ab-angle->ABCF B

Percentage Accurate: 54.5% → 68.1%

Time: 20.7s

Alternatives: 25

Speedup: 23.3×

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

Specification

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 54.5% 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: 68.1% accurate, 0.7× speedup

Math

\[\begin{array}{l} angle\_m = \left|angle\right| \\ angle\_s = \mathsf{copysign}\left(1, angle\right) \\ \begin{array}{l} t_0 := \sqrt[3]{{\pi}^{1.5}}\\ angle\_s \cdot \begin{array}{l} \mathbf{if}\;\frac{angle\_m}{180} \leq 10^{+135}:\\ \;\;\;\;\frac{1}{\frac{\frac{1}{b + a}}{\left(\left(b - a\right) \cdot 2\right) \cdot \sin \left(\frac{angle\_m}{\frac{180}{\pi}}\right)}} \cdot \cos \left(\frac{0.005555555555555556}{\frac{\frac{1}{angle\_m}}{\pi}}\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\left(b - a\right) \cdot \left(\left(b + a\right) \cdot \left(2 \cdot \sin \left(\frac{\pi}{\frac{180}{angle\_m}}\right)\right)\right)\right) \cdot \cos \left(\frac{0.005555555555555556}{\frac{\frac{1}{angle\_m}}{t\_0 \cdot t\_0}}\right)\\ \end{array} \end{array} \end{array} \]

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 (cbrt (pow PI 1.5))))
   (*
    angle_s
    (if (<= (/ angle_m 180.0) 1e+135)
      (*
       (/
        1.0
        (/ (/ 1.0 (+ b a)) (* (* (- b a) 2.0) (sin (/ angle_m (/ 180.0 PI))))))
       (cos (/ 0.005555555555555556 (/ (/ 1.0 angle_m) PI))))
      (*
       (* (- b a) (* (+ b a) (* 2.0 (sin (/ PI (/ 180.0 angle_m))))))
       (cos (/ 0.005555555555555556 (/ (/ 1.0 angle_m) (* t_0 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 = cbrt(pow(((double) M_PI), 1.5));
	double tmp;
	if ((angle_m / 180.0) <= 1e+135) {
		tmp = (1.0 / ((1.0 / (b + a)) / (((b - a) * 2.0) * sin((angle_m / (180.0 / ((double) M_PI))))))) * cos((0.005555555555555556 / ((1.0 / angle_m) / ((double) M_PI))));
	} else {
		tmp = ((b - a) * ((b + a) * (2.0 * sin((((double) M_PI) / (180.0 / angle_m)))))) * cos((0.005555555555555556 / ((1.0 / angle_m) / (t_0 * 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 = Math.cbrt(Math.pow(Math.PI, 1.5));
	double tmp;
	if ((angle_m / 180.0) <= 1e+135) {
		tmp = (1.0 / ((1.0 / (b + a)) / (((b - a) * 2.0) * Math.sin((angle_m / (180.0 / Math.PI)))))) * Math.cos((0.005555555555555556 / ((1.0 / angle_m) / Math.PI)));
	} else {
		tmp = ((b - a) * ((b + a) * (2.0 * Math.sin((Math.PI / (180.0 / angle_m)))))) * Math.cos((0.005555555555555556 / ((1.0 / angle_m) / (t_0 * 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 = cbrt((pi ^ 1.5))
	tmp = 0.0
	if (Float64(angle_m / 180.0) <= 1e+135)
		tmp = Float64(Float64(1.0 / Float64(Float64(1.0 / Float64(b + a)) / Float64(Float64(Float64(b - a) * 2.0) * sin(Float64(angle_m / Float64(180.0 / pi)))))) * cos(Float64(0.005555555555555556 / Float64(Float64(1.0 / angle_m) / pi))));
	else
		tmp = Float64(Float64(Float64(b - a) * Float64(Float64(b + a) * Float64(2.0 * sin(Float64(pi / Float64(180.0 / angle_m)))))) * cos(Float64(0.005555555555555556 / Float64(Float64(1.0 / angle_m) / Float64(t_0 * t_0)))));
	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[Power[N[Power[Pi, 1.5], $MachinePrecision], 1/3], $MachinePrecision]}, N[(angle$95$s * If[LessEqual[N[(angle$95$m / 180.0), $MachinePrecision], 1e+135], N[(N[(1.0 / N[(N[(1.0 / N[(b + a), $MachinePrecision]), $MachinePrecision] / N[(N[(N[(b - a), $MachinePrecision] * 2.0), $MachinePrecision] * N[Sin[N[(angle$95$m / N[(180.0 / Pi), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[Cos[N[(0.005555555555555556 / N[(N[(1.0 / angle$95$m), $MachinePrecision] / Pi), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(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] * N[Cos[N[(0.005555555555555556 / N[(N[(1.0 / angle$95$m), $MachinePrecision] / N[(t$95$0 * t$95$0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

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

   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. Add Preprocessing
   3. Step-by-step derivation

      1. *-commutative N/A

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

      2. pow2 N/A

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

      3. pow2 N/A

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

      4. associate-*l* N/A

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

      5. difference-of-squares N/A

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

      6. *-commutative N/A

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

      7. associate-*r/ N/A

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

      8. associate-*l* N/A

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

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

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

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

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

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

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

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

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

      13. associate-*r/ N/A

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

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

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

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

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

      16. clear-num N/A

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

   4. Applied egg-rr 73.7%

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

      1. clear-num N/A

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

      2. div-inv N/A

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

      3. *-un-lft-identity N/A

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

      4. div-inv N/A

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

      5. times-frac N/A

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

      6. metadata-eval N/A

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

      7. clear-num N/A

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

      8. un-div-inv N/A

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

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

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

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

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

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

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

      12. PI-lowering-PI.f64 73.8%

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

   6. Applied egg-rr 73.8%

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

   8. Applied egg-rr 74.3%

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

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

   1. Initial program 31.2%

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

      1. *-commutative N/A

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

      2. pow2 N/A

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

      3. pow2 N/A

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

      4. associate-*l* N/A

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

      5. difference-of-squares N/A

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

      6. *-commutative N/A

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

      7. associate-*r/ N/A

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

      8. associate-*l* N/A

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

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

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

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

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

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

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

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

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

      13. associate-*r/ N/A

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

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

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

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

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

      16. clear-num N/A

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

   4. Applied egg-rr 40.5%

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

      1. clear-num N/A

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

      2. div-inv N/A

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

      3. *-un-lft-identity N/A

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

      4. div-inv N/A

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

      5. times-frac N/A

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

      6. metadata-eval N/A

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

      7. clear-num N/A

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

      8. un-div-inv N/A

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

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

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

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

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

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

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

      12. PI-lowering-PI.f64 42.7%

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

   6. Applied egg-rr 42.7%

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

      1. add-cbrt-cube N/A

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

      2. add-sqr-sqrt N/A

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

      3. unswap-sqr N/A

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

      4. cbrt-prod N/A

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

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

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

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

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

      7. pow1 N/A

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

      8. pow1/2 N/A

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

      9. pow-prod-up N/A

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

      10. metadata-eval N/A

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

      11. metadata-eval N/A

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

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

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

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

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

      14. metadata-eval N/A

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

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

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

      16. pow1 N/A

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

      17. pow1/2 N/A

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

      18. pow-prod-up N/A

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

      19. metadata-eval N/A

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

      20. metadata-eval N/A

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

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

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

   8. Applied egg-rr 50.2%

      \[\leadsto \left(\left(b - a\right) \cdot \left(\left(b + a\right) \cdot \left(2 \cdot \sin \left(\frac{\pi}{\frac{180}{angle}}\right)\right)\right)\right) \cdot \cos \left(\frac{0.005555555555555556}{\frac{\frac{1}{angle}}{\color{blue}{\sqrt[3]{{\pi}^{1.5}} \cdot \sqrt[3]{{\pi}^{1.5}}}}}\right) \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 2: 68.3% accurate, 0.4× speedup

Math

\[\begin{array}{l} angle\_m = \left|angle\right| \\ angle\_s = \mathsf{copysign}\left(1, angle\right) \\ \begin{array}{l} t_0 := \frac{angle\_m}{180} \cdot \pi\\ t_1 := \sqrt[3]{{\pi}^{1.5}}\\ angle\_s \cdot \begin{array}{l} \mathbf{if}\;\left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin t\_0\right) \cdot \cos t\_0 \leq -\infty:\\ \;\;\;\;\left(\left(b - a\right) \cdot \left(\left(b + a\right) \cdot \left(2 \cdot \sin \left(\frac{\pi}{\frac{180}{angle\_m}}\right)\right)\right)\right) \cdot \cos \left(\frac{angle\_m}{180} \cdot \left(t\_1 \cdot t\_1\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\frac{\frac{1}{b + a}}{\left(\left(b - a\right) \cdot 2\right) \cdot \sin \left(\frac{angle\_m}{\frac{180}{\pi}}\right)}} \cdot \cos \left(\frac{0.005555555555555556}{\frac{\frac{1}{angle\_m}}{\pi}}\right)\\ \end{array} \end{array} \end{array} \]

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)) (t_1 (cbrt (pow PI 1.5))))
   (*
    angle_s
    (if (<=
         (* (* (* 2.0 (- (pow b 2.0) (pow a 2.0))) (sin t_0)) (cos t_0))
         (- INFINITY))
      (*
       (* (- b a) (* (+ b a) (* 2.0 (sin (/ PI (/ 180.0 angle_m))))))
       (cos (* (/ angle_m 180.0) (* t_1 t_1))))
      (*
       (/
        1.0
        (/ (/ 1.0 (+ b a)) (* (* (- b a) 2.0) (sin (/ angle_m (/ 180.0 PI))))))
       (cos (/ 0.005555555555555556 (/ (/ 1.0 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 = (angle_m / 180.0) * ((double) M_PI);
	double t_1 = cbrt(pow(((double) M_PI), 1.5));
	double tmp;
	if ((((2.0 * (pow(b, 2.0) - pow(a, 2.0))) * sin(t_0)) * cos(t_0)) <= -((double) INFINITY)) {
		tmp = ((b - a) * ((b + a) * (2.0 * sin((((double) M_PI) / (180.0 / angle_m)))))) * cos(((angle_m / 180.0) * (t_1 * t_1)));
	} else {
		tmp = (1.0 / ((1.0 / (b + a)) / (((b - a) * 2.0) * sin((angle_m / (180.0 / ((double) M_PI))))))) * cos((0.005555555555555556 / ((1.0 / 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 = (angle_m / 180.0) * Math.PI;
	double t_1 = Math.cbrt(Math.pow(Math.PI, 1.5));
	double tmp;
	if ((((2.0 * (Math.pow(b, 2.0) - Math.pow(a, 2.0))) * Math.sin(t_0)) * Math.cos(t_0)) <= -Double.POSITIVE_INFINITY) {
		tmp = ((b - a) * ((b + a) * (2.0 * Math.sin((Math.PI / (180.0 / angle_m)))))) * Math.cos(((angle_m / 180.0) * (t_1 * t_1)));
	} else {
		tmp = (1.0 / ((1.0 / (b + a)) / (((b - a) * 2.0) * Math.sin((angle_m / (180.0 / Math.PI)))))) * Math.cos((0.005555555555555556 / ((1.0 / 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(angle_m / 180.0) * pi)
	t_1 = cbrt((pi ^ 1.5))
	tmp = 0.0
	if (Float64(Float64(Float64(2.0 * Float64((b ^ 2.0) - (a ^ 2.0))) * sin(t_0)) * cos(t_0)) <= Float64(-Inf))
		tmp = Float64(Float64(Float64(b - a) * Float64(Float64(b + a) * Float64(2.0 * sin(Float64(pi / Float64(180.0 / angle_m)))))) * cos(Float64(Float64(angle_m / 180.0) * Float64(t_1 * t_1))));
	else
		tmp = Float64(Float64(1.0 / Float64(Float64(1.0 / Float64(b + a)) / Float64(Float64(Float64(b - a) * 2.0) * sin(Float64(angle_m / Float64(180.0 / pi)))))) * cos(Float64(0.005555555555555556 / Float64(Float64(1.0 / angle_m) / pi))));
	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]}, Block[{t$95$1 = N[Power[N[Power[Pi, 1.5], $MachinePrecision], 1/3], $MachinePrecision]}, N[(angle$95$s * If[LessEqual[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], (-Infinity)], N[(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] * N[Cos[N[(N[(angle$95$m / 180.0), $MachinePrecision] * N[(t$95$1 * t$95$1), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(N[(1.0 / N[(N[(1.0 / N[(b + a), $MachinePrecision]), $MachinePrecision] / N[(N[(N[(b - a), $MachinePrecision] * 2.0), $MachinePrecision] * N[Sin[N[(angle$95$m / N[(180.0 / Pi), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[Cos[N[(0.005555555555555556 / N[(N[(1.0 / angle$95$m), $MachinePrecision] / Pi), $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 38.5%

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

      1. *-commutative N/A

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

      2. pow2 N/A

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

      3. pow2 N/A

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

      4. associate-*l* N/A

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

      5. difference-of-squares N/A

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

      6. *-commutative N/A

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

      7. associate-*r/ N/A

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

      8. associate-*l* N/A

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

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

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

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

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

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

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

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

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

      13. associate-*r/ N/A

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

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

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

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

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

      16. clear-num N/A

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

   4. Applied egg-rr 81.6%

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

      1. add-cbrt-cube N/A

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

      2. add-sqr-sqrt N/A

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

      3. unswap-sqr N/A

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

      4. cbrt-prod N/A

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

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

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

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

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

      7. pow1 N/A

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

      8. pow1/2 N/A

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

      9. pow-prod-up N/A

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

      10. metadata-eval N/A

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

      11. metadata-eval N/A

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

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

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

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

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

      14. metadata-eval N/A

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

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

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

      16. pow1 N/A

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

      17. pow1/2 N/A

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

      18. pow-prod-up N/A

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

      19. metadata-eval N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{\_.f64}\left(b, a\right), \mathsf{*.f64}\left(\mathsf{+.f64}\left(b, a\right), \mathsf{*.f64}\left(2, \mathsf{sin.f64}\left(\mathsf{/.f64}\left(\mathsf{PI.f64}\left(\right), \mathsf{/.f64}\left(180, angle\right)\right)\right)\right)\right)\right), \mathsf{cos.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{cbrt.f64}\left(\mathsf{pow.f64}\left(\mathsf{PI.f64}\left(\right), \frac{3}{2}\right)\right), \mathsf{cbrt.f64}\left(\left({\mathsf{PI}\left(\right)}^{\frac{3}{2}}\right)\right)\right), \mathsf{/.f64}\left(angle, 180\right)\right)\right)\right) \]

      20. metadata-eval N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{\_.f64}\left(b, a\right), \mathsf{*.f64}\left(\mathsf{+.f64}\left(b, a\right), \mathsf{*.f64}\left(2, \mathsf{sin.f64}\left(\mathsf{/.f64}\left(\mathsf{PI.f64}\left(\right), \mathsf{/.f64}\left(180, angle\right)\right)\right)\right)\right)\right), \mathsf{cos.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{cbrt.f64}\left(\mathsf{pow.f64}\left(\mathsf{PI.f64}\left(\right), \frac{3}{2}\right)\right), \mathsf{cbrt.f64}\left(\left({\mathsf{PI}\left(\right)}^{\left(\frac{3}{2}\right)}\right)\right)\right), \mathsf{/.f64}\left(angle, 180\right)\right)\right)\right) \]

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

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

   6. Applied egg-rr 81.6%

      \[\leadsto \left(\left(b - a\right) \cdot \left(\left(b + a\right) \cdot \left(2 \cdot \sin \left(\frac{\pi}{\frac{180}{angle}}\right)\right)\right)\right) \cdot \cos \left(\color{blue}{\left(\sqrt[3]{{\pi}^{1.5}} \cdot \sqrt[3]{{\pi}^{1.5}}\right)} \cdot \frac{angle}{180}\right) \]

   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 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. Add Preprocessing
   3. Step-by-step derivation

      1. *-commutative N/A

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

      2. pow2 N/A

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

      3. pow2 N/A

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

      4. associate-*l* N/A

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

      5. difference-of-squares N/A

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

      6. *-commutative N/A

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

      7. associate-*r/ N/A

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

      8. associate-*l* N/A

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

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

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

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

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

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

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

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

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

      13. associate-*r/ N/A

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

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

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

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

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

      16. clear-num N/A

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

   4. Applied egg-rr 66.6%

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

      1. clear-num N/A

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

      2. div-inv N/A

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

      3. *-un-lft-identity N/A

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

      4. div-inv N/A

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

      5. times-frac N/A

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

      6. metadata-eval N/A

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

      7. clear-num N/A

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

      8. un-div-inv N/A

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

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

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

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

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

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

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

      12. PI-lowering-PI.f64 68.0%

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

   6. Applied egg-rr 68.0%

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

   8. Applied egg-rr 68.6%

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

4. Final simplification 70.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right) \cdot \cos \left(\frac{angle}{180} \cdot \pi\right) \leq -\infty:\\ \;\;\;\;\left(\left(b - a\right) \cdot \left(\left(b + a\right) \cdot \left(2 \cdot \sin \left(\frac{\pi}{\frac{180}{angle}}\right)\right)\right)\right) \cdot \cos \left(\frac{angle}{180} \cdot \left(\sqrt[3]{{\pi}^{1.5}} \cdot \sqrt[3]{{\pi}^{1.5}}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\frac{\frac{1}{b + a}}{\left(\left(b - a\right) \cdot 2\right) \cdot \sin \left(\frac{angle}{\frac{180}{\pi}}\right)}} \cdot \cos \left(\frac{0.005555555555555556}{\frac{\frac{1}{angle}}{\pi}}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 68.1% accurate, 0.6× speedup

Math

\[\begin{array}{l} angle\_m = \left|angle\right| \\ angle\_s = \mathsf{copysign}\left(1, angle\right) \\ \begin{array}{l} t_0 := \frac{angle\_m}{180} \cdot \pi\\ angle\_s \cdot \begin{array}{l} \mathbf{if}\;\left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin t\_0\right) \cdot \cos t\_0 \leq 5 \cdot 10^{+58}:\\ \;\;\;\;\left(\left(b - a\right) \cdot \left(\left(b + a\right) \cdot \left(2 \cdot \sin \left(\frac{\pi}{\frac{180}{angle\_m}}\right)\right)\right)\right) \cdot \cos \left(\frac{0.005555555555555556}{\frac{\frac{1}{angle\_m}}{\sqrt[3]{\pi \cdot \left(\pi \cdot \pi\right)}}}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\frac{\frac{1}{b + a}}{\left(\left(b - a\right) \cdot 2\right) \cdot \sin \left(\frac{angle\_m}{\frac{180}{\pi}}\right)}} \cdot \cos \left(\frac{0.005555555555555556}{\frac{\frac{1}{angle\_m}}{\pi}}\right)\\ \end{array} \end{array} \end{array} \]

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 (<=
         (* (* (* 2.0 (- (pow b 2.0) (pow a 2.0))) (sin t_0)) (cos t_0))
         5e+58)
      (*
       (* (- b a) (* (+ b a) (* 2.0 (sin (/ PI (/ 180.0 angle_m))))))
       (cos
        (/ 0.005555555555555556 (/ (/ 1.0 angle_m) (cbrt (* PI (* PI PI)))))))
      (*
       (/
        1.0
        (/ (/ 1.0 (+ b a)) (* (* (- b a) 2.0) (sin (/ angle_m (/ 180.0 PI))))))
       (cos (/ 0.005555555555555556 (/ (/ 1.0 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 = (angle_m / 180.0) * ((double) M_PI);
	double tmp;
	if ((((2.0 * (pow(b, 2.0) - pow(a, 2.0))) * sin(t_0)) * cos(t_0)) <= 5e+58) {
		tmp = ((b - a) * ((b + a) * (2.0 * sin((((double) M_PI) / (180.0 / angle_m)))))) * cos((0.005555555555555556 / ((1.0 / angle_m) / cbrt((((double) M_PI) * (((double) M_PI) * ((double) M_PI)))))));
	} else {
		tmp = (1.0 / ((1.0 / (b + a)) / (((b - a) * 2.0) * sin((angle_m / (180.0 / ((double) M_PI))))))) * cos((0.005555555555555556 / ((1.0 / 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 = (angle_m / 180.0) * Math.PI;
	double tmp;
	if ((((2.0 * (Math.pow(b, 2.0) - Math.pow(a, 2.0))) * Math.sin(t_0)) * Math.cos(t_0)) <= 5e+58) {
		tmp = ((b - a) * ((b + a) * (2.0 * Math.sin((Math.PI / (180.0 / angle_m)))))) * Math.cos((0.005555555555555556 / ((1.0 / angle_m) / Math.cbrt((Math.PI * (Math.PI * Math.PI))))));
	} else {
		tmp = (1.0 / ((1.0 / (b + a)) / (((b - a) * 2.0) * Math.sin((angle_m / (180.0 / Math.PI)))))) * Math.cos((0.005555555555555556 / ((1.0 / 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(angle_m / 180.0) * pi)
	tmp = 0.0
	if (Float64(Float64(Float64(2.0 * Float64((b ^ 2.0) - (a ^ 2.0))) * sin(t_0)) * cos(t_0)) <= 5e+58)
		tmp = Float64(Float64(Float64(b - a) * Float64(Float64(b + a) * Float64(2.0 * sin(Float64(pi / Float64(180.0 / angle_m)))))) * cos(Float64(0.005555555555555556 / Float64(Float64(1.0 / angle_m) / cbrt(Float64(pi * Float64(pi * pi)))))));
	else
		tmp = Float64(Float64(1.0 / Float64(Float64(1.0 / Float64(b + a)) / Float64(Float64(Float64(b - a) * 2.0) * sin(Float64(angle_m / Float64(180.0 / pi)))))) * cos(Float64(0.005555555555555556 / Float64(Float64(1.0 / angle_m) / pi))));
	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[(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], 5e+58], N[(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] * N[Cos[N[(0.005555555555555556 / N[(N[(1.0 / angle$95$m), $MachinePrecision] / N[Power[N[(Pi * N[(Pi * Pi), $MachinePrecision]), $MachinePrecision], 1/3], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(N[(1.0 / N[(N[(1.0 / N[(b + a), $MachinePrecision]), $MachinePrecision] / N[(N[(N[(b - a), $MachinePrecision] * 2.0), $MachinePrecision] * N[Sin[N[(angle$95$m / N[(180.0 / Pi), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[Cos[N[(0.005555555555555556 / N[(N[(1.0 / angle$95$m), $MachinePrecision] / Pi), $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))))) < 4.99999999999999986e58

   1. Initial program 64.2%

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

      1. *-commutative N/A

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

      2. pow2 N/A

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

      3. pow2 N/A

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

      4. associate-*l* N/A

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

      5. difference-of-squares N/A

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

      6. *-commutative N/A

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

      7. associate-*r/ N/A

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

      8. associate-*l* N/A

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

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

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

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

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

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

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

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

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

      13. associate-*r/ N/A

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

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

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

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

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

      16. clear-num N/A

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

   4. Applied egg-rr 72.8%

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

      1. clear-num N/A

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

      2. div-inv N/A

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

      3. *-un-lft-identity N/A

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

      4. div-inv N/A

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

      5. times-frac N/A

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

      6. metadata-eval N/A

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

      7. clear-num N/A

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

      8. un-div-inv N/A

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

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

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

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

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

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

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

      12. PI-lowering-PI.f64 71.3%

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

   6. Applied egg-rr 71.3%

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

      1. add-cbrt-cube N/A

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

      2. add-sqr-sqrt N/A

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

      3. unswap-sqr N/A

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

      4. cbrt-prod N/A

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

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

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

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

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

      7. pow1 N/A

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

      8. pow1/2 N/A

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

      9. pow-prod-up N/A

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

      10. metadata-eval N/A

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

      11. metadata-eval N/A

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

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

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

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

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

      14. metadata-eval N/A

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

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

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

      16. pow1 N/A

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

      17. pow1/2 N/A

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

      18. pow-prod-up N/A

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

      19. metadata-eval N/A

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

      20. metadata-eval N/A

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

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

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

   8. Applied egg-rr 73.8%

      \[\leadsto \left(\left(b - a\right) \cdot \left(\left(b + a\right) \cdot \left(2 \cdot \sin \left(\frac{\pi}{\frac{180}{angle}}\right)\right)\right)\right) \cdot \cos \left(\frac{0.005555555555555556}{\frac{\frac{1}{angle}}{\color{blue}{\sqrt[3]{{\pi}^{1.5}} \cdot \sqrt[3]{{\pi}^{1.5}}}}}\right) \]
   9. Step-by-step derivation

      1. cbrt-unprod N/A

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

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

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

      3. pow-sqr N/A

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

      4. metadata-eval N/A

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

      5. cube-unmult N/A

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

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

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

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

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

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

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

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

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

      10. PI-lowering-PI.f64 71.7%

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

   10. Applied egg-rr 71.7%

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

   if 4.99999999999999986e58 < (*.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 36.9%

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

      1. *-commutative N/A

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

      2. pow2 N/A

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

      3. pow2 N/A

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

      4. associate-*l* N/A

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

      5. difference-of-squares N/A

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

      6. *-commutative N/A

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

      7. associate-*r/ N/A

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

      8. associate-*l* N/A

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

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

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

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

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

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

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

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

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

      13. associate-*r/ N/A

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

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

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

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

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

      16. clear-num N/A

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

   4. Applied egg-rr 60.5%

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

      1. clear-num N/A

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

      2. div-inv N/A

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

      3. *-un-lft-identity N/A

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

      4. div-inv N/A

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

      5. times-frac N/A

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

      6. metadata-eval N/A

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

      7. clear-num N/A

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

      8. un-div-inv N/A

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

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

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

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

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

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

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

      12. PI-lowering-PI.f64 64.6%

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

   6. Applied egg-rr 64.6%

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

   8. Applied egg-rr 65.7%

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

4. Final simplification 69.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right) \cdot \cos \left(\frac{angle}{180} \cdot \pi\right) \leq 5 \cdot 10^{+58}:\\ \;\;\;\;\left(\left(b - a\right) \cdot \left(\left(b + a\right) \cdot \left(2 \cdot \sin \left(\frac{\pi}{\frac{180}{angle}}\right)\right)\right)\right) \cdot \cos \left(\frac{0.005555555555555556}{\frac{\frac{1}{angle}}{\sqrt[3]{\pi \cdot \left(\pi \cdot \pi\right)}}}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\frac{\frac{1}{b + a}}{\left(\left(b - a\right) \cdot 2\right) \cdot \sin \left(\frac{angle}{\frac{180}{\pi}}\right)}} \cdot \cos \left(\frac{0.005555555555555556}{\frac{\frac{1}{angle}}{\pi}}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 67.9% accurate, 1.8× speedup

Math

\[\begin{array}{l} angle\_m = \left|angle\right| \\ angle\_s = \mathsf{copysign}\left(1, angle\right) \\ angle\_s \cdot \begin{array}{l} \mathbf{if}\;\frac{angle\_m}{180} \leq 2 \cdot 10^{+143}:\\ \;\;\;\;\frac{1}{\frac{\frac{1}{b + a}}{\left(\left(b - a\right) \cdot 2\right) \cdot \sin \left(\frac{angle\_m}{\frac{180}{\pi}}\right)}} \cdot \cos \left(\frac{0.005555555555555556}{\frac{\frac{1}{angle\_m}}{\pi}}\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\left(b - a\right) \cdot \left(\left(b + a\right) \cdot \left(2 \cdot \sin \left(\frac{\pi}{\frac{180}{angle\_m}}\right)\right)\right)\right) \cdot \cos \left(\frac{angle\_m}{180} \cdot \pi\right)\\ \end{array} \end{array} \]

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) 2e+143)
    (*
     (/
      1.0
      (/ (/ 1.0 (+ b a)) (* (* (- b a) 2.0) (sin (/ angle_m (/ 180.0 PI))))))
     (cos (/ 0.005555555555555556 (/ (/ 1.0 angle_m) PI))))
    (*
     (* (- b a) (* (+ b a) (* 2.0 (sin (/ PI (/ 180.0 angle_m))))))
     (cos (* (/ angle_m 180.0) 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 tmp;
	if ((angle_m / 180.0) <= 2e+143) {
		tmp = (1.0 / ((1.0 / (b + a)) / (((b - a) * 2.0) * sin((angle_m / (180.0 / ((double) M_PI))))))) * cos((0.005555555555555556 / ((1.0 / angle_m) / ((double) M_PI))));
	} else {
		tmp = ((b - a) * ((b + a) * (2.0 * sin((((double) M_PI) / (180.0 / angle_m)))))) * cos(((angle_m / 180.0) * ((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 tmp;
	if ((angle_m / 180.0) <= 2e+143) {
		tmp = (1.0 / ((1.0 / (b + a)) / (((b - a) * 2.0) * Math.sin((angle_m / (180.0 / Math.PI)))))) * Math.cos((0.005555555555555556 / ((1.0 / angle_m) / Math.PI)));
	} else {
		tmp = ((b - a) * ((b + a) * (2.0 * Math.sin((Math.PI / (180.0 / angle_m)))))) * Math.cos(((angle_m / 180.0) * 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):
	tmp = 0
	if (angle_m / 180.0) <= 2e+143:
		tmp = (1.0 / ((1.0 / (b + a)) / (((b - a) * 2.0) * math.sin((angle_m / (180.0 / math.pi)))))) * math.cos((0.005555555555555556 / ((1.0 / angle_m) / math.pi)))
	else:
		tmp = ((b - a) * ((b + a) * (2.0 * math.sin((math.pi / (180.0 / angle_m)))))) * math.cos(((angle_m / 180.0) * 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)
	tmp = 0.0
	if (Float64(angle_m / 180.0) <= 2e+143)
		tmp = Float64(Float64(1.0 / Float64(Float64(1.0 / Float64(b + a)) / Float64(Float64(Float64(b - a) * 2.0) * sin(Float64(angle_m / Float64(180.0 / pi)))))) * cos(Float64(0.005555555555555556 / Float64(Float64(1.0 / angle_m) / pi))));
	else
		tmp = Float64(Float64(Float64(b - a) * Float64(Float64(b + a) * Float64(2.0 * sin(Float64(pi / Float64(180.0 / angle_m)))))) * cos(Float64(Float64(angle_m / 180.0) * 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)
	tmp = 0.0;
	if ((angle_m / 180.0) <= 2e+143)
		tmp = (1.0 / ((1.0 / (b + a)) / (((b - a) * 2.0) * sin((angle_m / (180.0 / pi)))))) * cos((0.005555555555555556 / ((1.0 / angle_m) / pi)));
	else
		tmp = ((b - a) * ((b + a) * (2.0 * sin((pi / (180.0 / angle_m)))))) * cos(((angle_m / 180.0) * 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_] := N[(angle$95$s * If[LessEqual[N[(angle$95$m / 180.0), $MachinePrecision], 2e+143], N[(N[(1.0 / N[(N[(1.0 / N[(b + a), $MachinePrecision]), $MachinePrecision] / N[(N[(N[(b - a), $MachinePrecision] * 2.0), $MachinePrecision] * N[Sin[N[(angle$95$m / N[(180.0 / Pi), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[Cos[N[(0.005555555555555556 / N[(N[(1.0 / angle$95$m), $MachinePrecision] / Pi), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(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] * N[Cos[N[(N[(angle$95$m / 180.0), $MachinePrecision] * Pi), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if (/.f64 angle #s(literal 180 binary64)) < 2e143

   1. Initial program 58.5%

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

      1. *-commutative N/A

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

      2. pow2 N/A

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

      3. pow2 N/A

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

      4. associate-*l* N/A

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

      5. difference-of-squares N/A

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

      6. *-commutative N/A

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

      7. associate-*r/ N/A

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

      8. associate-*l* N/A

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

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

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

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

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

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

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

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

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

      13. associate-*r/ N/A

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

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

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

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

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

      16. clear-num N/A

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

   4. Applied egg-rr 73.1%

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

      1. clear-num N/A

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

      2. div-inv N/A

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

      3. *-un-lft-identity N/A

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

      4. div-inv N/A

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

      5. times-frac N/A

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

      6. metadata-eval N/A

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

      7. clear-num N/A

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

      8. un-div-inv N/A

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

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

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

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

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

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

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

      12. PI-lowering-PI.f64 73.2%

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

   6. Applied egg-rr 73.2%

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

   8. Applied egg-rr 73.7%

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

   if 2e143 < (/.f64 angle #s(literal 180 binary64))

   1. Initial program 31.9%

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

      1. *-commutative N/A

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

      2. pow2 N/A

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

      3. pow2 N/A

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

      4. associate-*l* N/A

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

      5. difference-of-squares N/A

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

      6. *-commutative N/A

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

      7. associate-*r/ N/A

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

      8. associate-*l* N/A

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

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

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

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

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

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

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

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

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

      13. associate-*r/ N/A

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

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

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

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

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

      16. clear-num N/A

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

   4. Applied egg-rr 42.1%

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

4. Final simplification 69.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{angle}{180} \leq 2 \cdot 10^{+143}:\\ \;\;\;\;\frac{1}{\frac{\frac{1}{b + a}}{\left(\left(b - a\right) \cdot 2\right) \cdot \sin \left(\frac{angle}{\frac{180}{\pi}}\right)}} \cdot \cos \left(\frac{0.005555555555555556}{\frac{\frac{1}{angle}}{\pi}}\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\left(b - a\right) \cdot \left(\left(b + a\right) \cdot \left(2 \cdot \sin \left(\frac{\pi}{\frac{180}{angle}}\right)\right)\right)\right) \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 67.1% accurate, 1.8× speedup

Math

\[\begin{array}{l} angle\_m = \left|angle\right| \\ angle\_s = \mathsf{copysign}\left(1, angle\right) \\ \begin{array}{l} t_0 := \sin \left(\frac{\pi}{\frac{180}{angle\_m}}\right)\\ angle\_s \cdot \begin{array}{l} \mathbf{if}\;b \leq 9.4 \cdot 10^{-110}:\\ \;\;\;\;\left(b - a\right) \cdot \left(t\_0 \cdot \left(\left(b + a\right) \cdot 2\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\cos \left(\frac{0.005555555555555556}{\frac{\frac{1}{angle\_m}}{\pi}}\right) \cdot \left(\left(b - a\right) \cdot \left(\left(b + a\right) \cdot \left(2 \cdot t\_0\right)\right)\right)\\ \end{array} \end{array} \end{array} \]

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 (sin (/ PI (/ 180.0 angle_m)))))
   (*
    angle_s
    (if (<= b 9.4e-110)
      (* (- b a) (* t_0 (* (+ b a) 2.0)))
      (*
       (cos (/ 0.005555555555555556 (/ (/ 1.0 angle_m) PI)))
       (* (- b a) (* (+ b a) (* 2.0 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 = sin((((double) M_PI) / (180.0 / angle_m)));
	double tmp;
	if (b <= 9.4e-110) {
		tmp = (b - a) * (t_0 * ((b + a) * 2.0));
	} else {
		tmp = cos((0.005555555555555556 / ((1.0 / angle_m) / ((double) M_PI)))) * ((b - a) * ((b + a) * (2.0 * 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 = Math.sin((Math.PI / (180.0 / angle_m)));
	double tmp;
	if (b <= 9.4e-110) {
		tmp = (b - a) * (t_0 * ((b + a) * 2.0));
	} else {
		tmp = Math.cos((0.005555555555555556 / ((1.0 / angle_m) / Math.PI))) * ((b - a) * ((b + a) * (2.0 * 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 = math.sin((math.pi / (180.0 / angle_m)))
	tmp = 0
	if b <= 9.4e-110:
		tmp = (b - a) * (t_0 * ((b + a) * 2.0))
	else:
		tmp = math.cos((0.005555555555555556 / ((1.0 / angle_m) / math.pi))) * ((b - a) * ((b + a) * (2.0 * 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 = sin(Float64(pi / Float64(180.0 / angle_m)))
	tmp = 0.0
	if (b <= 9.4e-110)
		tmp = Float64(Float64(b - a) * Float64(t_0 * Float64(Float64(b + a) * 2.0)));
	else
		tmp = Float64(cos(Float64(0.005555555555555556 / Float64(Float64(1.0 / angle_m) / pi))) * Float64(Float64(b - a) * Float64(Float64(b + a) * Float64(2.0 * 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 = sin((pi / (180.0 / angle_m)));
	tmp = 0.0;
	if (b <= 9.4e-110)
		tmp = (b - a) * (t_0 * ((b + a) * 2.0));
	else
		tmp = cos((0.005555555555555556 / ((1.0 / angle_m) / pi))) * ((b - a) * ((b + a) * (2.0 * 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[Sin[N[(Pi / N[(180.0 / angle$95$m), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]}, N[(angle$95$s * If[LessEqual[b, 9.4e-110], N[(N[(b - a), $MachinePrecision] * N[(t$95$0 * N[(N[(b + a), $MachinePrecision] * 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[Cos[N[(0.005555555555555556 / N[(N[(1.0 / angle$95$m), $MachinePrecision] / Pi), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * N[(N[(b - a), $MachinePrecision] * N[(N[(b + a), $MachinePrecision] * N[(2.0 * t$95$0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b < 9.39999999999999983e-110

   1. Initial program 56.4%

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

   3. Taylor expanded in angle around 0

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

   5. Simplified 57.0%

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

      1. *-rgt-identity N/A

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

      2. *-commutative N/A

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

      3. pow2 N/A

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

      4. pow2 N/A

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

      5. associate-*l* N/A

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

      6. clear-num N/A

         \[\leadsto \left(b \cdot b - a \cdot a\right) \cdot \left(2 \cdot \sin \left(\mathsf{PI}\left(\right) \cdot \frac{1}{\frac{180}{angle}}\right)\right) \]

      7. div-inv N/A

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

      8. difference-of-squares N/A

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

      9. *-commutative N/A

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

      10. associate-*r* N/A

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

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

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

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

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

      13. associate-*r* N/A

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

      14. div-inv N/A

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

      15. clear-num N/A

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

   7. Applied egg-rr 68.2%

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

   if 9.39999999999999983e-110 < b

   1. Initial program 50.9%

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

      1. *-commutative N/A

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

      2. pow2 N/A

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

      3. pow2 N/A

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

      4. associate-*l* N/A

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

      5. difference-of-squares N/A

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

      6. *-commutative N/A

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

      7. associate-*r/ N/A

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

      8. associate-*l* N/A

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

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

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

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

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

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

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

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

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

      13. associate-*r/ N/A

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

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

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

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

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

      16. clear-num N/A

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

   4. Applied egg-rr 70.6%

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

      1. clear-num N/A

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

      2. div-inv N/A

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

      3. *-un-lft-identity N/A

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

      4. div-inv N/A

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

      5. times-frac N/A

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

      6. metadata-eval N/A

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

      7. clear-num N/A

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

      8. un-div-inv N/A

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

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

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

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

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

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

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

      12. PI-lowering-PI.f64 70.3%

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

   6. Applied egg-rr 70.3%

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

4. Final simplification 68.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq 9.4 \cdot 10^{-110}:\\ \;\;\;\;\left(b - a\right) \cdot \left(\sin \left(\frac{\pi}{\frac{180}{angle}}\right) \cdot \left(\left(b + a\right) \cdot 2\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\cos \left(\frac{0.005555555555555556}{\frac{\frac{1}{angle}}{\pi}}\right) \cdot \left(\left(b - a\right) \cdot \left(\left(b + a\right) \cdot \left(2 \cdot \sin \left(\frac{\pi}{\frac{180}{angle}}\right)\right)\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 68.1% accurate, 1.9× speedup

Math

\[\begin{array}{l} angle\_m = \left|angle\right| \\ angle\_s = \mathsf{copysign}\left(1, angle\right) \\ angle\_s \cdot \begin{array}{l} \mathbf{if}\;b \leq 5.4 \cdot 10^{+178}:\\ \;\;\;\;\left(\left(b - a\right) \cdot \left(\left(b + a\right) \cdot \left(2 \cdot \sin \left(\frac{\pi}{\frac{180}{angle\_m}}\right)\right)\right)\right) \cdot \cos \left(\frac{angle\_m}{180} \cdot \pi\right)\\ \mathbf{elif}\;b \leq 1.95 \cdot 10^{+288}:\\ \;\;\;\;\frac{1}{\frac{\frac{1}{b + a}}{\left(\left(b - a\right) \cdot 2\right) \cdot \sin \left(\frac{angle\_m}{\frac{180}{\pi}}\right)}}\\ \mathbf{else}:\\ \;\;\;\;\left(b \cdot b\right) \cdot \sin \left(0.011111111111111112 \cdot \left(angle\_m \cdot \pi\right)\right)\\ \end{array} \end{array} \]

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 (<= b 5.4e+178)
    (*
     (* (- b a) (* (+ b a) (* 2.0 (sin (/ PI (/ 180.0 angle_m))))))
     (cos (* (/ angle_m 180.0) PI)))
    (if (<= b 1.95e+288)
      (/
       1.0
       (/ (/ 1.0 (+ b a)) (* (* (- b a) 2.0) (sin (/ angle_m (/ 180.0 PI))))))
      (* (* b b) (sin (* 0.011111111111111112 (* 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 tmp;
	if (b <= 5.4e+178) {
		tmp = ((b - a) * ((b + a) * (2.0 * sin((((double) M_PI) / (180.0 / angle_m)))))) * cos(((angle_m / 180.0) * ((double) M_PI)));
	} else if (b <= 1.95e+288) {
		tmp = 1.0 / ((1.0 / (b + a)) / (((b - a) * 2.0) * sin((angle_m / (180.0 / ((double) M_PI))))));
	} else {
		tmp = (b * b) * sin((0.011111111111111112 * (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 tmp;
	if (b <= 5.4e+178) {
		tmp = ((b - a) * ((b + a) * (2.0 * Math.sin((Math.PI / (180.0 / angle_m)))))) * Math.cos(((angle_m / 180.0) * Math.PI));
	} else if (b <= 1.95e+288) {
		tmp = 1.0 / ((1.0 / (b + a)) / (((b - a) * 2.0) * Math.sin((angle_m / (180.0 / Math.PI)))));
	} else {
		tmp = (b * b) * Math.sin((0.011111111111111112 * (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):
	tmp = 0
	if b <= 5.4e+178:
		tmp = ((b - a) * ((b + a) * (2.0 * math.sin((math.pi / (180.0 / angle_m)))))) * math.cos(((angle_m / 180.0) * math.pi))
	elif b <= 1.95e+288:
		tmp = 1.0 / ((1.0 / (b + a)) / (((b - a) * 2.0) * math.sin((angle_m / (180.0 / math.pi)))))
	else:
		tmp = (b * b) * math.sin((0.011111111111111112 * (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)
	tmp = 0.0
	if (b <= 5.4e+178)
		tmp = Float64(Float64(Float64(b - a) * Float64(Float64(b + a) * Float64(2.0 * sin(Float64(pi / Float64(180.0 / angle_m)))))) * cos(Float64(Float64(angle_m / 180.0) * pi)));
	elseif (b <= 1.95e+288)
		tmp = Float64(1.0 / Float64(Float64(1.0 / Float64(b + a)) / Float64(Float64(Float64(b - a) * 2.0) * sin(Float64(angle_m / Float64(180.0 / pi))))));
	else
		tmp = Float64(Float64(b * b) * sin(Float64(0.011111111111111112 * 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)
	tmp = 0.0;
	if (b <= 5.4e+178)
		tmp = ((b - a) * ((b + a) * (2.0 * sin((pi / (180.0 / angle_m)))))) * cos(((angle_m / 180.0) * pi));
	elseif (b <= 1.95e+288)
		tmp = 1.0 / ((1.0 / (b + a)) / (((b - a) * 2.0) * sin((angle_m / (180.0 / pi)))));
	else
		tmp = (b * b) * sin((0.011111111111111112 * (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_] := N[(angle$95$s * If[LessEqual[b, 5.4e+178], N[(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] * N[Cos[N[(N[(angle$95$m / 180.0), $MachinePrecision] * Pi), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[b, 1.95e+288], N[(1.0 / N[(N[(1.0 / N[(b + a), $MachinePrecision]), $MachinePrecision] / N[(N[(N[(b - a), $MachinePrecision] * 2.0), $MachinePrecision] * N[Sin[N[(angle$95$m / N[(180.0 / Pi), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(b * b), $MachinePrecision] * N[Sin[N[(0.011111111111111112 * N[(angle$95$m * Pi), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]]), $MachinePrecision]

Derivation

1. Split input into 3 regimes

2. if b < 5.40000000000000036e178

   1. Initial program 56.9%

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

      1. *-commutative N/A

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

      2. pow2 N/A

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

      3. pow2 N/A

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

      4. associate-*l* N/A

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

      5. difference-of-squares N/A

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

      6. *-commutative N/A

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

      7. associate-*r/ N/A

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

      8. associate-*l* N/A

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

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

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

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

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

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

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

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

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

      13. associate-*r/ N/A

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

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

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

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

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

      16. clear-num N/A

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

   4. Applied egg-rr 67.4%

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

   if 5.40000000000000036e178 < b < 1.94999999999999989e288

   1. Initial program 28.0%

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

   3. Taylor expanded in angle around 0

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

   5. Simplified 41.1%

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

      1. pow2 N/A

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

      2. pow2 N/A

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

      3. difference-of-squares N/A

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

      4. *-commutative N/A

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

      5. flip3-+ N/A

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

      6. clear-num N/A

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

      7. clear-num N/A

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

      8. flip3-+ N/A

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

      9. un-div-inv N/A

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

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

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

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

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

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

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

      13. +-lowering-+.f64 54.3%

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

   7. Applied egg-rr 54.3%

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

      1. *-rgt-identity N/A

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

      2. associate-*r/ N/A

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

      3. associate-*l/ N/A

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

      4. clear-num N/A

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

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

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

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

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

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

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

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

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

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

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

      10. *-commutative N/A

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

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

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

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

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

      13. clear-num N/A

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

      14. div-inv N/A

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

      15. *-un-lft-identity N/A

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

      16. div-inv N/A

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

      17. times-frac N/A

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

   9. Applied egg-rr 86.9%

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

   if 1.94999999999999989e288 < b

   1. Initial program 75.0%

      \[\left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin \left(\pi \cdot \frac{angle}{180}\right)\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right) \]
   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. associate-*l* N/A

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

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

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

      4. associate-*r* N/A

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

      5. *-commutative N/A

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

      6. associate-*l* N/A

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

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

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

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

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

      9. associate-*r/ N/A

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

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

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

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

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

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

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

   3. Simplified 75.0%

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

      1. associate-*r* N/A

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

      2. *-commutative N/A

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

      3. associate-*r/ N/A

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

      4. associate-*r* N/A

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

      5. flip-- N/A

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

      6. clear-num N/A

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

      7. un-div-inv N/A

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

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

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

   6. Applied egg-rr 75.0%

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

   7. Taylor expanded in b around inf

      \[\leadsto \color{blue}{{b}^{2} \cdot \sin \left(\frac{1}{90} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)} \]
   8. 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. sin-lowering-sin.f64 N/A

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

      5. *-commutative N/A

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

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

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

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

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

      8. PI-lowering-PI.f64 100.0%

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

   9. Simplified 100.0%

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

4. Final simplification 69.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq 5.4 \cdot 10^{+178}:\\ \;\;\;\;\left(\left(b - a\right) \cdot \left(\left(b + a\right) \cdot \left(2 \cdot \sin \left(\frac{\pi}{\frac{180}{angle}}\right)\right)\right)\right) \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)\\ \mathbf{elif}\;b \leq 1.95 \cdot 10^{+288}:\\ \;\;\;\;\frac{1}{\frac{\frac{1}{b + a}}{\left(\left(b - a\right) \cdot 2\right) \cdot \sin \left(\frac{angle}{\frac{180}{\pi}}\right)}}\\ \mathbf{else}:\\ \;\;\;\;\left(b \cdot b\right) \cdot \sin \left(0.011111111111111112 \cdot \left(angle \cdot \pi\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 68.1% accurate, 3.3× speedup

Math

\[\begin{array}{l} angle\_m = \left|angle\right| \\ angle\_s = \mathsf{copysign}\left(1, angle\right) \\ angle\_s \cdot \begin{array}{l} \mathbf{if}\;\frac{angle\_m}{180} \leq 10^{+166}:\\ \;\;\;\;\left(b - a\right) \cdot \left(\left(b + a\right) \cdot \sin \left(0.011111111111111112 \cdot \left(angle\_m \cdot \pi\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(2 \cdot \frac{b - a}{\frac{1}{b + a}}\right) \cdot \sin \left(\frac{\frac{\pi}{180}}{\frac{1}{angle\_m}}\right)\\ \end{array} \end{array} \]

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+166)
    (* (- b a) (* (+ b a) (sin (* 0.011111111111111112 (* angle_m PI)))))
    (*
     (* 2.0 (/ (- b a) (/ 1.0 (+ b a))))
     (sin (/ (/ PI 180.0) (/ 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 tmp;
	if ((angle_m / 180.0) <= 1e+166) {
		tmp = (b - a) * ((b + a) * sin((0.011111111111111112 * (angle_m * ((double) M_PI)))));
	} else {
		tmp = (2.0 * ((b - a) / (1.0 / (b + a)))) * sin(((((double) M_PI) / 180.0) / (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 tmp;
	if ((angle_m / 180.0) <= 1e+166) {
		tmp = (b - a) * ((b + a) * Math.sin((0.011111111111111112 * (angle_m * Math.PI))));
	} else {
		tmp = (2.0 * ((b - a) / (1.0 / (b + a)))) * Math.sin(((Math.PI / 180.0) / (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):
	tmp = 0
	if (angle_m / 180.0) <= 1e+166:
		tmp = (b - a) * ((b + a) * math.sin((0.011111111111111112 * (angle_m * math.pi))))
	else:
		tmp = (2.0 * ((b - a) / (1.0 / (b + a)))) * math.sin(((math.pi / 180.0) / (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)
	tmp = 0.0
	if (Float64(angle_m / 180.0) <= 1e+166)
		tmp = Float64(Float64(b - a) * Float64(Float64(b + a) * sin(Float64(0.011111111111111112 * Float64(angle_m * pi)))));
	else
		tmp = Float64(Float64(2.0 * Float64(Float64(b - a) / Float64(1.0 / Float64(b + a)))) * sin(Float64(Float64(pi / 180.0) / 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)
	tmp = 0.0;
	if ((angle_m / 180.0) <= 1e+166)
		tmp = (b - a) * ((b + a) * sin((0.011111111111111112 * (angle_m * pi))));
	else
		tmp = (2.0 * ((b - a) / (1.0 / (b + a)))) * sin(((pi / 180.0) / (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_] := N[(angle$95$s * If[LessEqual[N[(angle$95$m / 180.0), $MachinePrecision], 1e+166], N[(N[(b - a), $MachinePrecision] * N[(N[(b + a), $MachinePrecision] * N[Sin[N[(0.011111111111111112 * N[(angle$95$m * Pi), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(2.0 * N[(N[(b - a), $MachinePrecision] / N[(1.0 / N[(b + a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[Sin[N[(N[(Pi / 180.0), $MachinePrecision] / N[(1.0 / angle$95$m), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if (/.f64 angle #s(literal 180 binary64)) < 9.9999999999999994e165

   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. associate-*l* N/A

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

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

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

      4. associate-*r* N/A

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

      5. *-commutative N/A

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

      6. associate-*l* N/A

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

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

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

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

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

      9. associate-*r/ N/A

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

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

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

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

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

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

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

   3. Simplified 58.8%

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

      1. associate-*r* N/A

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

      2. *-commutative N/A

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

      3. associate-*r/ N/A

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

      4. associate-*r* N/A

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

      5. *-commutative N/A

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

      6. difference-of-squares N/A

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

      7. *-commutative N/A

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

      8. associate-*l* N/A

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

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

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

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

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

   6. Applied egg-rr 72.6%

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

   if 9.9999999999999994e165 < (/.f64 angle #s(literal 180 binary64))

