot \cos \color{blue}{\left(\frac{{\left(\sqrt{\mathsf{PI}\left(\right)}\right)}^{2}}{\frac{180}{angle}}\right)}\right)\right) \]

   12. Applied egg-rr 83.2%

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

4. Final simplification 70.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \leq 4.1 \cdot 10^{+186}:\\ \;\;\;\;\left(\left(b - a\right) \cdot \cos \left(\frac{\pi}{\frac{180}{angle}}\right)\right) \cdot \left(\left(b + a\right) \cdot \left(2 \cdot \sin \left(\frac{1}{\frac{180}{angle \cdot \pi}}\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(b - a\right) \cdot \left(\left(2 \cdot \sin \left(\frac{angle}{\frac{180}{\pi}}\right)\right) \cdot \left(\left(b + a\right) \cdot \cos \left(\frac{angle}{\frac{180}{\pi}}\right)\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 67.7% accurate, 1.8× speedup

Math

\[\begin{array}{l} angle\_m = \left|angle\right| \\ angle\_s = \mathsf{copysign}\left(1, angle\right) \\ \begin{array}{l} t_0 := \frac{angle\_m}{\frac{180}{\pi}}\\ angle\_s \cdot \begin{array}{l} \mathbf{if}\;\frac{angle\_m}{180} \leq 10^{+156}:\\ \;\;\;\;\left(b - a\right) \cdot \left(\left(2 \cdot \sin t\_0\right) \cdot \left(\left(b + a\right) \cdot \cos t\_0\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(b - a\right) \cdot \left(\left(b + a\right) \cdot \left(2 \cdot \sin \left(\frac{\pi}{\frac{180}{angle\_m}}\right)\right)\right)\\ \end{array} \end{array} \end{array} \]

FPCore

angle\_m = (fabs.f64 angle)
angle\_s = (copysign.f64 #s(literal 1 binary64) angle)
(FPCore (angle_s a b angle_m)
 :precision binary64
 (let* ((t_0 (/ angle_m (/ 180.0 PI))))
   (*
    angle_s
    (if (<= (/ angle_m 180.0) 1e+156)
      (* (- b a) (* (* 2.0 (sin t_0)) (* (+ b a) (cos t_0))))
      (* (- b a) (* (+ b a) (* 2.0 (sin (/ PI (/ 180.0 angle_m))))))))))

C

angle\_m = fabs(angle);
angle\_s = copysign(1.0, angle);
double code(double angle_s, double a, double b, double angle_m) {
	double t_0 = angle_m / (180.0 / ((double) M_PI));
	double tmp;
	if ((angle_m / 180.0) <= 1e+156) {
		tmp = (b - a) * ((2.0 * sin(t_0)) * ((b + a) * cos(t_0)));
	} else {
		tmp = (b - a) * ((b + a) * (2.0 * sin((((double) M_PI) / (180.0 / angle_m)))));
	}
	return angle_s * tmp;
}

Java

angle\_m = Math.abs(angle);
angle\_s = Math.copySign(1.0, angle);
public static double code(double angle_s, double a, double b, double angle_m) {
	double t_0 = angle_m / (180.0 / Math.PI);
	double tmp;
	if ((angle_m / 180.0) <= 1e+156) {
		tmp = (b - a) * ((2.0 * Math.sin(t_0)) * ((b + a) * Math.cos(t_0)));
	} else {
		tmp = (b - a) * ((b + a) * (2.0 * Math.sin((Math.PI / (180.0 / angle_m)))));
	}
	return angle_s * tmp;
}

Python

angle\_m = math.fabs(angle)
angle\_s = math.copysign(1.0, angle)
def code(angle_s, a, b, angle_m):
	t_0 = angle_m / (180.0 / math.pi)
	tmp = 0
	if (angle_m / 180.0) <= 1e+156:
		tmp = (b - a) * ((2.0 * math.sin(t_0)) * ((b + a) * math.cos(t_0)))
	else:
		tmp = (b - a) * ((b + a) * (2.0 * math.sin((math.pi / (180.0 / angle_m)))))
	return angle_s * tmp

Julia

angle\_m = abs(angle)
angle\_s = copysign(1.0, angle)
function code(angle_s, a, b, angle_m)
	t_0 = Float64(angle_m / Float64(180.0 / pi))
	tmp = 0.0
	if (Float64(angle_m / 180.0) <= 1e+156)
		tmp = Float64(Float64(b - a) * Float64(Float64(2.0 * sin(t_0)) * Float64(Float64(b + a) * cos(t_0))));
	else
		tmp = Float64(Float64(b - a) * Float64(Float64(b + a) * Float64(2.0 * sin(Float64(pi / Float64(180.0 / angle_m))))));
	end
	return Float64(angle_s * tmp)
end

MATLAB

angle\_m = abs(angle);
angle\_s = sign(angle) * abs(1.0);
function tmp_2 = code(angle_s, a, b, angle_m)
	t_0 = angle_m / (180.0 / pi);
	tmp = 0.0;
	if ((angle_m / 180.0) <= 1e+156)
		tmp = (b - a) * ((2.0 * sin(t_0)) * ((b + a) * cos(t_0)));
	else
		tmp = (b - a) * ((b + a) * (2.0 * sin((pi / (180.0 / angle_m)))));
	end
	tmp_2 = angle_s * tmp;
end

Wolfram

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_, b_, angle$95$m_] := Block[{t$95$0 = N[(angle$95$m / N[(180.0 / Pi), $MachinePrecision]), $MachinePrecision]}, N[(angle$95$s * If[LessEqual[N[(angle$95$m / 180.0), $MachinePrecision], 1e+156], N[(N[(b - a), $MachinePrecision] * N[(N[(2.0 * N[Sin[t$95$0], $MachinePrecision]), $MachinePrecision] * N[(N[(b + a), $MachinePrecision] * N[Cos[t$95$0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(b - a), $MachinePrecision] * N[(N[(b + a), $MachinePrecision] * N[(2.0 * N[Sin[N[(Pi / N[(180.0 / angle$95$m), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 angle #s(literal 180 binary64)) < 9.9999999999999998e155

   1. Initial program 57.0%

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

      1. associate-*l* N/A

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

      2. *-commutative N/A

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

      3. associate-*l* N/A

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

      4. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\sin \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right), \color{blue}{\left(\cos \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right) \cdot \left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right)\right)}\right) \]

      5. sin-lowering-sin.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sin.f64}\left(\left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right)\right), \left(\color{blue}{\cos \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right)} \cdot \left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right)\right)\right) \]

      6. associate-*r/ N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sin.f64}\left(\left(\frac{\mathsf{PI}\left(\right) \cdot angle}{180}\right)\right), \left(\cos \color{blue}{\left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right)} \cdot \left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right)\right)\right) \]

      7. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sin.f64}\left(\mathsf{/.f64}\left(\left(\mathsf{PI}\left(\right) \cdot angle\right), 180\right)\right), \left(\cos \color{blue}{\left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right)} \cdot \left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right)\right)\right) \]

      8. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sin.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{PI}\left(\right), angle\right), 180\right)\right), \left(\cos \left(\color{blue}{\mathsf{PI}\left(\right)} \cdot \frac{angle}{180}\right) \cdot \left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right)\right)\right) \]

      9. PI-lowering-PI.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sin.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{PI.f64}\left(\right), angle\right), 180\right)\right), \left(\cos \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right) \cdot \left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right)\right)\right) \]

      10. *-commutative N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{sin.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{PI.f64}\left(\right), angle\right), 180\right)\right), \left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \color{blue}{\cos \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right)}\right)\right) \]

      11. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{sin.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{PI.f64}\left(\right), angle\right), 180\right)\right), \mathsf{*.f64}\left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right), \color{blue}{\cos \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right)}\right)\right) \]

   3. Simplified 56.8%

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

      1. associate-*r/ N/A

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

      2. associate-*l* N/A

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

      3. associate-*r* N/A

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

      4. difference-of-squares N/A

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

      5. associate-*l* N/A

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

      6. associate-*r* N/A

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

      7. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\left(\left(\sin \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right) \cdot 2\right) \cdot \left(b + a\right)\right), \color{blue}{\left(\left(b - a\right) \cdot \cos \left(\frac{\mathsf{PI}\left(\right) \cdot angle}{180}\right)\right)}\right) \]

   6. Applied egg-rr 72.2%

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

      1. add-sqr-sqrt N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(2, \mathsf{sin.f64}\left(\mathsf{/.f64}\left(\left(\sqrt{\mathsf{PI}\left(\right)} \cdot \sqrt{\mathsf{PI}\left(\right)}\right), \mathsf{/.f64}\left(180, angle\right)\right)\right)\right), \mathsf{+.f64}\left(b, a\right)\right), \mathsf{*.f64}\left(\mathsf{cos.f64}\left(\mathsf{/.f64}\left(\mathsf{PI.f64}\left(\right), \mathsf{/.f64}\left(180, angle\right)\right)\right), \mathsf{\_.f64}\left(b, a\right)\right)\right) \]

      2. pow2 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(2, \mathsf{sin.f64}\left(\mathsf{/.f64}\left(\left({\left(\sqrt{\mathsf{PI}\left(\right)}\right)}^{2}\right), \mathsf{/.f64}\left(180, angle\right)\right)\right)\right), \mathsf{+.f64}\left(b, a\right)\right), \mathsf{*.f64}\left(\mathsf{cos.f64}\left(\mathsf{/.f64}\left(\mathsf{PI.f64}\left(\right), \mathsf{/.f64}\left(180, angle\right)\right)\right), \mathsf{\_.f64}\left(b, a\right)\right)\right) \]

      3. pow-lowering-pow.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(2, \mathsf{sin.f64}\left(\mathsf{/.f64}\left(\mathsf{pow.f64}\left(\left(\sqrt{\mathsf{PI}\left(\right)}\right), 2\right), \mathsf{/.f64}\left(180, angle\right)\right)\right)\right), \mathsf{+.f64}\left(b, a\right)\right), \mathsf{*.f64}\left(\mathsf{cos.f64}\left(\mathsf{/.f64}\left(\mathsf{PI.f64}\left(\right), \mathsf{/.f64}\left(180, angle\right)\right)\right), \mathsf{\_.f64}\left(b, a\right)\right)\right) \]

      4. sqrt-lowering-sqrt.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(2, \mathsf{sin.f64}\left(\mathsf{/.f64}\left(\mathsf{pow.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{PI}\left(\right)\right), 2\right), \mathsf{/.f64}\left(180, angle\right)\right)\right)\right), \mathsf{+.f64}\left(b, a\right)\right), \mathsf{*.f64}\left(\mathsf{cos.f64}\left(\mathsf{/.f64}\left(\mathsf{PI.f64}\left(\right), \mathsf{/.f64}\left(180, angle\right)\right)\right), \mathsf{\_.f64}\left(b, a\right)\right)\right) \]

      5. PI-lowering-PI.f64 73.0%

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(2, \mathsf{sin.f64}\left(\mathsf{/.f64}\left(\mathsf{pow.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{PI.f64}\left(\right)\right), 2\right), \mathsf{/.f64}\left(180, angle\right)\right)\right)\right), \mathsf{+.f64}\left(b, a\right)\right), \mathsf{*.f64}\left(\mathsf{cos.f64}\left(\mathsf{/.f64}\left(\mathsf{PI.f64}\left(\right), \mathsf{/.f64}\left(180, angle\right)\right)\right), \mathsf{\_.f64}\left(b, a\right)\right)\right) \]

   8. Applied egg-rr 73.0%

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

      1. add-sqr-sqrt N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(2, \mathsf{sin.f64}\left(\mathsf{/.f64}\left(\mathsf{pow.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{PI.f64}\left(\right)\right), 2\right), \mathsf{/.f64}\left(180, angle\right)\right)\right)\right), \mathsf{+.f64}\left(b, a\right)\right), \mathsf{*.f64}\left(\mathsf{cos.f64}\left(\mathsf{/.f64}\left(\left(\sqrt{\mathsf{PI}\left(\right)} \cdot \sqrt{\mathsf{PI}\left(\right)}\right), \mathsf{/.f64}\left(180, angle\right)\right)\right), \mathsf{\_.f64}\left(b, a\right)\right)\right) \]

      2. unpow2 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(2, \mathsf{sin.f64}\left(\mathsf{/.f64}\left(\mathsf{pow.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{PI.f64}\left(\right)\right), 2\right), \mathsf{/.f64}\left(180, angle\right)\right)\right)\right), \mathsf{+.f64}\left(b, a\right)\right), \mathsf{*.f64}\left(\mathsf{cos.f64}\left(\mathsf{/.f64}\left(\left({\left(\sqrt{\mathsf{PI}\left(\right)}\right)}^{2}\right), \mathsf{/.f64}\left(180, angle\right)\right)\right), \mathsf{\_.f64}\left(b, a\right)\right)\right) \]

      3. pow-lowering-pow.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(2, \mathsf{sin.f64}\left(\mathsf{/.f64}\left(\mathsf{pow.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{PI.f64}\left(\right)\right), 2\right), \mathsf{/.f64}\left(180, angle\right)\right)\right)\right), \mathsf{+.f64}\left(b, a\right)\right), \mathsf{*.f64}\left(\mathsf{cos.f64}\left(\mathsf{/.f64}\left(\mathsf{pow.f64}\left(\left(\sqrt{\mathsf{PI}\left(\right)}\right), 2\right), \mathsf{/.f64}\left(180, angle\right)\right)\right), \mathsf{\_.f64}\left(b, a\right)\right)\right) \]

      4. sqrt-lowering-sqrt.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(2, \mathsf{sin.f64}\left(\mathsf{/.f64}\left(\mathsf{pow.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{PI.f64}\left(\right)\right), 2\right), \mathsf{/.f64}\left(180, angle\right)\right)\right)\right), \mathsf{+.f64}\left(b, a\right)\right), \mathsf{*.f64}\left(\mathsf{cos.f64}\left(\mathsf{/.f64}\left(\mathsf{pow.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{PI}\left(\right)\right), 2\right), \mathsf{/.f64}\left(180, angle\right)\right)\right), \mathsf{\_.f64}\left(b, a\right)\right)\right) \]

      5. PI-lowering-PI.f64 73.2%

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(2, \mathsf{sin.f64}\left(\mathsf{/.f64}\left(\mathsf{pow.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{PI.f64}\left(\right)\right), 2\right), \mathsf{/.f64}\left(180, angle\right)\right)\right)\right), \mathsf{+.f64}\left(b, a\right)\right), \mathsf{*.f64}\left(\mathsf{cos.f64}\left(\mathsf{/.f64}\left(\mathsf{pow.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{PI.f64}\left(\right)\right), 2\right), \mathsf{/.f64}\left(180, angle\right)\right)\right), \mathsf{\_.f64}\left(b, a\right)\right)\right) \]

   10. Applied egg-rr 73.2%

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

       1. associate-*r* N/A

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

       2. *-commutative N/A

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

       3. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\left(b - a\right), \color{blue}{\left(\left(\left(2 \cdot \sin \left(\frac{{\left(\sqrt{\mathsf{PI}\left(\right)}\right)}^{2}}{\frac{180}{angle}}\right)\right) \cdot \left(b + a\right)\right) \cdot \cos \left(\frac{{\left(\sqrt{\mathsf{PI}\left(\right)}\right)}^{2}}{\frac{180}{angle}}\right)\right)}\right) \]

       4. --lowering--.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{\_.f64}\left(b, a\right), \left(\color{blue}{\left(\left(2 \cdot \sin \left(\frac{{\left(\sqrt{\mathsf{PI}\left(\right)}\right)}^{2}}{\frac{180}{angle}}\right)\right) \cdot \left(b + a\right)\right)} \cdot \cos \left(\frac{{\left(\sqrt{\mathsf{PI}\left(\right)}\right)}^{2}}{\frac{180}{angle}}\right)\right)\right) \]

       5. associate-*l* N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{\_.f64}\left(b, a\right), \left(\left(2 \cdot \left(\sin \left(\frac{{\left(\sqrt{\mathsf{PI}\left(\right)}\right)}^{2}}{\frac{180}{angle}}\right) \cdot \left(b + a\right)\right)\right) \cdot \cos \color{blue}{\left(\frac{{\left(\sqrt{\mathsf{PI}\left(\right)}\right)}^{2}}{\frac{180}{angle}}\right)}\right)\right) \]

       6. unpow2 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{\_.f64}\left(b, a\right), \left(\left(2 \cdot \left(\sin \left(\frac{\sqrt{\mathsf{PI}\left(\right)} \cdot \sqrt{\mathsf{PI}\left(\right)}}{\frac{180}{angle}}\right) \cdot \left(b + a\right)\right)\right) \cdot \cos \left(\frac{{\left(\sqrt{\mathsf{PI}\left(\right)}\right)}^{2}}{\frac{180}{angle}}\right)\right)\right) \]

       7. add-sqr-sqrt N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{\_.f64}\left(b, a\right), \left(\left(2 \cdot \left(\sin \left(\frac{\mathsf{PI}\left(\right)}{\frac{180}{angle}}\right) \cdot \left(b + a\right)\right)\right) \cdot \cos \left(\frac{{\left(\sqrt{\mathsf{PI}\left(\right)}\right)}^{2}}{\frac{180}{angle}}\right)\right)\right) \]

       8. associate-*l* N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{\_.f64}\left(b, a\right), \left(\left(\left(2 \cdot \sin \left(\frac{\mathsf{PI}\left(\right)}{\frac{180}{angle}}\right)\right) \cdot \left(b + a\right)\right) \cdot \cos \color{blue}{\left(\frac{{\left(\sqrt{\mathsf{PI}\left(\right)}\right)}^{2}}{\frac{180}{angle}}\right)}\right)\right) \]

   12. Applied egg-rr 72.9%

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

   if 9.9999999999999998e155 < (/.f64 angle #s(literal 180 binary64))

   1. Initial program 32.2%

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

      1. associate-*l* N/A

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

      2. *-commutative N/A

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

      3. associate-*l* N/A

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

      4. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\sin \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right), \color{blue}{\left(\cos \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right) \cdot \left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right)\right)}\right) \]

      5. sin-lowering-sin.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sin.f64}\left(\left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right)\right), \left(\color{blue}{\cos \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right)} \cdot \left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right)\right)\right) \]

      6. associate-*r/ N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sin.f64}\left(\left(\frac{\mathsf{PI}\left(\right) \cdot angle}{180}\right)\right), \left(\cos \color{blue}{\left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right)} \cdot \left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right)\right)\right) \]

      7. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sin.f64}\left(\mathsf{/.f64}\left(\left(\mathsf{PI}\left(\right) \cdot angle\right), 180\right)\right), \left(\cos \color{blue}{\left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right)} \cdot \left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right)\right)\right) \]

      8. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sin.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{PI}\left(\right), angle\right), 180\right)\right), \left(\cos \left(\color{blue}{\mathsf{PI}\left(\right)} \cdot \frac{angle}{180}\right) \cdot \left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right)\right)\right) \]

      9. PI-lowering-PI.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sin.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{PI.f64}\left(\right), angle\right), 180\right)\right), \left(\cos \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right) \cdot \left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right)\right)\right) \]

      10. *-commutative N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{sin.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{PI.f64}\left(\right), angle\right), 180\right)\right), \left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \color{blue}{\cos \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right)}\right)\right) \]

      11. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{sin.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{PI.f64}\left(\right), angle\right), 180\right)\right), \mathsf{*.f64}\left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right), \color{blue}{\cos \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right)}\right)\right) \]

   3. Simplified 35.3%

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

      1. associate-*r/ N/A

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

      2. associate-*l* N/A

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

      3. associate-*r* N/A

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

      4. difference-of-squares N/A

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

      5. associate-*l* N/A

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

      6. associate-*r* N/A

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

      7. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\left(\left(\sin \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right) \cdot 2\right) \cdot \left(b + a\right)\right), \color{blue}{\left(\left(b - a\right) \cdot \cos \left(\frac{\mathsf{PI}\left(\right) \cdot angle}{180}\right)\right)}\right) \]

   6. Applied egg-rr 35.4%

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

      1. clear-num N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(2, \mathsf{sin.f64}\left(\mathsf{/.f64}\left(\mathsf{PI.f64}\left(\right), \mathsf{/.f64}\left(180, angle\right)\right)\right)\right), \mathsf{+.f64}\left(b, a\right)\right), \mathsf{*.f64}\left(\mathsf{cos.f64}\left(\left(\frac{\mathsf{PI}\left(\right)}{\frac{1}{\frac{angle}{180}}}\right)\right), \mathsf{\_.f64}\left(b, a\right)\right)\right) \]

      2. add-sqr-sqrt N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(2, \mathsf{sin.f64}\left(\mathsf{/.f64}\left(\mathsf{PI.f64}\left(\right), \mathsf{/.f64}\left(180, angle\right)\right)\right)\right), \mathsf{+.f64}\left(b, a\right)\right), \mathsf{*.f64}\left(\mathsf{cos.f64}\left(\left(\frac{\sqrt{\mathsf{PI}\left(\right)} \cdot \sqrt{\mathsf{PI}\left(\right)}}{\frac{1}{\frac{angle}{180}}}\right)\right), \mathsf{\_.f64}\left(b, a\right)\right)\right) \]

      3. frac-2neg N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(2, \mathsf{sin.f64}\left(\mathsf{/.f64}\left(\mathsf{PI.f64}\left(\right), \mathsf{/.f64}\left(180, angle\right)\right)\right)\right), \mathsf{+.f64}\left(b, a\right)\right), \mathsf{*.f64}\left(\mathsf{cos.f64}\left(\left(\frac{\sqrt{\mathsf{PI}\left(\right)} \cdot \sqrt{\mathsf{PI}\left(\right)}}{\frac{1}{\frac{\mathsf{neg}\left(angle\right)}{\mathsf{neg}\left(180\right)}}}\right)\right), \mathsf{\_.f64}\left(b, a\right)\right)\right) \]

      4. associate-/r/ N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(2, \mathsf{sin.f64}\left(\mathsf{/.f64}\left(\mathsf{PI.f64}\left(\right), \mathsf{/.f64}\left(180, angle\right)\right)\right)\right), \mathsf{+.f64}\left(b, a\right)\right), \mathsf{*.f64}\left(\mathsf{cos.f64}\left(\left(\frac{\sqrt{\mathsf{PI}\left(\right)} \cdot \sqrt{\mathsf{PI}\left(\right)}}{\frac{1}{\mathsf{neg}\left(angle\right)} \cdot \left(\mathsf{neg}\left(180\right)\right)}\right)\right), \mathsf{\_.f64}\left(b, a\right)\right)\right) \]

      5. times-frac N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(2, \mathsf{sin.f64}\left(\mathsf{/.f64}\left(\mathsf{PI.f64}\left(\right), \mathsf{/.f64}\left(180, angle\right)\right)\right)\right), \mathsf{+.f64}\left(b, a\right)\right), \mathsf{*.f64}\left(\mathsf{cos.f64}\left(\left(\frac{\sqrt{\mathsf{PI}\left(\right)}}{\frac{1}{\mathsf{neg}\left(angle\right)}} \cdot \frac{\sqrt{\mathsf{PI}\left(\right)}}{\mathsf{neg}\left(180\right)}\right)\right), \mathsf{\_.f64}\left(b, a\right)\right)\right) \]

      6. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(2, \mathsf{sin.f64}\left(\mathsf{/.f64}\left(\mathsf{PI.f64}\left(\right), \mathsf{/.f64}\left(180, angle\right)\right)\right)\right), \mathsf{+.f64}\left(b, a\right)\right), \mathsf{*.f64}\left(\mathsf{cos.f64}\left(\mathsf{*.f64}\left(\left(\frac{\sqrt{\mathsf{PI}\left(\right)}}{\frac{1}{\mathsf{neg}\left(angle\right)}}\right), \left(\frac{\sqrt{\mathsf{PI}\left(\right)}}{\mathsf{neg}\left(180\right)}\right)\right)\right), \mathsf{\_.f64}\left(b, a\right)\right)\right) \]

      7. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(2, \mathsf{sin.f64}\left(\mathsf{/.f64}\left(\mathsf{PI.f64}\left(\right), \mathsf{/.f64}\left(180, angle\right)\right)\right)\right), \mathsf{+.f64}\left(b, a\right)\right), \mathsf{*.f64}\left(\mathsf{cos.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(\left(\sqrt{\mathsf{PI}\left(\right)}\right), \left(\frac{1}{\mathsf{neg}\left(angle\right)}\right)\right), \left(\frac{\sqrt{\mathsf{PI}\left(\right)}}{\mathsf{neg}\left(180\right)}\right)\right)\right), \mathsf{\_.f64}\left(b, a\right)\right)\right) \]

      8. sqrt-lowering-sqrt.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(2, \mathsf{sin.f64}\left(\mathsf{/.f64}\left(\mathsf{PI.f64}\left(\right), \mathsf{/.f64}\left(180, angle\right)\right)\right)\right), \mathsf{+.f64}\left(b, a\right)\right), \mathsf{*.f64}\left(\mathsf{cos.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{PI}\left(\right)\right), \left(\frac{1}{\mathsf{neg}\left(angle\right)}\right)\right), \left(\frac{\sqrt{\mathsf{PI}\left(\right)}}{\mathsf{neg}\left(180\right)}\right)\right)\right), \mathsf{\_.f64}\left(b, a\right)\right)\right) \]

      9. PI-lowering-PI.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(2, \mathsf{sin.f64}\left(\mathsf{/.f64}\left(\mathsf{PI.f64}\left(\right), \mathsf{/.f64}\left(180, angle\right)\right)\right)\right), \mathsf{+.f64}\left(b, a\right)\right), \mathsf{*.f64}\left(\mathsf{cos.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{PI.f64}\left(\right)\right), \left(\frac{1}{\mathsf{neg}\left(angle\right)}\right)\right), \left(\frac{\sqrt{\mathsf{PI}\left(\right)}}{\mathsf{neg}\left(180\right)}\right)\right)\right), \mathsf{\_.f64}\left(b, a\right)\right)\right) \]

      10. frac-2neg N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(2, \mathsf{sin.f64}\left(\mathsf{/.f64}\left(\mathsf{PI.f64}\left(\right), \mathsf{/.f64}\left(180, angle\right)\right)\right)\right), \mathsf{+.f64}\left(b, a\right)\right), \mathsf{*.f64}\left(\mathsf{cos.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{PI.f64}\left(\right)\right), \left(\frac{\mathsf{neg}\left(1\right)}{\mathsf{neg}\left(\left(\mathsf{neg}\left(angle\right)\right)\right)}\right)\right), \left(\frac{\sqrt{\mathsf{PI}\left(\right)}}{\mathsf{neg}\left(180\right)}\right)\right)\right), \mathsf{\_.f64}\left(b, a\right)\right)\right) \]

      11. metadata-eval N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(2, \mathsf{sin.f64}\left(\mathsf{/.f64}\left(\mathsf{PI.f64}\left(\right), \mathsf{/.f64}\left(180, angle\right)\right)\right)\right), \mathsf{+.f64}\left(b, a\right)\right), \mathsf{*.f64}\left(\mathsf{cos.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{PI.f64}\left(\right)\right), \left(\frac{-1}{\mathsf{neg}\left(\left(\mathsf{neg}\left(angle\right)\right)\right)}\right)\right), \left(\frac{\sqrt{\mathsf{PI}\left(\right)}}{\mathsf{neg}\left(180\right)}\right)\right)\right), \mathsf{\_.f64}\left(b, a\right)\right)\right) \]

      12. remove-double-neg N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(2, \mathsf{sin.f64}\left(\mathsf{/.f64}\left(\mathsf{PI.f64}\left(\right), \mathsf{/.f64}\left(180, angle\right)\right)\right)\right), \mathsf{+.f64}\left(b, a\right)\right), \mathsf{*.f64}\left(\mathsf{cos.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{PI.f64}\left(\right)\right), \left(\frac{-1}{angle}\right)\right), \left(\frac{\sqrt{\mathsf{PI}\left(\right)}}{\mathsf{neg}\left(180\right)}\right)\right)\right), \mathsf{\_.f64}\left(b, a\right)\right)\right) \]

      13. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(2, \mathsf{sin.f64}\left(\mathsf{/.f64}\left(\mathsf{PI.f64}\left(\right), \mathsf{/.f64}\left(180, angle\right)\right)\right)\right), \mathsf{+.f64}\left(b, a\right)\right), \mathsf{*.f64}\left(\mathsf{cos.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{PI.f64}\left(\right)\right), \mathsf{/.f64}\left(-1, angle\right)\right), \left(\frac{\sqrt{\mathsf{PI}\left(\right)}}{\mathsf{neg}\left(180\right)}\right)\right)\right), \mathsf{\_.f64}\left(b, a\right)\right)\right) \]

