ab-angle->ABCF B

Percentage Accurate: 53.3% → 66.3%

Time: 19.5s

Alternatives: 18

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: 53.3% 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: 66.3% accurate, 0.7× speedup

Math

\[\begin{array}{l} a_m = \left|a\right| \\ b_m = \left|b\right| \\ \begin{array}{l} t_0 := \sqrt[3]{{\pi}^{1.5}}\\ t_1 := \frac{\pi}{\frac{180}{angle}}\\ t_2 := \left(\pi \cdot angle\right) \cdot 0.005555555555555556\\ \mathbf{if}\;a\_m \leq 1.4 \cdot 10^{+14}:\\ \;\;\;\;\left(\left(a\_m + b\_m\right) \cdot \left(\left(b\_m - a\_m\right) \cdot \sin \left(\frac{\frac{\pi}{180}}{\frac{1}{angle}}\right)\right)\right) \cdot \left(2 \cdot \cos t\_1\right)\\ \mathbf{elif}\;a\_m \leq 2.2 \cdot 10^{+224}:\\ \;\;\;\;\left(\left(a\_m + b\_m\right) \cdot \left(\left(b\_m - a\_m\right) \cdot \sin t\_1\right)\right) \cdot \left(2 \cdot \cos \left(\frac{t\_0 \cdot t\_0}{\frac{180}{angle}}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(a\_m \cdot -2\right) \cdot \left(\cos t\_2 \cdot \sin t\_2\right)}{\frac{1}{a\_m + b\_m}}\\ \end{array} \end{array} \]

FPCore

a_m = (fabs.f64 a)
b_m = (fabs.f64 b)
(FPCore (a_m b_m angle)
 :precision binary64
 (let* ((t_0 (cbrt (pow PI 1.5)))
        (t_1 (/ PI (/ 180.0 angle)))
        (t_2 (* (* PI angle) 0.005555555555555556)))
   (if (<= a_m 1.4e+14)
     (*
      (* (+ a_m b_m) (* (- b_m a_m) (sin (/ (/ PI 180.0) (/ 1.0 angle)))))
      (* 2.0 (cos t_1)))
     (if (<= a_m 2.2e+224)
       (*
        (* (+ a_m b_m) (* (- b_m a_m) (sin t_1)))
        (* 2.0 (cos (/ (* t_0 t_0) (/ 180.0 angle)))))
       (/ (* (* a_m -2.0) (* (cos t_2) (sin t_2))) (/ 1.0 (+ a_m b_m)))))))

C

a_m = fabs(a);
b_m = fabs(b);
double code(double a_m, double b_m, double angle) {
	double t_0 = cbrt(pow(((double) M_PI), 1.5));
	double t_1 = ((double) M_PI) / (180.0 / angle);
	double t_2 = (((double) M_PI) * angle) * 0.005555555555555556;
	double tmp;
	if (a_m <= 1.4e+14) {
		tmp = ((a_m + b_m) * ((b_m - a_m) * sin(((((double) M_PI) / 180.0) / (1.0 / angle))))) * (2.0 * cos(t_1));
	} else if (a_m <= 2.2e+224) {
		tmp = ((a_m + b_m) * ((b_m - a_m) * sin(t_1))) * (2.0 * cos(((t_0 * t_0) / (180.0 / angle))));
	} else {
		tmp = ((a_m * -2.0) * (cos(t_2) * sin(t_2))) / (1.0 / (a_m + b_m));
	}
	return tmp;
}

Java

a_m = Math.abs(a);
b_m = Math.abs(b);
public static double code(double a_m, double b_m, double angle) {
	double t_0 = Math.cbrt(Math.pow(Math.PI, 1.5));
	double t_1 = Math.PI / (180.0 / angle);
	double t_2 = (Math.PI * angle) * 0.005555555555555556;
	double tmp;
	if (a_m <= 1.4e+14) {
		tmp = ((a_m + b_m) * ((b_m - a_m) * Math.sin(((Math.PI / 180.0) / (1.0 / angle))))) * (2.0 * Math.cos(t_1));
	} else if (a_m <= 2.2e+224) {
		tmp = ((a_m + b_m) * ((b_m - a_m) * Math.sin(t_1))) * (2.0 * Math.cos(((t_0 * t_0) / (180.0 / angle))));
	} else {
		tmp = ((a_m * -2.0) * (Math.cos(t_2) * Math.sin(t_2))) / (1.0 / (a_m + b_m));
	}
	return tmp;
}

Julia

a_m = abs(a)
b_m = abs(b)
function code(a_m, b_m, angle)
	t_0 = cbrt((pi ^ 1.5))
	t_1 = Float64(pi / Float64(180.0 / angle))
	t_2 = Float64(Float64(pi * angle) * 0.005555555555555556)
	tmp = 0.0
	if (a_m <= 1.4e+14)
		tmp = Float64(Float64(Float64(a_m + b_m) * Float64(Float64(b_m - a_m) * sin(Float64(Float64(pi / 180.0) / Float64(1.0 / angle))))) * Float64(2.0 * cos(t_1)));
	elseif (a_m <= 2.2e+224)
		tmp = Float64(Float64(Float64(a_m + b_m) * Float64(Float64(b_m - a_m) * sin(t_1))) * Float64(2.0 * cos(Float64(Float64(t_0 * t_0) / Float64(180.0 / angle)))));
	else
		tmp = Float64(Float64(Float64(a_m * -2.0) * Float64(cos(t_2) * sin(t_2))) / Float64(1.0 / Float64(a_m + b_m)));
	end
	return tmp
end

Wolfram

a_m = N[Abs[a], $MachinePrecision]
b_m = N[Abs[b], $MachinePrecision]
code[a$95$m_, b$95$m_, angle_] := Block[{t$95$0 = N[Power[N[Power[Pi, 1.5], $MachinePrecision], 1/3], $MachinePrecision]}, Block[{t$95$1 = N[(Pi / N[(180.0 / angle), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(N[(Pi * angle), $MachinePrecision] * 0.005555555555555556), $MachinePrecision]}, If[LessEqual[a$95$m, 1.4e+14], N[(N[(N[(a$95$m + b$95$m), $MachinePrecision] * N[(N[(b$95$m - a$95$m), $MachinePrecision] * N[Sin[N[(N[(Pi / 180.0), $MachinePrecision] / N[(1.0 / angle), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(2.0 * N[Cos[t$95$1], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[a$95$m, 2.2e+224], N[(N[(N[(a$95$m + b$95$m), $MachinePrecision] * N[(N[(b$95$m - a$95$m), $MachinePrecision] * N[Sin[t$95$1], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(2.0 * N[Cos[N[(N[(t$95$0 * t$95$0), $MachinePrecision] / N[(180.0 / angle), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(a$95$m * -2.0), $MachinePrecision] * N[(N[Cos[t$95$2], $MachinePrecision] * N[Sin[t$95$2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(1.0 / N[(a$95$m + b$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]]

Derivation

1. Split input into 3 regimes

2. if a < 1.4e14

   1. Initial program 59.2%

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

      1. associate-*l* N/A

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

      2. *-commutative N/A

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

      3. associate-*l* N/A

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

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

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

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

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

      6. associate-*r/ N/A

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

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

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

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

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

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

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

      10. *-commutative N/A

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

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

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

   3. Simplified 61.3%

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

      1. *-commutative N/A

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

      2. *-commutative N/A

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

      3. associate-*r* N/A

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

      4. associate-*r/ N/A

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

      5. associate-*r* N/A

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

      6. pow2 N/A

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

      7. pow2 N/A

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

      8. *-commutative N/A

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

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

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

   6. Applied egg-rr 69.7%

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

      1. div-inv N/A

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

      2. associate-/r* N/A

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

      3. div-inv N/A

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

      4. metadata-eval N/A

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

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

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

      6. metadata-eval N/A

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

      7. div-inv N/A

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

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

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

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

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

      10. /-lowering-/.f64 68.9%

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

   8. Applied egg-rr 68.9%

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

   if 1.4e14 < a < 2.2e224

   1. Initial program 72.9%

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

      1. associate-*l* N/A

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

      2. *-commutative N/A

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

      3. associate-*l* N/A

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

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

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

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

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

      6. associate-*r/ N/A

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

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

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

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

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

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

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

      10. *-commutative N/A

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

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

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

   3. Simplified 73.7%

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

      1. *-commutative N/A

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

      2. *-commutative N/A

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

      3. associate-*r* N/A

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

      4. associate-*r/ N/A

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

      5. associate-*r* N/A

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

      6. pow2 N/A

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

      7. pow2 N/A

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

      8. *-commutative N/A

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

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

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

   6. Applied egg-rr 78.7%

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

      1. add-cbrt-cube N/A

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

      2. add-sqr-sqrt N/A

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

      3. unswap-sqr N/A

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

      4. cbrt-prod N/A

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

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

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

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

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

      7. pow1 N/A

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

      8. pow1/2 N/A

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

      9. pow-prod-up N/A

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

      10. metadata-eval N/A

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

      11. metadata-eval N/A

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

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

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

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

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

      14. metadata-eval N/A

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

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

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

      16. pow1 N/A

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

      17. pow1/2 N/A

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

      18. pow-prod-up N/A

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

      19. metadata-eval N/A

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

      20. metadata-eval N/A

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

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

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

   8. Applied egg-rr 79.4%

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

   if 2.2e224 < a

   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. *-commutative N/A

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

      3. associate-*l* N/A

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

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

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

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

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

      6. associate-*r/ N/A

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

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

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

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

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

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

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

      10. *-commutative N/A

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

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

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

   3. Simplified 61.8%

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

      1. *-commutative N/A

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

      2. *-commutative N/A

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

      3. associate-*r* N/A

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

      4. associate-*r/ N/A

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

      5. associate-*r* N/A

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

      6. pow2 N/A

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

      7. pow2 N/A

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

      8. *-commutative N/A

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

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

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

   6. Applied egg-rr 94.4%

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

      1. associate-*l* N/A

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

      2. flip-+ N/A

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

      3. associate-*l/ N/A

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

      4. flip-- N/A

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

      5. div-inv N/A

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

      6. times-frac N/A

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

   8. Applied egg-rr 94.4%

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

      1. *-lft-identity N/A

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

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

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

   10. Applied egg-rr 94.4%

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

   11. Taylor expanded in b around 0

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

       1. associate-*r* N/A

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

       12. PI-lowering-PI.f64 94.4%

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

   13. Simplified 94.4%

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

4. Final simplification 72.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \leq 1.4 \cdot 10^{+14}:\\ \;\;\;\;\left(\left(a + b\right) \cdot \left(\left(b - a\right) \cdot \sin \left(\frac{\frac{\pi}{180}}{\frac{1}{angle}}\right)\right)\right) \cdot \left(2 \cdot \cos \left(\frac{\pi}{\frac{180}{angle}}\right)\right)\\ \mathbf{elif}\;a \leq 2.2 \cdot 10^{+224}:\\ \;\;\;\;\left(\left(a + b\right) \cdot \left(\left(b - a\right) \cdot \sin \left(\frac{\pi}{\frac{180}{angle}}\right)\right)\right) \cdot \left(2 \cdot \cos \left(\frac{\sqrt[3]{{\pi}^{1.5}} \cdot \sqrt[3]{{\pi}^{1.5}}}{\frac{180}{angle}}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(a \cdot -2\right) \cdot \left(\cos \left(\left(\pi \cdot angle\right) \cdot 0.005555555555555556\right) \cdot \sin \left(\left(\pi \cdot angle\right) \cdot 0.005555555555555556\right)\right)}{\frac{1}{a + b}}\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 63.0% accurate, 0.5× speedup

Math

\[\begin{array}{l} a_m = \left|a\right| \\ b_m = \left|b\right| \\ \begin{array}{l} t_0 := {\left(e^{-1}\right)}^{\left(\frac{\log \left(\frac{180}{\pi \cdot angle}\right)}{2}\right)}\\ t_1 := \sqrt[3]{{\pi}^{1.5}}\\ t_2 := \frac{1}{a\_m + b\_m}\\ \mathbf{if}\;a\_m \leq 3.3 \cdot 10^{+250}:\\ \;\;\;\;\frac{\left(\sin \left(\frac{\pi \cdot angle}{180}\right) \cdot \left(\left(b\_m - a\_m\right) \cdot 2\right)\right) \cdot \cos \left(\frac{angle \cdot \left(t\_1 \cdot t\_1\right)}{180}\right)}{t\_2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(2 \cdot \left(\left(b\_m - a\_m\right) \cdot \sin \left(\frac{\pi}{\frac{180}{angle}}\right)\right)\right) \cdot \cos \left(t\_0 \cdot t\_0\right)}{t\_2}\\ \end{array} \end{array} \]

FPCore

a_m = (fabs.f64 a)
b_m = (fabs.f64 b)
(FPCore (a_m b_m angle)
 :precision binary64
 (let* ((t_0 (pow (exp -1.0) (/ (log (/ 180.0 (* PI angle))) 2.0)))
        (t_1 (cbrt (pow PI 1.5)))
        (t_2 (/ 1.0 (+ a_m b_m))))
   (if (<= a_m 3.3e+250)
     (/
      (*
       (* (sin (/ (* PI angle) 180.0)) (* (- b_m a_m) 2.0))
       (cos (/ (* angle (* t_1 t_1)) 180.0)))
      t_2)
     (/
      (*
       (* 2.0 (* (- b_m a_m) (sin (/ PI (/ 180.0 angle)))))
       (cos (* t_0 t_0)))
      t_2))))

C

a_m = fabs(a);
b_m = fabs(b);
double code(double a_m, double b_m, double angle) {
	double t_0 = pow(exp(-1.0), (log((180.0 / (((double) M_PI) * angle))) / 2.0));
	double t_1 = cbrt(pow(((double) M_PI), 1.5));
	double t_2 = 1.0 / (a_m + b_m);
	double tmp;
	if (a_m <= 3.3e+250) {
		tmp = ((sin(((((double) M_PI) * angle) / 180.0)) * ((b_m - a_m) * 2.0)) * cos(((angle * (t_1 * t_1)) / 180.0))) / t_2;
	} else {
		tmp = ((2.0 * ((b_m - a_m) * sin((((double) M_PI) / (180.0 / angle))))) * cos((t_0 * t_0))) / t_2;
	}
	return tmp;
}

Java

a_m = Math.abs(a);
b_m = Math.abs(b);
public static double code(double a_m, double b_m, double angle) {
	double t_0 = Math.pow(Math.exp(-1.0), (Math.log((180.0 / (Math.PI * angle))) / 2.0));
	double t_1 = Math.cbrt(Math.pow(Math.PI, 1.5));
	double t_2 = 1.0 / (a_m + b_m);
	double tmp;
	if (a_m <= 3.3e+250) {
		tmp = ((Math.sin(((Math.PI * angle) / 180.0)) * ((b_m - a_m) * 2.0)) * Math.cos(((angle * (t_1 * t_1)) / 180.0))) / t_2;
	} else {
		tmp = ((2.0 * ((b_m - a_m) * Math.sin((Math.PI / (180.0 / angle))))) * Math.cos((t_0 * t_0))) / t_2;
	}
	return tmp;
}

Julia

a_m = abs(a)
b_m = abs(b)
function code(a_m, b_m, angle)
	t_0 = exp(-1.0) ^ Float64(log(Float64(180.0 / Float64(pi * angle))) / 2.0)
	t_1 = cbrt((pi ^ 1.5))
	t_2 = Float64(1.0 / Float64(a_m + b_m))
	tmp = 0.0
	if (a_m <= 3.3e+250)
		tmp = Float64(Float64(Float64(sin(Float64(Float64(pi * angle) / 180.0)) * Float64(Float64(b_m - a_m) * 2.0)) * cos(Float64(Float64(angle * Float64(t_1 * t_1)) / 180.0))) / t_2);
	else
		tmp = Float64(Float64(Float64(2.0 * Float64(Float64(b_m - a_m) * sin(Float64(pi / Float64(180.0 / angle))))) * cos(Float64(t_0 * t_0))) / t_2);
	end
	return tmp
end

Wolfram

a_m = N[Abs[a], $MachinePrecision]
b_m = N[Abs[b], $MachinePrecision]
code[a$95$m_, b$95$m_, angle_] := Block[{t$95$0 = N[Power[N[Exp[-1.0], $MachinePrecision], N[(N[Log[N[(180.0 / N[(Pi * angle), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / 2.0), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$1 = N[Power[N[Power[Pi, 1.5], $MachinePrecision], 1/3], $MachinePrecision]}, Block[{t$95$2 = N[(1.0 / N[(a$95$m + b$95$m), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[a$95$m, 3.3e+250], N[(N[(N[(N[Sin[N[(N[(Pi * angle), $MachinePrecision] / 180.0), $MachinePrecision]], $MachinePrecision] * N[(N[(b$95$m - a$95$m), $MachinePrecision] * 2.0), $MachinePrecision]), $MachinePrecision] * N[Cos[N[(N[(angle * N[(t$95$1 * t$95$1), $MachinePrecision]), $MachinePrecision] / 180.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / t$95$2), $MachinePrecision], N[(N[(N[(2.0 * N[(N[(b$95$m - a$95$m), $MachinePrecision] * N[Sin[N[(Pi / N[(180.0 / angle), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[Cos[N[(t$95$0 * t$95$0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / t$95$2), $MachinePrecision]]]]]