   1. Initial program 29.3%

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

   3. Taylor expanded in angle around 0

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

   5. Simplified 38.5%

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

      1. pow2 N/A

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

      2. pow2 N/A

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

      3. difference-of-squares N/A

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

      4. *-commutative N/A

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

      5. flip3-+ N/A

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

      6. clear-num N/A

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

      7. clear-num N/A

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

      8. flip3-+ N/A

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

      9. un-div-inv N/A

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

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

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

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

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

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

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

      13. +-lowering-+.f64 41.7%

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

   7. Applied egg-rr 41.7%

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

      1. clear-num N/A

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

      2. div-inv N/A

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

      3. div-inv N/A

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

      4. associate-/r* N/A

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

      5. div-inv N/A

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

      6. metadata-eval N/A

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

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

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

      8. metadata-eval N/A

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

      9. div-inv N/A

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

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

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

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

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

      12. /-lowering-/.f64 47.0%

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

   9. Applied egg-rr 47.0%

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

4. Final simplification 69.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{angle}{180} \leq 10^{+166}:\\ \;\;\;\;\left(b - a\right) \cdot \left(\left(b + a\right) \cdot \sin \left(0.011111111111111112 \cdot \left(angle \cdot \pi\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(2 \cdot \frac{b - a}{\frac{1}{b + a}}\right) \cdot \sin \left(\frac{\frac{\pi}{180}}{\frac{1}{angle}}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 67.9% accurate, 3.4× speedup

Math

\[\begin{array}{l} angle\_m = \left|angle\right| \\ angle\_s = \mathsf{copysign}\left(1, angle\right) \\ angle\_s \cdot \begin{array}{l} \mathbf{if}\;\frac{angle\_m}{180} \leq 1.5 \cdot 10^{+143}:\\ \;\;\;\;\left(b - a\right) \cdot \left(\left(b + a\right) \cdot \sin \left(0.011111111111111112 \cdot \left(angle\_m \cdot \pi\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(b - a\right) \cdot \left(\sin \left(\frac{angle\_m}{\frac{180}{\pi}}\right) \cdot \left(\left(b + a\right) \cdot 2\right)\right)\\ \end{array} \end{array} \]

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) 1.5e+143)
    (* (- b a) (* (+ b a) (sin (* 0.011111111111111112 (* angle_m PI)))))
    (* (- b a) (* (sin (/ angle_m (/ 180.0 PI))) (* (+ b a) 2.0))))))

C

angle\_m = fabs(angle);
angle\_s = copysign(1.0, angle);
double code(double angle_s, double a, double b, double angle_m) {
	double tmp;
	if ((angle_m / 180.0) <= 1.5e+143) {
		tmp = (b - a) * ((b + a) * sin((0.011111111111111112 * (angle_m * ((double) M_PI)))));
	} else {
		tmp = (b - a) * (sin((angle_m / (180.0 / ((double) M_PI)))) * ((b + a) * 2.0));
	}
	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 tmp;
	if ((angle_m / 180.0) <= 1.5e+143) {
		tmp = (b - a) * ((b + a) * Math.sin((0.011111111111111112 * (angle_m * Math.PI))));
	} else {
		tmp = (b - a) * (Math.sin((angle_m / (180.0 / Math.PI))) * ((b + a) * 2.0));
	}
	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):
	tmp = 0
	if (angle_m / 180.0) <= 1.5e+143:
		tmp = (b - a) * ((b + a) * math.sin((0.011111111111111112 * (angle_m * math.pi))))
	else:
		tmp = (b - a) * (math.sin((angle_m / (180.0 / math.pi))) * ((b + a) * 2.0))
	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) <= 1.5e+143)
		tmp = Float64(Float64(b - a) * Float64(Float64(b + a) * sin(Float64(0.011111111111111112 * Float64(angle_m * pi)))));
	else
		tmp = Float64(Float64(b - a) * Float64(sin(Float64(angle_m / Float64(180.0 / pi))) * Float64(Float64(b + a) * 2.0)));
	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)
	tmp = 0.0;
	if ((angle_m / 180.0) <= 1.5e+143)
		tmp = (b - a) * ((b + a) * sin((0.011111111111111112 * (angle_m * pi))));
	else
		tmp = (b - a) * (sin((angle_m / (180.0 / pi))) * ((b + a) * 2.0));
	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_] := N[(angle$95$s * If[LessEqual[N[(angle$95$m / 180.0), $MachinePrecision], 1.5e+143], N[(N[(b - a), $MachinePrecision] * N[(N[(b + a), $MachinePrecision] * N[Sin[N[(0.011111111111111112 * N[(angle$95$m * Pi), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(b - a), $MachinePrecision] * N[(N[Sin[N[(angle$95$m / N[(180.0 / Pi), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * N[(N[(b + a), $MachinePrecision] * 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if (/.f64 angle #s(literal 180 binary64)) < 1.5e143

   1. Initial program 58.7%

      \[\left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin \left(\pi \cdot \frac{angle}{180}\right)\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right) \]
   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. associate-*l* N/A

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

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

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

      4. associate-*r* N/A

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

      5. *-commutative N/A

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

      6. associate-*l* N/A

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

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

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

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

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

      9. associate-*r/ N/A

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

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

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

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

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

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

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

   3. Simplified 59.5%

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

      1. associate-*r* N/A

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

      2. *-commutative N/A

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

      3. associate-*r/ N/A

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

      4. associate-*r* N/A

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

      5. *-commutative N/A

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

      6. difference-of-squares N/A

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

      7. *-commutative N/A

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

      8. associate-*l* N/A

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

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

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

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

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

   6. Applied egg-rr 73.0%

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

   if 1.5e143 < (/.f64 angle #s(literal 180 binary64))

   1. Initial program 31.5%

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

   3. Taylor expanded in angle around 0

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

   5. Simplified 36.6%

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

      1. pow2 N/A

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

      2. pow2 N/A

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

      3. difference-of-squares N/A

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

      4. *-commutative N/A

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

      5. flip3-+ N/A

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

      6. clear-num N/A

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

      7. clear-num N/A

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

      8. flip3-+ N/A

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

      9. un-div-inv N/A

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

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

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

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

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

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

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

      13. +-lowering-+.f64 41.8%

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

   7. Applied egg-rr 41.8%

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

   9. Applied egg-rr 38.7%

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

4. Final simplification 67.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{angle}{180} \leq 1.5 \cdot 10^{+143}:\\ \;\;\;\;\left(b - a\right) \cdot \left(\left(b + a\right) \cdot \sin \left(0.011111111111111112 \cdot \left(angle \cdot \pi\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(b - a\right) \cdot \left(\sin \left(\frac{angle}{\frac{180}{\pi}}\right) \cdot \left(\left(b + a\right) \cdot 2\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 67.2% accurate, 3.5× speedup

Math

\[\begin{array}{l} angle\_m = \left|angle\right| \\ angle\_s = \mathsf{copysign}\left(1, angle\right) \\ angle\_s \cdot \begin{array}{l} \mathbf{if}\;b \leq 8.5 \cdot 10^{-111}:\\ \;\;\;\;\left(b - a\right) \cdot \left(\sin \left(\frac{\pi}{\frac{180}{angle\_m}}\right) \cdot \left(\left(b + a\right) \cdot 2\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(b - a\right) \cdot \left(\left(b + a\right) \cdot \sin \left(0.011111111111111112 \cdot \left(angle\_m \cdot \pi\right)\right)\right)\\ \end{array} \end{array} \]

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 (<= b 8.5e-111)
    (* (- b a) (* (sin (/ PI (/ 180.0 angle_m))) (* (+ b a) 2.0)))
    (* (- b a) (* (+ b a) (sin (* 0.011111111111111112 (* 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 tmp;
	if (b <= 8.5e-111) {
		tmp = (b - a) * (sin((((double) M_PI) / (180.0 / angle_m))) * ((b + a) * 2.0));
	} else {
		tmp = (b - a) * ((b + a) * sin((0.011111111111111112 * (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 tmp;
	if (b <= 8.5e-111) {
		tmp = (b - a) * (Math.sin((Math.PI / (180.0 / angle_m))) * ((b + a) * 2.0));
	} else {
		tmp = (b - a) * ((b + a) * Math.sin((0.011111111111111112 * (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):
	tmp = 0
	if b <= 8.5e-111:
		tmp = (b - a) * (math.sin((math.pi / (180.0 / angle_m))) * ((b + a) * 2.0))
	else:
		tmp = (b - a) * ((b + a) * math.sin((0.011111111111111112 * (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)
	tmp = 0.0
	if (b <= 8.5e-111)
		tmp = Float64(Float64(b - a) * Float64(sin(Float64(pi / Float64(180.0 / angle_m))) * Float64(Float64(b + a) * 2.0)));
	else
		tmp = Float64(Float64(b - a) * Float64(Float64(b + a) * sin(Float64(0.011111111111111112 * 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)
	tmp = 0.0;
	if (b <= 8.5e-111)
		tmp = (b - a) * (sin((pi / (180.0 / angle_m))) * ((b + a) * 2.0));
	else
		tmp = (b - a) * ((b + a) * sin((0.011111111111111112 * (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_] := N[(angle$95$s * If[LessEqual[b, 8.5e-111], N[(N[(b - a), $MachinePrecision] * N[(N[Sin[N[(Pi / N[(180.0 / angle$95$m), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * N[(N[(b + a), $MachinePrecision] * 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(b - a), $MachinePrecision] * N[(N[(b + a), $MachinePrecision] * N[Sin[N[(0.011111111111111112 * N[(angle$95$m * Pi), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if b < 8.5000000000000003e-111

   1. Initial program 56.4%

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

   3. Taylor expanded in angle around 0

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

   5. Simplified 57.0%

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

      1. *-rgt-identity N/A

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

      2. *-commutative N/A

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

      3. pow2 N/A

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

      4. pow2 N/A

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

      5. associate-*l* N/A

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

      6. clear-num N/A

         \[\leadsto \left(b \cdot b - a \cdot a\right) \cdot \left(2 \cdot \sin \left(\mathsf{PI}\left(\right) \cdot \frac{1}{\frac{180}{angle}}\right)\right) \]

      7. div-inv N/A

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

      8. difference-of-squares N/A

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

      9. *-commutative N/A

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

      10. associate-*r* N/A

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

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

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

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

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

      13. associate-*r* N/A

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

      14. div-inv N/A

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

      15. clear-num N/A

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

   7. Applied egg-rr 68.2%

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

   if 8.5000000000000003e-111 < b

   1. Initial program 50.9%

      \[\left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin \left(\pi \cdot \frac{angle}{180}\right)\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right) \]
   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. associate-*l* N/A

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

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

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

      4. associate-*r* N/A

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

      5. *-commutative N/A

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

      6. associate-*l* N/A

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

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

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

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

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

      9. associate-*r/ N/A

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

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

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

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

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

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

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

   3. Simplified 51.2%

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

      1. associate-*r* N/A

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

      2. *-commutative N/A

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

      3. associate-*r/ N/A

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

      4. associate-*r* N/A

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

      5. *-commutative N/A

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

      6. difference-of-squares N/A

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

      7. *-commutative N/A

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

      8. associate-*l* N/A

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

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

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

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

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

   6. Applied egg-rr 68.9%

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

4. Final simplification 68.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq 8.5 \cdot 10^{-111}:\\ \;\;\;\;\left(b - a\right) \cdot \left(\sin \left(\frac{\pi}{\frac{180}{angle}}\right) \cdot \left(\left(b + a\right) \cdot 2\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(b - a\right) \cdot \left(\left(b + a\right) \cdot \sin \left(0.011111111111111112 \cdot \left(angle \cdot \pi\right)\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 50.3% accurate, 3.6× speedup

Math

\[\begin{array}{l} angle\_m = \left|angle\right| \\ angle\_s = \mathsf{copysign}\left(1, angle\right) \\ angle\_s \cdot \begin{array}{l} \mathbf{if}\;a \leq 1.8 \cdot 10^{-166}:\\ \;\;\;\;\frac{\sin \left(0.011111111111111112 \cdot \left(angle\_m \cdot \pi\right)\right)}{\frac{1}{b \cdot b}}\\ \mathbf{else}:\\ \;\;\;\;\left(b + a\right) \cdot \left(\left(b - a\right) \cdot \left(\pi \cdot \left(angle\_m \cdot 0.011111111111111112\right)\right)\right)\\ \end{array} \end{array} \]

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 (<= a 1.8e-166)
    (/ (sin (* 0.011111111111111112 (* angle_m PI))) (/ 1.0 (* b b)))
    (* (+ b a) (* (- b a) (* 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 (a <= 1.8e-166) {
		tmp = sin((0.011111111111111112 * (angle_m * ((double) M_PI)))) / (1.0 / (b * b));
	} else {
		tmp = (b + a) * ((b - a) * (((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 (a <= 1.8e-166) {
		tmp = Math.sin((0.011111111111111112 * (angle_m * Math.PI))) / (1.0 / (b * b));
	} else {
		tmp = (b + a) * ((b - a) * (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 a <= 1.8e-166:
		tmp = math.sin((0.011111111111111112 * (angle_m * math.pi))) / (1.0 / (b * b))
	else:
		tmp = (b + a) * ((b - a) * (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 (a <= 1.8e-166)
		tmp = Float64(sin(Float64(0.011111111111111112 * Float64(angle_m * pi))) / Float64(1.0 / Float64(b * b)));
	else
		tmp = Float64(Float64(b + a) * Float64(Float64(b - a) * 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 (a <= 1.8e-166)
		tmp = sin((0.011111111111111112 * (angle_m * pi))) / (1.0 / (b * b));
	else
		tmp = (b + a) * ((b - a) * (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[a, 1.8e-166], N[(N[Sin[N[(0.011111111111111112 * N[(angle$95$m * Pi), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / N[(1.0 / N[(b * b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(b + a), $MachinePrecision] * N[(N[(b - a), $MachinePrecision] * N[(Pi * N[(angle$95$m * 0.011111111111111112), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if a < 1.8e-166

   1. Initial program 57.2%

      \[\left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin \left(\pi \cdot \frac{angle}{180}\right)\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right) \]
   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. associate-*l* N/A

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

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

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

      4. associate-*r* N/A

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

      5. *-commutative N/A

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

      6. associate-*l* N/A

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

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

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

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

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

      9. associate-*r/ N/A

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

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

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

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

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

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

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

   3. Simplified 58.2%

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

      1. associate-*r* N/A

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

      2. *-commutative N/A

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

      3. associate-*r/ N/A

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

      4. associate-*r* N/A

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

      5. flip-- N/A

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

      6. clear-num N/A

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

      7. un-div-inv N/A

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

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

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

   6. Applied egg-rr 57.3%

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

   7. Taylor expanded in b around inf

      \[\leadsto \mathsf{/.f64}\left(\mathsf{sin.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{PI.f64}\left(\right), angle\right), \frac{1}{90}\right)\right), \mathsf{/.f64}\left(1, \color{blue}{\left({b}^{2}\right)}\right)\right) \]
   8. Step-by-step derivation

      1. unpow2 N/A

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

      2. *-lowering-*.f64 47.6%

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

   9. Simplified 47.6%

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

   if 1.8e-166 < a

   1. Initial program 50.4%

      \[\left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin \left(\pi \cdot \frac{angle}{180}\right)\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right) \]
   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. associate-*l* N/A

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

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

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

      4. associate-*r* N/A

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

      5. *-commutative N/A

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

      6. associate-*l* N/A

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

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

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

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

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

      9. associate-*r/ N/A

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

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

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

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

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

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

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

   3. Simplified 51.7%

      \[\leadsto \color{blue}{2 \cdot \left(\sin \left(\frac{\pi \cdot angle}{180}\right) \cdot \left(\left(b \cdot b - a \cdot a\right) \cdot \cos \left(\frac{\pi \cdot angle}{180}\right)\right)\right)} \]
   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. *-commutative N/A

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

      2. associate-*r* N/A

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

      3. *-commutative N/A

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

      4. associate-*r* N/A

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

      5. associate-*r* N/A

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

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

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

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

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

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

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

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

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

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

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

      11. unpow2 N/A

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

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

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

      13. unpow2 N/A

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

      14. *-lowering-*.f64 48.5%

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

   7. Simplified 48.5%

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

      1. *-commutative N/A

         \[\leadsto \left(b \cdot b - a \cdot a\right) \cdot \color{blue}{\left(angle \cdot \left(\frac{1}{90} \cdot \mathsf{PI}\left(\right)\right)\right)} \]

      2. difference-of-squares N/A

         \[\leadsto \left(\left(b + a\right) \cdot \left(b - a\right)\right) \cdot \left(\color{blue}{angle} \cdot \left(\frac{1}{90} \cdot \mathsf{PI}\left(\right)\right)\right) \]

      3. associate-*l* N/A

         \[\leadsto \left(b + a\right) \cdot \color{blue}{\left(\left(b - a\right) \cdot \left(angle \cdot \left(\frac{1}{90} \cdot \mathsf{PI}\left(\right)\right)\right)\right)} \]

      4. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\left(b + a\right), \color{blue}{\left(\left(b - a\right) \cdot \left(angle \cdot \left(\frac{1}{90} \cdot \mathsf{PI}\left(\right)\right)\right)\right)}\right) \]

      5. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(b, a\right), \left(\color{blue}{\left(b - a\right)} \cdot \left(angle \cdot \left(\frac{1}{90} \cdot \mathsf{PI}\left(\right)\right)\right)\right)\right) \]

      6. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(b, a\right), \mathsf{*.f64}\left(\left(b - a\right), \color{blue}{\left(angle \cdot \left(\frac{1}{90} \cdot \mathsf{PI}\left(\right)\right)\right)}\right)\right) \]

      7. --lowering--.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(b, a\right), \mathsf{*.f64}\left(\mathsf{\_.f64}\left(b, a\right), \left(\color{blue}{angle} \cdot \left(\frac{1}{90} \cdot \mathsf{PI}\left(\right)\right)\right)\right)\right) \]

      8. associate-*r* N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(b, a\right), \mathsf{*.f64}\left(\mathsf{\_.f64}\left(b, a\right), \left(\left(angle \cdot \frac{1}{90}\right) \cdot \color{blue}{\mathsf{PI}\left(\right)}\right)\right)\right) \]

      9. *-commutative N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(b, a\right), \mathsf{*.f64}\left(\mathsf{\_.f64}\left(b, a\right), \left(\mathsf{PI}\left(\right) \cdot \color{blue}{\left(angle \cdot \frac{1}{90}\right)}\right)\right)\right) \]

      10. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(b, a\right), \mathsf{*.f64}\left(\mathsf{\_.f64}\left(b, a\right), \mathsf{*.f64}\left(\mathsf{PI}\left(\right), \color{blue}{\left(angle \cdot \frac{1}{90}\right)}\right)\right)\right) \]

      11. PI-lowering-PI.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(b, a\right), \mathsf{*.f64}\left(\mathsf{\_.f64}\left(b, a\right), \mathsf{*.f64}\left(\mathsf{PI.f64}\left(\right), \left(\color{blue}{angle} \cdot \frac{1}{90}\right)\right)\right)\right) \]

      12. *-lowering-*.f64 61.3%

          \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(b, a\right), \mathsf{*.f64}\left(\mathsf{\_.f64}\left(b, a\right), \mathsf{*.f64}\left(\mathsf{PI.f64}\left(\right), \mathsf{*.f64}\left(angle, \color{blue}{\frac{1}{90}}\right)\right)\right)\right) \]

   9. Applied egg-rr 61.3%

      \[\leadsto \color{blue}{\left(b + a\right) \cdot \left(\left(b - a\right) \cdot \left(\pi \cdot \left(angle \cdot 0.011111111111111112\right)\right)\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 52.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \leq 1.8 \cdot 10^{-166}:\\ \;\;\;\;\frac{\sin \left(0.011111111111111112 \cdot \left(angle \cdot \pi\right)\right)}{\frac{1}{b \cdot b}}\\ \mathbf{else}:\\ \;\;\;\;\left(b + a\right) \cdot \left(\left(b - a\right) \cdot \left(\pi \cdot \left(angle \cdot 0.011111111111111112\right)\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 50.3% accurate, 3.7× speedup

Math

\[\begin{array}{l} angle\_m = \left|angle\right| \\ angle\_s = \mathsf{copysign}\left(1, angle\right) \\ angle\_s \cdot \begin{array}{l} \mathbf{if}\;a \leq 1.25 \cdot 10^{-166}:\\ \;\;\;\;\left(b \cdot b\right) \cdot \sin \left(0.011111111111111112 \cdot \left(angle\_m \cdot \pi\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(b + a\right) \cdot \left(\left(b - a\right) \cdot \left(\pi \cdot \left(angle\_m \cdot 0.011111111111111112\right)\right)\right)\\ \end{array} \end{array} \]

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 (<= a 1.25e-166)
    (* (* b b) (sin (* 0.011111111111111112 (* angle_m PI))))
    (* (+ b a) (* (- b a) (* 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 (a <= 1.25e-166) {
		tmp = (b * b) * sin((0.011111111111111112 * (angle_m * ((double) M_PI))));
	} else {
		tmp = (b + a) * ((b - a) * (((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 (a <= 1.25e-166) {
		tmp = (b * b) * Math.sin((0.011111111111111112 * (angle_m * Math.PI)));
	} else {
		tmp = (b + a) * ((b - a) * (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 a <= 1.25e-166:
		tmp = (b * b) * math.sin((0.011111111111111112 * (angle_m * math.pi)))
	else:
		tmp = (b + a) * ((b - a) * (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 (a <= 1.25e-166)
		tmp = Float64(Float64(b * b) * sin(Float64(0.011111111111111112 * Float64(angle_m * pi))));
	else
		tmp = Float64(Float64(b + a) * Float64(Float64(b - a) * 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 (a <= 1.25e-166)
		tmp = (b * b) * sin((0.011111111111111112 * (angle_m * pi)));
	else
		tmp = (b + a) * ((b - a) * (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[a, 1.25e-166], N[(N[(b * b), $MachinePrecision] * N[Sin[N[(0.011111111111111112 * N[(angle$95$m * Pi), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(N[(b + a), $MachinePrecision] * N[(N[(b - a), $MachinePrecision] * N[(Pi * N[(angle$95$m * 0.011111111111111112), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if a < 1.25e-166

   1. Initial program 57.2%

      \[\left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin \left(\pi \cdot \frac{angle}{180}\right)\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right) \]
   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. associate-*l* N/A

         \[\leadsto 2 \cdot \color{blue}{\left(\left({b}^{2} - {a}^{2}\right) \cdot \left(\sin \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right) \cdot \cos \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right)\right)\right)} \]

      3. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(2, \color{blue}{\left(\left({b}^{2} - {a}^{2}\right) \cdot \left(\sin \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right) \cdot \cos \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right)\right)\right)}\right) \]

      4. associate-*r* N/A

         \[\leadsto \mathsf{*.f64}\left(2, \left(\left(\left({b}^{2} - {a}^{2}\right) \cdot \sin \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right)\right) \cdot \color{blue}{\cos \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right)}\right)\right) \]

      5. *-commutative N/A

         \[\leadsto \mathsf{*.f64}\left(2, \left(\left(\sin \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right) \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \cos \color{blue}{\left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right)}\right)\right) \]

      6. associate-*l* N/A

         \[\leadsto \mathsf{*.f64}\left(2, \left(\sin \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right) \cdot \color{blue}{\left(\left({b}^{2} - {a}^{2}\right) \cdot \cos \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right)\right)}\right)\right) \]

      7. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(2, \mathsf{*.f64}\left(\sin \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right), \color{blue}{\left(\left({b}^{2} - {a}^{2}\right) \cdot \cos \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right)\right)}\right)\right) \]

      8. sin-lowering-sin.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(2, \mathsf{*.f64}\left(\mathsf{sin.f64}\left(\left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right)\right), \left(\color{blue}{\left({b}^{2} - {a}^{2}\right)} \cdot \cos \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right)\right)\right)\right) \]

      9. associate-*r/ N/A

         \[\leadsto \mathsf{*.f64}\left(2, \mathsf{*.f64}\left(\mathsf{sin.f64}\left(\left(\frac{\mathsf{PI}\left(\right) \cdot angle}{180}\right)\right), \left(\left(\color{blue}{{b}^{2}} - {a}^{2}\right) \cdot \cos \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right)\right)\right)\right) \]

      10. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(2, \mathsf{*.f64}\left(\mathsf{sin.f64}\left(\mathsf{/.f64}\left(\left(\mathsf{PI}\left(\right) \cdot angle\right), 180\right)\right), \left(\left(\color{blue}{{b}^{2}} - {a}^{2}\right) \cdot \cos \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right)\right)\right)\right) \]

      11. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(2, \mathsf{*.f64}\left(\mathsf{sin.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{PI}\left(\right), angle\right), 180\right)\right), \left(\left({\color{blue}{b}}^{2} - {a}^{2}\right) \cdot \cos \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right)\right)\right)\right) \]

      12. PI-lowering-PI.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(2, \mathsf{*.f64}\left(\mathsf{sin.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{PI.f64}\left(\right), angle\right), 180\right)\right), \left(\left({b}^{2} - {a}^{2}\right) \cdot \cos \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right)\right)\right)\right) \]

   3. Simplified 58.2%

      \[\leadsto \color{blue}{2 \cdot \left(\sin \left(\frac{\pi \cdot angle}{180}\right) \cdot \left(\left(b \cdot b - a \cdot a\right) \cdot \cos \left(\frac{\pi \cdot angle}{180}\right)\right)\right)} \]
   4. Add Preprocessing
   5. Step-by-step derivation

      1. associate-*r* N/A

         \[\leadsto \left(2 \cdot \sin \left(\frac{\mathsf{PI}\left(\right) \cdot angle}{180}\right)\right) \cdot \color{blue}{\left(\left(b \cdot b - a \cdot a\right) \cdot \cos \left(\frac{\mathsf{PI}\left(\right) \cdot angle}{180}\right)\right)} \]

      2. *-commutative N/A

         \[\leadsto \left(2 \cdot \sin \left(\frac{\mathsf{PI}\left(\right) \cdot angle}{180}\right)\right) \cdot \left(\cos \left(\frac{\mathsf{PI}\left(\right) \cdot angle}{180}\right) \cdot \color{blue}{\left(b \cdot b - a \cdot a\right)}\right) \]

      3. associate-*r/ N/A

         \[\leadsto \left(2 \cdot \sin \left(\frac{\mathsf{PI}\left(\right) \cdot angle}{180}\right)\right) \cdot \left(\cos \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right) \cdot \left(\color{blue}{b} \cdot b - a \cdot a\right)\right) \]

      4. associate-*r* N/A

         \[\leadsto \left(\left(2 \cdot \sin \left(\frac{\mathsf{PI}\left(\right) \cdot angle}{180}\right)\right) \cdot \cos \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right)\right) \cdot \color{blue}{\left(b \cdot b - a \cdot a\right)} \]

      5. flip-- N/A

         \[\leadsto \left(\left(2 \cdot \sin \left(\frac{\mathsf{PI}\left(\right) \cdot angle}{180}\right)\right) \cdot \cos \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right)\right) \cdot \frac{\left(b \cdot b\right) \cdot \left(b \cdot b\right) - \left(a \cdot a\right) \cdot \left(a \cdot a\right)}{\color{blue}{b \cdot b + a \cdot a}} \]

      6. clear-num N/A

         \[\leadsto \left(\left(2 \cdot \sin \left(\frac{\mathsf{PI}\left(\right) \cdot angle}{180}\right)\right) \cdot \cos \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right)\right) \cdot \frac{1}{\color{blue}{\frac{b \cdot b + a \cdot a}{\left(b \cdot b\right) \cdot \left(b \cdot b\right) - \left(a \cdot a\right) \cdot \left(a \cdot a\right)}}} \]

      7. un-div-inv N/A

         \[\leadsto \frac{\left(2 \cdot \sin \left(\frac{\mathsf{PI}\left(\right) \cdot angle}{180}\right)\right) \cdot \cos \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right)}{\color{blue}{\frac{b \cdot b + a \cdot a}{\left(b \cdot b\right) \cdot \left(b \cdot b\right) - \left(a \cdot a\right) \cdot \left(a \cdot a\right)}}} \]

      8. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\left(\left(2 \cdot \sin \left(\frac{\mathsf{PI}\left(\right) \cdot angle}{180}\right)\right) \cdot \cos \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right)\right), \color{blue}{\left(\frac{b \cdot b + a \cdot a}{\left(b \cdot b\right) \cdot \left(b \cdot b\right) - \left(a \cdot a\right) \cdot \left(a \cdot a\right)}\right)}\right) \]

   6. Applied egg-rr 57.3%

      \[\leadsto \color{blue}{\frac{\sin \left(\left(\pi \cdot angle\right) \cdot 0.011111111111111112\right)}{\frac{1}{b \cdot b - a \cdot a}}} \]

   7. Taylor expanded in b around inf

      \[\leadsto \color{blue}{{b}^{2} \cdot \sin \left(\frac{1}{90} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)} \]
   8. 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. sin-lowering-sin.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(b, b\right), \mathsf{sin.f64}\left(\left(\frac{1}{90} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)\right)\right) \]

      5. *-commutative N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(b, b\right), \mathsf{sin.f64}\left(\left(\left(angle \cdot \mathsf{PI}\left(\right)\right) \cdot \frac{1}{90}\right)\right)\right) \]

      6. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(b, b\right), \mathsf{sin.f64}\left(\mathsf{*.f64}\left(\left(angle \cdot \mathsf{PI}\left(\right)\right), \frac{1}{90}\right)\right)\right) \]

      7. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(b, b\right), \mathsf{sin.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(angle, \mathsf{PI}\left(\right)\right), \frac{1}{90}\right)\right)\right) \]

      8. PI-lowering-PI.f64 47.6%

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(b, b\right), \mathsf{sin.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(angle, \mathsf{PI.f64}\left(\right)\right), \frac{1}{90}\right)\right)\right) \]

   9. Simplified 47.6%

      \[\leadsto \color{blue}{\left(b \cdot b\right) \cdot \sin \left(\left(angle \cdot \pi\right) \cdot 0.011111111111111112\right)} \]

   if 1.25e-166 < a

   1. Initial program 50.4%

      \[\left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin \left(\pi \cdot \frac{angle}{180}\right)\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right) \]
   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. associate-*l* N/A

         \[\leadsto 2 \cdot \color{blue}{\left(\left({b}^{2} - {a}^{2}\right) \cdot \left(\sin \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right) \cdot \cos \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right)\right)\right)} \]

      3. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(2, \color{blue}{\left(\left({b}^{2} - {a}^{2}\right) \cdot \left(\sin \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right) \cdot \cos \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right)\right)\right)}\right) \]

      4. associate-*r* N/A

         \[\leadsto \mathsf{*.f64}\left(2, \left(\left(\left({b}^{2} - {a}^{2}\right) \cdot \sin \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right)\right) \cdot \color{blue}{\cos \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right)}\right)\right) \]

      5. *-commutative N/A

         \[\leadsto \mathsf{*.f64}\left(2, \left(\left(\sin \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right) \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \cos \color{blue}{\left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right)}\right)\right) \]

      6. associate-*l* N/A

         \[\leadsto \mathsf{*.f64}\left(2, \left(\sin \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right) \cdot \color{blue}{\left(\left({b}^{2} - {a}^{2}\right) \cdot \cos \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right)\right)}\right)\right) \]

      7. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(2, \mathsf{*.f64}\left(\sin \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right), \color{blue}{\left(\left({b}^{2} - {a}^{2}\right) \cdot \cos \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right)\right)}\right)\right) \]

      8. sin-lowering-sin.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(2, \mathsf{*.f64}\left(\mathsf{sin.f64}\left(\left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right)\right), \left(\color{blue}{\left({b}^{2} - {a}^{2}\right)} \cdot \cos \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right)\right)\right)\right) \]

      9. associate-*r/ N/A

         \[\leadsto \mathsf{*.f64}\left(2, \mathsf{*.f64}\left(\mathsf{sin.f64}\left(\left(\frac{\mathsf{PI}\left(\right) \cdot angle}{180}\right)\right), \left(\left(\color{blue}{{b}^{2}} - {a}^{2}\right) \cdot \cos \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right)\right)\right)\right) \]

      10. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(2, \mathsf{*.f64}\left(\mathsf{sin.f64}\left(\mathsf{/.f64}\left(\left(\mathsf{PI}\left(\right) \cdot angle\right), 180\right)\right), \left(\left(\color{blue}{{b}^{2}} - {a}^{2}\right) \cdot \cos \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right)\right)\right)\right) \]

      11. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(2, \mathsf{*.f64}\left(\mathsf{sin.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{PI}\left(\right), angle\right), 180\right)\right), \left(\left({\color{blue}{b}}^{2} - {a}^{2}\right) \cdot \cos \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right)\right)\right)\right) \]

      12. PI-lowering-PI.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(2, \mathsf{*.f64}\left(\mathsf{sin.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{PI.f64}\left(\right), angle\right), 180\right)\right), \left(\left({b}^{2} - {a}^{2}\right) \cdot \cos \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right)\right)\right)\right) \]

   3. Simplified 51.7%

      \[\leadsto \color{blue}{2 \cdot \left(\sin \left(\frac{\pi \cdot angle}{180}\right) \cdot \left(\left(b \cdot b - a \cdot a\right) \cdot \cos \left(\frac{\pi \cdot angle}{180}\right)\right)\right)} \]
   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. *-commutative N/A

         \[\leadsto \left(angle \cdot \left(\mathsf{PI}\left(\right) \cdot \left({b}^{2} - {a}^{2}\right)\right)\right) \cdot \color{blue}{\frac{1}{90}} \]

      2. associate-*r* N/A

         \[\leadsto angle \cdot \color{blue}{\left(\left(\mathsf{PI}\left(\right) \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \frac{1}{90}\right)} \]

      3. *-commutative N/A

         \[\leadsto angle \cdot \left(\frac{1}{90} \cdot \color{blue}{\left(\mathsf{PI}\left(\right) \cdot \left({b}^{2} - {a}^{2}\right)\right)}\right) \]

      4. associate-*r* N/A

         \[\leadsto angle \cdot \left(\left(\frac{1}{90} \cdot \mathsf{PI}\left(\right)\right) \cdot \color{blue}{\left({b}^{2} - {a}^{2}\right)}\right) \]

      5. associate-*r* N/A

         \[\leadsto \left(angle \cdot \left(\frac{1}{90} \cdot \mathsf{PI}\left(\right)\right)\right) \cdot \color{blue}{\left({b}^{2} - {a}^{2}\right)} \]

      6. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\left(angle \cdot \left(\frac{1}{90} \cdot \mathsf{PI}\left(\right)\right)\right), \color{blue}{\left({b}^{2} - {a}^{2}\right)}\right) \]

      7. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(angle, \left(\frac{1}{90} \cdot \mathsf{PI}\left(\right)\right)\right), \left(\color{blue}{{b}^{2}} - {a}^{2}\right)\right) \]

      8. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(angle, \mathsf{*.f64}\left(\frac{1}{90}, \mathsf{PI}\left(\right)\right)\right), \left({b}^{\color{blue}{2}} - {a}^{2}\right)\right) \]

      9. PI-lowering-PI.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(angle, \mathsf{*.f64}\left(\frac{1}{90}, \mathsf{PI.f64}\left(\right)\right)\right), \left({b}^{2} - {a}^{2}\right)\right) \]

      10. --lowering--.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(angle, \mathsf{*.f64}\left(\frac{1}{90}, \mathsf{PI.f64}\left(\right)\right)\right), \mathsf{\_.f64}\left(\left({b}^{2}\right), \color{blue}{\left({a}^{2}\right)}\right)\right) \]

      11. unpow2 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(angle, \mathsf{*.f64}\left(\frac{1}{90}, \mathsf{PI.f64}\left(\right)\right)\right), \mathsf{\_.f64}\left(\left(b \cdot b\right), \left({\color{blue}{a}}^{2}\right)\right)\right) \]

      12. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(angle, \mathsf{*.f64}\left(\frac{1}{90}, \mathsf{PI.f64}\left(\right)\right)\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(b, b\right), \left({\color{blue}{a}}^{2}\right)\right)\right) \]

      13. unpow2 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(angle, \mathsf{*.f64}\left(\frac{1}{90}, \mathsf{PI.f64}\left(\right)\right)\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(b, b\right), \left(a \cdot \color{blue}{a}\right)\right)\right) \]

      14. *-lowering-*.f64 48.5%

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(angle, \mathsf{*.f64}\left(\frac{1}{90}, \mathsf{PI.f64}\left(\right)\right)\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(b, b\right), \mathsf{*.f64}\left(a, \color{blue}{a}\right)\right)\right) \]

   7. Simplified 48.5%

      \[\leadsto \color{blue}{\left(angle \cdot \left(0.011111111111111112 \cdot \pi\right)\right) \cdot \left(b \cdot b - a \cdot a\right)} \]
   8. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \left(b \cdot b - a \cdot a\right) \cdot \color{blue}{\left(angle \cdot \left(\frac{1}{90} \cdot \mathsf{PI}\left(\right)\right)\right)} \]

      2. difference-of-squares N/A

         \[\leadsto \left(\left(b + a\right) \cdot \left(b - a\right)\right) \cdot \left(\color{blue}{angle} \cdot \left(\frac{1}{90} \cdot \mathsf{PI}\left(\right)\right)\right) \]

      3. associate-*l* N/A

         \[\leadsto \left(b + a\right) \cdot \color{blue}{\left(\left(b - a\right) \cdot \left(angle \cdot \left(\frac{1}{90} \cdot \mathsf{PI}\left(\right)\right)\right)\right)} \]

      4. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\left(b + a\right), \color{blue}{\left(\left(b - a\right) \cdot \left(angle \cdot \left(\frac{1}{90} \cdot \mathsf{PI}\left(\right)\right)\right)\right)}\right) \]

      5. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(b, a\right), \left(\color{blue}{\left(b - a\right)} \cdot \left(angle \cdot \left(\frac{1}{90} \cdot \mathsf{PI}\left(\right)\right)\right)\right)\right) \]

      6. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(b, a\right), \mathsf{*.f64}\left(\left(b - a\right), \color{blue}{\left(angle \cdot \left(\frac{1}{90} \cdot \mathsf{PI}\left(\right)\right)\right)}\right)\right) \]

      7. --lowering--.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(b, a\right), \mathsf{*.f64}\left(\mathsf{\_.f64}\left(b, a\right), \left(\color{blue}{angle} \cdot \left(\frac{1}{90} \cdot \mathsf{PI}\left(\right)\right)\right)\right)\right) \]

      8. associate-*r* N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(b, a\right), \mathsf{*.f64}\left(\mathsf{\_.f64}\left(b, a\right), \left(\left(angle \cdot \frac{1}{90}\right) \cdot \color{blue}{\mathsf{PI}\left(\right)}\right)\right)\right) \]

      9. *-commutative N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(b, a\right), \mathsf{*.f64}\left(\mathsf{\_.f64}\left(b, a\right), \left(\mathsf{PI}\left(\right) \cdot \color{blue}{\left(angle \cdot \frac{1}{90}\right)}\right)\right)\right) \]

      10. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(b, a\right), \mathsf{*.f64}\left(\mathsf{\_.f64}\left(b, a\right), \mathsf{*.f64}\left(\mathsf{PI}\left(\right), \color{blue}{\left(angle \cdot \frac{1}{90}\right)}\right)\right)\right) \]

      11. PI-lowering-PI.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(b, a\right), \mathsf{*.f64}\left(\mathsf{\_.f64}\left(b, a\right), \mathsf{*.f64}\left(\mathsf{PI.f64}\left(\right), \left(\color{blue}{angle} \cdot \frac{1}{90}\right)\right)\right)\right) \]

      12. *-lowering-*.f64 61.3%

          \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(b, a\right), \mathsf{*.f64}\left(\mathsf{\_.f64}\left(b, a\right), \mathsf{*.f64}\left(\mathsf{PI.f64}\left(\right), \mathsf{*.f64}\left(angle, \color{blue}{\frac{1}{90}}\right)\right)\right)\right) \]

   9. Applied egg-rr 61.3%

      \[\leadsto \color{blue}{\left(b + a\right) \cdot \left(\left(b - a\right) \cdot \left(\pi \cdot \left(angle \cdot 0.011111111111111112\right)\right)\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 52.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \leq 1.25 \cdot 10^{-166}:\\ \;\;\;\;\left(b \cdot b\right) \cdot \sin \left(0.011111111111111112 \cdot \left(angle \cdot \pi\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(b + a\right) \cdot \left(\left(b - a\right) \cdot \left(\pi \cdot \left(angle \cdot 0.011111111111111112\right)\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 12: 68.2% accurate, 3.7× speedup

Math

\[\begin{array}{l} angle\_m = \left|angle\right| \\ angle\_s = \mathsf{copysign}\left(1, angle\right) \\ angle\_s \cdot \left(\left(b - a\right) \cdot \left(\left(b + a\right) \cdot \sin \left(0.011111111111111112 \cdot \left(angle\_m \cdot \pi\right)\right)\right)\right) \end{array} \]

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 (* 0.011111111111111112 (* 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) {
	return angle_s * ((b - a) * ((b + a) * sin((0.011111111111111112 * (angle_m * ((double) M_PI))))));
}

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((0.011111111111111112 * (angle_m * Math.PI)))));
}

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((0.011111111111111112 * (angle_m * math.pi)))))

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(0.011111111111111112 * Float64(angle_m * pi))))))
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((0.011111111111111112 * (angle_m * pi)))));
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[(0.011111111111111112 * N[(angle$95$m * Pi), $MachinePrecision]), $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. associate-*l* N/A

      \[\leadsto 2 \cdot \color{blue}{\left(\left({b}^{2} - {a}^{2}\right) \cdot \left(\sin \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right) \cdot \cos \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right)\right)\right)} \]

   3. *-lowering-*.f64 N/A

      \[\leadsto \mathsf{*.f64}\left(2, \color{blue}{\left(\left({b}^{2} - {a}^{2}\right) \cdot \left(\sin \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right) \cdot \cos \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right)\right)\right)}\right) \]

   4. associate-*r* N/A

      \[\leadsto \mathsf{*.f64}\left(2, \left(\left(\left({b}^{2} - {a}^{2}\right) \cdot \sin \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right)\right) \cdot \color{blue}{\cos \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right)}\right)\right) \]

   5. *-commutative N/A

      \[\leadsto \mathsf{*.f64}\left(2, \left(\left(\sin \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right) \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \cos \color{blue}{\left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right)}\right)\right) \]

   6. associate-*l* N/A

      \[\leadsto \mathsf{*.f64}\left(2, \left(\sin \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right) \cdot \color{blue}{\left(\left({b}^{2} - {a}^{2}\right) \cdot \cos \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right)\right)}\right)\right) \]

   7. *-lowering-*.f64 N/A

      \[\leadsto \mathsf{*.f64}\left(2, \mathsf{*.f64}\left(\sin \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right), \color{blue}{\left(\left({b}^{2} - {a}^{2}\right) \cdot \cos \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right)\right)}\right)\right) \]

   8. sin-lowering-sin.f64 N/A

      \[\leadsto \mathsf{*.f64}\left(2, \mathsf{*.f64}\left(\mathsf{sin.f64}\left(\left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right)\right), \left(\color{blue}{\left({b}^{2} - {a}^{2}\right)} \cdot \cos \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right)\right)\right)\right) \]

   9. associate-*r/ N/A

      \[\leadsto \mathsf{*.f64}\left(2, \mathsf{*.f64}\left(\mathsf{sin.f64}\left(\left(\frac{\mathsf{PI}\left(\right) \cdot angle}{180}\right)\right), \left(\left(\color{blue}{{b}^{2}} - {a}^{2}\right) \cdot \cos \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right)\right)\right)\right) \]

   10. /-lowering-/.f64 N/A

       \[\leadsto \mathsf{*.f64}\left(2, \mathsf{*.f64}\left(\mathsf{sin.f64}\left(\mathsf{/.f64}\left(\left(\mathsf{PI}\left(\right) \cdot angle\right), 180\right)\right), \left(\left(\color{blue}{{b}^{2}} - {a}^{2}\right) \cdot \cos \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right)\right)\right)\right) \]

   11. *-lowering-*.f64 N/A

       \[\leadsto \mathsf{*.f64}\left(2, \mathsf{*.f64}\left(\mathsf{sin.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{PI}\left(\right), angle\right), 180\right)\right), \left(\left({\color{blue}{b}}^{2} - {a}^{2}\right) \cdot \cos \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right)\right)\right)\right) \]

   12. PI-lowering-PI.f64 N/A

       \[\leadsto \mathsf{*.f64}\left(2, \mathsf{*.f64}\left(\mathsf{sin.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{PI.f64}\left(\right), angle\right), 180\right)\right), \left(\left({b}^{2} - {a}^{2}\right) \cdot \cos \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right)\right)\right)\right) \]

3. Simplified 55.7%

   \[\leadsto \color{blue}{2 \cdot \left(\sin \left(\frac{\pi \cdot angle}{180}\right) \cdot \left(\left(b \cdot b - a \cdot a\right) \cdot \cos \left(\frac{\pi \cdot angle}{180}\right)\right)\right)} \]
4. Add Preprocessing
5. Step-by-step derivation

   1. associate-*r* N/A

      \[\leadsto \left(2 \cdot \sin \left(\frac{\mathsf{PI}\left(\right) \cdot angle}{180}\right)\right) \cdot \color{blue}{\left(\left(b \cdot b - a \cdot a\right) \cdot \cos \left(\frac{\mathsf{PI}\left(\right) \cdot angle}{180}\right)\right)} \]

   2. *-commutative N/A

      \[\leadsto \left(2 \cdot \sin \left(\frac{\mathsf{PI}\left(\right) \cdot angle}{180}\right)\right) \cdot \left(\cos \left(\frac{\mathsf{PI}\left(\right) \cdot angle}{180}\right) \cdot \color{blue}{\left(b \cdot b - a \cdot a\right)}\right) \]

   3. associate-*r/ N/A

      \[\leadsto \left(2 \cdot \sin \left(\frac{\mathsf{PI}\left(\right) \cdot angle}{180}\right)\right) \cdot \left(\cos \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right) \cdot \left(\color{blue}{b} \cdot b - a \cdot a\right)\right) \]

   4. associate-*r* N/A

      \[\leadsto \left(\left(2 \cdot \sin \left(\frac{\mathsf{PI}\left(\right) \cdot angle}{180}\right)\right) \cdot \cos \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right)\right) \cdot \color{blue}{\left(b \cdot b - a \cdot a\right)} \]

   5. *-commutative N/A

      \[\leadsto \left(b \cdot b - a \cdot a\right) \cdot \color{blue}{\left(\left(2 \cdot \sin \left(\frac{\mathsf{PI}\left(\right) \cdot angle}{180}\right)\right) \cdot \cos \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right)\right)} \]

   6. difference-of-squares N/A

      \[\leadsto \left(\left(b + a\right) \cdot \left(b - a\right)\right) \cdot \left(\color{blue}{\left(2 \cdot \sin \left(\frac{\mathsf{PI}\left(\right) \cdot angle}{180}\right)\right)} \cdot \cos \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right)\right) \]

   7. *-commutative N/A

      \[\leadsto \left(\left(b - a\right) \cdot \left(b + a\right)\right) \cdot \left(\color{blue}{\left(2 \cdot \sin \left(\frac{\mathsf{PI}\left(\right) \cdot angle}{180}\right)\right)} \cdot \cos \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right)\right) \]

   8. associate-*l* N/A

      \[\leadsto \left(b - a\right) \cdot \color{blue}{\left(\left(b + a\right) \cdot \left(\left(2 \cdot \sin \left(\frac{\mathsf{PI}\left(\right) \cdot angle}{180}\right)\right) \cdot \cos \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right)\right)\right)} \]

   9. *-lowering-*.f64 N/A

      \[\leadsto \mathsf{*.f64}\left(\left(b - a\right), \color{blue}{\left(\left(b + a\right) \cdot \left(\left(2 \cdot \sin \left(\frac{\mathsf{PI}\left(\right) \cdot angle}{180}\right)\right) \cdot \cos \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right)\right)\right)}\right) \]

   10. --lowering--.f64 N/A

       \[\leadsto \mathsf{*.f64}\left(\mathsf{\_.f64}\left(b, a\right), \left(\color{blue}{\left(b + a\right)} \cdot \left(\left(2 \cdot \sin \left(\frac{\mathsf{PI}\left(\right) \cdot angle}{180}\right)\right) \cdot \cos \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right)\right)\right)\right) \]

6. Applied egg-rr 67.9%

   \[\leadsto \color{blue}{\left(b - a\right) \cdot \left(\left(b + a\right) \cdot \sin \left(\left(\pi \cdot angle\right) \cdot 0.011111111111111112\right)\right)} \]

7. Final simplification 67.9%

   \[\leadsto \left(b - a\right) \cdot \left(\left(b + a\right) \cdot \sin \left(0.011111111111111112 \cdot \left(angle \cdot \pi\right)\right)\right) \]
8. Add Preprocessing

Alternative 13: 68.5% accurate, 3.7× speedup

Math

\[\begin{array}{l} angle\_m = \left|angle\right| \\ angle\_s = \mathsf{copysign}\left(1, angle\right) \\ angle\_s \cdot \left(\left(b - a\right) \cdot \left(\left(b + a\right) \cdot \sin \left(angle\_m \cdot \left(\pi \cdot 0.011111111111111112\right)\right)\right)\right) \end{array} \]

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(angle_m * Float64(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[(angle$95$m * N[(Pi * 0.011111111111111112), $MachinePrecision]), $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. associate-*l* N/A

      \[\leadsto 2 \cdot \color{blue}{\left(\left({b}^{2} - {a}^{2}\right) \cdot \left(\sin \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right) \cdot \cos \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right)\right)\right)} \]

   3. *-lowering-*.f64 N/A

      \[\leadsto \mathsf{*.f64}\left(2, \color{blue}{\left(\left({b}^{2} - {a}^{2}\right) \cdot \left(\sin \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right) \cdot \cos \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right)\right)\right)}\right) \]

   4. associate-*r* N/A

      \[\leadsto \mathsf{*.f64}\left(2, \left(\left(\left({b}^{2} - {a}^{2}\right) \cdot \sin \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right)\right) \cdot \color{blue}{\cos \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right)}\right)\right) \]

   5. *-commutative N/A

      \[\leadsto \mathsf{*.f64}\left(2, \left(\left(\sin \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right) \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \cos \color{blue}{\left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right)}\right)\right) \]

   6. associate-*l* N/A

      \[\leadsto \mathsf{*.f64}\left(2, \left(\sin \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right) \cdot \color{blue}{\left(\left({b}^{2} - {a}^{2}\right) \cdot \cos \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right)\right)}\right)\right) \]

   7. *-lowering-*.f64 N/A

      \[\leadsto \mathsf{*.f64}\left(2, \mathsf{*.f64}\left(\sin \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right), \color{blue}{\left(\left({b}^{2} - {a}^{2}\right) \cdot \cos \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right)\right)}\right)\right) \]

   8. sin-lowering-sin.f64 N/A

      \[\leadsto \mathsf{*.f64}\left(2, \mathsf{*.f64}\left(\mathsf{sin.f64}\left(\left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right)\right), \left(\color{blue}{\left({b}^{2} - {a}^{2}\right)} \cdot \cos \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right)\right)\right)\right) \]

   9. associate-*r/ N/A

      \[\leadsto \mathsf{*.f64}\left(2, \mathsf{*.f64}\left(\mathsf{sin.f64}\left(\left(\frac{\mathsf{PI}\left(\right) \cdot angle}{180}\right)\right), \left(\left(\color{blue}{{b}^{2}} - {a}^{2}\right) \cdot \cos \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right)\right)\right)\right) \]

   10. /-lowering-/.f64 N/A

       \[\leadsto \mathsf{*.f64}\left(2, \mathsf{*.f64}\left(\mathsf{sin.f64}\left(\mathsf{/.f64}\left(\left(\mathsf{PI}\left(\right) \cdot angle\right), 180\right)\right), \left(\left(\color{blue}{{b}^{2}} - {a}^{2}\right) \cdot \cos \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right)\right)\right)\right) \]

   11. *-lowering-*.f64 N/A

       \[\leadsto \mathsf{*.f64}\left(2, \mathsf{*.f64}\left(\mathsf{sin.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{PI}\left(\right), angle\right), 180\right)\right), \left(\left({\color{blue}{b}}^{2} - {a}^{2}\right) \cdot \cos \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right)\right)\right)\right) \]

   12. PI-lowering-PI.f64 N/A

       \[\leadsto \mathsf{*.f64}\left(2, \mathsf{*.f64}\left(\mathsf{sin.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{PI.f64}\left(\right), angle\right), 180\right)\right), \left(\left({b}^{2} - {a}^{2}\right) \cdot \cos \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right)\right)\right)\right) \]

3. Simplified 55.7%

   \[\leadsto \color{blue}{2 \cdot \left(\sin \left(\frac{\pi \cdot angle}{180}\right) \cdot \left(\left(b \cdot b - a \cdot a\right) \cdot \cos \left(\frac{\pi \cdot angle}{180}\right)\right)\right)} \]
4. Add Preprocessing
5. Step-by-step derivation