      14. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(2, \mathsf{sin.f64}\left(\mathsf{/.f64}\left(\mathsf{PI.f64}\left(\right), \mathsf{/.f64}\left(180, angle\right)\right)\right)\right), \mathsf{+.f64}\left(b, a\right)\right), \mathsf{*.f64}\left(\mathsf{cos.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{PI.f64}\left(\right)\right), \mathsf{/.f64}\left(-1, angle\right)\right), \mathsf{/.f64}\left(\left(\sqrt{\mathsf{PI}\left(\right)}\right), \left(\mathsf{neg}\left(180\right)\right)\right)\right)\right), \mathsf{\_.f64}\left(b, a\right)\right)\right) \]

      15. sqrt-lowering-sqrt.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(2, \mathsf{sin.f64}\left(\mathsf{/.f64}\left(\mathsf{PI.f64}\left(\right), \mathsf{/.f64}\left(180, angle\right)\right)\right)\right), \mathsf{+.f64}\left(b, a\right)\right), \mathsf{*.f64}\left(\mathsf{cos.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{PI.f64}\left(\right)\right), \mathsf{/.f64}\left(-1, angle\right)\right), \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{PI}\left(\right)\right), \left(\mathsf{neg}\left(180\right)\right)\right)\right)\right), \mathsf{\_.f64}\left(b, a\right)\right)\right) \]

      16. PI-lowering-PI.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(2, \mathsf{sin.f64}\left(\mathsf{/.f64}\left(\mathsf{PI.f64}\left(\right), \mathsf{/.f64}\left(180, angle\right)\right)\right)\right), \mathsf{+.f64}\left(b, a\right)\right), \mathsf{*.f64}\left(\mathsf{cos.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{PI.f64}\left(\right)\right), \mathsf{/.f64}\left(-1, angle\right)\right), \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{PI.f64}\left(\right)\right), \left(\mathsf{neg}\left(180\right)\right)\right)\right)\right), \mathsf{\_.f64}\left(b, a\right)\right)\right) \]

      17. metadata-eval 48.0%

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(2, \mathsf{sin.f64}\left(\mathsf{/.f64}\left(\mathsf{PI.f64}\left(\right), \mathsf{/.f64}\left(180, angle\right)\right)\right)\right), \mathsf{+.f64}\left(b, a\right)\right), \mathsf{*.f64}\left(\mathsf{cos.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{PI.f64}\left(\right)\right), \mathsf{/.f64}\left(-1, angle\right)\right), \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{PI.f64}\left(\right)\right), -180\right)\right)\right), \mathsf{\_.f64}\left(b, a\right)\right)\right) \]

   8. Applied egg-rr 48.0%

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

   9. Taylor expanded in angle around 0

      \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(2, \mathsf{sin.f64}\left(\mathsf{/.f64}\left(\mathsf{PI.f64}\left(\right), \mathsf{/.f64}\left(180, angle\right)\right)\right)\right), \mathsf{+.f64}\left(b, a\right)\right), \color{blue}{\left(b - a\right)}\right) \]
   10. Step-by-step derivation

       1. --lowering--.f64 39.7%

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(2, \mathsf{sin.f64}\left(\mathsf{/.f64}\left(\mathsf{PI.f64}\left(\right), \mathsf{/.f64}\left(180, angle\right)\right)\right)\right), \mathsf{+.f64}\left(b, a\right)\right), \mathsf{\_.f64}\left(b, \color{blue}{a}\right)\right) \]

   11. Simplified 39.7%

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

4. Final simplification 69.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{angle}{180} \leq 10^{+156}:\\ \;\;\;\;\left(b - a\right) \cdot \left(\left(2 \cdot \sin \left(\frac{angle}{\frac{180}{\pi}}\right)\right) \cdot \left(\left(b + a\right) \cdot \cos \left(\frac{angle}{\frac{180}{\pi}}\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(b - a\right) \cdot \left(\left(b + a\right) \cdot \left(2 \cdot \sin \left(\frac{\pi}{\frac{180}{angle}}\right)\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 67.1% accurate, 3.2× speedup

Math

\[\begin{array}{l} angle\_m = \left|angle\right| \\ angle\_s = \mathsf{copysign}\left(1, angle\right) \\ \begin{array}{l} t_0 := \left(angle\_m \cdot \pi\right) \cdot 0.011111111111111112\\ angle\_s \cdot \begin{array}{l} \mathbf{if}\;\frac{angle\_m}{180} \leq 5 \cdot 10^{+109}:\\ \;\;\;\;\left(b - a\right) \cdot \left(\left(b + a\right) \cdot \sin t\_0\right)\\ \mathbf{elif}\;\frac{angle\_m}{180} \leq 10^{+156}:\\ \;\;\;\;\left(b \cdot b - a \cdot a\right) \cdot t\_0\\ \mathbf{else}:\\ \;\;\;\;\left(b - a\right) \cdot \left(\left(b + a\right) \cdot \left(2 \cdot \sin \left(\frac{\pi}{\frac{180}{angle\_m}}\right)\right)\right)\\ \end{array} \end{array} \end{array} \]

FPCore

angle\_m = (fabs.f64 angle)
angle\_s = (copysign.f64 #s(literal 1 binary64) angle)
(FPCore (angle_s a b angle_m)
 :precision binary64
 (let* ((t_0 (* (* angle_m PI) 0.011111111111111112)))
   (*
    angle_s
    (if (<= (/ angle_m 180.0) 5e+109)
      (* (- b a) (* (+ b a) (sin t_0)))
      (if (<= (/ angle_m 180.0) 1e+156)
        (* (- (* b b) (* a a)) t_0)
        (* (- b a) (* (+ b a) (* 2.0 (sin (/ PI (/ 180.0 angle_m)))))))))))

C

angle\_m = fabs(angle);
angle\_s = copysign(1.0, angle);
double code(double angle_s, double a, double b, double angle_m) {
	double t_0 = (angle_m * ((double) M_PI)) * 0.011111111111111112;
	double tmp;
	if ((angle_m / 180.0) <= 5e+109) {
		tmp = (b - a) * ((b + a) * sin(t_0));
	} else if ((angle_m / 180.0) <= 1e+156) {
		tmp = ((b * b) - (a * a)) * t_0;
	} else {
		tmp = (b - a) * ((b + a) * (2.0 * sin((((double) M_PI) / (180.0 / angle_m)))));
	}
	return angle_s * tmp;
}

Java

angle\_m = Math.abs(angle);
angle\_s = Math.copySign(1.0, angle);
public static double code(double angle_s, double a, double b, double angle_m) {
	double t_0 = (angle_m * Math.PI) * 0.011111111111111112;
	double tmp;
	if ((angle_m / 180.0) <= 5e+109) {
		tmp = (b - a) * ((b + a) * Math.sin(t_0));
	} else if ((angle_m / 180.0) <= 1e+156) {
		tmp = ((b * b) - (a * a)) * t_0;
	} else {
		tmp = (b - a) * ((b + a) * (2.0 * Math.sin((Math.PI / (180.0 / angle_m)))));
	}
	return angle_s * tmp;
}

Python

angle\_m = math.fabs(angle)
angle\_s = math.copysign(1.0, angle)
def code(angle_s, a, b, angle_m):
	t_0 = (angle_m * math.pi) * 0.011111111111111112
	tmp = 0
	if (angle_m / 180.0) <= 5e+109:
		tmp = (b - a) * ((b + a) * math.sin(t_0))
	elif (angle_m / 180.0) <= 1e+156:
		tmp = ((b * b) - (a * a)) * t_0
	else:
		tmp = (b - a) * ((b + a) * (2.0 * math.sin((math.pi / (180.0 / angle_m)))))
	return angle_s * tmp

Julia

angle\_m = abs(angle)
angle\_s = copysign(1.0, angle)
function code(angle_s, a, b, angle_m)
	t_0 = Float64(Float64(angle_m * pi) * 0.011111111111111112)
	tmp = 0.0
	if (Float64(angle_m / 180.0) <= 5e+109)
		tmp = Float64(Float64(b - a) * Float64(Float64(b + a) * sin(t_0)));
	elseif (Float64(angle_m / 180.0) <= 1e+156)
		tmp = Float64(Float64(Float64(b * b) - Float64(a * a)) * t_0);
	else
		tmp = Float64(Float64(b - a) * Float64(Float64(b + a) * Float64(2.0 * sin(Float64(pi / Float64(180.0 / angle_m))))));
	end
	return Float64(angle_s * tmp)
end

MATLAB

angle\_m = abs(angle);
angle\_s = sign(angle) * abs(1.0);
function tmp_2 = code(angle_s, a, b, angle_m)
	t_0 = (angle_m * pi) * 0.011111111111111112;
	tmp = 0.0;
	if ((angle_m / 180.0) <= 5e+109)
		tmp = (b - a) * ((b + a) * sin(t_0));
	elseif ((angle_m / 180.0) <= 1e+156)
		tmp = ((b * b) - (a * a)) * t_0;
	else
		tmp = (b - a) * ((b + a) * (2.0 * sin((pi / (180.0 / angle_m)))));
	end
	tmp_2 = angle_s * tmp;
end

Wolfram

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_, b_, angle$95$m_] := Block[{t$95$0 = N[(N[(angle$95$m * Pi), $MachinePrecision] * 0.011111111111111112), $MachinePrecision]}, N[(angle$95$s * If[LessEqual[N[(angle$95$m / 180.0), $MachinePrecision], 5e+109], N[(N[(b - a), $MachinePrecision] * N[(N[(b + a), $MachinePrecision] * N[Sin[t$95$0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[N[(angle$95$m / 180.0), $MachinePrecision], 1e+156], N[(N[(N[(b * b), $MachinePrecision] - N[(a * a), $MachinePrecision]), $MachinePrecision] * t$95$0), $MachinePrecision], N[(N[(b - a), $MachinePrecision] * N[(N[(b + a), $MachinePrecision] * N[(2.0 * N[Sin[N[(Pi / N[(180.0 / angle$95$m), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]), $MachinePrecision]]

Derivation

1. Split input into 3 regimes

2. if (/.f64 angle #s(literal 180 binary64)) < 5.0000000000000001e109

   1. Initial program 58.1%

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

      1. associate-*l* N/A

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

      2. *-commutative N/A

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

      3. associate-*l* N/A

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

      4. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\sin \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right), \color{blue}{\left(\cos \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right) \cdot \left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right)\right)}\right) \]

      5. sin-lowering-sin.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sin.f64}\left(\left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right)\right), \left(\color{blue}{\cos \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right)} \cdot \left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right)\right)\right) \]

      6. associate-*r/ N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sin.f64}\left(\left(\frac{\mathsf{PI}\left(\right) \cdot angle}{180}\right)\right), \left(\cos \color{blue}{\left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right)} \cdot \left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right)\right)\right) \]

      7. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sin.f64}\left(\mathsf{/.f64}\left(\left(\mathsf{PI}\left(\right) \cdot angle\right), 180\right)\right), \left(\cos \color{blue}{\left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right)} \cdot \left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right)\right)\right) \]

      8. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sin.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{PI}\left(\right), angle\right), 180\right)\right), \left(\cos \left(\color{blue}{\mathsf{PI}\left(\right)} \cdot \frac{angle}{180}\right) \cdot \left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right)\right)\right) \]

      9. PI-lowering-PI.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sin.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{PI.f64}\left(\right), angle\right), 180\right)\right), \left(\cos \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right) \cdot \left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right)\right)\right) \]

      10. *-commutative N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{sin.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{PI.f64}\left(\right), angle\right), 180\right)\right), \left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \color{blue}{\cos \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right)}\right)\right) \]

      11. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{sin.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{PI.f64}\left(\right), angle\right), 180\right)\right), \mathsf{*.f64}\left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right), \color{blue}{\cos \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right)}\right)\right) \]

   3. Simplified 58.3%

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

      1. associate-*r/ N/A

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

      2. *-commutative N/A

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

      3. associate-*r* N/A

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

      4. associate-*r/ N/A

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

      5. associate-*r* N/A

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

      6. *-commutative N/A

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

      7. *-commutative N/A

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

      8. difference-of-squares N/A

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

   6. Applied egg-rr 74.2%

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

   if 5.0000000000000001e109 < (/.f64 angle #s(literal 180 binary64)) < 9.9999999999999998e155

   1. Initial program 27.6%

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

      1. associate-*l* N/A

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

      2. *-commutative N/A

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

      3. associate-*l* N/A

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

      4. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\sin \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right), \color{blue}{\left(\cos \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right) \cdot \left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right)\right)}\right) \]

      5. sin-lowering-sin.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sin.f64}\left(\left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right)\right), \left(\color{blue}{\cos \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right)} \cdot \left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right)\right)\right) \]

      6. associate-*r/ N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sin.f64}\left(\left(\frac{\mathsf{PI}\left(\right) \cdot angle}{180}\right)\right), \left(\cos \color{blue}{\left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right)} \cdot \left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right)\right)\right) \]

      7. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sin.f64}\left(\mathsf{/.f64}\left(\left(\mathsf{PI}\left(\right) \cdot angle\right), 180\right)\right), \left(\cos \color{blue}{\left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right)} \cdot \left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right)\right)\right) \]

      8. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sin.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{PI}\left(\right), angle\right), 180\right)\right), \left(\cos \left(\color{blue}{\mathsf{PI}\left(\right)} \cdot \frac{angle}{180}\right) \cdot \left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right)\right)\right) \]

      9. PI-lowering-PI.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sin.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{PI.f64}\left(\right), angle\right), 180\right)\right), \left(\cos \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right) \cdot \left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right)\right)\right) \]

      10. *-commutative N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{sin.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{PI.f64}\left(\right), angle\right), 180\right)\right), \left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \color{blue}{\cos \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right)}\right)\right) \]

      11. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{sin.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{PI.f64}\left(\right), angle\right), 180\right)\right), \mathsf{*.f64}\left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right), \color{blue}{\cos \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right)}\right)\right) \]

   3. Simplified 15.1%

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

   5. Taylor expanded in angle around 0

      \[\leadsto \color{blue}{\frac{1}{90} \cdot \left(angle \cdot \left(\mathsf{PI}\left(\right) \cdot \left({b}^{2} - {a}^{2}\right)\right)\right)} \]
   6. Step-by-step derivation

      1. associate-*r* N/A

         \[\leadsto \frac{1}{90} \cdot \left(\left(angle \cdot \mathsf{PI}\left(\right)\right) \cdot \color{blue}{\left({b}^{2} - {a}^{2}\right)}\right) \]

      2. associate-*r* N/A

         \[\leadsto \left(\frac{1}{90} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right) \cdot \color{blue}{\left({b}^{2} - {a}^{2}\right)} \]

      3. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\left(\frac{1}{90} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right), \color{blue}{\left({b}^{2} - {a}^{2}\right)}\right) \]

      4. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{1}{90}, \left(angle \cdot \mathsf{PI}\left(\right)\right)\right), \left(\color{blue}{{b}^{2}} - {a}^{2}\right)\right) \]

      5. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{1}{90}, \mathsf{*.f64}\left(angle, \mathsf{PI}\left(\right)\right)\right), \left({b}^{\color{blue}{2}} - {a}^{2}\right)\right) \]

      6. PI-lowering-PI.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{1}{90}, \mathsf{*.f64}\left(angle, \mathsf{PI.f64}\left(\right)\right)\right), \left({b}^{2} - {a}^{2}\right)\right) \]

      7. --lowering--.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{1}{90}, \mathsf{*.f64}\left(angle, \mathsf{PI.f64}\left(\right)\right)\right), \mathsf{\_.f64}\left(\left({b}^{2}\right), \color{blue}{\left({a}^{2}\right)}\right)\right) \]

      8. unpow2 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{1}{90}, \mathsf{*.f64}\left(angle, \mathsf{PI.f64}\left(\right)\right)\right), \mathsf{\_.f64}\left(\left(b \cdot b\right), \left({\color{blue}{a}}^{2}\right)\right)\right) \]

      9. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{1}{90}, \mathsf{*.f64}\left(angle, \mathsf{PI.f64}\left(\right)\right)\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(b, b\right), \left({\color{blue}{a}}^{2}\right)\right)\right) \]

      10. unpow2 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{1}{90}, \mathsf{*.f64}\left(angle, \mathsf{PI.f64}\left(\right)\right)\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(b, b\right), \left(a \cdot \color{blue}{a}\right)\right)\right) \]

      11. *-lowering-*.f64 51.9%

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{1}{90}, \mathsf{*.f64}\left(angle, \mathsf{PI.f64}\left(\right)\right)\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(b, b\right), \mathsf{*.f64}\left(a, \color{blue}{a}\right)\right)\right) \]

   7. Simplified 51.9%

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

   if 9.9999999999999998e155 < (/.f64 angle #s(literal 180 binary64))

   1. Initial program 32.2%

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

      1. associate-*l* N/A

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

      2. *-commutative N/A

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

      3. associate-*l* N/A

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

      4. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\sin \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right), \color{blue}{\left(\cos \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right) \cdot \left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right)\right)}\right) \]

      5. sin-lowering-sin.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sin.f64}\left(\left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right)\right), \left(\color{blue}{\cos \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right)} \cdot \left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right)\right)\right) \]

      6. associate-*r/ N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sin.f64}\left(\left(\frac{\mathsf{PI}\left(\right) \cdot angle}{180}\right)\right), \left(\cos \color{blue}{\left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right)} \cdot \left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right)\right)\right) \]

      7. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sin.f64}\left(\mathsf{/.f64}\left(\left(\mathsf{PI}\left(\right) \cdot angle\right), 180\right)\right), \left(\cos \color{blue}{\left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right)} \cdot \left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right)\right)\right) \]

      8. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sin.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{PI}\left(\right), angle\right), 180\right)\right), \left(\cos \left(\color{blue}{\mathsf{PI}\left(\right)} \cdot \frac{angle}{180}\right) \cdot \left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right)\right)\right) \]

      9. PI-lowering-PI.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sin.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{PI.f64}\left(\right), angle\right), 180\right)\right), \left(\cos \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right) \cdot \left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right)\right)\right) \]

      10. *-commutative N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{sin.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{PI.f64}\left(\right), angle\right), 180\right)\right), \left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \color{blue}{\cos \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right)}\right)\right) \]

      11. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{sin.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{PI.f64}\left(\right), angle\right), 180\right)\right), \mathsf{*.f64}\left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right), \color{blue}{\cos \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right)}\right)\right) \]

   3. Simplified 35.3%

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

      1. associate-*r/ N/A

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

      2. associate-*l* N/A

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

      3. associate-*r* N/A

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

      4. difference-of-squares N/A

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

      5. associate-*l* N/A

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

      6. associate-*r* N/A

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

      7. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\left(\left(\sin \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right) \cdot 2\right) \cdot \left(b + a\right)\right), \color{blue}{\left(\left(b - a\right) \cdot \cos \left(\frac{\mathsf{PI}\left(\right) \cdot angle}{180}\right)\right)}\right) \]

   6. Applied egg-rr 35.4%

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

      1. clear-num N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(2, \mathsf{sin.f64}\left(\mathsf{/.f64}\left(\mathsf{PI.f64}\left(\right), \mathsf{/.f64}\left(180, angle\right)\right)\right)\right), \mathsf{+.f64}\left(b, a\right)\right), \mathsf{*.f64}\left(\mathsf{cos.f64}\left(\left(\frac{\mathsf{PI}\left(\right)}{\frac{1}{\frac{angle}{180}}}\right)\right), \mathsf{\_.f64}\left(b, a\right)\right)\right) \]

      2. add-sqr-sqrt N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(2, \mathsf{sin.f64}\left(\mathsf{/.f64}\left(\mathsf{PI.f64}\left(\right), \mathsf{/.f64}\left(180, angle\right)\right)\right)\right), \mathsf{+.f64}\left(b, a\right)\right), \mathsf{*.f64}\left(\mathsf{cos.f64}\left(\left(\frac{\sqrt{\mathsf{PI}\left(\right)} \cdot \sqrt{\mathsf{PI}\left(\right)}}{\frac{1}{\frac{angle}{180}}}\right)\right), \mathsf{\_.f64}\left(b, a\right)\right)\right) \]

      3. frac-2neg N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(2, \mathsf{sin.f64}\left(\mathsf{/.f64}\left(\mathsf{PI.f64}\left(\right), \mathsf{/.f64}\left(180, angle\right)\right)\right)\right), \mathsf{+.f64}\left(b, a\right)\right), \mathsf{*.f64}\left(\mathsf{cos.f64}\left(\left(\frac{\sqrt{\mathsf{PI}\left(\right)} \cdot \sqrt{\mathsf{PI}\left(\right)}}{\frac{1}{\frac{\mathsf{neg}\left(angle\right)}{\mathsf{neg}\left(180\right)}}}\right)\right), \mathsf{\_.f64}\left(b, a\right)\right)\right) \]

      4. associate-/r/ N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(2, \mathsf{sin.f64}\left(\mathsf{/.f64}\left(\mathsf{PI.f64}\left(\right), \mathsf{/.f64}\left(180, angle\right)\right)\right)\right), \mathsf{+.f64}\left(b, a\right)\right), \mathsf{*.f64}\left(\mathsf{cos.f64}\left(\left(\frac{\sqrt{\mathsf{PI}\left(\right)} \cdot \sqrt{\mathsf{PI}\left(\right)}}{\frac{1}{\mathsf{neg}\left(angle\right)} \cdot \left(\mathsf{neg}\left(180\right)\right)}\right)\right), \mathsf{\_.f64}\left(b, a\right)\right)\right) \]

      5. times-frac N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(2, \mathsf{sin.f64}\left(\mathsf{/.f64}\left(\mathsf{PI.f64}\left(\right), \mathsf{/.f64}\left(180, angle\right)\right)\right)\right), \mathsf{+.f64}\left(b, a\right)\right), \mathsf{*.f64}\left(\mathsf{cos.f64}\left(\left(\frac{\sqrt{\mathsf{PI}\left(\right)}}{\frac{1}{\mathsf{neg}\left(angle\right)}} \cdot \frac{\sqrt{\mathsf{PI}\left(\right)}}{\mathsf{neg}\left(180\right)}\right)\right), \mathsf{\_.f64}\left(b, a\right)\right)\right) \]

      6. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(2, \mathsf{sin.f64}\left(\mathsf{/.f64}\left(\mathsf{PI.f64}\left(\right), \mathsf{/.f64}\left(180, angle\right)\right)\right)\right), \mathsf{+.f64}\left(b, a\right)\right), \mathsf{*.f64}\left(\mathsf{cos.f64}\left(\mathsf{*.f64}\left(\left(\frac{\sqrt{\mathsf{PI}\left(\right)}}{\frac{1}{\mathsf{neg}\left(angle\right)}}\right), \left(\frac{\sqrt{\mathsf{PI}\left(\right)}}{\mathsf{neg}\left(180\right)}\right)\right)\right), \mathsf{\_.f64}\left(b, a\right)\right)\right) \]

      7. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(2, \mathsf{sin.f64}\left(\mathsf{/.f64}\left(\mathsf{PI.f64}\left(\right), \mathsf{/.f64}\left(180, angle\right)\right)\right)\right), \mathsf{+.f64}\left(b, a\right)\right), \mathsf{*.f64}\left(\mathsf{cos.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(\left(\sqrt{\mathsf{PI}\left(\right)}\right), \left(\frac{1}{\mathsf{neg}\left(angle\right)}\right)\right), \left(\frac{\sqrt{\mathsf{PI}\left(\right)}}{\mathsf{neg}\left(180\right)}\right)\right)\right), \mathsf{\_.f64}\left(b, a\right)\right)\right) \]

      8. sqrt-lowering-sqrt.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(2, \mathsf{sin.f64}\left(\mathsf{/.f64}\left(\mathsf{PI.f64}\left(\right), \mathsf{/.f64}\left(180, angle\right)\right)\right)\right), \mathsf{+.f64}\left(b, a\right)\right), \mathsf{*.f64}\left(\mathsf{cos.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{PI}\left(\right)\right), \left(\frac{1}{\mathsf{neg}\left(angle\right)}\right)\right), \left(\frac{\sqrt{\mathsf{PI}\left(\right)}}{\mathsf{neg}\left(180\right)}\right)\right)\right), \mathsf{\_.f64}\left(b, a\right)\right)\right) \]

      9. PI-lowering-PI.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(2, \mathsf{sin.f64}\left(\mathsf{/.f64}\left(\mathsf{PI.f64}\left(\right), \mathsf{/.f64}\left(180, angle\right)\right)\right)\right), \mathsf{+.f64}\left(b, a\right)\right), \mathsf{*.f64}\left(\mathsf{cos.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{PI.f64}\left(\right)\right), \left(\frac{1}{\mathsf{neg}\left(angle\right)}\right)\right), \left(\frac{\sqrt{\mathsf{PI}\left(\right)}}{\mathsf{neg}\left(180\right)}\right)\right)\right), \mathsf{\_.f64}\left(b, a\right)\right)\right) \]

      10. frac-2neg N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(2, \mathsf{sin.f64}\left(\mathsf{/.f64}\left(\mathsf{PI.f64}\left(\right), \mathsf{/.f64}\left(180, angle\right)\right)\right)\right), \mathsf{+.f64}\left(b, a\right)\right), \mathsf{*.f64}\left(\mathsf{cos.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{PI.f64}\left(\right)\right), \left(\frac{\mathsf{neg}\left(1\right)}{\mathsf{neg}\left(\left(\mathsf{neg}\left(angle\right)\right)\right)}\right)\right), \left(\frac{\sqrt{\mathsf{PI}\left(\right)}}{\mathsf{neg}\left(180\right)}\right)\right)\right), \mathsf{\_.f64}\left(b, a\right)\right)\right) \]

      11. metadata-eval N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(2, \mathsf{sin.f64}\left(\mathsf{/.f64}\left(\mathsf{PI.f64}\left(\right), \mathsf{/.f64}\left(180, angle\right)\right)\right)\right), \mathsf{+.f64}\left(b, a\right)\right), \mathsf{*.f64}\left(\mathsf{cos.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{PI.f64}\left(\right)\right), \left(\frac{-1}{\mathsf{neg}\left(\left(\mathsf{neg}\left(angle\right)\right)\right)}\right)\right), \left(\frac{\sqrt{\mathsf{PI}\left(\right)}}{\mathsf{neg}\left(180\right)}\right)\right)\right), \mathsf{\_.f64}\left(b, a\right)\right)\right) \]

      12. remove-double-neg N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(2, \mathsf{sin.f64}\left(\mathsf{/.f64}\left(\mathsf{PI.f64}\left(\right), \mathsf{/.f64}\left(180, angle\right)\right)\right)\right), \mathsf{+.f64}\left(b, a\right)\right), \mathsf{*.f64}\left(\mathsf{cos.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{PI.f64}\left(\right)\right), \left(\frac{-1}{angle}\right)\right), \left(\frac{\sqrt{\mathsf{PI}\left(\right)}}{\mathsf{neg}\left(180\right)}\right)\right)\right), \mathsf{\_.f64}\left(b, a\right)\right)\right) \]

      13. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(2, \mathsf{sin.f64}\left(\mathsf{/.f64}\left(\mathsf{PI.f64}\left(\right), \mathsf{/.f64}\left(180, angle\right)\right)\right)\right), \mathsf{+.f64}\left(b, a\right)\right), \mathsf{*.f64}\left(\mathsf{cos.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{PI.f64}\left(\right)\right), \mathsf{/.f64}\left(-1, angle\right)\right), \left(\frac{\sqrt{\mathsf{PI}\left(\right)}}{\mathsf{neg}\left(180\right)}\right)\right)\right), \mathsf{\_.f64}\left(b, a\right)\right)\right) \]

      14. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(2, \mathsf{sin.f64}\left(\mathsf{/.f64}\left(\mathsf{PI.f64}\left(\right), \mathsf{/.f64}\left(180, angle\right)\right)\right)\right), \mathsf{+.f64}\left(b, a\right)\right), \mathsf{*.f64}\left(\mathsf{cos.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{PI.f64}\left(\right)\right), \mathsf{/.f64}\left(-1, angle\right)\right), \mathsf{/.f64}\left(\left(\sqrt{\mathsf{PI}\left(\right)}\right), \left(\mathsf{neg}\left(180\right)\right)\right)\right)\right), \mathsf{\_.f64}\left(b, a\right)\right)\right) \]

      15. sqrt-lowering-sqrt.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(2, \mathsf{sin.f64}\left(\mathsf{/.f64}\left(\mathsf{PI.f64}\left(\right), \mathsf{/.f64}\left(180, angle\right)\right)\right)\right), \mathsf{+.f64}\left(b, a\right)\right), \mathsf{*.f64}\left(\mathsf{cos.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{PI.f64}\left(\right)\right), \mathsf{/.f64}\left(-1, angle\right)\right), \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{PI}\left(\right)\right), \left(\mathsf{neg}\left(180\right)\right)\right)\right)\right), \mathsf{\_.f64}\left(b, a\right)\right)\right) \]