Derivation

1. Split input into 2 regimes

2. if a < 3.30000000000000006e250

   1. Initial program 60.8%

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

      1. associate-*l* N/A

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

      2. *-commutative N/A

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

      3. associate-*l* N/A

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

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

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

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

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

      6. associate-*r/ N/A

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

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

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

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

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

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

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

      10. *-commutative N/A

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

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

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

   3. Simplified 62.6%

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

      1. *-commutative N/A

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

      2. *-commutative N/A

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

      3. associate-*r* N/A

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

      4. associate-*r/ N/A

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

      5. associate-*r* N/A

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

      6. pow2 N/A

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

      7. pow2 N/A

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

      8. *-commutative N/A

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

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

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

   6. Applied egg-rr 71.5%

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

      1. associate-*l* N/A

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

      2. flip-+ N/A

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

      3. associate-*l/ N/A

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

      4. flip-- N/A

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

      5. div-inv N/A

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

      6. times-frac N/A

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

   8. Applied egg-rr 71.5%

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

      1. *-lft-identity N/A

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

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

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

   10. Applied egg-rr 71.5%

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

       1. add-cbrt-cube N/A

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

       2. add-sqr-sqrt N/A

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

       3. unswap-sqr N/A

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

       4. cbrt-prod N/A

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

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

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

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

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

       7. pow1 N/A

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

       8. pow1/2 N/A

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

       9. pow-prod-up N/A

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

       10. metadata-eval N/A

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

       11. metadata-eval N/A

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

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

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

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

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

       14. metadata-eval N/A

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

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

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

       16. pow1 N/A

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

       17. pow1/2 N/A

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

       18. pow-prod-up N/A

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

       19. metadata-eval N/A

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

       20. metadata-eval N/A

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

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

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

   12. Applied egg-rr 71.5%

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

   if 3.30000000000000006e250 < a

   1. Initial program 64.3%

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

      1. associate-*l* N/A

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

      2. *-commutative N/A

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

      3. associate-*l* N/A

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

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

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

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

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

      6. associate-*r/ N/A

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

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

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

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

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

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

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

      10. *-commutative N/A

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

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

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

   3. Simplified 71.4%

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

      1. *-commutative N/A

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

      2. *-commutative N/A

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

      3. associate-*r* N/A

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

      4. associate-*r/ N/A

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

      5. associate-*r* N/A

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

      6. pow2 N/A

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

      7. pow2 N/A

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

      8. *-commutative N/A

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

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

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

   6. Applied egg-rr 92.9%

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

      1. associate-*l* N/A

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

      2. flip-+ N/A

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

      3. associate-*l/ N/A

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

      4. flip-- N/A

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

      5. div-inv N/A

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

      6. times-frac N/A

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

   8. Applied egg-rr 92.9%

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

      1. clear-num N/A

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

      2. inv-pow N/A

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

      3. exp-to-pow N/A

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

      4. *-commutative N/A

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

      5. exp-prod N/A

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

      6. sqr-pow N/A

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

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

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

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

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

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

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

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

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

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

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

      12. associate-/l/ N/A

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

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

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

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

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

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

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

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

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

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

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

   10. Applied egg-rr 50.0%

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

4. Final simplification 70.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \leq 3.3 \cdot 10^{+250}:\\ \;\;\;\;\frac{\left(\sin \left(\frac{\pi \cdot angle}{180}\right) \cdot \left(\left(b - a\right) \cdot 2\right)\right) \cdot \cos \left(\frac{angle \cdot \left(\sqrt[3]{{\pi}^{1.5}} \cdot \sqrt[3]{{\pi}^{1.5}}\right)}{180}\right)}{\frac{1}{a + b}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(2 \cdot \left(\left(b - a\right) \cdot \sin \left(\frac{\pi}{\frac{180}{angle}}\right)\right)\right) \cdot \cos \left({\left(e^{-1}\right)}^{\left(\frac{\log \left(\frac{180}{\pi \cdot angle}\right)}{2}\right)} \cdot {\left(e^{-1}\right)}^{\left(\frac{\log \left(\frac{180}{\pi \cdot angle}\right)}{2}\right)}\right)}{\frac{1}{a + b}}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 61.5% accurate, 0.8× speedup

Math

\[\begin{array}{l} a_m = \left|a\right| \\ b_m = \left|b\right| \\ \begin{array}{l} \mathbf{if}\;b\_m \leq 10^{+215}:\\ \;\;\;\;\left(a\_m + b\_m\right) \cdot \left(\left(b\_m - a\_m\right) \cdot \sin \left(\left(\pi \cdot angle\right) \cdot 0.011111111111111112\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\left(a\_m + b\_m\right) \cdot \left(\left(b\_m - a\_m\right) \cdot \sin \left(\frac{\frac{\pi}{180}}{\frac{1}{angle}}\right)\right)\right) \cdot \left(2 \cdot \cos \left({\left(e^{-1}\right)}^{\log \left(\frac{180}{\pi \cdot angle}\right)}\right)\right)\\ \end{array} \end{array} \]

FPCore

a_m = (fabs.f64 a)
b_m = (fabs.f64 b)
(FPCore (a_m b_m angle)
 :precision binary64
 (if (<= b_m 1e+215)
   (* (+ a_m b_m) (* (- b_m a_m) (sin (* (* PI angle) 0.011111111111111112))))
   (*
    (* (+ a_m b_m) (* (- b_m a_m) (sin (/ (/ PI 180.0) (/ 1.0 angle)))))
    (* 2.0 (cos (pow (exp -1.0) (log (/ 180.0 (* PI angle)))))))))

C

a_m = fabs(a);
b_m = fabs(b);
double code(double a_m, double b_m, double angle) {
	double tmp;
	if (b_m <= 1e+215) {
		tmp = (a_m + b_m) * ((b_m - a_m) * sin(((((double) M_PI) * angle) * 0.011111111111111112)));
	} else {
		tmp = ((a_m + b_m) * ((b_m - a_m) * sin(((((double) M_PI) / 180.0) / (1.0 / angle))))) * (2.0 * cos(pow(exp(-1.0), log((180.0 / (((double) M_PI) * angle))))));
	}
	return tmp;
}

Java

a_m = Math.abs(a);
b_m = Math.abs(b);
public static double code(double a_m, double b_m, double angle) {
	double tmp;
	if (b_m <= 1e+215) {
		tmp = (a_m + b_m) * ((b_m - a_m) * Math.sin(((Math.PI * angle) * 0.011111111111111112)));
	} else {
		tmp = ((a_m + b_m) * ((b_m - a_m) * Math.sin(((Math.PI / 180.0) / (1.0 / angle))))) * (2.0 * Math.cos(Math.pow(Math.exp(-1.0), Math.log((180.0 / (Math.PI * angle))))));
	}
	return tmp;
}

Python

a_m = math.fabs(a)
b_m = math.fabs(b)
def code(a_m, b_m, angle):
	tmp = 0
	if b_m <= 1e+215:
		tmp = (a_m + b_m) * ((b_m - a_m) * math.sin(((math.pi * angle) * 0.011111111111111112)))
	else:
		tmp = ((a_m + b_m) * ((b_m - a_m) * math.sin(((math.pi / 180.0) / (1.0 / angle))))) * (2.0 * math.cos(math.pow(math.exp(-1.0), math.log((180.0 / (math.pi * angle))))))
	return tmp

Julia

a_m = abs(a)
b_m = abs(b)
function code(a_m, b_m, angle)
	tmp = 0.0
	if (b_m <= 1e+215)
		tmp = Float64(Float64(a_m + b_m) * Float64(Float64(b_m - a_m) * sin(Float64(Float64(pi * angle) * 0.011111111111111112))));
	else
		tmp = Float64(Float64(Float64(a_m + b_m) * Float64(Float64(b_m - a_m) * sin(Float64(Float64(pi / 180.0) / Float64(1.0 / angle))))) * Float64(2.0 * cos((exp(-1.0) ^ log(Float64(180.0 / Float64(pi * angle)))))));
	end
	return tmp
end

MATLAB

a_m = abs(a);
b_m = abs(b);
function tmp_2 = code(a_m, b_m, angle)
	tmp = 0.0;
	if (b_m <= 1e+215)
		tmp = (a_m + b_m) * ((b_m - a_m) * sin(((pi * angle) * 0.011111111111111112)));
	else
		tmp = ((a_m + b_m) * ((b_m - a_m) * sin(((pi / 180.0) / (1.0 / angle))))) * (2.0 * cos((exp(-1.0) ^ log((180.0 / (pi * angle))))));
	end
	tmp_2 = tmp;
end

Wolfram

a_m = N[Abs[a], $MachinePrecision]
b_m = N[Abs[b], $MachinePrecision]
code[a$95$m_, b$95$m_, angle_] := If[LessEqual[b$95$m, 1e+215], N[(N[(a$95$m + b$95$m), $MachinePrecision] * N[(N[(b$95$m - a$95$m), $MachinePrecision] * N[Sin[N[(N[(Pi * angle), $MachinePrecision] * 0.011111111111111112), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(a$95$m + b$95$m), $MachinePrecision] * N[(N[(b$95$m - a$95$m), $MachinePrecision] * N[Sin[N[(N[(Pi / 180.0), $MachinePrecision] / N[(1.0 / angle), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(2.0 * N[Cos[N[Power[N[Exp[-1.0], $MachinePrecision], N[Log[N[(180.0 / N[(Pi * angle), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]], $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b < 9.99999999999999907e214

   1. Initial program 62.0%

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

      1. associate-*l* N/A

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

      2. *-commutative N/A

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

      3. associate-*l* N/A

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

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

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

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

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

      6. associate-*r/ N/A

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

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

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

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

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

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

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

      10. *-commutative N/A

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

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

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

   3. Simplified 63.4%

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

      1. associate-*r/ N/A

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

      2. *-commutative N/A

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

      3. associate-*r* N/A

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

      4. associate-*r/ N/A

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

      5. associate-*r* N/A

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

      6. *-commutative N/A

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

      7. *-commutative N/A

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

      8. difference-of-squares N/A

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

      9. associate-*l* N/A

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

   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 9.99999999999999907e214 < b

   1. Initial program 43.4%

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

      1. associate-*l* N/A

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

      2. *-commutative N/A

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

      3. associate-*l* N/A

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

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

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

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

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

      6. associate-*r/ N/A

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

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

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

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

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

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

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

      10. *-commutative N/A

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

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

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

   3. Simplified 57.7%

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

      1. *-commutative N/A

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

      2. *-commutative N/A

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

      3. associate-*r* N/A

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

      4. associate-*r/ N/A

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

      5. associate-*r* N/A

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

      6. pow2 N/A

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

      7. pow2 N/A

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

      8. *-commutative N/A

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

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

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

   6. Applied egg-rr 71.4%

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

      1. div-inv N/A

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

      2. associate-/r* N/A

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

      3. div-inv N/A

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

      4. metadata-eval N/A

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

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

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

      6. metadata-eval N/A

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

      7. div-inv N/A

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

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

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

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

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

      10. /-lowering-/.f64 71.4%

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

   8. Applied egg-rr 71.4%

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

      1. clear-num N/A

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

      2. inv-pow N/A

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

      3. exp-to-pow N/A

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

      4. *-commutative N/A

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

      5. exp-prod N/A

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

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

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

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

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

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

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

      9. associate-/l/ N/A

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

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

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

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

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

      12. PI-lowering-PI.f64 35.7%

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

   10. Applied egg-rr 35.7%

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

4. Final simplification 70.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq 10^{+215}:\\ \;\;\;\;\left(a + b\right) \cdot \left(\left(b - a\right) \cdot \sin \left(\left(\pi \cdot angle\right) \cdot 0.011111111111111112\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\left(a + b\right) \cdot \left(\left(b - a\right) \cdot \sin \left(\frac{\frac{\pi}{180}}{\frac{1}{angle}}\right)\right)\right) \cdot \left(2 \cdot \cos \left({\left(e^{-1}\right)}^{\log \left(\frac{180}{\pi \cdot angle}\right)}\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 65.3% accurate, 1.3× speedup

Math

\[\begin{array}{l} a_m = \left|a\right| \\ b_m = \left|b\right| \\ \begin{array}{l} t_0 := 2 \cdot \cos \left(\frac{\pi}{\frac{180}{angle}}\right)\\ \mathbf{if}\;{a\_m}^{2} \leq 5 \cdot 10^{+293}:\\ \;\;\;\;\left(\left(a\_m + b\_m\right) \cdot \left(\left(b\_m - a\_m\right) \cdot \sin \left(\frac{\frac{\pi}{180}}{\frac{1}{angle}}\right)\right)\right) \cdot t\_0\\ \mathbf{else}:\\ \;\;\;\;t\_0 \cdot \left(\left(a\_m + b\_m\right) \cdot \left(\sin \left(\left(\pi \cdot angle\right) \cdot 0.005555555555555556\right) \cdot \left(0 - a\_m\right)\right)\right)\\ \end{array} \end{array} \]

FPCore

a_m = (fabs.f64 a)
b_m = (fabs.f64 b)
(FPCore (a_m b_m angle)
 :precision binary64
 (let* ((t_0 (* 2.0 (cos (/ PI (/ 180.0 angle))))))
   (if (<= (pow a_m 2.0) 5e+293)
     (*
      (* (+ a_m b_m) (* (- b_m a_m) (sin (/ (/ PI 180.0) (/ 1.0 angle)))))
      t_0)
     (*
      t_0
      (*
       (+ a_m b_m)
       (* (sin (* (* PI angle) 0.005555555555555556)) (- 0.0 a_m)))))))

C

a_m = fabs(a);
b_m = fabs(b);
double code(double a_m, double b_m, double angle) {
	double t_0 = 2.0 * cos((((double) M_PI) / (180.0 / angle)));
	double tmp;
	if (pow(a_m, 2.0) <= 5e+293) {
		tmp = ((a_m + b_m) * ((b_m - a_m) * sin(((((double) M_PI) / 180.0) / (1.0 / angle))))) * t_0;
	} else {
		tmp = t_0 * ((a_m + b_m) * (sin(((((double) M_PI) * angle) * 0.005555555555555556)) * (0.0 - a_m)));
	}
	return tmp;
}

Java

a_m = Math.abs(a);
b_m = Math.abs(b);
public static double code(double a_m, double b_m, double angle) {
	double t_0 = 2.0 * Math.cos((Math.PI / (180.0 / angle)));
	double tmp;
	if (Math.pow(a_m, 2.0) <= 5e+293) {
		tmp = ((a_m + b_m) * ((b_m - a_m) * Math.sin(((Math.PI / 180.0) / (1.0 / angle))))) * t_0;
	} else {
		tmp = t_0 * ((a_m + b_m) * (Math.sin(((Math.PI * angle) * 0.005555555555555556)) * (0.0 - a_m)));
	}
	return tmp;
}

Python

a_m = math.fabs(a)
b_m = math.fabs(b)
def code(a_m, b_m, angle):
	t_0 = 2.0 * math.cos((math.pi / (180.0 / angle)))
	tmp = 0
	if math.pow(a_m, 2.0) <= 5e+293:
		tmp = ((a_m + b_m) * ((b_m - a_m) * math.sin(((math.pi / 180.0) / (1.0 / angle))))) * t_0
	else:
		tmp = t_0 * ((a_m + b_m) * (math.sin(((math.pi * angle) * 0.005555555555555556)) * (0.0 - a_m)))
	return tmp

Julia

a_m = abs(a)
b_m = abs(b)
function code(a_m, b_m, angle)
	t_0 = Float64(2.0 * cos(Float64(pi / Float64(180.0 / angle))))
	tmp = 0.0
	if ((a_m ^ 2.0) <= 5e+293)
		tmp = Float64(Float64(Float64(a_m + b_m) * Float64(Float64(b_m - a_m) * sin(Float64(Float64(pi / 180.0) / Float64(1.0 / angle))))) * t_0);
	else
		tmp = Float64(t_0 * Float64(Float64(a_m + b_m) * Float64(sin(Float64(Float64(pi * angle) * 0.005555555555555556)) * Float64(0.0 - a_m))));
	end
	return tmp
end