   1. associate-*r* N/A

      \[\leadsto \left(2 \cdot \sin \left(\frac{\mathsf{PI}\left(\right) \cdot angle}{180}\right)\right) \cdot \color{blue}{\left(\left(b \cdot b - a \cdot a\right) \cdot \cos \left(\frac{\mathsf{PI}\left(\right) \cdot angle}{180}\right)\right)} \]

   2. *-commutative N/A

      \[\leadsto \left(2 \cdot \sin \left(\frac{\mathsf{PI}\left(\right) \cdot angle}{180}\right)\right) \cdot \left(\cos \left(\frac{\mathsf{PI}\left(\right) \cdot angle}{180}\right) \cdot \color{blue}{\left(b \cdot b - a \cdot a\right)}\right) \]

   3. associate-*r/ N/A

      \[\leadsto \left(2 \cdot \sin \left(\frac{\mathsf{PI}\left(\right) \cdot angle}{180}\right)\right) \cdot \left(\cos \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right) \cdot \left(\color{blue}{b} \cdot b - a \cdot a\right)\right) \]

   4. associate-*r* N/A

      \[\leadsto \left(\left(2 \cdot \sin \left(\frac{\mathsf{PI}\left(\right) \cdot angle}{180}\right)\right) \cdot \cos \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right)\right) \cdot \color{blue}{\left(b \cdot b - a \cdot a\right)} \]

   5. flip-- N/A

      \[\leadsto \left(\left(2 \cdot \sin \left(\frac{\mathsf{PI}\left(\right) \cdot angle}{180}\right)\right) \cdot \cos \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right)\right) \cdot \frac{\left(b \cdot b\right) \cdot \left(b \cdot b\right) - \left(a \cdot a\right) \cdot \left(a \cdot a\right)}{\color{blue}{b \cdot b + a \cdot a}} \]

   6. clear-num N/A

      \[\leadsto \left(\left(2 \cdot \sin \left(\frac{\mathsf{PI}\left(\right) \cdot angle}{180}\right)\right) \cdot \cos \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right)\right) \cdot \frac{1}{\color{blue}{\frac{b \cdot b + a \cdot a}{\left(b \cdot b\right) \cdot \left(b \cdot b\right) - \left(a \cdot a\right) \cdot \left(a \cdot a\right)}}} \]

   7. un-div-inv N/A

      \[\leadsto \frac{\left(2 \cdot \sin \left(\frac{\mathsf{PI}\left(\right) \cdot angle}{180}\right)\right) \cdot \cos \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right)}{\color{blue}{\frac{b \cdot b + a \cdot a}{\left(b \cdot b\right) \cdot \left(b \cdot b\right) - \left(a \cdot a\right) \cdot \left(a \cdot a\right)}}} \]

   8. /-lowering-/.f64 N/A

      \[\leadsto \mathsf{/.f64}\left(\left(\left(2 \cdot \sin \left(\frac{\mathsf{PI}\left(\right) \cdot angle}{180}\right)\right) \cdot \cos \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right)\right), \color{blue}{\left(\frac{b \cdot b + a \cdot a}{\left(b \cdot b\right) \cdot \left(b \cdot b\right) - \left(a \cdot a\right) \cdot \left(a \cdot a\right)}\right)}\right) \]

6. Applied egg-rr 55.2%

   \[\leadsto \color{blue}{\frac{\sin \left(\left(\pi \cdot angle\right) \cdot 0.011111111111111112\right)}{\frac{1}{b \cdot b - a \cdot a}}} \]
7. Step-by-step derivation

   1. associate-/r/ N/A

      \[\leadsto \frac{\sin \left(\left(\mathsf{PI}\left(\right) \cdot angle\right) \cdot \frac{1}{90}\right)}{1} \cdot \color{blue}{\left(b \cdot b - a \cdot a\right)} \]

   2. /-rgt-identity N/A

      \[\leadsto \sin \left(\left(\mathsf{PI}\left(\right) \cdot angle\right) \cdot \frac{1}{90}\right) \cdot \left(\color{blue}{b \cdot b} - a \cdot a\right) \]

   3. difference-of-squares N/A

      \[\leadsto \sin \left(\left(\mathsf{PI}\left(\right) \cdot angle\right) \cdot \frac{1}{90}\right) \cdot \left(\left(b + a\right) \cdot \color{blue}{\left(b - a\right)}\right) \]

   4. associate-*r* N/A

      \[\leadsto \left(\sin \left(\left(\mathsf{PI}\left(\right) \cdot angle\right) \cdot \frac{1}{90}\right) \cdot \left(b + a\right)\right) \cdot \color{blue}{\left(b - a\right)} \]

   5. *-lowering-*.f64 N/A

      \[\leadsto \mathsf{*.f64}\left(\left(\sin \left(\left(\mathsf{PI}\left(\right) \cdot angle\right) \cdot \frac{1}{90}\right) \cdot \left(b + a\right)\right), \color{blue}{\left(b - a\right)}\right) \]

   6. *-lowering-*.f64 N/A

      \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\sin \left(\left(\mathsf{PI}\left(\right) \cdot angle\right) \cdot \frac{1}{90}\right), \left(b + a\right)\right), \left(\color{blue}{b} - a\right)\right) \]

   7. sin-lowering-sin.f64 N/A

      \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{sin.f64}\left(\left(\left(\mathsf{PI}\left(\right) \cdot angle\right) \cdot \frac{1}{90}\right)\right), \left(b + a\right)\right), \left(b - a\right)\right) \]

   8. *-commutative N/A

      \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{sin.f64}\left(\left(\left(angle \cdot \mathsf{PI}\left(\right)\right) \cdot \frac{1}{90}\right)\right), \left(b + a\right)\right), \left(b - a\right)\right) \]

   9. associate-*l* N/A

      \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{sin.f64}\left(\left(angle \cdot \left(\mathsf{PI}\left(\right) \cdot \frac{1}{90}\right)\right)\right), \left(b + a\right)\right), \left(b - a\right)\right) \]

   10. *-lowering-*.f64 N/A

       \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{sin.f64}\left(\mathsf{*.f64}\left(angle, \left(\mathsf{PI}\left(\right) \cdot \frac{1}{90}\right)\right)\right), \left(b + a\right)\right), \left(b - a\right)\right) \]

   11. *-lowering-*.f64 N/A

       \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{sin.f64}\left(\mathsf{*.f64}\left(angle, \mathsf{*.f64}\left(\mathsf{PI}\left(\right), \frac{1}{90}\right)\right)\right), \left(b + a\right)\right), \left(b - a\right)\right) \]

   12. PI-lowering-PI.f64 N/A

       \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{sin.f64}\left(\mathsf{*.f64}\left(angle, \mathsf{*.f64}\left(\mathsf{PI.f64}\left(\right), \frac{1}{90}\right)\right)\right), \left(b + a\right)\right), \left(b - a\right)\right) \]

   13. +-lowering-+.f64 N/A

       \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{sin.f64}\left(\mathsf{*.f64}\left(angle, \mathsf{*.f64}\left(\mathsf{PI.f64}\left(\right), \frac{1}{90}\right)\right)\right), \mathsf{+.f64}\left(b, a\right)\right), \left(b - a\right)\right) \]

   14. --lowering--.f64 67.3%

       \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{sin.f64}\left(\mathsf{*.f64}\left(angle, \mathsf{*.f64}\left(\mathsf{PI.f64}\left(\right), \frac{1}{90}\right)\right)\right), \mathsf{+.f64}\left(b, a\right)\right), \mathsf{\_.f64}\left(b, \color{blue}{a}\right)\right) \]

8. Applied egg-rr 67.3%

   \[\leadsto \color{blue}{\left(\sin \left(angle \cdot \left(\pi \cdot 0.011111111111111112\right)\right) \cdot \left(b + a\right)\right) \cdot \left(b - a\right)} \]

9. Final simplification 67.3%

   \[\leadsto \left(b - a\right) \cdot \left(\left(b + a\right) \cdot \sin \left(angle \cdot \left(\pi \cdot 0.011111111111111112\right)\right)\right) \]
10. Add Preprocessing

Alternative 14: 63.3% accurate, 10.7× speedup

Math

\[\begin{array}{l} angle\_m = \left|angle\right| \\ angle\_s = \mathsf{copysign}\left(1, angle\right) \\ angle\_s \cdot \begin{array}{l} \mathbf{if}\;angle\_m \leq 2.05 \cdot 10^{+49}:\\ \;\;\;\;\left(b + a\right) \cdot \left(\left(b - a\right) \cdot \left(\pi \cdot \left(angle\_m \cdot 0.011111111111111112\right)\right)\right)\\ \mathbf{elif}\;angle\_m \leq 8.5 \cdot 10^{+188}:\\ \;\;\;\;\left(2 \cdot \frac{b - a}{\frac{1}{b + a}}\right) \cdot \left(angle\_m \cdot \left(\pi \cdot 0.005555555555555556 + \left(angle\_m \cdot angle\_m\right) \cdot \left(\pi \cdot \left(\left(\pi \cdot \pi\right) \cdot -2.8577960676726107 \cdot 10^{-8}\right)\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(angle\_m \cdot \left(\pi \cdot 0.011111111111111112\right)\right) \cdot \left(\left(b \cdot b\right) \cdot \left(1 - \frac{\frac{a \cdot a}{b}}{b}\right)\right)\\ \end{array} \end{array} \]

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 2.05e+49)
    (* (+ b a) (* (- b a) (* PI (* angle_m 0.011111111111111112))))
    (if (<= angle_m 8.5e+188)
      (*
       (* 2.0 (/ (- b a) (/ 1.0 (+ b a))))
       (*
        angle_m
        (+
         (* PI 0.005555555555555556)
         (* (* angle_m angle_m) (* PI (* (* PI PI) -2.8577960676726107e-8))))))
      (*
       (* angle_m (* PI 0.011111111111111112))
       (* (* b b) (- 1.0 (/ (/ (* a a) b) b))))))))

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 <= 2.05e+49) {
		tmp = (b + a) * ((b - a) * (((double) M_PI) * (angle_m * 0.011111111111111112)));
	} else if (angle_m <= 8.5e+188) {
		tmp = (2.0 * ((b - a) / (1.0 / (b + a)))) * (angle_m * ((((double) M_PI) * 0.005555555555555556) + ((angle_m * angle_m) * (((double) M_PI) * ((((double) M_PI) * ((double) M_PI)) * -2.8577960676726107e-8)))));
	} else {
		tmp = (angle_m * (((double) M_PI) * 0.011111111111111112)) * ((b * b) * (1.0 - (((a * a) / b) / b)));
	}
	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 <= 2.05e+49) {
		tmp = (b + a) * ((b - a) * (Math.PI * (angle_m * 0.011111111111111112)));
	} else if (angle_m <= 8.5e+188) {
		tmp = (2.0 * ((b - a) / (1.0 / (b + a)))) * (angle_m * ((Math.PI * 0.005555555555555556) + ((angle_m * angle_m) * (Math.PI * ((Math.PI * Math.PI) * -2.8577960676726107e-8)))));
	} else {
		tmp = (angle_m * (Math.PI * 0.011111111111111112)) * ((b * b) * (1.0 - (((a * a) / b) / b)));
	}
	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 <= 2.05e+49:
		tmp = (b + a) * ((b - a) * (math.pi * (angle_m * 0.011111111111111112)))
	elif angle_m <= 8.5e+188:
		tmp = (2.0 * ((b - a) / (1.0 / (b + a)))) * (angle_m * ((math.pi * 0.005555555555555556) + ((angle_m * angle_m) * (math.pi * ((math.pi * math.pi) * -2.8577960676726107e-8)))))
	else:
		tmp = (angle_m * (math.pi * 0.011111111111111112)) * ((b * b) * (1.0 - (((a * a) / b) / b)))
	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 <= 2.05e+49)
		tmp = Float64(Float64(b + a) * Float64(Float64(b - a) * Float64(pi * Float64(angle_m * 0.011111111111111112))));
	elseif (angle_m <= 8.5e+188)
		tmp = Float64(Float64(2.0 * Float64(Float64(b - a) / Float64(1.0 / Float64(b + a)))) * Float64(angle_m * Float64(Float64(pi * 0.005555555555555556) + Float64(Float64(angle_m * angle_m) * Float64(pi * Float64(Float64(pi * pi) * -2.8577960676726107e-8))))));
	else
		tmp = Float64(Float64(angle_m * Float64(pi * 0.011111111111111112)) * Float64(Float64(b * b) * Float64(1.0 - Float64(Float64(Float64(a * a) / b) / b))));
	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 <= 2.05e+49)
		tmp = (b + a) * ((b - a) * (pi * (angle_m * 0.011111111111111112)));
	elseif (angle_m <= 8.5e+188)
		tmp = (2.0 * ((b - a) / (1.0 / (b + a)))) * (angle_m * ((pi * 0.005555555555555556) + ((angle_m * angle_m) * (pi * ((pi * pi) * -2.8577960676726107e-8)))));
	else
		tmp = (angle_m * (pi * 0.011111111111111112)) * ((b * b) * (1.0 - (((a * a) / b) / b)));
	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, 2.05e+49], N[(N[(b + a), $MachinePrecision] * N[(N[(b - a), $MachinePrecision] * N[(Pi * N[(angle$95$m * 0.011111111111111112), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[angle$95$m, 8.5e+188], N[(N[(2.0 * N[(N[(b - a), $MachinePrecision] / N[(1.0 / N[(b + a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(angle$95$m * N[(N[(Pi * 0.005555555555555556), $MachinePrecision] + N[(N[(angle$95$m * angle$95$m), $MachinePrecision] * N[(Pi * N[(N[(Pi * Pi), $MachinePrecision] * -2.8577960676726107e-8), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(angle$95$m * N[(Pi * 0.011111111111111112), $MachinePrecision]), $MachinePrecision] * N[(N[(b * b), $MachinePrecision] * N[(1.0 - N[(N[(N[(a * a), $MachinePrecision] / b), $MachinePrecision] / b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]), $MachinePrecision]

Derivation

1. Split input into 3 regimes

2. if angle < 2.05e49

   1. Initial program 61.8%

      \[\left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin \left(\pi \cdot \frac{angle}{180}\right)\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right) \]
   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. associate-*l* N/A

         \[\leadsto 2 \cdot \color{blue}{\left(\left({b}^{2} - {a}^{2}\right) \cdot \left(\sin \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right) \cdot \cos \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right)\right)\right)} \]

      3. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(2, \color{blue}{\left(\left({b}^{2} - {a}^{2}\right) \cdot \left(\sin \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right) \cdot \cos \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right)\right)\right)}\right) \]

      4. associate-*r* N/A

         \[\leadsto \mathsf{*.f64}\left(2, \left(\left(\left({b}^{2} - {a}^{2}\right) \cdot \sin \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right)\right) \cdot \color{blue}{\cos \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right)}\right)\right) \]

      5. *-commutative N/A

         \[\leadsto \mathsf{*.f64}\left(2, \left(\left(\sin \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right) \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \cos \color{blue}{\left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right)}\right)\right) \]

      6. associate-*l* N/A

         \[\leadsto \mathsf{*.f64}\left(2, \left(\sin \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right) \cdot \color{blue}{\left(\left({b}^{2} - {a}^{2}\right) \cdot \cos \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right)\right)}\right)\right) \]

      7. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(2, \mathsf{*.f64}\left(\sin \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right), \color{blue}{\left(\left({b}^{2} - {a}^{2}\right) \cdot \cos \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right)\right)}\right)\right) \]

      8. sin-lowering-sin.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(2, \mathsf{*.f64}\left(\mathsf{sin.f64}\left(\left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right)\right), \left(\color{blue}{\left({b}^{2} - {a}^{2}\right)} \cdot \cos \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right)\right)\right)\right) \]

      9. associate-*r/ N/A

         \[\leadsto \mathsf{*.f64}\left(2, \mathsf{*.f64}\left(\mathsf{sin.f64}\left(\left(\frac{\mathsf{PI}\left(\right) \cdot angle}{180}\right)\right), \left(\left(\color{blue}{{b}^{2}} - {a}^{2}\right) \cdot \cos \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right)\right)\right)\right) \]

      10. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(2, \mathsf{*.f64}\left(\mathsf{sin.f64}\left(\mathsf{/.f64}\left(\left(\mathsf{PI}\left(\right) \cdot angle\right), 180\right)\right), \left(\left(\color{blue}{{b}^{2}} - {a}^{2}\right) \cdot \cos \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right)\right)\right)\right) \]

      11. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(2, \mathsf{*.f64}\left(\mathsf{sin.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{PI}\left(\right), angle\right), 180\right)\right), \left(\left({\color{blue}{b}}^{2} - {a}^{2}\right) \cdot \cos \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right)\right)\right)\right) \]

      12. PI-lowering-PI.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(2, \mathsf{*.f64}\left(\mathsf{sin.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{PI.f64}\left(\right), angle\right), 180\right)\right), \left(\left({b}^{2} - {a}^{2}\right) \cdot \cos \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right)\right)\right)\right) \]

   3. Simplified 62.6%

      \[\leadsto \color{blue}{2 \cdot \left(\sin \left(\frac{\pi \cdot angle}{180}\right) \cdot \left(\left(b \cdot b - a \cdot a\right) \cdot \cos \left(\frac{\pi \cdot angle}{180}\right)\right)\right)} \]
   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. *-commutative N/A

         \[\leadsto \left(angle \cdot \left(\mathsf{PI}\left(\right) \cdot \left({b}^{2} - {a}^{2}\right)\right)\right) \cdot \color{blue}{\frac{1}{90}} \]

      2. associate-*r* N/A

         \[\leadsto angle \cdot \color{blue}{\left(\left(\mathsf{PI}\left(\right) \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \frac{1}{90}\right)} \]

      3. *-commutative N/A

         \[\leadsto angle \cdot \left(\frac{1}{90} \cdot \color{blue}{\left(\mathsf{PI}\left(\right) \cdot \left({b}^{2} - {a}^{2}\right)\right)}\right) \]

      4. associate-*r* N/A

         \[\leadsto angle \cdot \left(\left(\frac{1}{90} \cdot \mathsf{PI}\left(\right)\right) \cdot \color{blue}{\left({b}^{2} - {a}^{2}\right)}\right) \]

      5. associate-*r* N/A

         \[\leadsto \left(angle \cdot \left(\frac{1}{90} \cdot \mathsf{PI}\left(\right)\right)\right) \cdot \color{blue}{\left({b}^{2} - {a}^{2}\right)} \]

      6. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\left(angle \cdot \left(\frac{1}{90} \cdot \mathsf{PI}\left(\right)\right)\right), \color{blue}{\left({b}^{2} - {a}^{2}\right)}\right) \]

      7. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(angle, \left(\frac{1}{90} \cdot \mathsf{PI}\left(\right)\right)\right), \left(\color{blue}{{b}^{2}} - {a}^{2}\right)\right) \]

      8. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(angle, \mathsf{*.f64}\left(\frac{1}{90}, \mathsf{PI}\left(\right)\right)\right), \left({b}^{\color{blue}{2}} - {a}^{2}\right)\right) \]

      9. PI-lowering-PI.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(angle, \mathsf{*.f64}\left(\frac{1}{90}, \mathsf{PI.f64}\left(\right)\right)\right), \left({b}^{2} - {a}^{2}\right)\right) \]

      10. --lowering--.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(angle, \mathsf{*.f64}\left(\frac{1}{90}, \mathsf{PI.f64}\left(\right)\right)\right), \mathsf{\_.f64}\left(\left({b}^{2}\right), \color{blue}{\left({a}^{2}\right)}\right)\right) \]

      11. unpow2 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(angle, \mathsf{*.f64}\left(\frac{1}{90}, \mathsf{PI.f64}\left(\right)\right)\right), \mathsf{\_.f64}\left(\left(b \cdot b\right), \left({\color{blue}{a}}^{2}\right)\right)\right) \]

      12. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(angle, \mathsf{*.f64}\left(\frac{1}{90}, \mathsf{PI.f64}\left(\right)\right)\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(b, b\right), \left({\color{blue}{a}}^{2}\right)\right)\right) \]

      13. unpow2 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(angle, \mathsf{*.f64}\left(\frac{1}{90}, \mathsf{PI.f64}\left(\right)\right)\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(b, b\right), \left(a \cdot \color{blue}{a}\right)\right)\right) \]

      14. *-lowering-*.f64 59.8%

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(angle, \mathsf{*.f64}\left(\frac{1}{90}, \mathsf{PI.f64}\left(\right)\right)\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(b, b\right), \mathsf{*.f64}\left(a, \color{blue}{a}\right)\right)\right) \]

   7. Simplified 59.8%

      \[\leadsto \color{blue}{\left(angle \cdot \left(0.011111111111111112 \cdot \pi\right)\right) \cdot \left(b \cdot b - a \cdot a\right)} \]
   8. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \left(b \cdot b - a \cdot a\right) \cdot \color{blue}{\left(angle \cdot \left(\frac{1}{90} \cdot \mathsf{PI}\left(\right)\right)\right)} \]

      2. difference-of-squares N/A

         \[\leadsto \left(\left(b + a\right) \cdot \left(b - a\right)\right) \cdot \left(\color{blue}{angle} \cdot \left(\frac{1}{90} \cdot \mathsf{PI}\left(\right)\right)\right) \]

      3. associate-*l* N/A

         \[\leadsto \left(b + a\right) \cdot \color{blue}{\left(\left(b - a\right) \cdot \left(angle \cdot \left(\frac{1}{90} \cdot \mathsf{PI}\left(\right)\right)\right)\right)} \]

      4. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\left(b + a\right), \color{blue}{\left(\left(b - a\right) \cdot \left(angle \cdot \left(\frac{1}{90} \cdot \mathsf{PI}\left(\right)\right)\right)\right)}\right) \]

      5. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(b, a\right), \left(\color{blue}{\left(b - a\right)} \cdot \left(angle \cdot \left(\frac{1}{90} \cdot \mathsf{PI}\left(\right)\right)\right)\right)\right) \]

      6. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(b, a\right), \mathsf{*.f64}\left(\left(b - a\right), \color{blue}{\left(angle \cdot \left(\frac{1}{90} \cdot \mathsf{PI}\left(\right)\right)\right)}\right)\right) \]

      7. --lowering--.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(b, a\right), \mathsf{*.f64}\left(\mathsf{\_.f64}\left(b, a\right), \left(\color{blue}{angle} \cdot \left(\frac{1}{90} \cdot \mathsf{PI}\left(\right)\right)\right)\right)\right) \]

      8. associate-*r* N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(b, a\right), \mathsf{*.f64}\left(\mathsf{\_.f64}\left(b, a\right), \left(\left(angle \cdot \frac{1}{90}\right) \cdot \color{blue}{\mathsf{PI}\left(\right)}\right)\right)\right) \]

      9. *-commutative N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(b, a\right), \mathsf{*.f64}\left(\mathsf{\_.f64}\left(b, a\right), \left(\mathsf{PI}\left(\right) \cdot \color{blue}{\left(angle \cdot \frac{1}{90}\right)}\right)\right)\right) \]

      10. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(b, a\right), \mathsf{*.f64}\left(\mathsf{\_.f64}\left(b, a\right), \mathsf{*.f64}\left(\mathsf{PI}\left(\right), \color{blue}{\left(angle \cdot \frac{1}{90}\right)}\right)\right)\right) \]

      11. PI-lowering-PI.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(b, a\right), \mathsf{*.f64}\left(\mathsf{\_.f64}\left(b, a\right), \mathsf{*.f64}\left(\mathsf{PI.f64}\left(\right), \left(\color{blue}{angle} \cdot \frac{1}{90}\right)\right)\right)\right) \]

      12. *-lowering-*.f64 70.8%

          \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(b, a\right), \mathsf{*.f64}\left(\mathsf{\_.f64}\left(b, a\right), \mathsf{*.f64}\left(\mathsf{PI.f64}\left(\right), \mathsf{*.f64}\left(angle, \color{blue}{\frac{1}{90}}\right)\right)\right)\right) \]

   9. Applied egg-rr 70.8%

      \[\leadsto \color{blue}{\left(b + a\right) \cdot \left(\left(b - a\right) \cdot \left(\pi \cdot \left(angle \cdot 0.011111111111111112\right)\right)\right)} \]

   if 2.05e49 < angle < 8.49999999999999958e188

   1. Initial program 30.1%

      \[\left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin \left(\pi \cdot \frac{angle}{180}\right)\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right) \]
   2. Add Preprocessing

   3. Taylor expanded in angle around 0

      \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(2, \mathsf{\_.f64}\left(\mathsf{pow.f64}\left(b, 2\right), \mathsf{pow.f64}\left(a, 2\right)\right)\right), \mathsf{sin.f64}\left(\mathsf{*.f64}\left(\mathsf{PI.f64}\left(\right), \mathsf{/.f64}\left(angle, 180\right)\right)\right)\right), \color{blue}{1}\right) \]
   4. Step-by-step derivation

   5. Simplified 26.2%

      \[\leadsto \left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin \left(\pi \cdot \frac{angle}{180}\right)\right) \cdot \color{blue}{1} \]
   6. Step-by-step derivation

      1. pow2 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(2, \left(b \cdot b - {a}^{2}\right)\right), \mathsf{sin.f64}\left(\mathsf{*.f64}\left(\mathsf{PI.f64}\left(\right), \mathsf{/.f64}\left(angle, 180\right)\right)\right)\right), 1\right) \]

      2. pow2 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(2, \left(b \cdot b - a \cdot a\right)\right), \mathsf{sin.f64}\left(\mathsf{*.f64}\left(\mathsf{PI.f64}\left(\right), \mathsf{/.f64}\left(angle, 180\right)\right)\right)\right), 1\right) \]

      3. difference-of-squares N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(2, \left(\left(b + a\right) \cdot \left(b - a\right)\right)\right), \mathsf{sin.f64}\left(\mathsf{*.f64}\left(\mathsf{PI.f64}\left(\right), \mathsf{/.f64}\left(angle, 180\right)\right)\right)\right), 1\right) \]

      4. *-commutative N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(2, \left(\left(b - a\right) \cdot \left(b + a\right)\right)\right), \mathsf{sin.f64}\left(\mathsf{*.f64}\left(\mathsf{PI.f64}\left(\right), \mathsf{/.f64}\left(angle, 180\right)\right)\right)\right), 1\right) \]

      5. flip3-+ N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(2, \left(\left(b - a\right) \cdot \frac{{b}^{3} + {a}^{3}}{b \cdot b + \left(a \cdot a - b \cdot a\right)}\right)\right), \mathsf{sin.f64}\left(\mathsf{*.f64}\left(\mathsf{PI.f64}\left(\right), \mathsf{/.f64}\left(angle, 180\right)\right)\right)\right), 1\right) \]

      6. clear-num N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(2, \left(\left(b - a\right) \cdot \frac{1}{\frac{b \cdot b + \left(a \cdot a - b \cdot a\right)}{{b}^{3} + {a}^{3}}}\right)\right), \mathsf{sin.f64}\left(\mathsf{*.f64}\left(\mathsf{PI.f64}\left(\right), \mathsf{/.f64}\left(angle, 180\right)\right)\right)\right), 1\right) \]

      7. clear-num N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(2, \left(\left(b - a\right) \cdot \frac{1}{\frac{1}{\frac{{b}^{3} + {a}^{3}}{b \cdot b + \left(a \cdot a - b \cdot a\right)}}}\right)\right), \mathsf{sin.f64}\left(\mathsf{*.f64}\left(\mathsf{PI.f64}\left(\right), \mathsf{/.f64}\left(angle, 180\right)\right)\right)\right), 1\right) \]

      8. flip3-+ N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(2, \left(\left(b - a\right) \cdot \frac{1}{\frac{1}{b + a}}\right)\right), \mathsf{sin.f64}\left(\mathsf{*.f64}\left(\mathsf{PI.f64}\left(\right), \mathsf{/.f64}\left(angle, 180\right)\right)\right)\right), 1\right) \]

      9. un-div-inv N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(2, \left(\frac{b - a}{\frac{1}{b + a}}\right)\right), \mathsf{sin.f64}\left(\mathsf{*.f64}\left(\mathsf{PI.f64}\left(\right), \mathsf{/.f64}\left(angle, 180\right)\right)\right)\right), 1\right) \]

      10. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(2, \mathsf{/.f64}\left(\left(b - a\right), \left(\frac{1}{b + a}\right)\right)\right), \mathsf{sin.f64}\left(\mathsf{*.f64}\left(\mathsf{PI.f64}\left(\right), \mathsf{/.f64}\left(angle, 180\right)\right)\right)\right), 1\right) \]

      11. --lowering--.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(2, \mathsf{/.f64}\left(\mathsf{\_.f64}\left(b, a\right), \left(\frac{1}{b + a}\right)\right)\right), \mathsf{sin.f64}\left(\mathsf{*.f64}\left(\mathsf{PI.f64}\left(\right), \mathsf{/.f64}\left(angle, 180\right)\right)\right)\right), 1\right) \]

      12. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(2, \mathsf{/.f64}\left(\mathsf{\_.f64}\left(b, a\right), \mathsf{/.f64}\left(1, \left(b + a\right)\right)\right)\right), \mathsf{sin.f64}\left(\mathsf{*.f64}\left(\mathsf{PI.f64}\left(\right), \mathsf{/.f64}\left(angle, 180\right)\right)\right)\right), 1\right) \]

      13. +-lowering-+.f64 29.5%

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(2, \mathsf{/.f64}\left(\mathsf{\_.f64}\left(b, a\right), \mathsf{/.f64}\left(1, \mathsf{+.f64}\left(b, a\right)\right)\right)\right), \mathsf{sin.f64}\left(\mathsf{*.f64}\left(\mathsf{PI.f64}\left(\right), \mathsf{/.f64}\left(angle, 180\right)\right)\right)\right), 1\right) \]

   7. Applied egg-rr 29.5%

      \[\leadsto \left(\left(2 \cdot \color{blue}{\frac{b - a}{\frac{1}{b + a}}}\right) \cdot \sin \left(\pi \cdot \frac{angle}{180}\right)\right) \cdot 1 \]

   8. Taylor expanded in angle around 0

      \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(2, \mathsf{/.f64}\left(\mathsf{\_.f64}\left(b, a\right), \mathsf{/.f64}\left(1, \mathsf{+.f64}\left(b, a\right)\right)\right)\right), \color{blue}{\left(angle \cdot \left(\frac{-1}{34992000} \cdot \left({angle}^{2} \cdot {\mathsf{PI}\left(\right)}^{3}\right) + \frac{1}{180} \cdot \mathsf{PI}\left(\right)\right)\right)}\right), 1\right) \]
   9. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(2, \mathsf{/.f64}\left(\mathsf{\_.f64}\left(b, a\right), \mathsf{/.f64}\left(1, \mathsf{+.f64}\left(b, a\right)\right)\right)\right), \left(angle \cdot \left(\frac{1}{180} \cdot \mathsf{PI}\left(\right) + \frac{-1}{34992000} \cdot \left({angle}^{2} \cdot {\mathsf{PI}\left(\right)}^{3}\right)\right)\right)\right), 1\right) \]

      2. *-commutative N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(2, \mathsf{/.f64}\left(\mathsf{\_.f64}\left(b, a\right), \mathsf{/.f64}\left(1, \mathsf{+.f64}\left(b, a\right)\right)\right)\right), \left(angle \cdot \left(\frac{1}{180} \cdot \mathsf{PI}\left(\right) + \left({angle}^{2} \cdot {\mathsf{PI}\left(\right)}^{3}\right) \cdot \frac{-1}{34992000}\right)\right)\right), 1\right) \]

      3. associate-*r* N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(2, \mathsf{/.f64}\left(\mathsf{\_.f64}\left(b, a\right), \mathsf{/.f64}\left(1, \mathsf{+.f64}\left(b, a\right)\right)\right)\right), \left(angle \cdot \left(\frac{1}{180} \cdot \mathsf{PI}\left(\right) + {angle}^{2} \cdot \left({\mathsf{PI}\left(\right)}^{3} \cdot \frac{-1}{34992000}\right)\right)\right)\right), 1\right) \]

      4. *-commutative N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(2, \mathsf{/.f64}\left(\mathsf{\_.f64}\left(b, a\right), \mathsf{/.f64}\left(1, \mathsf{+.f64}\left(b, a\right)\right)\right)\right), \left(angle \cdot \left(\frac{1}{180} \cdot \mathsf{PI}\left(\right) + {angle}^{2} \cdot \left(\frac{-1}{34992000} \cdot {\mathsf{PI}\left(\right)}^{3}\right)\right)\right)\right), 1\right) \]

      5. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(2, \mathsf{/.f64}\left(\mathsf{\_.f64}\left(b, a\right), \mathsf{/.f64}\left(1, \mathsf{+.f64}\left(b, a\right)\right)\right)\right), \mathsf{*.f64}\left(angle, \left(\frac{1}{180} \cdot \mathsf{PI}\left(\right) + {angle}^{2} \cdot \left(\frac{-1}{34992000} \cdot {\mathsf{PI}\left(\right)}^{3}\right)\right)\right)\right), 1\right) \]

      6. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(2, \mathsf{/.f64}\left(\mathsf{\_.f64}\left(b, a\right), \mathsf{/.f64}\left(1, \mathsf{+.f64}\left(b, a\right)\right)\right)\right), \mathsf{*.f64}\left(angle, \mathsf{+.f64}\left(\left(\frac{1}{180} \cdot \mathsf{PI}\left(\right)\right), \left({angle}^{2} \cdot \left(\frac{-1}{34992000} \cdot {\mathsf{PI}\left(\right)}^{3}\right)\right)\right)\right)\right), 1\right) \]

      7. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(2, \mathsf{/.f64}\left(\mathsf{\_.f64}\left(b, a\right), \mathsf{/.f64}\left(1, \mathsf{+.f64}\left(b, a\right)\right)\right)\right), \mathsf{*.f64}\left(angle, \mathsf{+.f64}\left(\mathsf{*.f64}\left(\frac{1}{180}, \mathsf{PI}\left(\right)\right), \left({angle}^{2} \cdot \left(\frac{-1}{34992000} \cdot {\mathsf{PI}\left(\right)}^{3}\right)\right)\right)\right)\right), 1\right) \]

      8. PI-lowering-PI.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(2, \mathsf{/.f64}\left(\mathsf{\_.f64}\left(b, a\right), \mathsf{/.f64}\left(1, \mathsf{+.f64}\left(b, a\right)\right)\right)\right), \mathsf{*.f64}\left(angle, \mathsf{+.f64}\left(\mathsf{*.f64}\left(\frac{1}{180}, \mathsf{PI.f64}\left(\right)\right), \left({angle}^{2} \cdot \left(\frac{-1}{34992000} \cdot {\mathsf{PI}\left(\right)}^{3}\right)\right)\right)\right)\right), 1\right) \]

      9. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(2, \mathsf{/.f64}\left(\mathsf{\_.f64}\left(b, a\right), \mathsf{/.f64}\left(1, \mathsf{+.f64}\left(b, a\right)\right)\right)\right), \mathsf{*.f64}\left(angle, \mathsf{+.f64}\left(\mathsf{*.f64}\left(\frac{1}{180}, \mathsf{PI.f64}\left(\right)\right), \mathsf{*.f64}\left(\left({angle}^{2}\right), \left(\frac{-1}{34992000} \cdot {\mathsf{PI}\left(\right)}^{3}\right)\right)\right)\right)\right), 1\right) \]

      10. unpow2 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(2, \mathsf{/.f64}\left(\mathsf{\_.f64}\left(b, a\right), \mathsf{/.f64}\left(1, \mathsf{+.f64}\left(b, a\right)\right)\right)\right), \mathsf{*.f64}\left(angle, \mathsf{+.f64}\left(\mathsf{*.f64}\left(\frac{1}{180}, \mathsf{PI.f64}\left(\right)\right), \mathsf{*.f64}\left(\left(angle \cdot angle\right), \left(\frac{-1}{34992000} \cdot {\mathsf{PI}\left(\right)}^{3}\right)\right)\right)\right)\right), 1\right) \]

      11. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(2, \mathsf{/.f64}\left(\mathsf{\_.f64}\left(b, a\right), \mathsf{/.f64}\left(1, \mathsf{+.f64}\left(b, a\right)\right)\right)\right), \mathsf{*.f64}\left(angle, \mathsf{+.f64}\left(\mathsf{*.f64}\left(\frac{1}{180}, \mathsf{PI.f64}\left(\right)\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(angle, angle\right), \left(\frac{-1}{34992000} \cdot {\mathsf{PI}\left(\right)}^{3}\right)\right)\right)\right)\right), 1\right) \]

      12. *-commutative N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(2, \mathsf{/.f64}\left(\mathsf{\_.f64}\left(b, a\right), \mathsf{/.f64}\left(1, \mathsf{+.f64}\left(b, a\right)\right)\right)\right), \mathsf{*.f64}\left(angle, \mathsf{+.f64}\left(\mathsf{*.f64}\left(\frac{1}{180}, \mathsf{PI.f64}\left(\right)\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(angle, angle\right), \left({\mathsf{PI}\left(\right)}^{3} \cdot \frac{-1}{34992000}\right)\right)\right)\right)\right), 1\right) \]

      13. cube-mult N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(2, \mathsf{/.f64}\left(\mathsf{\_.f64}\left(b, a\right), \mathsf{/.f64}\left(1, \mathsf{+.f64}\left(b, a\right)\right)\right)\right), \mathsf{*.f64}\left(angle, \mathsf{+.f64}\left(\mathsf{*.f64}\left(\frac{1}{180}, \mathsf{PI.f64}\left(\right)\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(angle, angle\right), \left(\left(\mathsf{PI}\left(\right) \cdot \left(\mathsf{PI}\left(\right) \cdot \mathsf{PI}\left(\right)\right)\right) \cdot \frac{-1}{34992000}\right)\right)\right)\right)\right), 1\right) \]

      14. unpow2 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(2, \mathsf{/.f64}\left(\mathsf{\_.f64}\left(b, a\right), \mathsf{/.f64}\left(1, \mathsf{+.f64}\left(b, a\right)\right)\right)\right), \mathsf{*.f64}\left(angle, \mathsf{+.f64}\left(\mathsf{*.f64}\left(\frac{1}{180}, \mathsf{PI.f64}\left(\right)\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(angle, angle\right), \left(\left(\mathsf{PI}\left(\right) \cdot {\mathsf{PI}\left(\right)}^{2}\right) \cdot \frac{-1}{34992000}\right)\right)\right)\right)\right), 1\right) \]

      15. associate-*l* N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(2, \mathsf{/.f64}\left(\mathsf{\_.f64}\left(b, a\right), \mathsf{/.f64}\left(1, \mathsf{+.f64}\left(b, a\right)\right)\right)\right), \mathsf{*.f64}\left(angle, \mathsf{+.f64}\left(\mathsf{*.f64}\left(\frac{1}{180}, \mathsf{PI.f64}\left(\right)\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(angle, angle\right), \left(\mathsf{PI}\left(\right) \cdot \left({\mathsf{PI}\left(\right)}^{2} \cdot \frac{-1}{34992000}\right)\right)\right)\right)\right)\right), 1\right) \]

      16. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(2, \mathsf{/.f64}\left(\mathsf{\_.f64}\left(b, a\right), \mathsf{/.f64}\left(1, \mathsf{+.f64}\left(b, a\right)\right)\right)\right), \mathsf{*.f64}\left(angle, \mathsf{+.f64}\left(\mathsf{*.f64}\left(\frac{1}{180}, \mathsf{PI.f64}\left(\right)\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(angle, angle\right), \mathsf{*.f64}\left(\mathsf{PI}\left(\right), \left({\mathsf{PI}\left(\right)}^{2} \cdot \frac{-1}{34992000}\right)\right)\right)\right)\right)\right), 1\right) \]

      17. PI-lowering-PI.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(2, \mathsf{/.f64}\left(\mathsf{\_.f64}\left(b, a\right), \mathsf{/.f64}\left(1, \mathsf{+.f64}\left(b, a\right)\right)\right)\right), \mathsf{*.f64}\left(angle, \mathsf{+.f64}\left(\mathsf{*.f64}\left(\frac{1}{180}, \mathsf{PI.f64}\left(\right)\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(angle, angle\right), \mathsf{*.f64}\left(\mathsf{PI.f64}\left(\right), \left({\mathsf{PI}\left(\right)}^{2} \cdot \frac{-1}{34992000}\right)\right)\right)\right)\right)\right), 1\right) \]

      18. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(2, \mathsf{/.f64}\left(\mathsf{\_.f64}\left(b, a\right), \mathsf{/.f64}\left(1, \mathsf{+.f64}\left(b, a\right)\right)\right)\right), \mathsf{*.f64}\left(angle, \mathsf{+.f64}\left(\mathsf{*.f64}\left(\frac{1}{180}, \mathsf{PI.f64}\left(\right)\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(angle, angle\right), \mathsf{*.f64}\left(\mathsf{PI.f64}\left(\right), \mathsf{*.f64}\left(\left({\mathsf{PI}\left(\right)}^{2}\right), \frac{-1}{34992000}\right)\right)\right)\right)\right)\right), 1\right) \]

   10. Simplified 28.2%

       \[\leadsto \left(\left(2 \cdot \frac{b - a}{\frac{1}{b + a}}\right) \cdot \color{blue}{\left(angle \cdot \left(0.005555555555555556 \cdot \pi + \left(angle \cdot angle\right) \cdot \left(\pi \cdot \left(\left(\pi \cdot \pi\right) \cdot -2.8577960676726107 \cdot 10^{-8}\right)\right)\right)\right)}\right) \cdot 1 \]

   if 8.49999999999999958e188 < angle

   1. Initial program 31.2%

      \[\left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin \left(\pi \cdot \frac{angle}{180}\right)\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right) \]
   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. associate-*l* N/A

         \[\leadsto 2 \cdot \color{blue}{\left(\left({b}^{2} - {a}^{2}\right) \cdot \left(\sin \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right) \cdot \cos \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right)\right)\right)} \]

      3. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(2, \color{blue}{\left(\left({b}^{2} - {a}^{2}\right) \cdot \left(\sin \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right) \cdot \cos \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right)\right)\right)}\right) \]

      4. associate-*r* N/A

         \[\leadsto \mathsf{*.f64}\left(2, \left(\left(\left({b}^{2} - {a}^{2}\right) \cdot \sin \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right)\right) \cdot \color{blue}{\cos \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right)}\right)\right) \]

      5. *-commutative N/A

         \[\leadsto \mathsf{*.f64}\left(2, \left(\left(\sin \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right) \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \cos \color{blue}{\left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right)}\right)\right) \]

      6. associate-*l* N/A

         \[\leadsto \mathsf{*.f64}\left(2, \left(\sin \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right) \cdot \color{blue}{\left(\left({b}^{2} - {a}^{2}\right) \cdot \cos \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right)\right)}\right)\right) \]

      7. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(2, \mathsf{*.f64}\left(\sin \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right), \color{blue}{\left(\left({b}^{2} - {a}^{2}\right) \cdot \cos \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right)\right)}\right)\right) \]

      8. sin-lowering-sin.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(2, \mathsf{*.f64}\left(\mathsf{sin.f64}\left(\left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right)\right), \left(\color{blue}{\left({b}^{2} - {a}^{2}\right)} \cdot \cos \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right)\right)\right)\right) \]

      9. associate-*r/ N/A

         \[\leadsto \mathsf{*.f64}\left(2, \mathsf{*.f64}\left(\mathsf{sin.f64}\left(\left(\frac{\mathsf{PI}\left(\right) \cdot angle}{180}\right)\right), \left(\left(\color{blue}{{b}^{2}} - {a}^{2}\right) \cdot \cos \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right)\right)\right)\right) \]

      10. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(2, \mathsf{*.f64}\left(\mathsf{sin.f64}\left(\mathsf{/.f64}\left(\left(\mathsf{PI}\left(\right) \cdot angle\right), 180\right)\right), \left(\left(\color{blue}{{b}^{2}} - {a}^{2}\right) \cdot \cos \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right)\right)\right)\right) \]

      11. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(2, \mathsf{*.f64}\left(\mathsf{sin.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{PI}\left(\right), angle\right), 180\right)\right), \left(\left({\color{blue}{b}}^{2} - {a}^{2}\right) \cdot \cos \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right)\right)\right)\right) \]

      12. PI-lowering-PI.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(2, \mathsf{*.f64}\left(\mathsf{sin.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{PI.f64}\left(\right), angle\right), 180\right)\right), \left(\left({b}^{2} - {a}^{2}\right) \cdot \cos \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right)\right)\right)\right) \]

   3. Simplified 35.5%

      \[\leadsto \color{blue}{2 \cdot \left(\sin \left(\frac{\pi \cdot angle}{180}\right) \cdot \left(\left(b \cdot b - a \cdot a\right) \cdot \cos \left(\frac{\pi \cdot angle}{180}\right)\right)\right)} \]
   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. *-commutative N/A

         \[\leadsto \left(angle \cdot \left(\mathsf{PI}\left(\right) \cdot \left({b}^{2} - {a}^{2}\right)\right)\right) \cdot \color{blue}{\frac{1}{90}} \]

      2. associate-*r* N/A

         \[\leadsto angle \cdot \color{blue}{\left(\left(\mathsf{PI}\left(\right) \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \frac{1}{90}\right)} \]

      3. *-commutative N/A

         \[\leadsto angle \cdot \left(\frac{1}{90} \cdot \color{blue}{\left(\mathsf{PI}\left(\right) \cdot \left({b}^{2} - {a}^{2}\right)\right)}\right) \]

      4. associate-*r* N/A

         \[\leadsto angle \cdot \left(\left(\frac{1}{90} \cdot \mathsf{PI}\left(\right)\right) \cdot \color{blue}{\left({b}^{2} - {a}^{2}\right)}\right) \]

      5. associate-*r* N/A

         \[\leadsto \left(angle \cdot \left(\frac{1}{90} \cdot \mathsf{PI}\left(\right)\right)\right) \cdot \color{blue}{\left({b}^{2} - {a}^{2}\right)} \]

      6. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\left(angle \cdot \left(\frac{1}{90} \cdot \mathsf{PI}\left(\right)\right)\right), \color{blue}{\left({b}^{2} - {a}^{2}\right)}\right) \]

      7. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(angle, \left(\frac{1}{90} \cdot \mathsf{PI}\left(\right)\right)\right), \left(\color{blue}{{b}^{2}} - {a}^{2}\right)\right) \]

      8. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(angle, \mathsf{*.f64}\left(\frac{1}{90}, \mathsf{PI}\left(\right)\right)\right), \left({b}^{\color{blue}{2}} - {a}^{2}\right)\right) \]

      9. PI-lowering-PI.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(angle, \mathsf{*.f64}\left(\frac{1}{90}, \mathsf{PI.f64}\left(\right)\right)\right), \left({b}^{2} - {a}^{2}\right)\right) \]

      10. --lowering--.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(angle, \mathsf{*.f64}\left(\frac{1}{90}, \mathsf{PI.f64}\left(\right)\right)\right), \mathsf{\_.f64}\left(\left({b}^{2}\right), \color{blue}{\left({a}^{2}\right)}\right)\right) \]

      11. unpow2 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(angle, \mathsf{*.f64}\left(\frac{1}{90}, \mathsf{PI.f64}\left(\right)\right)\right), \mathsf{\_.f64}\left(\left(b \cdot b\right), \left({\color{blue}{a}}^{2}\right)\right)\right) \]

      12. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(angle, \mathsf{*.f64}\left(\frac{1}{90}, \mathsf{PI.f64}\left(\right)\right)\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(b, b\right), \left({\color{blue}{a}}^{2}\right)\right)\right) \]

      13. unpow2 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(angle, \mathsf{*.f64}\left(\frac{1}{90}, \mathsf{PI.f64}\left(\right)\right)\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(b, b\right), \left(a \cdot \color{blue}{a}\right)\right)\right) \]

      14. *-lowering-*.f64 36.2%

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(angle, \mathsf{*.f64}\left(\frac{1}{90}, \mathsf{PI.f64}\left(\right)\right)\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(b, b\right), \mathsf{*.f64}\left(a, \color{blue}{a}\right)\right)\right) \]

   7. Simplified 36.2%

      \[\leadsto \color{blue}{\left(angle \cdot \left(0.011111111111111112 \cdot \pi\right)\right) \cdot \left(b \cdot b - a \cdot a\right)} \]

   8. Taylor expanded in b around inf

      \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(angle, \mathsf{*.f64}\left(\frac{1}{90}, \mathsf{PI.f64}\left(\right)\right)\right), \color{blue}{\left({b}^{2} \cdot \left(1 + -1 \cdot \frac{{a}^{2}}{{b}^{2}}\right)\right)}\right) \]
   9. Step-by-step derivation

      1. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(angle, \mathsf{*.f64}\left(\frac{1}{90}, \mathsf{PI.f64}\left(\right)\right)\right), \mathsf{*.f64}\left(\left({b}^{2}\right), \color{blue}{\left(1 + -1 \cdot \frac{{a}^{2}}{{b}^{2}}\right)}\right)\right) \]

      2. unpow2 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(angle, \mathsf{*.f64}\left(\frac{1}{90}, \mathsf{PI.f64}\left(\right)\right)\right), \mathsf{*.f64}\left(\left(b \cdot b\right), \left(\color{blue}{1} + -1 \cdot \frac{{a}^{2}}{{b}^{2}}\right)\right)\right) \]

      3. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(angle, \mathsf{*.f64}\left(\frac{1}{90}, \mathsf{PI.f64}\left(\right)\right)\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(b, b\right), \left(\color{blue}{1} + -1 \cdot \frac{{a}^{2}}{{b}^{2}}\right)\right)\right) \]

      4. mul-1-neg N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(angle, \mathsf{*.f64}\left(\frac{1}{90}, \mathsf{PI.f64}\left(\right)\right)\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(b, b\right), \left(1 + \left(\mathsf{neg}\left(\frac{{a}^{2}}{{b}^{2}}\right)\right)\right)\right)\right) \]

      5. unsub-neg N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(angle, \mathsf{*.f64}\left(\frac{1}{90}, \mathsf{PI.f64}\left(\right)\right)\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(b, b\right), \left(1 - \color{blue}{\frac{{a}^{2}}{{b}^{2}}}\right)\right)\right) \]

      6. --lowering--.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(angle, \mathsf{*.f64}\left(\frac{1}{90}, \mathsf{PI.f64}\left(\right)\right)\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(b, b\right), \mathsf{\_.f64}\left(1, \color{blue}{\left(\frac{{a}^{2}}{{b}^{2}}\right)}\right)\right)\right) \]

      7. unpow2 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(angle, \mathsf{*.f64}\left(\frac{1}{90}, \mathsf{PI.f64}\left(\right)\right)\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(b, b\right), \mathsf{\_.f64}\left(1, \left(\frac{{a}^{2}}{b \cdot \color{blue}{b}}\right)\right)\right)\right) \]

      8. associate-/r* N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(angle, \mathsf{*.f64}\left(\frac{1}{90}, \mathsf{PI.f64}\left(\right)\right)\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(b, b\right), \mathsf{\_.f64}\left(1, \left(\frac{\frac{{a}^{2}}{b}}{\color{blue}{b}}\right)\right)\right)\right) \]

      9. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(angle, \mathsf{*.f64}\left(\frac{1}{90}, \mathsf{PI.f64}\left(\right)\right)\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(b, b\right), \mathsf{\_.f64}\left(1, \mathsf{/.f64}\left(\left(\frac{{a}^{2}}{b}\right), \color{blue}{b}\right)\right)\right)\right) \]

      10. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(angle, \mathsf{*.f64}\left(\frac{1}{90}, \mathsf{PI.f64}\left(\right)\right)\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(b, b\right), \mathsf{\_.f64}\left(1, \mathsf{/.f64}\left(\mathsf{/.f64}\left(\left({a}^{2}\right), b\right), b\right)\right)\right)\right) \]

      11. unpow2 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(angle, \mathsf{*.f64}\left(\frac{1}{90}, \mathsf{PI.f64}\left(\right)\right)\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(b, b\right), \mathsf{\_.f64}\left(1, \mathsf{/.f64}\left(\mathsf{/.f64}\left(\left(a \cdot a\right), b\right), b\right)\right)\right)\right) \]

      12. *-lowering-*.f64 32.4%

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(angle, \mathsf{*.f64}\left(\frac{1}{90}, \mathsf{PI.f64}\left(\right)\right)\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(b, b\right), \mathsf{\_.f64}\left(1, \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(a, a\right), b\right), b\right)\right)\right)\right) \]

   10. Simplified 32.4%

       \[\leadsto \left(angle \cdot \left(0.011111111111111112 \cdot \pi\right)\right) \cdot \color{blue}{\left(\left(b \cdot b\right) \cdot \left(1 - \frac{\frac{a \cdot a}{b}}{b}\right)\right)} \]
3. Recombined 3 regimes into one program.