      16. PI-lowering-PI.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(2, \mathsf{sin.f64}\left(\mathsf{/.f64}\left(\mathsf{PI.f64}\left(\right), \mathsf{/.f64}\left(180, angle\right)\right)\right)\right), \mathsf{+.f64}\left(b, a\right)\right), \mathsf{*.f64}\left(\mathsf{cos.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{PI.f64}\left(\right)\right), \mathsf{/.f64}\left(-1, angle\right)\right), \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{PI.f64}\left(\right)\right), \left(\mathsf{neg}\left(180\right)\right)\right)\right)\right), \mathsf{\_.f64}\left(b, a\right)\right)\right) \]

      17. metadata-eval 48.0%

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(2, \mathsf{sin.f64}\left(\mathsf{/.f64}\left(\mathsf{PI.f64}\left(\right), \mathsf{/.f64}\left(180, angle\right)\right)\right)\right), \mathsf{+.f64}\left(b, a\right)\right), \mathsf{*.f64}\left(\mathsf{cos.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{PI.f64}\left(\right)\right), \mathsf{/.f64}\left(-1, angle\right)\right), \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{PI.f64}\left(\right)\right), -180\right)\right)\right), \mathsf{\_.f64}\left(b, a\right)\right)\right) \]

   8. Applied egg-rr 48.0%

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

   9. Taylor expanded in angle around 0

      \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(2, \mathsf{sin.f64}\left(\mathsf{/.f64}\left(\mathsf{PI.f64}\left(\right), \mathsf{/.f64}\left(180, angle\right)\right)\right)\right), \mathsf{+.f64}\left(b, a\right)\right), \color{blue}{\left(b - a\right)}\right) \]
   10. Step-by-step derivation

       1. --lowering--.f64 39.7%

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(2, \mathsf{sin.f64}\left(\mathsf{/.f64}\left(\mathsf{PI.f64}\left(\right), \mathsf{/.f64}\left(180, angle\right)\right)\right)\right), \mathsf{+.f64}\left(b, a\right)\right), \mathsf{\_.f64}\left(b, \color{blue}{a}\right)\right) \]

   11. Simplified 39.7%

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

4. Final simplification 70.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{angle}{180} \leq 5 \cdot 10^{+109}:\\ \;\;\;\;\left(b - a\right) \cdot \left(\left(b + a\right) \cdot \sin \left(\left(angle \cdot \pi\right) \cdot 0.011111111111111112\right)\right)\\ \mathbf{elif}\;\frac{angle}{180} \leq 10^{+156}:\\ \;\;\;\;\left(b \cdot b - a \cdot a\right) \cdot \left(\left(angle \cdot \pi\right) \cdot 0.011111111111111112\right)\\ \mathbf{else}:\\ \;\;\;\;\left(b - a\right) \cdot \left(\left(b + a\right) \cdot \left(2 \cdot \sin \left(\frac{\pi}{\frac{180}{angle}}\right)\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 66.0% accurate, 3.6× speedup

Math

\[\begin{array}{l} angle\_m = \left|angle\right| \\ angle\_s = \mathsf{copysign}\left(1, angle\right) \\ angle\_s \cdot \begin{array}{l} \mathbf{if}\;angle\_m \leq 85:\\ \;\;\;\;\left(b + a\right) \cdot \left(\left(b - a\right) \cdot \left(\left(angle\_m \cdot \pi\right) \cdot 0.011111111111111112\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(b \cdot b - a \cdot a\right) \cdot \sin \left(\pi \cdot \left(angle\_m \cdot 0.011111111111111112\right)\right)\\ \end{array} \end{array} \]

FPCore

angle\_m = (fabs.f64 angle)
angle\_s = (copysign.f64 #s(literal 1 binary64) angle)
(FPCore (angle_s a b angle_m)
 :precision binary64
 (*
  angle_s
  (if (<= angle_m 85.0)
    (* (+ b a) (* (- b a) (* (* angle_m PI) 0.011111111111111112)))
    (* (- (* b b) (* a a)) (sin (* PI (* angle_m 0.011111111111111112)))))))

C

angle\_m = fabs(angle);
angle\_s = copysign(1.0, angle);
double code(double angle_s, double a, double b, double angle_m) {
	double tmp;
	if (angle_m <= 85.0) {
		tmp = (b + a) * ((b - a) * ((angle_m * ((double) M_PI)) * 0.011111111111111112));
	} else {
		tmp = ((b * b) - (a * a)) * sin((((double) M_PI) * (angle_m * 0.011111111111111112)));
	}
	return angle_s * tmp;
}

Java

angle\_m = Math.abs(angle);
angle\_s = Math.copySign(1.0, angle);
public static double code(double angle_s, double a, double b, double angle_m) {
	double tmp;
	if (angle_m <= 85.0) {
		tmp = (b + a) * ((b - a) * ((angle_m * Math.PI) * 0.011111111111111112));
	} else {
		tmp = ((b * b) - (a * a)) * Math.sin((Math.PI * (angle_m * 0.011111111111111112)));
	}
	return angle_s * tmp;
}

Python

angle\_m = math.fabs(angle)
angle\_s = math.copysign(1.0, angle)
def code(angle_s, a, b, angle_m):
	tmp = 0
	if angle_m <= 85.0:
		tmp = (b + a) * ((b - a) * ((angle_m * math.pi) * 0.011111111111111112))
	else:
		tmp = ((b * b) - (a * a)) * math.sin((math.pi * (angle_m * 0.011111111111111112)))
	return angle_s * tmp

Julia

angle\_m = abs(angle)
angle\_s = copysign(1.0, angle)
function code(angle_s, a, b, angle_m)
	tmp = 0.0
	if (angle_m <= 85.0)
		tmp = Float64(Float64(b + a) * Float64(Float64(b - a) * Float64(Float64(angle_m * pi) * 0.011111111111111112)));
	else
		tmp = Float64(Float64(Float64(b * b) - Float64(a * a)) * sin(Float64(pi * Float64(angle_m * 0.011111111111111112))));
	end
	return Float64(angle_s * tmp)
end

MATLAB

angle\_m = abs(angle);
angle\_s = sign(angle) * abs(1.0);
function tmp_2 = code(angle_s, a, b, angle_m)
	tmp = 0.0;
	if (angle_m <= 85.0)
		tmp = (b + a) * ((b - a) * ((angle_m * pi) * 0.011111111111111112));
	else
		tmp = ((b * b) - (a * a)) * sin((pi * (angle_m * 0.011111111111111112)));
	end
	tmp_2 = angle_s * tmp;
end

Wolfram

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_, b_, angle$95$m_] := N[(angle$95$s * If[LessEqual[angle$95$m, 85.0], N[(N[(b + a), $MachinePrecision] * N[(N[(b - a), $MachinePrecision] * N[(N[(angle$95$m * Pi), $MachinePrecision] * 0.011111111111111112), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(b * b), $MachinePrecision] - N[(a * a), $MachinePrecision]), $MachinePrecision] * N[Sin[N[(Pi * N[(angle$95$m * 0.011111111111111112), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if angle < 85

   1. Initial program 61.3%

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

      1. associate-*l* N/A

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

      2. *-commutative N/A

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

      3. associate-*l* N/A

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

      4. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\sin \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right), \color{blue}{\left(\cos \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right) \cdot \left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right)\right)}\right) \]

      5. sin-lowering-sin.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sin.f64}\left(\left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right)\right), \left(\color{blue}{\cos \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right)} \cdot \left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right)\right)\right) \]

      6. associate-*r/ N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sin.f64}\left(\left(\frac{\mathsf{PI}\left(\right) \cdot angle}{180}\right)\right), \left(\cos \color{blue}{\left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right)} \cdot \left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right)\right)\right) \]

      7. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sin.f64}\left(\mathsf{/.f64}\left(\left(\mathsf{PI}\left(\right) \cdot angle\right), 180\right)\right), \left(\cos \color{blue}{\left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right)} \cdot \left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right)\right)\right) \]

      8. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sin.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{PI}\left(\right), angle\right), 180\right)\right), \left(\cos \left(\color{blue}{\mathsf{PI}\left(\right)} \cdot \frac{angle}{180}\right) \cdot \left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right)\right)\right) \]

      9. PI-lowering-PI.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sin.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{PI.f64}\left(\right), angle\right), 180\right)\right), \left(\cos \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right) \cdot \left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right)\right)\right) \]

      10. *-commutative N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{sin.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{PI.f64}\left(\right), angle\right), 180\right)\right), \left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \color{blue}{\cos \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right)}\right)\right) \]

      11. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{sin.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{PI.f64}\left(\right), angle\right), 180\right)\right), \mathsf{*.f64}\left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right), \color{blue}{\cos \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right)}\right)\right) \]

   3. Simplified 61.4%

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

   5. Taylor expanded in angle around 0

      \[\leadsto \color{blue}{\frac{1}{90} \cdot \left(angle \cdot \left(\mathsf{PI}\left(\right) \cdot \left({b}^{2} - {a}^{2}\right)\right)\right)} \]
   6. Step-by-step derivation

      1. associate-*r* N/A

         \[\leadsto \frac{1}{90} \cdot \left(\left(angle \cdot \mathsf{PI}\left(\right)\right) \cdot \color{blue}{\left({b}^{2} - {a}^{2}\right)}\right) \]

      2. associate-*r* N/A

         \[\leadsto \left(\frac{1}{90} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right) \cdot \color{blue}{\left({b}^{2} - {a}^{2}\right)} \]

      3. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\left(\frac{1}{90} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right), \color{blue}{\left({b}^{2} - {a}^{2}\right)}\right) \]

      4. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{1}{90}, \left(angle \cdot \mathsf{PI}\left(\right)\right)\right), \left(\color{blue}{{b}^{2}} - {a}^{2}\right)\right) \]

      5. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{1}{90}, \mathsf{*.f64}\left(angle, \mathsf{PI}\left(\right)\right)\right), \left({b}^{\color{blue}{2}} - {a}^{2}\right)\right) \]

      6. PI-lowering-PI.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{1}{90}, \mathsf{*.f64}\left(angle, \mathsf{PI.f64}\left(\right)\right)\right), \left({b}^{2} - {a}^{2}\right)\right) \]

      7. --lowering--.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{1}{90}, \mathsf{*.f64}\left(angle, \mathsf{PI.f64}\left(\right)\right)\right), \mathsf{\_.f64}\left(\left({b}^{2}\right), \color{blue}{\left({a}^{2}\right)}\right)\right) \]

      8. unpow2 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{1}{90}, \mathsf{*.f64}\left(angle, \mathsf{PI.f64}\left(\right)\right)\right), \mathsf{\_.f64}\left(\left(b \cdot b\right), \left({\color{blue}{a}}^{2}\right)\right)\right) \]

      9. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{1}{90}, \mathsf{*.f64}\left(angle, \mathsf{PI.f64}\left(\right)\right)\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(b, b\right), \left({\color{blue}{a}}^{2}\right)\right)\right) \]

      10. unpow2 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{1}{90}, \mathsf{*.f64}\left(angle, \mathsf{PI.f64}\left(\right)\right)\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(b, b\right), \left(a \cdot \color{blue}{a}\right)\right)\right) \]

      11. *-lowering-*.f64 58.1%

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{1}{90}, \mathsf{*.f64}\left(angle, \mathsf{PI.f64}\left(\right)\right)\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(b, b\right), \mathsf{*.f64}\left(a, \color{blue}{a}\right)\right)\right) \]

   7. Simplified 58.1%

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

      1. *-commutative N/A

         \[\leadsto \left(b \cdot b - a \cdot a\right) \cdot \color{blue}{\left(\frac{1}{90} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)} \]

      2. difference-of-squares N/A

         \[\leadsto \left(\left(b + a\right) \cdot \left(b - a\right)\right) \cdot \left(\color{blue}{\frac{1}{90}} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right) \]

      3. associate-*l* N/A

         \[\leadsto \left(b + a\right) \cdot \color{blue}{\left(\left(b - a\right) \cdot \left(\frac{1}{90} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)\right)} \]

      4. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\left(b + a\right), \color{blue}{\left(\left(b - a\right) \cdot \left(\frac{1}{90} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)\right)}\right) \]

      5. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(b, a\right), \left(\color{blue}{\left(b - a\right)} \cdot \left(\frac{1}{90} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)\right)\right) \]

      6. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(b, a\right), \mathsf{*.f64}\left(\left(b - a\right), \color{blue}{\left(\frac{1}{90} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}\right)\right) \]

      7. --lowering--.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(b, a\right), \mathsf{*.f64}\left(\mathsf{\_.f64}\left(b, a\right), \left(\color{blue}{\frac{1}{90}} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)\right)\right) \]

      8. *-commutative N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(b, a\right), \mathsf{*.f64}\left(\mathsf{\_.f64}\left(b, a\right), \left(\frac{1}{90} \cdot \left(\mathsf{PI}\left(\right) \cdot \color{blue}{angle}\right)\right)\right)\right) \]

      9. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(b, a\right), \mathsf{*.f64}\left(\mathsf{\_.f64}\left(b, a\right), \mathsf{*.f64}\left(\frac{1}{90}, \color{blue}{\left(\mathsf{PI}\left(\right) \cdot angle\right)}\right)\right)\right) \]

      10. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(b, a\right), \mathsf{*.f64}\left(\mathsf{\_.f64}\left(b, a\right), \mathsf{*.f64}\left(\frac{1}{90}, \mathsf{*.f64}\left(\mathsf{PI}\left(\right), \color{blue}{angle}\right)\right)\right)\right) \]

      11. PI-lowering-PI.f64 75.7%

          \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(b, a\right), \mathsf{*.f64}\left(\mathsf{\_.f64}\left(b, a\right), \mathsf{*.f64}\left(\frac{1}{90}, \mathsf{*.f64}\left(\mathsf{PI.f64}\left(\right), angle\right)\right)\right)\right) \]

   9. Applied egg-rr 75.7%

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

   if 85 < angle

   1. Initial program 32.4%

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

      1. associate-*l* N/A

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

      2. *-commutative N/A

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

      3. associate-*l* N/A

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

      4. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\sin \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right), \color{blue}{\left(\cos \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right) \cdot \left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right)\right)}\right) \]

      5. sin-lowering-sin.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sin.f64}\left(\left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right)\right), \left(\color{blue}{\cos \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right)} \cdot \left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right)\right)\right) \]

      6. associate-*r/ N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sin.f64}\left(\left(\frac{\mathsf{PI}\left(\right) \cdot angle}{180}\right)\right), \left(\cos \color{blue}{\left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right)} \cdot \left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right)\right)\right) \]

      7. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sin.f64}\left(\mathsf{/.f64}\left(\left(\mathsf{PI}\left(\right) \cdot angle\right), 180\right)\right), \left(\cos \color{blue}{\left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right)} \cdot \left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right)\right)\right) \]

      8. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sin.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{PI}\left(\right), angle\right), 180\right)\right), \left(\cos \left(\color{blue}{\mathsf{PI}\left(\right)} \cdot \frac{angle}{180}\right) \cdot \left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right)\right)\right) \]

      9. PI-lowering-PI.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sin.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{PI.f64}\left(\right), angle\right), 180\right)\right), \left(\cos \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right) \cdot \left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right)\right)\right) \]

      10. *-commutative N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{sin.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{PI.f64}\left(\right), angle\right), 180\right)\right), \left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \color{blue}{\cos \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right)}\right)\right) \]

      11. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{sin.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{PI.f64}\left(\right), angle\right), 180\right)\right), \mathsf{*.f64}\left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right), \color{blue}{\cos \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right)}\right)\right) \]

   3. Simplified 32.3%

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

   6. Applied egg-rr 5.3%

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

      1. *-commutative N/A

         \[\leadsto \frac{\sin \left(\left(\mathsf{PI}\left(\right) \cdot angle\right) \cdot \frac{1}{90}\right) \cdot \left(b \cdot \left(b \cdot \left(b \cdot b\right)\right) - a \cdot \left(a \cdot \left(a \cdot a\right)\right)\right)}{\color{blue}{b \cdot b} + a \cdot a} \]

      2. associate-/l* N/A

         \[\leadsto \sin \left(\left(\mathsf{PI}\left(\right) \cdot angle\right) \cdot \frac{1}{90}\right) \cdot \color{blue}{\frac{b \cdot \left(b \cdot \left(b \cdot b\right)\right) - a \cdot \left(a \cdot \left(a \cdot a\right)\right)}{b \cdot b + a \cdot a}} \]

      3. associate-*r* N/A

         \[\leadsto \sin \left(\left(\mathsf{PI}\left(\right) \cdot angle\right) \cdot \frac{1}{90}\right) \cdot \frac{\left(b \cdot b\right) \cdot \left(b \cdot b\right) - a \cdot \left(a \cdot \left(a \cdot a\right)\right)}{\color{blue}{b} \cdot b + a \cdot a} \]

      4. associate-*r* N/A

         \[\leadsto \sin \left(\left(\mathsf{PI}\left(\right) \cdot angle\right) \cdot \frac{1}{90}\right) \cdot \frac{\left(b \cdot b\right) \cdot \left(b \cdot b\right) - \left(a \cdot a\right) \cdot \left(a \cdot a\right)}{b \cdot \color{blue}{b} + a \cdot a} \]

      5. flip-- N/A

         \[\leadsto \sin \left(\left(\mathsf{PI}\left(\right) \cdot angle\right) \cdot \frac{1}{90}\right) \cdot \left(b \cdot b - \color{blue}{a \cdot a}\right) \]

      6. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\sin \left(\left(\mathsf{PI}\left(\right) \cdot angle\right) \cdot \frac{1}{90}\right), \color{blue}{\left(b \cdot b - a \cdot a\right)}\right) \]

      7. sin-lowering-sin.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sin.f64}\left(\left(\left(\mathsf{PI}\left(\right) \cdot angle\right) \cdot \frac{1}{90}\right)\right), \left(\color{blue}{b \cdot b} - a \cdot a\right)\right) \]

      8. associate-*l* N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sin.f64}\left(\left(\mathsf{PI}\left(\right) \cdot \left(angle \cdot \frac{1}{90}\right)\right)\right), \left(\color{blue}{b} \cdot b - a \cdot a\right)\right) \]

      9. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sin.f64}\left(\mathsf{*.f64}\left(\mathsf{PI}\left(\right), \left(angle \cdot \frac{1}{90}\right)\right)\right), \left(\color{blue}{b} \cdot b - a \cdot a\right)\right) \]

      10. PI-lowering-PI.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{sin.f64}\left(\mathsf{*.f64}\left(\mathsf{PI.f64}\left(\right), \left(angle \cdot \frac{1}{90}\right)\right)\right), \left(b \cdot b - a \cdot a\right)\right) \]

      11. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{sin.f64}\left(\mathsf{*.f64}\left(\mathsf{PI.f64}\left(\right), \mathsf{*.f64}\left(angle, \frac{1}{90}\right)\right)\right), \left(b \cdot b - a \cdot a\right)\right) \]

      12. --lowering--.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{sin.f64}\left(\mathsf{*.f64}\left(\mathsf{PI.f64}\left(\right), \mathsf{*.f64}\left(angle, \frac{1}{90}\right)\right)\right), \mathsf{\_.f64}\left(\left(b \cdot b\right), \color{blue}{\left(a \cdot a\right)}\right)\right) \]

      13. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{sin.f64}\left(\mathsf{*.f64}\left(\mathsf{PI.f64}\left(\right), \mathsf{*.f64}\left(angle, \frac{1}{90}\right)\right)\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(b, b\right), \left(\color{blue}{a} \cdot a\right)\right)\right) \]

      14. *-lowering-*.f64 33.5%

          \[\leadsto \mathsf{*.f64}\left(\mathsf{sin.f64}\left(\mathsf{*.f64}\left(\mathsf{PI.f64}\left(\right), \mathsf{*.f64}\left(angle, \frac{1}{90}\right)\right)\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(b, b\right), \mathsf{*.f64}\left(a, \color{blue}{a}\right)\right)\right) \]

   8. Applied egg-rr 33.5%

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

4. Final simplification 66.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;angle \leq 85:\\ \;\;\;\;\left(b + a\right) \cdot \left(\left(b - a\right) \cdot \left(\left(angle \cdot \pi\right) \cdot 0.011111111111111112\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(b \cdot b - a \cdot a\right) \cdot \sin \left(\pi \cdot \left(angle \cdot 0.011111111111111112\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 49.6% accurate, 3.7× speedup

Math

\[\begin{array}{l} angle\_m = \left|angle\right| \\ angle\_s = \mathsf{copysign}\left(1, angle\right) \\ \begin{array}{l} t_0 := \left(angle\_m \cdot \pi\right) \cdot 0.011111111111111112\\ angle\_s \cdot \begin{array}{l} \mathbf{if}\;a \leq 4.5 \cdot 10^{-118}:\\ \;\;\;\;\left(b \cdot b\right) \cdot \sin t\_0\\ \mathbf{else}:\\ \;\;\;\;\left(b + a\right) \cdot \left(\left(b - a\right) \cdot t\_0\right)\\ \end{array} \end{array} \end{array} \]

FPCore

angle\_m = (fabs.f64 angle)
angle\_s = (copysign.f64 #s(literal 1 binary64) angle)
(FPCore (angle_s a b angle_m)
 :precision binary64
 (let* ((t_0 (* (* angle_m PI) 0.011111111111111112)))
   (*
    angle_s
    (if (<= a 4.5e-118) (* (* b b) (sin t_0)) (* (+ b a) (* (- b a) t_0))))))

C

angle\_m = fabs(angle);
angle\_s = copysign(1.0, angle);
double code(double angle_s, double a, double b, double angle_m) {
	double t_0 = (angle_m * ((double) M_PI)) * 0.011111111111111112;
	double tmp;
	if (a <= 4.5e-118) {
		tmp = (b * b) * sin(t_0);
	} else {
		tmp = (b + a) * ((b - a) * t_0);
	}
	return angle_s * tmp;
}

Java

angle\_m = Math.abs(angle);
angle\_s = Math.copySign(1.0, angle);
public static double code(double angle_s, double a, double b, double angle_m) {
	double t_0 = (angle_m * Math.PI) * 0.011111111111111112;
	double tmp;
	if (a <= 4.5e-118) {
		tmp = (b * b) * Math.sin(t_0);
	} else {
		tmp = (b + a) * ((b - a) * t_0);
	}
	return angle_s * tmp;
}

Python

angle\_m = math.fabs(angle)
angle\_s = math.copysign(1.0, angle)
def code(angle_s, a, b, angle_m):
	t_0 = (angle_m * math.pi) * 0.011111111111111112
	tmp = 0
	if a <= 4.5e-118:
		tmp = (b * b) * math.sin(t_0)
	else:
		tmp = (b + a) * ((b - a) * t_0)
	return angle_s * tmp

Julia

angle\_m = abs(angle)
angle\_s = copysign(1.0, angle)
function code(angle_s, a, b, angle_m)
	t_0 = Float64(Float64(angle_m * pi) * 0.011111111111111112)
	tmp = 0.0
	if (a <= 4.5e-118)
		tmp = Float64(Float64(b * b) * sin(t_0));
	else
		tmp = Float64(Float64(b + a) * Float64(Float64(b - a) * t_0));
	end
	return Float64(angle_s * tmp)
end

MATLAB

angle\_m = abs(angle);
angle\_s = sign(angle) * abs(1.0);
function tmp_2 = code(angle_s, a, b, angle_m)
	t_0 = (angle_m * pi) * 0.011111111111111112;
	tmp = 0.0;
	if (a <= 4.5e-118)
		tmp = (b * b) * sin(t_0);
	else
		tmp = (b + a) * ((b - a) * t_0);
	end
	tmp_2 = angle_s * tmp;
end

Wolfram

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_, b_, angle$95$m_] := Block[{t$95$0 = N[(N[(angle$95$m * Pi), $MachinePrecision] * 0.011111111111111112), $MachinePrecision]}, N[(angle$95$s * If[LessEqual[a, 4.5e-118], N[(N[(b * b), $MachinePrecision] * N[Sin[t$95$0], $MachinePrecision]), $MachinePrecision], N[(N[(b + a), $MachinePrecision] * N[(N[(b - a), $MachinePrecision] * t$95$0), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if a < 4.5e-118

   1. Initial program 58.6%

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

      1. associate-*l* N/A

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

      2. *-commutative N/A

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

      3. associate-*l* N/A

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

      4. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\sin \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right), \color{blue}{\left(\cos \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right) \cdot \left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right)\right)}\right) \]

      5. sin-lowering-sin.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sin.f64}\left(\left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right)\right), \left(\color{blue}{\cos \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right)} \cdot \left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right)\right)\right) \]

      6. associate-*r/ N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sin.f64}\left(\left(\frac{\mathsf{PI}\left(\right) \cdot angle}{180}\right)\right), \left(\cos \color{blue}{\left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right)} \cdot \left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right)\right)\right) \]

      7. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sin.f64}\left(\mathsf{/.f64}\left(\left(\mathsf{PI}\left(\right) \cdot angle\right), 180\right)\right), \left(\cos \color{blue}{\left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right)} \cdot \left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right)\right)\right) \]

      8. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sin.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{PI}\left(\right), angle\right), 180\right)\right), \left(\cos \left(\color{blue}{\mathsf{PI}\left(\right)} \cdot \frac{angle}{180}\right) \cdot \left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right)\right)\right) \]

      9. PI-lowering-PI.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sin.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{PI.f64}\left(\right), angle\right), 180\right)\right), \left(\cos \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right) \cdot \left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right)\right)\right) \]

      10. *-commutative N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{sin.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{PI.f64}\left(\right), angle\right), 180\right)\right), \left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \color{blue}{\cos \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right)}\right)\right) \]

      11. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{sin.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{PI.f64}\left(\right), angle\right), 180\right)\right), \mathsf{*.f64}\left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right), \color{blue}{\cos \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right)}\right)\right) \]

   3. Simplified 58.0%

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

   6. Applied egg-rr 20.1%

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

      1. *-commutative N/A

         \[\leadsto \frac{\sin \left(\left(\mathsf{PI}\left(\right) \cdot angle\right) \cdot \frac{1}{90}\right) \cdot \left(b \cdot \left(b \cdot \left(b \cdot b\right)\right) - a \cdot \left(a \cdot \left(a \cdot a\right)\right)\right)}{\color{blue}{b \cdot b} + a \cdot a} \]

      2. flip-+ N/A

         \[\leadsto \frac{\sin \left(\left(\mathsf{PI}\left(\right) \cdot angle\right) \cdot \frac{1}{90}\right) \cdot \left(b \cdot \left(b \cdot \left(b \cdot b\right)\right) - a \cdot \left(a \cdot \left(a \cdot a\right)\right)\right)}{\frac{\left(b \cdot b\right) \cdot \left(b \cdot b\right) - \left(a \cdot a\right) \cdot \left(a \cdot a\right)}{\color{blue}{b \cdot b - a \cdot a}}} \]

      3. associate-*r* N/A

         \[\leadsto \frac{\sin \left(\left(\mathsf{PI}\left(\right) \cdot angle\right) \cdot \frac{1}{90}\right) \cdot \left(b \cdot \left(b \cdot \left(b \cdot b\right)\right) - a \cdot \left(a \cdot \left(a \cdot a\right)\right)\right)}{\frac{b \cdot \left(b \cdot \left(b \cdot b\right)\right) - \left(a \cdot a\right) \cdot \left(a \cdot a\right)}{\color{blue}{b} \cdot b - a \cdot a}} \]

      4. associate-*r* N/A

         \[\leadsto \frac{\sin \left(\left(\mathsf{PI}\left(\right) \cdot angle\right) \cdot \frac{1}{90}\right) \cdot \left(b \cdot \left(b \cdot \left(b \cdot b\right)\right) - a \cdot \left(a \cdot \left(a \cdot a\right)\right)\right)}{\frac{b \cdot \left(b \cdot \left(b \cdot b\right)\right) - a \cdot \left(a \cdot \left(a \cdot a\right)\right)}{b \cdot \color{blue}{b} - a \cdot a}} \]

      5. div-inv N/A

         \[\leadsto \frac{\sin \left(\left(\mathsf{PI}\left(\right) \cdot angle\right) \cdot \frac{1}{90}\right) \cdot \left(b \cdot \left(b \cdot \left(b \cdot b\right)\right) - a \cdot \left(a \cdot \left(a \cdot a\right)\right)\right)}{\left(b \cdot \left(b \cdot \left(b \cdot b\right)\right) - a \cdot \left(a \cdot \left(a \cdot a\right)\right)\right) \cdot \color{blue}{\frac{1}{b \cdot b - a \cdot a}}} \]

      6. times-frac N/A

         \[\leadsto \frac{\sin \left(\left(\mathsf{PI}\left(\right) \cdot angle\right) \cdot \frac{1}{90}\right)}{b \cdot \left(b \cdot \left(b \cdot b\right)\right) - a \cdot \left(a \cdot \left(a \cdot a\right)\right)} \cdot \color{blue}{\frac{b \cdot \left(b \cdot \left(b \cdot b\right)\right) - a \cdot \left(a \cdot \left(a \cdot a\right)\right)}{\frac{1}{b \cdot b - a \cdot a}}} \]