MATLAB

a_m = abs(a);
b_m = abs(b);
function tmp_2 = code(a_m, b_m, angle)
	t_0 = 2.0 * cos((pi / (180.0 / angle)));
	tmp = 0.0;
	if ((a_m ^ 2.0) <= 5e+293)
		tmp = ((a_m + b_m) * ((b_m - a_m) * sin(((pi / 180.0) / (1.0 / angle))))) * t_0;
	else
		tmp = t_0 * ((a_m + b_m) * (sin(((pi * angle) * 0.005555555555555556)) * (0.0 - a_m)));
	end
	tmp_2 = tmp;
end

Wolfram

a_m = N[Abs[a], $MachinePrecision]
b_m = N[Abs[b], $MachinePrecision]
code[a$95$m_, b$95$m_, angle_] := Block[{t$95$0 = N[(2.0 * N[Cos[N[(Pi / N[(180.0 / angle), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[Power[a$95$m, 2.0], $MachinePrecision], 5e+293], N[(N[(N[(a$95$m + b$95$m), $MachinePrecision] * N[(N[(b$95$m - a$95$m), $MachinePrecision] * N[Sin[N[(N[(Pi / 180.0), $MachinePrecision] / N[(1.0 / angle), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * t$95$0), $MachinePrecision], N[(t$95$0 * N[(N[(a$95$m + b$95$m), $MachinePrecision] * N[(N[Sin[N[(N[(Pi * angle), $MachinePrecision] * 0.005555555555555556), $MachinePrecision]], $MachinePrecision] * N[(0.0 - a$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (pow.f64 a #s(literal 2 binary64)) < 5.00000000000000033e293

   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. Step-by-step derivation

      1. associate-*l* N/A

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

      2. *-commutative N/A

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

      3. associate-*l* N/A

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

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

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

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

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

      6. associate-*r/ N/A

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

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

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

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

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

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

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

      10. *-commutative N/A

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

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

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

   3. Simplified 66.1%

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

      1. *-commutative N/A

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

      2. *-commutative N/A

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

      3. associate-*r* N/A

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

      4. associate-*r/ N/A

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

      5. associate-*r* N/A

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

      6. pow2 N/A

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

      7. pow2 N/A

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

      8. *-commutative N/A

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

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

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

   6. Applied egg-rr 70.4%

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

      1. div-inv N/A

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

      2. associate-/r* N/A

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

      3. div-inv N/A

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

      4. metadata-eval N/A

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

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

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

      6. metadata-eval N/A

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

      7. div-inv N/A

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

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

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

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

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

      10. /-lowering-/.f64 71.2%

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

   8. Applied egg-rr 71.2%

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

   if 5.00000000000000033e293 < (pow.f64 a #s(literal 2 binary64))

   1. Initial program 51.8%

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

      1. associate-*l* N/A

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

      2. *-commutative N/A

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

      3. associate-*l* N/A

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

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

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

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

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

      6. associate-*r/ N/A

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

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

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

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

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

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

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

      10. *-commutative N/A

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

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

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

   3. Simplified 54.7%

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

      1. *-commutative N/A

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

      2. *-commutative N/A

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

      3. associate-*r* N/A

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

      4. associate-*r/ N/A

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

      5. associate-*r* N/A

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

      6. pow2 N/A

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

      7. pow2 N/A

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

      8. *-commutative N/A

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

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

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

   6. Applied egg-rr 79.1%

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

   7. Taylor expanded in b around 0

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

      1. mul-1-neg N/A

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

      2. *-commutative N/A

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

      3. distribute-rgt-neg-in N/A

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

      4. mul-1-neg N/A

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

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

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

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

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

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

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

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

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

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

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

      10. mul-1-neg N/A

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

      11. neg-sub0 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(b, a\right), \mathsf{*.f64}\left(\mathsf{sin.f64}\left(\mathsf{*.f64}\left(\frac{1}{180}, \mathsf{*.f64}\left(angle, \mathsf{PI.f64}\left(\right)\right)\right)\right), \left(0 - a\right)\right)\right), \mathsf{*.f64}\left(2, \mathsf{cos.f64}\left(\mathsf{/.f64}\left(\mathsf{PI.f64}\left(\right), \mathsf{/.f64}\left(180, angle\right)\right)\right)\right)\right) \]

      12. --lowering--.f64 82.0%

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(b, a\right), \mathsf{*.f64}\left(\mathsf{sin.f64}\left(\mathsf{*.f64}\left(\frac{1}{180}, \mathsf{*.f64}\left(angle, \mathsf{PI.f64}\left(\right)\right)\right)\right), \mathsf{\_.f64}\left(0, a\right)\right)\right), \mathsf{*.f64}\left(2, \mathsf{cos.f64}\left(\mathsf{/.f64}\left(\mathsf{PI.f64}\left(\right), \mathsf{/.f64}\left(180, angle\right)\right)\right)\right)\right) \]

   9. Simplified 82.0%

      \[\leadsto \left(\left(b + a\right) \cdot \color{blue}{\left(\sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right) \cdot \left(0 - a\right)\right)}\right) \cdot \left(2 \cdot \cos \left(\frac{\pi}{\frac{180}{angle}}\right)\right) \]
3. Recombined 2 regimes into one program.

4. Final simplification 74.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;{a}^{2} \leq 5 \cdot 10^{+293}:\\ \;\;\;\;\left(\left(a + b\right) \cdot \left(\left(b - a\right) \cdot \sin \left(\frac{\frac{\pi}{180}}{\frac{1}{angle}}\right)\right)\right) \cdot \left(2 \cdot \cos \left(\frac{\pi}{\frac{180}{angle}}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(2 \cdot \cos \left(\frac{\pi}{\frac{180}{angle}}\right)\right) \cdot \left(\left(a + b\right) \cdot \left(\sin \left(\left(\pi \cdot angle\right) \cdot 0.005555555555555556\right) \cdot \left(0 - a\right)\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 66.8% accurate, 1.9× speedup

Math

\[\begin{array}{l} a_m = \left|a\right| \\ b_m = \left|b\right| \\ \begin{array}{l} t_0 := \frac{\pi}{\frac{180}{angle}}\\ t_1 := \left(\pi \cdot angle\right) \cdot 0.005555555555555556\\ \mathbf{if}\;a\_m \leq 1.82 \cdot 10^{+226}:\\ \;\;\;\;\left(2 \cdot \cos t\_0\right) \cdot \left(\left(a\_m + b\_m\right) \cdot \left(\left(b\_m - a\_m\right) \cdot \sin t\_0\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(a\_m \cdot -2\right) \cdot \left(\cos t\_1 \cdot \sin t\_1\right)}{\frac{1}{a\_m + b\_m}}\\ \end{array} \end{array} \]

FPCore

a_m = (fabs.f64 a)
b_m = (fabs.f64 b)
(FPCore (a_m b_m angle)
 :precision binary64
 (let* ((t_0 (/ PI (/ 180.0 angle)))
        (t_1 (* (* PI angle) 0.005555555555555556)))
   (if (<= a_m 1.82e+226)
     (* (* 2.0 (cos t_0)) (* (+ a_m b_m) (* (- b_m a_m) (sin t_0))))
     (/ (* (* a_m -2.0) (* (cos t_1) (sin t_1))) (/ 1.0 (+ a_m b_m))))))

C

a_m = fabs(a);
b_m = fabs(b);
double code(double a_m, double b_m, double angle) {
	double t_0 = ((double) M_PI) / (180.0 / angle);
	double t_1 = (((double) M_PI) * angle) * 0.005555555555555556;
	double tmp;
	if (a_m <= 1.82e+226) {
		tmp = (2.0 * cos(t_0)) * ((a_m + b_m) * ((b_m - a_m) * sin(t_0)));
	} else {
		tmp = ((a_m * -2.0) * (cos(t_1) * sin(t_1))) / (1.0 / (a_m + b_m));
	}
	return tmp;
}

Java

a_m = Math.abs(a);
b_m = Math.abs(b);
public static double code(double a_m, double b_m, double angle) {
	double t_0 = Math.PI / (180.0 / angle);
	double t_1 = (Math.PI * angle) * 0.005555555555555556;
	double tmp;
	if (a_m <= 1.82e+226) {
		tmp = (2.0 * Math.cos(t_0)) * ((a_m + b_m) * ((b_m - a_m) * Math.sin(t_0)));
	} else {
		tmp = ((a_m * -2.0) * (Math.cos(t_1) * Math.sin(t_1))) / (1.0 / (a_m + b_m));
	}
	return tmp;
}

Python

a_m = math.fabs(a)
b_m = math.fabs(b)
def code(a_m, b_m, angle):
	t_0 = math.pi / (180.0 / angle)
	t_1 = (math.pi * angle) * 0.005555555555555556
	tmp = 0
	if a_m <= 1.82e+226:
		tmp = (2.0 * math.cos(t_0)) * ((a_m + b_m) * ((b_m - a_m) * math.sin(t_0)))
	else:
		tmp = ((a_m * -2.0) * (math.cos(t_1) * math.sin(t_1))) / (1.0 / (a_m + b_m))
	return tmp

Julia

a_m = abs(a)
b_m = abs(b)
function code(a_m, b_m, angle)
	t_0 = Float64(pi / Float64(180.0 / angle))
	t_1 = Float64(Float64(pi * angle) * 0.005555555555555556)
	tmp = 0.0
	if (a_m <= 1.82e+226)
		tmp = Float64(Float64(2.0 * cos(t_0)) * Float64(Float64(a_m + b_m) * Float64(Float64(b_m - a_m) * sin(t_0))));
	else
		tmp = Float64(Float64(Float64(a_m * -2.0) * Float64(cos(t_1) * sin(t_1))) / Float64(1.0 / Float64(a_m + b_m)));
	end
	return tmp
end

MATLAB

a_m = abs(a);
b_m = abs(b);
function tmp_2 = code(a_m, b_m, angle)
	t_0 = pi / (180.0 / angle);
	t_1 = (pi * angle) * 0.005555555555555556;
	tmp = 0.0;
	if (a_m <= 1.82e+226)
		tmp = (2.0 * cos(t_0)) * ((a_m + b_m) * ((b_m - a_m) * sin(t_0)));
	else
		tmp = ((a_m * -2.0) * (cos(t_1) * sin(t_1))) / (1.0 / (a_m + b_m));
	end
	tmp_2 = tmp;
end

Wolfram

a_m = N[Abs[a], $MachinePrecision]
b_m = N[Abs[b], $MachinePrecision]
code[a$95$m_, b$95$m_, angle_] := Block[{t$95$0 = N[(Pi / N[(180.0 / angle), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[(Pi * angle), $MachinePrecision] * 0.005555555555555556), $MachinePrecision]}, If[LessEqual[a$95$m, 1.82e+226], N[(N[(2.0 * N[Cos[t$95$0], $MachinePrecision]), $MachinePrecision] * N[(N[(a$95$m + b$95$m), $MachinePrecision] * N[(N[(b$95$m - a$95$m), $MachinePrecision] * N[Sin[t$95$0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(a$95$m * -2.0), $MachinePrecision] * N[(N[Cos[t$95$1], $MachinePrecision] * N[Sin[t$95$1], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(1.0 / N[(a$95$m + b$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 2 regimes

2. if a < 1.8200000000000001e226

   1. Initial program 61.3%

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

      1. associate-*l* N/A

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

      2. *-commutative N/A

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

      3. associate-*l* N/A

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

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

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

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

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

      6. associate-*r/ N/A

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

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

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

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

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

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

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

      10. *-commutative N/A

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

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

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

   3. Simplified 63.2%

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

      1. *-commutative N/A

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

      2. *-commutative N/A

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

      3. associate-*r* N/A

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

      4. associate-*r/ N/A

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

      5. associate-*r* N/A

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

      6. pow2 N/A

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

      7. pow2 N/A

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

      8. *-commutative N/A

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

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

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

   6. Applied egg-rr 71.1%

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

   if 1.8200000000000001e226 < a

   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. *-commutative N/A

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

      3. associate-*l* N/A

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

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

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

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

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

      6. associate-*r/ N/A

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

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

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

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

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

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

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

      10. *-commutative N/A

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

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

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

   3. Simplified 61.8%

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

      1. *-commutative N/A

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

      2. *-commutative N/A

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

      3. associate-*r* N/A

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

      4. associate-*r/ N/A

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

      5. associate-*r* N/A

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

      6. pow2 N/A

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

      7. pow2 N/A

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

      8. *-commutative N/A

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

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

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

   6. Applied egg-rr 94.4%

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

      1. associate-*l* N/A

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

      2. flip-+ N/A

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

      3. associate-*l/ N/A

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

      4. flip-- N/A

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

      5. div-inv N/A

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

      6. times-frac N/A

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

   8. Applied egg-rr 94.4%

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

      1. *-lft-identity N/A

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

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

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

   10. Applied egg-rr 94.4%

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

   11. Taylor expanded in b around 0

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

       1. associate-*r* N/A

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

       12. PI-lowering-PI.f64 94.4%

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

   13. Simplified 94.4%

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

4. Final simplification 72.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \leq 1.82 \cdot 10^{+226}:\\ \;\;\;\;\left(2 \cdot \cos \left(\frac{\pi}{\frac{180}{angle}}\right)\right) \cdot \left(\left(a + b\right) \cdot \left(\left(b - a\right) \cdot \sin \left(\frac{\pi}{\frac{180}{angle}}\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(a \cdot -2\right) \cdot \left(\cos \left(\left(\pi \cdot angle\right) \cdot 0.005555555555555556\right) \cdot \sin \left(\left(\pi \cdot angle\right) \cdot 0.005555555555555556\right)\right)}{\frac{1}{a + b}}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 67.1% accurate, 1.9× speedup

Math

\[\begin{array}{l} a_m = \left|a\right| \\ b_m = \left|b\right| \\ \begin{array}{l} t_0 := \frac{\pi}{\frac{180}{angle}}\\ t_1 := 2 \cdot \cos t\_0\\ \mathbf{if}\;a\_m \leq 3.1 \cdot 10^{+226}:\\ \;\;\;\;t\_1 \cdot \left(\left(a\_m + b\_m\right) \cdot \left(\left(b\_m - a\_m\right) \cdot \sin t\_0\right)\right)\\ \mathbf{else}:\\ \;\;\;\;t\_1 \cdot \left(\left(a\_m + b\_m\right) \cdot \left(\sin \left(\left(\pi \cdot angle\right) \cdot 0.005555555555555556\right) \cdot \left(0 - a\_m\right)\right)\right)\\ \end{array} \end{array} \]

FPCore

a_m = (fabs.f64 a)
b_m = (fabs.f64 b)
(FPCore (a_m b_m angle)
 :precision binary64
 (let* ((t_0 (/ PI (/ 180.0 angle))) (t_1 (* 2.0 (cos t_0))))
   (if (<= a_m 3.1e+226)
     (* t_1 (* (+ a_m b_m) (* (- b_m a_m) (sin t_0))))
     (*
      t_1
      (*
       (+ a_m b_m)
       (* (sin (* (* PI angle) 0.005555555555555556)) (- 0.0 a_m)))))))

C

a_m = fabs(a);
b_m = fabs(b);
double code(double a_m, double b_m, double angle) {
	double t_0 = ((double) M_PI) / (180.0 / angle);
	double t_1 = 2.0 * cos(t_0);
	double tmp;
	if (a_m <= 3.1e+226) {
		tmp = t_1 * ((a_m + b_m) * ((b_m - a_m) * sin(t_0)));
	} else {
		tmp = t_1 * ((a_m + b_m) * (sin(((((double) M_PI) * angle) * 0.005555555555555556)) * (0.0 - a_m)));
	}
	return tmp;
}

Java

a_m = Math.abs(a);
b_m = Math.abs(b);
public static double code(double a_m, double b_m, double angle) {
	double t_0 = Math.PI / (180.0 / angle);
	double t_1 = 2.0 * Math.cos(t_0);
	double tmp;
	if (a_m <= 3.1e+226) {
		tmp = t_1 * ((a_m + b_m) * ((b_m - a_m) * Math.sin(t_0)));
	} else {
		tmp = t_1 * ((a_m + b_m) * (Math.sin(((Math.PI * angle) * 0.005555555555555556)) * (0.0 - a_m)));
	}
	return tmp;
}

Python

a_m = math.fabs(a)
b_m = math.fabs(b)
def code(a_m, b_m, angle):
	t_0 = math.pi / (180.0 / angle)
	t_1 = 2.0 * math.cos(t_0)
	tmp = 0
	if a_m <= 3.1e+226:
		tmp = t_1 * ((a_m + b_m) * ((b_m - a_m) * math.sin(t_0)))
	else:
		tmp = t_1 * ((a_m + b_m) * (math.sin(((math.pi * angle) * 0.005555555555555556)) * (0.0 - a_m)))
	return tmp