4. Final simplification 61.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;angle \leq 2.05 \cdot 10^{+49}:\\ \;\;\;\;\left(b + a\right) \cdot \left(\left(b - a\right) \cdot \left(\pi \cdot \left(angle \cdot 0.011111111111111112\right)\right)\right)\\ \mathbf{elif}\;angle \leq 8.5 \cdot 10^{+188}:\\ \;\;\;\;\left(2 \cdot \frac{b - a}{\frac{1}{b + a}}\right) \cdot \left(angle \cdot \left(\pi \cdot 0.005555555555555556 + \left(angle \cdot angle\right) \cdot \left(\pi \cdot \left(\left(\pi \cdot \pi\right) \cdot -2.8577960676726107 \cdot 10^{-8}\right)\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(angle \cdot \left(\pi \cdot 0.011111111111111112\right)\right) \cdot \left(\left(b \cdot b\right) \cdot \left(1 - \frac{\frac{a \cdot a}{b}}{b}\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 15: 63.2% accurate, 11.3× speedup

Math

\[\begin{array}{l} angle\_m = \left|angle\right| \\ angle\_s = \mathsf{copysign}\left(1, angle\right) \\ \begin{array}{l} t_0 := b \cdot b - a \cdot a\\ angle\_s \cdot \begin{array}{l} \mathbf{if}\;angle\_m \leq 2.05 \cdot 10^{+49}:\\ \;\;\;\;\left(b + a\right) \cdot \left(\left(b - a\right) \cdot \left(\pi \cdot \left(angle\_m \cdot 0.011111111111111112\right)\right)\right)\\ \mathbf{elif}\;angle\_m \leq 4 \cdot 10^{+151}:\\ \;\;\;\;\frac{angle\_m \cdot \left(\pi \cdot 0.011111111111111112 + \left(\pi \cdot \left(\pi \cdot \pi\right)\right) \cdot \left(\left(angle\_m \cdot angle\_m\right) \cdot -2.2862368541380886 \cdot 10^{-7}\right)\right)}{\frac{1}{t\_0}}\\ \mathbf{else}:\\ \;\;\;\;\left(angle\_m \cdot \left(\pi \cdot 0.011111111111111112\right)\right) \cdot \left(\frac{1}{b - a} \cdot \left(\left(b - a\right) \cdot t\_0\right)\right)\\ \end{array} \end{array} \end{array} \]

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 (- (* b b) (* a a))))
   (*
    angle_s
    (if (<= angle_m 2.05e+49)
      (* (+ b a) (* (- b a) (* PI (* angle_m 0.011111111111111112))))
      (if (<= angle_m 4e+151)
        (/
         (*
          angle_m
          (+
           (* PI 0.011111111111111112)
           (*
            (* PI (* PI PI))
            (* (* angle_m angle_m) -2.2862368541380886e-7))))
         (/ 1.0 t_0))
        (*
         (* angle_m (* PI 0.011111111111111112))
         (* (/ 1.0 (- 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 = (b * b) - (a * a);
	double tmp;
	if (angle_m <= 2.05e+49) {
		tmp = (b + a) * ((b - a) * (((double) M_PI) * (angle_m * 0.011111111111111112)));
	} else if (angle_m <= 4e+151) {
		tmp = (angle_m * ((((double) M_PI) * 0.011111111111111112) + ((((double) M_PI) * (((double) M_PI) * ((double) M_PI))) * ((angle_m * angle_m) * -2.2862368541380886e-7)))) / (1.0 / t_0);
	} else {
		tmp = (angle_m * (((double) M_PI) * 0.011111111111111112)) * ((1.0 / (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 = (b * b) - (a * a);
	double tmp;
	if (angle_m <= 2.05e+49) {
		tmp = (b + a) * ((b - a) * (Math.PI * (angle_m * 0.011111111111111112)));
	} else if (angle_m <= 4e+151) {
		tmp = (angle_m * ((Math.PI * 0.011111111111111112) + ((Math.PI * (Math.PI * Math.PI)) * ((angle_m * angle_m) * -2.2862368541380886e-7)))) / (1.0 / t_0);
	} else {
		tmp = (angle_m * (Math.PI * 0.011111111111111112)) * ((1.0 / (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 = (b * b) - (a * a)
	tmp = 0
	if angle_m <= 2.05e+49:
		tmp = (b + a) * ((b - a) * (math.pi * (angle_m * 0.011111111111111112)))
	elif angle_m <= 4e+151:
		tmp = (angle_m * ((math.pi * 0.011111111111111112) + ((math.pi * (math.pi * math.pi)) * ((angle_m * angle_m) * -2.2862368541380886e-7)))) / (1.0 / t_0)
	else:
		tmp = (angle_m * (math.pi * 0.011111111111111112)) * ((1.0 / (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(b * b) - Float64(a * a))
	tmp = 0.0
	if (angle_m <= 2.05e+49)
		tmp = Float64(Float64(b + a) * Float64(Float64(b - a) * Float64(pi * Float64(angle_m * 0.011111111111111112))));
	elseif (angle_m <= 4e+151)
		tmp = Float64(Float64(angle_m * Float64(Float64(pi * 0.011111111111111112) + Float64(Float64(pi * Float64(pi * pi)) * Float64(Float64(angle_m * angle_m) * -2.2862368541380886e-7)))) / Float64(1.0 / t_0));
	else
		tmp = Float64(Float64(angle_m * Float64(pi * 0.011111111111111112)) * Float64(Float64(1.0 / 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 = (b * b) - (a * a);
	tmp = 0.0;
	if (angle_m <= 2.05e+49)
		tmp = (b + a) * ((b - a) * (pi * (angle_m * 0.011111111111111112)));
	elseif (angle_m <= 4e+151)
		tmp = (angle_m * ((pi * 0.011111111111111112) + ((pi * (pi * pi)) * ((angle_m * angle_m) * -2.2862368541380886e-7)))) / (1.0 / t_0);
	else
		tmp = (angle_m * (pi * 0.011111111111111112)) * ((1.0 / (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[(b * b), $MachinePrecision] - N[(a * a), $MachinePrecision]), $MachinePrecision]}, N[(angle$95$s * If[LessEqual[angle$95$m, 2.05e+49], N[(N[(b + a), $MachinePrecision] * N[(N[(b - a), $MachinePrecision] * N[(Pi * N[(angle$95$m * 0.011111111111111112), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[angle$95$m, 4e+151], N[(N[(angle$95$m * N[(N[(Pi * 0.011111111111111112), $MachinePrecision] + N[(N[(Pi * N[(Pi * Pi), $MachinePrecision]), $MachinePrecision] * N[(N[(angle$95$m * angle$95$m), $MachinePrecision] * -2.2862368541380886e-7), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(1.0 / t$95$0), $MachinePrecision]), $MachinePrecision], N[(N[(angle$95$m * N[(Pi * 0.011111111111111112), $MachinePrecision]), $MachinePrecision] * N[(N[(1.0 / N[(b - a), $MachinePrecision]), $MachinePrecision] * N[(N[(b - a), $MachinePrecision] * t$95$0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]), $MachinePrecision]]

Derivation

1. Split input into 3 regimes

2. if angle < 2.05e49

   1. Initial program 61.8%

      \[\left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin \left(\pi \cdot \frac{angle}{180}\right)\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right) \]
   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. associate-*l* N/A

         \[\leadsto 2 \cdot \color{blue}{\left(\left({b}^{2} - {a}^{2}\right) \cdot \left(\sin \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right) \cdot \cos \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right)\right)\right)} \]

      3. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(2, \color{blue}{\left(\left({b}^{2} - {a}^{2}\right) \cdot \left(\sin \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right) \cdot \cos \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right)\right)\right)}\right) \]

      4. associate-*r* N/A

         \[\leadsto \mathsf{*.f64}\left(2, \left(\left(\left({b}^{2} - {a}^{2}\right) \cdot \sin \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right)\right) \cdot \color{blue}{\cos \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right)}\right)\right) \]

      5. *-commutative N/A

         \[\leadsto \mathsf{*.f64}\left(2, \left(\left(\sin \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right) \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \cos \color{blue}{\left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right)}\right)\right) \]

      6. associate-*l* N/A

         \[\leadsto \mathsf{*.f64}\left(2, \left(\sin \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right) \cdot \color{blue}{\left(\left({b}^{2} - {a}^{2}\right) \cdot \cos \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right)\right)}\right)\right) \]

      7. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(2, \mathsf{*.f64}\left(\sin \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right), \color{blue}{\left(\left({b}^{2} - {a}^{2}\right) \cdot \cos \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right)\right)}\right)\right) \]

      8. sin-lowering-sin.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(2, \mathsf{*.f64}\left(\mathsf{sin.f64}\left(\left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right)\right), \left(\color{blue}{\left({b}^{2} - {a}^{2}\right)} \cdot \cos \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right)\right)\right)\right) \]

      9. associate-*r/ N/A

         \[\leadsto \mathsf{*.f64}\left(2, \mathsf{*.f64}\left(\mathsf{sin.f64}\left(\left(\frac{\mathsf{PI}\left(\right) \cdot angle}{180}\right)\right), \left(\left(\color{blue}{{b}^{2}} - {a}^{2}\right) \cdot \cos \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right)\right)\right)\right) \]

      10. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(2, \mathsf{*.f64}\left(\mathsf{sin.f64}\left(\mathsf{/.f64}\left(\left(\mathsf{PI}\left(\right) \cdot angle\right), 180\right)\right), \left(\left(\color{blue}{{b}^{2}} - {a}^{2}\right) \cdot \cos \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right)\right)\right)\right) \]

      11. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(2, \mathsf{*.f64}\left(\mathsf{sin.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{PI}\left(\right), angle\right), 180\right)\right), \left(\left({\color{blue}{b}}^{2} - {a}^{2}\right) \cdot \cos \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right)\right)\right)\right) \]

      12. PI-lowering-PI.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(2, \mathsf{*.f64}\left(\mathsf{sin.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{PI.f64}\left(\right), angle\right), 180\right)\right), \left(\left({b}^{2} - {a}^{2}\right) \cdot \cos \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right)\right)\right)\right) \]

   3. Simplified 62.6%

      \[\leadsto \color{blue}{2 \cdot \left(\sin \left(\frac{\pi \cdot angle}{180}\right) \cdot \left(\left(b \cdot b - a \cdot a\right) \cdot \cos \left(\frac{\pi \cdot angle}{180}\right)\right)\right)} \]
   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. *-commutative N/A

         \[\leadsto \left(angle \cdot \left(\mathsf{PI}\left(\right) \cdot \left({b}^{2} - {a}^{2}\right)\right)\right) \cdot \color{blue}{\frac{1}{90}} \]

      2. associate-*r* N/A

         \[\leadsto angle \cdot \color{blue}{\left(\left(\mathsf{PI}\left(\right) \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \frac{1}{90}\right)} \]

      3. *-commutative N/A

         \[\leadsto angle \cdot \left(\frac{1}{90} \cdot \color{blue}{\left(\mathsf{PI}\left(\right) \cdot \left({b}^{2} - {a}^{2}\right)\right)}\right) \]

      4. associate-*r* N/A

         \[\leadsto angle \cdot \left(\left(\frac{1}{90} \cdot \mathsf{PI}\left(\right)\right) \cdot \color{blue}{\left({b}^{2} - {a}^{2}\right)}\right) \]

      5. associate-*r* N/A

         \[\leadsto \left(angle \cdot \left(\frac{1}{90} \cdot \mathsf{PI}\left(\right)\right)\right) \cdot \color{blue}{\left({b}^{2} - {a}^{2}\right)} \]

      6. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\left(angle \cdot \left(\frac{1}{90} \cdot \mathsf{PI}\left(\right)\right)\right), \color{blue}{\left({b}^{2} - {a}^{2}\right)}\right) \]

      7. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(angle, \left(\frac{1}{90} \cdot \mathsf{PI}\left(\right)\right)\right), \left(\color{blue}{{b}^{2}} - {a}^{2}\right)\right) \]

      8. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(angle, \mathsf{*.f64}\left(\frac{1}{90}, \mathsf{PI}\left(\right)\right)\right), \left({b}^{\color{blue}{2}} - {a}^{2}\right)\right) \]

      9. PI-lowering-PI.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(angle, \mathsf{*.f64}\left(\frac{1}{90}, \mathsf{PI.f64}\left(\right)\right)\right), \left({b}^{2} - {a}^{2}\right)\right) \]

      10. --lowering--.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(angle, \mathsf{*.f64}\left(\frac{1}{90}, \mathsf{PI.f64}\left(\right)\right)\right), \mathsf{\_.f64}\left(\left({b}^{2}\right), \color{blue}{\left({a}^{2}\right)}\right)\right) \]

      11. unpow2 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(angle, \mathsf{*.f64}\left(\frac{1}{90}, \mathsf{PI.f64}\left(\right)\right)\right), \mathsf{\_.f64}\left(\left(b \cdot b\right), \left({\color{blue}{a}}^{2}\right)\right)\right) \]

      12. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(angle, \mathsf{*.f64}\left(\frac{1}{90}, \mathsf{PI.f64}\left(\right)\right)\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(b, b\right), \left({\color{blue}{a}}^{2}\right)\right)\right) \]

      13. unpow2 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(angle, \mathsf{*.f64}\left(\frac{1}{90}, \mathsf{PI.f64}\left(\right)\right)\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(b, b\right), \left(a \cdot \color{blue}{a}\right)\right)\right) \]

      14. *-lowering-*.f64 59.8%

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(angle, \mathsf{*.f64}\left(\frac{1}{90}, \mathsf{PI.f64}\left(\right)\right)\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(b, b\right), \mathsf{*.f64}\left(a, \color{blue}{a}\right)\right)\right) \]

   7. Simplified 59.8%

      \[\leadsto \color{blue}{\left(angle \cdot \left(0.011111111111111112 \cdot \pi\right)\right) \cdot \left(b \cdot b - a \cdot a\right)} \]
   8. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \left(b \cdot b - a \cdot a\right) \cdot \color{blue}{\left(angle \cdot \left(\frac{1}{90} \cdot \mathsf{PI}\left(\right)\right)\right)} \]

      2. difference-of-squares N/A

         \[\leadsto \left(\left(b + a\right) \cdot \left(b - a\right)\right) \cdot \left(\color{blue}{angle} \cdot \left(\frac{1}{90} \cdot \mathsf{PI}\left(\right)\right)\right) \]

      3. associate-*l* N/A

         \[\leadsto \left(b + a\right) \cdot \color{blue}{\left(\left(b - a\right) \cdot \left(angle \cdot \left(\frac{1}{90} \cdot \mathsf{PI}\left(\right)\right)\right)\right)} \]

      4. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\left(b + a\right), \color{blue}{\left(\left(b - a\right) \cdot \left(angle \cdot \left(\frac{1}{90} \cdot \mathsf{PI}\left(\right)\right)\right)\right)}\right) \]

      5. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(b, a\right), \left(\color{blue}{\left(b - a\right)} \cdot \left(angle \cdot \left(\frac{1}{90} \cdot \mathsf{PI}\left(\right)\right)\right)\right)\right) \]

      6. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(b, a\right), \mathsf{*.f64}\left(\left(b - a\right), \color{blue}{\left(angle \cdot \left(\frac{1}{90} \cdot \mathsf{PI}\left(\right)\right)\right)}\right)\right) \]

      7. --lowering--.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(b, a\right), \mathsf{*.f64}\left(\mathsf{\_.f64}\left(b, a\right), \left(\color{blue}{angle} \cdot \left(\frac{1}{90} \cdot \mathsf{PI}\left(\right)\right)\right)\right)\right) \]

      8. associate-*r* N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(b, a\right), \mathsf{*.f64}\left(\mathsf{\_.f64}\left(b, a\right), \left(\left(angle \cdot \frac{1}{90}\right) \cdot \color{blue}{\mathsf{PI}\left(\right)}\right)\right)\right) \]

      9. *-commutative N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(b, a\right), \mathsf{*.f64}\left(\mathsf{\_.f64}\left(b, a\right), \left(\mathsf{PI}\left(\right) \cdot \color{blue}{\left(angle \cdot \frac{1}{90}\right)}\right)\right)\right) \]

      10. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(b, a\right), \mathsf{*.f64}\left(\mathsf{\_.f64}\left(b, a\right), \mathsf{*.f64}\left(\mathsf{PI}\left(\right), \color{blue}{\left(angle \cdot \frac{1}{90}\right)}\right)\right)\right) \]

      11. PI-lowering-PI.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(b, a\right), \mathsf{*.f64}\left(\mathsf{\_.f64}\left(b, a\right), \mathsf{*.f64}\left(\mathsf{PI.f64}\left(\right), \left(\color{blue}{angle} \cdot \frac{1}{90}\right)\right)\right)\right) \]

      12. *-lowering-*.f64 70.8%

          \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(b, a\right), \mathsf{*.f64}\left(\mathsf{\_.f64}\left(b, a\right), \mathsf{*.f64}\left(\mathsf{PI.f64}\left(\right), \mathsf{*.f64}\left(angle, \color{blue}{\frac{1}{90}}\right)\right)\right)\right) \]

   9. Applied egg-rr 70.8%

      \[\leadsto \color{blue}{\left(b + a\right) \cdot \left(\left(b - a\right) \cdot \left(\pi \cdot \left(angle \cdot 0.011111111111111112\right)\right)\right)} \]

   if 2.05e49 < angle < 4.00000000000000007e151

   1. Initial program 27.2%

      \[\left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin \left(\pi \cdot \frac{angle}{180}\right)\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right) \]
   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. associate-*l* N/A

         \[\leadsto 2 \cdot \color{blue}{\left(\left({b}^{2} - {a}^{2}\right) \cdot \left(\sin \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right) \cdot \cos \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right)\right)\right)} \]

      3. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(2, \color{blue}{\left(\left({b}^{2} - {a}^{2}\right) \cdot \left(\sin \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right) \cdot \cos \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right)\right)\right)}\right) \]

      4. associate-*r* N/A

         \[\leadsto \mathsf{*.f64}\left(2, \left(\left(\left({b}^{2} - {a}^{2}\right) \cdot \sin \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right)\right) \cdot \color{blue}{\cos \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right)}\right)\right) \]

      5. *-commutative N/A

         \[\leadsto \mathsf{*.f64}\left(2, \left(\left(\sin \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right) \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \cos \color{blue}{\left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right)}\right)\right) \]

      6. associate-*l* N/A

         \[\leadsto \mathsf{*.f64}\left(2, \left(\sin \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right) \cdot \color{blue}{\left(\left({b}^{2} - {a}^{2}\right) \cdot \cos \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right)\right)}\right)\right) \]

      7. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(2, \mathsf{*.f64}\left(\sin \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right), \color{blue}{\left(\left({b}^{2} - {a}^{2}\right) \cdot \cos \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right)\right)}\right)\right) \]

      8. sin-lowering-sin.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(2, \mathsf{*.f64}\left(\mathsf{sin.f64}\left(\left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right)\right), \left(\color{blue}{\left({b}^{2} - {a}^{2}\right)} \cdot \cos \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right)\right)\right)\right) \]

      9. associate-*r/ N/A

         \[\leadsto \mathsf{*.f64}\left(2, \mathsf{*.f64}\left(\mathsf{sin.f64}\left(\left(\frac{\mathsf{PI}\left(\right) \cdot angle}{180}\right)\right), \left(\left(\color{blue}{{b}^{2}} - {a}^{2}\right) \cdot \cos \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right)\right)\right)\right) \]

      10. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(2, \mathsf{*.f64}\left(\mathsf{sin.f64}\left(\mathsf{/.f64}\left(\left(\mathsf{PI}\left(\right) \cdot angle\right), 180\right)\right), \left(\left(\color{blue}{{b}^{2}} - {a}^{2}\right) \cdot \cos \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right)\right)\right)\right) \]

      11. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(2, \mathsf{*.f64}\left(\mathsf{sin.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{PI}\left(\right), angle\right), 180\right)\right), \left(\left({\color{blue}{b}}^{2} - {a}^{2}\right) \cdot \cos \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right)\right)\right)\right) \]

      12. PI-lowering-PI.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(2, \mathsf{*.f64}\left(\mathsf{sin.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{PI.f64}\left(\right), angle\right), 180\right)\right), \left(\left({b}^{2} - {a}^{2}\right) \cdot \cos \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right)\right)\right)\right) \]

   3. Simplified 26.9%

      \[\leadsto \color{blue}{2 \cdot \left(\sin \left(\frac{\pi \cdot angle}{180}\right) \cdot \left(\left(b \cdot b - a \cdot a\right) \cdot \cos \left(\frac{\pi \cdot angle}{180}\right)\right)\right)} \]
   4. Add Preprocessing
   5. Step-by-step derivation

      1. associate-*r* N/A

         \[\leadsto \left(2 \cdot \sin \left(\frac{\mathsf{PI}\left(\right) \cdot angle}{180}\right)\right) \cdot \color{blue}{\left(\left(b \cdot b - a \cdot a\right) \cdot \cos \left(\frac{\mathsf{PI}\left(\right) \cdot angle}{180}\right)\right)} \]

      2. *-commutative N/A

         \[\leadsto \left(2 \cdot \sin \left(\frac{\mathsf{PI}\left(\right) \cdot angle}{180}\right)\right) \cdot \left(\cos \left(\frac{\mathsf{PI}\left(\right) \cdot angle}{180}\right) \cdot \color{blue}{\left(b \cdot b - a \cdot a\right)}\right) \]

      3. associate-*r/ N/A

         \[\leadsto \left(2 \cdot \sin \left(\frac{\mathsf{PI}\left(\right) \cdot angle}{180}\right)\right) \cdot \left(\cos \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right) \cdot \left(\color{blue}{b} \cdot b - a \cdot a\right)\right) \]

      4. associate-*r* N/A

         \[\leadsto \left(\left(2 \cdot \sin \left(\frac{\mathsf{PI}\left(\right) \cdot angle}{180}\right)\right) \cdot \cos \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right)\right) \cdot \color{blue}{\left(b \cdot b - a \cdot a\right)} \]

      5. flip-- N/A

         \[\leadsto \left(\left(2 \cdot \sin \left(\frac{\mathsf{PI}\left(\right) \cdot angle}{180}\right)\right) \cdot \cos \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right)\right) \cdot \frac{\left(b \cdot b\right) \cdot \left(b \cdot b\right) - \left(a \cdot a\right) \cdot \left(a \cdot a\right)}{\color{blue}{b \cdot b + a \cdot a}} \]

      6. clear-num N/A

         \[\leadsto \left(\left(2 \cdot \sin \left(\frac{\mathsf{PI}\left(\right) \cdot angle}{180}\right)\right) \cdot \cos \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right)\right) \cdot \frac{1}{\color{blue}{\frac{b \cdot b + a \cdot a}{\left(b \cdot b\right) \cdot \left(b \cdot b\right) - \left(a \cdot a\right) \cdot \left(a \cdot a\right)}}} \]

      7. un-div-inv N/A

         \[\leadsto \frac{\left(2 \cdot \sin \left(\frac{\mathsf{PI}\left(\right) \cdot angle}{180}\right)\right) \cdot \cos \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right)}{\color{blue}{\frac{b \cdot b + a \cdot a}{\left(b \cdot b\right) \cdot \left(b \cdot b\right) - \left(a \cdot a\right) \cdot \left(a \cdot a\right)}}} \]

      8. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\left(\left(2 \cdot \sin \left(\frac{\mathsf{PI}\left(\right) \cdot angle}{180}\right)\right) \cdot \cos \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right)\right), \color{blue}{\left(\frac{b \cdot b + a \cdot a}{\left(b \cdot b\right) \cdot \left(b \cdot b\right) - \left(a \cdot a\right) \cdot \left(a \cdot a\right)}\right)}\right) \]

   6. Applied egg-rr 31.3%

      \[\leadsto \color{blue}{\frac{\sin \left(\left(\pi \cdot angle\right) \cdot 0.011111111111111112\right)}{\frac{1}{b \cdot b - a \cdot a}}} \]

   7. Taylor expanded in angle around 0

      \[\leadsto \mathsf{/.f64}\left(\color{blue}{\left(angle \cdot \left(\frac{-1}{4374000} \cdot \left({angle}^{2} \cdot {\mathsf{PI}\left(\right)}^{3}\right) + \frac{1}{90} \cdot \mathsf{PI}\left(\right)\right)\right)}, \mathsf{/.f64}\left(1, \mathsf{\_.f64}\left(\mathsf{*.f64}\left(b, b\right), \mathsf{*.f64}\left(a, a\right)\right)\right)\right) \]
   8. Step-by-step derivation

      1. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(angle, \left(\frac{-1}{4374000} \cdot \left({angle}^{2} \cdot {\mathsf{PI}\left(\right)}^{3}\right) + \frac{1}{90} \cdot \mathsf{PI}\left(\right)\right)\right), \mathsf{/.f64}\left(\color{blue}{1}, \mathsf{\_.f64}\left(\mathsf{*.f64}\left(b, b\right), \mathsf{*.f64}\left(a, a\right)\right)\right)\right) \]

      2. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(angle, \mathsf{+.f64}\left(\left(\frac{-1}{4374000} \cdot \left({angle}^{2} \cdot {\mathsf{PI}\left(\right)}^{3}\right)\right), \left(\frac{1}{90} \cdot \mathsf{PI}\left(\right)\right)\right)\right), \mathsf{/.f64}\left(1, \mathsf{\_.f64}\left(\mathsf{*.f64}\left(b, b\right), \mathsf{*.f64}\left(a, a\right)\right)\right)\right) \]

      3. associate-*r* N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(angle, \mathsf{+.f64}\left(\left(\left(\frac{-1}{4374000} \cdot {angle}^{2}\right) \cdot {\mathsf{PI}\left(\right)}^{3}\right), \left(\frac{1}{90} \cdot \mathsf{PI}\left(\right)\right)\right)\right), \mathsf{/.f64}\left(1, \mathsf{\_.f64}\left(\mathsf{*.f64}\left(b, b\right), \mathsf{*.f64}\left(a, a\right)\right)\right)\right) \]

      4. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(angle, \mathsf{+.f64}\left(\mathsf{*.f64}\left(\left(\frac{-1}{4374000} \cdot {angle}^{2}\right), \left({\mathsf{PI}\left(\right)}^{3}\right)\right), \left(\frac{1}{90} \cdot \mathsf{PI}\left(\right)\right)\right)\right), \mathsf{/.f64}\left(1, \mathsf{\_.f64}\left(\mathsf{*.f64}\left(b, b\right), \mathsf{*.f64}\left(a, a\right)\right)\right)\right) \]

      5. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(angle, \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{-1}{4374000}, \left({angle}^{2}\right)\right), \left({\mathsf{PI}\left(\right)}^{3}\right)\right), \left(\frac{1}{90} \cdot \mathsf{PI}\left(\right)\right)\right)\right), \mathsf{/.f64}\left(1, \mathsf{\_.f64}\left(\mathsf{*.f64}\left(b, b\right), \mathsf{*.f64}\left(a, a\right)\right)\right)\right) \]

      6. unpow2 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(angle, \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{-1}{4374000}, \left(angle \cdot angle\right)\right), \left({\mathsf{PI}\left(\right)}^{3}\right)\right), \left(\frac{1}{90} \cdot \mathsf{PI}\left(\right)\right)\right)\right), \mathsf{/.f64}\left(1, \mathsf{\_.f64}\left(\mathsf{*.f64}\left(b, b\right), \mathsf{*.f64}\left(a, a\right)\right)\right)\right) \]

      7. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(angle, \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{-1}{4374000}, \mathsf{*.f64}\left(angle, angle\right)\right), \left({\mathsf{PI}\left(\right)}^{3}\right)\right), \left(\frac{1}{90} \cdot \mathsf{PI}\left(\right)\right)\right)\right), \mathsf{/.f64}\left(1, \mathsf{\_.f64}\left(\mathsf{*.f64}\left(b, b\right), \mathsf{*.f64}\left(a, a\right)\right)\right)\right) \]

      8. cube-mult N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(angle, \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{-1}{4374000}, \mathsf{*.f64}\left(angle, angle\right)\right), \left(\mathsf{PI}\left(\right) \cdot \left(\mathsf{PI}\left(\right) \cdot \mathsf{PI}\left(\right)\right)\right)\right), \left(\frac{1}{90} \cdot \mathsf{PI}\left(\right)\right)\right)\right), \mathsf{/.f64}\left(1, \mathsf{\_.f64}\left(\mathsf{*.f64}\left(b, b\right), \mathsf{*.f64}\left(a, a\right)\right)\right)\right) \]

      9. unpow2 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(angle, \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{-1}{4374000}, \mathsf{*.f64}\left(angle, angle\right)\right), \left(\mathsf{PI}\left(\right) \cdot {\mathsf{PI}\left(\right)}^{2}\right)\right), \left(\frac{1}{90} \cdot \mathsf{PI}\left(\right)\right)\right)\right), \mathsf{/.f64}\left(1, \mathsf{\_.f64}\left(\mathsf{*.f64}\left(b, b\right), \mathsf{*.f64}\left(a, a\right)\right)\right)\right) \]

      10. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(angle, \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{-1}{4374000}, \mathsf{*.f64}\left(angle, angle\right)\right), \mathsf{*.f64}\left(\mathsf{PI}\left(\right), \left({\mathsf{PI}\left(\right)}^{2}\right)\right)\right), \left(\frac{1}{90} \cdot \mathsf{PI}\left(\right)\right)\right)\right), \mathsf{/.f64}\left(1, \mathsf{\_.f64}\left(\mathsf{*.f64}\left(b, b\right), \mathsf{*.f64}\left(a, a\right)\right)\right)\right) \]

      11. PI-lowering-PI.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(angle, \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{-1}{4374000}, \mathsf{*.f64}\left(angle, angle\right)\right), \mathsf{*.f64}\left(\mathsf{PI.f64}\left(\right), \left({\mathsf{PI}\left(\right)}^{2}\right)\right)\right), \left(\frac{1}{90} \cdot \mathsf{PI}\left(\right)\right)\right)\right), \mathsf{/.f64}\left(1, \mathsf{\_.f64}\left(\mathsf{*.f64}\left(b, b\right), \mathsf{*.f64}\left(a, a\right)\right)\right)\right) \]

      12. unpow2 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(angle, \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{-1}{4374000}, \mathsf{*.f64}\left(angle, angle\right)\right), \mathsf{*.f64}\left(\mathsf{PI.f64}\left(\right), \left(\mathsf{PI}\left(\right) \cdot \mathsf{PI}\left(\right)\right)\right)\right), \left(\frac{1}{90} \cdot \mathsf{PI}\left(\right)\right)\right)\right), \mathsf{/.f64}\left(1, \mathsf{\_.f64}\left(\mathsf{*.f64}\left(b, b\right), \mathsf{*.f64}\left(a, a\right)\right)\right)\right) \]

      13. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(angle, \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{-1}{4374000}, \mathsf{*.f64}\left(angle, angle\right)\right), \mathsf{*.f64}\left(\mathsf{PI.f64}\left(\right), \mathsf{*.f64}\left(\mathsf{PI}\left(\right), \mathsf{PI}\left(\right)\right)\right)\right), \left(\frac{1}{90} \cdot \mathsf{PI}\left(\right)\right)\right)\right), \mathsf{/.f64}\left(1, \mathsf{\_.f64}\left(\mathsf{*.f64}\left(b, b\right), \mathsf{*.f64}\left(a, a\right)\right)\right)\right) \]

      14. PI-lowering-PI.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(angle, \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{-1}{4374000}, \mathsf{*.f64}\left(angle, angle\right)\right), \mathsf{*.f64}\left(\mathsf{PI.f64}\left(\right), \mathsf{*.f64}\left(\mathsf{PI.f64}\left(\right), \mathsf{PI}\left(\right)\right)\right)\right), \left(\frac{1}{90} \cdot \mathsf{PI}\left(\right)\right)\right)\right), \mathsf{/.f64}\left(1, \mathsf{\_.f64}\left(\mathsf{*.f64}\left(b, b\right), \mathsf{*.f64}\left(a, a\right)\right)\right)\right) \]

      15. PI-lowering-PI.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(angle, \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{-1}{4374000}, \mathsf{*.f64}\left(angle, angle\right)\right), \mathsf{*.f64}\left(\mathsf{PI.f64}\left(\right), \mathsf{*.f64}\left(\mathsf{PI.f64}\left(\right), \mathsf{PI.f64}\left(\right)\right)\right)\right), \left(\frac{1}{90} \cdot \mathsf{PI}\left(\right)\right)\right)\right), \mathsf{/.f64}\left(1, \mathsf{\_.f64}\left(\mathsf{*.f64}\left(b, b\right), \mathsf{*.f64}\left(a, a\right)\right)\right)\right) \]

      16. *-commutative N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(angle, \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{-1}{4374000}, \mathsf{*.f64}\left(angle, angle\right)\right), \mathsf{*.f64}\left(\mathsf{PI.f64}\left(\right), \mathsf{*.f64}\left(\mathsf{PI.f64}\left(\right), \mathsf{PI.f64}\left(\right)\right)\right)\right), \left(\mathsf{PI}\left(\right) \cdot \frac{1}{90}\right)\right)\right), \mathsf{/.f64}\left(1, \mathsf{\_.f64}\left(\mathsf{*.f64}\left(b, b\right), \mathsf{*.f64}\left(a, a\right)\right)\right)\right) \]

      17. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(angle, \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{-1}{4374000}, \mathsf{*.f64}\left(angle, angle\right)\right), \mathsf{*.f64}\left(\mathsf{PI.f64}\left(\right), \mathsf{*.f64}\left(\mathsf{PI.f64}\left(\right), \mathsf{PI.f64}\left(\right)\right)\right)\right), \mathsf{*.f64}\left(\mathsf{PI}\left(\right), \frac{1}{90}\right)\right)\right), \mathsf{/.f64}\left(1, \mathsf{\_.f64}\left(\mathsf{*.f64}\left(b, b\right), \mathsf{*.f64}\left(a, a\right)\right)\right)\right) \]

      18. PI-lowering-PI.f64 20.2%

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(angle, \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{-1}{4374000}, \mathsf{*.f64}\left(angle, angle\right)\right), \mathsf{*.f64}\left(\mathsf{PI.f64}\left(\right), \mathsf{*.f64}\left(\mathsf{PI.f64}\left(\right), \mathsf{PI.f64}\left(\right)\right)\right)\right), \mathsf{*.f64}\left(\mathsf{PI.f64}\left(\right), \frac{1}{90}\right)\right)\right), \mathsf{/.f64}\left(1, \mathsf{\_.f64}\left(\mathsf{*.f64}\left(b, b\right), \mathsf{*.f64}\left(a, a\right)\right)\right)\right) \]

   9. Simplified 20.2%

      \[\leadsto \frac{\color{blue}{angle \cdot \left(\left(-2.2862368541380886 \cdot 10^{-7} \cdot \left(angle \cdot angle\right)\right) \cdot \left(\pi \cdot \left(\pi \cdot \pi\right)\right) + \pi \cdot 0.011111111111111112\right)}}{\frac{1}{b \cdot b - a \cdot a}} \]

   if 4.00000000000000007e151 < angle

   1. Initial program 32.6%

      \[\left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin \left(\pi \cdot \frac{angle}{180}\right)\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right) \]
   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. associate-*l* N/A

         \[\leadsto 2 \cdot \color{blue}{\left(\left({b}^{2} - {a}^{2}\right) \cdot \left(\sin \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right) \cdot \cos \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right)\right)\right)} \]

      3. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(2, \color{blue}{\left(\left({b}^{2} - {a}^{2}\right) \cdot \left(\sin \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right) \cdot \cos \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right)\right)\right)}\right) \]

      4. associate-*r* N/A

         \[\leadsto \mathsf{*.f64}\left(2, \left(\left(\left({b}^{2} - {a}^{2}\right) \cdot \sin \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right)\right) \cdot \color{blue}{\cos \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right)}\right)\right) \]

      5. *-commutative N/A

         \[\leadsto \mathsf{*.f64}\left(2, \left(\left(\sin \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right) \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \cos \color{blue}{\left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right)}\right)\right) \]

      6. associate-*l* N/A

         \[\leadsto \mathsf{*.f64}\left(2, \left(\sin \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right) \cdot \color{blue}{\left(\left({b}^{2} - {a}^{2}\right) \cdot \cos \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right)\right)}\right)\right) \]

      7. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(2, \mathsf{*.f64}\left(\sin \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right), \color{blue}{\left(\left({b}^{2} - {a}^{2}\right) \cdot \cos \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right)\right)}\right)\right) \]

      8. sin-lowering-sin.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(2, \mathsf{*.f64}\left(\mathsf{sin.f64}\left(\left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right)\right), \left(\color{blue}{\left({b}^{2} - {a}^{2}\right)} \cdot \cos \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right)\right)\right)\right) \]

      9. associate-*r/ N/A

         \[\leadsto \mathsf{*.f64}\left(2, \mathsf{*.f64}\left(\mathsf{sin.f64}\left(\left(\frac{\mathsf{PI}\left(\right) \cdot angle}{180}\right)\right), \left(\left(\color{blue}{{b}^{2}} - {a}^{2}\right) \cdot \cos \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right)\right)\right)\right) \]

      10. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(2, \mathsf{*.f64}\left(\mathsf{sin.f64}\left(\mathsf{/.f64}\left(\left(\mathsf{PI}\left(\right) \cdot angle\right), 180\right)\right), \left(\left(\color{blue}{{b}^{2}} - {a}^{2}\right) \cdot \cos \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right)\right)\right)\right) \]

      11. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(2, \mathsf{*.f64}\left(\mathsf{sin.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{PI}\left(\right), angle\right), 180\right)\right), \left(\left({\color{blue}{b}}^{2} - {a}^{2}\right) \cdot \cos \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right)\right)\right)\right) \]

      12. PI-lowering-PI.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(2, \mathsf{*.f64}\left(\mathsf{sin.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{PI.f64}\left(\right), angle\right), 180\right)\right), \left(\left({b}^{2} - {a}^{2}\right) \cdot \cos \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right)\right)\right)\right) \]

   3. Simplified 36.1%

      \[\leadsto \color{blue}{2 \cdot \left(\sin \left(\frac{\pi \cdot angle}{180}\right) \cdot \left(\left(b \cdot b - a \cdot a\right) \cdot \cos \left(\frac{\pi \cdot angle}{180}\right)\right)\right)} \]
   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. *-commutative N/A

         \[\leadsto \left(angle \cdot \left(\mathsf{PI}\left(\right) \cdot \left({b}^{2} - {a}^{2}\right)\right)\right) \cdot \color{blue}{\frac{1}{90}} \]

      2. associate-*r* N/A

         \[\leadsto angle \cdot \color{blue}{\left(\left(\mathsf{PI}\left(\right) \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \frac{1}{90}\right)} \]

      3. *-commutative N/A

         \[\leadsto angle \cdot \left(\frac{1}{90} \cdot \color{blue}{\left(\mathsf{PI}\left(\right) \cdot \left({b}^{2} - {a}^{2}\right)\right)}\right) \]

      4. associate-*r* N/A

         \[\leadsto angle \cdot \left(\left(\frac{1}{90} \cdot \mathsf{PI}\left(\right)\right) \cdot \color{blue}{\left({b}^{2} - {a}^{2}\right)}\right) \]

      5. associate-*r* N/A

         \[\leadsto \left(angle \cdot \left(\frac{1}{90} \cdot \mathsf{PI}\left(\right)\right)\right) \cdot \color{blue}{\left({b}^{2} - {a}^{2}\right)} \]

      6. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\left(angle \cdot \left(\frac{1}{90} \cdot \mathsf{PI}\left(\right)\right)\right), \color{blue}{\left({b}^{2} - {a}^{2}\right)}\right) \]

      7. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(angle, \left(\frac{1}{90} \cdot \mathsf{PI}\left(\right)\right)\right), \left(\color{blue}{{b}^{2}} - {a}^{2}\right)\right) \]

      8. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(angle, \mathsf{*.f64}\left(\frac{1}{90}, \mathsf{PI}\left(\right)\right)\right), \left({b}^{\color{blue}{2}} - {a}^{2}\right)\right) \]

      9. PI-lowering-PI.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(angle, \mathsf{*.f64}\left(\frac{1}{90}, \mathsf{PI.f64}\left(\right)\right)\right), \left({b}^{2} - {a}^{2}\right)\right) \]

      10. --lowering--.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(angle, \mathsf{*.f64}\left(\frac{1}{90}, \mathsf{PI.f64}\left(\right)\right)\right), \mathsf{\_.f64}\left(\left({b}^{2}\right), \color{blue}{\left({a}^{2}\right)}\right)\right) \]

      11. unpow2 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(angle, \mathsf{*.f64}\left(\frac{1}{90}, \mathsf{PI.f64}\left(\right)\right)\right), \mathsf{\_.f64}\left(\left(b \cdot b\right), \left({\color{blue}{a}}^{2}\right)\right)\right) \]

      12. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(angle, \mathsf{*.f64}\left(\frac{1}{90}, \mathsf{PI.f64}\left(\right)\right)\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(b, b\right), \left({\color{blue}{a}}^{2}\right)\right)\right) \]

      13. unpow2 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(angle, \mathsf{*.f64}\left(\frac{1}{90}, \mathsf{PI.f64}\left(\right)\right)\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(b, b\right), \left(a \cdot \color{blue}{a}\right)\right)\right) \]

      14. *-lowering-*.f64 39.3%

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(angle, \mathsf{*.f64}\left(\frac{1}{90}, \mathsf{PI.f64}\left(\right)\right)\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(b, b\right), \mathsf{*.f64}\left(a, \color{blue}{a}\right)\right)\right) \]

   7. Simplified 39.3%

      \[\leadsto \color{blue}{\left(angle \cdot \left(0.011111111111111112 \cdot \pi\right)\right) \cdot \left(b \cdot b - a \cdot a\right)} \]
   8. Step-by-step derivation

      1. difference-of-squares N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(angle, \mathsf{*.f64}\left(\frac{1}{90}, \mathsf{PI.f64}\left(\right)\right)\right), \left(\left(b + a\right) \cdot \color{blue}{\left(b - a\right)}\right)\right) \]

      2. flip-- N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(angle, \mathsf{*.f64}\left(\frac{1}{90}, \mathsf{PI.f64}\left(\right)\right)\right), \left(\left(b + a\right) \cdot \frac{b \cdot b - a \cdot a}{\color{blue}{b + a}}\right)\right) \]

      3. associate-*r/ N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(angle, \mathsf{*.f64}\left(\frac{1}{90}, \mathsf{PI.f64}\left(\right)\right)\right), \left(\frac{\left(b + a\right) \cdot \left(b \cdot b - a \cdot a\right)}{\color{blue}{b + a}}\right)\right) \]

      4. flip-+ N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(angle, \mathsf{*.f64}\left(\frac{1}{90}, \mathsf{PI.f64}\left(\right)\right)\right), \left(\frac{\left(b + a\right) \cdot \left(b \cdot b - a \cdot a\right)}{\frac{b \cdot b - a \cdot a}{\color{blue}{b - a}}}\right)\right) \]

      5. div-inv N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(angle, \mathsf{*.f64}\left(\frac{1}{90}, \mathsf{PI.f64}\left(\right)\right)\right), \left(\frac{\left(b + a\right) \cdot \left(b \cdot b - a \cdot a\right)}{\left(b \cdot b - a \cdot a\right) \cdot \color{blue}{\frac{1}{b - a}}}\right)\right) \]

      6. times-frac N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(angle, \mathsf{*.f64}\left(\frac{1}{90}, \mathsf{PI.f64}\left(\right)\right)\right), \left(\frac{b + a}{b \cdot b - a \cdot a} \cdot \color{blue}{\frac{b \cdot b - a \cdot a}{\frac{1}{b - a}}}\right)\right) \]

      7. clear-num N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(angle, \mathsf{*.f64}\left(\frac{1}{90}, \mathsf{PI.f64}\left(\right)\right)\right), \left(\frac{1}{\frac{b \cdot b - a \cdot a}{b + a}} \cdot \frac{\color{blue}{b \cdot b - a \cdot a}}{\frac{1}{b - a}}\right)\right) \]

      8. flip-- N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(angle, \mathsf{*.f64}\left(\frac{1}{90}, \mathsf{PI.f64}\left(\right)\right)\right), \left(\frac{1}{b - a} \cdot \frac{b \cdot b - \color{blue}{a \cdot a}}{\frac{1}{b - a}}\right)\right) \]

      9. un-div-inv N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(angle, \mathsf{*.f64}\left(\frac{1}{90}, \mathsf{PI.f64}\left(\right)\right)\right), \left(\frac{1}{b - a} \cdot \left(\left(b \cdot b - a \cdot a\right) \cdot \color{blue}{\frac{1}{\frac{1}{b - a}}}\right)\right)\right) \]

      10. flip3-- N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(angle, \mathsf{*.f64}\left(\frac{1}{90}, \mathsf{PI.f64}\left(\right)\right)\right), \left(\frac{1}{b - a} \cdot \left(\left(b \cdot b - a \cdot a\right) \cdot \frac{1}{\frac{1}{\frac{{b}^{3} - {a}^{3}}{\color{blue}{b \cdot b + \left(a \cdot a + b \cdot a\right)}}}}\right)\right)\right) \]

      11. clear-num N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(angle, \mathsf{*.f64}\left(\frac{1}{90}, \mathsf{PI.f64}\left(\right)\right)\right), \left(\frac{1}{b - a} \cdot \left(\left(b \cdot b - a \cdot a\right) \cdot \frac{1}{\frac{b \cdot b + \left(a \cdot a + b \cdot a\right)}{\color{blue}{{b}^{3} - {a}^{3}}}}\right)\right)\right) \]

      12. clear-num N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(angle, \mathsf{*.f64}\left(\frac{1}{90}, \mathsf{PI.f64}\left(\right)\right)\right), \left(\frac{1}{b - a} \cdot \left(\left(b \cdot b - a \cdot a\right) \cdot \frac{{b}^{3} - {a}^{3}}{\color{blue}{b \cdot b + \left(a \cdot a + b \cdot a\right)}}\right)\right)\right) \]

      13. flip3-- N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(angle, \mathsf{*.f64}\left(\frac{1}{90}, \mathsf{PI.f64}\left(\right)\right)\right), \left(\frac{1}{b - a} \cdot \left(\left(b \cdot b - a \cdot a\right) \cdot \left(b - \color{blue}{a}\right)\right)\right)\right) \]

      14. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(angle, \mathsf{*.f64}\left(\frac{1}{90}, \mathsf{PI.f64}\left(\right)\right)\right), \mathsf{*.f64}\left(\left(\frac{1}{b - a}\right), \color{blue}{\left(\left(b \cdot b - a \cdot a\right) \cdot \left(b - a\right)\right)}\right)\right) \]

   9. Applied egg-rr 41.3%

      \[\leadsto \left(angle \cdot \left(0.011111111111111112 \cdot \pi\right)\right) \cdot \color{blue}{\left(\frac{1}{b - a} \cdot \left(\left(b \cdot b - a \cdot a\right) \cdot \left(b - a\right)\right)\right)} \]
3. Recombined 3 regimes into one program.