      7. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\left(\frac{\sin \left(\left(\mathsf{PI}\left(\right) \cdot angle\right) \cdot \frac{1}{90}\right)}{b \cdot \left(b \cdot \left(b \cdot b\right)\right) - a \cdot \left(a \cdot \left(a \cdot a\right)\right)}\right), \color{blue}{\left(\frac{b \cdot \left(b \cdot \left(b \cdot b\right)\right) - a \cdot \left(a \cdot \left(a \cdot a\right)\right)}{\frac{1}{b \cdot b - a \cdot a}}\right)}\right) \]

   8. Applied egg-rr 7.8%

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

   9. Taylor expanded in b around inf

      \[\leadsto \color{blue}{{b}^{2} \cdot \sin \left(\frac{1}{90} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)} \]
   10. Step-by-step derivation

       1. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\left({b}^{2}\right), \color{blue}{\sin \left(\frac{1}{90} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}\right) \]

       2. unpow2 N/A

          \[\leadsto \mathsf{*.f64}\left(\left(b \cdot b\right), \sin \color{blue}{\left(\frac{1}{90} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}\right) \]

       3. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(b, b\right), \sin \color{blue}{\left(\frac{1}{90} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}\right) \]

       4. *-commutative N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(b, b\right), \sin \left(\left(angle \cdot \mathsf{PI}\left(\right)\right) \cdot \frac{1}{90}\right)\right) \]

       5. associate-*r* N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(b, b\right), \sin \left(angle \cdot \left(\mathsf{PI}\left(\right) \cdot \frac{1}{90}\right)\right)\right) \]

       6. *-commutative N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(b, b\right), \sin \left(angle \cdot \left(\frac{1}{90} \cdot \mathsf{PI}\left(\right)\right)\right)\right) \]

       7. sin-lowering-sin.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(b, b\right), \mathsf{sin.f64}\left(\left(angle \cdot \left(\frac{1}{90} \cdot \mathsf{PI}\left(\right)\right)\right)\right)\right) \]

       8. *-commutative N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(b, b\right), \mathsf{sin.f64}\left(\left(angle \cdot \left(\mathsf{PI}\left(\right) \cdot \frac{1}{90}\right)\right)\right)\right) \]

       9. associate-*r* N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(b, b\right), \mathsf{sin.f64}\left(\left(\left(angle \cdot \mathsf{PI}\left(\right)\right) \cdot \frac{1}{90}\right)\right)\right) \]

       10. *-commutative N/A

           \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(b, b\right), \mathsf{sin.f64}\left(\left(\frac{1}{90} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)\right)\right) \]

       11. *-lowering-*.f64 N/A

           \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(b, b\right), \mathsf{sin.f64}\left(\mathsf{*.f64}\left(\frac{1}{90}, \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)\right)\right) \]

       12. *-lowering-*.f64 N/A

           \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(b, b\right), \mathsf{sin.f64}\left(\mathsf{*.f64}\left(\frac{1}{90}, \mathsf{*.f64}\left(angle, \mathsf{PI}\left(\right)\right)\right)\right)\right) \]

       13. PI-lowering-PI.f64 45.6%

           \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(b, b\right), \mathsf{sin.f64}\left(\mathsf{*.f64}\left(\frac{1}{90}, \mathsf{*.f64}\left(angle, \mathsf{PI.f64}\left(\right)\right)\right)\right)\right) \]

   11. Simplified 45.6%

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

   if 4.5e-118 < a

   1. Initial program 46.1%

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

      1. associate-*l* N/A

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

      2. *-commutative N/A

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

      3. associate-*l* N/A

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

      4. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\sin \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right), \color{blue}{\left(\cos \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right) \cdot \left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right)\right)}\right) \]

      5. sin-lowering-sin.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sin.f64}\left(\left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right)\right), \left(\color{blue}{\cos \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right)} \cdot \left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right)\right)\right) \]

      6. associate-*r/ N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sin.f64}\left(\left(\frac{\mathsf{PI}\left(\right) \cdot angle}{180}\right)\right), \left(\cos \color{blue}{\left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right)} \cdot \left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right)\right)\right) \]

      7. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sin.f64}\left(\mathsf{/.f64}\left(\left(\mathsf{PI}\left(\right) \cdot angle\right), 180\right)\right), \left(\cos \color{blue}{\left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right)} \cdot \left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right)\right)\right) \]

      8. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sin.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{PI}\left(\right), angle\right), 180\right)\right), \left(\cos \left(\color{blue}{\mathsf{PI}\left(\right)} \cdot \frac{angle}{180}\right) \cdot \left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right)\right)\right) \]

      9. PI-lowering-PI.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sin.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{PI.f64}\left(\right), angle\right), 180\right)\right), \left(\cos \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right) \cdot \left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right)\right)\right) \]

      10. *-commutative N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{sin.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{PI.f64}\left(\right), angle\right), 180\right)\right), \left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \color{blue}{\cos \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right)}\right)\right) \]

      11. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{sin.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{PI.f64}\left(\right), angle\right), 180\right)\right), \mathsf{*.f64}\left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right), \color{blue}{\cos \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right)}\right)\right) \]

   3. Simplified 47.6%

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

   5. Taylor expanded in angle around 0

      \[\leadsto \color{blue}{\frac{1}{90} \cdot \left(angle \cdot \left(\mathsf{PI}\left(\right) \cdot \left({b}^{2} - {a}^{2}\right)\right)\right)} \]
   6. Step-by-step derivation

      1. associate-*r* N/A

         \[\leadsto \frac{1}{90} \cdot \left(\left(angle \cdot \mathsf{PI}\left(\right)\right) \cdot \color{blue}{\left({b}^{2} - {a}^{2}\right)}\right) \]

      2. associate-*r* N/A

         \[\leadsto \left(\frac{1}{90} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right) \cdot \color{blue}{\left({b}^{2} - {a}^{2}\right)} \]

      3. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\left(\frac{1}{90} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right), \color{blue}{\left({b}^{2} - {a}^{2}\right)}\right) \]

      4. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{1}{90}, \left(angle \cdot \mathsf{PI}\left(\right)\right)\right), \left(\color{blue}{{b}^{2}} - {a}^{2}\right)\right) \]

      5. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{1}{90}, \mathsf{*.f64}\left(angle, \mathsf{PI}\left(\right)\right)\right), \left({b}^{\color{blue}{2}} - {a}^{2}\right)\right) \]

      6. PI-lowering-PI.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{1}{90}, \mathsf{*.f64}\left(angle, \mathsf{PI.f64}\left(\right)\right)\right), \left({b}^{2} - {a}^{2}\right)\right) \]

      7. --lowering--.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{1}{90}, \mathsf{*.f64}\left(angle, \mathsf{PI.f64}\left(\right)\right)\right), \mathsf{\_.f64}\left(\left({b}^{2}\right), \color{blue}{\left({a}^{2}\right)}\right)\right) \]

      8. unpow2 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{1}{90}, \mathsf{*.f64}\left(angle, \mathsf{PI.f64}\left(\right)\right)\right), \mathsf{\_.f64}\left(\left(b \cdot b\right), \left({\color{blue}{a}}^{2}\right)\right)\right) \]

      9. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{1}{90}, \mathsf{*.f64}\left(angle, \mathsf{PI.f64}\left(\right)\right)\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(b, b\right), \left({\color{blue}{a}}^{2}\right)\right)\right) \]

      10. unpow2 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{1}{90}, \mathsf{*.f64}\left(angle, \mathsf{PI.f64}\left(\right)\right)\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(b, b\right), \left(a \cdot \color{blue}{a}\right)\right)\right) \]

      11. *-lowering-*.f64 45.8%

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{1}{90}, \mathsf{*.f64}\left(angle, \mathsf{PI.f64}\left(\right)\right)\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(b, b\right), \mathsf{*.f64}\left(a, \color{blue}{a}\right)\right)\right) \]

   7. Simplified 45.8%

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

      1. *-commutative N/A

         \[\leadsto \left(b \cdot b - a \cdot a\right) \cdot \color{blue}{\left(\frac{1}{90} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)} \]

      2. difference-of-squares N/A

         \[\leadsto \left(\left(b + a\right) \cdot \left(b - a\right)\right) \cdot \left(\color{blue}{\frac{1}{90}} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right) \]

      3. associate-*l* N/A

         \[\leadsto \left(b + a\right) \cdot \color{blue}{\left(\left(b - a\right) \cdot \left(\frac{1}{90} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)\right)} \]

      4. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\left(b + a\right), \color{blue}{\left(\left(b - a\right) \cdot \left(\frac{1}{90} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)\right)}\right) \]

      5. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(b, a\right), \left(\color{blue}{\left(b - a\right)} \cdot \left(\frac{1}{90} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)\right)\right) \]

      6. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(b, a\right), \mathsf{*.f64}\left(\left(b - a\right), \color{blue}{\left(\frac{1}{90} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}\right)\right) \]

      7. --lowering--.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(b, a\right), \mathsf{*.f64}\left(\mathsf{\_.f64}\left(b, a\right), \left(\color{blue}{\frac{1}{90}} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)\right)\right) \]

      8. *-commutative N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(b, a\right), \mathsf{*.f64}\left(\mathsf{\_.f64}\left(b, a\right), \left(\frac{1}{90} \cdot \left(\mathsf{PI}\left(\right) \cdot \color{blue}{angle}\right)\right)\right)\right) \]

      9. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(b, a\right), \mathsf{*.f64}\left(\mathsf{\_.f64}\left(b, a\right), \mathsf{*.f64}\left(\frac{1}{90}, \color{blue}{\left(\mathsf{PI}\left(\right) \cdot angle\right)}\right)\right)\right) \]

      10. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(b, a\right), \mathsf{*.f64}\left(\mathsf{\_.f64}\left(b, a\right), \mathsf{*.f64}\left(\frac{1}{90}, \mathsf{*.f64}\left(\mathsf{PI}\left(\right), \color{blue}{angle}\right)\right)\right)\right) \]

      11. PI-lowering-PI.f64 67.4%

          \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(b, a\right), \mathsf{*.f64}\left(\mathsf{\_.f64}\left(b, a\right), \mathsf{*.f64}\left(\frac{1}{90}, \mathsf{*.f64}\left(\mathsf{PI.f64}\left(\right), angle\right)\right)\right)\right) \]

   9. Applied egg-rr 67.4%

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

4. Final simplification 52.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \leq 4.5 \cdot 10^{-118}:\\ \;\;\;\;\left(b \cdot b\right) \cdot \sin \left(\left(angle \cdot \pi\right) \cdot 0.011111111111111112\right)\\ \mathbf{else}:\\ \;\;\;\;\left(b + a\right) \cdot \left(\left(b - a\right) \cdot \left(\left(angle \cdot \pi\right) \cdot 0.011111111111111112\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 12: 67.8% accurate, 3.7× speedup

Math

\[\begin{array}{l} angle\_m = \left|angle\right| \\ angle\_s = \mathsf{copysign}\left(1, angle\right) \\ angle\_s \cdot \left(\left(b - a\right) \cdot \left(\left(b + a\right) \cdot \sin \left(\left(angle\_m \cdot \pi\right) \cdot 0.011111111111111112\right)\right)\right) \end{array} \]

FPCore

angle\_m = (fabs.f64 angle)
angle\_s = (copysign.f64 #s(literal 1 binary64) angle)
(FPCore (angle_s a b angle_m)
 :precision binary64
 (*
  angle_s
  (* (- b a) (* (+ b a) (sin (* (* angle_m PI) 0.011111111111111112))))))

C

angle\_m = fabs(angle);
angle\_s = copysign(1.0, angle);
double code(double angle_s, double a, double b, double angle_m) {
	return angle_s * ((b - a) * ((b + a) * sin(((angle_m * ((double) M_PI)) * 0.011111111111111112))));
}

Java

angle\_m = Math.abs(angle);
angle\_s = Math.copySign(1.0, angle);
public static double code(double angle_s, double a, double b, double angle_m) {
	return angle_s * ((b - a) * ((b + a) * Math.sin(((angle_m * Math.PI) * 0.011111111111111112))));
}

Python

angle\_m = math.fabs(angle)
angle\_s = math.copysign(1.0, angle)
def code(angle_s, a, b, angle_m):
	return angle_s * ((b - a) * ((b + a) * math.sin(((angle_m * math.pi) * 0.011111111111111112))))

Julia

angle\_m = abs(angle)
angle\_s = copysign(1.0, angle)
function code(angle_s, a, b, angle_m)
	return Float64(angle_s * Float64(Float64(b - a) * Float64(Float64(b + a) * sin(Float64(Float64(angle_m * pi) * 0.011111111111111112)))))
end

MATLAB

angle\_m = abs(angle);
angle\_s = sign(angle) * abs(1.0);
function tmp = code(angle_s, a, b, angle_m)
	tmp = angle_s * ((b - a) * ((b + a) * sin(((angle_m * pi) * 0.011111111111111112))));
end

Wolfram

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_, b_, angle$95$m_] := N[(angle$95$s * N[(N[(b - a), $MachinePrecision] * N[(N[(b + a), $MachinePrecision] * N[Sin[N[(N[(angle$95$m * Pi), $MachinePrecision] * 0.011111111111111112), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 54.6%

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

   1. associate-*l* N/A

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

   2. *-commutative N/A

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

   3. associate-*l* N/A

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

   4. *-lowering-*.f64 N/A

      \[\leadsto \mathsf{*.f64}\left(\sin \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right), \color{blue}{\left(\cos \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right) \cdot \left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right)\right)}\right) \]

   5. sin-lowering-sin.f64 N/A

      \[\leadsto \mathsf{*.f64}\left(\mathsf{sin.f64}\left(\left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right)\right), \left(\color{blue}{\cos \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right)} \cdot \left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right)\right)\right) \]

   6. associate-*r/ N/A

      \[\leadsto \mathsf{*.f64}\left(\mathsf{sin.f64}\left(\left(\frac{\mathsf{PI}\left(\right) \cdot angle}{180}\right)\right), \left(\cos \color{blue}{\left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right)} \cdot \left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right)\right)\right) \]

   7. /-lowering-/.f64 N/A

      \[\leadsto \mathsf{*.f64}\left(\mathsf{sin.f64}\left(\mathsf{/.f64}\left(\left(\mathsf{PI}\left(\right) \cdot angle\right), 180\right)\right), \left(\cos \color{blue}{\left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right)} \cdot \left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right)\right)\right) \]

   8. *-lowering-*.f64 N/A

      \[\leadsto \mathsf{*.f64}\left(\mathsf{sin.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{PI}\left(\right), angle\right), 180\right)\right), \left(\cos \left(\color{blue}{\mathsf{PI}\left(\right)} \cdot \frac{angle}{180}\right) \cdot \left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right)\right)\right) \]

   9. PI-lowering-PI.f64 N/A

      \[\leadsto \mathsf{*.f64}\left(\mathsf{sin.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{PI.f64}\left(\right), angle\right), 180\right)\right), \left(\cos \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right) \cdot \left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right)\right)\right) \]

   10. *-commutative N/A

       \[\leadsto \mathsf{*.f64}\left(\mathsf{sin.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{PI.f64}\left(\right), angle\right), 180\right)\right), \left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \color{blue}{\cos \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right)}\right)\right) \]

   11. *-lowering-*.f64 N/A

       \[\leadsto \mathsf{*.f64}\left(\mathsf{sin.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{PI.f64}\left(\right), angle\right), 180\right)\right), \mathsf{*.f64}\left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right), \color{blue}{\cos \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right)}\right)\right) \]

3. Simplified 54.7%

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

   1. associate-*r/ N/A

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

   2. *-commutative N/A

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

   3. associate-*r* N/A

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

   4. associate-*r/ N/A

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

   5. associate-*r* N/A

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

   6. *-commutative N/A

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

   7. *-commutative N/A

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

   8. difference-of-squares N/A

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

6. Applied egg-rr 68.2%

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

7. Final simplification 68.2%

   \[\leadsto \left(b - a\right) \cdot \left(\left(b + a\right) \cdot \sin \left(\left(angle \cdot \pi\right) \cdot 0.011111111111111112\right)\right) \]
8. Add Preprocessing

Alternative 13: 54.2% accurate, 23.3× speedup

Math

\[\begin{array}{l} angle\_m = \left|angle\right| \\ angle\_s = \mathsf{copysign}\left(1, angle\right) \\ angle\_s \cdot \begin{array}{l} \mathbf{if}\;a \leq 1.35 \cdot 10^{+154}:\\ \;\;\;\;\left(angle\_m \cdot \pi\right) \cdot \left(0.011111111111111112 \cdot \left(b \cdot b - a \cdot a\right)\right)\\ \mathbf{else}:\\ \;\;\;\;a \cdot \left(\left(a \cdot \left(angle\_m \cdot \pi\right)\right) \cdot -0.011111111111111112\right)\\ \end{array} \end{array} \]

FPCore

angle\_m = (fabs.f64 angle)
angle\_s = (copysign.f64 #s(literal 1 binary64) angle)
(FPCore (angle_s a b angle_m)
 :precision binary64
 (*
  angle_s
  (if (<= a 1.35e+154)
    (* (* angle_m PI) (* 0.011111111111111112 (- (* b b) (* a a))))
    (* a (* (* a (* angle_m PI)) -0.011111111111111112)))))

C

angle\_m = fabs(angle);
angle\_s = copysign(1.0, angle);
double code(double angle_s, double a, double b, double angle_m) {
	double tmp;
	if (a <= 1.35e+154) {
		tmp = (angle_m * ((double) M_PI)) * (0.011111111111111112 * ((b * b) - (a * a)));
	} else {
		tmp = a * ((a * (angle_m * ((double) M_PI))) * -0.011111111111111112);
	}
	return angle_s * tmp;
}

Java

angle\_m = Math.abs(angle);
angle\_s = Math.copySign(1.0, angle);
public static double code(double angle_s, double a, double b, double angle_m) {
	double tmp;
	if (a <= 1.35e+154) {
		tmp = (angle_m * Math.PI) * (0.011111111111111112 * ((b * b) - (a * a)));
	} else {
		tmp = a * ((a * (angle_m * Math.PI)) * -0.011111111111111112);
	}
	return angle_s * tmp;
}

Python

angle\_m = math.fabs(angle)
angle\_s = math.copysign(1.0, angle)
def code(angle_s, a, b, angle_m):
	tmp = 0
	if a <= 1.35e+154:
		tmp = (angle_m * math.pi) * (0.011111111111111112 * ((b * b) - (a * a)))
	else:
		tmp = a * ((a * (angle_m * math.pi)) * -0.011111111111111112)
	return angle_s * tmp

Julia

angle\_m = abs(angle)
angle\_s = copysign(1.0, angle)
function code(angle_s, a, b, angle_m)
	tmp = 0.0
	if (a <= 1.35e+154)
		tmp = Float64(Float64(angle_m * pi) * Float64(0.011111111111111112 * Float64(Float64(b * b) - Float64(a * a))));
	else
		tmp = Float64(a * Float64(Float64(a * Float64(angle_m * pi)) * -0.011111111111111112));
	end
	return Float64(angle_s * tmp)
end

MATLAB

angle\_m = abs(angle);
angle\_s = sign(angle) * abs(1.0);
function tmp_2 = code(angle_s, a, b, angle_m)
	tmp = 0.0;
	if (a <= 1.35e+154)
		tmp = (angle_m * pi) * (0.011111111111111112 * ((b * b) - (a * a)));
	else
		tmp = a * ((a * (angle_m * pi)) * -0.011111111111111112);
	end
	tmp_2 = angle_s * tmp;
end

Wolfram

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_, b_, angle$95$m_] := N[(angle$95$s * If[LessEqual[a, 1.35e+154], N[(N[(angle$95$m * Pi), $MachinePrecision] * N[(0.011111111111111112 * N[(N[(b * b), $MachinePrecision] - N[(a * a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(a * N[(N[(a * N[(angle$95$m * Pi), $MachinePrecision]), $MachinePrecision] * -0.011111111111111112), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if a < 1.35000000000000003e154

   1. Initial program 56.7%

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

      1. associate-*l* N/A

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

      2. *-commutative N/A

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

      3. associate-*l* N/A

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

      4. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\sin \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right), \color{blue}{\left(\cos \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right) \cdot \left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right)\right)}\right) \]

      5. sin-lowering-sin.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sin.f64}\left(\left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right)\right), \left(\color{blue}{\cos \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right)} \cdot \left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right)\right)\right) \]

      6. associate-*r/ N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sin.f64}\left(\left(\frac{\mathsf{PI}\left(\right) \cdot angle}{180}\right)\right), \left(\cos \color{blue}{\left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right)} \cdot \left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right)\right)\right) \]

      7. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sin.f64}\left(\mathsf{/.f64}\left(\left(\mathsf{PI}\left(\right) \cdot angle\right), 180\right)\right), \left(\cos \color{blue}{\left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right)} \cdot \left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right)\right)\right) \]

      8. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sin.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{PI}\left(\right), angle\right), 180\right)\right), \left(\cos \left(\color{blue}{\mathsf{PI}\left(\right)} \cdot \frac{angle}{180}\right) \cdot \left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right)\right)\right) \]

      9. PI-lowering-PI.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sin.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{PI.f64}\left(\right), angle\right), 180\right)\right), \left(\cos \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right) \cdot \left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right)\right)\right) \]

      10. *-commutative N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{sin.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{PI.f64}\left(\right), angle\right), 180\right)\right), \left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \color{blue}{\cos \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right)}\right)\right) \]

      11. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{sin.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{PI.f64}\left(\right), angle\right), 180\right)\right), \mathsf{*.f64}\left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right), \color{blue}{\cos \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right)}\right)\right) \]

   3. Simplified 56.8%

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

   5. Taylor expanded in angle around 0

      \[\leadsto \color{blue}{\frac{1}{90} \cdot \left(angle \cdot \left(\mathsf{PI}\left(\right) \cdot \left({b}^{2} - {a}^{2}\right)\right)\right)} \]
   6. Step-by-step derivation

      1. associate-*r* N/A

         \[\leadsto \frac{1}{90} \cdot \left(\left(angle \cdot \mathsf{PI}\left(\right)\right) \cdot \color{blue}{\left({b}^{2} - {a}^{2}\right)}\right) \]

      2. associate-*r* N/A

         \[\leadsto \left(\frac{1}{90} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right) \cdot \color{blue}{\left({b}^{2} - {a}^{2}\right)} \]

      3. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\left(\frac{1}{90} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right), \color{blue}{\left({b}^{2} - {a}^{2}\right)}\right) \]

      4. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{1}{90}, \left(angle \cdot \mathsf{PI}\left(\right)\right)\right), \left(\color{blue}{{b}^{2}} - {a}^{2}\right)\right) \]

      5. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{1}{90}, \mathsf{*.f64}\left(angle, \mathsf{PI}\left(\right)\right)\right), \left({b}^{\color{blue}{2}} - {a}^{2}\right)\right) \]

      6. PI-lowering-PI.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{1}{90}, \mathsf{*.f64}\left(angle, \mathsf{PI.f64}\left(\right)\right)\right), \left({b}^{2} - {a}^{2}\right)\right) \]

      7. --lowering--.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{1}{90}, \mathsf{*.f64}\left(angle, \mathsf{PI.f64}\left(\right)\right)\right), \mathsf{\_.f64}\left(\left({b}^{2}\right), \color{blue}{\left({a}^{2}\right)}\right)\right) \]

      8. unpow2 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{1}{90}, \mathsf{*.f64}\left(angle, \mathsf{PI.f64}\left(\right)\right)\right), \mathsf{\_.f64}\left(\left(b \cdot b\right), \left({\color{blue}{a}}^{2}\right)\right)\right) \]

      9. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{1}{90}, \mathsf{*.f64}\left(angle, \mathsf{PI.f64}\left(\right)\right)\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(b, b\right), \left({\color{blue}{a}}^{2}\right)\right)\right) \]

      10. unpow2 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{1}{90}, \mathsf{*.f64}\left(angle, \mathsf{PI.f64}\left(\right)\right)\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(b, b\right), \left(a \cdot \color{blue}{a}\right)\right)\right) \]

      11. *-lowering-*.f64 53.6%

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{1}{90}, \mathsf{*.f64}\left(angle, \mathsf{PI.f64}\left(\right)\right)\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(b, b\right), \mathsf{*.f64}\left(a, \color{blue}{a}\right)\right)\right) \]

   7. Simplified 53.6%

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

      1. *-commutative N/A

         \[\leadsto \left(b \cdot b - a \cdot a\right) \cdot \color{blue}{\left(\frac{1}{90} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)} \]

      2. *-commutative N/A

         \[\leadsto \left(b \cdot b - a \cdot a\right) \cdot \left(\frac{1}{90} \cdot \left(\mathsf{PI}\left(\right) \cdot \color{blue}{angle}\right)\right) \]

      3. associate-*r* N/A

         \[\leadsto \left(\left(b \cdot b - a \cdot a\right) \cdot \frac{1}{90}\right) \cdot \color{blue}{\left(\mathsf{PI}\left(\right) \cdot angle\right)} \]

      4. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\left(\left(b \cdot b - a \cdot a\right) \cdot \frac{1}{90}\right), \color{blue}{\left(\mathsf{PI}\left(\right) \cdot angle\right)}\right) \]

      5. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\left(b \cdot b - a \cdot a\right), \frac{1}{90}\right), \left(\color{blue}{\mathsf{PI}\left(\right)} \cdot angle\right)\right) \]

      6. --lowering--.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{\_.f64}\left(\left(b \cdot b\right), \left(a \cdot a\right)\right), \frac{1}{90}\right), \left(\mathsf{PI}\left(\right) \cdot angle\right)\right) \]

      7. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(b, b\right), \left(a \cdot a\right)\right), \frac{1}{90}\right), \left(\mathsf{PI}\left(\right) \cdot angle\right)\right) \]

      8. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(b, b\right), \mathsf{*.f64}\left(a, a\right)\right), \frac{1}{90}\right), \left(\mathsf{PI}\left(\right) \cdot angle\right)\right) \]

      9. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(b, b\right), \mathsf{*.f64}\left(a, a\right)\right), \frac{1}{90}\right), \mathsf{*.f64}\left(\mathsf{PI}\left(\right), \color{blue}{angle}\right)\right) \]

      10. PI-lowering-PI.f64 54.0%

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(b, b\right), \mathsf{*.f64}\left(a, a\right)\right), \frac{1}{90}\right), \mathsf{*.f64}\left(\mathsf{PI.f64}\left(\right), angle\right)\right) \]

   9. Applied egg-rr 54.0%

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

   if 1.35000000000000003e154 < a

   1. Initial program 35.5%

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

      1. associate-*l* N/A

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

      2. *-commutative N/A

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

      3. associate-*l* N/A

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

      4. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\sin \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right), \color{blue}{\left(\cos \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right) \cdot \left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right)\right)}\right) \]

      5. sin-lowering-sin.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sin.f64}\left(\left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right)\right), \left(\color{blue}{\cos \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right)} \cdot \left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right)\right)\right) \]

      6. associate-*r/ N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sin.f64}\left(\left(\frac{\mathsf{PI}\left(\right) \cdot angle}{180}\right)\right), \left(\cos \color{blue}{\left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right)} \cdot \left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right)\right)\right) \]

      7. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sin.f64}\left(\mathsf{/.f64}\left(\left(\mathsf{PI}\left(\right) \cdot angle\right), 180\right)\right), \left(\cos \color{blue}{\left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right)} \cdot \left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right)\right)\right) \]

      8. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sin.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{PI}\left(\right), angle\right), 180\right)\right), \left(\cos \left(\color{blue}{\mathsf{PI}\left(\right)} \cdot \frac{angle}{180}\right) \cdot \left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right)\right)\right) \]

      9. PI-lowering-PI.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sin.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{PI.f64}\left(\right), angle\right), 180\right)\right), \left(\cos \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right) \cdot \left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right)\right)\right) \]

      10. *-commutative N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{sin.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{PI.f64}\left(\right), angle\right), 180\right)\right), \left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \color{blue}{\cos \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right)}\right)\right) \]

      11. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{sin.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{PI.f64}\left(\right), angle\right), 180\right)\right), \mathsf{*.f64}\left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right), \color{blue}{\cos \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right)}\right)\right) \]

   3. Simplified 35.5%

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

   5. Taylor expanded in angle around 0

      \[\leadsto \color{blue}{\frac{1}{90} \cdot \left(angle \cdot \left(\mathsf{PI}\left(\right) \cdot \left({b}^{2} - {a}^{2}\right)\right)\right)} \]
   6. Step-by-step derivation

      1. associate-*r* N/A

         \[\leadsto \frac{1}{90} \cdot \left(\left(angle \cdot \mathsf{PI}\left(\right)\right) \cdot \color{blue}{\left({b}^{2} - {a}^{2}\right)}\right) \]