Julia

a_m = abs(a)
b_m = abs(b)
function code(a_m, b_m, angle)
	t_0 = Float64(pi / Float64(180.0 / angle))
	t_1 = Float64(2.0 * cos(t_0))
	tmp = 0.0
	if (a_m <= 3.1e+226)
		tmp = Float64(t_1 * Float64(Float64(a_m + b_m) * Float64(Float64(b_m - a_m) * sin(t_0))));
	else
		tmp = Float64(t_1 * Float64(Float64(a_m + b_m) * Float64(sin(Float64(Float64(pi * angle) * 0.005555555555555556)) * Float64(0.0 - a_m))));
	end
	return tmp
end

MATLAB

a_m = abs(a);
b_m = abs(b);
function tmp_2 = code(a_m, b_m, angle)
	t_0 = pi / (180.0 / angle);
	t_1 = 2.0 * cos(t_0);
	tmp = 0.0;
	if (a_m <= 3.1e+226)
		tmp = t_1 * ((a_m + b_m) * ((b_m - a_m) * sin(t_0)));
	else
		tmp = t_1 * ((a_m + b_m) * (sin(((pi * angle) * 0.005555555555555556)) * (0.0 - a_m)));
	end
	tmp_2 = tmp;
end

Wolfram

a_m = N[Abs[a], $MachinePrecision]
b_m = N[Abs[b], $MachinePrecision]
code[a$95$m_, b$95$m_, angle_] := Block[{t$95$0 = N[(Pi / N[(180.0 / angle), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(2.0 * N[Cos[t$95$0], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[a$95$m, 3.1e+226], N[(t$95$1 * N[(N[(a$95$m + b$95$m), $MachinePrecision] * N[(N[(b$95$m - a$95$m), $MachinePrecision] * N[Sin[t$95$0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(t$95$1 * N[(N[(a$95$m + b$95$m), $MachinePrecision] * N[(N[Sin[N[(N[(Pi * angle), $MachinePrecision] * 0.005555555555555556), $MachinePrecision]], $MachinePrecision] * N[(0.0 - a$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 2 regimes

2. if a < 3.09999999999999976e226

   1. Initial program 61.3%

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

      1. associate-*l* N/A

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

      2. *-commutative N/A

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

      3. associate-*l* N/A

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

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

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

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

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

      6. associate-*r/ N/A

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

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

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

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

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

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

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

      10. *-commutative N/A

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

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

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

   3. Simplified 63.2%

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

      1. *-commutative N/A

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

      2. *-commutative N/A

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

      3. associate-*r* N/A

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

      4. associate-*r/ N/A

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

      5. associate-*r* N/A

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

      6. pow2 N/A

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

      7. pow2 N/A

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

      8. *-commutative N/A

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

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

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

   6. Applied egg-rr 71.1%

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

   if 3.09999999999999976e226 < a

   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. *-commutative N/A

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

      3. associate-*l* N/A

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

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

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

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

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

      6. associate-*r/ N/A

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

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

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

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

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

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

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

      10. *-commutative N/A

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

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

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

   3. Simplified 61.8%

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

      1. *-commutative N/A

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

      2. *-commutative N/A

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

      3. associate-*r* N/A

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

      4. associate-*r/ N/A

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

      5. associate-*r* N/A

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

      6. pow2 N/A

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

      7. pow2 N/A

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

      8. *-commutative N/A

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

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

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

   6. Applied egg-rr 94.4%

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

   7. Taylor expanded in b around 0

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

      1. mul-1-neg N/A

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

      2. *-commutative N/A

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

      3. distribute-rgt-neg-in N/A

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

      4. mul-1-neg N/A

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

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

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

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

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

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

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

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

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

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

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

      10. mul-1-neg N/A

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

      11. neg-sub0 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(b, a\right), \mathsf{*.f64}\left(\mathsf{sin.f64}\left(\mathsf{*.f64}\left(\frac{1}{180}, \mathsf{*.f64}\left(angle, \mathsf{PI.f64}\left(\right)\right)\right)\right), \left(0 - a\right)\right)\right), \mathsf{*.f64}\left(2, \mathsf{cos.f64}\left(\mathsf{/.f64}\left(\mathsf{PI.f64}\left(\right), \mathsf{/.f64}\left(180, angle\right)\right)\right)\right)\right) \]

      12. --lowering--.f64 94.4%

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(b, a\right), \mathsf{*.f64}\left(\mathsf{sin.f64}\left(\mathsf{*.f64}\left(\frac{1}{180}, \mathsf{*.f64}\left(angle, \mathsf{PI.f64}\left(\right)\right)\right)\right), \mathsf{\_.f64}\left(0, a\right)\right)\right), \mathsf{*.f64}\left(2, \mathsf{cos.f64}\left(\mathsf{/.f64}\left(\mathsf{PI.f64}\left(\right), \mathsf{/.f64}\left(180, angle\right)\right)\right)\right)\right) \]

   9. Simplified 94.4%

      \[\leadsto \left(\left(b + a\right) \cdot \color{blue}{\left(\sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right) \cdot \left(0 - a\right)\right)}\right) \cdot \left(2 \cdot \cos \left(\frac{\pi}{\frac{180}{angle}}\right)\right) \]
3. Recombined 2 regimes into one program.

4. Final simplification 72.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \leq 3.1 \cdot 10^{+226}:\\ \;\;\;\;\left(2 \cdot \cos \left(\frac{\pi}{\frac{180}{angle}}\right)\right) \cdot \left(\left(a + b\right) \cdot \left(\left(b - a\right) \cdot \sin \left(\frac{\pi}{\frac{180}{angle}}\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(2 \cdot \cos \left(\frac{\pi}{\frac{180}{angle}}\right)\right) \cdot \left(\left(a + b\right) \cdot \left(\sin \left(\left(\pi \cdot angle\right) \cdot 0.005555555555555556\right) \cdot \left(0 - a\right)\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 62.0% accurate, 3.4× speedup

Math

\[\begin{array}{l} a_m = \left|a\right| \\ b_m = \left|b\right| \\ \begin{array}{l} \mathbf{if}\;\frac{angle}{180} \leq 10^{+205}:\\ \;\;\;\;\left(a\_m + b\_m\right) \cdot \left(\left(b\_m - a\_m\right) \cdot \left(\left(\pi \cdot angle\right) \cdot 0.011111111111111112\right)\right)\\ \mathbf{elif}\;\frac{angle}{180} \leq 5 \cdot 10^{+261}:\\ \;\;\;\;\left(\left(angle \cdot 0.011111111111111112\right) \cdot \left(\pi \cdot \left(b\_m \cdot b\_m - a\_m \cdot a\_m\right)\right)\right) \cdot \left(1 + \left(-1.54320987654321 \cdot 10^{-5} \cdot \left(angle \cdot angle\right)\right) \cdot \left(\pi \cdot \pi\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(2 \cdot \left(b\_m \cdot b\_m\right)\right) \cdot \sin \left(\pi \cdot \frac{angle}{180}\right)\\ \end{array} \end{array} \]

FPCore

a_m = (fabs.f64 a)
b_m = (fabs.f64 b)
(FPCore (a_m b_m angle)
 :precision binary64
 (if (<= (/ angle 180.0) 1e+205)
   (* (+ a_m b_m) (* (- b_m a_m) (* (* PI angle) 0.011111111111111112)))
   (if (<= (/ angle 180.0) 5e+261)
     (*
      (* (* angle 0.011111111111111112) (* PI (- (* b_m b_m) (* a_m a_m))))
      (+ 1.0 (* (* -1.54320987654321e-5 (* angle angle)) (* PI PI))))
     (* (* 2.0 (* b_m b_m)) (sin (* PI (/ angle 180.0)))))))

C

a_m = fabs(a);
b_m = fabs(b);
double code(double a_m, double b_m, double angle) {
	double tmp;
	if ((angle / 180.0) <= 1e+205) {
		tmp = (a_m + b_m) * ((b_m - a_m) * ((((double) M_PI) * angle) * 0.011111111111111112));
	} else if ((angle / 180.0) <= 5e+261) {
		tmp = ((angle * 0.011111111111111112) * (((double) M_PI) * ((b_m * b_m) - (a_m * a_m)))) * (1.0 + ((-1.54320987654321e-5 * (angle * angle)) * (((double) M_PI) * ((double) M_PI))));
	} else {
		tmp = (2.0 * (b_m * b_m)) * sin((((double) M_PI) * (angle / 180.0)));
	}
	return tmp;
}

Java

a_m = Math.abs(a);
b_m = Math.abs(b);
public static double code(double a_m, double b_m, double angle) {
	double tmp;
	if ((angle / 180.0) <= 1e+205) {
		tmp = (a_m + b_m) * ((b_m - a_m) * ((Math.PI * angle) * 0.011111111111111112));
	} else if ((angle / 180.0) <= 5e+261) {
		tmp = ((angle * 0.011111111111111112) * (Math.PI * ((b_m * b_m) - (a_m * a_m)))) * (1.0 + ((-1.54320987654321e-5 * (angle * angle)) * (Math.PI * Math.PI)));
	} else {
		tmp = (2.0 * (b_m * b_m)) * Math.sin((Math.PI * (angle / 180.0)));
	}
	return tmp;
}

Python

a_m = math.fabs(a)
b_m = math.fabs(b)
def code(a_m, b_m, angle):
	tmp = 0
	if (angle / 180.0) <= 1e+205:
		tmp = (a_m + b_m) * ((b_m - a_m) * ((math.pi * angle) * 0.011111111111111112))
	elif (angle / 180.0) <= 5e+261:
		tmp = ((angle * 0.011111111111111112) * (math.pi * ((b_m * b_m) - (a_m * a_m)))) * (1.0 + ((-1.54320987654321e-5 * (angle * angle)) * (math.pi * math.pi)))
	else:
		tmp = (2.0 * (b_m * b_m)) * math.sin((math.pi * (angle / 180.0)))
	return tmp

Julia

a_m = abs(a)
b_m = abs(b)
function code(a_m, b_m, angle)
	tmp = 0.0
	if (Float64(angle / 180.0) <= 1e+205)
		tmp = Float64(Float64(a_m + b_m) * Float64(Float64(b_m - a_m) * Float64(Float64(pi * angle) * 0.011111111111111112)));
	elseif (Float64(angle / 180.0) <= 5e+261)
		tmp = Float64(Float64(Float64(angle * 0.011111111111111112) * Float64(pi * Float64(Float64(b_m * b_m) - Float64(a_m * a_m)))) * Float64(1.0 + Float64(Float64(-1.54320987654321e-5 * Float64(angle * angle)) * Float64(pi * pi))));
	else
		tmp = Float64(Float64(2.0 * Float64(b_m * b_m)) * sin(Float64(pi * Float64(angle / 180.0))));
	end
	return tmp
end

MATLAB

a_m = abs(a);
b_m = abs(b);
function tmp_2 = code(a_m, b_m, angle)
	tmp = 0.0;
	if ((angle / 180.0) <= 1e+205)
		tmp = (a_m + b_m) * ((b_m - a_m) * ((pi * angle) * 0.011111111111111112));
	elseif ((angle / 180.0) <= 5e+261)
		tmp = ((angle * 0.011111111111111112) * (pi * ((b_m * b_m) - (a_m * a_m)))) * (1.0 + ((-1.54320987654321e-5 * (angle * angle)) * (pi * pi)));
	else
		tmp = (2.0 * (b_m * b_m)) * sin((pi * (angle / 180.0)));
	end
	tmp_2 = tmp;
end

Wolfram

a_m = N[Abs[a], $MachinePrecision]
b_m = N[Abs[b], $MachinePrecision]
code[a$95$m_, b$95$m_, angle_] := If[LessEqual[N[(angle / 180.0), $MachinePrecision], 1e+205], N[(N[(a$95$m + b$95$m), $MachinePrecision] * N[(N[(b$95$m - a$95$m), $MachinePrecision] * N[(N[(Pi * angle), $MachinePrecision] * 0.011111111111111112), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[N[(angle / 180.0), $MachinePrecision], 5e+261], N[(N[(N[(angle * 0.011111111111111112), $MachinePrecision] * N[(Pi * N[(N[(b$95$m * b$95$m), $MachinePrecision] - N[(a$95$m * a$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(1.0 + N[(N[(-1.54320987654321e-5 * N[(angle * angle), $MachinePrecision]), $MachinePrecision] * N[(Pi * Pi), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(2.0 * N[(b$95$m * b$95$m), $MachinePrecision]), $MachinePrecision] * N[Sin[N[(Pi * N[(angle / 180.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if (/.f64 angle #s(literal 180 binary64)) < 1.00000000000000002e205

   1. Initial program 63.1%

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

      1. associate-*l* N/A

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

      2. *-commutative N/A

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

      3. associate-*l* N/A

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

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

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

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

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

      6. associate-*r/ N/A

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

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

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

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

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

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

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

      10. *-commutative N/A

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

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

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

   3. Simplified 64.6%

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

   5. Taylor expanded in angle around 0

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

      1. associate-*r* N/A

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

      2. associate-*r* N/A

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

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

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

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

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

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

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

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

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

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

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

      8. unpow2 N/A

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

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

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

      10. unpow2 N/A

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

      11. *-lowering-*.f64 61.7%

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

   7. Simplified 61.7%

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

      1. *-commutative N/A

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

      2. difference-of-squares N/A

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

      3. associate-*l* N/A

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

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

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

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

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

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

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

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

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

      8. *-commutative N/A

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

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

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

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

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

      11. PI-lowering-PI.f64 71.2%

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

   9. Applied egg-rr 71.2%

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

   if 1.00000000000000002e205 < (/.f64 angle #s(literal 180 binary64)) < 5.0000000000000001e261

   1. Initial program 33.9%

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

      1. pow2 N/A

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

      2. pow2 N/A

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

      3. flip-- N/A

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

      4. clear-num N/A

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

      5. un-div-inv N/A

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

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

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

      7. clear-num N/A

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

      8. flip-- N/A

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

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

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

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

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

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

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

      12. *-lowering-*.f64 33.9%

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

   4. Applied egg-rr 33.9%

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

   5. Taylor expanded in angle around 0

      \[\leadsto \mathsf{*.f64}\left(\color{blue}{\left(\frac{1}{90} \cdot \left(angle \cdot \left(\mathsf{PI}\left(\right) \cdot \left({b}^{2} - {a}^{2}\right)\right)\right)\right)}, \mathsf{cos.f64}\left(\mathsf{*.f64}\left(\mathsf{PI.f64}\left(\right), \mathsf{/.f64}\left(angle, 180\right)\right)\right)\right) \]
   6. Step-by-step derivation

      1. associate-*r* N/A

         \[\leadsto \mathsf{*.f64}\left(\left(\left(\frac{1}{90} \cdot angle\right) \cdot \left(\mathsf{PI}\left(\right) \cdot \left({b}^{2} - {a}^{2}\right)\right)\right), \mathsf{cos.f64}\left(\color{blue}{\mathsf{*.f64}\left(\mathsf{PI.f64}\left(\right), \mathsf{/.f64}\left(angle, 180\right)\right)}\right)\right) \]

      2. unpow2 N/A

         \[\leadsto \mathsf{*.f64}\left(\left(\left(\frac{1}{90} \cdot angle\right) \cdot \left(\mathsf{PI}\left(\right) \cdot \left(b \cdot b - {a}^{2}\right)\right)\right), \mathsf{cos.f64}\left(\mathsf{*.f64}\left(\mathsf{PI.f64}\left(\right), \mathsf{/.f64}\left(angle, 180\right)\right)\right)\right) \]

      3. unpow2 N/A

         \[\leadsto \mathsf{*.f64}\left(\left(\left(\frac{1}{90} \cdot angle\right) \cdot \left(\mathsf{PI}\left(\right) \cdot \left(b \cdot b - a \cdot a\right)\right)\right), \mathsf{cos.f64}\left(\mathsf{*.f64}\left(\mathsf{PI.f64}\left(\right), \mathsf{/.f64}\left(angle, 180\right)\right)\right)\right) \]

      4. difference-of-squares N/A

         \[\leadsto \mathsf{*.f64}\left(\left(\left(\frac{1}{90} \cdot angle\right) \cdot \left(\mathsf{PI}\left(\right) \cdot \left(\left(b + a\right) \cdot \left(b - a\right)\right)\right)\right), \mathsf{cos.f64}\left(\mathsf{*.f64}\left(\mathsf{PI.f64}\left(\right), \mathsf{/.f64}\left(angle, \color{blue}{180}\right)\right)\right)\right) \]