4. Final simplification 62.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;angle \leq 2.05 \cdot 10^{+49}:\\ \;\;\;\;\left(b + a\right) \cdot \left(\left(b - a\right) \cdot \left(\pi \cdot \left(angle \cdot 0.011111111111111112\right)\right)\right)\\ \mathbf{elif}\;angle \leq 4 \cdot 10^{+151}:\\ \;\;\;\;\frac{angle \cdot \left(\pi \cdot 0.011111111111111112 + \left(\pi \cdot \left(\pi \cdot \pi\right)\right) \cdot \left(\left(angle \cdot angle\right) \cdot -2.2862368541380886 \cdot 10^{-7}\right)\right)}{\frac{1}{b \cdot b - a \cdot a}}\\ \mathbf{else}:\\ \;\;\;\;\left(angle \cdot \left(\pi \cdot 0.011111111111111112\right)\right) \cdot \left(\frac{1}{b - a} \cdot \left(\left(b - a\right) \cdot \left(b \cdot b - a \cdot a\right)\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 16: 63.5% accurate, 17.4× speedup

Math

\[\begin{array}{l} angle\_m = \left|angle\right| \\ angle\_s = \mathsf{copysign}\left(1, angle\right) \\ angle\_s \cdot \begin{array}{l} \mathbf{if}\;angle\_m \leq 1.45 \cdot 10^{-9}:\\ \;\;\;\;\left(b + a\right) \cdot \left(\left(b - a\right) \cdot \left(\pi \cdot \left(angle\_m \cdot 0.011111111111111112\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(angle\_m \cdot \left(\pi \cdot 0.011111111111111112\right)\right) \cdot \left(\left(b \cdot b\right) \cdot \left(1 - \frac{\frac{a \cdot a}{b}}{b}\right)\right)\\ \end{array} \end{array} \]

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 1.45e-9)
    (* (+ b a) (* (- b a) (* PI (* angle_m 0.011111111111111112))))
    (*
     (* angle_m (* PI 0.011111111111111112))
     (* (* b b) (- 1.0 (/ (/ (* a a) b) b)))))))

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 <= 1.45e-9) {
		tmp = (b + a) * ((b - a) * (((double) M_PI) * (angle_m * 0.011111111111111112)));
	} else {
		tmp = (angle_m * (((double) M_PI) * 0.011111111111111112)) * ((b * b) * (1.0 - (((a * a) / b) / b)));
	}
	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 <= 1.45e-9) {
		tmp = (b + a) * ((b - a) * (Math.PI * (angle_m * 0.011111111111111112)));
	} else {
		tmp = (angle_m * (Math.PI * 0.011111111111111112)) * ((b * b) * (1.0 - (((a * a) / b) / b)));
	}
	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 <= 1.45e-9:
		tmp = (b + a) * ((b - a) * (math.pi * (angle_m * 0.011111111111111112)))
	else:
		tmp = (angle_m * (math.pi * 0.011111111111111112)) * ((b * b) * (1.0 - (((a * a) / b) / b)))
	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 <= 1.45e-9)
		tmp = Float64(Float64(b + a) * Float64(Float64(b - a) * Float64(pi * Float64(angle_m * 0.011111111111111112))));
	else
		tmp = Float64(Float64(angle_m * Float64(pi * 0.011111111111111112)) * Float64(Float64(b * b) * Float64(1.0 - Float64(Float64(Float64(a * a) / b) / b))));
	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 <= 1.45e-9)
		tmp = (b + a) * ((b - a) * (pi * (angle_m * 0.011111111111111112)));
	else
		tmp = (angle_m * (pi * 0.011111111111111112)) * ((b * b) * (1.0 - (((a * a) / b) / b)));
	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, 1.45e-9], N[(N[(b + a), $MachinePrecision] * N[(N[(b - a), $MachinePrecision] * N[(Pi * N[(angle$95$m * 0.011111111111111112), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(angle$95$m * N[(Pi * 0.011111111111111112), $MachinePrecision]), $MachinePrecision] * N[(N[(b * b), $MachinePrecision] * N[(1.0 - N[(N[(N[(a * a), $MachinePrecision] / b), $MachinePrecision] / b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if angle < 1.44999999999999996e-9

   1. Initial program 60.9%

      \[\left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin \left(\pi \cdot \frac{angle}{180}\right)\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right) \]
   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. associate-*l* N/A

         \[\leadsto 2 \cdot \color{blue}{\left(\left({b}^{2} - {a}^{2}\right) \cdot \left(\sin \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right) \cdot \cos \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right)\right)\right)} \]

      3. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(2, \color{blue}{\left(\left({b}^{2} - {a}^{2}\right) \cdot \left(\sin \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right) \cdot \cos \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right)\right)\right)}\right) \]

      4. associate-*r* N/A

         \[\leadsto \mathsf{*.f64}\left(2, \left(\left(\left({b}^{2} - {a}^{2}\right) \cdot \sin \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right)\right) \cdot \color{blue}{\cos \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right)}\right)\right) \]

      5. *-commutative N/A

         \[\leadsto \mathsf{*.f64}\left(2, \left(\left(\sin \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right) \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \cos \color{blue}{\left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right)}\right)\right) \]

      6. associate-*l* N/A

         \[\leadsto \mathsf{*.f64}\left(2, \left(\sin \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right) \cdot \color{blue}{\left(\left({b}^{2} - {a}^{2}\right) \cdot \cos \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right)\right)}\right)\right) \]

      7. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(2, \mathsf{*.f64}\left(\sin \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right), \color{blue}{\left(\left({b}^{2} - {a}^{2}\right) \cdot \cos \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right)\right)}\right)\right) \]

      8. sin-lowering-sin.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(2, \mathsf{*.f64}\left(\mathsf{sin.f64}\left(\left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right)\right), \left(\color{blue}{\left({b}^{2} - {a}^{2}\right)} \cdot \cos \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right)\right)\right)\right) \]

      9. associate-*r/ N/A

         \[\leadsto \mathsf{*.f64}\left(2, \mathsf{*.f64}\left(\mathsf{sin.f64}\left(\left(\frac{\mathsf{PI}\left(\right) \cdot angle}{180}\right)\right), \left(\left(\color{blue}{{b}^{2}} - {a}^{2}\right) \cdot \cos \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right)\right)\right)\right) \]

      10. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(2, \mathsf{*.f64}\left(\mathsf{sin.f64}\left(\mathsf{/.f64}\left(\left(\mathsf{PI}\left(\right) \cdot angle\right), 180\right)\right), \left(\left(\color{blue}{{b}^{2}} - {a}^{2}\right) \cdot \cos \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right)\right)\right)\right) \]

      11. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(2, \mathsf{*.f64}\left(\mathsf{sin.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{PI}\left(\right), angle\right), 180\right)\right), \left(\left({\color{blue}{b}}^{2} - {a}^{2}\right) \cdot \cos \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right)\right)\right)\right) \]

      12. PI-lowering-PI.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(2, \mathsf{*.f64}\left(\mathsf{sin.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{PI.f64}\left(\right), angle\right), 180\right)\right), \left(\left({b}^{2} - {a}^{2}\right) \cdot \cos \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right)\right)\right)\right) \]

   3. Simplified 61.8%

      \[\leadsto \color{blue}{2 \cdot \left(\sin \left(\frac{\pi \cdot angle}{180}\right) \cdot \left(\left(b \cdot b - a \cdot a\right) \cdot \cos \left(\frac{\pi \cdot angle}{180}\right)\right)\right)} \]
   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. *-commutative N/A

         \[\leadsto \left(angle \cdot \left(\mathsf{PI}\left(\right) \cdot \left({b}^{2} - {a}^{2}\right)\right)\right) \cdot \color{blue}{\frac{1}{90}} \]

      2. associate-*r* N/A

         \[\leadsto angle \cdot \color{blue}{\left(\left(\mathsf{PI}\left(\right) \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \frac{1}{90}\right)} \]

      3. *-commutative N/A

         \[\leadsto angle \cdot \left(\frac{1}{90} \cdot \color{blue}{\left(\mathsf{PI}\left(\right) \cdot \left({b}^{2} - {a}^{2}\right)\right)}\right) \]

      4. associate-*r* N/A

         \[\leadsto angle \cdot \left(\left(\frac{1}{90} \cdot \mathsf{PI}\left(\right)\right) \cdot \color{blue}{\left({b}^{2} - {a}^{2}\right)}\right) \]

      5. associate-*r* N/A

         \[\leadsto \left(angle \cdot \left(\frac{1}{90} \cdot \mathsf{PI}\left(\right)\right)\right) \cdot \color{blue}{\left({b}^{2} - {a}^{2}\right)} \]

      6. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\left(angle \cdot \left(\frac{1}{90} \cdot \mathsf{PI}\left(\right)\right)\right), \color{blue}{\left({b}^{2} - {a}^{2}\right)}\right) \]

      7. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(angle, \left(\frac{1}{90} \cdot \mathsf{PI}\left(\right)\right)\right), \left(\color{blue}{{b}^{2}} - {a}^{2}\right)\right) \]

      8. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(angle, \mathsf{*.f64}\left(\frac{1}{90}, \mathsf{PI}\left(\right)\right)\right), \left({b}^{\color{blue}{2}} - {a}^{2}\right)\right) \]

      9. PI-lowering-PI.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(angle, \mathsf{*.f64}\left(\frac{1}{90}, \mathsf{PI.f64}\left(\right)\right)\right), \left({b}^{2} - {a}^{2}\right)\right) \]

      10. --lowering--.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(angle, \mathsf{*.f64}\left(\frac{1}{90}, \mathsf{PI.f64}\left(\right)\right)\right), \mathsf{\_.f64}\left(\left({b}^{2}\right), \color{blue}{\left({a}^{2}\right)}\right)\right) \]

      11. unpow2 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(angle, \mathsf{*.f64}\left(\frac{1}{90}, \mathsf{PI.f64}\left(\right)\right)\right), \mathsf{\_.f64}\left(\left(b \cdot b\right), \left({\color{blue}{a}}^{2}\right)\right)\right) \]

      12. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(angle, \mathsf{*.f64}\left(\frac{1}{90}, \mathsf{PI.f64}\left(\right)\right)\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(b, b\right), \left({\color{blue}{a}}^{2}\right)\right)\right) \]

      13. unpow2 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(angle, \mathsf{*.f64}\left(\frac{1}{90}, \mathsf{PI.f64}\left(\right)\right)\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(b, b\right), \left(a \cdot \color{blue}{a}\right)\right)\right) \]

      14. *-lowering-*.f64 59.8%

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(angle, \mathsf{*.f64}\left(\frac{1}{90}, \mathsf{PI.f64}\left(\right)\right)\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(b, b\right), \mathsf{*.f64}\left(a, \color{blue}{a}\right)\right)\right) \]

   7. Simplified 59.8%

      \[\leadsto \color{blue}{\left(angle \cdot \left(0.011111111111111112 \cdot \pi\right)\right) \cdot \left(b \cdot b - a \cdot a\right)} \]
   8. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \left(b \cdot b - a \cdot a\right) \cdot \color{blue}{\left(angle \cdot \left(\frac{1}{90} \cdot \mathsf{PI}\left(\right)\right)\right)} \]

      2. difference-of-squares N/A

         \[\leadsto \left(\left(b + a\right) \cdot \left(b - a\right)\right) \cdot \left(\color{blue}{angle} \cdot \left(\frac{1}{90} \cdot \mathsf{PI}\left(\right)\right)\right) \]

      3. associate-*l* N/A

         \[\leadsto \left(b + a\right) \cdot \color{blue}{\left(\left(b - a\right) \cdot \left(angle \cdot \left(\frac{1}{90} \cdot \mathsf{PI}\left(\right)\right)\right)\right)} \]

      4. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\left(b + a\right), \color{blue}{\left(\left(b - a\right) \cdot \left(angle \cdot \left(\frac{1}{90} \cdot \mathsf{PI}\left(\right)\right)\right)\right)}\right) \]

      5. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(b, a\right), \left(\color{blue}{\left(b - a\right)} \cdot \left(angle \cdot \left(\frac{1}{90} \cdot \mathsf{PI}\left(\right)\right)\right)\right)\right) \]

      6. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(b, a\right), \mathsf{*.f64}\left(\left(b - a\right), \color{blue}{\left(angle \cdot \left(\frac{1}{90} \cdot \mathsf{PI}\left(\right)\right)\right)}\right)\right) \]

      7. --lowering--.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(b, a\right), \mathsf{*.f64}\left(\mathsf{\_.f64}\left(b, a\right), \left(\color{blue}{angle} \cdot \left(\frac{1}{90} \cdot \mathsf{PI}\left(\right)\right)\right)\right)\right) \]

      8. associate-*r* N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(b, a\right), \mathsf{*.f64}\left(\mathsf{\_.f64}\left(b, a\right), \left(\left(angle \cdot \frac{1}{90}\right) \cdot \color{blue}{\mathsf{PI}\left(\right)}\right)\right)\right) \]

      9. *-commutative N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(b, a\right), \mathsf{*.f64}\left(\mathsf{\_.f64}\left(b, a\right), \left(\mathsf{PI}\left(\right) \cdot \color{blue}{\left(angle \cdot \frac{1}{90}\right)}\right)\right)\right) \]

      10. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(b, a\right), \mathsf{*.f64}\left(\mathsf{\_.f64}\left(b, a\right), \mathsf{*.f64}\left(\mathsf{PI}\left(\right), \color{blue}{\left(angle \cdot \frac{1}{90}\right)}\right)\right)\right) \]

      11. PI-lowering-PI.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(b, a\right), \mathsf{*.f64}\left(\mathsf{\_.f64}\left(b, a\right), \mathsf{*.f64}\left(\mathsf{PI.f64}\left(\right), \left(\color{blue}{angle} \cdot \frac{1}{90}\right)\right)\right)\right) \]

      12. *-lowering-*.f64 71.1%

          \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(b, a\right), \mathsf{*.f64}\left(\mathsf{\_.f64}\left(b, a\right), \mathsf{*.f64}\left(\mathsf{PI.f64}\left(\right), \mathsf{*.f64}\left(angle, \color{blue}{\frac{1}{90}}\right)\right)\right)\right) \]

   9. Applied egg-rr 71.1%

      \[\leadsto \color{blue}{\left(b + a\right) \cdot \left(\left(b - a\right) \cdot \left(\pi \cdot \left(angle \cdot 0.011111111111111112\right)\right)\right)} \]

   if 1.44999999999999996e-9 < angle

   1. Initial program 36.3%

      \[\left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin \left(\pi \cdot \frac{angle}{180}\right)\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right) \]
   2. Step-by-step derivation

      1. associate-*l* 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. associate-*l* N/A

         \[\leadsto 2 \cdot \color{blue}{\left(\left({b}^{2} - {a}^{2}\right) \cdot \left(\sin \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right) \cdot \cos \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right)\right)\right)} \]

      3. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(2, \color{blue}{\left(\left({b}^{2} - {a}^{2}\right) \cdot \left(\sin \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right) \cdot \cos \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right)\right)\right)}\right) \]

      4. associate-*r* N/A

         \[\leadsto \mathsf{*.f64}\left(2, \left(\left(\left({b}^{2} - {a}^{2}\right) \cdot \sin \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right)\right) \cdot \color{blue}{\cos \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right)}\right)\right) \]

      5. *-commutative N/A

         \[\leadsto \mathsf{*.f64}\left(2, \left(\left(\sin \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right) \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \cos \color{blue}{\left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right)}\right)\right) \]

      6. associate-*l* N/A

         \[\leadsto \mathsf{*.f64}\left(2, \left(\sin \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right) \cdot \color{blue}{\left(\left({b}^{2} - {a}^{2}\right) \cdot \cos \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right)\right)}\right)\right) \]

      7. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(2, \mathsf{*.f64}\left(\sin \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right), \color{blue}{\left(\left({b}^{2} - {a}^{2}\right) \cdot \cos \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right)\right)}\right)\right) \]

      8. sin-lowering-sin.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(2, \mathsf{*.f64}\left(\mathsf{sin.f64}\left(\left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right)\right), \left(\color{blue}{\left({b}^{2} - {a}^{2}\right)} \cdot \cos \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right)\right)\right)\right) \]

      9. associate-*r/ N/A

         \[\leadsto \mathsf{*.f64}\left(2, \mathsf{*.f64}\left(\mathsf{sin.f64}\left(\left(\frac{\mathsf{PI}\left(\right) \cdot angle}{180}\right)\right), \left(\left(\color{blue}{{b}^{2}} - {a}^{2}\right) \cdot \cos \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right)\right)\right)\right) \]

      10. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(2, \mathsf{*.f64}\left(\mathsf{sin.f64}\left(\mathsf{/.f64}\left(\left(\mathsf{PI}\left(\right) \cdot angle\right), 180\right)\right), \left(\left(\color{blue}{{b}^{2}} - {a}^{2}\right) \cdot \cos \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right)\right)\right)\right) \]

      11. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(2, \mathsf{*.f64}\left(\mathsf{sin.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{PI}\left(\right), angle\right), 180\right)\right), \left(\left({\color{blue}{b}}^{2} - {a}^{2}\right) \cdot \cos \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right)\right)\right)\right) \]

      12. PI-lowering-PI.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(2, \mathsf{*.f64}\left(\mathsf{sin.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{PI.f64}\left(\right), angle\right), 180\right)\right), \left(\left({b}^{2} - {a}^{2}\right) \cdot \cos \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right)\right)\right)\right) \]

   3. Simplified 38.1%

      \[\leadsto \color{blue}{2 \cdot \left(\sin \left(\frac{\pi \cdot angle}{180}\right) \cdot \left(\left(b \cdot b - a \cdot a\right) \cdot \cos \left(\frac{\pi \cdot angle}{180}\right)\right)\right)} \]
   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. *-commutative N/A

         \[\leadsto \left(angle \cdot \left(\mathsf{PI}\left(\right) \cdot \left({b}^{2} - {a}^{2}\right)\right)\right) \cdot \color{blue}{\frac{1}{90}} \]

      2. associate-*r* N/A

         \[\leadsto angle \cdot \color{blue}{\left(\left(\mathsf{PI}\left(\right) \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \frac{1}{90}\right)} \]

      3. *-commutative N/A

         \[\leadsto angle \cdot \left(\frac{1}{90} \cdot \color{blue}{\left(\mathsf{PI}\left(\right) \cdot \left({b}^{2} - {a}^{2}\right)\right)}\right) \]

      4. associate-*r* N/A

         \[\leadsto angle \cdot \left(\left(\frac{1}{90} \cdot \mathsf{PI}\left(\right)\right) \cdot \color{blue}{\left({b}^{2} - {a}^{2}\right)}\right) \]

      5. associate-*r* N/A

         \[\leadsto \left(angle \cdot \left(\frac{1}{90} \cdot \mathsf{PI}\left(\right)\right)\right) \cdot \color{blue}{\left({b}^{2} - {a}^{2}\right)} \]

      6. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\left(angle \cdot \left(\frac{1}{90} \cdot \mathsf{PI}\left(\right)\right)\right), \color{blue}{\left({b}^{2} - {a}^{2}\right)}\right) \]

      7. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(angle, \left(\frac{1}{90} \cdot \mathsf{PI}\left(\right)\right)\right), \left(\color{blue}{{b}^{2}} - {a}^{2}\right)\right) \]

      8. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(angle, \mathsf{*.f64}\left(\frac{1}{90}, \mathsf{PI}\left(\right)\right)\right), \left({b}^{\color{blue}{2}} - {a}^{2}\right)\right) \]

      9. PI-lowering-PI.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(angle, \mathsf{*.f64}\left(\frac{1}{90}, \mathsf{PI.f64}\left(\right)\right)\right), \left({b}^{2} - {a}^{2}\right)\right) \]

      10. --lowering--.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(angle, \mathsf{*.f64}\left(\frac{1}{90}, \mathsf{PI.f64}\left(\right)\right)\right), \mathsf{\_.f64}\left(\left({b}^{2}\right), \color{blue}{\left({a}^{2}\right)}\right)\right) \]

      11. unpow2 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(angle, \mathsf{*.f64}\left(\frac{1}{90}, \mathsf{PI.f64}\left(\right)\right)\right), \mathsf{\_.f64}\left(\left(b \cdot b\right), \left({\color{blue}{a}}^{2}\right)\right)\right) \]

      12. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(angle, \mathsf{*.f64}\left(\frac{1}{90}, \mathsf{PI.f64}\left(\right)\right)\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(b, b\right), \left({\color{blue}{a}}^{2}\right)\right)\right) \]

      13. unpow2 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(angle, \mathsf{*.f64}\left(\frac{1}{90}, \mathsf{PI.f64}\left(\right)\right)\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(b, b\right), \left(a \cdot \color{blue}{a}\right)\right)\right) \]

      14. *-lowering-*.f64 33.6%

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(angle, \mathsf{*.f64}\left(\frac{1}{90}, \mathsf{PI.f64}\left(\right)\right)\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(b, b\right), \mathsf{*.f64}\left(a, \color{blue}{a}\right)\right)\right) \]

   7. Simplified 33.6%

      \[\leadsto \color{blue}{\left(angle \cdot \left(0.011111111111111112 \cdot \pi\right)\right) \cdot \left(b \cdot b - a \cdot a\right)} \]

   8. Taylor expanded in b around inf

      \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(angle, \mathsf{*.f64}\left(\frac{1}{90}, \mathsf{PI.f64}\left(\right)\right)\right), \color{blue}{\left({b}^{2} \cdot \left(1 + -1 \cdot \frac{{a}^{2}}{{b}^{2}}\right)\right)}\right) \]
   9. Step-by-step derivation

      1. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(angle, \mathsf{*.f64}\left(\frac{1}{90}, \mathsf{PI.f64}\left(\right)\right)\right), \mathsf{*.f64}\left(\left({b}^{2}\right), \color{blue}{\left(1 + -1 \cdot \frac{{a}^{2}}{{b}^{2}}\right)}\right)\right) \]

      2. unpow2 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(angle, \mathsf{*.f64}\left(\frac{1}{90}, \mathsf{PI.f64}\left(\right)\right)\right), \mathsf{*.f64}\left(\left(b \cdot b\right), \left(\color{blue}{1} + -1 \cdot \frac{{a}^{2}}{{b}^{2}}\right)\right)\right) \]

      3. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(angle, \mathsf{*.f64}\left(\frac{1}{90}, \mathsf{PI.f64}\left(\right)\right)\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(b, b\right), \left(\color{blue}{1} + -1 \cdot \frac{{a}^{2}}{{b}^{2}}\right)\right)\right) \]

      4. mul-1-neg N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(angle, \mathsf{*.f64}\left(\frac{1}{90}, \mathsf{PI.f64}\left(\right)\right)\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(b, b\right), \left(1 + \left(\mathsf{neg}\left(\frac{{a}^{2}}{{b}^{2}}\right)\right)\right)\right)\right) \]

      5. unsub-neg N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(angle, \mathsf{*.f64}\left(\frac{1}{90}, \mathsf{PI.f64}\left(\right)\right)\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(b, b\right), \left(1 - \color{blue}{\frac{{a}^{2}}{{b}^{2}}}\right)\right)\right) \]

      6. --lowering--.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(angle, \mathsf{*.f64}\left(\frac{1}{90}, \mathsf{PI.f64}\left(\right)\right)\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(b, b\right), \mathsf{\_.f64}\left(1, \color{blue}{\left(\frac{{a}^{2}}{{b}^{2}}\right)}\right)\right)\right) \]

      7. unpow2 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(angle, \mathsf{*.f64}\left(\frac{1}{90}, \mathsf{PI.f64}\left(\right)\right)\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(b, b\right), \mathsf{\_.f64}\left(1, \left(\frac{{a}^{2}}{b \cdot \color{blue}{b}}\right)\right)\right)\right) \]

      8. associate-/r* N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(angle, \mathsf{*.f64}\left(\frac{1}{90}, \mathsf{PI.f64}\left(\right)\right)\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(b, b\right), \mathsf{\_.f64}\left(1, \left(\frac{\frac{{a}^{2}}{b}}{\color{blue}{b}}\right)\right)\right)\right) \]

      9. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(angle, \mathsf{*.f64}\left(\frac{1}{90}, \mathsf{PI.f64}\left(\right)\right)\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(b, b\right), \mathsf{\_.f64}\left(1, \mathsf{/.f64}\left(\left(\frac{{a}^{2}}{b}\right), \color{blue}{b}\right)\right)\right)\right) \]

      10. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(angle, \mathsf{*.f64}\left(\frac{1}{90}, \mathsf{PI.f64}\left(\right)\right)\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(b, b\right), \mathsf{\_.f64}\left(1, \mathsf{/.f64}\left(\mathsf{/.f64}\left(\left({a}^{2}\right), b\right), b\right)\right)\right)\right) \]

      11. unpow2 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(angle, \mathsf{*.f64}\left(\frac{1}{90}, \mathsf{PI.f64}\left(\right)\right)\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(b, b\right), \mathsf{\_.f64}\left(1, \mathsf{/.f64}\left(\mathsf{/.f64}\left(\left(a \cdot a\right), b\right), b\right)\right)\right)\right) \]

      12. *-lowering-*.f64 33.4%

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(angle, \mathsf{*.f64}\left(\frac{1}{90}, \mathsf{PI.f64}\left(\right)\right)\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(b, b\right), \mathsf{\_.f64}\left(1, \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(a, a\right), b\right), b\right)\right)\right)\right) \]

   10. Simplified 33.4%

       \[\leadsto \left(angle \cdot \left(0.011111111111111112 \cdot \pi\right)\right) \cdot \color{blue}{\left(\left(b \cdot b\right) \cdot \left(1 - \frac{\frac{a \cdot a}{b}}{b}\right)\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 61.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;angle \leq 1.45 \cdot 10^{-9}:\\ \;\;\;\;\left(b + a\right) \cdot \left(\left(b - a\right) \cdot \left(\pi \cdot \left(angle \cdot 0.011111111111111112\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(angle \cdot \left(\pi \cdot 0.011111111111111112\right)\right) \cdot \left(\left(b \cdot b\right) \cdot \left(1 - \frac{\frac{a \cdot a}{b}}{b}\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 17: 57.9% accurate, 20.9× speedup

Math

\[\begin{array}{l} angle\_m = \left|angle\right| \\ angle\_s = \mathsf{copysign}\left(1, angle\right) \\ \begin{array}{l} t_0 := \pi \cdot \left(angle\_m \cdot 0.011111111111111112\right)\\ angle\_s \cdot \begin{array}{l} \mathbf{if}\;a \leq 6.3 \cdot 10^{-162}:\\ \;\;\;\;\frac{t\_0}{\frac{1}{b \cdot b - a \cdot a}}\\ \mathbf{else}:\\ \;\;\;\;\left(b + a\right) \cdot \left(\left(b - a\right) \cdot t\_0\right)\\ \end{array} \end{array} \end{array} \]

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 (* PI (* angle_m 0.011111111111111112))))
   (*
    angle_s
    (if (<= a 6.3e-162)
      (/ t_0 (/ 1.0 (- (* b b) (* a a))))
      (* (+ 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 = ((double) M_PI) * (angle_m * 0.011111111111111112);
	double tmp;
	if (a <= 6.3e-162) {
		tmp = t_0 / (1.0 / ((b * b) - (a * a)));
	} 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 = Math.PI * (angle_m * 0.011111111111111112);
	double tmp;
	if (a <= 6.3e-162) {
		tmp = t_0 / (1.0 / ((b * b) - (a * a)));
	} 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 = math.pi * (angle_m * 0.011111111111111112)
	tmp = 0
	if a <= 6.3e-162:
		tmp = t_0 / (1.0 / ((b * b) - (a * a)))
	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(pi * Float64(angle_m * 0.011111111111111112))
	tmp = 0.0
	if (a <= 6.3e-162)
		tmp = Float64(t_0 / Float64(1.0 / Float64(Float64(b * b) - Float64(a * a))));
	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 = pi * (angle_m * 0.011111111111111112);
	tmp = 0.0;
	if (a <= 6.3e-162)
		tmp = t_0 / (1.0 / ((b * b) - (a * a)));
	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[(Pi * N[(angle$95$m * 0.011111111111111112), $MachinePrecision]), $MachinePrecision]}, N[(angle$95$s * If[LessEqual[a, 6.3e-162], N[(t$95$0 / N[(1.0 / N[(N[(b * b), $MachinePrecision] - N[(a * a), $MachinePrecision]), $MachinePrecision]), $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 < 6.3000000000000001e-162

   1. Initial program 57.1%

      \[\left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin \left(\pi \cdot \frac{angle}{180}\right)\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right) \]
   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. associate-*l* N/A

         \[\leadsto 2 \cdot \color{blue}{\left(\left({b}^{2} - {a}^{2}\right) \cdot \left(\sin \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right) \cdot \cos \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right)\right)\right)} \]

      3. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(2, \color{blue}{\left(\left({b}^{2} - {a}^{2}\right) \cdot \left(\sin \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right) \cdot \cos \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right)\right)\right)}\right) \]

      4. associate-*r* N/A

         \[\leadsto \mathsf{*.f64}\left(2, \left(\left(\left({b}^{2} - {a}^{2}\right) \cdot \sin \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right)\right) \cdot \color{blue}{\cos \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right)}\right)\right) \]

      5. *-commutative N/A

         \[\leadsto \mathsf{*.f64}\left(2, \left(\left(\sin \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right) \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \cos \color{blue}{\left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right)}\right)\right) \]

      6. associate-*l* N/A

         \[\leadsto \mathsf{*.f64}\left(2, \left(\sin \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right) \cdot \color{blue}{\left(\left({b}^{2} - {a}^{2}\right) \cdot \cos \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right)\right)}\right)\right) \]

      7. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(2, \mathsf{*.f64}\left(\sin \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right), \color{blue}{\left(\left({b}^{2} - {a}^{2}\right) \cdot \cos \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right)\right)}\right)\right) \]

      8. sin-lowering-sin.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(2, \mathsf{*.f64}\left(\mathsf{sin.f64}\left(\left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right)\right), \left(\color{blue}{\left({b}^{2} - {a}^{2}\right)} \cdot \cos \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right)\right)\right)\right) \]

      9. associate-*r/ N/A

         \[\leadsto \mathsf{*.f64}\left(2, \mathsf{*.f64}\left(\mathsf{sin.f64}\left(\left(\frac{\mathsf{PI}\left(\right) \cdot angle}{180}\right)\right), \left(\left(\color{blue}{{b}^{2}} - {a}^{2}\right) \cdot \cos \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right)\right)\right)\right) \]

      10. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(2, \mathsf{*.f64}\left(\mathsf{sin.f64}\left(\mathsf{/.f64}\left(\left(\mathsf{PI}\left(\right) \cdot angle\right), 180\right)\right), \left(\left(\color{blue}{{b}^{2}} - {a}^{2}\right) \cdot \cos \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right)\right)\right)\right) \]

      11. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(2, \mathsf{*.f64}\left(\mathsf{sin.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{PI}\left(\right), angle\right), 180\right)\right), \left(\left({\color{blue}{b}}^{2} - {a}^{2}\right) \cdot \cos \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right)\right)\right)\right) \]

      12. PI-lowering-PI.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(2, \mathsf{*.f64}\left(\mathsf{sin.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{PI.f64}\left(\right), angle\right), 180\right)\right), \left(\left({b}^{2} - {a}^{2}\right) \cdot \cos \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right)\right)\right)\right) \]

   3. Simplified 58.0%

      \[\leadsto \color{blue}{2 \cdot \left(\sin \left(\frac{\pi \cdot angle}{180}\right) \cdot \left(\left(b \cdot b - a \cdot a\right) \cdot \cos \left(\frac{\pi \cdot angle}{180}\right)\right)\right)} \]
   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. *-commutative N/A

         \[\leadsto \left(angle \cdot \left(\mathsf{PI}\left(\right) \cdot \left({b}^{2} - {a}^{2}\right)\right)\right) \cdot \color{blue}{\frac{1}{90}} \]

      2. associate-*r* N/A

         \[\leadsto angle \cdot \color{blue}{\left(\left(\mathsf{PI}\left(\right) \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \frac{1}{90}\right)} \]

      3. *-commutative N/A

         \[\leadsto angle \cdot \left(\frac{1}{90} \cdot \color{blue}{\left(\mathsf{PI}\left(\right) \cdot \left({b}^{2} - {a}^{2}\right)\right)}\right) \]

      4. associate-*r* N/A

         \[\leadsto angle \cdot \left(\left(\frac{1}{90} \cdot \mathsf{PI}\left(\right)\right) \cdot \color{blue}{\left({b}^{2} - {a}^{2}\right)}\right) \]

      5. associate-*r* N/A

         \[\leadsto \left(angle \cdot \left(\frac{1}{90} \cdot \mathsf{PI}\left(\right)\right)\right) \cdot \color{blue}{\left({b}^{2} - {a}^{2}\right)} \]

      6. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\left(angle \cdot \left(\frac{1}{90} \cdot \mathsf{PI}\left(\right)\right)\right), \color{blue}{\left({b}^{2} - {a}^{2}\right)}\right) \]

      7. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(angle, \left(\frac{1}{90} \cdot \mathsf{PI}\left(\right)\right)\right), \left(\color{blue}{{b}^{2}} - {a}^{2}\right)\right) \]

      8. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(angle, \mathsf{*.f64}\left(\frac{1}{90}, \mathsf{PI}\left(\right)\right)\right), \left({b}^{\color{blue}{2}} - {a}^{2}\right)\right) \]

      9. PI-lowering-PI.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(angle, \mathsf{*.f64}\left(\frac{1}{90}, \mathsf{PI.f64}\left(\right)\right)\right), \left({b}^{2} - {a}^{2}\right)\right) \]

      10. --lowering--.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(angle, \mathsf{*.f64}\left(\frac{1}{90}, \mathsf{PI.f64}\left(\right)\right)\right), \mathsf{\_.f64}\left(\left({b}^{2}\right), \color{blue}{\left({a}^{2}\right)}\right)\right) \]

      11. unpow2 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(angle, \mathsf{*.f64}\left(\frac{1}{90}, \mathsf{PI.f64}\left(\right)\right)\right), \mathsf{\_.f64}\left(\left(b \cdot b\right), \left({\color{blue}{a}}^{2}\right)\right)\right) \]

      12. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(angle, \mathsf{*.f64}\left(\frac{1}{90}, \mathsf{PI.f64}\left(\right)\right)\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(b, b\right), \left({\color{blue}{a}}^{2}\right)\right)\right) \]

      13. unpow2 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(angle, \mathsf{*.f64}\left(\frac{1}{90}, \mathsf{PI.f64}\left(\right)\right)\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(b, b\right), \left(a \cdot \color{blue}{a}\right)\right)\right) \]

      14. *-lowering-*.f64 56.4%

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(angle, \mathsf{*.f64}\left(\frac{1}{90}, \mathsf{PI.f64}\left(\right)\right)\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(b, b\right), \mathsf{*.f64}\left(a, \color{blue}{a}\right)\right)\right) \]

   7. Simplified 56.4%

      \[\leadsto \color{blue}{\left(angle \cdot \left(0.011111111111111112 \cdot \pi\right)\right) \cdot \left(b \cdot b - a \cdot a\right)} \]
   8. Step-by-step derivation

      1. flip-- N/A

         \[\leadsto \left(angle \cdot \left(\frac{1}{90} \cdot \mathsf{PI}\left(\right)\right)\right) \cdot \frac{\left(b \cdot b\right) \cdot \left(b \cdot b\right) - \left(a \cdot a\right) \cdot \left(a \cdot a\right)}{\color{blue}{b \cdot b + a \cdot a}} \]

      2. clear-num N/A

         \[\leadsto \left(angle \cdot \left(\frac{1}{90} \cdot \mathsf{PI}\left(\right)\right)\right) \cdot \frac{1}{\color{blue}{\frac{b \cdot b + a \cdot a}{\left(b \cdot b\right) \cdot \left(b \cdot b\right) - \left(a \cdot a\right) \cdot \left(a \cdot a\right)}}} \]

      3. un-div-inv N/A

         \[\leadsto \frac{angle \cdot \left(\frac{1}{90} \cdot \mathsf{PI}\left(\right)\right)}{\color{blue}{\frac{b \cdot b + a \cdot a}{\left(b \cdot b\right) \cdot \left(b \cdot b\right) - \left(a \cdot a\right) \cdot \left(a \cdot a\right)}}} \]

      4. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\left(angle \cdot \left(\frac{1}{90} \cdot \mathsf{PI}\left(\right)\right)\right), \color{blue}{\left(\frac{b \cdot b + a \cdot a}{\left(b \cdot b\right) \cdot \left(b \cdot b\right) - \left(a \cdot a\right) \cdot \left(a \cdot a\right)}\right)}\right) \]

      5. associate-*r* N/A

         \[\leadsto \mathsf{/.f64}\left(\left(\left(angle \cdot \frac{1}{90}\right) \cdot \mathsf{PI}\left(\right)\right), \left(\frac{\color{blue}{b \cdot b + a \cdot a}}{\left(b \cdot b\right) \cdot \left(b \cdot b\right) - \left(a \cdot a\right) \cdot \left(a \cdot a\right)}\right)\right) \]

      6. *-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(\left(\mathsf{PI}\left(\right) \cdot \left(angle \cdot \frac{1}{90}\right)\right), \left(\frac{\color{blue}{b \cdot b + a \cdot a}}{\left(b \cdot b\right) \cdot \left(b \cdot b\right) - \left(a \cdot a\right) \cdot \left(a \cdot a\right)}\right)\right) \]

      7. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{PI}\left(\right), \left(angle \cdot \frac{1}{90}\right)\right), \left(\frac{\color{blue}{b \cdot b + a \cdot a}}{\left(b \cdot b\right) \cdot \left(b \cdot b\right) - \left(a \cdot a\right) \cdot \left(a \cdot a\right)}\right)\right) \]

      8. PI-lowering-PI.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{PI.f64}\left(\right), \left(angle \cdot \frac{1}{90}\right)\right), \left(\frac{\color{blue}{b \cdot b} + a \cdot a}{\left(b \cdot b\right) \cdot \left(b \cdot b\right) - \left(a \cdot a\right) \cdot \left(a \cdot a\right)}\right)\right) \]

      9. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{PI.f64}\left(\right), \mathsf{*.f64}\left(angle, \frac{1}{90}\right)\right), \left(\frac{b \cdot b + \color{blue}{a \cdot a}}{\left(b \cdot b\right) \cdot \left(b \cdot b\right) - \left(a \cdot a\right) \cdot \left(a \cdot a\right)}\right)\right) \]

      10. clear-num N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{PI.f64}\left(\right), \mathsf{*.f64}\left(angle, \frac{1}{90}\right)\right), \left(\frac{1}{\color{blue}{\frac{\left(b \cdot b\right) \cdot \left(b \cdot b\right) - \left(a \cdot a\right) \cdot \left(a \cdot a\right)}{b \cdot b + a \cdot a}}}\right)\right) \]

      11. flip-- N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{PI.f64}\left(\right), \mathsf{*.f64}\left(angle, \frac{1}{90}\right)\right), \left(\frac{1}{b \cdot b - \color{blue}{a \cdot a}}\right)\right) \]

      12. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{PI.f64}\left(\right), \mathsf{*.f64}\left(angle, \frac{1}{90}\right)\right), \mathsf{/.f64}\left(1, \color{blue}{\left(b \cdot b - a \cdot a\right)}\right)\right) \]

      13. --lowering--.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{PI.f64}\left(\right), \mathsf{*.f64}\left(angle, \frac{1}{90}\right)\right), \mathsf{/.f64}\left(1, \mathsf{\_.f64}\left(\left(b \cdot b\right), \color{blue}{\left(a \cdot a\right)}\right)\right)\right) \]

      14. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{PI.f64}\left(\right), \mathsf{*.f64}\left(angle, \frac{1}{90}\right)\right), \mathsf{/.f64}\left(1, \mathsf{\_.f64}\left(\mathsf{*.f64}\left(b, b\right), \left(\color{blue}{a} \cdot a\right)\right)\right)\right) \]

      15. *-lowering-*.f64 57.0%

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{PI.f64}\left(\right), \mathsf{*.f64}\left(angle, \frac{1}{90}\right)\right), \mathsf{/.f64}\left(1, \mathsf{\_.f64}\left(\mathsf{*.f64}\left(b, b\right), \mathsf{*.f64}\left(a, \color{blue}{a}\right)\right)\right)\right) \]

   9. Applied egg-rr 57.0%

      \[\leadsto \color{blue}{\frac{\pi \cdot \left(angle \cdot 0.011111111111111112\right)}{\frac{1}{b \cdot b - a \cdot a}}} \]

   if 6.3000000000000001e-162 < a

   1. Initial program 50.4%

      \[\left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin \left(\pi \cdot \frac{angle}{180}\right)\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right) \]
   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. associate-*l* N/A

         \[\leadsto 2 \cdot \color{blue}{\left(\left({b}^{2} - {a}^{2}\right) \cdot \left(\sin \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right) \cdot \cos \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right)\right)\right)} \]

      3. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(2, \color{blue}{\left(\left({b}^{2} - {a}^{2}\right) \cdot \left(\sin \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right) \cdot \cos \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right)\right)\right)}\right) \]

      4. associate-*r* N/A

         \[\leadsto \mathsf{*.f64}\left(2, \left(\left(\left({b}^{2} - {a}^{2}\right) \cdot \sin \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right)\right) \cdot \color{blue}{\cos \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right)}\right)\right) \]

      5. *-commutative N/A

         \[\leadsto \mathsf{*.f64}\left(2, \left(\left(\sin \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right) \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \cos \color{blue}{\left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right)}\right)\right) \]

      6. associate-*l* N/A

         \[\leadsto \mathsf{*.f64}\left(2, \left(\sin \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right) \cdot \color{blue}{\left(\left({b}^{2} - {a}^{2}\right) \cdot \cos \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right)\right)}\right)\right) \]

      7. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(2, \mathsf{*.f64}\left(\sin \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right), \color{blue}{\left(\left({b}^{2} - {a}^{2}\right) \cdot \cos \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right)\right)}\right)\right) \]

      8. sin-lowering-sin.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(2, \mathsf{*.f64}\left(\mathsf{sin.f64}\left(\left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right)\right), \left(\color{blue}{\left({b}^{2} - {a}^{2}\right)} \cdot \cos \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right)\right)\right)\right) \]

      9. associate-*r/ N/A

         \[\leadsto \mathsf{*.f64}\left(2, \mathsf{*.f64}\left(\mathsf{sin.f64}\left(\left(\frac{\mathsf{PI}\left(\right) \cdot angle}{180}\right)\right), \left(\left(\color{blue}{{b}^{2}} - {a}^{2}\right) \cdot \cos \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right)\right)\right)\right) \]

      10. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(2, \mathsf{*.f64}\left(\mathsf{sin.f64}\left(\mathsf{/.f64}\left(\left(\mathsf{PI}\left(\right) \cdot angle\right), 180\right)\right), \left(\left(\color{blue}{{b}^{2}} - {a}^{2}\right) \cdot \cos \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right)\right)\right)\right) \]

      11. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(2, \mathsf{*.f64}\left(\mathsf{sin.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{PI}\left(\right), angle\right), 180\right)\right), \left(\left({\color{blue}{b}}^{2} - {a}^{2}\right) \cdot \cos \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right)\right)\right)\right) \]

      12. PI-lowering-PI.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(2, \mathsf{*.f64}\left(\mathsf{sin.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{PI.f64}\left(\right), angle\right), 180\right)\right), \left(\left({b}^{2} - {a}^{2}\right) \cdot \cos \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right)\right)\right)\right) \]

   3. Simplified 51.7%

      \[\leadsto \color{blue}{2 \cdot \left(\sin \left(\frac{\pi \cdot angle}{180}\right) \cdot \left(\left(b \cdot b - a \cdot a\right) \cdot \cos \left(\frac{\pi \cdot angle}{180}\right)\right)\right)} \]
   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. *-commutative N/A

         \[\leadsto \left(angle \cdot \left(\mathsf{PI}\left(\right) \cdot \left({b}^{2} - {a}^{2}\right)\right)\right) \cdot \color{blue}{\frac{1}{90}} \]

      2. associate-*r* N/A

         \[\leadsto angle \cdot \color{blue}{\left(\left(\mathsf{PI}\left(\right) \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \frac{1}{90}\right)} \]

      3. *-commutative N/A

         \[\leadsto angle \cdot \left(\frac{1}{90} \cdot \color{blue}{\left(\mathsf{PI}\left(\right) \cdot \left({b}^{2} - {a}^{2}\right)\right)}\right) \]

      4. associate-*r* N/A

         \[\leadsto angle \cdot \left(\left(\frac{1}{90} \cdot \mathsf{PI}\left(\right)\right) \cdot \color{blue}{\left({b}^{2} - {a}^{2}\right)}\right) \]

      5. associate-*r* N/A

         \[\leadsto \left(angle \cdot \left(\frac{1}{90} \cdot \mathsf{PI}\left(\right)\right)\right) \cdot \color{blue}{\left({b}^{2} - {a}^{2}\right)} \]

      6. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\left(angle \cdot \left(\frac{1}{90} \cdot \mathsf{PI}\left(\right)\right)\right), \color{blue}{\left({b}^{2} - {a}^{2}\right)}\right) \]

      7. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(angle, \left(\frac{1}{90} \cdot \mathsf{PI}\left(\right)\right)\right), \left(\color{blue}{{b}^{2}} - {a}^{2}\right)\right) \]

      8. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(angle, \mathsf{*.f64}\left(\frac{1}{90}, \mathsf{PI}\left(\right)\right)\right), \left({b}^{\color{blue}{2}} - {a}^{2}\right)\right) \]

      9. PI-lowering-PI.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(angle, \mathsf{*.f64}\left(\frac{1}{90}, \mathsf{PI.f64}\left(\right)\right)\right), \left({b}^{2} - {a}^{2}\right)\right) \]

      10. --lowering--.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(angle, \mathsf{*.f64}\left(\frac{1}{90}, \mathsf{PI.f64}\left(\right)\right)\right), \mathsf{\_.f64}\left(\left({b}^{2}\right), \color{blue}{\left({a}^{2}\right)}\right)\right) \]

      11. unpow2 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(angle, \mathsf{*.f64}\left(\frac{1}{90}, \mathsf{PI.f64}\left(\right)\right)\right), \mathsf{\_.f64}\left(\left(b \cdot b\right), \left({\color{blue}{a}}^{2}\right)\right)\right) \]

      12. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(angle, \mathsf{*.f64}\left(\frac{1}{90}, \mathsf{PI.f64}\left(\right)\right)\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(b, b\right), \left({\color{blue}{a}}^{2}\right)\right)\right) \]

      13. unpow2 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(angle, \mathsf{*.f64}\left(\frac{1}{90}, \mathsf{PI.f64}\left(\right)\right)\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(b, b\right), \left(a \cdot \color{blue}{a}\right)\right)\right) \]

      14. *-lowering-*.f64 47.5%

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(angle, \mathsf{*.f64}\left(\frac{1}{90}, \mathsf{PI.f64}\left(\right)\right)\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(b, b\right), \mathsf{*.f64}\left(a, \color{blue}{a}\right)\right)\right) \]

   7. Simplified 47.5%

      \[\leadsto \color{blue}{\left(angle \cdot \left(0.011111111111111112 \cdot \pi\right)\right) \cdot \left(b \cdot b - a \cdot a\right)} \]
   8. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \left(b \cdot b - a \cdot a\right) \cdot \color{blue}{\left(angle \cdot \left(\frac{1}{90} \cdot \mathsf{PI}\left(\right)\right)\right)} \]

      2. difference-of-squares N/A

         \[\leadsto \left(\left(b + a\right) \cdot \left(b - a\right)\right) \cdot \left(\color{blue}{angle} \cdot \left(\frac{1}{90} \cdot \mathsf{PI}\left(\right)\right)\right) \]

      3. associate-*l* N/A

         \[\leadsto \left(b + a\right) \cdot \color{blue}{\left(\left(b - a\right) \cdot \left(angle \cdot \left(\frac{1}{90} \cdot \mathsf{PI}\left(\right)\right)\right)\right)} \]

      4. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\left(b + a\right), \color{blue}{\left(\left(b - a\right) \cdot \left(angle \cdot \left(\frac{1}{90} \cdot \mathsf{PI}\left(\right)\right)\right)\right)}\right) \]

      5. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(b, a\right), \left(\color{blue}{\left(b - a\right)} \cdot \left(angle \cdot \left(\frac{1}{90} \cdot \mathsf{PI}\left(\right)\right)\right)\right)\right) \]

      6. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(b, a\right), \mathsf{*.f64}\left(\left(b - a\right), \color{blue}{\left(angle \cdot \left(\frac{1}{90} \cdot \mathsf{PI}\left(\right)\right)\right)}\right)\right) \]

      7. --lowering--.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(b, a\right), \mathsf{*.f64}\left(\mathsf{\_.f64}\left(b, a\right), \left(\color{blue}{angle} \cdot \left(\frac{1}{90} \cdot \mathsf{PI}\left(\right)\right)\right)\right)\right) \]

      8. associate-*r* N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(b, a\right), \mathsf{*.f64}\left(\mathsf{\_.f64}\left(b, a\right), \left(\left(angle \cdot \frac{1}{90}\right) \cdot \color{blue}{\mathsf{PI}\left(\right)}\right)\right)\right) \]

      9. *-commutative N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(b, a\right), \mathsf{*.f64}\left(\mathsf{\_.f64}\left(b, a\right), \left(\mathsf{PI}\left(\right) \cdot \color{blue}{\left(angle \cdot \frac{1}{90}\right)}\right)\right)\right) \]

      10. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(b, a\right), \mathsf{*.f64}\left(\mathsf{\_.f64}\left(b, a\right), \mathsf{*.f64}\left(\mathsf{PI}\left(\right), \color{blue}{\left(angle \cdot \frac{1}{90}\right)}\right)\right)\right) \]

      11. PI-lowering-PI.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(b, a\right), \mathsf{*.f64}\left(\mathsf{\_.f64}\left(b, a\right), \mathsf{*.f64}\left(\mathsf{PI.f64}\left(\right), \left(\color{blue}{angle} \cdot \frac{1}{90}\right)\right)\right)\right) \]

      12. *-lowering-*.f64 60.4%

          \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(b, a\right), \mathsf{*.f64}\left(\mathsf{\_.f64}\left(b, a\right), \mathsf{*.f64}\left(\mathsf{PI.f64}\left(\right), \mathsf{*.f64}\left(angle, \color{blue}{\frac{1}{90}}\right)\right)\right)\right) \]

   9. Applied egg-rr 60.4%

      \[\leadsto \color{blue}{\left(b + a\right) \cdot \left(\left(b - a\right) \cdot \left(\pi \cdot \left(angle \cdot 0.011111111111111112\right)\right)\right)} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 18: 65.2% accurate, 20.9× speedup

Math

\[\begin{array}{l} angle\_m = \left|angle\right| \\ angle\_s = \mathsf{copysign}\left(1, angle\right) \\ angle\_s \cdot \begin{array}{l} \mathbf{if}\;angle\_m \leq 1.7 \cdot 10^{-6}:\\ \;\;\;\;\left(b + a\right) \cdot \left(\left(b - a\right) \cdot \left(\pi \cdot \left(angle\_m \cdot 0.011111111111111112\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(angle\_m \cdot \left(\pi \cdot 0.011111111111111112\right)\right) \cdot \frac{b + a}{\frac{1}{b - a}}\\ \end{array} \end{array} \]

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 1.7e-6)
    (* (+ b a) (* (- b a) (* PI (* angle_m 0.011111111111111112))))
    (* (* angle_m (* PI 0.011111111111111112)) (/ (+ b a) (/ 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 tmp;
	if (angle_m <= 1.7e-6) {
		tmp = (b + a) * ((b - a) * (((double) M_PI) * (angle_m * 0.011111111111111112)));
	} else {
		tmp = (angle_m * (((double) M_PI) * 0.011111111111111112)) * ((b + a) / (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 tmp;
	if (angle_m <= 1.7e-6) {
		tmp = (b + a) * ((b - a) * (Math.PI * (angle_m * 0.011111111111111112)));
	} else {
		tmp = (angle_m * (Math.PI * 0.011111111111111112)) * ((b + a) / (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):
	tmp = 0
	if angle_m <= 1.7e-6:
		tmp = (b + a) * ((b - a) * (math.pi * (angle_m * 0.011111111111111112)))
	else:
		tmp = (angle_m * (math.pi * 0.011111111111111112)) * ((b + a) / (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)
	tmp = 0.0
	if (angle_m <= 1.7e-6)
		tmp = Float64(Float64(b + a) * Float64(Float64(b - a) * Float64(pi * Float64(angle_m * 0.011111111111111112))));
	else
		tmp = Float64(Float64(angle_m * Float64(pi * 0.011111111111111112)) * Float64(Float64(b + a) / 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)
	tmp = 0.0;
	if (angle_m <= 1.7e-6)
		tmp = (b + a) * ((b - a) * (pi * (angle_m * 0.011111111111111112)));
	else
		tmp = (angle_m * (pi * 0.011111111111111112)) * ((b + a) / (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_] := N[(angle$95$s * If[LessEqual[angle$95$m, 1.7e-6], N[(N[(b + a), $MachinePrecision] * N[(N[(b - a), $MachinePrecision] * N[(Pi * N[(angle$95$m * 0.011111111111111112), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(angle$95$m * N[(Pi * 0.011111111111111112), $MachinePrecision]), $MachinePrecision] * N[(N[(b + a), $MachinePrecision] / N[(1.0 / N[(b - a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if angle < 1.70000000000000003e-6

   1. Initial program 61.1%

      \[\left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin \left(\pi \cdot \frac{angle}{180}\right)\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right) \]
   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. associate-*l* N/A

         \[\leadsto 2 \cdot \color{blue}{\left(\left({b}^{2} - {a}^{2}\right) \cdot \left(\sin \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right) \cdot \cos \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right)\right)\right)} \]

      3. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(2, \color{blue}{\left(\left({b}^{2} - {a}^{2}\right) \cdot \left(\sin \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right) \cdot \cos \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right)\right)\right)}\right) \]

      4. associate-*r* N/A

         \[\leadsto \mathsf{*.f64}\left(2, \left(\left(\left({b}^{2} - {a}^{2}\right) \cdot \sin \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right)\right) \cdot \color{blue}{\cos \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right)}\right)\right) \]

      5. *-commutative N/A

         \[\leadsto \mathsf{*.f64}\left(2, \left(\left(\sin \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right) \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \cos \color{blue}{\left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right)}\right)\right) \]

      6. associate-*l* N/A

         \[\leadsto \mathsf{*.f64}\left(2, \left(\sin \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right) \cdot \color{blue}{\left(\left({b}^{2} - {a}^{2}\right) \cdot \cos \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right)\right)}\right)\right) \]

      7. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(2, \mathsf{*.f64}\left(\sin \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right), \color{blue}{\left(\left({b}^{2} - {a}^{2}\right) \cdot \cos \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right)\right)}\right)\right) \]

      8. sin-lowering-sin.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(2, \mathsf{*.f64}\left(\mathsf{sin.f64}\left(\left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right)\right), \left(\color{blue}{\left({b}^{2} - {a}^{2}\right)} \cdot \cos \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right)\right)\right)\right) \]

      9. associate-*r/ N/A

         \[\leadsto \mathsf{*.f64}\left(2, \mathsf{*.f64}\left(\mathsf{sin.f64}\left(\left(\frac{\mathsf{PI}\left(\right) \cdot angle}{180}\right)\right), \left(\left(\color{blue}{{b}^{2}} - {a}^{2}\right) \cdot \cos \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right)\right)\right)\right) \]

      10. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(2, \mathsf{*.f64}\left(\mathsf{sin.f64}\left(\mathsf{/.f64}\left(\left(\mathsf{PI}\left(\right) \cdot angle\right), 180\right)\right), \left(\left(\color{blue}{{b}^{2}} - {a}^{2}\right) \cdot \cos \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right)\right)\right)\right) \]

      11. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(2, \mathsf{*.f64}\left(\mathsf{sin.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{PI}\left(\right), angle\right), 180\right)\right), \left(\left({\color{blue}{b}}^{2} - {a}^{2}\right) \cdot \cos \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right)\right)\right)\right) \]

      12. PI-lowering-PI.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(2, \mathsf{*.f64}\left(\mathsf{sin.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{PI.f64}\left(\right), angle\right), 180\right)\right), \left(\left({b}^{2} - {a}^{2}\right) \cdot \cos \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right)\right)\right)\right) \]

   3. Simplified 62.0%

      \[\leadsto \color{blue}{2 \cdot \left(\sin \left(\frac{\pi \cdot angle}{180}\right) \cdot \left(\left(b \cdot b - a \cdot a\right) \cdot \cos \left(\frac{\pi \cdot angle}{180}\right)\right)\right)} \]
   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. *-commutative N/A

         \[\leadsto \left(angle \cdot \left(\mathsf{PI}\left(\right) \cdot \left({b}^{2} - {a}^{2}\right)\right)\right) \cdot \color{blue}{\frac{1}{90}} \]

      2. associate-*r* N/A

         \[\leadsto angle \cdot \color{blue}{\left(\left(\mathsf{PI}\left(\right) \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \frac{1}{90}\right)} \]

      3. *-commutative N/A

         \[\leadsto angle \cdot \left(\frac{1}{90} \cdot \color{blue}{\left(\mathsf{PI}\left(\right) \cdot \left({b}^{2} - {a}^{2}\right)\right)}\right) \]