      2. associate-*r* N/A

         \[\leadsto \left(\frac{1}{90} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right) \cdot \color{blue}{\left({b}^{2} - {a}^{2}\right)} \]

      3. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\left(\frac{1}{90} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right), \color{blue}{\left({b}^{2} - {a}^{2}\right)}\right) \]

      4. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{1}{90}, \left(angle \cdot \mathsf{PI}\left(\right)\right)\right), \left(\color{blue}{{b}^{2}} - {a}^{2}\right)\right) \]

      5. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{1}{90}, \mathsf{*.f64}\left(angle, \mathsf{PI}\left(\right)\right)\right), \left({b}^{\color{blue}{2}} - {a}^{2}\right)\right) \]

      6. PI-lowering-PI.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{1}{90}, \mathsf{*.f64}\left(angle, \mathsf{PI.f64}\left(\right)\right)\right), \left({b}^{2} - {a}^{2}\right)\right) \]

      7. --lowering--.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{1}{90}, \mathsf{*.f64}\left(angle, \mathsf{PI.f64}\left(\right)\right)\right), \mathsf{\_.f64}\left(\left({b}^{2}\right), \color{blue}{\left({a}^{2}\right)}\right)\right) \]

      8. unpow2 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{1}{90}, \mathsf{*.f64}\left(angle, \mathsf{PI.f64}\left(\right)\right)\right), \mathsf{\_.f64}\left(\left(b \cdot b\right), \left({\color{blue}{a}}^{2}\right)\right)\right) \]

      9. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{1}{90}, \mathsf{*.f64}\left(angle, \mathsf{PI.f64}\left(\right)\right)\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(b, b\right), \left({\color{blue}{a}}^{2}\right)\right)\right) \]

      10. unpow2 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{1}{90}, \mathsf{*.f64}\left(angle, \mathsf{PI.f64}\left(\right)\right)\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(b, b\right), \left(a \cdot \color{blue}{a}\right)\right)\right) \]

      11. *-lowering-*.f64 31.5%

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{1}{90}, \mathsf{*.f64}\left(angle, \mathsf{PI.f64}\left(\right)\right)\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(b, b\right), \mathsf{*.f64}\left(a, \color{blue}{a}\right)\right)\right) \]

   7. Simplified 31.5%

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

   8. Taylor expanded in b around 0

      \[\leadsto \color{blue}{\frac{-1}{90} \cdot \left({a}^{2} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)} \]
   9. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \left({a}^{2} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right) \cdot \color{blue}{\frac{-1}{90}} \]

      2. associate-*r* N/A

         \[\leadsto {a}^{2} \cdot \color{blue}{\left(\left(angle \cdot \mathsf{PI}\left(\right)\right) \cdot \frac{-1}{90}\right)} \]

      3. *-commutative N/A

         \[\leadsto {a}^{2} \cdot \left(\frac{-1}{90} \cdot \color{blue}{\left(angle \cdot \mathsf{PI}\left(\right)\right)}\right) \]

      4. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\left({a}^{2}\right), \color{blue}{\left(\frac{-1}{90} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}\right) \]

      5. unpow2 N/A

         \[\leadsto \mathsf{*.f64}\left(\left(a \cdot a\right), \left(\color{blue}{\frac{-1}{90}} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)\right) \]

      6. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(a, a\right), \left(\color{blue}{\frac{-1}{90}} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)\right) \]

      7. *-commutative N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(a, a\right), \left(\left(angle \cdot \mathsf{PI}\left(\right)\right) \cdot \color{blue}{\frac{-1}{90}}\right)\right) \]

      8. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(a, a\right), \mathsf{*.f64}\left(\left(angle \cdot \mathsf{PI}\left(\right)\right), \color{blue}{\frac{-1}{90}}\right)\right) \]

      9. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(a, a\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(angle, \mathsf{PI}\left(\right)\right), \frac{-1}{90}\right)\right) \]

      10. PI-lowering-PI.f64 35.5%

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(a, a\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(angle, \mathsf{PI.f64}\left(\right)\right), \frac{-1}{90}\right)\right) \]

   10. Simplified 35.5%

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

       1. associate-*l* N/A

          \[\leadsto a \cdot \color{blue}{\left(a \cdot \left(\left(angle \cdot \mathsf{PI}\left(\right)\right) \cdot \frac{-1}{90}\right)\right)} \]

       2. *-commutative N/A

          \[\leadsto \left(a \cdot \left(\left(angle \cdot \mathsf{PI}\left(\right)\right) \cdot \frac{-1}{90}\right)\right) \cdot \color{blue}{a} \]

       3. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\left(a \cdot \left(\left(angle \cdot \mathsf{PI}\left(\right)\right) \cdot \frac{-1}{90}\right)\right), \color{blue}{a}\right) \]

       4. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(a, \left(\left(angle \cdot \mathsf{PI}\left(\right)\right) \cdot \frac{-1}{90}\right)\right), a\right) \]

       5. *-commutative N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(a, \left(\frac{-1}{90} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)\right), a\right) \]

       6. *-commutative N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(a, \left(\frac{-1}{90} \cdot \left(\mathsf{PI}\left(\right) \cdot angle\right)\right)\right), a\right) \]

       7. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(a, \mathsf{*.f64}\left(\frac{-1}{90}, \left(\mathsf{PI}\left(\right) \cdot angle\right)\right)\right), a\right) \]

       8. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(a, \mathsf{*.f64}\left(\frac{-1}{90}, \mathsf{*.f64}\left(\mathsf{PI}\left(\right), angle\right)\right)\right), a\right) \]

       9. PI-lowering-PI.f64 67.8%

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(a, \mathsf{*.f64}\left(\frac{-1}{90}, \mathsf{*.f64}\left(\mathsf{PI.f64}\left(\right), angle\right)\right)\right), a\right) \]

   12. Applied egg-rr 67.8%

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

       1. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\left(a \cdot \left(\frac{-1}{90} \cdot \left(\mathsf{PI}\left(\right) \cdot angle\right)\right)\right), \color{blue}{a}\right) \]

       2. *-commutative N/A

          \[\leadsto \mathsf{*.f64}\left(\left(a \cdot \left(\left(\mathsf{PI}\left(\right) \cdot angle\right) \cdot \frac{-1}{90}\right)\right), a\right) \]

       3. associate-*r* N/A

          \[\leadsto \mathsf{*.f64}\left(\left(\left(a \cdot \left(\mathsf{PI}\left(\right) \cdot angle\right)\right) \cdot \frac{-1}{90}\right), a\right) \]

       4. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\left(a \cdot \left(\mathsf{PI}\left(\right) \cdot angle\right)\right), \frac{-1}{90}\right), a\right) \]

       5. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(a, \left(\mathsf{PI}\left(\right) \cdot angle\right)\right), \frac{-1}{90}\right), a\right) \]

       6. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(a, \mathsf{*.f64}\left(\mathsf{PI}\left(\right), angle\right)\right), \frac{-1}{90}\right), a\right) \]

       7. PI-lowering-PI.f64 67.9%

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(a, \mathsf{*.f64}\left(\mathsf{PI.f64}\left(\right), angle\right)\right), \frac{-1}{90}\right), a\right) \]

   14. Applied egg-rr 67.9%

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

4. Final simplification 55.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \leq 1.35 \cdot 10^{+154}:\\ \;\;\;\;\left(angle \cdot \pi\right) \cdot \left(0.011111111111111112 \cdot \left(b \cdot b - a \cdot a\right)\right)\\ \mathbf{else}:\\ \;\;\;\;a \cdot \left(\left(a \cdot \left(angle \cdot \pi\right)\right) \cdot -0.011111111111111112\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 14: 54.2% accurate, 23.3× speedup

Math

\[\begin{array}{l} angle\_m = \left|angle\right| \\ angle\_s = \mathsf{copysign}\left(1, angle\right) \\ angle\_s \cdot \begin{array}{l} \mathbf{if}\;a \leq 1.32 \cdot 10^{+154}:\\ \;\;\;\;\left(b \cdot b - a \cdot a\right) \cdot \left(\pi \cdot \left(angle\_m \cdot 0.011111111111111112\right)\right)\\ \mathbf{else}:\\ \;\;\;\;a \cdot \left(\left(a \cdot \left(angle\_m \cdot \pi\right)\right) \cdot -0.011111111111111112\right)\\ \end{array} \end{array} \]

FPCore

angle\_m = (fabs.f64 angle)
angle\_s = (copysign.f64 #s(literal 1 binary64) angle)
(FPCore (angle_s a b angle_m)
 :precision binary64
 (*
  angle_s
  (if (<= a 1.32e+154)
    (* (- (* b b) (* a a)) (* PI (* angle_m 0.011111111111111112)))
    (* a (* (* a (* angle_m PI)) -0.011111111111111112)))))

C

angle\_m = fabs(angle);
angle\_s = copysign(1.0, angle);
double code(double angle_s, double a, double b, double angle_m) {
	double tmp;
	if (a <= 1.32e+154) {
		tmp = ((b * b) - (a * a)) * (((double) M_PI) * (angle_m * 0.011111111111111112));
	} else {
		tmp = a * ((a * (angle_m * ((double) M_PI))) * -0.011111111111111112);
	}
	return angle_s * tmp;
}

Java

angle\_m = Math.abs(angle);
angle\_s = Math.copySign(1.0, angle);
public static double code(double angle_s, double a, double b, double angle_m) {
	double tmp;
	if (a <= 1.32e+154) {
		tmp = ((b * b) - (a * a)) * (Math.PI * (angle_m * 0.011111111111111112));
	} else {
		tmp = a * ((a * (angle_m * Math.PI)) * -0.011111111111111112);
	}
	return angle_s * tmp;
}

Python

angle\_m = math.fabs(angle)
angle\_s = math.copysign(1.0, angle)
def code(angle_s, a, b, angle_m):
	tmp = 0
	if a <= 1.32e+154:
		tmp = ((b * b) - (a * a)) * (math.pi * (angle_m * 0.011111111111111112))
	else:
		tmp = a * ((a * (angle_m * math.pi)) * -0.011111111111111112)
	return angle_s * tmp

Julia

angle\_m = abs(angle)
angle\_s = copysign(1.0, angle)
function code(angle_s, a, b, angle_m)
	tmp = 0.0
	if (a <= 1.32e+154)
		tmp = Float64(Float64(Float64(b * b) - Float64(a * a)) * Float64(pi * Float64(angle_m * 0.011111111111111112)));
	else
		tmp = Float64(a * Float64(Float64(a * Float64(angle_m * pi)) * -0.011111111111111112));
	end
	return Float64(angle_s * tmp)
end

MATLAB

angle\_m = abs(angle);
angle\_s = sign(angle) * abs(1.0);
function tmp_2 = code(angle_s, a, b, angle_m)
	tmp = 0.0;
	if (a <= 1.32e+154)
		tmp = ((b * b) - (a * a)) * (pi * (angle_m * 0.011111111111111112));
	else
		tmp = a * ((a * (angle_m * pi)) * -0.011111111111111112);
	end
	tmp_2 = angle_s * tmp;
end

Wolfram

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_, b_, angle$95$m_] := N[(angle$95$s * If[LessEqual[a, 1.32e+154], N[(N[(N[(b * b), $MachinePrecision] - N[(a * a), $MachinePrecision]), $MachinePrecision] * N[(Pi * N[(angle$95$m * 0.011111111111111112), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(a * N[(N[(a * N[(angle$95$m * Pi), $MachinePrecision]), $MachinePrecision] * -0.011111111111111112), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if a < 1.31999999999999998e154

   1. Initial program 56.7%

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

      1. associate-*l* N/A

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

      2. *-commutative N/A

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

      3. associate-*l* N/A

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

      4. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\sin \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right), \color{blue}{\left(\cos \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right) \cdot \left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right)\right)}\right) \]

      5. sin-lowering-sin.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sin.f64}\left(\left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right)\right), \left(\color{blue}{\cos \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right)} \cdot \left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right)\right)\right) \]

      6. associate-*r/ N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sin.f64}\left(\left(\frac{\mathsf{PI}\left(\right) \cdot angle}{180}\right)\right), \left(\cos \color{blue}{\left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right)} \cdot \left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right)\right)\right) \]

      7. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sin.f64}\left(\mathsf{/.f64}\left(\left(\mathsf{PI}\left(\right) \cdot angle\right), 180\right)\right), \left(\cos \color{blue}{\left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right)} \cdot \left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right)\right)\right) \]

      8. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sin.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{PI}\left(\right), angle\right), 180\right)\right), \left(\cos \left(\color{blue}{\mathsf{PI}\left(\right)} \cdot \frac{angle}{180}\right) \cdot \left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right)\right)\right) \]

      9. PI-lowering-PI.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sin.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{PI.f64}\left(\right), angle\right), 180\right)\right), \left(\cos \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right) \cdot \left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right)\right)\right) \]

      10. *-commutative N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{sin.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{PI.f64}\left(\right), angle\right), 180\right)\right), \left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \color{blue}{\cos \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right)}\right)\right) \]

      11. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{sin.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{PI.f64}\left(\right), angle\right), 180\right)\right), \mathsf{*.f64}\left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right), \color{blue}{\cos \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right)}\right)\right) \]

   3. Simplified 56.8%

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

   5. Taylor expanded in angle around 0

      \[\leadsto \color{blue}{\frac{1}{90} \cdot \left(angle \cdot \left(\mathsf{PI}\left(\right) \cdot \left({b}^{2} - {a}^{2}\right)\right)\right)} \]
   6. Step-by-step derivation

      1. associate-*r* N/A

         \[\leadsto \frac{1}{90} \cdot \left(\left(angle \cdot \mathsf{PI}\left(\right)\right) \cdot \color{blue}{\left({b}^{2} - {a}^{2}\right)}\right) \]

      2. associate-*r* N/A

         \[\leadsto \left(\frac{1}{90} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right) \cdot \color{blue}{\left({b}^{2} - {a}^{2}\right)} \]

      3. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\left(\frac{1}{90} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right), \color{blue}{\left({b}^{2} - {a}^{2}\right)}\right) \]

      4. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{1}{90}, \left(angle \cdot \mathsf{PI}\left(\right)\right)\right), \left(\color{blue}{{b}^{2}} - {a}^{2}\right)\right) \]

      5. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{1}{90}, \mathsf{*.f64}\left(angle, \mathsf{PI}\left(\right)\right)\right), \left({b}^{\color{blue}{2}} - {a}^{2}\right)\right) \]

      6. PI-lowering-PI.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{1}{90}, \mathsf{*.f64}\left(angle, \mathsf{PI.f64}\left(\right)\right)\right), \left({b}^{2} - {a}^{2}\right)\right) \]

      7. --lowering--.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{1}{90}, \mathsf{*.f64}\left(angle, \mathsf{PI.f64}\left(\right)\right)\right), \mathsf{\_.f64}\left(\left({b}^{2}\right), \color{blue}{\left({a}^{2}\right)}\right)\right) \]

      8. unpow2 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{1}{90}, \mathsf{*.f64}\left(angle, \mathsf{PI.f64}\left(\right)\right)\right), \mathsf{\_.f64}\left(\left(b \cdot b\right), \left({\color{blue}{a}}^{2}\right)\right)\right) \]

      9. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{1}{90}, \mathsf{*.f64}\left(angle, \mathsf{PI.f64}\left(\right)\right)\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(b, b\right), \left({\color{blue}{a}}^{2}\right)\right)\right) \]

      10. unpow2 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{1}{90}, \mathsf{*.f64}\left(angle, \mathsf{PI.f64}\left(\right)\right)\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(b, b\right), \left(a \cdot \color{blue}{a}\right)\right)\right) \]

      11. *-lowering-*.f64 53.6%

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{1}{90}, \mathsf{*.f64}\left(angle, \mathsf{PI.f64}\left(\right)\right)\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(b, b\right), \mathsf{*.f64}\left(a, \color{blue}{a}\right)\right)\right) \]

   7. Simplified 53.6%

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

      1. associate-*r* N/A

         \[\leadsto \mathsf{*.f64}\left(\left(\left(\frac{1}{90} \cdot angle\right) \cdot \mathsf{PI}\left(\right)\right), \mathsf{\_.f64}\left(\color{blue}{\mathsf{*.f64}\left(b, b\right)}, \mathsf{*.f64}\left(a, a\right)\right)\right) \]

      2. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\left(\frac{1}{90} \cdot angle\right), \mathsf{PI}\left(\right)\right), \mathsf{\_.f64}\left(\color{blue}{\mathsf{*.f64}\left(b, b\right)}, \mathsf{*.f64}\left(a, a\right)\right)\right) \]

      3. *-commutative N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\left(angle \cdot \frac{1}{90}\right), \mathsf{PI}\left(\right)\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\color{blue}{b}, b\right), \mathsf{*.f64}\left(a, a\right)\right)\right) \]

      4. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(angle, \frac{1}{90}\right), \mathsf{PI}\left(\right)\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\color{blue}{b}, b\right), \mathsf{*.f64}\left(a, a\right)\right)\right) \]

      5. PI-lowering-PI.f64 53.6%

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(angle, \frac{1}{90}\right), \mathsf{PI.f64}\left(\right)\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(b, \color{blue}{b}\right), \mathsf{*.f64}\left(a, a\right)\right)\right) \]

   9. Applied egg-rr 53.6%

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

   if 1.31999999999999998e154 < a

   1. Initial program 35.5%

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

      1. associate-*l* N/A

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

      2. *-commutative N/A

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

      3. associate-*l* N/A

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

      4. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\sin \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right), \color{blue}{\left(\cos \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right) \cdot \left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right)\right)}\right) \]

      5. sin-lowering-sin.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sin.f64}\left(\left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right)\right), \left(\color{blue}{\cos \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right)} \cdot \left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right)\right)\right) \]

      6. associate-*r/ N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sin.f64}\left(\left(\frac{\mathsf{PI}\left(\right) \cdot angle}{180}\right)\right), \left(\cos \color{blue}{\left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right)} \cdot \left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right)\right)\right) \]

      7. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sin.f64}\left(\mathsf{/.f64}\left(\left(\mathsf{PI}\left(\right) \cdot angle\right), 180\right)\right), \left(\cos \color{blue}{\left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right)} \cdot \left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right)\right)\right) \]

      8. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sin.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{PI}\left(\right), angle\right), 180\right)\right), \left(\cos \left(\color{blue}{\mathsf{PI}\left(\right)} \cdot \frac{angle}{180}\right) \cdot \left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right)\right)\right) \]

      9. PI-lowering-PI.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sin.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{PI.f64}\left(\right), angle\right), 180\right)\right), \left(\cos \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right) \cdot \left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right)\right)\right) \]

      10. *-commutative N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{sin.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{PI.f64}\left(\right), angle\right), 180\right)\right), \left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \color{blue}{\cos \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right)}\right)\right) \]

      11. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{sin.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{PI.f64}\left(\right), angle\right), 180\right)\right), \mathsf{*.f64}\left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right), \color{blue}{\cos \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right)}\right)\right) \]

   3. Simplified 35.5%

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

   5. Taylor expanded in angle around 0

      \[\leadsto \color{blue}{\frac{1}{90} \cdot \left(angle \cdot \left(\mathsf{PI}\left(\right) \cdot \left({b}^{2} - {a}^{2}\right)\right)\right)} \]
   6. Step-by-step derivation

      1. associate-*r* N/A

         \[\leadsto \frac{1}{90} \cdot \left(\left(angle \cdot \mathsf{PI}\left(\right)\right) \cdot \color{blue}{\left({b}^{2} - {a}^{2}\right)}\right) \]

      2. associate-*r* N/A

         \[\leadsto \left(\frac{1}{90} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right) \cdot \color{blue}{\left({b}^{2} - {a}^{2}\right)} \]

      3. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\left(\frac{1}{90} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right), \color{blue}{\left({b}^{2} - {a}^{2}\right)}\right) \]

      4. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{1}{90}, \left(angle \cdot \mathsf{PI}\left(\right)\right)\right), \left(\color{blue}{{b}^{2}} - {a}^{2}\right)\right) \]

      5. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{1}{90}, \mathsf{*.f64}\left(angle, \mathsf{PI}\left(\right)\right)\right), \left({b}^{\color{blue}{2}} - {a}^{2}\right)\right) \]

      6. PI-lowering-PI.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{1}{90}, \mathsf{*.f64}\left(angle, \mathsf{PI.f64}\left(\right)\right)\right), \left({b}^{2} - {a}^{2}\right)\right) \]

      7. --lowering--.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{1}{90}, \mathsf{*.f64}\left(angle, \mathsf{PI.f64}\left(\right)\right)\right), \mathsf{\_.f64}\left(\left({b}^{2}\right), \color{blue}{\left({a}^{2}\right)}\right)\right) \]

      8. unpow2 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{1}{90}, \mathsf{*.f64}\left(angle, \mathsf{PI.f64}\left(\right)\right)\right), \mathsf{\_.f64}\left(\left(b \cdot b\right), \left({\color{blue}{a}}^{2}\right)\right)\right) \]

      9. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{1}{90}, \mathsf{*.f64}\left(angle, \mathsf{PI.f64}\left(\right)\right)\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(b, b\right), \left({\color{blue}{a}}^{2}\right)\right)\right) \]

      10. unpow2 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{1}{90}, \mathsf{*.f64}\left(angle, \mathsf{PI.f64}\left(\right)\right)\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(b, b\right), \left(a \cdot \color{blue}{a}\right)\right)\right) \]

      11. *-lowering-*.f64 31.5%

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{1}{90}, \mathsf{*.f64}\left(angle, \mathsf{PI.f64}\left(\right)\right)\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(b, b\right), \mathsf{*.f64}\left(a, \color{blue}{a}\right)\right)\right) \]

   7. Simplified 31.5%

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

   8. Taylor expanded in b around 0

      \[\leadsto \color{blue}{\frac{-1}{90} \cdot \left({a}^{2} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)} \]
   9. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \left({a}^{2} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right) \cdot \color{blue}{\frac{-1}{90}} \]

      2. associate-*r* N/A

         \[\leadsto {a}^{2} \cdot \color{blue}{\left(\left(angle \cdot \mathsf{PI}\left(\right)\right) \cdot \frac{-1}{90}\right)} \]

      3. *-commutative N/A

         \[\leadsto {a}^{2} \cdot \left(\frac{-1}{90} \cdot \color{blue}{\left(angle \cdot \mathsf{PI}\left(\right)\right)}\right) \]

      4. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\left({a}^{2}\right), \color{blue}{\left(\frac{-1}{90} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}\right) \]

      5. unpow2 N/A

         \[\leadsto \mathsf{*.f64}\left(\left(a \cdot a\right), \left(\color{blue}{\frac{-1}{90}} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)\right) \]

      6. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(a, a\right), \left(\color{blue}{\frac{-1}{90}} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)\right) \]

      7. *-commutative N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(a, a\right), \left(\left(angle \cdot \mathsf{PI}\left(\right)\right) \cdot \color{blue}{\frac{-1}{90}}\right)\right) \]

      8. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(a, a\right), \mathsf{*.f64}\left(\left(angle \cdot \mathsf{PI}\left(\right)\right), \color{blue}{\frac{-1}{90}}\right)\right) \]

      9. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(a, a\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(angle, \mathsf{PI}\left(\right)\right), \frac{-1}{90}\right)\right) \]

      10. PI-lowering-PI.f64 35.5%

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(a, a\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(angle, \mathsf{PI.f64}\left(\right)\right), \frac{-1}{90}\right)\right) \]

   10. Simplified 35.5%

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

       1. associate-*l* N/A

          \[\leadsto a \cdot \color{blue}{\left(a \cdot \left(\left(angle \cdot \mathsf{PI}\left(\right)\right) \cdot \frac{-1}{90}\right)\right)} \]

       2. *-commutative N/A

          \[\leadsto \left(a \cdot \left(\left(angle \cdot \mathsf{PI}\left(\right)\right) \cdot \frac{-1}{90}\right)\right) \cdot \color{blue}{a} \]

       3. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\left(a \cdot \left(\left(angle \cdot \mathsf{PI}\left(\right)\right) \cdot \frac{-1}{90}\right)\right), \color{blue}{a}\right) \]

       4. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(a, \left(\left(angle \cdot \mathsf{PI}\left(\right)\right) \cdot \frac{-1}{90}\right)\right), a\right) \]

       5. *-commutative N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(a, \left(\frac{-1}{90} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)\right), a\right) \]

       6. *-commutative N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(a, \left(\frac{-1}{90} \cdot \left(\mathsf{PI}\left(\right) \cdot angle\right)\right)\right), a\right) \]

       7. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(a, \mathsf{*.f64}\left(\frac{-1}{90}, \left(\mathsf{PI}\left(\right) \cdot angle\right)\right)\right), a\right) \]

       8. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(a, \mathsf{*.f64}\left(\frac{-1}{90}, \mathsf{*.f64}\left(\mathsf{PI}\left(\right), angle\right)\right)\right), a\right) \]

       9. PI-lowering-PI.f64 67.8%

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(a, \mathsf{*.f64}\left(\frac{-1}{90}, \mathsf{*.f64}\left(\mathsf{PI.f64}\left(\right), angle\right)\right)\right), a\right) \]

   12. Applied egg-rr 67.8%

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

       1. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\left(a \cdot \left(\frac{-1}{90} \cdot \left(\mathsf{PI}\left(\right) \cdot angle\right)\right)\right), \color{blue}{a}\right) \]

       2. *-commutative N/A

          \[\leadsto \mathsf{*.f64}\left(\left(a \cdot \left(\left(\mathsf{PI}\left(\right) \cdot angle\right) \cdot \frac{-1}{90}\right)\right), a\right) \]

       3. associate-*r* N/A

          \[\leadsto \mathsf{*.f64}\left(\left(\left(a \cdot \left(\mathsf{PI}\left(\right) \cdot angle\right)\right) \cdot \frac{-1}{90}\right), a\right) \]

       4. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\left(a \cdot \left(\mathsf{PI}\left(\right) \cdot angle\right)\right), \frac{-1}{90}\right), a\right) \]

       5. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(a, \left(\mathsf{PI}\left(\right) \cdot angle\right)\right), \frac{-1}{90}\right), a\right) \]

       6. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(a, \mathsf{*.f64}\left(\mathsf{PI}\left(\right), angle\right)\right), \frac{-1}{90}\right), a\right) \]

       7. PI-lowering-PI.f64 67.9%

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(a, \mathsf{*.f64}\left(\mathsf{PI.f64}\left(\right), angle\right)\right), \frac{-1}{90}\right), a\right) \]

   14. Applied egg-rr 67.9%

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

4. Final simplification 55.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \leq 1.32 \cdot 10^{+154}:\\ \;\;\;\;\left(b \cdot b - a \cdot a\right) \cdot \left(\pi \cdot \left(angle \cdot 0.011111111111111112\right)\right)\\ \mathbf{else}:\\ \;\;\;\;a \cdot \left(\left(a \cdot \left(angle \cdot \pi\right)\right) \cdot -0.011111111111111112\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 15: 54.2% accurate, 23.3× speedup

Math

\[\begin{array}{l} angle\_m = \left|angle\right| \\ angle\_s = \mathsf{copysign}\left(1, angle\right) \\ angle\_s \cdot \begin{array}{l} \mathbf{if}\;a \leq 1.3 \cdot 10^{+154}:\\ \;\;\;\;\left(b \cdot b - a \cdot a\right) \cdot \left(\left(angle\_m \cdot \pi\right) \cdot 0.011111111111111112\right)\\ \mathbf{else}:\\ \;\;\;\;a \cdot \left(\left(a \cdot \left(angle\_m \cdot \pi\right)\right) \cdot -0.011111111111111112\right)\\ \end{array} \end{array} \]

FPCore

angle\_m = (fabs.f64 angle)
angle\_s = (copysign.f64 #s(literal 1 binary64) angle)
(FPCore (angle_s a b angle_m)
 :precision binary64
 (*
  angle_s
  (if (<= a 1.3e+154)
    (* (- (* b b) (* a a)) (* (* angle_m PI) 0.011111111111111112))
    (* a (* (* a (* angle_m PI)) -0.011111111111111112)))))

C

angle\_m = fabs(angle);
angle\_s = copysign(1.0, angle);
double code(double angle_s, double a, double b, double angle_m) {
	double tmp;
	if (a <= 1.3e+154) {
		tmp = ((b * b) - (a * a)) * ((angle_m * ((double) M_PI)) * 0.011111111111111112);
	} else {
		tmp = a * ((a * (angle_m * ((double) M_PI))) * -0.011111111111111112);
	}
	return angle_s * tmp;
}

Java

angle\_m = Math.abs(angle);
angle\_s = Math.copySign(1.0, angle);
public static double code(double angle_s, double a, double b, double angle_m) {
	double tmp;
	if (a <= 1.3e+154) {
		tmp = ((b * b) - (a * a)) * ((angle_m * Math.PI) * 0.011111111111111112);
	} else {
		tmp = a * ((a * (angle_m * Math.PI)) * -0.011111111111111112);
	}
	return angle_s * tmp;
}