      5. +-commutative N/A

         \[\leadsto \mathsf{*.f64}\left(\left(\left(\frac{1}{90} \cdot angle\right) \cdot \left(\mathsf{PI}\left(\right) \cdot \left(\left(a + b\right) \cdot \left(b - a\right)\right)\right)\right), \mathsf{cos.f64}\left(\mathsf{*.f64}\left(\mathsf{PI.f64}\left(\right), \mathsf{/.f64}\left(angle, 180\right)\right)\right)\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\left(\frac{1}{90} \cdot angle\right), \left(\mathsf{PI}\left(\right) \cdot \left(\left(a + b\right) \cdot \left(b - a\right)\right)\right)\right), \mathsf{cos.f64}\left(\color{blue}{\mathsf{*.f64}\left(\mathsf{PI.f64}\left(\right), \mathsf{/.f64}\left(angle, 180\right)\right)}\right)\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{1}{90}, angle\right), \left(\mathsf{PI}\left(\right) \cdot \left(\left(a + b\right) \cdot \left(b - a\right)\right)\right)\right), \mathsf{cos.f64}\left(\mathsf{*.f64}\left(\color{blue}{\mathsf{PI.f64}\left(\right)}, \mathsf{/.f64}\left(angle, 180\right)\right)\right)\right) \]

      8. +-commutative N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{1}{90}, angle\right), \left(\mathsf{PI}\left(\right) \cdot \left(\left(b + a\right) \cdot \left(b - a\right)\right)\right)\right), \mathsf{cos.f64}\left(\mathsf{*.f64}\left(\mathsf{PI.f64}\left(\right), \mathsf{/.f64}\left(angle, 180\right)\right)\right)\right) \]

      9. difference-of-squares N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{1}{90}, angle\right), \left(\mathsf{PI}\left(\right) \cdot \left(b \cdot b - a \cdot a\right)\right)\right), \mathsf{cos.f64}\left(\mathsf{*.f64}\left(\mathsf{PI.f64}\left(\right), \mathsf{/.f64}\left(angle, \color{blue}{180}\right)\right)\right)\right) \]

      10. unpow2 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{1}{90}, angle\right), \left(\mathsf{PI}\left(\right) \cdot \left({b}^{2} - a \cdot a\right)\right)\right), \mathsf{cos.f64}\left(\mathsf{*.f64}\left(\mathsf{PI.f64}\left(\right), \mathsf{/.f64}\left(angle, 180\right)\right)\right)\right) \]

      11. unpow2 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{1}{90}, angle\right), \left(\mathsf{PI}\left(\right) \cdot \left({b}^{2} - {a}^{2}\right)\right)\right), \mathsf{cos.f64}\left(\mathsf{*.f64}\left(\mathsf{PI.f64}\left(\right), \mathsf{/.f64}\left(angle, 180\right)\right)\right)\right) \]

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

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{1}{90}, angle\right), \mathsf{*.f64}\left(\mathsf{PI}\left(\right), \left({b}^{2} - {a}^{2}\right)\right)\right), \mathsf{cos.f64}\left(\mathsf{*.f64}\left(\mathsf{PI.f64}\left(\right), \color{blue}{\mathsf{/.f64}\left(angle, 180\right)}\right)\right)\right) \]

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

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{1}{90}, angle\right), \mathsf{*.f64}\left(\mathsf{PI.f64}\left(\right), \left({b}^{2} - {a}^{2}\right)\right)\right), \mathsf{cos.f64}\left(\mathsf{*.f64}\left(\mathsf{PI.f64}\left(\right), \mathsf{/.f64}\left(\color{blue}{angle}, 180\right)\right)\right)\right) \]

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

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{1}{90}, angle\right), \mathsf{*.f64}\left(\mathsf{PI.f64}\left(\right), \mathsf{\_.f64}\left(\left({b}^{2}\right), \left({a}^{2}\right)\right)\right)\right), \mathsf{cos.f64}\left(\mathsf{*.f64}\left(\mathsf{PI.f64}\left(\right), \mathsf{/.f64}\left(angle, \color{blue}{180}\right)\right)\right)\right) \]

      15. unpow2 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{1}{90}, angle\right), \mathsf{*.f64}\left(\mathsf{PI.f64}\left(\right), \mathsf{\_.f64}\left(\left(b \cdot b\right), \left({a}^{2}\right)\right)\right)\right), \mathsf{cos.f64}\left(\mathsf{*.f64}\left(\mathsf{PI.f64}\left(\right), \mathsf{/.f64}\left(angle, 180\right)\right)\right)\right) \]

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

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{1}{90}, angle\right), \mathsf{*.f64}\left(\mathsf{PI.f64}\left(\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(b, b\right), \left({a}^{2}\right)\right)\right)\right), \mathsf{cos.f64}\left(\mathsf{*.f64}\left(\mathsf{PI.f64}\left(\right), \mathsf{/.f64}\left(angle, 180\right)\right)\right)\right) \]

      17. unpow2 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{1}{90}, angle\right), \mathsf{*.f64}\left(\mathsf{PI.f64}\left(\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(b, b\right), \left(a \cdot a\right)\right)\right)\right), \mathsf{cos.f64}\left(\mathsf{*.f64}\left(\mathsf{PI.f64}\left(\right), \mathsf{/.f64}\left(angle, 180\right)\right)\right)\right) \]

      18. *-lowering-*.f64 19.4%

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{1}{90}, angle\right), \mathsf{*.f64}\left(\mathsf{PI.f64}\left(\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(b, b\right), \mathsf{*.f64}\left(a, a\right)\right)\right)\right), \mathsf{cos.f64}\left(\mathsf{*.f64}\left(\mathsf{PI.f64}\left(\right), \mathsf{/.f64}\left(angle, 180\right)\right)\right)\right) \]

   7. Simplified 19.4%

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

   8. Taylor expanded in angle around 0

      \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{1}{90}, angle\right), \mathsf{*.f64}\left(\mathsf{PI.f64}\left(\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(b, b\right), \mathsf{*.f64}\left(a, a\right)\right)\right)\right), \color{blue}{\left(1 + \frac{-1}{64800} \cdot \left({angle}^{2} \cdot {\mathsf{PI}\left(\right)}^{2}\right)\right)}\right) \]
   9. Step-by-step derivation

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

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{1}{90}, angle\right), \mathsf{*.f64}\left(\mathsf{PI.f64}\left(\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(b, b\right), \mathsf{*.f64}\left(a, a\right)\right)\right)\right), \mathsf{+.f64}\left(1, \color{blue}{\left(\frac{-1}{64800} \cdot \left({angle}^{2} \cdot {\mathsf{PI}\left(\right)}^{2}\right)\right)}\right)\right) \]

      2. associate-*r* N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{1}{90}, angle\right), \mathsf{*.f64}\left(\mathsf{PI.f64}\left(\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(b, b\right), \mathsf{*.f64}\left(a, a\right)\right)\right)\right), \mathsf{+.f64}\left(1, \left(\left(\frac{-1}{64800} \cdot {angle}^{2}\right) \cdot \color{blue}{{\mathsf{PI}\left(\right)}^{2}}\right)\right)\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{1}{90}, angle\right), \mathsf{*.f64}\left(\mathsf{PI.f64}\left(\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(b, b\right), \mathsf{*.f64}\left(a, a\right)\right)\right)\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\left(\frac{-1}{64800} \cdot {angle}^{2}\right), \color{blue}{\left({\mathsf{PI}\left(\right)}^{2}\right)}\right)\right)\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{1}{90}, angle\right), \mathsf{*.f64}\left(\mathsf{PI.f64}\left(\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(b, b\right), \mathsf{*.f64}\left(a, a\right)\right)\right)\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{-1}{64800}, \left({angle}^{2}\right)\right), \left({\color{blue}{\mathsf{PI}\left(\right)}}^{2}\right)\right)\right)\right) \]

      5. unpow2 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{1}{90}, angle\right), \mathsf{*.f64}\left(\mathsf{PI.f64}\left(\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(b, b\right), \mathsf{*.f64}\left(a, a\right)\right)\right)\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{-1}{64800}, \left(angle \cdot angle\right)\right), \left({\mathsf{PI}\left(\right)}^{2}\right)\right)\right)\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{1}{90}, angle\right), \mathsf{*.f64}\left(\mathsf{PI.f64}\left(\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(b, b\right), \mathsf{*.f64}\left(a, a\right)\right)\right)\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{-1}{64800}, \mathsf{*.f64}\left(angle, angle\right)\right), \left({\mathsf{PI}\left(\right)}^{2}\right)\right)\right)\right) \]

      7. unpow2 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{1}{90}, angle\right), \mathsf{*.f64}\left(\mathsf{PI.f64}\left(\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(b, b\right), \mathsf{*.f64}\left(a, a\right)\right)\right)\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{-1}{64800}, \mathsf{*.f64}\left(angle, angle\right)\right), \left(\mathsf{PI}\left(\right) \cdot \color{blue}{\mathsf{PI}\left(\right)}\right)\right)\right)\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{1}{90}, angle\right), \mathsf{*.f64}\left(\mathsf{PI.f64}\left(\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(b, b\right), \mathsf{*.f64}\left(a, a\right)\right)\right)\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{-1}{64800}, \mathsf{*.f64}\left(angle, angle\right)\right), \mathsf{*.f64}\left(\mathsf{PI}\left(\right), \color{blue}{\mathsf{PI}\left(\right)}\right)\right)\right)\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{1}{90}, angle\right), \mathsf{*.f64}\left(\mathsf{PI.f64}\left(\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(b, b\right), \mathsf{*.f64}\left(a, a\right)\right)\right)\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{-1}{64800}, \mathsf{*.f64}\left(angle, angle\right)\right), \mathsf{*.f64}\left(\mathsf{PI.f64}\left(\right), \mathsf{PI}\left(\right)\right)\right)\right)\right) \]

      10. PI-lowering-PI.f64 26.8%

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{1}{90}, angle\right), \mathsf{*.f64}\left(\mathsf{PI.f64}\left(\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(b, b\right), \mathsf{*.f64}\left(a, a\right)\right)\right)\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{-1}{64800}, \mathsf{*.f64}\left(angle, angle\right)\right), \mathsf{*.f64}\left(\mathsf{PI.f64}\left(\right), \mathsf{PI.f64}\left(\right)\right)\right)\right)\right) \]

   10. Simplified 26.8%

       \[\leadsto \left(\left(0.011111111111111112 \cdot angle\right) \cdot \left(\pi \cdot \left(b \cdot b - a \cdot a\right)\right)\right) \cdot \color{blue}{\left(1 + \left(-1.54320987654321 \cdot 10^{-5} \cdot \left(angle \cdot angle\right)\right) \cdot \left(\pi \cdot \pi\right)\right)} \]

   if 5.0000000000000001e261 < (/.f64 angle #s(literal 180 binary64))

   1. Initial program 24.3%

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

      1. pow2 N/A

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

      2. pow2 N/A

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

      3. flip-- N/A

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

      4. clear-num N/A

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

      5. un-div-inv N/A

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

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

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

      7. clear-num N/A

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

      8. flip-- N/A

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

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

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

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

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

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

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

      12. *-lowering-*.f64 24.3%

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

   4. Applied egg-rr 24.3%

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

   5. Taylor expanded in angle around 0

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

   7. Simplified 40.4%

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

   8. Taylor expanded in b around inf

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

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

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

      2. unpow2 N/A

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

      3. *-lowering-*.f64 62.2%

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

   10. Simplified 62.2%

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

4. Final simplification 69.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{angle}{180} \leq 10^{+205}:\\ \;\;\;\;\left(a + b\right) \cdot \left(\left(b - a\right) \cdot \left(\left(\pi \cdot angle\right) \cdot 0.011111111111111112\right)\right)\\ \mathbf{elif}\;\frac{angle}{180} \leq 5 \cdot 10^{+261}:\\ \;\;\;\;\left(\left(angle \cdot 0.011111111111111112\right) \cdot \left(\pi \cdot \left(b \cdot b - a \cdot a\right)\right)\right) \cdot \left(1 + \left(-1.54320987654321 \cdot 10^{-5} \cdot \left(angle \cdot angle\right)\right) \cdot \left(\pi \cdot \pi\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(2 \cdot \left(b \cdot b\right)\right) \cdot \sin \left(\pi \cdot \frac{angle}{180}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 62.0% accurate, 3.5× speedup

Math

\[\begin{array}{l} a_m = \left|a\right| \\ b_m = \left|b\right| \\ \begin{array}{l} \mathbf{if}\;angle \leq 2.2 \cdot 10^{+208}:\\ \;\;\;\;\left(a\_m + b\_m\right) \cdot \left(\left(b\_m - a\_m\right) \cdot \left(\left(\pi \cdot angle\right) \cdot 0.011111111111111112\right)\right)\\ \mathbf{elif}\;angle \leq 5.2 \cdot 10^{+263}:\\ \;\;\;\;\left(\left(angle \cdot 0.011111111111111112\right) \cdot \left(\pi \cdot \left(b\_m \cdot b\_m - a\_m \cdot a\_m\right)\right)\right) \cdot \left(1 + \left(-1.54320987654321 \cdot 10^{-5} \cdot \left(angle \cdot angle\right)\right) \cdot \left(\pi \cdot \pi\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\sin \left(\left(\pi \cdot angle\right) \cdot 0.005555555555555556\right) \cdot \left(2 \cdot \left(b\_m \cdot b\_m\right)\right)\\ \end{array} \end{array} \]

FPCore

a_m = (fabs.f64 a)
b_m = (fabs.f64 b)
(FPCore (a_m b_m angle)
 :precision binary64
 (if (<= angle 2.2e+208)
   (* (+ a_m b_m) (* (- b_m a_m) (* (* PI angle) 0.011111111111111112)))
   (if (<= angle 5.2e+263)
     (*
      (* (* angle 0.011111111111111112) (* PI (- (* b_m b_m) (* a_m a_m))))
      (+ 1.0 (* (* -1.54320987654321e-5 (* angle angle)) (* PI PI))))
     (* (sin (* (* PI angle) 0.005555555555555556)) (* 2.0 (* b_m b_m))))))

C

a_m = fabs(a);
b_m = fabs(b);
double code(double a_m, double b_m, double angle) {
	double tmp;
	if (angle <= 2.2e+208) {
		tmp = (a_m + b_m) * ((b_m - a_m) * ((((double) M_PI) * angle) * 0.011111111111111112));
	} else if (angle <= 5.2e+263) {
		tmp = ((angle * 0.011111111111111112) * (((double) M_PI) * ((b_m * b_m) - (a_m * a_m)))) * (1.0 + ((-1.54320987654321e-5 * (angle * angle)) * (((double) M_PI) * ((double) M_PI))));
	} else {
		tmp = sin(((((double) M_PI) * angle) * 0.005555555555555556)) * (2.0 * (b_m * b_m));
	}
	return tmp;
}

Java

a_m = Math.abs(a);
b_m = Math.abs(b);
public static double code(double a_m, double b_m, double angle) {
	double tmp;
	if (angle <= 2.2e+208) {
		tmp = (a_m + b_m) * ((b_m - a_m) * ((Math.PI * angle) * 0.011111111111111112));
	} else if (angle <= 5.2e+263) {
		tmp = ((angle * 0.011111111111111112) * (Math.PI * ((b_m * b_m) - (a_m * a_m)))) * (1.0 + ((-1.54320987654321e-5 * (angle * angle)) * (Math.PI * Math.PI)));
	} else {
		tmp = Math.sin(((Math.PI * angle) * 0.005555555555555556)) * (2.0 * (b_m * b_m));
	}
	return tmp;
}

Python

a_m = math.fabs(a)
b_m = math.fabs(b)
def code(a_m, b_m, angle):
	tmp = 0
	if angle <= 2.2e+208:
		tmp = (a_m + b_m) * ((b_m - a_m) * ((math.pi * angle) * 0.011111111111111112))
	elif angle <= 5.2e+263:
		tmp = ((angle * 0.011111111111111112) * (math.pi * ((b_m * b_m) - (a_m * a_m)))) * (1.0 + ((-1.54320987654321e-5 * (angle * angle)) * (math.pi * math.pi)))
	else:
		tmp = math.sin(((math.pi * angle) * 0.005555555555555556)) * (2.0 * (b_m * b_m))
	return tmp

Julia

a_m = abs(a)
b_m = abs(b)
function code(a_m, b_m, angle)
	tmp = 0.0
	if (angle <= 2.2e+208)
		tmp = Float64(Float64(a_m + b_m) * Float64(Float64(b_m - a_m) * Float64(Float64(pi * angle) * 0.011111111111111112)));
	elseif (angle <= 5.2e+263)
		tmp = Float64(Float64(Float64(angle * 0.011111111111111112) * Float64(pi * Float64(Float64(b_m * b_m) - Float64(a_m * a_m)))) * Float64(1.0 + Float64(Float64(-1.54320987654321e-5 * Float64(angle * angle)) * Float64(pi * pi))));
	else
		tmp = Float64(sin(Float64(Float64(pi * angle) * 0.005555555555555556)) * Float64(2.0 * Float64(b_m * b_m)));
	end
	return tmp
end