      4. associate-*r* N/A

         \[\leadsto angle \cdot \left(\left(\frac{1}{90} \cdot \mathsf{PI}\left(\right)\right) \cdot \color{blue}{\left({b}^{2} - {a}^{2}\right)}\right) \]

      5. associate-*r* N/A

         \[\leadsto \left(angle \cdot \left(\frac{1}{90} \cdot \mathsf{PI}\left(\right)\right)\right) \cdot \color{blue}{\left({b}^{2} - {a}^{2}\right)} \]

      6. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\left(angle \cdot \left(\frac{1}{90} \cdot \mathsf{PI}\left(\right)\right)\right), \color{blue}{\left({b}^{2} - {a}^{2}\right)}\right) \]

      7. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(angle, \left(\frac{1}{90} \cdot \mathsf{PI}\left(\right)\right)\right), \left(\color{blue}{{b}^{2}} - {a}^{2}\right)\right) \]

      8. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(angle, \mathsf{*.f64}\left(\frac{1}{90}, \mathsf{PI}\left(\right)\right)\right), \left({b}^{\color{blue}{2}} - {a}^{2}\right)\right) \]

      9. PI-lowering-PI.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(angle, \mathsf{*.f64}\left(\frac{1}{90}, \mathsf{PI.f64}\left(\right)\right)\right), \left({b}^{2} - {a}^{2}\right)\right) \]

      10. --lowering--.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(angle, \mathsf{*.f64}\left(\frac{1}{90}, \mathsf{PI.f64}\left(\right)\right)\right), \mathsf{\_.f64}\left(\left({b}^{2}\right), \color{blue}{\left({a}^{2}\right)}\right)\right) \]

      11. unpow2 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(angle, \mathsf{*.f64}\left(\frac{1}{90}, \mathsf{PI.f64}\left(\right)\right)\right), \mathsf{\_.f64}\left(\left(b \cdot b\right), \left({\color{blue}{a}}^{2}\right)\right)\right) \]

      12. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(angle, \mathsf{*.f64}\left(\frac{1}{90}, \mathsf{PI.f64}\left(\right)\right)\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(b, b\right), \left({\color{blue}{a}}^{2}\right)\right)\right) \]

      13. unpow2 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(angle, \mathsf{*.f64}\left(\frac{1}{90}, \mathsf{PI.f64}\left(\right)\right)\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(b, b\right), \left(a \cdot \color{blue}{a}\right)\right)\right) \]

      14. *-lowering-*.f64 60.0%

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(angle, \mathsf{*.f64}\left(\frac{1}{90}, \mathsf{PI.f64}\left(\right)\right)\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(b, b\right), \mathsf{*.f64}\left(a, \color{blue}{a}\right)\right)\right) \]

   7. Simplified 60.0%

      \[\leadsto \color{blue}{\left(angle \cdot \left(0.011111111111111112 \cdot \pi\right)\right) \cdot \left(b \cdot b - a \cdot a\right)} \]
   8. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \left(b \cdot b - a \cdot a\right) \cdot \color{blue}{\left(angle \cdot \left(\frac{1}{90} \cdot \mathsf{PI}\left(\right)\right)\right)} \]

      2. difference-of-squares N/A

         \[\leadsto \left(\left(b + a\right) \cdot \left(b - a\right)\right) \cdot \left(\color{blue}{angle} \cdot \left(\frac{1}{90} \cdot \mathsf{PI}\left(\right)\right)\right) \]

      3. associate-*l* N/A

         \[\leadsto \left(b + a\right) \cdot \color{blue}{\left(\left(b - a\right) \cdot \left(angle \cdot \left(\frac{1}{90} \cdot \mathsf{PI}\left(\right)\right)\right)\right)} \]

      4. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\left(b + a\right), \color{blue}{\left(\left(b - a\right) \cdot \left(angle \cdot \left(\frac{1}{90} \cdot \mathsf{PI}\left(\right)\right)\right)\right)}\right) \]

      5. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(b, a\right), \left(\color{blue}{\left(b - a\right)} \cdot \left(angle \cdot \left(\frac{1}{90} \cdot \mathsf{PI}\left(\right)\right)\right)\right)\right) \]

      6. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(b, a\right), \mathsf{*.f64}\left(\left(b - a\right), \color{blue}{\left(angle \cdot \left(\frac{1}{90} \cdot \mathsf{PI}\left(\right)\right)\right)}\right)\right) \]

      7. --lowering--.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(b, a\right), \mathsf{*.f64}\left(\mathsf{\_.f64}\left(b, a\right), \left(\color{blue}{angle} \cdot \left(\frac{1}{90} \cdot \mathsf{PI}\left(\right)\right)\right)\right)\right) \]

      8. associate-*r* N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(b, a\right), \mathsf{*.f64}\left(\mathsf{\_.f64}\left(b, a\right), \left(\left(angle \cdot \frac{1}{90}\right) \cdot \color{blue}{\mathsf{PI}\left(\right)}\right)\right)\right) \]

      9. *-commutative N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(b, a\right), \mathsf{*.f64}\left(\mathsf{\_.f64}\left(b, a\right), \left(\mathsf{PI}\left(\right) \cdot \color{blue}{\left(angle \cdot \frac{1}{90}\right)}\right)\right)\right) \]

      10. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(b, a\right), \mathsf{*.f64}\left(\mathsf{\_.f64}\left(b, a\right), \mathsf{*.f64}\left(\mathsf{PI}\left(\right), \color{blue}{\left(angle \cdot \frac{1}{90}\right)}\right)\right)\right) \]

      11. PI-lowering-PI.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(b, a\right), \mathsf{*.f64}\left(\mathsf{\_.f64}\left(b, a\right), \mathsf{*.f64}\left(\mathsf{PI.f64}\left(\right), \left(\color{blue}{angle} \cdot \frac{1}{90}\right)\right)\right)\right) \]

      12. *-lowering-*.f64 71.3%

          \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(b, a\right), \mathsf{*.f64}\left(\mathsf{\_.f64}\left(b, a\right), \mathsf{*.f64}\left(\mathsf{PI.f64}\left(\right), \mathsf{*.f64}\left(angle, \color{blue}{\frac{1}{90}}\right)\right)\right)\right) \]

   9. Applied egg-rr 71.3%

      \[\leadsto \color{blue}{\left(b + a\right) \cdot \left(\left(b - a\right) \cdot \left(\pi \cdot \left(angle \cdot 0.011111111111111112\right)\right)\right)} \]

   if 1.70000000000000003e-6 < angle

   1. Initial program 35.3%

      \[\left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin \left(\pi \cdot \frac{angle}{180}\right)\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right) \]
   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. associate-*l* N/A

         \[\leadsto 2 \cdot \color{blue}{\left(\left({b}^{2} - {a}^{2}\right) \cdot \left(\sin \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right) \cdot \cos \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right)\right)\right)} \]

      3. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(2, \color{blue}{\left(\left({b}^{2} - {a}^{2}\right) \cdot \left(\sin \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right) \cdot \cos \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right)\right)\right)}\right) \]

      4. associate-*r* N/A

         \[\leadsto \mathsf{*.f64}\left(2, \left(\left(\left({b}^{2} - {a}^{2}\right) \cdot \sin \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right)\right) \cdot \color{blue}{\cos \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right)}\right)\right) \]

      5. *-commutative N/A

         \[\leadsto \mathsf{*.f64}\left(2, \left(\left(\sin \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right) \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \cos \color{blue}{\left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right)}\right)\right) \]

      6. associate-*l* N/A

         \[\leadsto \mathsf{*.f64}\left(2, \left(\sin \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right) \cdot \color{blue}{\left(\left({b}^{2} - {a}^{2}\right) \cdot \cos \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right)\right)}\right)\right) \]

      7. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(2, \mathsf{*.f64}\left(\sin \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right), \color{blue}{\left(\left({b}^{2} - {a}^{2}\right) \cdot \cos \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right)\right)}\right)\right) \]

      8. sin-lowering-sin.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(2, \mathsf{*.f64}\left(\mathsf{sin.f64}\left(\left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right)\right), \left(\color{blue}{\left({b}^{2} - {a}^{2}\right)} \cdot \cos \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right)\right)\right)\right) \]

      9. associate-*r/ N/A

         \[\leadsto \mathsf{*.f64}\left(2, \mathsf{*.f64}\left(\mathsf{sin.f64}\left(\left(\frac{\mathsf{PI}\left(\right) \cdot angle}{180}\right)\right), \left(\left(\color{blue}{{b}^{2}} - {a}^{2}\right) \cdot \cos \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right)\right)\right)\right) \]

      10. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(2, \mathsf{*.f64}\left(\mathsf{sin.f64}\left(\mathsf{/.f64}\left(\left(\mathsf{PI}\left(\right) \cdot angle\right), 180\right)\right), \left(\left(\color{blue}{{b}^{2}} - {a}^{2}\right) \cdot \cos \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right)\right)\right)\right) \]

      11. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(2, \mathsf{*.f64}\left(\mathsf{sin.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{PI}\left(\right), angle\right), 180\right)\right), \left(\left({\color{blue}{b}}^{2} - {a}^{2}\right) \cdot \cos \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right)\right)\right)\right) \]

      12. PI-lowering-PI.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(2, \mathsf{*.f64}\left(\mathsf{sin.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{PI.f64}\left(\right), angle\right), 180\right)\right), \left(\left({b}^{2} - {a}^{2}\right) \cdot \cos \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right)\right)\right)\right) \]

   3. Simplified 37.2%

      \[\leadsto \color{blue}{2 \cdot \left(\sin \left(\frac{\pi \cdot angle}{180}\right) \cdot \left(\left(b \cdot b - a \cdot a\right) \cdot \cos \left(\frac{\pi \cdot angle}{180}\right)\right)\right)} \]
   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. *-commutative N/A

         \[\leadsto \left(angle \cdot \left(\mathsf{PI}\left(\right) \cdot \left({b}^{2} - {a}^{2}\right)\right)\right) \cdot \color{blue}{\frac{1}{90}} \]

      2. associate-*r* N/A

         \[\leadsto angle \cdot \color{blue}{\left(\left(\mathsf{PI}\left(\right) \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \frac{1}{90}\right)} \]

      3. *-commutative N/A

         \[\leadsto angle \cdot \left(\frac{1}{90} \cdot \color{blue}{\left(\mathsf{PI}\left(\right) \cdot \left({b}^{2} - {a}^{2}\right)\right)}\right) \]

      4. associate-*r* N/A

         \[\leadsto angle \cdot \left(\left(\frac{1}{90} \cdot \mathsf{PI}\left(\right)\right) \cdot \color{blue}{\left({b}^{2} - {a}^{2}\right)}\right) \]

      5. associate-*r* N/A

         \[\leadsto \left(angle \cdot \left(\frac{1}{90} \cdot \mathsf{PI}\left(\right)\right)\right) \cdot \color{blue}{\left({b}^{2} - {a}^{2}\right)} \]

      6. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\left(angle \cdot \left(\frac{1}{90} \cdot \mathsf{PI}\left(\right)\right)\right), \color{blue}{\left({b}^{2} - {a}^{2}\right)}\right) \]

      7. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(angle, \left(\frac{1}{90} \cdot \mathsf{PI}\left(\right)\right)\right), \left(\color{blue}{{b}^{2}} - {a}^{2}\right)\right) \]

      8. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(angle, \mathsf{*.f64}\left(\frac{1}{90}, \mathsf{PI}\left(\right)\right)\right), \left({b}^{\color{blue}{2}} - {a}^{2}\right)\right) \]

      9. PI-lowering-PI.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(angle, \mathsf{*.f64}\left(\frac{1}{90}, \mathsf{PI.f64}\left(\right)\right)\right), \left({b}^{2} - {a}^{2}\right)\right) \]

      10. --lowering--.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(angle, \mathsf{*.f64}\left(\frac{1}{90}, \mathsf{PI.f64}\left(\right)\right)\right), \mathsf{\_.f64}\left(\left({b}^{2}\right), \color{blue}{\left({a}^{2}\right)}\right)\right) \]

      11. unpow2 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(angle, \mathsf{*.f64}\left(\frac{1}{90}, \mathsf{PI.f64}\left(\right)\right)\right), \mathsf{\_.f64}\left(\left(b \cdot b\right), \left({\color{blue}{a}}^{2}\right)\right)\right) \]

      12. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(angle, \mathsf{*.f64}\left(\frac{1}{90}, \mathsf{PI.f64}\left(\right)\right)\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(b, b\right), \left({\color{blue}{a}}^{2}\right)\right)\right) \]

      13. unpow2 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(angle, \mathsf{*.f64}\left(\frac{1}{90}, \mathsf{PI.f64}\left(\right)\right)\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(b, b\right), \left(a \cdot \color{blue}{a}\right)\right)\right) \]

      14. *-lowering-*.f64 32.6%

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(angle, \mathsf{*.f64}\left(\frac{1}{90}, \mathsf{PI.f64}\left(\right)\right)\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(b, b\right), \mathsf{*.f64}\left(a, \color{blue}{a}\right)\right)\right) \]

   7. Simplified 32.6%

      \[\leadsto \color{blue}{\left(angle \cdot \left(0.011111111111111112 \cdot \pi\right)\right) \cdot \left(b \cdot b - a \cdot a\right)} \]
   8. Step-by-step derivation

      1. difference-of-squares N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(angle, \mathsf{*.f64}\left(\frac{1}{90}, \mathsf{PI.f64}\left(\right)\right)\right), \left(\left(b + a\right) \cdot \color{blue}{\left(b - a\right)}\right)\right) \]

      2. flip3-- N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(angle, \mathsf{*.f64}\left(\frac{1}{90}, \mathsf{PI.f64}\left(\right)\right)\right), \left(\left(b + a\right) \cdot \frac{{b}^{3} - {a}^{3}}{\color{blue}{b \cdot b + \left(a \cdot a + b \cdot a\right)}}\right)\right) \]

      3. clear-num N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(angle, \mathsf{*.f64}\left(\frac{1}{90}, \mathsf{PI.f64}\left(\right)\right)\right), \left(\left(b + a\right) \cdot \frac{1}{\color{blue}{\frac{b \cdot b + \left(a \cdot a + b \cdot a\right)}{{b}^{3} - {a}^{3}}}}\right)\right) \]

      4. clear-num N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(angle, \mathsf{*.f64}\left(\frac{1}{90}, \mathsf{PI.f64}\left(\right)\right)\right), \left(\left(b + a\right) \cdot \frac{1}{\frac{1}{\color{blue}{\frac{{b}^{3} - {a}^{3}}{b \cdot b + \left(a \cdot a + b \cdot a\right)}}}}\right)\right) \]

      5. flip3-- N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(angle, \mathsf{*.f64}\left(\frac{1}{90}, \mathsf{PI.f64}\left(\right)\right)\right), \left(\left(b + a\right) \cdot \frac{1}{\frac{1}{b - \color{blue}{a}}}\right)\right) \]

      6. un-div-inv N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(angle, \mathsf{*.f64}\left(\frac{1}{90}, \mathsf{PI.f64}\left(\right)\right)\right), \left(\frac{b + a}{\color{blue}{\frac{1}{b - a}}}\right)\right) \]

      7. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(angle, \mathsf{*.f64}\left(\frac{1}{90}, \mathsf{PI.f64}\left(\right)\right)\right), \mathsf{/.f64}\left(\left(b + a\right), \color{blue}{\left(\frac{1}{b - a}\right)}\right)\right) \]

      8. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(angle, \mathsf{*.f64}\left(\frac{1}{90}, \mathsf{PI.f64}\left(\right)\right)\right), \mathsf{/.f64}\left(\mathsf{+.f64}\left(b, a\right), \left(\frac{\color{blue}{1}}{b - a}\right)\right)\right) \]

      9. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(angle, \mathsf{*.f64}\left(\frac{1}{90}, \mathsf{PI.f64}\left(\right)\right)\right), \mathsf{/.f64}\left(\mathsf{+.f64}\left(b, a\right), \mathsf{/.f64}\left(1, \color{blue}{\left(b - a\right)}\right)\right)\right) \]

      10. --lowering--.f64 34.1%

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(angle, \mathsf{*.f64}\left(\frac{1}{90}, \mathsf{PI.f64}\left(\right)\right)\right), \mathsf{/.f64}\left(\mathsf{+.f64}\left(b, a\right), \mathsf{/.f64}\left(1, \mathsf{\_.f64}\left(b, \color{blue}{a}\right)\right)\right)\right) \]

   9. Applied egg-rr 34.1%

      \[\leadsto \left(angle \cdot \left(0.011111111111111112 \cdot \pi\right)\right) \cdot \color{blue}{\frac{b + a}{\frac{1}{b - a}}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 61.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;angle \leq 1.7 \cdot 10^{-6}:\\ \;\;\;\;\left(b + a\right) \cdot \left(\left(b - a\right) \cdot \left(\pi \cdot \left(angle \cdot 0.011111111111111112\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(angle \cdot \left(\pi \cdot 0.011111111111111112\right)\right) \cdot \frac{b + a}{\frac{1}{b - a}}\\ \end{array} \]
5. Add Preprocessing

Alternative 19: 49.6% accurate, 23.3× speedup

Math

\[\begin{array}{l} angle\_m = \left|angle\right| \\ angle\_s = \mathsf{copysign}\left(1, angle\right) \\ angle\_s \cdot \begin{array}{l} \mathbf{if}\;a \leq 3.6 \cdot 10^{-153}:\\ \;\;\;\;0.011111111111111112 \cdot \left(angle\_m \cdot \left(\pi \cdot \left(b \cdot b\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(b + a\right) \cdot \left(\left(b - a\right) \cdot \left(\pi \cdot \left(angle\_m \cdot 0.011111111111111112\right)\right)\right)\\ \end{array} \end{array} \]

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 (<= a 3.6e-153)
    (* 0.011111111111111112 (* angle_m (* PI (* b b))))
    (* (+ b a) (* (- b a) (* 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 (a <= 3.6e-153) {
		tmp = 0.011111111111111112 * (angle_m * (((double) M_PI) * (b * b)));
	} else {
		tmp = (b + a) * ((b - a) * (((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 (a <= 3.6e-153) {
		tmp = 0.011111111111111112 * (angle_m * (Math.PI * (b * b)));
	} else {
		tmp = (b + a) * ((b - a) * (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 a <= 3.6e-153:
		tmp = 0.011111111111111112 * (angle_m * (math.pi * (b * b)))
	else:
		tmp = (b + a) * ((b - a) * (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 (a <= 3.6e-153)
		tmp = Float64(0.011111111111111112 * Float64(angle_m * Float64(pi * Float64(b * b))));
	else
		tmp = Float64(Float64(b + a) * Float64(Float64(b - a) * 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 (a <= 3.6e-153)
		tmp = 0.011111111111111112 * (angle_m * (pi * (b * b)));
	else
		tmp = (b + a) * ((b - a) * (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[a, 3.6e-153], N[(0.011111111111111112 * N[(angle$95$m * N[(Pi * N[(b * b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(b + a), $MachinePrecision] * N[(N[(b - a), $MachinePrecision] * N[(Pi * N[(angle$95$m * 0.011111111111111112), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if a < 3.5999999999999998e-153

   1. Initial program 57.3%

      \[\left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin \left(\pi \cdot \frac{angle}{180}\right)\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right) \]
   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. associate-*l* N/A

         \[\leadsto 2 \cdot \color{blue}{\left(\left({b}^{2} - {a}^{2}\right) \cdot \left(\sin \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right) \cdot \cos \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right)\right)\right)} \]

      3. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(2, \color{blue}{\left(\left({b}^{2} - {a}^{2}\right) \cdot \left(\sin \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right) \cdot \cos \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right)\right)\right)}\right) \]

      4. associate-*r* N/A

         \[\leadsto \mathsf{*.f64}\left(2, \left(\left(\left({b}^{2} - {a}^{2}\right) \cdot \sin \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right)\right) \cdot \color{blue}{\cos \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right)}\right)\right) \]

      5. *-commutative N/A

         \[\leadsto \mathsf{*.f64}\left(2, \left(\left(\sin \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right) \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \cos \color{blue}{\left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right)}\right)\right) \]

      6. associate-*l* N/A

         \[\leadsto \mathsf{*.f64}\left(2, \left(\sin \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right) \cdot \color{blue}{\left(\left({b}^{2} - {a}^{2}\right) \cdot \cos \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right)\right)}\right)\right) \]

      7. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(2, \mathsf{*.f64}\left(\sin \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right), \color{blue}{\left(\left({b}^{2} - {a}^{2}\right) \cdot \cos \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right)\right)}\right)\right) \]

      8. sin-lowering-sin.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(2, \mathsf{*.f64}\left(\mathsf{sin.f64}\left(\left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right)\right), \left(\color{blue}{\left({b}^{2} - {a}^{2}\right)} \cdot \cos \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right)\right)\right)\right) \]

      9. associate-*r/ N/A

         \[\leadsto \mathsf{*.f64}\left(2, \mathsf{*.f64}\left(\mathsf{sin.f64}\left(\left(\frac{\mathsf{PI}\left(\right) \cdot angle}{180}\right)\right), \left(\left(\color{blue}{{b}^{2}} - {a}^{2}\right) \cdot \cos \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right)\right)\right)\right) \]

      10. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(2, \mathsf{*.f64}\left(\mathsf{sin.f64}\left(\mathsf{/.f64}\left(\left(\mathsf{PI}\left(\right) \cdot angle\right), 180\right)\right), \left(\left(\color{blue}{{b}^{2}} - {a}^{2}\right) \cdot \cos \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right)\right)\right)\right) \]

      11. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(2, \mathsf{*.f64}\left(\mathsf{sin.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{PI}\left(\right), angle\right), 180\right)\right), \left(\left({\color{blue}{b}}^{2} - {a}^{2}\right) \cdot \cos \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right)\right)\right)\right) \]

      12. PI-lowering-PI.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(2, \mathsf{*.f64}\left(\mathsf{sin.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{PI.f64}\left(\right), angle\right), 180\right)\right), \left(\left({b}^{2} - {a}^{2}\right) \cdot \cos \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right)\right)\right)\right) \]

   3. Simplified 58.3%

      \[\leadsto \color{blue}{2 \cdot \left(\sin \left(\frac{\pi \cdot angle}{180}\right) \cdot \left(\left(b \cdot b - a \cdot a\right) \cdot \cos \left(\frac{\pi \cdot angle}{180}\right)\right)\right)} \]
   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. *-commutative N/A

         \[\leadsto \left(angle \cdot \left(\mathsf{PI}\left(\right) \cdot \left({b}^{2} - {a}^{2}\right)\right)\right) \cdot \color{blue}{\frac{1}{90}} \]

      2. associate-*r* N/A

         \[\leadsto angle \cdot \color{blue}{\left(\left(\mathsf{PI}\left(\right) \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \frac{1}{90}\right)} \]

      3. *-commutative N/A

         \[\leadsto angle \cdot \left(\frac{1}{90} \cdot \color{blue}{\left(\mathsf{PI}\left(\right) \cdot \left({b}^{2} - {a}^{2}\right)\right)}\right) \]

      4. associate-*r* N/A

         \[\leadsto angle \cdot \left(\left(\frac{1}{90} \cdot \mathsf{PI}\left(\right)\right) \cdot \color{blue}{\left({b}^{2} - {a}^{2}\right)}\right) \]

      5. associate-*r* N/A

         \[\leadsto \left(angle \cdot \left(\frac{1}{90} \cdot \mathsf{PI}\left(\right)\right)\right) \cdot \color{blue}{\left({b}^{2} - {a}^{2}\right)} \]

      6. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\left(angle \cdot \left(\frac{1}{90} \cdot \mathsf{PI}\left(\right)\right)\right), \color{blue}{\left({b}^{2} - {a}^{2}\right)}\right) \]

      7. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(angle, \left(\frac{1}{90} \cdot \mathsf{PI}\left(\right)\right)\right), \left(\color{blue}{{b}^{2}} - {a}^{2}\right)\right) \]

      8. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(angle, \mathsf{*.f64}\left(\frac{1}{90}, \mathsf{PI}\left(\right)\right)\right), \left({b}^{\color{blue}{2}} - {a}^{2}\right)\right) \]

      9. PI-lowering-PI.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(angle, \mathsf{*.f64}\left(\frac{1}{90}, \mathsf{PI.f64}\left(\right)\right)\right), \left({b}^{2} - {a}^{2}\right)\right) \]

      10. --lowering--.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(angle, \mathsf{*.f64}\left(\frac{1}{90}, \mathsf{PI.f64}\left(\right)\right)\right), \mathsf{\_.f64}\left(\left({b}^{2}\right), \color{blue}{\left({a}^{2}\right)}\right)\right) \]

      11. unpow2 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(angle, \mathsf{*.f64}\left(\frac{1}{90}, \mathsf{PI.f64}\left(\right)\right)\right), \mathsf{\_.f64}\left(\left(b \cdot b\right), \left({\color{blue}{a}}^{2}\right)\right)\right) \]

      12. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(angle, \mathsf{*.f64}\left(\frac{1}{90}, \mathsf{PI.f64}\left(\right)\right)\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(b, b\right), \left({\color{blue}{a}}^{2}\right)\right)\right) \]

      13. unpow2 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(angle, \mathsf{*.f64}\left(\frac{1}{90}, \mathsf{PI.f64}\left(\right)\right)\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(b, b\right), \left(a \cdot \color{blue}{a}\right)\right)\right) \]

      14. *-lowering-*.f64 56.6%

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(angle, \mathsf{*.f64}\left(\frac{1}{90}, \mathsf{PI.f64}\left(\right)\right)\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(b, b\right), \mathsf{*.f64}\left(a, \color{blue}{a}\right)\right)\right) \]

   7. Simplified 56.6%

      \[\leadsto \color{blue}{\left(angle \cdot \left(0.011111111111111112 \cdot \pi\right)\right) \cdot \left(b \cdot b - a \cdot a\right)} \]

   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. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\frac{1}{90}, \color{blue}{\left(angle \cdot \left({b}^{2} \cdot \mathsf{PI}\left(\right)\right)\right)}\right) \]

      2. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\frac{1}{90}, \mathsf{*.f64}\left(angle, \color{blue}{\left({b}^{2} \cdot \mathsf{PI}\left(\right)\right)}\right)\right) \]

      3. *-commutative N/A

         \[\leadsto \mathsf{*.f64}\left(\frac{1}{90}, \mathsf{*.f64}\left(angle, \left(\mathsf{PI}\left(\right) \cdot \color{blue}{{b}^{2}}\right)\right)\right) \]

      4. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\frac{1}{90}, \mathsf{*.f64}\left(angle, \mathsf{*.f64}\left(\mathsf{PI}\left(\right), \color{blue}{\left({b}^{2}\right)}\right)\right)\right) \]

      5. PI-lowering-PI.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\frac{1}{90}, \mathsf{*.f64}\left(angle, \mathsf{*.f64}\left(\mathsf{PI.f64}\left(\right), \left({\color{blue}{b}}^{2}\right)\right)\right)\right) \]

      6. unpow2 N/A

         \[\leadsto \mathsf{*.f64}\left(\frac{1}{90}, \mathsf{*.f64}\left(angle, \mathsf{*.f64}\left(\mathsf{PI.f64}\left(\right), \left(b \cdot \color{blue}{b}\right)\right)\right)\right) \]

      7. *-lowering-*.f64 48.3%

         \[\leadsto \mathsf{*.f64}\left(\frac{1}{90}, \mathsf{*.f64}\left(angle, \mathsf{*.f64}\left(\mathsf{PI.f64}\left(\right), \mathsf{*.f64}\left(b, \color{blue}{b}\right)\right)\right)\right) \]

   10. Simplified 48.3%

       \[\leadsto \color{blue}{0.011111111111111112 \cdot \left(angle \cdot \left(\pi \cdot \left(b \cdot b\right)\right)\right)} \]

   if 3.5999999999999998e-153 < a

   1. Initial program 49.9%

      \[\left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin \left(\pi \cdot \frac{angle}{180}\right)\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right) \]
   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. associate-*l* N/A

         \[\leadsto 2 \cdot \color{blue}{\left(\left({b}^{2} - {a}^{2}\right) \cdot \left(\sin \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right) \cdot \cos \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right)\right)\right)} \]

      3. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(2, \color{blue}{\left(\left({b}^{2} - {a}^{2}\right) \cdot \left(\sin \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right) \cdot \cos \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right)\right)\right)}\right) \]

      4. associate-*r* N/A

         \[\leadsto \mathsf{*.f64}\left(2, \left(\left(\left({b}^{2} - {a}^{2}\right) \cdot \sin \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right)\right) \cdot \color{blue}{\cos \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right)}\right)\right) \]

      5. *-commutative N/A

         \[\leadsto \mathsf{*.f64}\left(2, \left(\left(\sin \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right) \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \cos \color{blue}{\left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right)}\right)\right) \]

      6. associate-*l* N/A

         \[\leadsto \mathsf{*.f64}\left(2, \left(\sin \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right) \cdot \color{blue}{\left(\left({b}^{2} - {a}^{2}\right) \cdot \cos \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right)\right)}\right)\right) \]

      7. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(2, \mathsf{*.f64}\left(\sin \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right), \color{blue}{\left(\left({b}^{2} - {a}^{2}\right) \cdot \cos \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right)\right)}\right)\right) \]

      8. sin-lowering-sin.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(2, \mathsf{*.f64}\left(\mathsf{sin.f64}\left(\left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right)\right), \left(\color{blue}{\left({b}^{2} - {a}^{2}\right)} \cdot \cos \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right)\right)\right)\right) \]

      9. associate-*r/ N/A

         \[\leadsto \mathsf{*.f64}\left(2, \mathsf{*.f64}\left(\mathsf{sin.f64}\left(\left(\frac{\mathsf{PI}\left(\right) \cdot angle}{180}\right)\right), \left(\left(\color{blue}{{b}^{2}} - {a}^{2}\right) \cdot \cos \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right)\right)\right)\right) \]

      10. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(2, \mathsf{*.f64}\left(\mathsf{sin.f64}\left(\mathsf{/.f64}\left(\left(\mathsf{PI}\left(\right) \cdot angle\right), 180\right)\right), \left(\left(\color{blue}{{b}^{2}} - {a}^{2}\right) \cdot \cos \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right)\right)\right)\right) \]

      11. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(2, \mathsf{*.f64}\left(\mathsf{sin.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{PI}\left(\right), angle\right), 180\right)\right), \left(\left({\color{blue}{b}}^{2} - {a}^{2}\right) \cdot \cos \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right)\right)\right)\right) \]

      12. PI-lowering-PI.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(2, \mathsf{*.f64}\left(\mathsf{sin.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{PI.f64}\left(\right), angle\right), 180\right)\right), \left(\left({b}^{2} - {a}^{2}\right) \cdot \cos \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right)\right)\right)\right) \]

   3. Simplified 51.3%

      \[\leadsto \color{blue}{2 \cdot \left(\sin \left(\frac{\pi \cdot angle}{180}\right) \cdot \left(\left(b \cdot b - a \cdot a\right) \cdot \cos \left(\frac{\pi \cdot angle}{180}\right)\right)\right)} \]
   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. *-commutative N/A

         \[\leadsto \left(angle \cdot \left(\mathsf{PI}\left(\right) \cdot \left({b}^{2} - {a}^{2}\right)\right)\right) \cdot \color{blue}{\frac{1}{90}} \]

      2. associate-*r* N/A

         \[\leadsto angle \cdot \color{blue}{\left(\left(\mathsf{PI}\left(\right) \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \frac{1}{90}\right)} \]

      3. *-commutative N/A

         \[\leadsto angle \cdot \left(\frac{1}{90} \cdot \color{blue}{\left(\mathsf{PI}\left(\right) \cdot \left({b}^{2} - {a}^{2}\right)\right)}\right) \]

      4. associate-*r* N/A

         \[\leadsto angle \cdot \left(\left(\frac{1}{90} \cdot \mathsf{PI}\left(\right)\right) \cdot \color{blue}{\left({b}^{2} - {a}^{2}\right)}\right) \]

      5. associate-*r* N/A

         \[\leadsto \left(angle \cdot \left(\frac{1}{90} \cdot \mathsf{PI}\left(\right)\right)\right) \cdot \color{blue}{\left({b}^{2} - {a}^{2}\right)} \]

      6. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\left(angle \cdot \left(\frac{1}{90} \cdot \mathsf{PI}\left(\right)\right)\right), \color{blue}{\left({b}^{2} - {a}^{2}\right)}\right) \]

      7. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(angle, \left(\frac{1}{90} \cdot \mathsf{PI}\left(\right)\right)\right), \left(\color{blue}{{b}^{2}} - {a}^{2}\right)\right) \]

      8. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(angle, \mathsf{*.f64}\left(\frac{1}{90}, \mathsf{PI}\left(\right)\right)\right), \left({b}^{\color{blue}{2}} - {a}^{2}\right)\right) \]

      9. PI-lowering-PI.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(angle, \mathsf{*.f64}\left(\frac{1}{90}, \mathsf{PI.f64}\left(\right)\right)\right), \left({b}^{2} - {a}^{2}\right)\right) \]

      10. --lowering--.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(angle, \mathsf{*.f64}\left(\frac{1}{90}, \mathsf{PI.f64}\left(\right)\right)\right), \mathsf{\_.f64}\left(\left({b}^{2}\right), \color{blue}{\left({a}^{2}\right)}\right)\right) \]

      11. unpow2 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(angle, \mathsf{*.f64}\left(\frac{1}{90}, \mathsf{PI.f64}\left(\right)\right)\right), \mathsf{\_.f64}\left(\left(b \cdot b\right), \left({\color{blue}{a}}^{2}\right)\right)\right) \]

      12. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(angle, \mathsf{*.f64}\left(\frac{1}{90}, \mathsf{PI.f64}\left(\right)\right)\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(b, b\right), \left({\color{blue}{a}}^{2}\right)\right)\right) \]

      13. unpow2 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(angle, \mathsf{*.f64}\left(\frac{1}{90}, \mathsf{PI.f64}\left(\right)\right)\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(b, b\right), \left(a \cdot \color{blue}{a}\right)\right)\right) \]

      14. *-lowering-*.f64 46.9%

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(angle, \mathsf{*.f64}\left(\frac{1}{90}, \mathsf{PI.f64}\left(\right)\right)\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(b, b\right), \mathsf{*.f64}\left(a, \color{blue}{a}\right)\right)\right) \]

   7. Simplified 46.9%

      \[\leadsto \color{blue}{\left(angle \cdot \left(0.011111111111111112 \cdot \pi\right)\right) \cdot \left(b \cdot b - a \cdot a\right)} \]
   8. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \left(b \cdot b - a \cdot a\right) \cdot \color{blue}{\left(angle \cdot \left(\frac{1}{90} \cdot \mathsf{PI}\left(\right)\right)\right)} \]

      2. difference-of-squares N/A

         \[\leadsto \left(\left(b + a\right) \cdot \left(b - a\right)\right) \cdot \left(\color{blue}{angle} \cdot \left(\frac{1}{90} \cdot \mathsf{PI}\left(\right)\right)\right) \]

      3. associate-*l* N/A

         \[\leadsto \left(b + a\right) \cdot \color{blue}{\left(\left(b - a\right) \cdot \left(angle \cdot \left(\frac{1}{90} \cdot \mathsf{PI}\left(\right)\right)\right)\right)} \]

      4. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\left(b + a\right), \color{blue}{\left(\left(b - a\right) \cdot \left(angle \cdot \left(\frac{1}{90} \cdot \mathsf{PI}\left(\right)\right)\right)\right)}\right) \]

      5. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(b, a\right), \left(\color{blue}{\left(b - a\right)} \cdot \left(angle \cdot \left(\frac{1}{90} \cdot \mathsf{PI}\left(\right)\right)\right)\right)\right) \]

      6. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(b, a\right), \mathsf{*.f64}\left(\left(b - a\right), \color{blue}{\left(angle \cdot \left(\frac{1}{90} \cdot \mathsf{PI}\left(\right)\right)\right)}\right)\right) \]

      7. --lowering--.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(b, a\right), \mathsf{*.f64}\left(\mathsf{\_.f64}\left(b, a\right), \left(\color{blue}{angle} \cdot \left(\frac{1}{90} \cdot \mathsf{PI}\left(\right)\right)\right)\right)\right) \]

      8. associate-*r* N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(b, a\right), \mathsf{*.f64}\left(\mathsf{\_.f64}\left(b, a\right), \left(\left(angle \cdot \frac{1}{90}\right) \cdot \color{blue}{\mathsf{PI}\left(\right)}\right)\right)\right) \]

      9. *-commutative N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(b, a\right), \mathsf{*.f64}\left(\mathsf{\_.f64}\left(b, a\right), \left(\mathsf{PI}\left(\right) \cdot \color{blue}{\left(angle \cdot \frac{1}{90}\right)}\right)\right)\right) \]

      10. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(b, a\right), \mathsf{*.f64}\left(\mathsf{\_.f64}\left(b, a\right), \mathsf{*.f64}\left(\mathsf{PI}\left(\right), \color{blue}{\left(angle \cdot \frac{1}{90}\right)}\right)\right)\right) \]

      11. PI-lowering-PI.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(b, a\right), \mathsf{*.f64}\left(\mathsf{\_.f64}\left(b, a\right), \mathsf{*.f64}\left(\mathsf{PI.f64}\left(\right), \left(\color{blue}{angle} \cdot \frac{1}{90}\right)\right)\right)\right) \]

      12. *-lowering-*.f64 60.0%

          \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(b, a\right), \mathsf{*.f64}\left(\mathsf{\_.f64}\left(b, a\right), \mathsf{*.f64}\left(\mathsf{PI.f64}\left(\right), \mathsf{*.f64}\left(angle, \color{blue}{\frac{1}{90}}\right)\right)\right)\right) \]

   9. Applied egg-rr 60.0%

      \[\leadsto \color{blue}{\left(b + a\right) \cdot \left(\left(b - a\right) \cdot \left(\pi \cdot \left(angle \cdot 0.011111111111111112\right)\right)\right)} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 20: 55.0% accurate, 23.3× speedup

Math

\[\begin{array}{l} angle\_m = \left|angle\right| \\ angle\_s = \mathsf{copysign}\left(1, angle\right) \\ angle\_s \cdot \begin{array}{l} \mathbf{if}\;a \leq 5.5 \cdot 10^{+97}:\\ \;\;\;\;\pi \cdot \left(\left(angle\_m \cdot 0.011111111111111112\right) \cdot \left(b \cdot b - a \cdot a\right)\right)\\ \mathbf{else}:\\ \;\;\;\;-0.011111111111111112 \cdot \left(a \cdot \left(\pi \cdot \left(angle\_m \cdot a\right)\right)\right)\\ \end{array} \end{array} \]

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 (<= a 5.5e+97)
    (* PI (* (* angle_m 0.011111111111111112) (- (* b b) (* a a))))
    (* -0.011111111111111112 (* a (* PI (* angle_m 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 tmp;
	if (a <= 5.5e+97) {
		tmp = ((double) M_PI) * ((angle_m * 0.011111111111111112) * ((b * b) - (a * a)));
	} else {
		tmp = -0.011111111111111112 * (a * (((double) M_PI) * (angle_m * 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 tmp;
	if (a <= 5.5e+97) {
		tmp = Math.PI * ((angle_m * 0.011111111111111112) * ((b * b) - (a * a)));
	} else {
		tmp = -0.011111111111111112 * (a * (Math.PI * (angle_m * 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):
	tmp = 0
	if a <= 5.5e+97:
		tmp = math.pi * ((angle_m * 0.011111111111111112) * ((b * b) - (a * a)))
	else:
		tmp = -0.011111111111111112 * (a * (math.pi * (angle_m * a)))
	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 <= 5.5e+97)
		tmp = Float64(pi * Float64(Float64(angle_m * 0.011111111111111112) * Float64(Float64(b * b) - Float64(a * a))));
	else
		tmp = Float64(-0.011111111111111112 * Float64(a * Float64(pi * Float64(angle_m * 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)
	tmp = 0.0;
	if (a <= 5.5e+97)
		tmp = pi * ((angle_m * 0.011111111111111112) * ((b * b) - (a * a)));
	else
		tmp = -0.011111111111111112 * (a * (pi * (angle_m * 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_] := N[(angle$95$s * If[LessEqual[a, 5.5e+97], N[(Pi * N[(N[(angle$95$m * 0.011111111111111112), $MachinePrecision] * N[(N[(b * b), $MachinePrecision] - N[(a * a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(-0.011111111111111112 * N[(a * N[(Pi * N[(angle$95$m * a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if a < 5.50000000000000021e97

   1. Initial program 56.2%

      \[\left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin \left(\pi \cdot \frac{angle}{180}\right)\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right) \]
   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. associate-*l* N/A

         \[\leadsto 2 \cdot \color{blue}{\left(\left({b}^{2} - {a}^{2}\right) \cdot \left(\sin \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right) \cdot \cos \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right)\right)\right)} \]

      3. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(2, \color{blue}{\left(\left({b}^{2} - {a}^{2}\right) \cdot \left(\sin \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right) \cdot \cos \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right)\right)\right)}\right) \]

      4. associate-*r* N/A

         \[\leadsto \mathsf{*.f64}\left(2, \left(\left(\left({b}^{2} - {a}^{2}\right) \cdot \sin \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right)\right) \cdot \color{blue}{\cos \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right)}\right)\right) \]

      5. *-commutative N/A

         \[\leadsto \mathsf{*.f64}\left(2, \left(\left(\sin \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right) \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \cos \color{blue}{\left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right)}\right)\right) \]

      6. associate-*l* N/A

         \[\leadsto \mathsf{*.f64}\left(2, \left(\sin \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right) \cdot \color{blue}{\left(\left({b}^{2} - {a}^{2}\right) \cdot \cos \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right)\right)}\right)\right) \]

      7. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(2, \mathsf{*.f64}\left(\sin \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right), \color{blue}{\left(\left({b}^{2} - {a}^{2}\right) \cdot \cos \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right)\right)}\right)\right) \]

      8. sin-lowering-sin.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(2, \mathsf{*.f64}\left(\mathsf{sin.f64}\left(\left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right)\right), \left(\color{blue}{\left({b}^{2} - {a}^{2}\right)} \cdot \cos \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right)\right)\right)\right) \]

      9. associate-*r/ N/A

         \[\leadsto \mathsf{*.f64}\left(2, \mathsf{*.f64}\left(\mathsf{sin.f64}\left(\left(\frac{\mathsf{PI}\left(\right) \cdot angle}{180}\right)\right), \left(\left(\color{blue}{{b}^{2}} - {a}^{2}\right) \cdot \cos \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right)\right)\right)\right) \]

      10. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(2, \mathsf{*.f64}\left(\mathsf{sin.f64}\left(\mathsf{/.f64}\left(\left(\mathsf{PI}\left(\right) \cdot angle\right), 180\right)\right), \left(\left(\color{blue}{{b}^{2}} - {a}^{2}\right) \cdot \cos \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right)\right)\right)\right) \]

      11. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(2, \mathsf{*.f64}\left(\mathsf{sin.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{PI}\left(\right), angle\right), 180\right)\right), \left(\left({\color{blue}{b}}^{2} - {a}^{2}\right) \cdot \cos \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right)\right)\right)\right) \]

      12. PI-lowering-PI.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(2, \mathsf{*.f64}\left(\mathsf{sin.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{PI.f64}\left(\right), angle\right), 180\right)\right), \left(\left({b}^{2} - {a}^{2}\right) \cdot \cos \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right)\right)\right)\right) \]

   3. Simplified 57.1%

      \[\leadsto \color{blue}{2 \cdot \left(\sin \left(\frac{\pi \cdot angle}{180}\right) \cdot \left(\left(b \cdot b - a \cdot a\right) \cdot \cos \left(\frac{\pi \cdot angle}{180}\right)\right)\right)} \]
   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. *-commutative N/A

         \[\leadsto \left(angle \cdot \left(\mathsf{PI}\left(\right) \cdot \left({b}^{2} - {a}^{2}\right)\right)\right) \cdot \color{blue}{\frac{1}{90}} \]

      2. associate-*r* N/A

         \[\leadsto angle \cdot \color{blue}{\left(\left(\mathsf{PI}\left(\right) \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \frac{1}{90}\right)} \]

      3. *-commutative N/A

         \[\leadsto angle \cdot \left(\frac{1}{90} \cdot \color{blue}{\left(\mathsf{PI}\left(\right) \cdot \left({b}^{2} - {a}^{2}\right)\right)}\right) \]

      4. associate-*r* N/A

         \[\leadsto angle \cdot \left(\left(\frac{1}{90} \cdot \mathsf{PI}\left(\right)\right) \cdot \color{blue}{\left({b}^{2} - {a}^{2}\right)}\right) \]

      5. associate-*r* N/A

         \[\leadsto \left(angle \cdot \left(\frac{1}{90} \cdot \mathsf{PI}\left(\right)\right)\right) \cdot \color{blue}{\left({b}^{2} - {a}^{2}\right)} \]

      6. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\left(angle \cdot \left(\frac{1}{90} \cdot \mathsf{PI}\left(\right)\right)\right), \color{blue}{\left({b}^{2} - {a}^{2}\right)}\right) \]

      7. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(angle, \left(\frac{1}{90} \cdot \mathsf{PI}\left(\right)\right)\right), \left(\color{blue}{{b}^{2}} - {a}^{2}\right)\right) \]

      8. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(angle, \mathsf{*.f64}\left(\frac{1}{90}, \mathsf{PI}\left(\right)\right)\right), \left({b}^{\color{blue}{2}} - {a}^{2}\right)\right) \]

      9. PI-lowering-PI.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(angle, \mathsf{*.f64}\left(\frac{1}{90}, \mathsf{PI.f64}\left(\right)\right)\right), \left({b}^{2} - {a}^{2}\right)\right) \]

      10. --lowering--.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(angle, \mathsf{*.f64}\left(\frac{1}{90}, \mathsf{PI.f64}\left(\right)\right)\right), \mathsf{\_.f64}\left(\left({b}^{2}\right), \color{blue}{\left({a}^{2}\right)}\right)\right) \]

      11. unpow2 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(angle, \mathsf{*.f64}\left(\frac{1}{90}, \mathsf{PI.f64}\left(\right)\right)\right), \mathsf{\_.f64}\left(\left(b \cdot b\right), \left({\color{blue}{a}}^{2}\right)\right)\right) \]

      12. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(angle, \mathsf{*.f64}\left(\frac{1}{90}, \mathsf{PI.f64}\left(\right)\right)\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(b, b\right), \left({\color{blue}{a}}^{2}\right)\right)\right) \]

      13. unpow2 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(angle, \mathsf{*.f64}\left(\frac{1}{90}, \mathsf{PI.f64}\left(\right)\right)\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(b, b\right), \left(a \cdot \color{blue}{a}\right)\right)\right) \]

      14. *-lowering-*.f64 55.5%

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(angle, \mathsf{*.f64}\left(\frac{1}{90}, \mathsf{PI.f64}\left(\right)\right)\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(b, b\right), \mathsf{*.f64}\left(a, \color{blue}{a}\right)\right)\right) \]

   7. Simplified 55.5%

      \[\leadsto \color{blue}{\left(angle \cdot \left(0.011111111111111112 \cdot \pi\right)\right) \cdot \left(b \cdot b - a \cdot a\right)} \]
   8. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \left(b \cdot b - a \cdot a\right) \cdot \color{blue}{\left(angle \cdot \left(\frac{1}{90} \cdot \mathsf{PI}\left(\right)\right)\right)} \]

      2. associate-*r* N/A

         \[\leadsto \left(b \cdot b - a \cdot a\right) \cdot \left(\left(angle \cdot \frac{1}{90}\right) \cdot \color{blue}{\mathsf{PI}\left(\right)}\right) \]

      3. associate-*r* N/A

         \[\leadsto \left(\left(b \cdot b - a \cdot a\right) \cdot \left(angle \cdot \frac{1}{90}\right)\right) \cdot \color{blue}{\mathsf{PI}\left(\right)} \]

      4. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\left(\left(b \cdot b - a \cdot a\right) \cdot \left(angle \cdot \frac{1}{90}\right)\right), \color{blue}{\mathsf{PI}\left(\right)}\right) \]

      5. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\left(b \cdot b - a \cdot a\right), \left(angle \cdot \frac{1}{90}\right)\right), \mathsf{PI}\left(\right)\right) \]

      6. --lowering--.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{\_.f64}\left(\left(b \cdot b\right), \left(a \cdot a\right)\right), \left(angle \cdot \frac{1}{90}\right)\right), \mathsf{PI}\left(\right)\right) \]

      7. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(b, b\right), \left(a \cdot a\right)\right), \left(angle \cdot \frac{1}{90}\right)\right), \mathsf{PI}\left(\right)\right) \]

      8. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(b, b\right), \mathsf{*.f64}\left(a, a\right)\right), \left(angle \cdot \frac{1}{90}\right)\right), \mathsf{PI}\left(\right)\right) \]

      9. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(b, b\right), \mathsf{*.f64}\left(a, a\right)\right), \mathsf{*.f64}\left(angle, \frac{1}{90}\right)\right), \mathsf{PI}\left(\right)\right) \]

      10. PI-lowering-PI.f64 55.5%

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(b, b\right), \mathsf{*.f64}\left(a, a\right)\right), \mathsf{*.f64}\left(angle, \frac{1}{90}\right)\right), \mathsf{PI.f64}\left(\right)\right) \]

   9. Applied egg-rr 55.5%

      \[\leadsto \color{blue}{\left(\left(b \cdot b - a \cdot a\right) \cdot \left(angle \cdot 0.011111111111111112\right)\right) \cdot \pi} \]

   if 5.50000000000000021e97 < a

   1. Initial program 45.4%

      \[\left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin \left(\pi \cdot \frac{angle}{180}\right)\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right) \]
   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. associate-*l* N/A

         \[\leadsto 2 \cdot \color{blue}{\left(\left({b}^{2} - {a}^{2}\right) \cdot \left(\sin \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right) \cdot \cos \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right)\right)\right)} \]

      3. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(2, \color{blue}{\left(\left({b}^{2} - {a}^{2}\right) \cdot \left(\sin \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right) \cdot \cos \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right)\right)\right)}\right) \]

      4. associate-*r* N/A

         \[\leadsto \mathsf{*.f64}\left(2, \left(\left(\left({b}^{2} - {a}^{2}\right) \cdot \sin \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right)\right) \cdot \color{blue}{\cos \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right)}\right)\right) \]

      5. *-commutative N/A

         \[\leadsto \mathsf{*.f64}\left(2, \left(\left(\sin \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right) \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \cos \color{blue}{\left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right)}\right)\right) \]

      6. associate-*l* N/A

         \[\leadsto \mathsf{*.f64}\left(2, \left(\sin \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right) \cdot \color{blue}{\left(\left({b}^{2} - {a}^{2}\right) \cdot \cos \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right)\right)}\right)\right) \]

      7. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(2, \mathsf{*.f64}\left(\sin \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right), \color{blue}{\left(\left({b}^{2} - {a}^{2}\right) \cdot \cos \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right)\right)}\right)\right) \]

      8. sin-lowering-sin.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(2, \mathsf{*.f64}\left(\mathsf{sin.f64}\left(\left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right)\right), \left(\color{blue}{\left({b}^{2} - {a}^{2}\right)} \cdot \cos \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right)\right)\right)\right) \]

      9. associate-*r/ N/A

         \[\leadsto \mathsf{*.f64}\left(2, \mathsf{*.f64}\left(\mathsf{sin.f64}\left(\left(\frac{\mathsf{PI}\left(\right) \cdot angle}{180}\right)\right), \left(\left(\color{blue}{{b}^{2}} - {a}^{2}\right) \cdot \cos \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right)\right)\right)\right) \]

      10. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(2, \mathsf{*.f64}\left(\mathsf{sin.f64}\left(\mathsf{/.f64}\left(\left(\mathsf{PI}\left(\right) \cdot angle\right), 180\right)\right), \left(\left(\color{blue}{{b}^{2}} - {a}^{2}\right) \cdot \cos \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right)\right)\right)\right) \]

      11. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(2, \mathsf{*.f64}\left(\mathsf{sin.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{PI}\left(\right), angle\right), 180\right)\right), \left(\left({\color{blue}{b}}^{2} - {a}^{2}\right) \cdot \cos \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right)\right)\right)\right) \]

      12. PI-lowering-PI.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(2, \mathsf{*.f64}\left(\mathsf{sin.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{PI.f64}\left(\right), angle\right), 180\right)\right), \left(\left({b}^{2} - {a}^{2}\right) \cdot \cos \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right)\right)\right)\right) \]