Python

angle\_m = math.fabs(angle)
angle\_s = math.copysign(1.0, angle)
def code(angle_s, a, b, angle_m):
	tmp = 0
	if a <= 1.3e+154:
		tmp = ((b * b) - (a * a)) * ((angle_m * math.pi) * 0.011111111111111112)
	else:
		tmp = a * ((a * (angle_m * math.pi)) * -0.011111111111111112)
	return angle_s * tmp

Julia

angle\_m = abs(angle)
angle\_s = copysign(1.0, angle)
function code(angle_s, a, b, angle_m)
	tmp = 0.0
	if (a <= 1.3e+154)
		tmp = Float64(Float64(Float64(b * b) - Float64(a * a)) * Float64(Float64(angle_m * pi) * 0.011111111111111112));
	else
		tmp = Float64(a * Float64(Float64(a * Float64(angle_m * pi)) * -0.011111111111111112));
	end
	return Float64(angle_s * tmp)
end

MATLAB

angle\_m = abs(angle);
angle\_s = sign(angle) * abs(1.0);
function tmp_2 = code(angle_s, a, b, angle_m)
	tmp = 0.0;
	if (a <= 1.3e+154)
		tmp = ((b * b) - (a * a)) * ((angle_m * pi) * 0.011111111111111112);
	else
		tmp = a * ((a * (angle_m * pi)) * -0.011111111111111112);
	end
	tmp_2 = angle_s * tmp;
end

Wolfram

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_, b_, angle$95$m_] := N[(angle$95$s * If[LessEqual[a, 1.3e+154], N[(N[(N[(b * b), $MachinePrecision] - N[(a * a), $MachinePrecision]), $MachinePrecision] * N[(N[(angle$95$m * Pi), $MachinePrecision] * 0.011111111111111112), $MachinePrecision]), $MachinePrecision], N[(a * N[(N[(a * N[(angle$95$m * Pi), $MachinePrecision]), $MachinePrecision] * -0.011111111111111112), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if a < 1.29999999999999994e154

   1. Initial program 56.7%

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

      1. associate-*l* N/A

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

      2. *-commutative N/A

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

      3. associate-*l* N/A

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

      4. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\sin \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right), \color{blue}{\left(\cos \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right) \cdot \left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right)\right)}\right) \]

      5. sin-lowering-sin.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sin.f64}\left(\left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right)\right), \left(\color{blue}{\cos \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right)} \cdot \left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right)\right)\right) \]

      6. associate-*r/ N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sin.f64}\left(\left(\frac{\mathsf{PI}\left(\right) \cdot angle}{180}\right)\right), \left(\cos \color{blue}{\left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right)} \cdot \left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right)\right)\right) \]

      7. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sin.f64}\left(\mathsf{/.f64}\left(\left(\mathsf{PI}\left(\right) \cdot angle\right), 180\right)\right), \left(\cos \color{blue}{\left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right)} \cdot \left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right)\right)\right) \]

      8. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sin.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{PI}\left(\right), angle\right), 180\right)\right), \left(\cos \left(\color{blue}{\mathsf{PI}\left(\right)} \cdot \frac{angle}{180}\right) \cdot \left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right)\right)\right) \]

      9. PI-lowering-PI.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sin.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{PI.f64}\left(\right), angle\right), 180\right)\right), \left(\cos \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right) \cdot \left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right)\right)\right) \]

      10. *-commutative N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{sin.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{PI.f64}\left(\right), angle\right), 180\right)\right), \left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \color{blue}{\cos \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right)}\right)\right) \]

      11. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{sin.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{PI.f64}\left(\right), angle\right), 180\right)\right), \mathsf{*.f64}\left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right), \color{blue}{\cos \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right)}\right)\right) \]

   3. Simplified 56.8%

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

   5. Taylor expanded in angle around 0

      \[\leadsto \color{blue}{\frac{1}{90} \cdot \left(angle \cdot \left(\mathsf{PI}\left(\right) \cdot \left({b}^{2} - {a}^{2}\right)\right)\right)} \]
   6. Step-by-step derivation

      1. associate-*r* N/A

         \[\leadsto \frac{1}{90} \cdot \left(\left(angle \cdot \mathsf{PI}\left(\right)\right) \cdot \color{blue}{\left({b}^{2} - {a}^{2}\right)}\right) \]

      2. associate-*r* N/A

         \[\leadsto \left(\frac{1}{90} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right) \cdot \color{blue}{\left({b}^{2} - {a}^{2}\right)} \]

      3. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\left(\frac{1}{90} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right), \color{blue}{\left({b}^{2} - {a}^{2}\right)}\right) \]

      4. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{1}{90}, \left(angle \cdot \mathsf{PI}\left(\right)\right)\right), \left(\color{blue}{{b}^{2}} - {a}^{2}\right)\right) \]

      5. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{1}{90}, \mathsf{*.f64}\left(angle, \mathsf{PI}\left(\right)\right)\right), \left({b}^{\color{blue}{2}} - {a}^{2}\right)\right) \]

      6. PI-lowering-PI.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{1}{90}, \mathsf{*.f64}\left(angle, \mathsf{PI.f64}\left(\right)\right)\right), \left({b}^{2} - {a}^{2}\right)\right) \]

      7. --lowering--.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{1}{90}, \mathsf{*.f64}\left(angle, \mathsf{PI.f64}\left(\right)\right)\right), \mathsf{\_.f64}\left(\left({b}^{2}\right), \color{blue}{\left({a}^{2}\right)}\right)\right) \]

      8. unpow2 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{1}{90}, \mathsf{*.f64}\left(angle, \mathsf{PI.f64}\left(\right)\right)\right), \mathsf{\_.f64}\left(\left(b \cdot b\right), \left({\color{blue}{a}}^{2}\right)\right)\right) \]

      9. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{1}{90}, \mathsf{*.f64}\left(angle, \mathsf{PI.f64}\left(\right)\right)\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(b, b\right), \left({\color{blue}{a}}^{2}\right)\right)\right) \]

      10. unpow2 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{1}{90}, \mathsf{*.f64}\left(angle, \mathsf{PI.f64}\left(\right)\right)\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(b, b\right), \left(a \cdot \color{blue}{a}\right)\right)\right) \]

      11. *-lowering-*.f64 53.6%

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{1}{90}, \mathsf{*.f64}\left(angle, \mathsf{PI.f64}\left(\right)\right)\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(b, b\right), \mathsf{*.f64}\left(a, \color{blue}{a}\right)\right)\right) \]

   7. Simplified 53.6%

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

   if 1.29999999999999994e154 < a

   1. Initial program 35.5%

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

      1. associate-*l* N/A

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

      2. *-commutative N/A

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

      3. associate-*l* N/A

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

      4. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\sin \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right), \color{blue}{\left(\cos \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right) \cdot \left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right)\right)}\right) \]

      5. sin-lowering-sin.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sin.f64}\left(\left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right)\right), \left(\color{blue}{\cos \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right)} \cdot \left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right)\right)\right) \]

      6. associate-*r/ N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sin.f64}\left(\left(\frac{\mathsf{PI}\left(\right) \cdot angle}{180}\right)\right), \left(\cos \color{blue}{\left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right)} \cdot \left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right)\right)\right) \]

      7. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sin.f64}\left(\mathsf{/.f64}\left(\left(\mathsf{PI}\left(\right) \cdot angle\right), 180\right)\right), \left(\cos \color{blue}{\left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right)} \cdot \left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right)\right)\right) \]

      8. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sin.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{PI}\left(\right), angle\right), 180\right)\right), \left(\cos \left(\color{blue}{\mathsf{PI}\left(\right)} \cdot \frac{angle}{180}\right) \cdot \left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right)\right)\right) \]

      9. PI-lowering-PI.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sin.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{PI.f64}\left(\right), angle\right), 180\right)\right), \left(\cos \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right) \cdot \left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right)\right)\right) \]

      10. *-commutative N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{sin.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{PI.f64}\left(\right), angle\right), 180\right)\right), \left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \color{blue}{\cos \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right)}\right)\right) \]

      11. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{sin.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{PI.f64}\left(\right), angle\right), 180\right)\right), \mathsf{*.f64}\left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right), \color{blue}{\cos \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right)}\right)\right) \]

   3. Simplified 35.5%

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

   5. Taylor expanded in angle around 0

      \[\leadsto \color{blue}{\frac{1}{90} \cdot \left(angle \cdot \left(\mathsf{PI}\left(\right) \cdot \left({b}^{2} - {a}^{2}\right)\right)\right)} \]
   6. Step-by-step derivation

      1. associate-*r* N/A

         \[\leadsto \frac{1}{90} \cdot \left(\left(angle \cdot \mathsf{PI}\left(\right)\right) \cdot \color{blue}{\left({b}^{2} - {a}^{2}\right)}\right) \]

      2. associate-*r* N/A

         \[\leadsto \left(\frac{1}{90} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right) \cdot \color{blue}{\left({b}^{2} - {a}^{2}\right)} \]

      3. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\left(\frac{1}{90} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right), \color{blue}{\left({b}^{2} - {a}^{2}\right)}\right) \]

      4. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{1}{90}, \left(angle \cdot \mathsf{PI}\left(\right)\right)\right), \left(\color{blue}{{b}^{2}} - {a}^{2}\right)\right) \]

      5. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{1}{90}, \mathsf{*.f64}\left(angle, \mathsf{PI}\left(\right)\right)\right), \left({b}^{\color{blue}{2}} - {a}^{2}\right)\right) \]

      6. PI-lowering-PI.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{1}{90}, \mathsf{*.f64}\left(angle, \mathsf{PI.f64}\left(\right)\right)\right), \left({b}^{2} - {a}^{2}\right)\right) \]

      7. --lowering--.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{1}{90}, \mathsf{*.f64}\left(angle, \mathsf{PI.f64}\left(\right)\right)\right), \mathsf{\_.f64}\left(\left({b}^{2}\right), \color{blue}{\left({a}^{2}\right)}\right)\right) \]

      8. unpow2 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{1}{90}, \mathsf{*.f64}\left(angle, \mathsf{PI.f64}\left(\right)\right)\right), \mathsf{\_.f64}\left(\left(b \cdot b\right), \left({\color{blue}{a}}^{2}\right)\right)\right) \]

      9. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{1}{90}, \mathsf{*.f64}\left(angle, \mathsf{PI.f64}\left(\right)\right)\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(b, b\right), \left({\color{blue}{a}}^{2}\right)\right)\right) \]

      10. unpow2 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{1}{90}, \mathsf{*.f64}\left(angle, \mathsf{PI.f64}\left(\right)\right)\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(b, b\right), \left(a \cdot \color{blue}{a}\right)\right)\right) \]

      11. *-lowering-*.f64 31.5%

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{1}{90}, \mathsf{*.f64}\left(angle, \mathsf{PI.f64}\left(\right)\right)\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(b, b\right), \mathsf{*.f64}\left(a, \color{blue}{a}\right)\right)\right) \]

   7. Simplified 31.5%

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

   8. Taylor expanded in b around 0

      \[\leadsto \color{blue}{\frac{-1}{90} \cdot \left({a}^{2} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)} \]
   9. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \left({a}^{2} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right) \cdot \color{blue}{\frac{-1}{90}} \]

      2. associate-*r* N/A

         \[\leadsto {a}^{2} \cdot \color{blue}{\left(\left(angle \cdot \mathsf{PI}\left(\right)\right) \cdot \frac{-1}{90}\right)} \]

      3. *-commutative N/A

         \[\leadsto {a}^{2} \cdot \left(\frac{-1}{90} \cdot \color{blue}{\left(angle \cdot \mathsf{PI}\left(\right)\right)}\right) \]

      4. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\left({a}^{2}\right), \color{blue}{\left(\frac{-1}{90} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}\right) \]

      5. unpow2 N/A

         \[\leadsto \mathsf{*.f64}\left(\left(a \cdot a\right), \left(\color{blue}{\frac{-1}{90}} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)\right) \]

      6. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(a, a\right), \left(\color{blue}{\frac{-1}{90}} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)\right) \]

      7. *-commutative N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(a, a\right), \left(\left(angle \cdot \mathsf{PI}\left(\right)\right) \cdot \color{blue}{\frac{-1}{90}}\right)\right) \]

      8. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(a, a\right), \mathsf{*.f64}\left(\left(angle \cdot \mathsf{PI}\left(\right)\right), \color{blue}{\frac{-1}{90}}\right)\right) \]

      9. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(a, a\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(angle, \mathsf{PI}\left(\right)\right), \frac{-1}{90}\right)\right) \]

      10. PI-lowering-PI.f64 35.5%

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(a, a\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(angle, \mathsf{PI.f64}\left(\right)\right), \frac{-1}{90}\right)\right) \]

   10. Simplified 35.5%

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

       1. associate-*l* N/A

          \[\leadsto a \cdot \color{blue}{\left(a \cdot \left(\left(angle \cdot \mathsf{PI}\left(\right)\right) \cdot \frac{-1}{90}\right)\right)} \]

       2. *-commutative N/A

          \[\leadsto \left(a \cdot \left(\left(angle \cdot \mathsf{PI}\left(\right)\right) \cdot \frac{-1}{90}\right)\right) \cdot \color{blue}{a} \]

       3. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\left(a \cdot \left(\left(angle \cdot \mathsf{PI}\left(\right)\right) \cdot \frac{-1}{90}\right)\right), \color{blue}{a}\right) \]

       4. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(a, \left(\left(angle \cdot \mathsf{PI}\left(\right)\right) \cdot \frac{-1}{90}\right)\right), a\right) \]

       5. *-commutative N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(a, \left(\frac{-1}{90} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)\right), a\right) \]

       6. *-commutative N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(a, \left(\frac{-1}{90} \cdot \left(\mathsf{PI}\left(\right) \cdot angle\right)\right)\right), a\right) \]

       7. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(a, \mathsf{*.f64}\left(\frac{-1}{90}, \left(\mathsf{PI}\left(\right) \cdot angle\right)\right)\right), a\right) \]

       8. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(a, \mathsf{*.f64}\left(\frac{-1}{90}, \mathsf{*.f64}\left(\mathsf{PI}\left(\right), angle\right)\right)\right), a\right) \]

       9. PI-lowering-PI.f64 67.8%

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(a, \mathsf{*.f64}\left(\frac{-1}{90}, \mathsf{*.f64}\left(\mathsf{PI.f64}\left(\right), angle\right)\right)\right), a\right) \]

   12. Applied egg-rr 67.8%

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

       1. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\left(a \cdot \left(\frac{-1}{90} \cdot \left(\mathsf{PI}\left(\right) \cdot angle\right)\right)\right), \color{blue}{a}\right) \]

       2. *-commutative N/A

          \[\leadsto \mathsf{*.f64}\left(\left(a \cdot \left(\left(\mathsf{PI}\left(\right) \cdot angle\right) \cdot \frac{-1}{90}\right)\right), a\right) \]

       3. associate-*r* N/A

          \[\leadsto \mathsf{*.f64}\left(\left(\left(a \cdot \left(\mathsf{PI}\left(\right) \cdot angle\right)\right) \cdot \frac{-1}{90}\right), a\right) \]

       4. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\left(a \cdot \left(\mathsf{PI}\left(\right) \cdot angle\right)\right), \frac{-1}{90}\right), a\right) \]

       5. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(a, \left(\mathsf{PI}\left(\right) \cdot angle\right)\right), \frac{-1}{90}\right), a\right) \]

       6. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(a, \mathsf{*.f64}\left(\mathsf{PI}\left(\right), angle\right)\right), \frac{-1}{90}\right), a\right) \]

       7. PI-lowering-PI.f64 67.9%

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(a, \mathsf{*.f64}\left(\mathsf{PI.f64}\left(\right), angle\right)\right), \frac{-1}{90}\right), a\right) \]

   14. Applied egg-rr 67.9%

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

4. Final simplification 55.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \leq 1.3 \cdot 10^{+154}:\\ \;\;\;\;\left(b \cdot b - a \cdot a\right) \cdot \left(\left(angle \cdot \pi\right) \cdot 0.011111111111111112\right)\\ \mathbf{else}:\\ \;\;\;\;a \cdot \left(\left(a \cdot \left(angle \cdot \pi\right)\right) \cdot -0.011111111111111112\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 16: 43.1% accurate, 29.9× speedup

Math

\[\begin{array}{l} angle\_m = \left|angle\right| \\ angle\_s = \mathsf{copysign}\left(1, angle\right) \\ angle\_s \cdot \begin{array}{l} \mathbf{if}\;a \leq 1.75 \cdot 10^{+18}:\\ \;\;\;\;\left(b \cdot b\right) \cdot \left(\left(angle\_m \cdot \pi\right) \cdot 0.011111111111111112\right)\\ \mathbf{else}:\\ \;\;\;\;a \cdot \left(\left(a \cdot \left(angle\_m \cdot \pi\right)\right) \cdot -0.011111111111111112\right)\\ \end{array} \end{array} \]

FPCore

angle\_m = (fabs.f64 angle)
angle\_s = (copysign.f64 #s(literal 1 binary64) angle)
(FPCore (angle_s a b angle_m)
 :precision binary64
 (*
  angle_s
  (if (<= a 1.75e+18)
    (* (* b b) (* (* angle_m PI) 0.011111111111111112))
    (* a (* (* a (* angle_m PI)) -0.011111111111111112)))))

C

angle\_m = fabs(angle);
angle\_s = copysign(1.0, angle);
double code(double angle_s, double a, double b, double angle_m) {
	double tmp;
	if (a <= 1.75e+18) {
		tmp = (b * b) * ((angle_m * ((double) M_PI)) * 0.011111111111111112);
	} else {
		tmp = a * ((a * (angle_m * ((double) M_PI))) * -0.011111111111111112);
	}
	return angle_s * tmp;
}

Java

angle\_m = Math.abs(angle);
angle\_s = Math.copySign(1.0, angle);
public static double code(double angle_s, double a, double b, double angle_m) {
	double tmp;
	if (a <= 1.75e+18) {
		tmp = (b * b) * ((angle_m * Math.PI) * 0.011111111111111112);
	} else {
		tmp = a * ((a * (angle_m * Math.PI)) * -0.011111111111111112);
	}
	return angle_s * tmp;
}

Python

angle\_m = math.fabs(angle)
angle\_s = math.copysign(1.0, angle)
def code(angle_s, a, b, angle_m):
	tmp = 0
	if a <= 1.75e+18:
		tmp = (b * b) * ((angle_m * math.pi) * 0.011111111111111112)
	else:
		tmp = a * ((a * (angle_m * math.pi)) * -0.011111111111111112)
	return angle_s * tmp

Julia

angle\_m = abs(angle)
angle\_s = copysign(1.0, angle)
function code(angle_s, a, b, angle_m)
	tmp = 0.0
	if (a <= 1.75e+18)
		tmp = Float64(Float64(b * b) * Float64(Float64(angle_m * pi) * 0.011111111111111112));
	else
		tmp = Float64(a * Float64(Float64(a * Float64(angle_m * pi)) * -0.011111111111111112));
	end
	return Float64(angle_s * tmp)
end

MATLAB

angle\_m = abs(angle);
angle\_s = sign(angle) * abs(1.0);
function tmp_2 = code(angle_s, a, b, angle_m)
	tmp = 0.0;
	if (a <= 1.75e+18)
		tmp = (b * b) * ((angle_m * pi) * 0.011111111111111112);
	else
		tmp = a * ((a * (angle_m * pi)) * -0.011111111111111112);
	end
	tmp_2 = angle_s * tmp;
end

Wolfram

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_, b_, angle$95$m_] := N[(angle$95$s * If[LessEqual[a, 1.75e+18], N[(N[(b * b), $MachinePrecision] * N[(N[(angle$95$m * Pi), $MachinePrecision] * 0.011111111111111112), $MachinePrecision]), $MachinePrecision], N[(a * N[(N[(a * N[(angle$95$m * Pi), $MachinePrecision]), $MachinePrecision] * -0.011111111111111112), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if a < 1.75e18

   1. Initial program 56.8%

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

      1. associate-*l* N/A

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

      2. *-commutative N/A

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

      3. associate-*l* N/A

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

      4. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\sin \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right), \color{blue}{\left(\cos \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right) \cdot \left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right)\right)}\right) \]

      5. sin-lowering-sin.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sin.f64}\left(\left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right)\right), \left(\color{blue}{\cos \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right)} \cdot \left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right)\right)\right) \]

      6. associate-*r/ N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sin.f64}\left(\left(\frac{\mathsf{PI}\left(\right) \cdot angle}{180}\right)\right), \left(\cos \color{blue}{\left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right)} \cdot \left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right)\right)\right) \]

      7. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sin.f64}\left(\mathsf{/.f64}\left(\left(\mathsf{PI}\left(\right) \cdot angle\right), 180\right)\right), \left(\cos \color{blue}{\left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right)} \cdot \left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right)\right)\right) \]

      8. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sin.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{PI}\left(\right), angle\right), 180\right)\right), \left(\cos \left(\color{blue}{\mathsf{PI}\left(\right)} \cdot \frac{angle}{180}\right) \cdot \left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right)\right)\right) \]

      9. PI-lowering-PI.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sin.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{PI.f64}\left(\right), angle\right), 180\right)\right), \left(\cos \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right) \cdot \left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right)\right)\right) \]

      10. *-commutative N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{sin.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{PI.f64}\left(\right), angle\right), 180\right)\right), \left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \color{blue}{\cos \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right)}\right)\right) \]

      11. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{sin.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{PI.f64}\left(\right), angle\right), 180\right)\right), \mathsf{*.f64}\left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right), \color{blue}{\cos \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right)}\right)\right) \]

   3. Simplified 56.8%

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

   5. Taylor expanded in angle around 0

      \[\leadsto \color{blue}{\frac{1}{90} \cdot \left(angle \cdot \left(\mathsf{PI}\left(\right) \cdot \left({b}^{2} - {a}^{2}\right)\right)\right)} \]
   6. Step-by-step derivation

      1. associate-*r* N/A

         \[\leadsto \frac{1}{90} \cdot \left(\left(angle \cdot \mathsf{PI}\left(\right)\right) \cdot \color{blue}{\left({b}^{2} - {a}^{2}\right)}\right) \]

      2. associate-*r* N/A

         \[\leadsto \left(\frac{1}{90} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right) \cdot \color{blue}{\left({b}^{2} - {a}^{2}\right)} \]

      3. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\left(\frac{1}{90} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right), \color{blue}{\left({b}^{2} - {a}^{2}\right)}\right) \]

      4. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{1}{90}, \left(angle \cdot \mathsf{PI}\left(\right)\right)\right), \left(\color{blue}{{b}^{2}} - {a}^{2}\right)\right) \]

      5. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{1}{90}, \mathsf{*.f64}\left(angle, \mathsf{PI}\left(\right)\right)\right), \left({b}^{\color{blue}{2}} - {a}^{2}\right)\right) \]

      6. PI-lowering-PI.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{1}{90}, \mathsf{*.f64}\left(angle, \mathsf{PI.f64}\left(\right)\right)\right), \left({b}^{2} - {a}^{2}\right)\right) \]

      7. --lowering--.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{1}{90}, \mathsf{*.f64}\left(angle, \mathsf{PI.f64}\left(\right)\right)\right), \mathsf{\_.f64}\left(\left({b}^{2}\right), \color{blue}{\left({a}^{2}\right)}\right)\right) \]

      8. unpow2 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{1}{90}, \mathsf{*.f64}\left(angle, \mathsf{PI.f64}\left(\right)\right)\right), \mathsf{\_.f64}\left(\left(b \cdot b\right), \left({\color{blue}{a}}^{2}\right)\right)\right) \]

      9. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{1}{90}, \mathsf{*.f64}\left(angle, \mathsf{PI.f64}\left(\right)\right)\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(b, b\right), \left({\color{blue}{a}}^{2}\right)\right)\right) \]

      10. unpow2 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{1}{90}, \mathsf{*.f64}\left(angle, \mathsf{PI.f64}\left(\right)\right)\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(b, b\right), \left(a \cdot \color{blue}{a}\right)\right)\right) \]

      11. *-lowering-*.f64 53.0%

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{1}{90}, \mathsf{*.f64}\left(angle, \mathsf{PI.f64}\left(\right)\right)\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(b, b\right), \mathsf{*.f64}\left(a, \color{blue}{a}\right)\right)\right) \]

   7. Simplified 53.0%

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

   8. Taylor expanded in b around inf

      \[\leadsto \color{blue}{\frac{1}{90} \cdot \left(angle \cdot \left({b}^{2} \cdot \mathsf{PI}\left(\right)\right)\right)} \]
   9. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \frac{1}{90} \cdot \left(angle \cdot \left(\mathsf{PI}\left(\right) \cdot \color{blue}{{b}^{2}}\right)\right) \]

      2. associate-*r* N/A

         \[\leadsto \frac{1}{90} \cdot \left(\left(angle \cdot \mathsf{PI}\left(\right)\right) \cdot \color{blue}{{b}^{2}}\right) \]

      3. associate-*l* N/A

         \[\leadsto \left(\frac{1}{90} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right) \cdot \color{blue}{{b}^{2}} \]

      4. *-commutative N/A

         \[\leadsto {b}^{2} \cdot \color{blue}{\left(\frac{1}{90} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)} \]

      5. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\left({b}^{2}\right), \color{blue}{\left(\frac{1}{90} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}\right) \]

      6. unpow2 N/A

         \[\leadsto \mathsf{*.f64}\left(\left(b \cdot b\right), \left(\color{blue}{\frac{1}{90}} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)\right) \]

      7. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(b, b\right), \left(\color{blue}{\frac{1}{90}} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)\right) \]

      8. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(b, b\right), \mathsf{*.f64}\left(\frac{1}{90}, \color{blue}{\left(angle \cdot \mathsf{PI}\left(\right)\right)}\right)\right) \]

      9. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(b, b\right), \mathsf{*.f64}\left(\frac{1}{90}, \mathsf{*.f64}\left(angle, \color{blue}{\mathsf{PI}\left(\right)}\right)\right)\right) \]

      10. PI-lowering-PI.f64 42.8%

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(b, b\right), \mathsf{*.f64}\left(\frac{1}{90}, \mathsf{*.f64}\left(angle, \mathsf{PI.f64}\left(\right)\right)\right)\right) \]

   10. Simplified 42.8%

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

   if 1.75e18 < a

   1. Initial program 45.3%

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

      1. associate-*l* N/A

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

      2. *-commutative N/A

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

      3. associate-*l* N/A

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

      4. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\sin \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right), \color{blue}{\left(\cos \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right) \cdot \left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right)\right)}\right) \]

      5. sin-lowering-sin.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sin.f64}\left(\left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right)\right), \left(\color{blue}{\cos \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right)} \cdot \left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right)\right)\right) \]

      6. associate-*r/ N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sin.f64}\left(\left(\frac{\mathsf{PI}\left(\right) \cdot angle}{180}\right)\right), \left(\cos \color{blue}{\left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right)} \cdot \left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right)\right)\right) \]

      7. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sin.f64}\left(\mathsf{/.f64}\left(\left(\mathsf{PI}\left(\right) \cdot angle\right), 180\right)\right), \left(\cos \color{blue}{\left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right)} \cdot \left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right)\right)\right) \]

      8. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sin.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{PI}\left(\right), angle\right), 180\right)\right), \left(\cos \left(\color{blue}{\mathsf{PI}\left(\right)} \cdot \frac{angle}{180}\right) \cdot \left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right)\right)\right) \]

      9. PI-lowering-PI.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sin.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{PI.f64}\left(\right), angle\right), 180\right)\right), \left(\cos \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right) \cdot \left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right)\right)\right) \]

      10. *-commutative N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{sin.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{PI.f64}\left(\right), angle\right), 180\right)\right), \left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \color{blue}{\cos \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right)}\right)\right) \]

      11. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{sin.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{PI.f64}\left(\right), angle\right), 180\right)\right), \mathsf{*.f64}\left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right), \color{blue}{\cos \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right)}\right)\right) \]

   3. Simplified 45.6%

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

   5. Taylor expanded in angle around 0

      \[\leadsto \color{blue}{\frac{1}{90} \cdot \left(angle \cdot \left(\mathsf{PI}\left(\right) \cdot \left({b}^{2} - {a}^{2}\right)\right)\right)} \]
   6. Step-by-step derivation

      1. associate-*r* N/A

         \[\leadsto \frac{1}{90} \cdot \left(\left(angle \cdot \mathsf{PI}\left(\right)\right) \cdot \color{blue}{\left({b}^{2} - {a}^{2}\right)}\right) \]

      2. associate-*r* N/A

         \[\leadsto \left(\frac{1}{90} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right) \cdot \color{blue}{\left({b}^{2} - {a}^{2}\right)} \]

      3. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\left(\frac{1}{90} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right), \color{blue}{\left({b}^{2} - {a}^{2}\right)}\right) \]