MATLAB

a_m = abs(a);
b_m = abs(b);
function tmp_2 = code(a_m, b_m, angle)
	tmp = 0.0;
	if (angle <= 2.2e+208)
		tmp = (a_m + b_m) * ((b_m - a_m) * ((pi * angle) * 0.011111111111111112));
	elseif (angle <= 5.2e+263)
		tmp = ((angle * 0.011111111111111112) * (pi * ((b_m * b_m) - (a_m * a_m)))) * (1.0 + ((-1.54320987654321e-5 * (angle * angle)) * (pi * pi)));
	else
		tmp = sin(((pi * angle) * 0.005555555555555556)) * (2.0 * (b_m * b_m));
	end
	tmp_2 = tmp;
end

Wolfram

a_m = N[Abs[a], $MachinePrecision]
b_m = N[Abs[b], $MachinePrecision]
code[a$95$m_, b$95$m_, angle_] := If[LessEqual[angle, 2.2e+208], N[(N[(a$95$m + b$95$m), $MachinePrecision] * N[(N[(b$95$m - a$95$m), $MachinePrecision] * N[(N[(Pi * angle), $MachinePrecision] * 0.011111111111111112), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[angle, 5.2e+263], N[(N[(N[(angle * 0.011111111111111112), $MachinePrecision] * N[(Pi * N[(N[(b$95$m * b$95$m), $MachinePrecision] - N[(a$95$m * a$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(1.0 + N[(N[(-1.54320987654321e-5 * N[(angle * angle), $MachinePrecision]), $MachinePrecision] * N[(Pi * Pi), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[Sin[N[(N[(Pi * angle), $MachinePrecision] * 0.005555555555555556), $MachinePrecision]], $MachinePrecision] * N[(2.0 * N[(b$95$m * b$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if angle < 2.20000000000000014e208

   1. Initial program 63.1%

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

      1. associate-*l* N/A

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

      2. *-commutative N/A

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

      3. associate-*l* N/A

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

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

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

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

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

      6. associate-*r/ N/A

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

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

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

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

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

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

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

      10. *-commutative N/A

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

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

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

   3. Simplified 64.6%

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

   5. Taylor expanded in angle around 0

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

      1. associate-*r* N/A

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

      2. associate-*r* N/A

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

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

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

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

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

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

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

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

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

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

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

      8. unpow2 N/A

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

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

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

      10. unpow2 N/A

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

      11. *-lowering-*.f64 61.7%

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

   7. Simplified 61.7%

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

      1. *-commutative N/A

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

      2. difference-of-squares N/A

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

      3. associate-*l* N/A

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

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

         \[\leadsto \mathsf{*.f64}\left(\left(b + a\right), \color{blue}{\left(\left(b - a\right) \cdot \left(\frac{1}{90} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)\right)}\right) \]

      5. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(b, a\right), \left(\color{blue}{\left(b - a\right)} \cdot \left(\frac{1}{90} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)\right)\right) \]

      6. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(b, a\right), \mathsf{*.f64}\left(\left(b - a\right), \color{blue}{\left(\frac{1}{90} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}\right)\right) \]

      7. --lowering--.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(b, a\right), \mathsf{*.f64}\left(\mathsf{\_.f64}\left(b, a\right), \left(\color{blue}{\frac{1}{90}} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)\right)\right) \]

      8. *-commutative N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(b, a\right), \mathsf{*.f64}\left(\mathsf{\_.f64}\left(b, a\right), \left(\frac{1}{90} \cdot \left(\mathsf{PI}\left(\right) \cdot \color{blue}{angle}\right)\right)\right)\right) \]

      9. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(b, a\right), \mathsf{*.f64}\left(\mathsf{\_.f64}\left(b, a\right), \mathsf{*.f64}\left(\frac{1}{90}, \color{blue}{\left(\mathsf{PI}\left(\right) \cdot angle\right)}\right)\right)\right) \]

      10. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(b, a\right), \mathsf{*.f64}\left(\mathsf{\_.f64}\left(b, a\right), \mathsf{*.f64}\left(\frac{1}{90}, \mathsf{*.f64}\left(\mathsf{PI}\left(\right), \color{blue}{angle}\right)\right)\right)\right) \]

      11. PI-lowering-PI.f64 71.2%

          \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(b, a\right), \mathsf{*.f64}\left(\mathsf{\_.f64}\left(b, a\right), \mathsf{*.f64}\left(\frac{1}{90}, \mathsf{*.f64}\left(\mathsf{PI.f64}\left(\right), angle\right)\right)\right)\right) \]

   9. Applied egg-rr 71.2%

      \[\leadsto \color{blue}{\left(b + a\right) \cdot \left(\left(b - a\right) \cdot \left(0.011111111111111112 \cdot \left(\pi \cdot angle\right)\right)\right)} \]

   if 2.20000000000000014e208 < angle < 5.2000000000000004e263

   1. Initial program 33.9%

      \[\left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin \left(\pi \cdot \frac{angle}{180}\right)\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right) \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. pow2 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\left(2 \cdot \left(b \cdot b - {a}^{2}\right)\right), \mathsf{sin.f64}\left(\mathsf{*.f64}\left(\mathsf{PI.f64}\left(\right), \mathsf{/.f64}\left(angle, 180\right)\right)\right)\right), \mathsf{cos.f64}\left(\mathsf{*.f64}\left(\mathsf{PI.f64}\left(\right), \mathsf{/.f64}\left(angle, 180\right)\right)\right)\right) \]

      2. pow2 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\left(2 \cdot \left(b \cdot b - a \cdot a\right)\right), \mathsf{sin.f64}\left(\mathsf{*.f64}\left(\mathsf{PI.f64}\left(\right), \mathsf{/.f64}\left(angle, 180\right)\right)\right)\right), \mathsf{cos.f64}\left(\mathsf{*.f64}\left(\mathsf{PI.f64}\left(\right), \mathsf{/.f64}\left(angle, 180\right)\right)\right)\right) \]

      3. flip-- N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\left(2 \cdot \frac{\left(b \cdot b\right) \cdot \left(b \cdot b\right) - \left(a \cdot a\right) \cdot \left(a \cdot a\right)}{b \cdot b + a \cdot a}\right), \mathsf{sin.f64}\left(\mathsf{*.f64}\left(\mathsf{PI.f64}\left(\right), \mathsf{/.f64}\left(angle, 180\right)\right)\right)\right), \mathsf{cos.f64}\left(\mathsf{*.f64}\left(\mathsf{PI.f64}\left(\right), \mathsf{/.f64}\left(angle, 180\right)\right)\right)\right) \]

      4. clear-num N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\left(2 \cdot \frac{1}{\frac{b \cdot b + a \cdot a}{\left(b \cdot b\right) \cdot \left(b \cdot b\right) - \left(a \cdot a\right) \cdot \left(a \cdot a\right)}}\right), \mathsf{sin.f64}\left(\mathsf{*.f64}\left(\mathsf{PI.f64}\left(\right), \mathsf{/.f64}\left(angle, 180\right)\right)\right)\right), \mathsf{cos.f64}\left(\mathsf{*.f64}\left(\mathsf{PI.f64}\left(\right), \mathsf{/.f64}\left(angle, 180\right)\right)\right)\right) \]

      5. un-div-inv N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\left(\frac{2}{\frac{b \cdot b + a \cdot a}{\left(b \cdot b\right) \cdot \left(b \cdot b\right) - \left(a \cdot a\right) \cdot \left(a \cdot a\right)}}\right), \mathsf{sin.f64}\left(\mathsf{*.f64}\left(\mathsf{PI.f64}\left(\right), \mathsf{/.f64}\left(angle, 180\right)\right)\right)\right), \mathsf{cos.f64}\left(\mathsf{*.f64}\left(\color{blue}{\mathsf{PI.f64}\left(\right)}, \mathsf{/.f64}\left(angle, 180\right)\right)\right)\right) \]

      6. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(2, \left(\frac{b \cdot b + a \cdot a}{\left(b \cdot b\right) \cdot \left(b \cdot b\right) - \left(a \cdot a\right) \cdot \left(a \cdot a\right)}\right)\right), \mathsf{sin.f64}\left(\mathsf{*.f64}\left(\mathsf{PI.f64}\left(\right), \mathsf{/.f64}\left(angle, 180\right)\right)\right)\right), \mathsf{cos.f64}\left(\mathsf{*.f64}\left(\color{blue}{\mathsf{PI.f64}\left(\right)}, \mathsf{/.f64}\left(angle, 180\right)\right)\right)\right) \]

      7. clear-num N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(2, \left(\frac{1}{\frac{\left(b \cdot b\right) \cdot \left(b \cdot b\right) - \left(a \cdot a\right) \cdot \left(a \cdot a\right)}{b \cdot b + a \cdot a}}\right)\right), \mathsf{sin.f64}\left(\mathsf{*.f64}\left(\mathsf{PI.f64}\left(\right), \mathsf{/.f64}\left(angle, 180\right)\right)\right)\right), \mathsf{cos.f64}\left(\mathsf{*.f64}\left(\mathsf{PI.f64}\left(\right), \mathsf{/.f64}\left(angle, 180\right)\right)\right)\right) \]

      8. flip-- N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(2, \left(\frac{1}{b \cdot b - a \cdot a}\right)\right), \mathsf{sin.f64}\left(\mathsf{*.f64}\left(\mathsf{PI.f64}\left(\right), \mathsf{/.f64}\left(angle, 180\right)\right)\right)\right), \mathsf{cos.f64}\left(\mathsf{*.f64}\left(\mathsf{PI.f64}\left(\right), \mathsf{/.f64}\left(angle, 180\right)\right)\right)\right) \]

      9. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(2, \mathsf{/.f64}\left(1, \left(b \cdot b - a \cdot a\right)\right)\right), \mathsf{sin.f64}\left(\mathsf{*.f64}\left(\mathsf{PI.f64}\left(\right), \mathsf{/.f64}\left(angle, 180\right)\right)\right)\right), \mathsf{cos.f64}\left(\mathsf{*.f64}\left(\mathsf{PI.f64}\left(\right), \mathsf{/.f64}\left(angle, 180\right)\right)\right)\right) \]

      10. --lowering--.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(2, \mathsf{/.f64}\left(1, \mathsf{\_.f64}\left(\left(b \cdot b\right), \left(a \cdot a\right)\right)\right)\right), \mathsf{sin.f64}\left(\mathsf{*.f64}\left(\mathsf{PI.f64}\left(\right), \mathsf{/.f64}\left(angle, 180\right)\right)\right)\right), \mathsf{cos.f64}\left(\mathsf{*.f64}\left(\mathsf{PI.f64}\left(\right), \mathsf{/.f64}\left(angle, 180\right)\right)\right)\right) \]

      11. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(2, \mathsf{/.f64}\left(1, \mathsf{\_.f64}\left(\mathsf{*.f64}\left(b, b\right), \left(a \cdot a\right)\right)\right)\right), \mathsf{sin.f64}\left(\mathsf{*.f64}\left(\mathsf{PI.f64}\left(\right), \mathsf{/.f64}\left(angle, 180\right)\right)\right)\right), \mathsf{cos.f64}\left(\mathsf{*.f64}\left(\mathsf{PI.f64}\left(\right), \mathsf{/.f64}\left(angle, 180\right)\right)\right)\right) \]

      12. *-lowering-*.f64 33.9%

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(2, \mathsf{/.f64}\left(1, \mathsf{\_.f64}\left(\mathsf{*.f64}\left(b, b\right), \mathsf{*.f64}\left(a, a\right)\right)\right)\right), \mathsf{sin.f64}\left(\mathsf{*.f64}\left(\mathsf{PI.f64}\left(\right), \mathsf{/.f64}\left(angle, 180\right)\right)\right)\right), \mathsf{cos.f64}\left(\mathsf{*.f64}\left(\mathsf{PI.f64}\left(\right), \mathsf{/.f64}\left(angle, 180\right)\right)\right)\right) \]

   4. Applied egg-rr 33.9%

      \[\leadsto \left(\color{blue}{\frac{2}{\frac{1}{b \cdot b - a \cdot a}}} \cdot \sin \left(\pi \cdot \frac{angle}{180}\right)\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right) \]

   5. Taylor expanded in angle around 0

      \[\leadsto \mathsf{*.f64}\left(\color{blue}{\left(\frac{1}{90} \cdot \left(angle \cdot \left(\mathsf{PI}\left(\right) \cdot \left({b}^{2} - {a}^{2}\right)\right)\right)\right)}, \mathsf{cos.f64}\left(\mathsf{*.f64}\left(\mathsf{PI.f64}\left(\right), \mathsf{/.f64}\left(angle, 180\right)\right)\right)\right) \]
   6. Step-by-step derivation

      1. associate-*r* N/A

         \[\leadsto \mathsf{*.f64}\left(\left(\left(\frac{1}{90} \cdot angle\right) \cdot \left(\mathsf{PI}\left(\right) \cdot \left({b}^{2} - {a}^{2}\right)\right)\right), \mathsf{cos.f64}\left(\color{blue}{\mathsf{*.f64}\left(\mathsf{PI.f64}\left(\right), \mathsf{/.f64}\left(angle, 180\right)\right)}\right)\right) \]

      2. unpow2 N/A

         \[\leadsto \mathsf{*.f64}\left(\left(\left(\frac{1}{90} \cdot angle\right) \cdot \left(\mathsf{PI}\left(\right) \cdot \left(b \cdot b - {a}^{2}\right)\right)\right), \mathsf{cos.f64}\left(\mathsf{*.f64}\left(\mathsf{PI.f64}\left(\right), \mathsf{/.f64}\left(angle, 180\right)\right)\right)\right) \]

      3. unpow2 N/A

         \[\leadsto \mathsf{*.f64}\left(\left(\left(\frac{1}{90} \cdot angle\right) \cdot \left(\mathsf{PI}\left(\right) \cdot \left(b \cdot b - a \cdot a\right)\right)\right), \mathsf{cos.f64}\left(\mathsf{*.f64}\left(\mathsf{PI.f64}\left(\right), \mathsf{/.f64}\left(angle, 180\right)\right)\right)\right) \]

      4. difference-of-squares N/A

         \[\leadsto \mathsf{*.f64}\left(\left(\left(\frac{1}{90} \cdot angle\right) \cdot \left(\mathsf{PI}\left(\right) \cdot \left(\left(b + a\right) \cdot \left(b - a\right)\right)\right)\right), \mathsf{cos.f64}\left(\mathsf{*.f64}\left(\mathsf{PI.f64}\left(\right), \mathsf{/.f64}\left(angle, \color{blue}{180}\right)\right)\right)\right) \]

      5. +-commutative N/A

         \[\leadsto \mathsf{*.f64}\left(\left(\left(\frac{1}{90} \cdot angle\right) \cdot \left(\mathsf{PI}\left(\right) \cdot \left(\left(a + b\right) \cdot \left(b - a\right)\right)\right)\right), \mathsf{cos.f64}\left(\mathsf{*.f64}\left(\mathsf{PI.f64}\left(\right), \mathsf{/.f64}\left(angle, 180\right)\right)\right)\right) \]

      6. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\left(\frac{1}{90} \cdot angle\right), \left(\mathsf{PI}\left(\right) \cdot \left(\left(a + b\right) \cdot \left(b - a\right)\right)\right)\right), \mathsf{cos.f64}\left(\color{blue}{\mathsf{*.f64}\left(\mathsf{PI.f64}\left(\right), \mathsf{/.f64}\left(angle, 180\right)\right)}\right)\right) \]

      7. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{1}{90}, angle\right), \left(\mathsf{PI}\left(\right) \cdot \left(\left(a + b\right) \cdot \left(b - a\right)\right)\right)\right), \mathsf{cos.f64}\left(\mathsf{*.f64}\left(\color{blue}{\mathsf{PI.f64}\left(\right)}, \mathsf{/.f64}\left(angle, 180\right)\right)\right)\right) \]

      8. +-commutative N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{1}{90}, angle\right), \left(\mathsf{PI}\left(\right) \cdot \left(\left(b + a\right) \cdot \left(b - a\right)\right)\right)\right), \mathsf{cos.f64}\left(\mathsf{*.f64}\left(\mathsf{PI.f64}\left(\right), \mathsf{/.f64}\left(angle, 180\right)\right)\right)\right) \]

      9. difference-of-squares N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{1}{90}, angle\right), \left(\mathsf{PI}\left(\right) \cdot \left(b \cdot b - a \cdot a\right)\right)\right), \mathsf{cos.f64}\left(\mathsf{*.f64}\left(\mathsf{PI.f64}\left(\right), \mathsf{/.f64}\left(angle, \color{blue}{180}\right)\right)\right)\right) \]