   3. Simplified 47.9%

      \[\leadsto \color{blue}{2 \cdot \left(\sin \left(\frac{\pi \cdot angle}{180}\right) \cdot \left(\left(b \cdot b - a \cdot a\right) \cdot \cos \left(\frac{\pi \cdot angle}{180}\right)\right)\right)} \]
   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. *-commutative N/A

         \[\leadsto \left(angle \cdot \left(\mathsf{PI}\left(\right) \cdot \left({b}^{2} - {a}^{2}\right)\right)\right) \cdot \color{blue}{\frac{1}{90}} \]

      2. associate-*r* N/A

         \[\leadsto angle \cdot \color{blue}{\left(\left(\mathsf{PI}\left(\right) \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \frac{1}{90}\right)} \]

      3. *-commutative N/A

         \[\leadsto angle \cdot \left(\frac{1}{90} \cdot \color{blue}{\left(\mathsf{PI}\left(\right) \cdot \left({b}^{2} - {a}^{2}\right)\right)}\right) \]

      4. associate-*r* N/A

         \[\leadsto angle \cdot \left(\left(\frac{1}{90} \cdot \mathsf{PI}\left(\right)\right) \cdot \color{blue}{\left({b}^{2} - {a}^{2}\right)}\right) \]

      5. associate-*r* N/A

         \[\leadsto \left(angle \cdot \left(\frac{1}{90} \cdot \mathsf{PI}\left(\right)\right)\right) \cdot \color{blue}{\left({b}^{2} - {a}^{2}\right)} \]

      6. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\left(angle \cdot \left(\frac{1}{90} \cdot \mathsf{PI}\left(\right)\right)\right), \color{blue}{\left({b}^{2} - {a}^{2}\right)}\right) \]

      7. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(angle, \left(\frac{1}{90} \cdot \mathsf{PI}\left(\right)\right)\right), \left(\color{blue}{{b}^{2}} - {a}^{2}\right)\right) \]

      8. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(angle, \mathsf{*.f64}\left(\frac{1}{90}, \mathsf{PI}\left(\right)\right)\right), \left({b}^{\color{blue}{2}} - {a}^{2}\right)\right) \]

      9. PI-lowering-PI.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(angle, \mathsf{*.f64}\left(\frac{1}{90}, \mathsf{PI.f64}\left(\right)\right)\right), \left({b}^{2} - {a}^{2}\right)\right) \]

      10. --lowering--.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(angle, \mathsf{*.f64}\left(\frac{1}{90}, \mathsf{PI.f64}\left(\right)\right)\right), \mathsf{\_.f64}\left(\left({b}^{2}\right), \color{blue}{\left({a}^{2}\right)}\right)\right) \]

      11. unpow2 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(angle, \mathsf{*.f64}\left(\frac{1}{90}, \mathsf{PI.f64}\left(\right)\right)\right), \mathsf{\_.f64}\left(\left(b \cdot b\right), \left({\color{blue}{a}}^{2}\right)\right)\right) \]

      12. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(angle, \mathsf{*.f64}\left(\frac{1}{90}, \mathsf{PI.f64}\left(\right)\right)\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(b, b\right), \left({\color{blue}{a}}^{2}\right)\right)\right) \]

      13. unpow2 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(angle, \mathsf{*.f64}\left(\frac{1}{90}, \mathsf{PI.f64}\left(\right)\right)\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(b, b\right), \left(a \cdot \color{blue}{a}\right)\right)\right) \]

      14. *-lowering-*.f64 39.3%

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(angle, \mathsf{*.f64}\left(\frac{1}{90}, \mathsf{PI.f64}\left(\right)\right)\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(b, b\right), \mathsf{*.f64}\left(a, \color{blue}{a}\right)\right)\right) \]

   7. Simplified 39.3%

      \[\leadsto \color{blue}{\left(angle \cdot \left(0.011111111111111112 \cdot \pi\right)\right) \cdot \left(b \cdot b - a \cdot a\right)} \]

   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. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\frac{-1}{90}, \color{blue}{\left({a}^{2} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}\right) \]

      2. associate-*r* N/A

         \[\leadsto \mathsf{*.f64}\left(\frac{-1}{90}, \left(\left({a}^{2} \cdot angle\right) \cdot \color{blue}{\mathsf{PI}\left(\right)}\right)\right) \]

      3. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\frac{-1}{90}, \mathsf{*.f64}\left(\left({a}^{2} \cdot angle\right), \color{blue}{\mathsf{PI}\left(\right)}\right)\right) \]

      4. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\frac{-1}{90}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\left({a}^{2}\right), angle\right), \mathsf{PI}\left(\right)\right)\right) \]

      5. unpow2 N/A

         \[\leadsto \mathsf{*.f64}\left(\frac{-1}{90}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\left(a \cdot a\right), angle\right), \mathsf{PI}\left(\right)\right)\right) \]

      6. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\frac{-1}{90}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(a, a\right), angle\right), \mathsf{PI}\left(\right)\right)\right) \]

      7. PI-lowering-PI.f64 59.6%

         \[\leadsto \mathsf{*.f64}\left(\frac{-1}{90}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(a, a\right), angle\right), \mathsf{PI.f64}\left(\right)\right)\right) \]

   10. Simplified 59.6%

       \[\leadsto \color{blue}{-0.011111111111111112 \cdot \left(\left(\left(a \cdot a\right) \cdot angle\right) \cdot \pi\right)} \]
   11. Step-by-step derivation

       1. associate-*l* N/A

          \[\leadsto \mathsf{*.f64}\left(\frac{-1}{90}, \left(\left(a \cdot \left(a \cdot angle\right)\right) \cdot \mathsf{PI}\left(\right)\right)\right) \]

       2. associate-*l* N/A

          \[\leadsto \mathsf{*.f64}\left(\frac{-1}{90}, \left(a \cdot \color{blue}{\left(\left(a \cdot angle\right) \cdot \mathsf{PI}\left(\right)\right)}\right)\right) \]

       3. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\frac{-1}{90}, \mathsf{*.f64}\left(a, \color{blue}{\left(\left(a \cdot angle\right) \cdot \mathsf{PI}\left(\right)\right)}\right)\right) \]

       4. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\frac{-1}{90}, \mathsf{*.f64}\left(a, \mathsf{*.f64}\left(\left(a \cdot angle\right), \color{blue}{\mathsf{PI}\left(\right)}\right)\right)\right) \]

       5. *-commutative N/A

          \[\leadsto \mathsf{*.f64}\left(\frac{-1}{90}, \mathsf{*.f64}\left(a, \mathsf{*.f64}\left(\left(angle \cdot a\right), \mathsf{PI}\left(\right)\right)\right)\right) \]

       6. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\frac{-1}{90}, \mathsf{*.f64}\left(a, \mathsf{*.f64}\left(\mathsf{*.f64}\left(angle, a\right), \mathsf{PI}\left(\right)\right)\right)\right) \]

       7. PI-lowering-PI.f64 64.4%

          \[\leadsto \mathsf{*.f64}\left(\frac{-1}{90}, \mathsf{*.f64}\left(a, \mathsf{*.f64}\left(\mathsf{*.f64}\left(angle, a\right), \mathsf{PI.f64}\left(\right)\right)\right)\right) \]

   12. Applied egg-rr 64.4%

       \[\leadsto -0.011111111111111112 \cdot \color{blue}{\left(a \cdot \left(\left(angle \cdot a\right) \cdot \pi\right)\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 56.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \leq 5.5 \cdot 10^{+97}:\\ \;\;\;\;\pi \cdot \left(\left(angle \cdot 0.011111111111111112\right) \cdot \left(b \cdot b - a \cdot a\right)\right)\\ \mathbf{else}:\\ \;\;\;\;-0.011111111111111112 \cdot \left(a \cdot \left(\pi \cdot \left(angle \cdot a\right)\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 21: 55.1% accurate, 23.3× speedup

Math

\[\begin{array}{l} angle\_m = \left|angle\right| \\ angle\_s = \mathsf{copysign}\left(1, angle\right) \\ angle\_s \cdot \begin{array}{l} \mathbf{if}\;a \leq 8 \cdot 10^{+97}:\\ \;\;\;\;angle\_m \cdot \left(\left(\pi \cdot 0.011111111111111112\right) \cdot \left(b \cdot b - a \cdot a\right)\right)\\ \mathbf{else}:\\ \;\;\;\;-0.011111111111111112 \cdot \left(a \cdot \left(\pi \cdot \left(angle\_m \cdot a\right)\right)\right)\\ \end{array} \end{array} \]

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 (<= a 8e+97)
    (* angle_m (* (* PI 0.011111111111111112) (- (* b b) (* a a))))
    (* -0.011111111111111112 (* a (* PI (* angle_m 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 tmp;
	if (a <= 8e+97) {
		tmp = angle_m * ((((double) M_PI) * 0.011111111111111112) * ((b * b) - (a * a)));
	} else {
		tmp = -0.011111111111111112 * (a * (((double) M_PI) * (angle_m * 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 tmp;
	if (a <= 8e+97) {
		tmp = angle_m * ((Math.PI * 0.011111111111111112) * ((b * b) - (a * a)));
	} else {
		tmp = -0.011111111111111112 * (a * (Math.PI * (angle_m * 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):
	tmp = 0
	if a <= 8e+97:
		tmp = angle_m * ((math.pi * 0.011111111111111112) * ((b * b) - (a * a)))
	else:
		tmp = -0.011111111111111112 * (a * (math.pi * (angle_m * a)))
	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 <= 8e+97)
		tmp = Float64(angle_m * Float64(Float64(pi * 0.011111111111111112) * Float64(Float64(b * b) - Float64(a * a))));
	else
		tmp = Float64(-0.011111111111111112 * Float64(a * Float64(pi * Float64(angle_m * 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)
	tmp = 0.0;
	if (a <= 8e+97)
		tmp = angle_m * ((pi * 0.011111111111111112) * ((b * b) - (a * a)));
	else
		tmp = -0.011111111111111112 * (a * (pi * (angle_m * 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_] := N[(angle$95$s * If[LessEqual[a, 8e+97], N[(angle$95$m * N[(N[(Pi * 0.011111111111111112), $MachinePrecision] * N[(N[(b * b), $MachinePrecision] - N[(a * a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(-0.011111111111111112 * N[(a * N[(Pi * N[(angle$95$m * a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if a < 8.0000000000000006e97

   1. Initial program 56.2%

      \[\left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin \left(\pi \cdot \frac{angle}{180}\right)\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right) \]
   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. associate-*l* N/A

         \[\leadsto 2 \cdot \color{blue}{\left(\left({b}^{2} - {a}^{2}\right) \cdot \left(\sin \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right) \cdot \cos \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right)\right)\right)} \]

      3. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(2, \color{blue}{\left(\left({b}^{2} - {a}^{2}\right) \cdot \left(\sin \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right) \cdot \cos \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right)\right)\right)}\right) \]

      4. associate-*r* N/A

         \[\leadsto \mathsf{*.f64}\left(2, \left(\left(\left({b}^{2} - {a}^{2}\right) \cdot \sin \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right)\right) \cdot \color{blue}{\cos \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right)}\right)\right) \]

      5. *-commutative N/A

         \[\leadsto \mathsf{*.f64}\left(2, \left(\left(\sin \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right) \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \cos \color{blue}{\left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right)}\right)\right) \]

      6. associate-*l* N/A

         \[\leadsto \mathsf{*.f64}\left(2, \left(\sin \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right) \cdot \color{blue}{\left(\left({b}^{2} - {a}^{2}\right) \cdot \cos \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right)\right)}\right)\right) \]

      7. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(2, \mathsf{*.f64}\left(\sin \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right), \color{blue}{\left(\left({b}^{2} - {a}^{2}\right) \cdot \cos \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right)\right)}\right)\right) \]

      8. sin-lowering-sin.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(2, \mathsf{*.f64}\left(\mathsf{sin.f64}\left(\left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right)\right), \left(\color{blue}{\left({b}^{2} - {a}^{2}\right)} \cdot \cos \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right)\right)\right)\right) \]

      9. associate-*r/ N/A

         \[\leadsto \mathsf{*.f64}\left(2, \mathsf{*.f64}\left(\mathsf{sin.f64}\left(\left(\frac{\mathsf{PI}\left(\right) \cdot angle}{180}\right)\right), \left(\left(\color{blue}{{b}^{2}} - {a}^{2}\right) \cdot \cos \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right)\right)\right)\right) \]

      10. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(2, \mathsf{*.f64}\left(\mathsf{sin.f64}\left(\mathsf{/.f64}\left(\left(\mathsf{PI}\left(\right) \cdot angle\right), 180\right)\right), \left(\left(\color{blue}{{b}^{2}} - {a}^{2}\right) \cdot \cos \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right)\right)\right)\right) \]

      11. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(2, \mathsf{*.f64}\left(\mathsf{sin.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{PI}\left(\right), angle\right), 180\right)\right), \left(\left({\color{blue}{b}}^{2} - {a}^{2}\right) \cdot \cos \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right)\right)\right)\right) \]

      12. PI-lowering-PI.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(2, \mathsf{*.f64}\left(\mathsf{sin.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{PI.f64}\left(\right), angle\right), 180\right)\right), \left(\left({b}^{2} - {a}^{2}\right) \cdot \cos \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right)\right)\right)\right) \]

   3. Simplified 57.1%

      \[\leadsto \color{blue}{2 \cdot \left(\sin \left(\frac{\pi \cdot angle}{180}\right) \cdot \left(\left(b \cdot b - a \cdot a\right) \cdot \cos \left(\frac{\pi \cdot angle}{180}\right)\right)\right)} \]
   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. *-commutative N/A

         \[\leadsto \left(angle \cdot \left(\mathsf{PI}\left(\right) \cdot \left({b}^{2} - {a}^{2}\right)\right)\right) \cdot \color{blue}{\frac{1}{90}} \]

      2. associate-*r* N/A

         \[\leadsto angle \cdot \color{blue}{\left(\left(\mathsf{PI}\left(\right) \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \frac{1}{90}\right)} \]

      3. *-commutative N/A

         \[\leadsto angle \cdot \left(\frac{1}{90} \cdot \color{blue}{\left(\mathsf{PI}\left(\right) \cdot \left({b}^{2} - {a}^{2}\right)\right)}\right) \]

      4. associate-*r* N/A

         \[\leadsto angle \cdot \left(\left(\frac{1}{90} \cdot \mathsf{PI}\left(\right)\right) \cdot \color{blue}{\left({b}^{2} - {a}^{2}\right)}\right) \]

      5. associate-*r* N/A

         \[\leadsto \left(angle \cdot \left(\frac{1}{90} \cdot \mathsf{PI}\left(\right)\right)\right) \cdot \color{blue}{\left({b}^{2} - {a}^{2}\right)} \]

      6. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\left(angle \cdot \left(\frac{1}{90} \cdot \mathsf{PI}\left(\right)\right)\right), \color{blue}{\left({b}^{2} - {a}^{2}\right)}\right) \]

      7. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(angle, \left(\frac{1}{90} \cdot \mathsf{PI}\left(\right)\right)\right), \left(\color{blue}{{b}^{2}} - {a}^{2}\right)\right) \]

      8. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(angle, \mathsf{*.f64}\left(\frac{1}{90}, \mathsf{PI}\left(\right)\right)\right), \left({b}^{\color{blue}{2}} - {a}^{2}\right)\right) \]

      9. PI-lowering-PI.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(angle, \mathsf{*.f64}\left(\frac{1}{90}, \mathsf{PI.f64}\left(\right)\right)\right), \left({b}^{2} - {a}^{2}\right)\right) \]

      10. --lowering--.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(angle, \mathsf{*.f64}\left(\frac{1}{90}, \mathsf{PI.f64}\left(\right)\right)\right), \mathsf{\_.f64}\left(\left({b}^{2}\right), \color{blue}{\left({a}^{2}\right)}\right)\right) \]

      11. unpow2 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(angle, \mathsf{*.f64}\left(\frac{1}{90}, \mathsf{PI.f64}\left(\right)\right)\right), \mathsf{\_.f64}\left(\left(b \cdot b\right), \left({\color{blue}{a}}^{2}\right)\right)\right) \]

      12. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(angle, \mathsf{*.f64}\left(\frac{1}{90}, \mathsf{PI.f64}\left(\right)\right)\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(b, b\right), \left({\color{blue}{a}}^{2}\right)\right)\right) \]

      13. unpow2 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(angle, \mathsf{*.f64}\left(\frac{1}{90}, \mathsf{PI.f64}\left(\right)\right)\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(b, b\right), \left(a \cdot \color{blue}{a}\right)\right)\right) \]

      14. *-lowering-*.f64 55.5%

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(angle, \mathsf{*.f64}\left(\frac{1}{90}, \mathsf{PI.f64}\left(\right)\right)\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(b, b\right), \mathsf{*.f64}\left(a, \color{blue}{a}\right)\right)\right) \]

   7. Simplified 55.5%

      \[\leadsto \color{blue}{\left(angle \cdot \left(0.011111111111111112 \cdot \pi\right)\right) \cdot \left(b \cdot b - a \cdot a\right)} \]
   8. Step-by-step derivation

      1. associate-*l* N/A

         \[\leadsto angle \cdot \color{blue}{\left(\left(\frac{1}{90} \cdot \mathsf{PI}\left(\right)\right) \cdot \left(b \cdot b - a \cdot a\right)\right)} \]

      2. *-commutative N/A

         \[\leadsto \left(\left(\frac{1}{90} \cdot \mathsf{PI}\left(\right)\right) \cdot \left(b \cdot b - a \cdot a\right)\right) \cdot \color{blue}{angle} \]

      3. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\left(\left(\frac{1}{90} \cdot \mathsf{PI}\left(\right)\right) \cdot \left(b \cdot b - a \cdot a\right)\right), \color{blue}{angle}\right) \]

      4. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\left(\frac{1}{90} \cdot \mathsf{PI}\left(\right)\right), \left(b \cdot b - a \cdot a\right)\right), angle\right) \]

      5. *-commutative N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\left(\mathsf{PI}\left(\right) \cdot \frac{1}{90}\right), \left(b \cdot b - a \cdot a\right)\right), angle\right) \]

      6. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{PI}\left(\right), \frac{1}{90}\right), \left(b \cdot b - a \cdot a\right)\right), angle\right) \]

      7. PI-lowering-PI.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{PI.f64}\left(\right), \frac{1}{90}\right), \left(b \cdot b - a \cdot a\right)\right), angle\right) \]

      8. --lowering--.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{PI.f64}\left(\right), \frac{1}{90}\right), \mathsf{\_.f64}\left(\left(b \cdot b\right), \left(a \cdot a\right)\right)\right), angle\right) \]

      9. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{PI.f64}\left(\right), \frac{1}{90}\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(b, b\right), \left(a \cdot a\right)\right)\right), angle\right) \]

      10. *-lowering-*.f64 55.5%

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{PI.f64}\left(\right), \frac{1}{90}\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(b, b\right), \mathsf{*.f64}\left(a, a\right)\right)\right), angle\right) \]

   9. Applied egg-rr 55.5%

      \[\leadsto \color{blue}{\left(\left(\pi \cdot 0.011111111111111112\right) \cdot \left(b \cdot b - a \cdot a\right)\right) \cdot angle} \]

   if 8.0000000000000006e97 < a

   1. Initial program 45.4%

      \[\left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin \left(\pi \cdot \frac{angle}{180}\right)\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right) \]
   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. associate-*l* N/A

         \[\leadsto 2 \cdot \color{blue}{\left(\left({b}^{2} - {a}^{2}\right) \cdot \left(\sin \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right) \cdot \cos \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right)\right)\right)} \]

      3. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(2, \color{blue}{\left(\left({b}^{2} - {a}^{2}\right) \cdot \left(\sin \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right) \cdot \cos \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right)\right)\right)}\right) \]

      4. associate-*r* N/A

         \[\leadsto \mathsf{*.f64}\left(2, \left(\left(\left({b}^{2} - {a}^{2}\right) \cdot \sin \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right)\right) \cdot \color{blue}{\cos \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right)}\right)\right) \]

      5. *-commutative N/A

         \[\leadsto \mathsf{*.f64}\left(2, \left(\left(\sin \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right) \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \cos \color{blue}{\left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right)}\right)\right) \]

      6. associate-*l* N/A

         \[\leadsto \mathsf{*.f64}\left(2, \left(\sin \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right) \cdot \color{blue}{\left(\left({b}^{2} - {a}^{2}\right) \cdot \cos \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right)\right)}\right)\right) \]

      7. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(2, \mathsf{*.f64}\left(\sin \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right), \color{blue}{\left(\left({b}^{2} - {a}^{2}\right) \cdot \cos \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right)\right)}\right)\right) \]

      8. sin-lowering-sin.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(2, \mathsf{*.f64}\left(\mathsf{sin.f64}\left(\left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right)\right), \left(\color{blue}{\left({b}^{2} - {a}^{2}\right)} \cdot \cos \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right)\right)\right)\right) \]

      9. associate-*r/ N/A

         \[\leadsto \mathsf{*.f64}\left(2, \mathsf{*.f64}\left(\mathsf{sin.f64}\left(\left(\frac{\mathsf{PI}\left(\right) \cdot angle}{180}\right)\right), \left(\left(\color{blue}{{b}^{2}} - {a}^{2}\right) \cdot \cos \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right)\right)\right)\right) \]

      10. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(2, \mathsf{*.f64}\left(\mathsf{sin.f64}\left(\mathsf{/.f64}\left(\left(\mathsf{PI}\left(\right) \cdot angle\right), 180\right)\right), \left(\left(\color{blue}{{b}^{2}} - {a}^{2}\right) \cdot \cos \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right)\right)\right)\right) \]

      11. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(2, \mathsf{*.f64}\left(\mathsf{sin.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{PI}\left(\right), angle\right), 180\right)\right), \left(\left({\color{blue}{b}}^{2} - {a}^{2}\right) \cdot \cos \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right)\right)\right)\right) \]

      12. PI-lowering-PI.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(2, \mathsf{*.f64}\left(\mathsf{sin.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{PI.f64}\left(\right), angle\right), 180\right)\right), \left(\left({b}^{2} - {a}^{2}\right) \cdot \cos \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right)\right)\right)\right) \]

   3. Simplified 47.9%

      \[\leadsto \color{blue}{2 \cdot \left(\sin \left(\frac{\pi \cdot angle}{180}\right) \cdot \left(\left(b \cdot b - a \cdot a\right) \cdot \cos \left(\frac{\pi \cdot angle}{180}\right)\right)\right)} \]
   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. *-commutative N/A

         \[\leadsto \left(angle \cdot \left(\mathsf{PI}\left(\right) \cdot \left({b}^{2} - {a}^{2}\right)\right)\right) \cdot \color{blue}{\frac{1}{90}} \]

      2. associate-*r* N/A

         \[\leadsto angle \cdot \color{blue}{\left(\left(\mathsf{PI}\left(\right) \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \frac{1}{90}\right)} \]

      3. *-commutative N/A

         \[\leadsto angle \cdot \left(\frac{1}{90} \cdot \color{blue}{\left(\mathsf{PI}\left(\right) \cdot \left({b}^{2} - {a}^{2}\right)\right)}\right) \]

      4. associate-*r* N/A

         \[\leadsto angle \cdot \left(\left(\frac{1}{90} \cdot \mathsf{PI}\left(\right)\right) \cdot \color{blue}{\left({b}^{2} - {a}^{2}\right)}\right) \]

      5. associate-*r* N/A

         \[\leadsto \left(angle \cdot \left(\frac{1}{90} \cdot \mathsf{PI}\left(\right)\right)\right) \cdot \color{blue}{\left({b}^{2} - {a}^{2}\right)} \]

      6. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\left(angle \cdot \left(\frac{1}{90} \cdot \mathsf{PI}\left(\right)\right)\right), \color{blue}{\left({b}^{2} - {a}^{2}\right)}\right) \]

      7. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(angle, \left(\frac{1}{90} \cdot \mathsf{PI}\left(\right)\right)\right), \left(\color{blue}{{b}^{2}} - {a}^{2}\right)\right) \]

      8. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(angle, \mathsf{*.f64}\left(\frac{1}{90}, \mathsf{PI}\left(\right)\right)\right), \left({b}^{\color{blue}{2}} - {a}^{2}\right)\right) \]

      9. PI-lowering-PI.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(angle, \mathsf{*.f64}\left(\frac{1}{90}, \mathsf{PI.f64}\left(\right)\right)\right), \left({b}^{2} - {a}^{2}\right)\right) \]

      10. --lowering--.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(angle, \mathsf{*.f64}\left(\frac{1}{90}, \mathsf{PI.f64}\left(\right)\right)\right), \mathsf{\_.f64}\left(\left({b}^{2}\right), \color{blue}{\left({a}^{2}\right)}\right)\right) \]

      11. unpow2 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(angle, \mathsf{*.f64}\left(\frac{1}{90}, \mathsf{PI.f64}\left(\right)\right)\right), \mathsf{\_.f64}\left(\left(b \cdot b\right), \left({\color{blue}{a}}^{2}\right)\right)\right) \]

      12. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(angle, \mathsf{*.f64}\left(\frac{1}{90}, \mathsf{PI.f64}\left(\right)\right)\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(b, b\right), \left({\color{blue}{a}}^{2}\right)\right)\right) \]

      13. unpow2 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(angle, \mathsf{*.f64}\left(\frac{1}{90}, \mathsf{PI.f64}\left(\right)\right)\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(b, b\right), \left(a \cdot \color{blue}{a}\right)\right)\right) \]

      14. *-lowering-*.f64 39.3%

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(angle, \mathsf{*.f64}\left(\frac{1}{90}, \mathsf{PI.f64}\left(\right)\right)\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(b, b\right), \mathsf{*.f64}\left(a, \color{blue}{a}\right)\right)\right) \]

   7. Simplified 39.3%

      \[\leadsto \color{blue}{\left(angle \cdot \left(0.011111111111111112 \cdot \pi\right)\right) \cdot \left(b \cdot b - a \cdot a\right)} \]

   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. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\frac{-1}{90}, \color{blue}{\left({a}^{2} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}\right) \]

      2. associate-*r* N/A

         \[\leadsto \mathsf{*.f64}\left(\frac{-1}{90}, \left(\left({a}^{2} \cdot angle\right) \cdot \color{blue}{\mathsf{PI}\left(\right)}\right)\right) \]

      3. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\frac{-1}{90}, \mathsf{*.f64}\left(\left({a}^{2} \cdot angle\right), \color{blue}{\mathsf{PI}\left(\right)}\right)\right) \]

      4. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\frac{-1}{90}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\left({a}^{2}\right), angle\right), \mathsf{PI}\left(\right)\right)\right) \]

      5. unpow2 N/A

         \[\leadsto \mathsf{*.f64}\left(\frac{-1}{90}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\left(a \cdot a\right), angle\right), \mathsf{PI}\left(\right)\right)\right) \]

      6. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\frac{-1}{90}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(a, a\right), angle\right), \mathsf{PI}\left(\right)\right)\right) \]

      7. PI-lowering-PI.f64 59.6%

         \[\leadsto \mathsf{*.f64}\left(\frac{-1}{90}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(a, a\right), angle\right), \mathsf{PI.f64}\left(\right)\right)\right) \]

   10. Simplified 59.6%

       \[\leadsto \color{blue}{-0.011111111111111112 \cdot \left(\left(\left(a \cdot a\right) \cdot angle\right) \cdot \pi\right)} \]
   11. Step-by-step derivation

       1. associate-*l* N/A

          \[\leadsto \mathsf{*.f64}\left(\frac{-1}{90}, \left(\left(a \cdot \left(a \cdot angle\right)\right) \cdot \mathsf{PI}\left(\right)\right)\right) \]

       2. associate-*l* N/A

          \[\leadsto \mathsf{*.f64}\left(\frac{-1}{90}, \left(a \cdot \color{blue}{\left(\left(a \cdot angle\right) \cdot \mathsf{PI}\left(\right)\right)}\right)\right) \]

       3. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\frac{-1}{90}, \mathsf{*.f64}\left(a, \color{blue}{\left(\left(a \cdot angle\right) \cdot \mathsf{PI}\left(\right)\right)}\right)\right) \]

       4. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\frac{-1}{90}, \mathsf{*.f64}\left(a, \mathsf{*.f64}\left(\left(a \cdot angle\right), \color{blue}{\mathsf{PI}\left(\right)}\right)\right)\right) \]

       5. *-commutative N/A

          \[\leadsto \mathsf{*.f64}\left(\frac{-1}{90}, \mathsf{*.f64}\left(a, \mathsf{*.f64}\left(\left(angle \cdot a\right), \mathsf{PI}\left(\right)\right)\right)\right) \]

       6. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\frac{-1}{90}, \mathsf{*.f64}\left(a, \mathsf{*.f64}\left(\mathsf{*.f64}\left(angle, a\right), \mathsf{PI}\left(\right)\right)\right)\right) \]

       7. PI-lowering-PI.f64 64.4%

          \[\leadsto \mathsf{*.f64}\left(\frac{-1}{90}, \mathsf{*.f64}\left(a, \mathsf{*.f64}\left(\mathsf{*.f64}\left(angle, a\right), \mathsf{PI.f64}\left(\right)\right)\right)\right) \]

   12. Applied egg-rr 64.4%

       \[\leadsto -0.011111111111111112 \cdot \color{blue}{\left(a \cdot \left(\left(angle \cdot a\right) \cdot \pi\right)\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 56.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \leq 8 \cdot 10^{+97}:\\ \;\;\;\;angle \cdot \left(\left(\pi \cdot 0.011111111111111112\right) \cdot \left(b \cdot b - a \cdot a\right)\right)\\ \mathbf{else}:\\ \;\;\;\;-0.011111111111111112 \cdot \left(a \cdot \left(\pi \cdot \left(angle \cdot a\right)\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 22: 55.1% accurate, 23.3× speedup

Math

\[\begin{array}{l} angle\_m = \left|angle\right| \\ angle\_s = \mathsf{copysign}\left(1, angle\right) \\ angle\_s \cdot \begin{array}{l} \mathbf{if}\;a \leq 8 \cdot 10^{+97}:\\ \;\;\;\;\left(angle\_m \cdot \left(\pi \cdot 0.011111111111111112\right)\right) \cdot \left(b \cdot b - a \cdot a\right)\\ \mathbf{else}:\\ \;\;\;\;-0.011111111111111112 \cdot \left(a \cdot \left(\pi \cdot \left(angle\_m \cdot a\right)\right)\right)\\ \end{array} \end{array} \]

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 (<= a 8e+97)
    (* (* angle_m (* PI 0.011111111111111112)) (- (* b b) (* a a)))
    (* -0.011111111111111112 (* a (* PI (* angle_m 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 tmp;
	if (a <= 8e+97) {
		tmp = (angle_m * (((double) M_PI) * 0.011111111111111112)) * ((b * b) - (a * a));
	} else {
		tmp = -0.011111111111111112 * (a * (((double) M_PI) * (angle_m * 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 tmp;
	if (a <= 8e+97) {
		tmp = (angle_m * (Math.PI * 0.011111111111111112)) * ((b * b) - (a * a));
	} else {
		tmp = -0.011111111111111112 * (a * (Math.PI * (angle_m * 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):
	tmp = 0
	if a <= 8e+97:
		tmp = (angle_m * (math.pi * 0.011111111111111112)) * ((b * b) - (a * a))
	else:
		tmp = -0.011111111111111112 * (a * (math.pi * (angle_m * a)))
	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 <= 8e+97)
		tmp = Float64(Float64(angle_m * Float64(pi * 0.011111111111111112)) * Float64(Float64(b * b) - Float64(a * a)));
	else
		tmp = Float64(-0.011111111111111112 * Float64(a * Float64(pi * Float64(angle_m * 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)
	tmp = 0.0;
	if (a <= 8e+97)
		tmp = (angle_m * (pi * 0.011111111111111112)) * ((b * b) - (a * a));
	else
		tmp = -0.011111111111111112 * (a * (pi * (angle_m * 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_] := N[(angle$95$s * If[LessEqual[a, 8e+97], N[(N[(angle$95$m * N[(Pi * 0.011111111111111112), $MachinePrecision]), $MachinePrecision] * N[(N[(b * b), $MachinePrecision] - N[(a * a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(-0.011111111111111112 * N[(a * N[(Pi * N[(angle$95$m * a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if a < 8.0000000000000006e97

   1. Initial program 56.2%

      \[\left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin \left(\pi \cdot \frac{angle}{180}\right)\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right) \]
   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. associate-*l* N/A

         \[\leadsto 2 \cdot \color{blue}{\left(\left({b}^{2} - {a}^{2}\right) \cdot \left(\sin \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right) \cdot \cos \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right)\right)\right)} \]

      3. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(2, \color{blue}{\left(\left({b}^{2} - {a}^{2}\right) \cdot \left(\sin \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right) \cdot \cos \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right)\right)\right)}\right) \]

      4. associate-*r* N/A

         \[\leadsto \mathsf{*.f64}\left(2, \left(\left(\left({b}^{2} - {a}^{2}\right) \cdot \sin \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right)\right) \cdot \color{blue}{\cos \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right)}\right)\right) \]

      5. *-commutative N/A

         \[\leadsto \mathsf{*.f64}\left(2, \left(\left(\sin \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right) \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \cos \color{blue}{\left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right)}\right)\right) \]

      6. associate-*l* N/A

         \[\leadsto \mathsf{*.f64}\left(2, \left(\sin \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right) \cdot \color{blue}{\left(\left({b}^{2} - {a}^{2}\right) \cdot \cos \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right)\right)}\right)\right) \]

      7. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(2, \mathsf{*.f64}\left(\sin \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right), \color{blue}{\left(\left({b}^{2} - {a}^{2}\right) \cdot \cos \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right)\right)}\right)\right) \]

      8. sin-lowering-sin.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(2, \mathsf{*.f64}\left(\mathsf{sin.f64}\left(\left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right)\right), \left(\color{blue}{\left({b}^{2} - {a}^{2}\right)} \cdot \cos \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right)\right)\right)\right) \]

      9. associate-*r/ N/A

         \[\leadsto \mathsf{*.f64}\left(2, \mathsf{*.f64}\left(\mathsf{sin.f64}\left(\left(\frac{\mathsf{PI}\left(\right) \cdot angle}{180}\right)\right), \left(\left(\color{blue}{{b}^{2}} - {a}^{2}\right) \cdot \cos \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right)\right)\right)\right) \]

      10. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(2, \mathsf{*.f64}\left(\mathsf{sin.f64}\left(\mathsf{/.f64}\left(\left(\mathsf{PI}\left(\right) \cdot angle\right), 180\right)\right), \left(\left(\color{blue}{{b}^{2}} - {a}^{2}\right) \cdot \cos \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right)\right)\right)\right) \]

      11. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(2, \mathsf{*.f64}\left(\mathsf{sin.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{PI}\left(\right), angle\right), 180\right)\right), \left(\left({\color{blue}{b}}^{2} - {a}^{2}\right) \cdot \cos \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right)\right)\right)\right) \]

      12. PI-lowering-PI.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(2, \mathsf{*.f64}\left(\mathsf{sin.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{PI.f64}\left(\right), angle\right), 180\right)\right), \left(\left({b}^{2} - {a}^{2}\right) \cdot \cos \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right)\right)\right)\right) \]

   3. Simplified 57.1%

      \[\leadsto \color{blue}{2 \cdot \left(\sin \left(\frac{\pi \cdot angle}{180}\right) \cdot \left(\left(b \cdot b - a \cdot a\right) \cdot \cos \left(\frac{\pi \cdot angle}{180}\right)\right)\right)} \]
   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. *-commutative N/A

         \[\leadsto \left(angle \cdot \left(\mathsf{PI}\left(\right) \cdot \left({b}^{2} - {a}^{2}\right)\right)\right) \cdot \color{blue}{\frac{1}{90}} \]

      2. associate-*r* N/A

         \[\leadsto angle \cdot \color{blue}{\left(\left(\mathsf{PI}\left(\right) \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \frac{1}{90}\right)} \]

      3. *-commutative N/A

         \[\leadsto angle \cdot \left(\frac{1}{90} \cdot \color{blue}{\left(\mathsf{PI}\left(\right) \cdot \left({b}^{2} - {a}^{2}\right)\right)}\right) \]

      4. associate-*r* N/A

         \[\leadsto angle \cdot \left(\left(\frac{1}{90} \cdot \mathsf{PI}\left(\right)\right) \cdot \color{blue}{\left({b}^{2} - {a}^{2}\right)}\right) \]

      5. associate-*r* N/A

         \[\leadsto \left(angle \cdot \left(\frac{1}{90} \cdot \mathsf{PI}\left(\right)\right)\right) \cdot \color{blue}{\left({b}^{2} - {a}^{2}\right)} \]

      6. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\left(angle \cdot \left(\frac{1}{90} \cdot \mathsf{PI}\left(\right)\right)\right), \color{blue}{\left({b}^{2} - {a}^{2}\right)}\right) \]

      7. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(angle, \left(\frac{1}{90} \cdot \mathsf{PI}\left(\right)\right)\right), \left(\color{blue}{{b}^{2}} - {a}^{2}\right)\right) \]

      8. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(angle, \mathsf{*.f64}\left(\frac{1}{90}, \mathsf{PI}\left(\right)\right)\right), \left({b}^{\color{blue}{2}} - {a}^{2}\right)\right) \]

      9. PI-lowering-PI.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(angle, \mathsf{*.f64}\left(\frac{1}{90}, \mathsf{PI.f64}\left(\right)\right)\right), \left({b}^{2} - {a}^{2}\right)\right) \]

      10. --lowering--.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(angle, \mathsf{*.f64}\left(\frac{1}{90}, \mathsf{PI.f64}\left(\right)\right)\right), \mathsf{\_.f64}\left(\left({b}^{2}\right), \color{blue}{\left({a}^{2}\right)}\right)\right) \]

      11. unpow2 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(angle, \mathsf{*.f64}\left(\frac{1}{90}, \mathsf{PI.f64}\left(\right)\right)\right), \mathsf{\_.f64}\left(\left(b \cdot b\right), \left({\color{blue}{a}}^{2}\right)\right)\right) \]

      12. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(angle, \mathsf{*.f64}\left(\frac{1}{90}, \mathsf{PI.f64}\left(\right)\right)\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(b, b\right), \left({\color{blue}{a}}^{2}\right)\right)\right) \]

      13. unpow2 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(angle, \mathsf{*.f64}\left(\frac{1}{90}, \mathsf{PI.f64}\left(\right)\right)\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(b, b\right), \left(a \cdot \color{blue}{a}\right)\right)\right) \]

      14. *-lowering-*.f64 55.5%

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(angle, \mathsf{*.f64}\left(\frac{1}{90}, \mathsf{PI.f64}\left(\right)\right)\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(b, b\right), \mathsf{*.f64}\left(a, \color{blue}{a}\right)\right)\right) \]

   7. Simplified 55.5%

      \[\leadsto \color{blue}{\left(angle \cdot \left(0.011111111111111112 \cdot \pi\right)\right) \cdot \left(b \cdot b - a \cdot a\right)} \]

   if 8.0000000000000006e97 < a

   1. Initial program 45.4%

      \[\left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin \left(\pi \cdot \frac{angle}{180}\right)\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right) \]
   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. associate-*l* N/A

         \[\leadsto 2 \cdot \color{blue}{\left(\left({b}^{2} - {a}^{2}\right) \cdot \left(\sin \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right) \cdot \cos \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right)\right)\right)} \]

      3. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(2, \color{blue}{\left(\left({b}^{2} - {a}^{2}\right) \cdot \left(\sin \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right) \cdot \cos \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right)\right)\right)}\right) \]

      4. associate-*r* N/A

         \[\leadsto \mathsf{*.f64}\left(2, \left(\left(\left({b}^{2} - {a}^{2}\right) \cdot \sin \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right)\right) \cdot \color{blue}{\cos \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right)}\right)\right) \]

      5. *-commutative N/A

         \[\leadsto \mathsf{*.f64}\left(2, \left(\left(\sin \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right) \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \cos \color{blue}{\left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right)}\right)\right) \]

      6. associate-*l* N/A

         \[\leadsto \mathsf{*.f64}\left(2, \left(\sin \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right) \cdot \color{blue}{\left(\left({b}^{2} - {a}^{2}\right) \cdot \cos \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right)\right)}\right)\right) \]

      7. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(2, \mathsf{*.f64}\left(\sin \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right), \color{blue}{\left(\left({b}^{2} - {a}^{2}\right) \cdot \cos \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right)\right)}\right)\right) \]

      8. sin-lowering-sin.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(2, \mathsf{*.f64}\left(\mathsf{sin.f64}\left(\left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right)\right), \left(\color{blue}{\left({b}^{2} - {a}^{2}\right)} \cdot \cos \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right)\right)\right)\right) \]

      9. associate-*r/ N/A

         \[\leadsto \mathsf{*.f64}\left(2, \mathsf{*.f64}\left(\mathsf{sin.f64}\left(\left(\frac{\mathsf{PI}\left(\right) \cdot angle}{180}\right)\right), \left(\left(\color{blue}{{b}^{2}} - {a}^{2}\right) \cdot \cos \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right)\right)\right)\right) \]

      10. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(2, \mathsf{*.f64}\left(\mathsf{sin.f64}\left(\mathsf{/.f64}\left(\left(\mathsf{PI}\left(\right) \cdot angle\right), 180\right)\right), \left(\left(\color{blue}{{b}^{2}} - {a}^{2}\right) \cdot \cos \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right)\right)\right)\right) \]

      11. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(2, \mathsf{*.f64}\left(\mathsf{sin.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{PI}\left(\right), angle\right), 180\right)\right), \left(\left({\color{blue}{b}}^{2} - {a}^{2}\right) \cdot \cos \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right)\right)\right)\right) \]

      12. PI-lowering-PI.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(2, \mathsf{*.f64}\left(\mathsf{sin.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{PI.f64}\left(\right), angle\right), 180\right)\right), \left(\left({b}^{2} - {a}^{2}\right) \cdot \cos \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right)\right)\right)\right) \]

   3. Simplified 47.9%

      \[\leadsto \color{blue}{2 \cdot \left(\sin \left(\frac{\pi \cdot angle}{180}\right) \cdot \left(\left(b \cdot b - a \cdot a\right) \cdot \cos \left(\frac{\pi \cdot angle}{180}\right)\right)\right)} \]
   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. *-commutative N/A

         \[\leadsto \left(angle \cdot \left(\mathsf{PI}\left(\right) \cdot \left({b}^{2} - {a}^{2}\right)\right)\right) \cdot \color{blue}{\frac{1}{90}} \]

      2. associate-*r* N/A

         \[\leadsto angle \cdot \color{blue}{\left(\left(\mathsf{PI}\left(\right) \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \frac{1}{90}\right)} \]

      3. *-commutative N/A

         \[\leadsto angle \cdot \left(\frac{1}{90} \cdot \color{blue}{\left(\mathsf{PI}\left(\right) \cdot \left({b}^{2} - {a}^{2}\right)\right)}\right) \]

      4. associate-*r* N/A

         \[\leadsto angle \cdot \left(\left(\frac{1}{90} \cdot \mathsf{PI}\left(\right)\right) \cdot \color{blue}{\left({b}^{2} - {a}^{2}\right)}\right) \]

      5. associate-*r* N/A

         \[\leadsto \left(angle \cdot \left(\frac{1}{90} \cdot \mathsf{PI}\left(\right)\right)\right) \cdot \color{blue}{\left({b}^{2} - {a}^{2}\right)} \]

      6. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\left(angle \cdot \left(\frac{1}{90} \cdot \mathsf{PI}\left(\right)\right)\right), \color{blue}{\left({b}^{2} - {a}^{2}\right)}\right) \]

      7. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(angle, \left(\frac{1}{90} \cdot \mathsf{PI}\left(\right)\right)\right), \left(\color{blue}{{b}^{2}} - {a}^{2}\right)\right) \]

      8. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(angle, \mathsf{*.f64}\left(\frac{1}{90}, \mathsf{PI}\left(\right)\right)\right), \left({b}^{\color{blue}{2}} - {a}^{2}\right)\right) \]

      9. PI-lowering-PI.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(angle, \mathsf{*.f64}\left(\frac{1}{90}, \mathsf{PI.f64}\left(\right)\right)\right), \left({b}^{2} - {a}^{2}\right)\right) \]

      10. --lowering--.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(angle, \mathsf{*.f64}\left(\frac{1}{90}, \mathsf{PI.f64}\left(\right)\right)\right), \mathsf{\_.f64}\left(\left({b}^{2}\right), \color{blue}{\left({a}^{2}\right)}\right)\right) \]

      11. unpow2 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(angle, \mathsf{*.f64}\left(\frac{1}{90}, \mathsf{PI.f64}\left(\right)\right)\right), \mathsf{\_.f64}\left(\left(b \cdot b\right), \left({\color{blue}{a}}^{2}\right)\right)\right) \]

      12. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(angle, \mathsf{*.f64}\left(\frac{1}{90}, \mathsf{PI.f64}\left(\right)\right)\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(b, b\right), \left({\color{blue}{a}}^{2}\right)\right)\right) \]

      13. unpow2 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(angle, \mathsf{*.f64}\left(\frac{1}{90}, \mathsf{PI.f64}\left(\right)\right)\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(b, b\right), \left(a \cdot \color{blue}{a}\right)\right)\right) \]

      14. *-lowering-*.f64 39.3%

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(angle, \mathsf{*.f64}\left(\frac{1}{90}, \mathsf{PI.f64}\left(\right)\right)\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(b, b\right), \mathsf{*.f64}\left(a, \color{blue}{a}\right)\right)\right) \]

   7. Simplified 39.3%

      \[\leadsto \color{blue}{\left(angle \cdot \left(0.011111111111111112 \cdot \pi\right)\right) \cdot \left(b \cdot b - a \cdot a\right)} \]

   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. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\frac{-1}{90}, \color{blue}{\left({a}^{2} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}\right) \]

      2. associate-*r* N/A

         \[\leadsto \mathsf{*.f64}\left(\frac{-1}{90}, \left(\left({a}^{2} \cdot angle\right) \cdot \color{blue}{\mathsf{PI}\left(\right)}\right)\right) \]

      3. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\frac{-1}{90}, \mathsf{*.f64}\left(\left({a}^{2} \cdot angle\right), \color{blue}{\mathsf{PI}\left(\right)}\right)\right) \]

      4. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\frac{-1}{90}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\left({a}^{2}\right), angle\right), \mathsf{PI}\left(\right)\right)\right) \]

      5. unpow2 N/A

         \[\leadsto \mathsf{*.f64}\left(\frac{-1}{90}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\left(a \cdot a\right), angle\right), \mathsf{PI}\left(\right)\right)\right) \]

      6. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\frac{-1}{90}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(a, a\right), angle\right), \mathsf{PI}\left(\right)\right)\right) \]

      7. PI-lowering-PI.f64 59.6%

         \[\leadsto \mathsf{*.f64}\left(\frac{-1}{90}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(a, a\right), angle\right), \mathsf{PI.f64}\left(\right)\right)\right) \]

   10. Simplified 59.6%

       \[\leadsto \color{blue}{-0.011111111111111112 \cdot \left(\left(\left(a \cdot a\right) \cdot angle\right) \cdot \pi\right)} \]
   11. Step-by-step derivation

       1. associate-*l* N/A

          \[\leadsto \mathsf{*.f64}\left(\frac{-1}{90}, \left(\left(a \cdot \left(a \cdot angle\right)\right) \cdot \mathsf{PI}\left(\right)\right)\right) \]

       2. associate-*l* N/A

          \[\leadsto \mathsf{*.f64}\left(\frac{-1}{90}, \left(a \cdot \color{blue}{\left(\left(a \cdot angle\right) \cdot \mathsf{PI}\left(\right)\right)}\right)\right) \]

       3. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\frac{-1}{90}, \mathsf{*.f64}\left(a, \color{blue}{\left(\left(a \cdot angle\right) \cdot \mathsf{PI}\left(\right)\right)}\right)\right) \]

       4. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\frac{-1}{90}, \mathsf{*.f64}\left(a, \mathsf{*.f64}\left(\left(a \cdot angle\right), \color{blue}{\mathsf{PI}\left(\right)}\right)\right)\right) \]

       5. *-commutative N/A

          \[\leadsto \mathsf{*.f64}\left(\frac{-1}{90}, \mathsf{*.f64}\left(a, \mathsf{*.f64}\left(\left(angle \cdot a\right), \mathsf{PI}\left(\right)\right)\right)\right) \]

       6. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\frac{-1}{90}, \mathsf{*.f64}\left(a, \mathsf{*.f64}\left(\mathsf{*.f64}\left(angle, a\right), \mathsf{PI}\left(\right)\right)\right)\right) \]

       7. PI-lowering-PI.f64 64.4%

          \[\leadsto \mathsf{*.f64}\left(\frac{-1}{90}, \mathsf{*.f64}\left(a, \mathsf{*.f64}\left(\mathsf{*.f64}\left(angle, a\right), \mathsf{PI.f64}\left(\right)\right)\right)\right) \]

   12. Applied egg-rr 64.4%

       \[\leadsto -0.011111111111111112 \cdot \color{blue}{\left(a \cdot \left(\left(angle \cdot a\right) \cdot \pi\right)\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 56.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \leq 8 \cdot 10^{+97}:\\ \;\;\;\;\left(angle \cdot \left(\pi \cdot 0.011111111111111112\right)\right) \cdot \left(b \cdot b - a \cdot a\right)\\ \mathbf{else}:\\ \;\;\;\;-0.011111111111111112 \cdot \left(a \cdot \left(\pi \cdot \left(angle \cdot a\right)\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 23: 44.3% accurate, 29.9× speedup

Math

\[\begin{array}{l} angle\_m = \left|angle\right| \\ angle\_s = \mathsf{copysign}\left(1, angle\right) \\ angle\_s \cdot \begin{array}{l} \mathbf{if}\;a \leq 5.5 \cdot 10^{+28}:\\ \;\;\;\;0.011111111111111112 \cdot \left(angle\_m \cdot \left(\pi \cdot \left(b \cdot b\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;-0.011111111111111112 \cdot \left(a \cdot \left(\pi \cdot \left(angle\_m \cdot a\right)\right)\right)\\ \end{array} \end{array} \]

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 (<= a 5.5e+28)
    (* 0.011111111111111112 (* angle_m (* PI (* b b))))
    (* -0.011111111111111112 (* a (* PI (* angle_m 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 tmp;
	if (a <= 5.5e+28) {
		tmp = 0.011111111111111112 * (angle_m * (((double) M_PI) * (b * b)));
	} else {
		tmp = -0.011111111111111112 * (a * (((double) M_PI) * (angle_m * 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 tmp;
	if (a <= 5.5e+28) {
		tmp = 0.011111111111111112 * (angle_m * (Math.PI * (b * b)));
	} else {
		tmp = -0.011111111111111112 * (a * (Math.PI * (angle_m * 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):
	tmp = 0
	if a <= 5.5e+28:
		tmp = 0.011111111111111112 * (angle_m * (math.pi * (b * b)))
	else:
		tmp = -0.011111111111111112 * (a * (math.pi * (angle_m * a)))
	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 <= 5.5e+28)
		tmp = Float64(0.011111111111111112 * Float64(angle_m * Float64(pi * Float64(b * b))));
	else
		tmp = Float64(-0.011111111111111112 * Float64(a * Float64(pi * Float64(angle_m * 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)
	tmp = 0.0;
	if (a <= 5.5e+28)
		tmp = 0.011111111111111112 * (angle_m * (pi * (b * b)));
	else
		tmp = -0.011111111111111112 * (a * (pi * (angle_m * 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_] := N[(angle$95$s * If[LessEqual[a, 5.5e+28], N[(0.011111111111111112 * N[(angle$95$m * N[(Pi * N[(b * b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(-0.011111111111111112 * N[(a * N[(Pi * N[(angle$95$m * a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if a < 5.5000000000000003e28

   1. Initial program 56.5%

      \[\left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin \left(\pi \cdot \frac{angle}{180}\right)\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right) \]
   2. Step-by-step derivation

      1. associate-*l* 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. associate-*l* N/A

         \[\leadsto 2 \cdot \color{blue}{\left(\left({b}^{2} - {a}^{2}\right) \cdot \left(\sin \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right) \cdot \cos \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right)\right)\right)} \]

      3. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(2, \color{blue}{\left(\left({b}^{2} - {a}^{2}\right) \cdot \left(\sin \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right) \cdot \cos \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right)\right)\right)}\right) \]

      4. associate-*r* N/A

         \[\leadsto \mathsf{*.f64}\left(2, \left(\left(\left({b}^{2} - {a}^{2}\right) \cdot \sin \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right)\right) \cdot \color{blue}{\cos \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right)}\right)\right) \]