      4. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{1}{90}, \left(angle \cdot \mathsf{PI}\left(\right)\right)\right), \left(\color{blue}{{b}^{2}} - {a}^{2}\right)\right) \]

      5. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{1}{90}, \mathsf{*.f64}\left(angle, \mathsf{PI}\left(\right)\right)\right), \left({b}^{\color{blue}{2}} - {a}^{2}\right)\right) \]

      6. PI-lowering-PI.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{1}{90}, \mathsf{*.f64}\left(angle, \mathsf{PI.f64}\left(\right)\right)\right), \left({b}^{2} - {a}^{2}\right)\right) \]

      7. --lowering--.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{1}{90}, \mathsf{*.f64}\left(angle, \mathsf{PI.f64}\left(\right)\right)\right), \mathsf{\_.f64}\left(\left({b}^{2}\right), \color{blue}{\left({a}^{2}\right)}\right)\right) \]

      8. unpow2 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{1}{90}, \mathsf{*.f64}\left(angle, \mathsf{PI.f64}\left(\right)\right)\right), \mathsf{\_.f64}\left(\left(b \cdot b\right), \left({\color{blue}{a}}^{2}\right)\right)\right) \]

      9. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{1}{90}, \mathsf{*.f64}\left(angle, \mathsf{PI.f64}\left(\right)\right)\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(b, b\right), \left({\color{blue}{a}}^{2}\right)\right)\right) \]

      10. unpow2 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{1}{90}, \mathsf{*.f64}\left(angle, \mathsf{PI.f64}\left(\right)\right)\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(b, b\right), \left(a \cdot \color{blue}{a}\right)\right)\right) \]

      11. *-lowering-*.f64 44.9%

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{1}{90}, \mathsf{*.f64}\left(angle, \mathsf{PI.f64}\left(\right)\right)\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(b, b\right), \mathsf{*.f64}\left(a, \color{blue}{a}\right)\right)\right) \]

   7. Simplified 44.9%

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

   8. Taylor expanded in b around 0

      \[\leadsto \color{blue}{\frac{-1}{90} \cdot \left({a}^{2} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)} \]
   9. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \left({a}^{2} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right) \cdot \color{blue}{\frac{-1}{90}} \]

      2. associate-*r* N/A

         \[\leadsto {a}^{2} \cdot \color{blue}{\left(\left(angle \cdot \mathsf{PI}\left(\right)\right) \cdot \frac{-1}{90}\right)} \]

      3. *-commutative N/A

         \[\leadsto {a}^{2} \cdot \left(\frac{-1}{90} \cdot \color{blue}{\left(angle \cdot \mathsf{PI}\left(\right)\right)}\right) \]

      4. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\left({a}^{2}\right), \color{blue}{\left(\frac{-1}{90} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}\right) \]

      5. unpow2 N/A

         \[\leadsto \mathsf{*.f64}\left(\left(a \cdot a\right), \left(\color{blue}{\frac{-1}{90}} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)\right) \]

      6. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(a, a\right), \left(\color{blue}{\frac{-1}{90}} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)\right) \]

      7. *-commutative N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(a, a\right), \left(\left(angle \cdot \mathsf{PI}\left(\right)\right) \cdot \color{blue}{\frac{-1}{90}}\right)\right) \]

      8. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(a, a\right), \mathsf{*.f64}\left(\left(angle \cdot \mathsf{PI}\left(\right)\right), \color{blue}{\frac{-1}{90}}\right)\right) \]

      9. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(a, a\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(angle, \mathsf{PI}\left(\right)\right), \frac{-1}{90}\right)\right) \]

      10. PI-lowering-PI.f64 40.6%

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(a, a\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(angle, \mathsf{PI.f64}\left(\right)\right), \frac{-1}{90}\right)\right) \]

   10. Simplified 40.6%

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

       1. associate-*l* N/A

          \[\leadsto a \cdot \color{blue}{\left(a \cdot \left(\left(angle \cdot \mathsf{PI}\left(\right)\right) \cdot \frac{-1}{90}\right)\right)} \]

       2. *-commutative N/A

          \[\leadsto \left(a \cdot \left(\left(angle \cdot \mathsf{PI}\left(\right)\right) \cdot \frac{-1}{90}\right)\right) \cdot \color{blue}{a} \]

       3. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\left(a \cdot \left(\left(angle \cdot \mathsf{PI}\left(\right)\right) \cdot \frac{-1}{90}\right)\right), \color{blue}{a}\right) \]

       4. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(a, \left(\left(angle \cdot \mathsf{PI}\left(\right)\right) \cdot \frac{-1}{90}\right)\right), a\right) \]

       5. *-commutative N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(a, \left(\frac{-1}{90} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)\right), a\right) \]

       6. *-commutative N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(a, \left(\frac{-1}{90} \cdot \left(\mathsf{PI}\left(\right) \cdot angle\right)\right)\right), a\right) \]

       7. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(a, \mathsf{*.f64}\left(\frac{-1}{90}, \left(\mathsf{PI}\left(\right) \cdot angle\right)\right)\right), a\right) \]

       8. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(a, \mathsf{*.f64}\left(\frac{-1}{90}, \mathsf{*.f64}\left(\mathsf{PI}\left(\right), angle\right)\right)\right), a\right) \]

       9. PI-lowering-PI.f64 57.0%

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(a, \mathsf{*.f64}\left(\frac{-1}{90}, \mathsf{*.f64}\left(\mathsf{PI.f64}\left(\right), angle\right)\right)\right), a\right) \]

   12. Applied egg-rr 57.0%

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

       1. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\left(a \cdot \left(\frac{-1}{90} \cdot \left(\mathsf{PI}\left(\right) \cdot angle\right)\right)\right), \color{blue}{a}\right) \]

       2. *-commutative N/A

          \[\leadsto \mathsf{*.f64}\left(\left(a \cdot \left(\left(\mathsf{PI}\left(\right) \cdot angle\right) \cdot \frac{-1}{90}\right)\right), a\right) \]

       3. associate-*r* N/A

          \[\leadsto \mathsf{*.f64}\left(\left(\left(a \cdot \left(\mathsf{PI}\left(\right) \cdot angle\right)\right) \cdot \frac{-1}{90}\right), a\right) \]

       4. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\left(a \cdot \left(\mathsf{PI}\left(\right) \cdot angle\right)\right), \frac{-1}{90}\right), a\right) \]

       5. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(a, \left(\mathsf{PI}\left(\right) \cdot angle\right)\right), \frac{-1}{90}\right), a\right) \]

       6. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(a, \mathsf{*.f64}\left(\mathsf{PI}\left(\right), angle\right)\right), \frac{-1}{90}\right), a\right) \]

       7. PI-lowering-PI.f64 57.1%

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(a, \mathsf{*.f64}\left(\mathsf{PI.f64}\left(\right), angle\right)\right), \frac{-1}{90}\right), a\right) \]

   14. Applied egg-rr 57.1%

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

4. Final simplification 45.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \leq 1.75 \cdot 10^{+18}:\\ \;\;\;\;\left(b \cdot b\right) \cdot \left(\left(angle \cdot \pi\right) \cdot 0.011111111111111112\right)\\ \mathbf{else}:\\ \;\;\;\;a \cdot \left(\left(a \cdot \left(angle \cdot \pi\right)\right) \cdot -0.011111111111111112\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 17: 43.1% accurate, 29.9× speedup

Math

\[\begin{array}{l} angle\_m = \left|angle\right| \\ angle\_s = \mathsf{copysign}\left(1, angle\right) \\ angle\_s \cdot \begin{array}{l} \mathbf{if}\;a \leq 2.8 \cdot 10^{+18}:\\ \;\;\;\;\left(b \cdot b\right) \cdot \left(\left(angle\_m \cdot \pi\right) \cdot 0.011111111111111112\right)\\ \mathbf{else}:\\ \;\;\;\;a \cdot \left(a \cdot \left(\left(angle\_m \cdot \pi\right) \cdot -0.011111111111111112\right)\right)\\ \end{array} \end{array} \]

FPCore

angle\_m = (fabs.f64 angle)
angle\_s = (copysign.f64 #s(literal 1 binary64) angle)
(FPCore (angle_s a b angle_m)
 :precision binary64
 (*
  angle_s
  (if (<= a 2.8e+18)
    (* (* b b) (* (* angle_m PI) 0.011111111111111112))
    (* a (* a (* (* angle_m PI) -0.011111111111111112))))))

C

angle\_m = fabs(angle);
angle\_s = copysign(1.0, angle);
double code(double angle_s, double a, double b, double angle_m) {
	double tmp;
	if (a <= 2.8e+18) {
		tmp = (b * b) * ((angle_m * ((double) M_PI)) * 0.011111111111111112);
	} else {
		tmp = a * (a * ((angle_m * ((double) M_PI)) * -0.011111111111111112));
	}
	return angle_s * tmp;
}

Java

angle\_m = Math.abs(angle);
angle\_s = Math.copySign(1.0, angle);
public static double code(double angle_s, double a, double b, double angle_m) {
	double tmp;
	if (a <= 2.8e+18) {
		tmp = (b * b) * ((angle_m * Math.PI) * 0.011111111111111112);
	} else {
		tmp = a * (a * ((angle_m * Math.PI) * -0.011111111111111112));
	}
	return angle_s * tmp;
}

Python

angle\_m = math.fabs(angle)
angle\_s = math.copysign(1.0, angle)
def code(angle_s, a, b, angle_m):
	tmp = 0
	if a <= 2.8e+18:
		tmp = (b * b) * ((angle_m * math.pi) * 0.011111111111111112)
	else:
		tmp = a * (a * ((angle_m * math.pi) * -0.011111111111111112))
	return angle_s * tmp

Julia

angle\_m = abs(angle)
angle\_s = copysign(1.0, angle)
function code(angle_s, a, b, angle_m)
	tmp = 0.0
	if (a <= 2.8e+18)
		tmp = Float64(Float64(b * b) * Float64(Float64(angle_m * pi) * 0.011111111111111112));
	else
		tmp = Float64(a * Float64(a * Float64(Float64(angle_m * pi) * -0.011111111111111112)));
	end
	return Float64(angle_s * tmp)
end

MATLAB

angle\_m = abs(angle);
angle\_s = sign(angle) * abs(1.0);
function tmp_2 = code(angle_s, a, b, angle_m)
	tmp = 0.0;
	if (a <= 2.8e+18)
		tmp = (b * b) * ((angle_m * pi) * 0.011111111111111112);
	else
		tmp = a * (a * ((angle_m * pi) * -0.011111111111111112));
	end
	tmp_2 = angle_s * tmp;
end

Wolfram

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_, b_, angle$95$m_] := N[(angle$95$s * If[LessEqual[a, 2.8e+18], N[(N[(b * b), $MachinePrecision] * N[(N[(angle$95$m * Pi), $MachinePrecision] * 0.011111111111111112), $MachinePrecision]), $MachinePrecision], N[(a * N[(a * N[(N[(angle$95$m * Pi), $MachinePrecision] * -0.011111111111111112), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if a < 2.8e18

   1. Initial program 56.8%

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

      1. associate-*l* N/A

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

      2. *-commutative N/A

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

      3. associate-*l* N/A

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

      4. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\sin \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right), \color{blue}{\left(\cos \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right) \cdot \left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right)\right)}\right) \]

      5. sin-lowering-sin.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sin.f64}\left(\left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right)\right), \left(\color{blue}{\cos \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right)} \cdot \left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right)\right)\right) \]

      6. associate-*r/ N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sin.f64}\left(\left(\frac{\mathsf{PI}\left(\right) \cdot angle}{180}\right)\right), \left(\cos \color{blue}{\left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right)} \cdot \left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right)\right)\right) \]

      7. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sin.f64}\left(\mathsf{/.f64}\left(\left(\mathsf{PI}\left(\right) \cdot angle\right), 180\right)\right), \left(\cos \color{blue}{\left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right)} \cdot \left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right)\right)\right) \]

      8. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sin.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{PI}\left(\right), angle\right), 180\right)\right), \left(\cos \left(\color{blue}{\mathsf{PI}\left(\right)} \cdot \frac{angle}{180}\right) \cdot \left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right)\right)\right) \]

      9. PI-lowering-PI.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sin.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{PI.f64}\left(\right), angle\right), 180\right)\right), \left(\cos \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right) \cdot \left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right)\right)\right) \]

      10. *-commutative N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{sin.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{PI.f64}\left(\right), angle\right), 180\right)\right), \left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \color{blue}{\cos \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right)}\right)\right) \]

      11. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{sin.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{PI.f64}\left(\right), angle\right), 180\right)\right), \mathsf{*.f64}\left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right), \color{blue}{\cos \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right)}\right)\right) \]

   3. Simplified 56.8%

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

   5. Taylor expanded in angle around 0

      \[\leadsto \color{blue}{\frac{1}{90} \cdot \left(angle \cdot \left(\mathsf{PI}\left(\right) \cdot \left({b}^{2} - {a}^{2}\right)\right)\right)} \]
   6. Step-by-step derivation

      1. associate-*r* N/A

         \[\leadsto \frac{1}{90} \cdot \left(\left(angle \cdot \mathsf{PI}\left(\right)\right) \cdot \color{blue}{\left({b}^{2} - {a}^{2}\right)}\right) \]

      2. associate-*r* N/A

         \[\leadsto \left(\frac{1}{90} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right) \cdot \color{blue}{\left({b}^{2} - {a}^{2}\right)} \]

      3. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\left(\frac{1}{90} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right), \color{blue}{\left({b}^{2} - {a}^{2}\right)}\right) \]

      4. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{1}{90}, \left(angle \cdot \mathsf{PI}\left(\right)\right)\right), \left(\color{blue}{{b}^{2}} - {a}^{2}\right)\right) \]

      5. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{1}{90}, \mathsf{*.f64}\left(angle, \mathsf{PI}\left(\right)\right)\right), \left({b}^{\color{blue}{2}} - {a}^{2}\right)\right) \]

      6. PI-lowering-PI.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{1}{90}, \mathsf{*.f64}\left(angle, \mathsf{PI.f64}\left(\right)\right)\right), \left({b}^{2} - {a}^{2}\right)\right) \]

      7. --lowering--.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{1}{90}, \mathsf{*.f64}\left(angle, \mathsf{PI.f64}\left(\right)\right)\right), \mathsf{\_.f64}\left(\left({b}^{2}\right), \color{blue}{\left({a}^{2}\right)}\right)\right) \]

      8. unpow2 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{1}{90}, \mathsf{*.f64}\left(angle, \mathsf{PI.f64}\left(\right)\right)\right), \mathsf{\_.f64}\left(\left(b \cdot b\right), \left({\color{blue}{a}}^{2}\right)\right)\right) \]

      9. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{1}{90}, \mathsf{*.f64}\left(angle, \mathsf{PI.f64}\left(\right)\right)\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(b, b\right), \left({\color{blue}{a}}^{2}\right)\right)\right) \]

      10. unpow2 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{1}{90}, \mathsf{*.f64}\left(angle, \mathsf{PI.f64}\left(\right)\right)\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(b, b\right), \left(a \cdot \color{blue}{a}\right)\right)\right) \]

      11. *-lowering-*.f64 53.0%

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{1}{90}, \mathsf{*.f64}\left(angle, \mathsf{PI.f64}\left(\right)\right)\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(b, b\right), \mathsf{*.f64}\left(a, \color{blue}{a}\right)\right)\right) \]

   7. Simplified 53.0%

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

   8. Taylor expanded in b around inf

      \[\leadsto \color{blue}{\frac{1}{90} \cdot \left(angle \cdot \left({b}^{2} \cdot \mathsf{PI}\left(\right)\right)\right)} \]
   9. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \frac{1}{90} \cdot \left(angle \cdot \left(\mathsf{PI}\left(\right) \cdot \color{blue}{{b}^{2}}\right)\right) \]

      2. associate-*r* N/A

         \[\leadsto \frac{1}{90} \cdot \left(\left(angle \cdot \mathsf{PI}\left(\right)\right) \cdot \color{blue}{{b}^{2}}\right) \]

      3. associate-*l* N/A

         \[\leadsto \left(\frac{1}{90} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right) \cdot \color{blue}{{b}^{2}} \]

      4. *-commutative N/A

         \[\leadsto {b}^{2} \cdot \color{blue}{\left(\frac{1}{90} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)} \]

      5. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\left({b}^{2}\right), \color{blue}{\left(\frac{1}{90} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}\right) \]

      6. unpow2 N/A

         \[\leadsto \mathsf{*.f64}\left(\left(b \cdot b\right), \left(\color{blue}{\frac{1}{90}} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)\right) \]

      7. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(b, b\right), \left(\color{blue}{\frac{1}{90}} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)\right) \]

      8. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(b, b\right), \mathsf{*.f64}\left(\frac{1}{90}, \color{blue}{\left(angle \cdot \mathsf{PI}\left(\right)\right)}\right)\right) \]

      9. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(b, b\right), \mathsf{*.f64}\left(\frac{1}{90}, \mathsf{*.f64}\left(angle, \color{blue}{\mathsf{PI}\left(\right)}\right)\right)\right) \]

      10. PI-lowering-PI.f64 42.8%

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(b, b\right), \mathsf{*.f64}\left(\frac{1}{90}, \mathsf{*.f64}\left(angle, \mathsf{PI.f64}\left(\right)\right)\right)\right) \]

   10. Simplified 42.8%

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

   if 2.8e18 < a

   1. Initial program 45.3%

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

      1. associate-*l* N/A

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

      2. *-commutative N/A

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

      3. associate-*l* N/A

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

      4. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\sin \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right), \color{blue}{\left(\cos \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right) \cdot \left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right)\right)}\right) \]

      5. sin-lowering-sin.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sin.f64}\left(\left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right)\right), \left(\color{blue}{\cos \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right)} \cdot \left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right)\right)\right) \]

      6. associate-*r/ N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sin.f64}\left(\left(\frac{\mathsf{PI}\left(\right) \cdot angle}{180}\right)\right), \left(\cos \color{blue}{\left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right)} \cdot \left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right)\right)\right) \]

      7. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sin.f64}\left(\mathsf{/.f64}\left(\left(\mathsf{PI}\left(\right) \cdot angle\right), 180\right)\right), \left(\cos \color{blue}{\left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right)} \cdot \left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right)\right)\right) \]

      8. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sin.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{PI}\left(\right), angle\right), 180\right)\right), \left(\cos \left(\color{blue}{\mathsf{PI}\left(\right)} \cdot \frac{angle}{180}\right) \cdot \left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right)\right)\right) \]

      9. PI-lowering-PI.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sin.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{PI.f64}\left(\right), angle\right), 180\right)\right), \left(\cos \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right) \cdot \left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right)\right)\right) \]

      10. *-commutative N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{sin.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{PI.f64}\left(\right), angle\right), 180\right)\right), \left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \color{blue}{\cos \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right)}\right)\right) \]

      11. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{sin.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{PI.f64}\left(\right), angle\right), 180\right)\right), \mathsf{*.f64}\left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right), \color{blue}{\cos \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right)}\right)\right) \]

   3. Simplified 45.6%

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

   5. Taylor expanded in angle around 0

      \[\leadsto \color{blue}{\frac{1}{90} \cdot \left(angle \cdot \left(\mathsf{PI}\left(\right) \cdot \left({b}^{2} - {a}^{2}\right)\right)\right)} \]
   6. Step-by-step derivation

      1. associate-*r* N/A

         \[\leadsto \frac{1}{90} \cdot \left(\left(angle \cdot \mathsf{PI}\left(\right)\right) \cdot \color{blue}{\left({b}^{2} - {a}^{2}\right)}\right) \]

      2. associate-*r* N/A

         \[\leadsto \left(\frac{1}{90} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right) \cdot \color{blue}{\left({b}^{2} - {a}^{2}\right)} \]

      3. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\left(\frac{1}{90} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right), \color{blue}{\left({b}^{2} - {a}^{2}\right)}\right) \]

      4. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{1}{90}, \left(angle \cdot \mathsf{PI}\left(\right)\right)\right), \left(\color{blue}{{b}^{2}} - {a}^{2}\right)\right) \]

      5. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{1}{90}, \mathsf{*.f64}\left(angle, \mathsf{PI}\left(\right)\right)\right), \left({b}^{\color{blue}{2}} - {a}^{2}\right)\right) \]

      6. PI-lowering-PI.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{1}{90}, \mathsf{*.f64}\left(angle, \mathsf{PI.f64}\left(\right)\right)\right), \left({b}^{2} - {a}^{2}\right)\right) \]

      7. --lowering--.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{1}{90}, \mathsf{*.f64}\left(angle, \mathsf{PI.f64}\left(\right)\right)\right), \mathsf{\_.f64}\left(\left({b}^{2}\right), \color{blue}{\left({a}^{2}\right)}\right)\right) \]

      8. unpow2 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{1}{90}, \mathsf{*.f64}\left(angle, \mathsf{PI.f64}\left(\right)\right)\right), \mathsf{\_.f64}\left(\left(b \cdot b\right), \left({\color{blue}{a}}^{2}\right)\right)\right) \]

      9. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{1}{90}, \mathsf{*.f64}\left(angle, \mathsf{PI.f64}\left(\right)\right)\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(b, b\right), \left({\color{blue}{a}}^{2}\right)\right)\right) \]

      10. unpow2 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{1}{90}, \mathsf{*.f64}\left(angle, \mathsf{PI.f64}\left(\right)\right)\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(b, b\right), \left(a \cdot \color{blue}{a}\right)\right)\right) \]

      11. *-lowering-*.f64 44.9%

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{1}{90}, \mathsf{*.f64}\left(angle, \mathsf{PI.f64}\left(\right)\right)\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(b, b\right), \mathsf{*.f64}\left(a, \color{blue}{a}\right)\right)\right) \]

   7. Simplified 44.9%

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

   8. Taylor expanded in b around 0

      \[\leadsto \color{blue}{\frac{-1}{90} \cdot \left({a}^{2} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)} \]
   9. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \left({a}^{2} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right) \cdot \color{blue}{\frac{-1}{90}} \]

      2. associate-*r* N/A

         \[\leadsto {a}^{2} \cdot \color{blue}{\left(\left(angle \cdot \mathsf{PI}\left(\right)\right) \cdot \frac{-1}{90}\right)} \]

      3. *-commutative N/A

         \[\leadsto {a}^{2} \cdot \left(\frac{-1}{90} \cdot \color{blue}{\left(angle \cdot \mathsf{PI}\left(\right)\right)}\right) \]

      4. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\left({a}^{2}\right), \color{blue}{\left(\frac{-1}{90} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}\right) \]

      5. unpow2 N/A

         \[\leadsto \mathsf{*.f64}\left(\left(a \cdot a\right), \left(\color{blue}{\frac{-1}{90}} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)\right) \]

      6. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(a, a\right), \left(\color{blue}{\frac{-1}{90}} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)\right) \]

      7. *-commutative N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(a, a\right), \left(\left(angle \cdot \mathsf{PI}\left(\right)\right) \cdot \color{blue}{\frac{-1}{90}}\right)\right) \]

      8. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(a, a\right), \mathsf{*.f64}\left(\left(angle \cdot \mathsf{PI}\left(\right)\right), \color{blue}{\frac{-1}{90}}\right)\right) \]

      9. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(a, a\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(angle, \mathsf{PI}\left(\right)\right), \frac{-1}{90}\right)\right) \]

      10. PI-lowering-PI.f64 40.6%

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(a, a\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(angle, \mathsf{PI.f64}\left(\right)\right), \frac{-1}{90}\right)\right) \]

   10. Simplified 40.6%

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

       1. associate-*l* N/A

          \[\leadsto a \cdot \color{blue}{\left(a \cdot \left(\left(angle \cdot \mathsf{PI}\left(\right)\right) \cdot \frac{-1}{90}\right)\right)} \]

       2. *-commutative N/A

          \[\leadsto \left(a \cdot \left(\left(angle \cdot \mathsf{PI}\left(\right)\right) \cdot \frac{-1}{90}\right)\right) \cdot \color{blue}{a} \]

       3. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\left(a \cdot \left(\left(angle \cdot \mathsf{PI}\left(\right)\right) \cdot \frac{-1}{90}\right)\right), \color{blue}{a}\right) \]

       4. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(a, \left(\left(angle \cdot \mathsf{PI}\left(\right)\right) \cdot \frac{-1}{90}\right)\right), a\right) \]

       5. *-commutative N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(a, \left(\frac{-1}{90} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)\right), a\right) \]

       6. *-commutative N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(a, \left(\frac{-1}{90} \cdot \left(\mathsf{PI}\left(\right) \cdot angle\right)\right)\right), a\right) \]

       7. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(a, \mathsf{*.f64}\left(\frac{-1}{90}, \left(\mathsf{PI}\left(\right) \cdot angle\right)\right)\right), a\right) \]

       8. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(a, \mathsf{*.f64}\left(\frac{-1}{90}, \mathsf{*.f64}\left(\mathsf{PI}\left(\right), angle\right)\right)\right), a\right) \]

       9. PI-lowering-PI.f64 57.0%

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(a, \mathsf{*.f64}\left(\frac{-1}{90}, \mathsf{*.f64}\left(\mathsf{PI.f64}\left(\right), angle\right)\right)\right), a\right) \]

   12. Applied egg-rr 57.0%

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

4. Final simplification 45.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \leq 2.8 \cdot 10^{+18}:\\ \;\;\;\;\left(b \cdot b\right) \cdot \left(\left(angle \cdot \pi\right) \cdot 0.011111111111111112\right)\\ \mathbf{else}:\\ \;\;\;\;a \cdot \left(a \cdot \left(\left(angle \cdot \pi\right) \cdot -0.011111111111111112\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 18: 62.9% accurate, 32.2× speedup

Math

\[\begin{array}{l} angle\_m = \left|angle\right| \\ angle\_s = \mathsf{copysign}\left(1, angle\right) \\ angle\_s \cdot \left(\left(b + a\right) \cdot \left(\left(b - a\right) \cdot \left(\left(angle\_m \cdot \pi\right) \cdot 0.011111111111111112\right)\right)\right) \end{array} \]

FPCore

angle\_m = (fabs.f64 angle)
angle\_s = (copysign.f64 #s(literal 1 binary64) angle)
(FPCore (angle_s a b angle_m)
 :precision binary64
 (* angle_s (* (+ b a) (* (- b a) (* (* angle_m PI) 0.011111111111111112)))))

C

angle\_m = fabs(angle);
angle\_s = copysign(1.0, angle);
double code(double angle_s, double a, double b, double angle_m) {
	return angle_s * ((b + a) * ((b - a) * ((angle_m * ((double) M_PI)) * 0.011111111111111112)));
}

Java

angle\_m = Math.abs(angle);
angle\_s = Math.copySign(1.0, angle);
public static double code(double angle_s, double a, double b, double angle_m) {
	return angle_s * ((b + a) * ((b - a) * ((angle_m * Math.PI) * 0.011111111111111112)));
}

Python

angle\_m = math.fabs(angle)
angle\_s = math.copysign(1.0, angle)
def code(angle_s, a, b, angle_m):
	return angle_s * ((b + a) * ((b - a) * ((angle_m * math.pi) * 0.011111111111111112)))

Julia

angle\_m = abs(angle)
angle\_s = copysign(1.0, angle)
function code(angle_s, a, b, angle_m)
	return Float64(angle_s * Float64(Float64(b + a) * Float64(Float64(b - a) * Float64(Float64(angle_m * pi) * 0.011111111111111112))))
end

MATLAB

angle\_m = abs(angle);
angle\_s = sign(angle) * abs(1.0);
function tmp = code(angle_s, a, b, angle_m)
	tmp = angle_s * ((b + a) * ((b - a) * ((angle_m * pi) * 0.011111111111111112)));
end

Wolfram

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_, b_, angle$95$m_] := N[(angle$95$s * N[(N[(b + a), $MachinePrecision] * N[(N[(b - a), $MachinePrecision] * N[(N[(angle$95$m * Pi), $MachinePrecision] * 0.011111111111111112), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 54.6%

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

   1. associate-*l* N/A

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

   2. *-commutative N/A

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

   3. associate-*l* N/A

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

   4. *-lowering-*.f64 N/A

      \[\leadsto \mathsf{*.f64}\left(\sin \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right), \color{blue}{\left(\cos \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right) \cdot \left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right)\right)}\right) \]

   5. sin-lowering-sin.f64 N/A

      \[\leadsto \mathsf{*.f64}\left(\mathsf{sin.f64}\left(\left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right)\right), \left(\color{blue}{\cos \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right)} \cdot \left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right)\right)\right) \]

   6. associate-*r/ N/A

      \[\leadsto \mathsf{*.f64}\left(\mathsf{sin.f64}\left(\left(\frac{\mathsf{PI}\left(\right) \cdot angle}{180}\right)\right), \left(\cos \color{blue}{\left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right)} \cdot \left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right)\right)\right) \]