      10. unpow2 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{1}{90}, angle\right), \left(\mathsf{PI}\left(\right) \cdot \left({b}^{2} - a \cdot a\right)\right)\right), \mathsf{cos.f64}\left(\mathsf{*.f64}\left(\mathsf{PI.f64}\left(\right), \mathsf{/.f64}\left(angle, 180\right)\right)\right)\right) \]

      11. unpow2 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{1}{90}, angle\right), \left(\mathsf{PI}\left(\right) \cdot \left({b}^{2} - {a}^{2}\right)\right)\right), \mathsf{cos.f64}\left(\mathsf{*.f64}\left(\mathsf{PI.f64}\left(\right), \mathsf{/.f64}\left(angle, 180\right)\right)\right)\right) \]

      12. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{1}{90}, angle\right), \mathsf{*.f64}\left(\mathsf{PI}\left(\right), \left({b}^{2} - {a}^{2}\right)\right)\right), \mathsf{cos.f64}\left(\mathsf{*.f64}\left(\mathsf{PI.f64}\left(\right), \color{blue}{\mathsf{/.f64}\left(angle, 180\right)}\right)\right)\right) \]

      13. PI-lowering-PI.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{1}{90}, angle\right), \mathsf{*.f64}\left(\mathsf{PI.f64}\left(\right), \left({b}^{2} - {a}^{2}\right)\right)\right), \mathsf{cos.f64}\left(\mathsf{*.f64}\left(\mathsf{PI.f64}\left(\right), \mathsf{/.f64}\left(\color{blue}{angle}, 180\right)\right)\right)\right) \]

      14. --lowering--.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{1}{90}, angle\right), \mathsf{*.f64}\left(\mathsf{PI.f64}\left(\right), \mathsf{\_.f64}\left(\left({b}^{2}\right), \left({a}^{2}\right)\right)\right)\right), \mathsf{cos.f64}\left(\mathsf{*.f64}\left(\mathsf{PI.f64}\left(\right), \mathsf{/.f64}\left(angle, \color{blue}{180}\right)\right)\right)\right) \]

      15. unpow2 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{1}{90}, angle\right), \mathsf{*.f64}\left(\mathsf{PI.f64}\left(\right), \mathsf{\_.f64}\left(\left(b \cdot b\right), \left({a}^{2}\right)\right)\right)\right), \mathsf{cos.f64}\left(\mathsf{*.f64}\left(\mathsf{PI.f64}\left(\right), \mathsf{/.f64}\left(angle, 180\right)\right)\right)\right) \]

      16. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{1}{90}, angle\right), \mathsf{*.f64}\left(\mathsf{PI.f64}\left(\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(b, b\right), \left({a}^{2}\right)\right)\right)\right), \mathsf{cos.f64}\left(\mathsf{*.f64}\left(\mathsf{PI.f64}\left(\right), \mathsf{/.f64}\left(angle, 180\right)\right)\right)\right) \]

      17. unpow2 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{1}{90}, angle\right), \mathsf{*.f64}\left(\mathsf{PI.f64}\left(\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(b, b\right), \left(a \cdot a\right)\right)\right)\right), \mathsf{cos.f64}\left(\mathsf{*.f64}\left(\mathsf{PI.f64}\left(\right), \mathsf{/.f64}\left(angle, 180\right)\right)\right)\right) \]

      18. *-lowering-*.f64 19.4%

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{1}{90}, angle\right), \mathsf{*.f64}\left(\mathsf{PI.f64}\left(\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(b, b\right), \mathsf{*.f64}\left(a, a\right)\right)\right)\right), \mathsf{cos.f64}\left(\mathsf{*.f64}\left(\mathsf{PI.f64}\left(\right), \mathsf{/.f64}\left(angle, 180\right)\right)\right)\right) \]

   7. Simplified 19.4%

      \[\leadsto \color{blue}{\left(\left(0.011111111111111112 \cdot angle\right) \cdot \left(\pi \cdot \left(b \cdot b - a \cdot a\right)\right)\right)} \cdot \cos \left(\pi \cdot \frac{angle}{180}\right) \]

   8. Taylor expanded in angle around 0

      \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{1}{90}, angle\right), \mathsf{*.f64}\left(\mathsf{PI.f64}\left(\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(b, b\right), \mathsf{*.f64}\left(a, a\right)\right)\right)\right), \color{blue}{\left(1 + \frac{-1}{64800} \cdot \left({angle}^{2} \cdot {\mathsf{PI}\left(\right)}^{2}\right)\right)}\right) \]
   9. Step-by-step derivation

      1. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{1}{90}, angle\right), \mathsf{*.f64}\left(\mathsf{PI.f64}\left(\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(b, b\right), \mathsf{*.f64}\left(a, a\right)\right)\right)\right), \mathsf{+.f64}\left(1, \color{blue}{\left(\frac{-1}{64800} \cdot \left({angle}^{2} \cdot {\mathsf{PI}\left(\right)}^{2}\right)\right)}\right)\right) \]

      2. associate-*r* N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{1}{90}, angle\right), \mathsf{*.f64}\left(\mathsf{PI.f64}\left(\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(b, b\right), \mathsf{*.f64}\left(a, a\right)\right)\right)\right), \mathsf{+.f64}\left(1, \left(\left(\frac{-1}{64800} \cdot {angle}^{2}\right) \cdot \color{blue}{{\mathsf{PI}\left(\right)}^{2}}\right)\right)\right) \]

      3. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{1}{90}, angle\right), \mathsf{*.f64}\left(\mathsf{PI.f64}\left(\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(b, b\right), \mathsf{*.f64}\left(a, a\right)\right)\right)\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\left(\frac{-1}{64800} \cdot {angle}^{2}\right), \color{blue}{\left({\mathsf{PI}\left(\right)}^{2}\right)}\right)\right)\right) \]

      4. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{1}{90}, angle\right), \mathsf{*.f64}\left(\mathsf{PI.f64}\left(\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(b, b\right), \mathsf{*.f64}\left(a, a\right)\right)\right)\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{-1}{64800}, \left({angle}^{2}\right)\right), \left({\color{blue}{\mathsf{PI}\left(\right)}}^{2}\right)\right)\right)\right) \]

      5. unpow2 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{1}{90}, angle\right), \mathsf{*.f64}\left(\mathsf{PI.f64}\left(\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(b, b\right), \mathsf{*.f64}\left(a, a\right)\right)\right)\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{-1}{64800}, \left(angle \cdot angle\right)\right), \left({\mathsf{PI}\left(\right)}^{2}\right)\right)\right)\right) \]

      6. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{1}{90}, angle\right), \mathsf{*.f64}\left(\mathsf{PI.f64}\left(\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(b, b\right), \mathsf{*.f64}\left(a, a\right)\right)\right)\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{-1}{64800}, \mathsf{*.f64}\left(angle, angle\right)\right), \left({\mathsf{PI}\left(\right)}^{2}\right)\right)\right)\right) \]

      7. unpow2 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{1}{90}, angle\right), \mathsf{*.f64}\left(\mathsf{PI.f64}\left(\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(b, b\right), \mathsf{*.f64}\left(a, a\right)\right)\right)\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{-1}{64800}, \mathsf{*.f64}\left(angle, angle\right)\right), \left(\mathsf{PI}\left(\right) \cdot \color{blue}{\mathsf{PI}\left(\right)}\right)\right)\right)\right) \]

      8. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{1}{90}, angle\right), \mathsf{*.f64}\left(\mathsf{PI.f64}\left(\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(b, b\right), \mathsf{*.f64}\left(a, a\right)\right)\right)\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{-1}{64800}, \mathsf{*.f64}\left(angle, angle\right)\right), \mathsf{*.f64}\left(\mathsf{PI}\left(\right), \color{blue}{\mathsf{PI}\left(\right)}\right)\right)\right)\right) \]

      9. PI-lowering-PI.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{1}{90}, angle\right), \mathsf{*.f64}\left(\mathsf{PI.f64}\left(\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(b, b\right), \mathsf{*.f64}\left(a, a\right)\right)\right)\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{-1}{64800}, \mathsf{*.f64}\left(angle, angle\right)\right), \mathsf{*.f64}\left(\mathsf{PI.f64}\left(\right), \mathsf{PI}\left(\right)\right)\right)\right)\right) \]

      10. PI-lowering-PI.f64 26.8%

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{1}{90}, angle\right), \mathsf{*.f64}\left(\mathsf{PI.f64}\left(\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(b, b\right), \mathsf{*.f64}\left(a, a\right)\right)\right)\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{-1}{64800}, \mathsf{*.f64}\left(angle, angle\right)\right), \mathsf{*.f64}\left(\mathsf{PI.f64}\left(\right), \mathsf{PI.f64}\left(\right)\right)\right)\right)\right) \]

   10. Simplified 26.8%

       \[\leadsto \left(\left(0.011111111111111112 \cdot angle\right) \cdot \left(\pi \cdot \left(b \cdot b - a \cdot a\right)\right)\right) \cdot \color{blue}{\left(1 + \left(-1.54320987654321 \cdot 10^{-5} \cdot \left(angle \cdot angle\right)\right) \cdot \left(\pi \cdot \pi\right)\right)} \]

   if 5.2000000000000004e263 < angle

   1. Initial program 24.3%

      \[\left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin \left(\pi \cdot \frac{angle}{180}\right)\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right) \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. pow2 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\left(2 \cdot \left(b \cdot b - {a}^{2}\right)\right), \mathsf{sin.f64}\left(\mathsf{*.f64}\left(\mathsf{PI.f64}\left(\right), \mathsf{/.f64}\left(angle, 180\right)\right)\right)\right), \mathsf{cos.f64}\left(\mathsf{*.f64}\left(\mathsf{PI.f64}\left(\right), \mathsf{/.f64}\left(angle, 180\right)\right)\right)\right) \]

      2. pow2 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\left(2 \cdot \left(b \cdot b - a \cdot a\right)\right), \mathsf{sin.f64}\left(\mathsf{*.f64}\left(\mathsf{PI.f64}\left(\right), \mathsf{/.f64}\left(angle, 180\right)\right)\right)\right), \mathsf{cos.f64}\left(\mathsf{*.f64}\left(\mathsf{PI.f64}\left(\right), \mathsf{/.f64}\left(angle, 180\right)\right)\right)\right) \]

      3. flip-- N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\left(2 \cdot \frac{\left(b \cdot b\right) \cdot \left(b \cdot b\right) - \left(a \cdot a\right) \cdot \left(a \cdot a\right)}{b \cdot b + a \cdot a}\right), \mathsf{sin.f64}\left(\mathsf{*.f64}\left(\mathsf{PI.f64}\left(\right), \mathsf{/.f64}\left(angle, 180\right)\right)\right)\right), \mathsf{cos.f64}\left(\mathsf{*.f64}\left(\mathsf{PI.f64}\left(\right), \mathsf{/.f64}\left(angle, 180\right)\right)\right)\right) \]

      4. clear-num N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\left(2 \cdot \frac{1}{\frac{b \cdot b + a \cdot a}{\left(b \cdot b\right) \cdot \left(b \cdot b\right) - \left(a \cdot a\right) \cdot \left(a \cdot a\right)}}\right), \mathsf{sin.f64}\left(\mathsf{*.f64}\left(\mathsf{PI.f64}\left(\right), \mathsf{/.f64}\left(angle, 180\right)\right)\right)\right), \mathsf{cos.f64}\left(\mathsf{*.f64}\left(\mathsf{PI.f64}\left(\right), \mathsf{/.f64}\left(angle, 180\right)\right)\right)\right) \]

      5. un-div-inv N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\left(\frac{2}{\frac{b \cdot b + a \cdot a}{\left(b \cdot b\right) \cdot \left(b \cdot b\right) - \left(a \cdot a\right) \cdot \left(a \cdot a\right)}}\right), \mathsf{sin.f64}\left(\mathsf{*.f64}\left(\mathsf{PI.f64}\left(\right), \mathsf{/.f64}\left(angle, 180\right)\right)\right)\right), \mathsf{cos.f64}\left(\mathsf{*.f64}\left(\color{blue}{\mathsf{PI.f64}\left(\right)}, \mathsf{/.f64}\left(angle, 180\right)\right)\right)\right) \]

      6. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(2, \left(\frac{b \cdot b + a \cdot a}{\left(b \cdot b\right) \cdot \left(b \cdot b\right) - \left(a \cdot a\right) \cdot \left(a \cdot a\right)}\right)\right), \mathsf{sin.f64}\left(\mathsf{*.f64}\left(\mathsf{PI.f64}\left(\right), \mathsf{/.f64}\left(angle, 180\right)\right)\right)\right), \mathsf{cos.f64}\left(\mathsf{*.f64}\left(\color{blue}{\mathsf{PI.f64}\left(\right)}, \mathsf{/.f64}\left(angle, 180\right)\right)\right)\right) \]

      7. clear-num N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(2, \left(\frac{1}{\frac{\left(b \cdot b\right) \cdot \left(b \cdot b\right) - \left(a \cdot a\right) \cdot \left(a \cdot a\right)}{b \cdot b + a \cdot a}}\right)\right), \mathsf{sin.f64}\left(\mathsf{*.f64}\left(\mathsf{PI.f64}\left(\right), \mathsf{/.f64}\left(angle, 180\right)\right)\right)\right), \mathsf{cos.f64}\left(\mathsf{*.f64}\left(\mathsf{PI.f64}\left(\right), \mathsf{/.f64}\left(angle, 180\right)\right)\right)\right) \]

      8. flip-- N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(2, \left(\frac{1}{b \cdot b - a \cdot a}\right)\right), \mathsf{sin.f64}\left(\mathsf{*.f64}\left(\mathsf{PI.f64}\left(\right), \mathsf{/.f64}\left(angle, 180\right)\right)\right)\right), \mathsf{cos.f64}\left(\mathsf{*.f64}\left(\mathsf{PI.f64}\left(\right), \mathsf{/.f64}\left(angle, 180\right)\right)\right)\right) \]

      9. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(2, \mathsf{/.f64}\left(1, \left(b \cdot b - a \cdot a\right)\right)\right), \mathsf{sin.f64}\left(\mathsf{*.f64}\left(\mathsf{PI.f64}\left(\right), \mathsf{/.f64}\left(angle, 180\right)\right)\right)\right), \mathsf{cos.f64}\left(\mathsf{*.f64}\left(\mathsf{PI.f64}\left(\right), \mathsf{/.f64}\left(angle, 180\right)\right)\right)\right) \]

      10. --lowering--.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(2, \mathsf{/.f64}\left(1, \mathsf{\_.f64}\left(\left(b \cdot b\right), \left(a \cdot a\right)\right)\right)\right), \mathsf{sin.f64}\left(\mathsf{*.f64}\left(\mathsf{PI.f64}\left(\right), \mathsf{/.f64}\left(angle, 180\right)\right)\right)\right), \mathsf{cos.f64}\left(\mathsf{*.f64}\left(\mathsf{PI.f64}\left(\right), \mathsf{/.f64}\left(angle, 180\right)\right)\right)\right) \]

      11. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(2, \mathsf{/.f64}\left(1, \mathsf{\_.f64}\left(\mathsf{*.f64}\left(b, b\right), \left(a \cdot a\right)\right)\right)\right), \mathsf{sin.f64}\left(\mathsf{*.f64}\left(\mathsf{PI.f64}\left(\right), \mathsf{/.f64}\left(angle, 180\right)\right)\right)\right), \mathsf{cos.f64}\left(\mathsf{*.f64}\left(\mathsf{PI.f64}\left(\right), \mathsf{/.f64}\left(angle, 180\right)\right)\right)\right) \]

      12. *-lowering-*.f64 24.3%

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(2, \mathsf{/.f64}\left(1, \mathsf{\_.f64}\left(\mathsf{*.f64}\left(b, b\right), \mathsf{*.f64}\left(a, a\right)\right)\right)\right), \mathsf{sin.f64}\left(\mathsf{*.f64}\left(\mathsf{PI.f64}\left(\right), \mathsf{/.f64}\left(angle, 180\right)\right)\right)\right), \mathsf{cos.f64}\left(\mathsf{*.f64}\left(\mathsf{PI.f64}\left(\right), \mathsf{/.f64}\left(angle, 180\right)\right)\right)\right) \]

   4. Applied egg-rr 24.3%

      \[\leadsto \left(\color{blue}{\frac{2}{\frac{1}{b \cdot b - a \cdot a}}} \cdot \sin \left(\pi \cdot \frac{angle}{180}\right)\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right) \]