      5. *-commutative N/A

         \[\leadsto \mathsf{*.f64}\left(2, \left(\left(\sin \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right) \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \cos \color{blue}{\left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right)}\right)\right) \]

      6. associate-*l* N/A

         \[\leadsto \mathsf{*.f64}\left(2, \left(\sin \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right) \cdot \color{blue}{\left(\left({b}^{2} - {a}^{2}\right) \cdot \cos \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right)\right)}\right)\right) \]

      7. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(2, \mathsf{*.f64}\left(\sin \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right), \color{blue}{\left(\left({b}^{2} - {a}^{2}\right) \cdot \cos \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right)\right)}\right)\right) \]

      8. sin-lowering-sin.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(2, \mathsf{*.f64}\left(\mathsf{sin.f64}\left(\left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right)\right), \left(\color{blue}{\left({b}^{2} - {a}^{2}\right)} \cdot \cos \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right)\right)\right)\right) \]

      9. associate-*r/ N/A

         \[\leadsto \mathsf{*.f64}\left(2, \mathsf{*.f64}\left(\mathsf{sin.f64}\left(\left(\frac{\mathsf{PI}\left(\right) \cdot angle}{180}\right)\right), \left(\left(\color{blue}{{b}^{2}} - {a}^{2}\right) \cdot \cos \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right)\right)\right)\right) \]

      10. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(2, \mathsf{*.f64}\left(\mathsf{sin.f64}\left(\mathsf{/.f64}\left(\left(\mathsf{PI}\left(\right) \cdot angle\right), 180\right)\right), \left(\left(\color{blue}{{b}^{2}} - {a}^{2}\right) \cdot \cos \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right)\right)\right)\right) \]

      11. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(2, \mathsf{*.f64}\left(\mathsf{sin.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{PI}\left(\right), angle\right), 180\right)\right), \left(\left({\color{blue}{b}}^{2} - {a}^{2}\right) \cdot \cos \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right)\right)\right)\right) \]

      12. PI-lowering-PI.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(2, \mathsf{*.f64}\left(\mathsf{sin.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{PI.f64}\left(\right), angle\right), 180\right)\right), \left(\left({b}^{2} - {a}^{2}\right) \cdot \cos \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right)\right)\right)\right) \]

   3. Simplified 57.4%

      \[\leadsto \color{blue}{2 \cdot \left(\sin \left(\frac{\pi \cdot angle}{180}\right) \cdot \left(\left(b \cdot b - a \cdot a\right) \cdot \cos \left(\frac{\pi \cdot angle}{180}\right)\right)\right)} \]
   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. *-commutative N/A

         \[\leadsto \left(angle \cdot \left(\mathsf{PI}\left(\right) \cdot \left({b}^{2} - {a}^{2}\right)\right)\right) \cdot \color{blue}{\frac{1}{90}} \]

      2. associate-*r* N/A

         \[\leadsto angle \cdot \color{blue}{\left(\left(\mathsf{PI}\left(\right) \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \frac{1}{90}\right)} \]

      3. *-commutative N/A

         \[\leadsto angle \cdot \left(\frac{1}{90} \cdot \color{blue}{\left(\mathsf{PI}\left(\right) \cdot \left({b}^{2} - {a}^{2}\right)\right)}\right) \]

      4. associate-*r* N/A

         \[\leadsto angle \cdot \left(\left(\frac{1}{90} \cdot \mathsf{PI}\left(\right)\right) \cdot \color{blue}{\left({b}^{2} - {a}^{2}\right)}\right) \]

      5. associate-*r* N/A

         \[\leadsto \left(angle \cdot \left(\frac{1}{90} \cdot \mathsf{PI}\left(\right)\right)\right) \cdot \color{blue}{\left({b}^{2} - {a}^{2}\right)} \]

      6. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\left(angle \cdot \left(\frac{1}{90} \cdot \mathsf{PI}\left(\right)\right)\right), \color{blue}{\left({b}^{2} - {a}^{2}\right)}\right) \]

      7. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(angle, \left(\frac{1}{90} \cdot \mathsf{PI}\left(\right)\right)\right), \left(\color{blue}{{b}^{2}} - {a}^{2}\right)\right) \]

      8. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(angle, \mathsf{*.f64}\left(\frac{1}{90}, \mathsf{PI}\left(\right)\right)\right), \left({b}^{\color{blue}{2}} - {a}^{2}\right)\right) \]

      9. PI-lowering-PI.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(angle, \mathsf{*.f64}\left(\frac{1}{90}, \mathsf{PI.f64}\left(\right)\right)\right), \left({b}^{2} - {a}^{2}\right)\right) \]

      10. --lowering--.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(angle, \mathsf{*.f64}\left(\frac{1}{90}, \mathsf{PI.f64}\left(\right)\right)\right), \mathsf{\_.f64}\left(\left({b}^{2}\right), \color{blue}{\left({a}^{2}\right)}\right)\right) \]

      11. unpow2 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(angle, \mathsf{*.f64}\left(\frac{1}{90}, \mathsf{PI.f64}\left(\right)\right)\right), \mathsf{\_.f64}\left(\left(b \cdot b\right), \left({\color{blue}{a}}^{2}\right)\right)\right) \]

      12. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(angle, \mathsf{*.f64}\left(\frac{1}{90}, \mathsf{PI.f64}\left(\right)\right)\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(b, b\right), \left({\color{blue}{a}}^{2}\right)\right)\right) \]

      13. unpow2 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(angle, \mathsf{*.f64}\left(\frac{1}{90}, \mathsf{PI.f64}\left(\right)\right)\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(b, b\right), \left(a \cdot \color{blue}{a}\right)\right)\right) \]

      14. *-lowering-*.f64 55.8%

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(angle, \mathsf{*.f64}\left(\frac{1}{90}, \mathsf{PI.f64}\left(\right)\right)\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(b, b\right), \mathsf{*.f64}\left(a, \color{blue}{a}\right)\right)\right) \]

   7. Simplified 55.8%

      \[\leadsto \color{blue}{\left(angle \cdot \left(0.011111111111111112 \cdot \pi\right)\right) \cdot \left(b \cdot b - a \cdot a\right)} \]

   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. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\frac{1}{90}, \color{blue}{\left(angle \cdot \left({b}^{2} \cdot \mathsf{PI}\left(\right)\right)\right)}\right) \]

      2. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\frac{1}{90}, \mathsf{*.f64}\left(angle, \color{blue}{\left({b}^{2} \cdot \mathsf{PI}\left(\right)\right)}\right)\right) \]

      3. *-commutative N/A

         \[\leadsto \mathsf{*.f64}\left(\frac{1}{90}, \mathsf{*.f64}\left(angle, \left(\mathsf{PI}\left(\right) \cdot \color{blue}{{b}^{2}}\right)\right)\right) \]

      4. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\frac{1}{90}, \mathsf{*.f64}\left(angle, \mathsf{*.f64}\left(\mathsf{PI}\left(\right), \color{blue}{\left({b}^{2}\right)}\right)\right)\right) \]

      5. PI-lowering-PI.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\frac{1}{90}, \mathsf{*.f64}\left(angle, \mathsf{*.f64}\left(\mathsf{PI.f64}\left(\right), \left({\color{blue}{b}}^{2}\right)\right)\right)\right) \]

      6. unpow2 N/A

         \[\leadsto \mathsf{*.f64}\left(\frac{1}{90}, \mathsf{*.f64}\left(angle, \mathsf{*.f64}\left(\mathsf{PI.f64}\left(\right), \left(b \cdot \color{blue}{b}\right)\right)\right)\right) \]

      7. *-lowering-*.f64 47.9%

         \[\leadsto \mathsf{*.f64}\left(\frac{1}{90}, \mathsf{*.f64}\left(angle, \mathsf{*.f64}\left(\mathsf{PI.f64}\left(\right), \mathsf{*.f64}\left(b, \color{blue}{b}\right)\right)\right)\right) \]

   10. Simplified 47.9%

       \[\leadsto \color{blue}{0.011111111111111112 \cdot \left(angle \cdot \left(\pi \cdot \left(b \cdot b\right)\right)\right)} \]

   if 5.5000000000000003e28 < a

   1. Initial program 46.9%

      \[\left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin \left(\pi \cdot \frac{angle}{180}\right)\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right) \]
   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. associate-*l* N/A

         \[\leadsto 2 \cdot \color{blue}{\left(\left({b}^{2} - {a}^{2}\right) \cdot \left(\sin \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right) \cdot \cos \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right)\right)\right)} \]

      3. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(2, \color{blue}{\left(\left({b}^{2} - {a}^{2}\right) \cdot \left(\sin \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right) \cdot \cos \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right)\right)\right)}\right) \]

      4. associate-*r* N/A

         \[\leadsto \mathsf{*.f64}\left(2, \left(\left(\left({b}^{2} - {a}^{2}\right) \cdot \sin \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right)\right) \cdot \color{blue}{\cos \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right)}\right)\right) \]

      5. *-commutative N/A

         \[\leadsto \mathsf{*.f64}\left(2, \left(\left(\sin \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right) \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \cos \color{blue}{\left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right)}\right)\right) \]

      6. associate-*l* N/A

         \[\leadsto \mathsf{*.f64}\left(2, \left(\sin \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right) \cdot \color{blue}{\left(\left({b}^{2} - {a}^{2}\right) \cdot \cos \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right)\right)}\right)\right) \]

      7. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(2, \mathsf{*.f64}\left(\sin \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right), \color{blue}{\left(\left({b}^{2} - {a}^{2}\right) \cdot \cos \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right)\right)}\right)\right) \]

      8. sin-lowering-sin.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(2, \mathsf{*.f64}\left(\mathsf{sin.f64}\left(\left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right)\right), \left(\color{blue}{\left({b}^{2} - {a}^{2}\right)} \cdot \cos \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right)\right)\right)\right) \]

      9. associate-*r/ N/A

         \[\leadsto \mathsf{*.f64}\left(2, \mathsf{*.f64}\left(\mathsf{sin.f64}\left(\left(\frac{\mathsf{PI}\left(\right) \cdot angle}{180}\right)\right), \left(\left(\color{blue}{{b}^{2}} - {a}^{2}\right) \cdot \cos \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right)\right)\right)\right) \]

      10. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(2, \mathsf{*.f64}\left(\mathsf{sin.f64}\left(\mathsf{/.f64}\left(\left(\mathsf{PI}\left(\right) \cdot angle\right), 180\right)\right), \left(\left(\color{blue}{{b}^{2}} - {a}^{2}\right) \cdot \cos \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right)\right)\right)\right) \]

      11. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(2, \mathsf{*.f64}\left(\mathsf{sin.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{PI}\left(\right), angle\right), 180\right)\right), \left(\left({\color{blue}{b}}^{2} - {a}^{2}\right) \cdot \cos \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right)\right)\right)\right) \]

      12. PI-lowering-PI.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(2, \mathsf{*.f64}\left(\mathsf{sin.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{PI.f64}\left(\right), angle\right), 180\right)\right), \left(\left({b}^{2} - {a}^{2}\right) \cdot \cos \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right)\right)\right)\right) \]

   3. Simplified 49.0%

      \[\leadsto \color{blue}{2 \cdot \left(\sin \left(\frac{\pi \cdot angle}{180}\right) \cdot \left(\left(b \cdot b - a \cdot a\right) \cdot \cos \left(\frac{\pi \cdot angle}{180}\right)\right)\right)} \]
   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. *-commutative N/A

         \[\leadsto \left(angle \cdot \left(\mathsf{PI}\left(\right) \cdot \left({b}^{2} - {a}^{2}\right)\right)\right) \cdot \color{blue}{\frac{1}{90}} \]

      2. associate-*r* N/A

         \[\leadsto angle \cdot \color{blue}{\left(\left(\mathsf{PI}\left(\right) \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \frac{1}{90}\right)} \]

      3. *-commutative N/A

         \[\leadsto angle \cdot \left(\frac{1}{90} \cdot \color{blue}{\left(\mathsf{PI}\left(\right) \cdot \left({b}^{2} - {a}^{2}\right)\right)}\right) \]

      4. associate-*r* N/A

         \[\leadsto angle \cdot \left(\left(\frac{1}{90} \cdot \mathsf{PI}\left(\right)\right) \cdot \color{blue}{\left({b}^{2} - {a}^{2}\right)}\right) \]

      5. associate-*r* N/A

         \[\leadsto \left(angle \cdot \left(\frac{1}{90} \cdot \mathsf{PI}\left(\right)\right)\right) \cdot \color{blue}{\left({b}^{2} - {a}^{2}\right)} \]

      6. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\left(angle \cdot \left(\frac{1}{90} \cdot \mathsf{PI}\left(\right)\right)\right), \color{blue}{\left({b}^{2} - {a}^{2}\right)}\right) \]

      7. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(angle, \left(\frac{1}{90} \cdot \mathsf{PI}\left(\right)\right)\right), \left(\color{blue}{{b}^{2}} - {a}^{2}\right)\right) \]

      8. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(angle, \mathsf{*.f64}\left(\frac{1}{90}, \mathsf{PI}\left(\right)\right)\right), \left({b}^{\color{blue}{2}} - {a}^{2}\right)\right) \]

      9. PI-lowering-PI.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(angle, \mathsf{*.f64}\left(\frac{1}{90}, \mathsf{PI.f64}\left(\right)\right)\right), \left({b}^{2} - {a}^{2}\right)\right) \]

      10. --lowering--.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(angle, \mathsf{*.f64}\left(\frac{1}{90}, \mathsf{PI.f64}\left(\right)\right)\right), \mathsf{\_.f64}\left(\left({b}^{2}\right), \color{blue}{\left({a}^{2}\right)}\right)\right) \]

      11. unpow2 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(angle, \mathsf{*.f64}\left(\frac{1}{90}, \mathsf{PI.f64}\left(\right)\right)\right), \mathsf{\_.f64}\left(\left(b \cdot b\right), \left({\color{blue}{a}}^{2}\right)\right)\right) \]

      12. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(angle, \mathsf{*.f64}\left(\frac{1}{90}, \mathsf{PI.f64}\left(\right)\right)\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(b, b\right), \left({\color{blue}{a}}^{2}\right)\right)\right) \]

      13. unpow2 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(angle, \mathsf{*.f64}\left(\frac{1}{90}, \mathsf{PI.f64}\left(\right)\right)\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(b, b\right), \left(a \cdot \color{blue}{a}\right)\right)\right) \]

      14. *-lowering-*.f64 42.0%

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(angle, \mathsf{*.f64}\left(\frac{1}{90}, \mathsf{PI.f64}\left(\right)\right)\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(b, b\right), \mathsf{*.f64}\left(a, \color{blue}{a}\right)\right)\right) \]

   7. Simplified 42.0%

      \[\leadsto \color{blue}{\left(angle \cdot \left(0.011111111111111112 \cdot \pi\right)\right) \cdot \left(b \cdot b - a \cdot a\right)} \]

   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. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\frac{-1}{90}, \color{blue}{\left({a}^{2} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}\right) \]

      2. associate-*r* N/A

         \[\leadsto \mathsf{*.f64}\left(\frac{-1}{90}, \left(\left({a}^{2} \cdot angle\right) \cdot \color{blue}{\mathsf{PI}\left(\right)}\right)\right) \]

      3. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\frac{-1}{90}, \mathsf{*.f64}\left(\left({a}^{2} \cdot angle\right), \color{blue}{\mathsf{PI}\left(\right)}\right)\right) \]

      4. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\frac{-1}{90}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\left({a}^{2}\right), angle\right), \mathsf{PI}\left(\right)\right)\right) \]

      5. unpow2 N/A

         \[\leadsto \mathsf{*.f64}\left(\frac{-1}{90}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\left(a \cdot a\right), angle\right), \mathsf{PI}\left(\right)\right)\right) \]

      6. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\frac{-1}{90}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(a, a\right), angle\right), \mathsf{PI}\left(\right)\right)\right) \]

      7. PI-lowering-PI.f64 57.6%

         \[\leadsto \mathsf{*.f64}\left(\frac{-1}{90}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(a, a\right), angle\right), \mathsf{PI.f64}\left(\right)\right)\right) \]

   10. Simplified 57.6%

       \[\leadsto \color{blue}{-0.011111111111111112 \cdot \left(\left(\left(a \cdot a\right) \cdot angle\right) \cdot \pi\right)} \]
   11. Step-by-step derivation

       1. associate-*l* N/A

          \[\leadsto \mathsf{*.f64}\left(\frac{-1}{90}, \left(\left(a \cdot \left(a \cdot angle\right)\right) \cdot \mathsf{PI}\left(\right)\right)\right) \]

       2. associate-*l* N/A

          \[\leadsto \mathsf{*.f64}\left(\frac{-1}{90}, \left(a \cdot \color{blue}{\left(\left(a \cdot angle\right) \cdot \mathsf{PI}\left(\right)\right)}\right)\right) \]

       3. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\frac{-1}{90}, \mathsf{*.f64}\left(a, \color{blue}{\left(\left(a \cdot angle\right) \cdot \mathsf{PI}\left(\right)\right)}\right)\right) \]

       4. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\frac{-1}{90}, \mathsf{*.f64}\left(a, \mathsf{*.f64}\left(\left(a \cdot angle\right), \color{blue}{\mathsf{PI}\left(\right)}\right)\right)\right) \]

       5. *-commutative N/A

          \[\leadsto \mathsf{*.f64}\left(\frac{-1}{90}, \mathsf{*.f64}\left(a, \mathsf{*.f64}\left(\left(angle \cdot a\right), \mathsf{PI}\left(\right)\right)\right)\right) \]

       6. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\frac{-1}{90}, \mathsf{*.f64}\left(a, \mathsf{*.f64}\left(\mathsf{*.f64}\left(angle, a\right), \mathsf{PI}\left(\right)\right)\right)\right) \]

       7. PI-lowering-PI.f64 61.3%

          \[\leadsto \mathsf{*.f64}\left(\frac{-1}{90}, \mathsf{*.f64}\left(a, \mathsf{*.f64}\left(\mathsf{*.f64}\left(angle, a\right), \mathsf{PI.f64}\left(\right)\right)\right)\right) \]

   12. Applied egg-rr 61.3%

       \[\leadsto -0.011111111111111112 \cdot \color{blue}{\left(a \cdot \left(\left(angle \cdot a\right) \cdot \pi\right)\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 50.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \leq 5.5 \cdot 10^{+28}:\\ \;\;\;\;0.011111111111111112 \cdot \left(angle \cdot \left(\pi \cdot \left(b \cdot b\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;-0.011111111111111112 \cdot \left(a \cdot \left(\pi \cdot \left(angle \cdot a\right)\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 24: 40.4% accurate, 29.9× speedup

Math

\[\begin{array}{l} angle\_m = \left|angle\right| \\ angle\_s = \mathsf{copysign}\left(1, angle\right) \\ angle\_s \cdot \begin{array}{l} \mathbf{if}\;angle\_m \leq 10^{-87}:\\ \;\;\;\;-0.011111111111111112 \cdot \left(a \cdot \left(\pi \cdot \left(angle\_m \cdot a\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;-0.011111111111111112 \cdot \left(\pi \cdot \left(angle\_m \cdot \left(a \cdot a\right)\right)\right)\\ \end{array} \end{array} \]

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 1e-87)
    (* -0.011111111111111112 (* a (* PI (* angle_m a))))
    (* -0.011111111111111112 (* PI (* angle_m (* a 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 tmp;
	if (angle_m <= 1e-87) {
		tmp = -0.011111111111111112 * (a * (((double) M_PI) * (angle_m * a)));
	} else {
		tmp = -0.011111111111111112 * (((double) M_PI) * (angle_m * (a * 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 tmp;
	if (angle_m <= 1e-87) {
		tmp = -0.011111111111111112 * (a * (Math.PI * (angle_m * a)));
	} else {
		tmp = -0.011111111111111112 * (Math.PI * (angle_m * (a * 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):
	tmp = 0
	if angle_m <= 1e-87:
		tmp = -0.011111111111111112 * (a * (math.pi * (angle_m * a)))
	else:
		tmp = -0.011111111111111112 * (math.pi * (angle_m * (a * a)))
	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 <= 1e-87)
		tmp = Float64(-0.011111111111111112 * Float64(a * Float64(pi * Float64(angle_m * a))));
	else
		tmp = Float64(-0.011111111111111112 * Float64(pi * Float64(angle_m * Float64(a * 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)
	tmp = 0.0;
	if (angle_m <= 1e-87)
		tmp = -0.011111111111111112 * (a * (pi * (angle_m * a)));
	else
		tmp = -0.011111111111111112 * (pi * (angle_m * (a * 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_] := N[(angle$95$s * If[LessEqual[angle$95$m, 1e-87], N[(-0.011111111111111112 * N[(a * N[(Pi * N[(angle$95$m * a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(-0.011111111111111112 * N[(Pi * N[(angle$95$m * N[(a * a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if angle < 1.00000000000000002e-87

   1. Initial program 58.7%

      \[\left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin \left(\pi \cdot \frac{angle}{180}\right)\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right) \]
   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. associate-*l* N/A

         \[\leadsto 2 \cdot \color{blue}{\left(\left({b}^{2} - {a}^{2}\right) \cdot \left(\sin \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right) \cdot \cos \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right)\right)\right)} \]

      3. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(2, \color{blue}{\left(\left({b}^{2} - {a}^{2}\right) \cdot \left(\sin \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right) \cdot \cos \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right)\right)\right)}\right) \]

      4. associate-*r* N/A

         \[\leadsto \mathsf{*.f64}\left(2, \left(\left(\left({b}^{2} - {a}^{2}\right) \cdot \sin \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right)\right) \cdot \color{blue}{\cos \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right)}\right)\right) \]

      5. *-commutative N/A

         \[\leadsto \mathsf{*.f64}\left(2, \left(\left(\sin \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right) \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \cos \color{blue}{\left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right)}\right)\right) \]

      6. associate-*l* N/A

         \[\leadsto \mathsf{*.f64}\left(2, \left(\sin \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right) \cdot \color{blue}{\left(\left({b}^{2} - {a}^{2}\right) \cdot \cos \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right)\right)}\right)\right) \]

      7. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(2, \mathsf{*.f64}\left(\sin \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right), \color{blue}{\left(\left({b}^{2} - {a}^{2}\right) \cdot \cos \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right)\right)}\right)\right) \]

      8. sin-lowering-sin.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(2, \mathsf{*.f64}\left(\mathsf{sin.f64}\left(\left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right)\right), \left(\color{blue}{\left({b}^{2} - {a}^{2}\right)} \cdot \cos \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right)\right)\right)\right) \]

      9. associate-*r/ N/A

         \[\leadsto \mathsf{*.f64}\left(2, \mathsf{*.f64}\left(\mathsf{sin.f64}\left(\left(\frac{\mathsf{PI}\left(\right) \cdot angle}{180}\right)\right), \left(\left(\color{blue}{{b}^{2}} - {a}^{2}\right) \cdot \cos \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right)\right)\right)\right) \]

      10. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(2, \mathsf{*.f64}\left(\mathsf{sin.f64}\left(\mathsf{/.f64}\left(\left(\mathsf{PI}\left(\right) \cdot angle\right), 180\right)\right), \left(\left(\color{blue}{{b}^{2}} - {a}^{2}\right) \cdot \cos \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right)\right)\right)\right) \]

      11. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(2, \mathsf{*.f64}\left(\mathsf{sin.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{PI}\left(\right), angle\right), 180\right)\right), \left(\left({\color{blue}{b}}^{2} - {a}^{2}\right) \cdot \cos \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right)\right)\right)\right) \]

      12. PI-lowering-PI.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(2, \mathsf{*.f64}\left(\mathsf{sin.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{PI.f64}\left(\right), angle\right), 180\right)\right), \left(\left({b}^{2} - {a}^{2}\right) \cdot \cos \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right)\right)\right)\right) \]

   3. Simplified 59.6%

      \[\leadsto \color{blue}{2 \cdot \left(\sin \left(\frac{\pi \cdot angle}{180}\right) \cdot \left(\left(b \cdot b - a \cdot a\right) \cdot \cos \left(\frac{\pi \cdot angle}{180}\right)\right)\right)} \]
   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. *-commutative N/A

         \[\leadsto \left(angle \cdot \left(\mathsf{PI}\left(\right) \cdot \left({b}^{2} - {a}^{2}\right)\right)\right) \cdot \color{blue}{\frac{1}{90}} \]

      2. associate-*r* N/A

         \[\leadsto angle \cdot \color{blue}{\left(\left(\mathsf{PI}\left(\right) \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \frac{1}{90}\right)} \]

      3. *-commutative N/A

         \[\leadsto angle \cdot \left(\frac{1}{90} \cdot \color{blue}{\left(\mathsf{PI}\left(\right) \cdot \left({b}^{2} - {a}^{2}\right)\right)}\right) \]

      4. associate-*r* N/A

         \[\leadsto angle \cdot \left(\left(\frac{1}{90} \cdot \mathsf{PI}\left(\right)\right) \cdot \color{blue}{\left({b}^{2} - {a}^{2}\right)}\right) \]

      5. associate-*r* N/A

         \[\leadsto \left(angle \cdot \left(\frac{1}{90} \cdot \mathsf{PI}\left(\right)\right)\right) \cdot \color{blue}{\left({b}^{2} - {a}^{2}\right)} \]

      6. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\left(angle \cdot \left(\frac{1}{90} \cdot \mathsf{PI}\left(\right)\right)\right), \color{blue}{\left({b}^{2} - {a}^{2}\right)}\right) \]

      7. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(angle, \left(\frac{1}{90} \cdot \mathsf{PI}\left(\right)\right)\right), \left(\color{blue}{{b}^{2}} - {a}^{2}\right)\right) \]

      8. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(angle, \mathsf{*.f64}\left(\frac{1}{90}, \mathsf{PI}\left(\right)\right)\right), \left({b}^{\color{blue}{2}} - {a}^{2}\right)\right) \]

      9. PI-lowering-PI.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(angle, \mathsf{*.f64}\left(\frac{1}{90}, \mathsf{PI.f64}\left(\right)\right)\right), \left({b}^{2} - {a}^{2}\right)\right) \]

      10. --lowering--.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(angle, \mathsf{*.f64}\left(\frac{1}{90}, \mathsf{PI.f64}\left(\right)\right)\right), \mathsf{\_.f64}\left(\left({b}^{2}\right), \color{blue}{\left({a}^{2}\right)}\right)\right) \]

      11. unpow2 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(angle, \mathsf{*.f64}\left(\frac{1}{90}, \mathsf{PI.f64}\left(\right)\right)\right), \mathsf{\_.f64}\left(\left(b \cdot b\right), \left({\color{blue}{a}}^{2}\right)\right)\right) \]

      12. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(angle, \mathsf{*.f64}\left(\frac{1}{90}, \mathsf{PI.f64}\left(\right)\right)\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(b, b\right), \left({\color{blue}{a}}^{2}\right)\right)\right) \]

      13. unpow2 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(angle, \mathsf{*.f64}\left(\frac{1}{90}, \mathsf{PI.f64}\left(\right)\right)\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(b, b\right), \left(a \cdot \color{blue}{a}\right)\right)\right) \]

      14. *-lowering-*.f64 57.4%

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(angle, \mathsf{*.f64}\left(\frac{1}{90}, \mathsf{PI.f64}\left(\right)\right)\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(b, b\right), \mathsf{*.f64}\left(a, \color{blue}{a}\right)\right)\right) \]

   7. Simplified 57.4%

      \[\leadsto \color{blue}{\left(angle \cdot \left(0.011111111111111112 \cdot \pi\right)\right) \cdot \left(b \cdot b - a \cdot a\right)} \]

   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. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\frac{-1}{90}, \color{blue}{\left({a}^{2} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}\right) \]

      2. associate-*r* N/A

         \[\leadsto \mathsf{*.f64}\left(\frac{-1}{90}, \left(\left({a}^{2} \cdot angle\right) \cdot \color{blue}{\mathsf{PI}\left(\right)}\right)\right) \]

      3. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\frac{-1}{90}, \mathsf{*.f64}\left(\left({a}^{2} \cdot angle\right), \color{blue}{\mathsf{PI}\left(\right)}\right)\right) \]

      4. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\frac{-1}{90}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\left({a}^{2}\right), angle\right), \mathsf{PI}\left(\right)\right)\right) \]

      5. unpow2 N/A

         \[\leadsto \mathsf{*.f64}\left(\frac{-1}{90}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\left(a \cdot a\right), angle\right), \mathsf{PI}\left(\right)\right)\right) \]

      6. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\frac{-1}{90}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(a, a\right), angle\right), \mathsf{PI}\left(\right)\right)\right) \]

      7. PI-lowering-PI.f64 36.8%

         \[\leadsto \mathsf{*.f64}\left(\frac{-1}{90}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(a, a\right), angle\right), \mathsf{PI.f64}\left(\right)\right)\right) \]

   10. Simplified 36.8%

       \[\leadsto \color{blue}{-0.011111111111111112 \cdot \left(\left(\left(a \cdot a\right) \cdot angle\right) \cdot \pi\right)} \]
   11. Step-by-step derivation

       1. associate-*l* N/A

          \[\leadsto \mathsf{*.f64}\left(\frac{-1}{90}, \left(\left(a \cdot \left(a \cdot angle\right)\right) \cdot \mathsf{PI}\left(\right)\right)\right) \]

       2. associate-*l* N/A

          \[\leadsto \mathsf{*.f64}\left(\frac{-1}{90}, \left(a \cdot \color{blue}{\left(\left(a \cdot angle\right) \cdot \mathsf{PI}\left(\right)\right)}\right)\right) \]

       3. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\frac{-1}{90}, \mathsf{*.f64}\left(a, \color{blue}{\left(\left(a \cdot angle\right) \cdot \mathsf{PI}\left(\right)\right)}\right)\right) \]

       4. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\frac{-1}{90}, \mathsf{*.f64}\left(a, \mathsf{*.f64}\left(\left(a \cdot angle\right), \color{blue}{\mathsf{PI}\left(\right)}\right)\right)\right) \]

       5. *-commutative N/A

          \[\leadsto \mathsf{*.f64}\left(\frac{-1}{90}, \mathsf{*.f64}\left(a, \mathsf{*.f64}\left(\left(angle \cdot a\right), \mathsf{PI}\left(\right)\right)\right)\right) \]

       6. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\frac{-1}{90}, \mathsf{*.f64}\left(a, \mathsf{*.f64}\left(\mathsf{*.f64}\left(angle, a\right), \mathsf{PI}\left(\right)\right)\right)\right) \]

       7. PI-lowering-PI.f64 37.8%

          \[\leadsto \mathsf{*.f64}\left(\frac{-1}{90}, \mathsf{*.f64}\left(a, \mathsf{*.f64}\left(\mathsf{*.f64}\left(angle, a\right), \mathsf{PI.f64}\left(\right)\right)\right)\right) \]

   12. Applied egg-rr 37.8%

       \[\leadsto -0.011111111111111112 \cdot \color{blue}{\left(a \cdot \left(\left(angle \cdot a\right) \cdot \pi\right)\right)} \]

   if 1.00000000000000002e-87 < angle

   1. Initial program 46.5%

      \[\left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin \left(\pi \cdot \frac{angle}{180}\right)\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right) \]
   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. associate-*l* N/A

         \[\leadsto 2 \cdot \color{blue}{\left(\left({b}^{2} - {a}^{2}\right) \cdot \left(\sin \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right) \cdot \cos \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right)\right)\right)} \]

      3. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(2, \color{blue}{\left(\left({b}^{2} - {a}^{2}\right) \cdot \left(\sin \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right) \cdot \cos \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right)\right)\right)}\right) \]

      4. associate-*r* N/A

         \[\leadsto \mathsf{*.f64}\left(2, \left(\left(\left({b}^{2} - {a}^{2}\right) \cdot \sin \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right)\right) \cdot \color{blue}{\cos \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right)}\right)\right) \]

      5. *-commutative N/A

         \[\leadsto \mathsf{*.f64}\left(2, \left(\left(\sin \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right) \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \cos \color{blue}{\left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right)}\right)\right) \]

      6. associate-*l* N/A

         \[\leadsto \mathsf{*.f64}\left(2, \left(\sin \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right) \cdot \color{blue}{\left(\left({b}^{2} - {a}^{2}\right) \cdot \cos \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right)\right)}\right)\right) \]

      7. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(2, \mathsf{*.f64}\left(\sin \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right), \color{blue}{\left(\left({b}^{2} - {a}^{2}\right) \cdot \cos \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right)\right)}\right)\right) \]

      8. sin-lowering-sin.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(2, \mathsf{*.f64}\left(\mathsf{sin.f64}\left(\left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right)\right), \left(\color{blue}{\left({b}^{2} - {a}^{2}\right)} \cdot \cos \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right)\right)\right)\right) \]

      9. associate-*r/ N/A

         \[\leadsto \mathsf{*.f64}\left(2, \mathsf{*.f64}\left(\mathsf{sin.f64}\left(\left(\frac{\mathsf{PI}\left(\right) \cdot angle}{180}\right)\right), \left(\left(\color{blue}{{b}^{2}} - {a}^{2}\right) \cdot \cos \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right)\right)\right)\right) \]

      10. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(2, \mathsf{*.f64}\left(\mathsf{sin.f64}\left(\mathsf{/.f64}\left(\left(\mathsf{PI}\left(\right) \cdot angle\right), 180\right)\right), \left(\left(\color{blue}{{b}^{2}} - {a}^{2}\right) \cdot \cos \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right)\right)\right)\right) \]

      11. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(2, \mathsf{*.f64}\left(\mathsf{sin.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{PI}\left(\right), angle\right), 180\right)\right), \left(\left({\color{blue}{b}}^{2} - {a}^{2}\right) \cdot \cos \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right)\right)\right)\right) \]

      12. PI-lowering-PI.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(2, \mathsf{*.f64}\left(\mathsf{sin.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{PI.f64}\left(\right), angle\right), 180\right)\right), \left(\left({b}^{2} - {a}^{2}\right) \cdot \cos \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right)\right)\right)\right) \]

   3. Simplified 47.9%

      \[\leadsto \color{blue}{2 \cdot \left(\sin \left(\frac{\pi \cdot angle}{180}\right) \cdot \left(\left(b \cdot b - a \cdot a\right) \cdot \cos \left(\frac{\pi \cdot angle}{180}\right)\right)\right)} \]
   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. *-commutative N/A

         \[\leadsto \left(angle \cdot \left(\mathsf{PI}\left(\right) \cdot \left({b}^{2} - {a}^{2}\right)\right)\right) \cdot \color{blue}{\frac{1}{90}} \]

      2. associate-*r* N/A

         \[\leadsto angle \cdot \color{blue}{\left(\left(\mathsf{PI}\left(\right) \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \frac{1}{90}\right)} \]

      3. *-commutative N/A

         \[\leadsto angle \cdot \left(\frac{1}{90} \cdot \color{blue}{\left(\mathsf{PI}\left(\right) \cdot \left({b}^{2} - {a}^{2}\right)\right)}\right) \]

      4. associate-*r* N/A

         \[\leadsto angle \cdot \left(\left(\frac{1}{90} \cdot \mathsf{PI}\left(\right)\right) \cdot \color{blue}{\left({b}^{2} - {a}^{2}\right)}\right) \]

      5. associate-*r* N/A

         \[\leadsto \left(angle \cdot \left(\frac{1}{90} \cdot \mathsf{PI}\left(\right)\right)\right) \cdot \color{blue}{\left({b}^{2} - {a}^{2}\right)} \]

      6. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\left(angle \cdot \left(\frac{1}{90} \cdot \mathsf{PI}\left(\right)\right)\right), \color{blue}{\left({b}^{2} - {a}^{2}\right)}\right) \]

      7. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(angle, \left(\frac{1}{90} \cdot \mathsf{PI}\left(\right)\right)\right), \left(\color{blue}{{b}^{2}} - {a}^{2}\right)\right) \]

      8. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(angle, \mathsf{*.f64}\left(\frac{1}{90}, \mathsf{PI}\left(\right)\right)\right), \left({b}^{\color{blue}{2}} - {a}^{2}\right)\right) \]

      9. PI-lowering-PI.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(angle, \mathsf{*.f64}\left(\frac{1}{90}, \mathsf{PI.f64}\left(\right)\right)\right), \left({b}^{2} - {a}^{2}\right)\right) \]

      10. --lowering--.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(angle, \mathsf{*.f64}\left(\frac{1}{90}, \mathsf{PI.f64}\left(\right)\right)\right), \mathsf{\_.f64}\left(\left({b}^{2}\right), \color{blue}{\left({a}^{2}\right)}\right)\right) \]

      11. unpow2 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(angle, \mathsf{*.f64}\left(\frac{1}{90}, \mathsf{PI.f64}\left(\right)\right)\right), \mathsf{\_.f64}\left(\left(b \cdot b\right), \left({\color{blue}{a}}^{2}\right)\right)\right) \]

      12. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(angle, \mathsf{*.f64}\left(\frac{1}{90}, \mathsf{PI.f64}\left(\right)\right)\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(b, b\right), \left({\color{blue}{a}}^{2}\right)\right)\right) \]

      13. unpow2 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(angle, \mathsf{*.f64}\left(\frac{1}{90}, \mathsf{PI.f64}\left(\right)\right)\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(b, b\right), \left(a \cdot \color{blue}{a}\right)\right)\right) \]

      14. *-lowering-*.f64 44.4%

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(angle, \mathsf{*.f64}\left(\frac{1}{90}, \mathsf{PI.f64}\left(\right)\right)\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(b, b\right), \mathsf{*.f64}\left(a, \color{blue}{a}\right)\right)\right) \]

   7. Simplified 44.4%

      \[\leadsto \color{blue}{\left(angle \cdot \left(0.011111111111111112 \cdot \pi\right)\right) \cdot \left(b \cdot b - a \cdot a\right)} \]

   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. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\frac{-1}{90}, \color{blue}{\left({a}^{2} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}\right) \]

      2. associate-*r* N/A

         \[\leadsto \mathsf{*.f64}\left(\frac{-1}{90}, \left(\left({a}^{2} \cdot angle\right) \cdot \color{blue}{\mathsf{PI}\left(\right)}\right)\right) \]

      3. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\frac{-1}{90}, \mathsf{*.f64}\left(\left({a}^{2} \cdot angle\right), \color{blue}{\mathsf{PI}\left(\right)}\right)\right) \]

      4. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\frac{-1}{90}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\left({a}^{2}\right), angle\right), \mathsf{PI}\left(\right)\right)\right) \]

      5. unpow2 N/A

         \[\leadsto \mathsf{*.f64}\left(\frac{-1}{90}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\left(a \cdot a\right), angle\right), \mathsf{PI}\left(\right)\right)\right) \]

      6. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\frac{-1}{90}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(a, a\right), angle\right), \mathsf{PI}\left(\right)\right)\right) \]

      7. PI-lowering-PI.f64 29.8%

         \[\leadsto \mathsf{*.f64}\left(\frac{-1}{90}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(a, a\right), angle\right), \mathsf{PI.f64}\left(\right)\right)\right) \]

   10. Simplified 29.8%

       \[\leadsto \color{blue}{-0.011111111111111112 \cdot \left(\left(\left(a \cdot a\right) \cdot angle\right) \cdot \pi\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 35.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;angle \leq 10^{-87}:\\ \;\;\;\;-0.011111111111111112 \cdot \left(a \cdot \left(\pi \cdot \left(angle \cdot a\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;-0.011111111111111112 \cdot \left(\pi \cdot \left(angle \cdot \left(a \cdot a\right)\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 25: 39.4% accurate, 46.6× speedup

Math

\[\begin{array}{l} angle\_m = \left|angle\right| \\ angle\_s = \mathsf{copysign}\left(1, angle\right) \\ angle\_s \cdot \left(-0.011111111111111112 \cdot \left(a \cdot \left(\pi \cdot \left(angle\_m \cdot a\right)\right)\right)\right) \end{array} \]

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 (* -0.011111111111111112 (* a (* PI (* angle_m a))))))

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 * (-0.011111111111111112 * (a * (((double) M_PI) * (angle_m * a))));
}

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 * (-0.011111111111111112 * (a * (Math.PI * (angle_m * a))));
}

Python

angle\_m = math.fabs(angle)
angle\_s = math.copysign(1.0, angle)
def code(angle_s, a, b, angle_m):
	return angle_s * (-0.011111111111111112 * (a * (math.pi * (angle_m * a))))

Julia

angle\_m = abs(angle)
angle\_s = copysign(1.0, angle)
function code(angle_s, a, b, angle_m)
	return Float64(angle_s * Float64(-0.011111111111111112 * Float64(a * Float64(pi * Float64(angle_m * a)))))
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 * (-0.011111111111111112 * (a * (pi * (angle_m * a))));
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[(-0.011111111111111112 * N[(a * N[(Pi * N[(angle$95$m * a), $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. associate-*l* N/A

      \[\leadsto 2 \cdot \color{blue}{\left(\left({b}^{2} - {a}^{2}\right) \cdot \left(\sin \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right) \cdot \cos \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right)\right)\right)} \]

   3. *-lowering-*.f64 N/A

      \[\leadsto \mathsf{*.f64}\left(2, \color{blue}{\left(\left({b}^{2} - {a}^{2}\right) \cdot \left(\sin \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right) \cdot \cos \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right)\right)\right)}\right) \]

   4. associate-*r* N/A

      \[\leadsto \mathsf{*.f64}\left(2, \left(\left(\left({b}^{2} - {a}^{2}\right) \cdot \sin \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right)\right) \cdot \color{blue}{\cos \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right)}\right)\right) \]

   5. *-commutative N/A

      \[\leadsto \mathsf{*.f64}\left(2, \left(\left(\sin \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right) \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \cos \color{blue}{\left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right)}\right)\right) \]

   6. associate-*l* N/A

      \[\leadsto \mathsf{*.f64}\left(2, \left(\sin \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right) \cdot \color{blue}{\left(\left({b}^{2} - {a}^{2}\right) \cdot \cos \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right)\right)}\right)\right) \]

   7. *-lowering-*.f64 N/A

      \[\leadsto \mathsf{*.f64}\left(2, \mathsf{*.f64}\left(\sin \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right), \color{blue}{\left(\left({b}^{2} - {a}^{2}\right) \cdot \cos \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right)\right)}\right)\right) \]

   8. sin-lowering-sin.f64 N/A

      \[\leadsto \mathsf{*.f64}\left(2, \mathsf{*.f64}\left(\mathsf{sin.f64}\left(\left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right)\right), \left(\color{blue}{\left({b}^{2} - {a}^{2}\right)} \cdot \cos \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right)\right)\right)\right) \]

   9. associate-*r/ N/A

      \[\leadsto \mathsf{*.f64}\left(2, \mathsf{*.f64}\left(\mathsf{sin.f64}\left(\left(\frac{\mathsf{PI}\left(\right) \cdot angle}{180}\right)\right), \left(\left(\color{blue}{{b}^{2}} - {a}^{2}\right) \cdot \cos \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right)\right)\right)\right) \]

   10. /-lowering-/.f64 N/A

       \[\leadsto \mathsf{*.f64}\left(2, \mathsf{*.f64}\left(\mathsf{sin.f64}\left(\mathsf{/.f64}\left(\left(\mathsf{PI}\left(\right) \cdot angle\right), 180\right)\right), \left(\left(\color{blue}{{b}^{2}} - {a}^{2}\right) \cdot \cos \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right)\right)\right)\right) \]

   11. *-lowering-*.f64 N/A

       \[\leadsto \mathsf{*.f64}\left(2, \mathsf{*.f64}\left(\mathsf{sin.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{PI}\left(\right), angle\right), 180\right)\right), \left(\left({\color{blue}{b}}^{2} - {a}^{2}\right) \cdot \cos \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right)\right)\right)\right) \]

   12. PI-lowering-PI.f64 N/A

       \[\leadsto \mathsf{*.f64}\left(2, \mathsf{*.f64}\left(\mathsf{sin.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{PI.f64}\left(\right), angle\right), 180\right)\right), \left(\left({b}^{2} - {a}^{2}\right) \cdot \cos \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right)\right)\right)\right) \]

3. Simplified 55.7%

   \[\leadsto \color{blue}{2 \cdot \left(\sin \left(\frac{\pi \cdot angle}{180}\right) \cdot \left(\left(b \cdot b - a \cdot a\right) \cdot \cos \left(\frac{\pi \cdot angle}{180}\right)\right)\right)} \]
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. *-commutative N/A

      \[\leadsto \left(angle \cdot \left(\mathsf{PI}\left(\right) \cdot \left({b}^{2} - {a}^{2}\right)\right)\right) \cdot \color{blue}{\frac{1}{90}} \]

   2. associate-*r* N/A

      \[\leadsto angle \cdot \color{blue}{\left(\left(\mathsf{PI}\left(\right) \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \frac{1}{90}\right)} \]

   3. *-commutative N/A

      \[\leadsto angle \cdot \left(\frac{1}{90} \cdot \color{blue}{\left(\mathsf{PI}\left(\right) \cdot \left({b}^{2} - {a}^{2}\right)\right)}\right) \]

   4. associate-*r* N/A

      \[\leadsto angle \cdot \left(\left(\frac{1}{90} \cdot \mathsf{PI}\left(\right)\right) \cdot \color{blue}{\left({b}^{2} - {a}^{2}\right)}\right) \]

   5. associate-*r* N/A

      \[\leadsto \left(angle \cdot \left(\frac{1}{90} \cdot \mathsf{PI}\left(\right)\right)\right) \cdot \color{blue}{\left({b}^{2} - {a}^{2}\right)} \]

   6. *-lowering-*.f64 N/A

      \[\leadsto \mathsf{*.f64}\left(\left(angle \cdot \left(\frac{1}{90} \cdot \mathsf{PI}\left(\right)\right)\right), \color{blue}{\left({b}^{2} - {a}^{2}\right)}\right) \]

   7. *-lowering-*.f64 N/A

      \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(angle, \left(\frac{1}{90} \cdot \mathsf{PI}\left(\right)\right)\right), \left(\color{blue}{{b}^{2}} - {a}^{2}\right)\right) \]

   8. *-lowering-*.f64 N/A

      \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(angle, \mathsf{*.f64}\left(\frac{1}{90}, \mathsf{PI}\left(\right)\right)\right), \left({b}^{\color{blue}{2}} - {a}^{2}\right)\right) \]

   9. PI-lowering-PI.f64 N/A

      \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(angle, \mathsf{*.f64}\left(\frac{1}{90}, \mathsf{PI.f64}\left(\right)\right)\right), \left({b}^{2} - {a}^{2}\right)\right) \]

   10. --lowering--.f64 N/A

       \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(angle, \mathsf{*.f64}\left(\frac{1}{90}, \mathsf{PI.f64}\left(\right)\right)\right), \mathsf{\_.f64}\left(\left({b}^{2}\right), \color{blue}{\left({a}^{2}\right)}\right)\right) \]

   11. unpow2 N/A

       \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(angle, \mathsf{*.f64}\left(\frac{1}{90}, \mathsf{PI.f64}\left(\right)\right)\right), \mathsf{\_.f64}\left(\left(b \cdot b\right), \left({\color{blue}{a}}^{2}\right)\right)\right) \]

   12. *-lowering-*.f64 N/A

       \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(angle, \mathsf{*.f64}\left(\frac{1}{90}, \mathsf{PI.f64}\left(\right)\right)\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(b, b\right), \left({\color{blue}{a}}^{2}\right)\right)\right) \]

   13. unpow2 N/A

       \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(angle, \mathsf{*.f64}\left(\frac{1}{90}, \mathsf{PI.f64}\left(\right)\right)\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(b, b\right), \left(a \cdot \color{blue}{a}\right)\right)\right) \]

   14. *-lowering-*.f64 53.0%

       \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(angle, \mathsf{*.f64}\left(\frac{1}{90}, \mathsf{PI.f64}\left(\right)\right)\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(b, b\right), \mathsf{*.f64}\left(a, \color{blue}{a}\right)\right)\right) \]

7. Simplified 53.0%

   \[\leadsto \color{blue}{\left(angle \cdot \left(0.011111111111111112 \cdot \pi\right)\right) \cdot \left(b \cdot b - a \cdot a\right)} \]

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. *-lowering-*.f64 N/A

      \[\leadsto \mathsf{*.f64}\left(\frac{-1}{90}, \color{blue}{\left({a}^{2} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}\right) \]

   2. associate-*r* N/A

      \[\leadsto \mathsf{*.f64}\left(\frac{-1}{90}, \left(\left({a}^{2} \cdot angle\right) \cdot \color{blue}{\mathsf{PI}\left(\right)}\right)\right) \]

   3. *-lowering-*.f64 N/A

      \[\leadsto \mathsf{*.f64}\left(\frac{-1}{90}, \mathsf{*.f64}\left(\left({a}^{2} \cdot angle\right), \color{blue}{\mathsf{PI}\left(\right)}\right)\right) \]

   4. *-lowering-*.f64 N/A

      \[\leadsto \mathsf{*.f64}\left(\frac{-1}{90}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\left({a}^{2}\right), angle\right), \mathsf{PI}\left(\right)\right)\right) \]

   5. unpow2 N/A

      \[\leadsto \mathsf{*.f64}\left(\frac{-1}{90}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\left(a \cdot a\right), angle\right), \mathsf{PI}\left(\right)\right)\right) \]

   6. *-lowering-*.f64 N/A

      \[\leadsto \mathsf{*.f64}\left(\frac{-1}{90}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(a, a\right), angle\right), \mathsf{PI}\left(\right)\right)\right) \]

   7. PI-lowering-PI.f64 34.5%

      \[\leadsto \mathsf{*.f64}\left(\frac{-1}{90}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(a, a\right), angle\right), \mathsf{PI.f64}\left(\right)\right)\right) \]

10. Simplified 34.5%

    \[\leadsto \color{blue}{-0.011111111111111112 \cdot \left(\left(\left(a \cdot a\right) \cdot angle\right) \cdot \pi\right)} \]
11. Step-by-step derivation

    1. associate-*l* N/A

       \[\leadsto \mathsf{*.f64}\left(\frac{-1}{90}, \left(\left(a \cdot \left(a \cdot angle\right)\right) \cdot \mathsf{PI}\left(\right)\right)\right) \]

    2. associate-*l* N/A

       \[\leadsto \mathsf{*.f64}\left(\frac{-1}{90}, \left(a \cdot \color{blue}{\left(\left(a \cdot angle\right) \cdot \mathsf{PI}\left(\right)\right)}\right)\right) \]

    3. *-lowering-*.f64 N/A

       \[\leadsto \mathsf{*.f64}\left(\frac{-1}{90}, \mathsf{*.f64}\left(a, \color{blue}{\left(\left(a \cdot angle\right) \cdot \mathsf{PI}\left(\right)\right)}\right)\right) \]

    4. *-lowering-*.f64 N/A

       \[\leadsto \mathsf{*.f64}\left(\frac{-1}{90}, \mathsf{*.f64}\left(a, \mathsf{*.f64}\left(\left(a \cdot angle\right), \color{blue}{\mathsf{PI}\left(\right)}\right)\right)\right) \]

    5. *-commutative N/A

       \[\leadsto \mathsf{*.f64}\left(\frac{-1}{90}, \mathsf{*.f64}\left(a, \mathsf{*.f64}\left(\left(angle \cdot a\right), \mathsf{PI}\left(\right)\right)\right)\right) \]

    6. *-lowering-*.f64 N/A

       \[\leadsto \mathsf{*.f64}\left(\frac{-1}{90}, \mathsf{*.f64}\left(a, \mathsf{*.f64}\left(\mathsf{*.f64}\left(angle, a\right), \mathsf{PI}\left(\right)\right)\right)\right) \]

    7. PI-lowering-PI.f64 32.8%

       \[\leadsto \mathsf{*.f64}\left(\frac{-1}{90}, \mathsf{*.f64}\left(a, \mathsf{*.f64}\left(\mathsf{*.f64}\left(angle, a\right), \mathsf{PI.f64}\left(\right)\right)\right)\right) \]

12. Applied egg-rr 32.8%

    \[\leadsto -0.011111111111111112 \cdot \color{blue}{\left(a \cdot \left(\left(angle \cdot a\right) \cdot \pi\right)\right)} \]

13. Final simplification 32.8%

    \[\leadsto -0.011111111111111112 \cdot \left(a \cdot \left(\pi \cdot \left(angle \cdot a\right)\right)\right) \]
14. Add Preprocessing

Reproduce

herbie shell --seed 2024170 
(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)))))