   7. /-lowering-/.f64 N/A

      \[\leadsto \mathsf{*.f64}\left(\mathsf{sin.f64}\left(\mathsf{/.f64}\left(\left(\mathsf{PI}\left(\right) \cdot angle\right), 180\right)\right), \left(\cos \color{blue}{\left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right)} \cdot \left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right)\right)\right) \]

   8. *-lowering-*.f64 N/A

      \[\leadsto \mathsf{*.f64}\left(\mathsf{sin.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{PI}\left(\right), angle\right), 180\right)\right), \left(\cos \left(\color{blue}{\mathsf{PI}\left(\right)} \cdot \frac{angle}{180}\right) \cdot \left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right)\right)\right) \]

   9. PI-lowering-PI.f64 N/A

      \[\leadsto \mathsf{*.f64}\left(\mathsf{sin.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{PI.f64}\left(\right), angle\right), 180\right)\right), \left(\cos \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right) \cdot \left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right)\right)\right) \]

   10. *-commutative N/A

       \[\leadsto \mathsf{*.f64}\left(\mathsf{sin.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{PI.f64}\left(\right), angle\right), 180\right)\right), \left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \color{blue}{\cos \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right)}\right)\right) \]

   11. *-lowering-*.f64 N/A

       \[\leadsto \mathsf{*.f64}\left(\mathsf{sin.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{PI.f64}\left(\right), angle\right), 180\right)\right), \mathsf{*.f64}\left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right), \color{blue}{\cos \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right)}\right)\right) \]

3. Simplified 54.7%

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

5. Taylor expanded in angle around 0

   \[\leadsto \color{blue}{\frac{1}{90} \cdot \left(angle \cdot \left(\mathsf{PI}\left(\right) \cdot \left({b}^{2} - {a}^{2}\right)\right)\right)} \]
6. Step-by-step derivation

   1. associate-*r* N/A

      \[\leadsto \frac{1}{90} \cdot \left(\left(angle \cdot \mathsf{PI}\left(\right)\right) \cdot \color{blue}{\left({b}^{2} - {a}^{2}\right)}\right) \]

   2. associate-*r* N/A

      \[\leadsto \left(\frac{1}{90} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right) \cdot \color{blue}{\left({b}^{2} - {a}^{2}\right)} \]

   3. *-lowering-*.f64 N/A

      \[\leadsto \mathsf{*.f64}\left(\left(\frac{1}{90} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right), \color{blue}{\left({b}^{2} - {a}^{2}\right)}\right) \]

   4. *-lowering-*.f64 N/A

      \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{1}{90}, \left(angle \cdot \mathsf{PI}\left(\right)\right)\right), \left(\color{blue}{{b}^{2}} - {a}^{2}\right)\right) \]

   5. *-lowering-*.f64 N/A

      \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{1}{90}, \mathsf{*.f64}\left(angle, \mathsf{PI}\left(\right)\right)\right), \left({b}^{\color{blue}{2}} - {a}^{2}\right)\right) \]

   6. PI-lowering-PI.f64 N/A

      \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{1}{90}, \mathsf{*.f64}\left(angle, \mathsf{PI.f64}\left(\right)\right)\right), \left({b}^{2} - {a}^{2}\right)\right) \]

   7. --lowering--.f64 N/A

      \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{1}{90}, \mathsf{*.f64}\left(angle, \mathsf{PI.f64}\left(\right)\right)\right), \mathsf{\_.f64}\left(\left({b}^{2}\right), \color{blue}{\left({a}^{2}\right)}\right)\right) \]

   8. unpow2 N/A

      \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{1}{90}, \mathsf{*.f64}\left(angle, \mathsf{PI.f64}\left(\right)\right)\right), \mathsf{\_.f64}\left(\left(b \cdot b\right), \left({\color{blue}{a}}^{2}\right)\right)\right) \]

   9. *-lowering-*.f64 N/A

      \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{1}{90}, \mathsf{*.f64}\left(angle, \mathsf{PI.f64}\left(\right)\right)\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(b, b\right), \left({\color{blue}{a}}^{2}\right)\right)\right) \]

   10. unpow2 N/A

       \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{1}{90}, \mathsf{*.f64}\left(angle, \mathsf{PI.f64}\left(\right)\right)\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(b, b\right), \left(a \cdot \color{blue}{a}\right)\right)\right) \]

   11. *-lowering-*.f64 51.4%

       \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{1}{90}, \mathsf{*.f64}\left(angle, \mathsf{PI.f64}\left(\right)\right)\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(b, b\right), \mathsf{*.f64}\left(a, \color{blue}{a}\right)\right)\right) \]

7. Simplified 51.4%

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

   1. *-commutative N/A

      \[\leadsto \left(b \cdot b - a \cdot a\right) \cdot \color{blue}{\left(\frac{1}{90} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)} \]

   2. difference-of-squares N/A

      \[\leadsto \left(\left(b + a\right) \cdot \left(b - a\right)\right) \cdot \left(\color{blue}{\frac{1}{90}} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right) \]

   3. associate-*l* N/A

      \[\leadsto \left(b + a\right) \cdot \color{blue}{\left(\left(b - a\right) \cdot \left(\frac{1}{90} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)\right)} \]

   4. *-lowering-*.f64 N/A

      \[\leadsto \mathsf{*.f64}\left(\left(b + a\right), \color{blue}{\left(\left(b - a\right) \cdot \left(\frac{1}{90} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)\right)}\right) \]

   5. +-lowering-+.f64 N/A

      \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(b, a\right), \left(\color{blue}{\left(b - a\right)} \cdot \left(\frac{1}{90} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)\right)\right) \]

   6. *-lowering-*.f64 N/A

      \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(b, a\right), \mathsf{*.f64}\left(\left(b - a\right), \color{blue}{\left(\frac{1}{90} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}\right)\right) \]

   7. --lowering--.f64 N/A

      \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(b, a\right), \mathsf{*.f64}\left(\mathsf{\_.f64}\left(b, a\right), \left(\color{blue}{\frac{1}{90}} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)\right)\right) \]

   8. *-commutative N/A

      \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(b, a\right), \mathsf{*.f64}\left(\mathsf{\_.f64}\left(b, a\right), \left(\frac{1}{90} \cdot \left(\mathsf{PI}\left(\right) \cdot \color{blue}{angle}\right)\right)\right)\right) \]

   9. *-lowering-*.f64 N/A

      \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(b, a\right), \mathsf{*.f64}\left(\mathsf{\_.f64}\left(b, a\right), \mathsf{*.f64}\left(\frac{1}{90}, \color{blue}{\left(\mathsf{PI}\left(\right) \cdot angle\right)}\right)\right)\right) \]

   10. *-lowering-*.f64 N/A

       \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(b, a\right), \mathsf{*.f64}\left(\mathsf{\_.f64}\left(b, a\right), \mathsf{*.f64}\left(\frac{1}{90}, \mathsf{*.f64}\left(\mathsf{PI}\left(\right), \color{blue}{angle}\right)\right)\right)\right) \]

   11. PI-lowering-PI.f64 64.3%

       \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(b, a\right), \mathsf{*.f64}\left(\mathsf{\_.f64}\left(b, a\right), \mathsf{*.f64}\left(\frac{1}{90}, \mathsf{*.f64}\left(\mathsf{PI.f64}\left(\right), angle\right)\right)\right)\right) \]

9. Applied egg-rr 64.3%

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

10. Final simplification 64.3%

    \[\leadsto \left(b + a\right) \cdot \left(\left(b - a\right) \cdot \left(\left(angle \cdot \pi\right) \cdot 0.011111111111111112\right)\right) \]
11. Add Preprocessing

Alternative 19: 38.0% accurate, 46.6× speedup

Math

\[\begin{array}{l} angle\_m = \left|angle\right| \\ angle\_s = \mathsf{copysign}\left(1, angle\right) \\ angle\_s \cdot \left(a \cdot \left(a \cdot \left(\left(angle\_m \cdot \pi\right) \cdot -0.011111111111111112\right)\right)\right) \end{array} \]

FPCore

angle\_m = (fabs.f64 angle)
angle\_s = (copysign.f64 #s(literal 1 binary64) angle)
(FPCore (angle_s a b angle_m)
 :precision binary64
 (* angle_s (* a (* a (* (* angle_m PI) -0.011111111111111112)))))

C

angle\_m = fabs(angle);
angle\_s = copysign(1.0, angle);
double code(double angle_s, double a, double b, double angle_m) {
	return angle_s * (a * (a * ((angle_m * ((double) M_PI)) * -0.011111111111111112)));
}

Java

angle\_m = Math.abs(angle);
angle\_s = Math.copySign(1.0, angle);
public static double code(double angle_s, double a, double b, double angle_m) {
	return angle_s * (a * (a * ((angle_m * Math.PI) * -0.011111111111111112)));
}

Python

angle\_m = math.fabs(angle)
angle\_s = math.copysign(1.0, angle)
def code(angle_s, a, b, angle_m):
	return angle_s * (a * (a * ((angle_m * math.pi) * -0.011111111111111112)))

Julia

angle\_m = abs(angle)
angle\_s = copysign(1.0, angle)
function code(angle_s, a, b, angle_m)
	return Float64(angle_s * Float64(a * Float64(a * Float64(Float64(angle_m * pi) * -0.011111111111111112))))
end

MATLAB

angle\_m = abs(angle);
angle\_s = sign(angle) * abs(1.0);
function tmp = code(angle_s, a, b, angle_m)
	tmp = angle_s * (a * (a * ((angle_m * pi) * -0.011111111111111112)));
end

Wolfram

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_, b_, angle$95$m_] := N[(angle$95$s * N[(a * N[(a * N[(N[(angle$95$m * Pi), $MachinePrecision] * -0.011111111111111112), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 54.6%

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

   1. associate-*l* N/A

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

   2. *-commutative N/A

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

   3. associate-*l* N/A

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

   4. *-lowering-*.f64 N/A

      \[\leadsto \mathsf{*.f64}\left(\sin \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right), \color{blue}{\left(\cos \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right) \cdot \left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right)\right)}\right) \]

   5. sin-lowering-sin.f64 N/A

      \[\leadsto \mathsf{*.f64}\left(\mathsf{sin.f64}\left(\left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right)\right), \left(\color{blue}{\cos \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right)} \cdot \left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right)\right)\right) \]

   6. associate-*r/ N/A

      \[\leadsto \mathsf{*.f64}\left(\mathsf{sin.f64}\left(\left(\frac{\mathsf{PI}\left(\right) \cdot angle}{180}\right)\right), \left(\cos \color{blue}{\left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right)} \cdot \left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right)\right)\right) \]

   7. /-lowering-/.f64 N/A

      \[\leadsto \mathsf{*.f64}\left(\mathsf{sin.f64}\left(\mathsf{/.f64}\left(\left(\mathsf{PI}\left(\right) \cdot angle\right), 180\right)\right), \left(\cos \color{blue}{\left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right)} \cdot \left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right)\right)\right) \]

   8. *-lowering-*.f64 N/A

      \[\leadsto \mathsf{*.f64}\left(\mathsf{sin.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{PI}\left(\right), angle\right), 180\right)\right), \left(\cos \left(\color{blue}{\mathsf{PI}\left(\right)} \cdot \frac{angle}{180}\right) \cdot \left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right)\right)\right) \]

   9. PI-lowering-PI.f64 N/A

      \[\leadsto \mathsf{*.f64}\left(\mathsf{sin.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{PI.f64}\left(\right), angle\right), 180\right)\right), \left(\cos \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right) \cdot \left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right)\right)\right) \]

   10. *-commutative N/A

       \[\leadsto \mathsf{*.f64}\left(\mathsf{sin.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{PI.f64}\left(\right), angle\right), 180\right)\right), \left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \color{blue}{\cos \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right)}\right)\right) \]

   11. *-lowering-*.f64 N/A

       \[\leadsto \mathsf{*.f64}\left(\mathsf{sin.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{PI.f64}\left(\right), angle\right), 180\right)\right), \mathsf{*.f64}\left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right), \color{blue}{\cos \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right)}\right)\right) \]

3. Simplified 54.7%

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

5. Taylor expanded in angle around 0

   \[\leadsto \color{blue}{\frac{1}{90} \cdot \left(angle \cdot \left(\mathsf{PI}\left(\right) \cdot \left({b}^{2} - {a}^{2}\right)\right)\right)} \]
6. Step-by-step derivation

   1. associate-*r* N/A

      \[\leadsto \frac{1}{90} \cdot \left(\left(angle \cdot \mathsf{PI}\left(\right)\right) \cdot \color{blue}{\left({b}^{2} - {a}^{2}\right)}\right) \]

   2. associate-*r* N/A

      \[\leadsto \left(\frac{1}{90} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right) \cdot \color{blue}{\left({b}^{2} - {a}^{2}\right)} \]

   3. *-lowering-*.f64 N/A

      \[\leadsto \mathsf{*.f64}\left(\left(\frac{1}{90} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right), \color{blue}{\left({b}^{2} - {a}^{2}\right)}\right) \]

   4. *-lowering-*.f64 N/A

      \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{1}{90}, \left(angle \cdot \mathsf{PI}\left(\right)\right)\right), \left(\color{blue}{{b}^{2}} - {a}^{2}\right)\right) \]

   5. *-lowering-*.f64 N/A

      \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{1}{90}, \mathsf{*.f64}\left(angle, \mathsf{PI}\left(\right)\right)\right), \left({b}^{\color{blue}{2}} - {a}^{2}\right)\right) \]

   6. PI-lowering-PI.f64 N/A

      \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{1}{90}, \mathsf{*.f64}\left(angle, \mathsf{PI.f64}\left(\right)\right)\right), \left({b}^{2} - {a}^{2}\right)\right) \]

   7. --lowering--.f64 N/A

      \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{1}{90}, \mathsf{*.f64}\left(angle, \mathsf{PI.f64}\left(\right)\right)\right), \mathsf{\_.f64}\left(\left({b}^{2}\right), \color{blue}{\left({a}^{2}\right)}\right)\right) \]

   8. unpow2 N/A

      \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{1}{90}, \mathsf{*.f64}\left(angle, \mathsf{PI.f64}\left(\right)\right)\right), \mathsf{\_.f64}\left(\left(b \cdot b\right), \left({\color{blue}{a}}^{2}\right)\right)\right) \]

   9. *-lowering-*.f64 N/A

      \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{1}{90}, \mathsf{*.f64}\left(angle, \mathsf{PI.f64}\left(\right)\right)\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(b, b\right), \left({\color{blue}{a}}^{2}\right)\right)\right) \]

   10. unpow2 N/A

       \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{1}{90}, \mathsf{*.f64}\left(angle, \mathsf{PI.f64}\left(\right)\right)\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(b, b\right), \left(a \cdot \color{blue}{a}\right)\right)\right) \]

   11. *-lowering-*.f64 51.4%

       \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{1}{90}, \mathsf{*.f64}\left(angle, \mathsf{PI.f64}\left(\right)\right)\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(b, b\right), \mathsf{*.f64}\left(a, \color{blue}{a}\right)\right)\right) \]

7. Simplified 51.4%

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

8. Taylor expanded in b around 0

   \[\leadsto \color{blue}{\frac{-1}{90} \cdot \left({a}^{2} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)} \]
9. Step-by-step derivation

   1. *-commutative N/A

      \[\leadsto \left({a}^{2} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right) \cdot \color{blue}{\frac{-1}{90}} \]

   2. associate-*r* N/A

      \[\leadsto {a}^{2} \cdot \color{blue}{\left(\left(angle \cdot \mathsf{PI}\left(\right)\right) \cdot \frac{-1}{90}\right)} \]

   3. *-commutative N/A

      \[\leadsto {a}^{2} \cdot \left(\frac{-1}{90} \cdot \color{blue}{\left(angle \cdot \mathsf{PI}\left(\right)\right)}\right) \]

   4. *-lowering-*.f64 N/A

      \[\leadsto \mathsf{*.f64}\left(\left({a}^{2}\right), \color{blue}{\left(\frac{-1}{90} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}\right) \]

   5. unpow2 N/A

      \[\leadsto \mathsf{*.f64}\left(\left(a \cdot a\right), \left(\color{blue}{\frac{-1}{90}} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)\right) \]

   6. *-lowering-*.f64 N/A

      \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(a, a\right), \left(\color{blue}{\frac{-1}{90}} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)\right) \]

   7. *-commutative N/A

      \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(a, a\right), \left(\left(angle \cdot \mathsf{PI}\left(\right)\right) \cdot \color{blue}{\frac{-1}{90}}\right)\right) \]

   8. *-lowering-*.f64 N/A

      \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(a, a\right), \mathsf{*.f64}\left(\left(angle \cdot \mathsf{PI}\left(\right)\right), \color{blue}{\frac{-1}{90}}\right)\right) \]

   9. *-lowering-*.f64 N/A

      \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(a, a\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(angle, \mathsf{PI}\left(\right)\right), \frac{-1}{90}\right)\right) \]

   10. PI-lowering-PI.f64 31.0%

       \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(a, a\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(angle, \mathsf{PI.f64}\left(\right)\right), \frac{-1}{90}\right)\right) \]

10. Simplified 31.0%

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

    1. associate-*l* N/A

       \[\leadsto a \cdot \color{blue}{\left(a \cdot \left(\left(angle \cdot \mathsf{PI}\left(\right)\right) \cdot \frac{-1}{90}\right)\right)} \]

    2. *-commutative N/A

       \[\leadsto \left(a \cdot \left(\left(angle \cdot \mathsf{PI}\left(\right)\right) \cdot \frac{-1}{90}\right)\right) \cdot \color{blue}{a} \]

    3. *-lowering-*.f64 N/A

       \[\leadsto \mathsf{*.f64}\left(\left(a \cdot \left(\left(angle \cdot \mathsf{PI}\left(\right)\right) \cdot \frac{-1}{90}\right)\right), \color{blue}{a}\right) \]

    4. *-lowering-*.f64 N/A

       \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(a, \left(\left(angle \cdot \mathsf{PI}\left(\right)\right) \cdot \frac{-1}{90}\right)\right), a\right) \]

    5. *-commutative N/A

       \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(a, \left(\frac{-1}{90} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)\right), a\right) \]

    6. *-commutative N/A

       \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(a, \left(\frac{-1}{90} \cdot \left(\mathsf{PI}\left(\right) \cdot angle\right)\right)\right), a\right) \]

    7. *-lowering-*.f64 N/A

       \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(a, \mathsf{*.f64}\left(\frac{-1}{90}, \left(\mathsf{PI}\left(\right) \cdot angle\right)\right)\right), a\right) \]

    8. *-lowering-*.f64 N/A

       \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(a, \mathsf{*.f64}\left(\frac{-1}{90}, \mathsf{*.f64}\left(\mathsf{PI}\left(\right), angle\right)\right)\right), a\right) \]

    9. PI-lowering-PI.f64 36.4%

       \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(a, \mathsf{*.f64}\left(\frac{-1}{90}, \mathsf{*.f64}\left(\mathsf{PI.f64}\left(\right), angle\right)\right)\right), a\right) \]

12. Applied egg-rr 36.4%

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

13. Final simplification 36.4%

    \[\leadsto a \cdot \left(a \cdot \left(\left(angle \cdot \pi\right) \cdot -0.011111111111111112\right)\right) \]
14. Add Preprocessing

Alternative 20: 34.1% accurate, 46.6× speedup

Math

\[\begin{array}{l} angle\_m = \left|angle\right| \\ angle\_s = \mathsf{copysign}\left(1, angle\right) \\ angle\_s \cdot \left(\left(a \cdot a\right) \cdot \left(\left(angle\_m \cdot \pi\right) \cdot -0.011111111111111112\right)\right) \end{array} \]

FPCore

angle\_m = (fabs.f64 angle)
angle\_s = (copysign.f64 #s(literal 1 binary64) angle)
(FPCore (angle_s a b angle_m)
 :precision binary64
 (* angle_s (* (* a a) (* (* angle_m PI) -0.011111111111111112))))

C

angle\_m = fabs(angle);
angle\_s = copysign(1.0, angle);
double code(double angle_s, double a, double b, double angle_m) {
	return angle_s * ((a * a) * ((angle_m * ((double) M_PI)) * -0.011111111111111112));
}

Java

angle\_m = Math.abs(angle);
angle\_s = Math.copySign(1.0, angle);
public static double code(double angle_s, double a, double b, double angle_m) {
	return angle_s * ((a * a) * ((angle_m * Math.PI) * -0.011111111111111112));
}

Python

angle\_m = math.fabs(angle)
angle\_s = math.copysign(1.0, angle)
def code(angle_s, a, b, angle_m):
	return angle_s * ((a * a) * ((angle_m * math.pi) * -0.011111111111111112))

Julia

angle\_m = abs(angle)
angle\_s = copysign(1.0, angle)
function code(angle_s, a, b, angle_m)
	return Float64(angle_s * Float64(Float64(a * a) * Float64(Float64(angle_m * pi) * -0.011111111111111112)))
end

MATLAB

angle\_m = abs(angle);
angle\_s = sign(angle) * abs(1.0);
function tmp = code(angle_s, a, b, angle_m)
	tmp = angle_s * ((a * a) * ((angle_m * pi) * -0.011111111111111112));
end

Wolfram

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_, b_, angle$95$m_] := N[(angle$95$s * N[(N[(a * a), $MachinePrecision] * N[(N[(angle$95$m * Pi), $MachinePrecision] * -0.011111111111111112), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 54.6%

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

   1. associate-*l* N/A

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

   2. *-commutative N/A

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

   3. associate-*l* N/A

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

   4. *-lowering-*.f64 N/A

      \[\leadsto \mathsf{*.f64}\left(\sin \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right), \color{blue}{\left(\cos \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right) \cdot \left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right)\right)}\right) \]

   5. sin-lowering-sin.f64 N/A

      \[\leadsto \mathsf{*.f64}\left(\mathsf{sin.f64}\left(\left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right)\right), \left(\color{blue}{\cos \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right)} \cdot \left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right)\right)\right) \]

   6. associate-*r/ N/A

      \[\leadsto \mathsf{*.f64}\left(\mathsf{sin.f64}\left(\left(\frac{\mathsf{PI}\left(\right) \cdot angle}{180}\right)\right), \left(\cos \color{blue}{\left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right)} \cdot \left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right)\right)\right) \]

   7. /-lowering-/.f64 N/A

      \[\leadsto \mathsf{*.f64}\left(\mathsf{sin.f64}\left(\mathsf{/.f64}\left(\left(\mathsf{PI}\left(\right) \cdot angle\right), 180\right)\right), \left(\cos \color{blue}{\left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right)} \cdot \left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right)\right)\right) \]

   8. *-lowering-*.f64 N/A

      \[\leadsto \mathsf{*.f64}\left(\mathsf{sin.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{PI}\left(\right), angle\right), 180\right)\right), \left(\cos \left(\color{blue}{\mathsf{PI}\left(\right)} \cdot \frac{angle}{180}\right) \cdot \left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right)\right)\right) \]

   9. PI-lowering-PI.f64 N/A

      \[\leadsto \mathsf{*.f64}\left(\mathsf{sin.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{PI.f64}\left(\right), angle\right), 180\right)\right), \left(\cos \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right) \cdot \left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right)\right)\right) \]

   10. *-commutative N/A

       \[\leadsto \mathsf{*.f64}\left(\mathsf{sin.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{PI.f64}\left(\right), angle\right), 180\right)\right), \left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \color{blue}{\cos \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right)}\right)\right) \]

   11. *-lowering-*.f64 N/A

       \[\leadsto \mathsf{*.f64}\left(\mathsf{sin.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{PI.f64}\left(\right), angle\right), 180\right)\right), \mathsf{*.f64}\left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right), \color{blue}{\cos \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right)}\right)\right) \]

3. Simplified 54.7%

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

5. Taylor expanded in angle around 0

   \[\leadsto \color{blue}{\frac{1}{90} \cdot \left(angle \cdot \left(\mathsf{PI}\left(\right) \cdot \left({b}^{2} - {a}^{2}\right)\right)\right)} \]
6. Step-by-step derivation

   1. associate-*r* N/A

      \[\leadsto \frac{1}{90} \cdot \left(\left(angle \cdot \mathsf{PI}\left(\right)\right) \cdot \color{blue}{\left({b}^{2} - {a}^{2}\right)}\right) \]

   2. associate-*r* N/A

      \[\leadsto \left(\frac{1}{90} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right) \cdot \color{blue}{\left({b}^{2} - {a}^{2}\right)} \]

   3. *-lowering-*.f64 N/A

      \[\leadsto \mathsf{*.f64}\left(\left(\frac{1}{90} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right), \color{blue}{\left({b}^{2} - {a}^{2}\right)}\right) \]

   4. *-lowering-*.f64 N/A

      \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{1}{90}, \left(angle \cdot \mathsf{PI}\left(\right)\right)\right), \left(\color{blue}{{b}^{2}} - {a}^{2}\right)\right) \]

   5. *-lowering-*.f64 N/A

      \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{1}{90}, \mathsf{*.f64}\left(angle, \mathsf{PI}\left(\right)\right)\right), \left({b}^{\color{blue}{2}} - {a}^{2}\right)\right) \]

   6. PI-lowering-PI.f64 N/A

      \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{1}{90}, \mathsf{*.f64}\left(angle, \mathsf{PI.f64}\left(\right)\right)\right), \left({b}^{2} - {a}^{2}\right)\right) \]

   7. --lowering--.f64 N/A

      \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{1}{90}, \mathsf{*.f64}\left(angle, \mathsf{PI.f64}\left(\right)\right)\right), \mathsf{\_.f64}\left(\left({b}^{2}\right), \color{blue}{\left({a}^{2}\right)}\right)\right) \]

   8. unpow2 N/A

      \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{1}{90}, \mathsf{*.f64}\left(angle, \mathsf{PI.f64}\left(\right)\right)\right), \mathsf{\_.f64}\left(\left(b \cdot b\right), \left({\color{blue}{a}}^{2}\right)\right)\right) \]

   9. *-lowering-*.f64 N/A

      \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{1}{90}, \mathsf{*.f64}\left(angle, \mathsf{PI.f64}\left(\right)\right)\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(b, b\right), \left({\color{blue}{a}}^{2}\right)\right)\right) \]

   10. unpow2 N/A

       \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{1}{90}, \mathsf{*.f64}\left(angle, \mathsf{PI.f64}\left(\right)\right)\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(b, b\right), \left(a \cdot \color{blue}{a}\right)\right)\right) \]

   11. *-lowering-*.f64 51.4%

       \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{1}{90}, \mathsf{*.f64}\left(angle, \mathsf{PI.f64}\left(\right)\right)\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(b, b\right), \mathsf{*.f64}\left(a, \color{blue}{a}\right)\right)\right) \]

7. Simplified 51.4%

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

8. Taylor expanded in b around 0

   \[\leadsto \color{blue}{\frac{-1}{90} \cdot \left({a}^{2} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)} \]
9. Step-by-step derivation

   1. *-commutative N/A

      \[\leadsto \left({a}^{2} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right) \cdot \color{blue}{\frac{-1}{90}} \]

   2. associate-*r* N/A

      \[\leadsto {a}^{2} \cdot \color{blue}{\left(\left(angle \cdot \mathsf{PI}\left(\right)\right) \cdot \frac{-1}{90}\right)} \]

   3. *-commutative N/A

      \[\leadsto {a}^{2} \cdot \left(\frac{-1}{90} \cdot \color{blue}{\left(angle \cdot \mathsf{PI}\left(\right)\right)}\right) \]

   4. *-lowering-*.f64 N/A

      \[\leadsto \mathsf{*.f64}\left(\left({a}^{2}\right), \color{blue}{\left(\frac{-1}{90} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}\right) \]

   5. unpow2 N/A

      \[\leadsto \mathsf{*.f64}\left(\left(a \cdot a\right), \left(\color{blue}{\frac{-1}{90}} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)\right) \]

   6. *-lowering-*.f64 N/A

      \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(a, a\right), \left(\color{blue}{\frac{-1}{90}} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)\right) \]

   7. *-commutative N/A

      \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(a, a\right), \left(\left(angle \cdot \mathsf{PI}\left(\right)\right) \cdot \color{blue}{\frac{-1}{90}}\right)\right) \]

   8. *-lowering-*.f64 N/A

      \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(a, a\right), \mathsf{*.f64}\left(\left(angle \cdot \mathsf{PI}\left(\right)\right), \color{blue}{\frac{-1}{90}}\right)\right) \]

   9. *-lowering-*.f64 N/A

      \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(a, a\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(angle, \mathsf{PI}\left(\right)\right), \frac{-1}{90}\right)\right) \]

   10. PI-lowering-PI.f64 31.0%

       \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(a, a\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(angle, \mathsf{PI.f64}\left(\right)\right), \frac{-1}{90}\right)\right) \]

10. Simplified 31.0%

    \[\leadsto \color{blue}{\left(a \cdot a\right) \cdot \left(\left(angle \cdot \pi\right) \cdot -0.011111111111111112\right)} \]
11. Add Preprocessing

Reproduce

herbie shell --seed 2024148 
(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)))))