   5. Taylor expanded in angle around 0

      \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(2, \mathsf{/.f64}\left(1, \mathsf{\_.f64}\left(\mathsf{*.f64}\left(b, b\right), \mathsf{*.f64}\left(a, a\right)\right)\right)\right), \mathsf{sin.f64}\left(\mathsf{*.f64}\left(\mathsf{PI.f64}\left(\right), \mathsf{/.f64}\left(angle, 180\right)\right)\right)\right), \color{blue}{1}\right) \]
   6. Step-by-step derivation

   7. Simplified 40.4%

      \[\leadsto \left(\frac{2}{\frac{1}{b \cdot b - a \cdot a}} \cdot \sin \left(\pi \cdot \frac{angle}{180}\right)\right) \cdot \color{blue}{1} \]

   8. Taylor expanded in b around inf

      \[\leadsto \color{blue}{2 \cdot \left({b}^{2} \cdot \sin \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)\right)} \]
   9. Step-by-step derivation

      1. associate-*r* N/A

         \[\leadsto \left(2 \cdot {b}^{2}\right) \cdot \color{blue}{\sin \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)} \]

      2. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\left(2 \cdot {b}^{2}\right), \color{blue}{\sin \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}\right) \]

      3. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(2, \left({b}^{2}\right)\right), \sin \color{blue}{\left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}\right) \]

      4. unpow2 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(2, \left(b \cdot b\right)\right), \sin \left(\frac{1}{180} \cdot \color{blue}{\left(angle \cdot \mathsf{PI}\left(\right)\right)}\right)\right) \]

      5. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(2, \mathsf{*.f64}\left(b, b\right)\right), \sin \left(\frac{1}{180} \cdot \color{blue}{\left(angle \cdot \mathsf{PI}\left(\right)\right)}\right)\right) \]

      6. sin-lowering-sin.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(2, \mathsf{*.f64}\left(b, b\right)\right), \mathsf{sin.f64}\left(\left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)\right)\right) \]

      7. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(2, \mathsf{*.f64}\left(b, b\right)\right), \mathsf{sin.f64}\left(\mathsf{*.f64}\left(\frac{1}{180}, \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)\right)\right) \]

      8. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(2, \mathsf{*.f64}\left(b, b\right)\right), \mathsf{sin.f64}\left(\mathsf{*.f64}\left(\frac{1}{180}, \mathsf{*.f64}\left(angle, \mathsf{PI}\left(\right)\right)\right)\right)\right) \]

      9. PI-lowering-PI.f64 62.2%

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(2, \mathsf{*.f64}\left(b, b\right)\right), \mathsf{sin.f64}\left(\mathsf{*.f64}\left(\frac{1}{180}, \mathsf{*.f64}\left(angle, \mathsf{PI.f64}\left(\right)\right)\right)\right)\right) \]

   10. Simplified 62.2%

       \[\leadsto \color{blue}{\left(2 \cdot \left(b \cdot b\right)\right) \cdot \sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)} \]
3. Recombined 3 regimes into one program.

4. Final simplification 69.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;angle \leq 2.2 \cdot 10^{+208}:\\ \;\;\;\;\left(a + b\right) \cdot \left(\left(b - a\right) \cdot \left(\left(\pi \cdot angle\right) \cdot 0.011111111111111112\right)\right)\\ \mathbf{elif}\;angle \leq 5.2 \cdot 10^{+263}:\\ \;\;\;\;\left(\left(angle \cdot 0.011111111111111112\right) \cdot \left(\pi \cdot \left(b \cdot b - a \cdot a\right)\right)\right) \cdot \left(1 + \left(-1.54320987654321 \cdot 10^{-5} \cdot \left(angle \cdot angle\right)\right) \cdot \left(\pi \cdot \pi\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\sin \left(\left(\pi \cdot angle\right) \cdot 0.005555555555555556\right) \cdot \left(2 \cdot \left(b \cdot b\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 67.4% accurate, 3.7× speedup

Math

\[\begin{array}{l} a_m = \left|a\right| \\ b_m = \left|b\right| \\ \left(a\_m + b\_m\right) \cdot \left(\left(b\_m - a\_m\right) \cdot \sin \left(\left(\pi \cdot angle\right) \cdot 0.011111111111111112\right)\right) \end{array} \]

FPCore

a_m = (fabs.f64 a)
b_m = (fabs.f64 b)
(FPCore (a_m b_m angle)
 :precision binary64
 (* (+ a_m b_m) (* (- b_m a_m) (sin (* (* PI angle) 0.011111111111111112)))))

C

a_m = fabs(a);
b_m = fabs(b);
double code(double a_m, double b_m, double angle) {
	return (a_m + b_m) * ((b_m - a_m) * sin(((((double) M_PI) * angle) * 0.011111111111111112)));
}

Java

a_m = Math.abs(a);
b_m = Math.abs(b);
public static double code(double a_m, double b_m, double angle) {
	return (a_m + b_m) * ((b_m - a_m) * Math.sin(((Math.PI * angle) * 0.011111111111111112)));
}

Python

a_m = math.fabs(a)
b_m = math.fabs(b)
def code(a_m, b_m, angle):
	return (a_m + b_m) * ((b_m - a_m) * math.sin(((math.pi * angle) * 0.011111111111111112)))

Julia

a_m = abs(a)
b_m = abs(b)
function code(a_m, b_m, angle)
	return Float64(Float64(a_m + b_m) * Float64(Float64(b_m - a_m) * sin(Float64(Float64(pi * angle) * 0.011111111111111112))))
end

MATLAB

a_m = abs(a);
b_m = abs(b);
function tmp = code(a_m, b_m, angle)
	tmp = (a_m + b_m) * ((b_m - a_m) * sin(((pi * angle) * 0.011111111111111112)));
end

Wolfram

a_m = N[Abs[a], $MachinePrecision]
b_m = N[Abs[b], $MachinePrecision]
code[a$95$m_, b$95$m_, angle_] := N[(N[(a$95$m + b$95$m), $MachinePrecision] * N[(N[(b$95$m - a$95$m), $MachinePrecision] * N[Sin[N[(N[(Pi * angle), $MachinePrecision] * 0.011111111111111112), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 61.0%

   \[\left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin \left(\pi \cdot \frac{angle}{180}\right)\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right) \]
2. Step-by-step derivation

   1. associate-*l* N/A

      \[\leadsto \left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \color{blue}{\left(\sin \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right) \cdot \cos \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right)\right)} \]

   2. *-commutative N/A

      \[\leadsto \left(\sin \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right) \cdot \cos \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right)\right) \cdot \color{blue}{\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right)} \]

   3. associate-*l* N/A

      \[\leadsto \sin \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right) \cdot \color{blue}{\left(\cos \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right) \cdot \left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right)\right)} \]

   4. *-lowering-*.f64 N/A

      \[\leadsto \mathsf{*.f64}\left(\sin \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right), \color{blue}{\left(\cos \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right) \cdot \left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right)\right)}\right) \]

   5. sin-lowering-sin.f64 N/A

      \[\leadsto \mathsf{*.f64}\left(\mathsf{sin.f64}\left(\left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right)\right), \left(\color{blue}{\cos \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right)} \cdot \left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right)\right)\right) \]

   6. associate-*r/ N/A

      \[\leadsto \mathsf{*.f64}\left(\mathsf{sin.f64}\left(\left(\frac{\mathsf{PI}\left(\right) \cdot angle}{180}\right)\right), \left(\cos \color{blue}{\left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right)} \cdot \left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right)\right)\right) \]

   7. /-lowering-/.f64 N/A

      \[\leadsto \mathsf{*.f64}\left(\mathsf{sin.f64}\left(\mathsf{/.f64}\left(\left(\mathsf{PI}\left(\right) \cdot angle\right), 180\right)\right), \left(\cos \color{blue}{\left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right)} \cdot \left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right)\right)\right) \]

   8. *-lowering-*.f64 N/A

      \[\leadsto \mathsf{*.f64}\left(\mathsf{sin.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{PI}\left(\right), angle\right), 180\right)\right), \left(\cos \left(\color{blue}{\mathsf{PI}\left(\right)} \cdot \frac{angle}{180}\right) \cdot \left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right)\right)\right) \]

   9. PI-lowering-PI.f64 N/A

      \[\leadsto \mathsf{*.f64}\left(\mathsf{sin.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{PI.f64}\left(\right), angle\right), 180\right)\right), \left(\cos \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right) \cdot \left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right)\right)\right) \]

   10. *-commutative N/A

       \[\leadsto \mathsf{*.f64}\left(\mathsf{sin.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{PI.f64}\left(\right), angle\right), 180\right)\right), \left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \color{blue}{\cos \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right)}\right)\right) \]

   11. *-lowering-*.f64 N/A

       \[\leadsto \mathsf{*.f64}\left(\mathsf{sin.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{PI.f64}\left(\right), angle\right), 180\right)\right), \mathsf{*.f64}\left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right), \color{blue}{\cos \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right)}\right)\right) \]

3. Simplified 63.1%

   \[\leadsto \color{blue}{\sin \left(\frac{\pi \cdot angle}{180}\right) \cdot \left(\left(2 \cdot \left(b \cdot b - a \cdot a\right)\right) \cdot \cos \left(\frac{\pi \cdot angle}{180}\right)\right)} \]
4. Add Preprocessing
5. Step-by-step derivation

   1. associate-*r/ N/A

      \[\leadsto \sin \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right) \cdot \left(\left(\color{blue}{2} \cdot \left(b \cdot b - a \cdot a\right)\right) \cdot \cos \left(\frac{\mathsf{PI}\left(\right) \cdot angle}{180}\right)\right) \]

   2. *-commutative N/A

      \[\leadsto \sin \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right) \cdot \left(\cos \left(\frac{\mathsf{PI}\left(\right) \cdot angle}{180}\right) \cdot \color{blue}{\left(2 \cdot \left(b \cdot b - a \cdot a\right)\right)}\right) \]

   3. associate-*r* N/A

      \[\leadsto \left(\sin \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right) \cdot \cos \left(\frac{\mathsf{PI}\left(\right) \cdot angle}{180}\right)\right) \cdot \color{blue}{\left(2 \cdot \left(b \cdot b - a \cdot a\right)\right)} \]

   4. associate-*r/ N/A

      \[\leadsto \left(\sin \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right) \cdot \cos \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right)\right) \cdot \left(2 \cdot \left(b \cdot b - a \cdot a\right)\right) \]

   5. associate-*r* N/A

      \[\leadsto \left(\left(\sin \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right) \cdot \cos \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right)\right) \cdot 2\right) \cdot \color{blue}{\left(b \cdot b - a \cdot a\right)} \]

   6. *-commutative N/A

      \[\leadsto \left(2 \cdot \left(\sin \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right) \cdot \cos \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right)\right)\right) \cdot \left(\color{blue}{b \cdot b} - a \cdot a\right) \]

   7. *-commutative N/A

      \[\leadsto \left(b \cdot b - a \cdot a\right) \cdot \color{blue}{\left(2 \cdot \left(\sin \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right) \cdot \cos \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right)\right)\right)} \]

   8. difference-of-squares N/A

      \[\leadsto \left(\left(b + a\right) \cdot \left(b - a\right)\right) \cdot \left(\color{blue}{2} \cdot \left(\sin \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right) \cdot \cos \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right)\right)\right) \]

   9. associate-*l* N/A

      \[\leadsto \left(b + a\right) \cdot \color{blue}{\left(\left(b - a\right) \cdot \left(2 \cdot \left(\sin \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right) \cdot \cos \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right)\right)\right)\right)} \]

6. Applied egg-rr 72.1%

   \[\leadsto \color{blue}{\left(b + a\right) \cdot \left(\left(b - a\right) \cdot \sin \left(\left(\pi \cdot angle\right) \cdot 0.011111111111111112\right)\right)} \]

7. Final simplification 72.1%

   \[\leadsto \left(a + b\right) \cdot \left(\left(b - a\right) \cdot \sin \left(\left(\pi \cdot angle\right) \cdot 0.011111111111111112\right)\right) \]
8. Add Preprocessing

Alternative 10: 62.1% accurate, 11.0× speedup

Math

\[\begin{array}{l} a_m = \left|a\right| \\ b_m = \left|b\right| \\ \begin{array}{l} \mathbf{if}\;angle \leq 2.2 \cdot 10^{+208}:\\ \;\;\;\;\left(a\_m + b\_m\right) \cdot \left(\left(b\_m - a\_m\right) \cdot \left(\left(\pi \cdot angle\right) \cdot 0.011111111111111112\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{angle \cdot \left(0.011111111111111112 \cdot \left(\pi \cdot \left(b\_m - a\_m\right)\right) + \left(2 \cdot \left(angle \cdot angle\right)\right) \cdot \left(\left(\left(b\_m - a\_m\right) \cdot \left(\pi \cdot \left(\pi \cdot \pi\right)\right)\right) \cdot -1.1431184270690443 \cdot 10^{-7}\right)\right)}{\frac{1}{a\_m + b\_m}}\\ \end{array} \end{array} \]

FPCore

a_m = (fabs.f64 a)
b_m = (fabs.f64 b)
(FPCore (a_m b_m angle)
 :precision binary64
 (if (<= angle 2.2e+208)
   (* (+ a_m b_m) (* (- b_m a_m) (* (* PI angle) 0.011111111111111112)))
   (/
    (*
     angle
     (+
      (* 0.011111111111111112 (* PI (- b_m a_m)))
      (*
       (* 2.0 (* angle angle))
       (* (* (- b_m a_m) (* PI (* PI PI))) -1.1431184270690443e-7))))
    (/ 1.0 (+ a_m b_m)))))

C

a_m = fabs(a);
b_m = fabs(b);
double code(double a_m, double b_m, double angle) {
	double tmp;
	if (angle <= 2.2e+208) {
		tmp = (a_m + b_m) * ((b_m - a_m) * ((((double) M_PI) * angle) * 0.011111111111111112));
	} else {
		tmp = (angle * ((0.011111111111111112 * (((double) M_PI) * (b_m - a_m))) + ((2.0 * (angle * angle)) * (((b_m - a_m) * (((double) M_PI) * (((double) M_PI) * ((double) M_PI)))) * -1.1431184270690443e-7)))) / (1.0 / (a_m + b_m));
	}
	return tmp;
}

Java

a_m = Math.abs(a);
b_m = Math.abs(b);
public static double code(double a_m, double b_m, double angle) {
	double tmp;
	if (angle <= 2.2e+208) {
		tmp = (a_m + b_m) * ((b_m - a_m) * ((Math.PI * angle) * 0.011111111111111112));
	} else {
		tmp = (angle * ((0.011111111111111112 * (Math.PI * (b_m - a_m))) + ((2.0 * (angle * angle)) * (((b_m - a_m) * (Math.PI * (Math.PI * Math.PI))) * -1.1431184270690443e-7)))) / (1.0 / (a_m + b_m));
	}
	return tmp;
}

Python

a_m = math.fabs(a)
b_m = math.fabs(b)
def code(a_m, b_m, angle):
	tmp = 0
	if angle <= 2.2e+208:
		tmp = (a_m + b_m) * ((b_m - a_m) * ((math.pi * angle) * 0.011111111111111112))
	else:
		tmp = (angle * ((0.011111111111111112 * (math.pi * (b_m - a_m))) + ((2.0 * (angle * angle)) * (((b_m - a_m) * (math.pi * (math.pi * math.pi))) * -1.1431184270690443e-7)))) / (1.0 / (a_m + b_m))
	return tmp

Julia

a_m = abs(a)
b_m = abs(b)
function code(a_m, b_m, angle)
	tmp = 0.0
	if (angle <= 2.2e+208)
		tmp = Float64(Float64(a_m + b_m) * Float64(Float64(b_m - a_m) * Float64(Float64(pi * angle) * 0.011111111111111112)));
	else
		tmp = Float64(Float64(angle * Float64(Float64(0.011111111111111112 * Float64(pi * Float64(b_m - a_m))) + Float64(Float64(2.0 * Float64(angle * angle)) * Float64(Float64(Float64(b_m - a_m) * Float64(pi * Float64(pi * pi))) * -1.1431184270690443e-7)))) / Float64(1.0 / Float64(a_m + b_m)));
	end
	return tmp
end

MATLAB

a_m = abs(a);
b_m = abs(b);
function tmp_2 = code(a_m, b_m, angle)
	tmp = 0.0;
	if (angle <= 2.2e+208)
		tmp = (a_m + b_m) * ((b_m - a_m) * ((pi * angle) * 0.011111111111111112));
	else
		tmp = (angle * ((0.011111111111111112 * (pi * (b_m - a_m))) + ((2.0 * (angle * angle)) * (((b_m - a_m) * (pi * (pi * pi))) * -1.1431184270690443e-7)))) / (1.0 / (a_m + b_m));
	end
	tmp_2 = tmp;
end

Wolfram

a_m = N[Abs[a], $MachinePrecision]
b_m = N[Abs[b], $MachinePrecision]
code[a$95$m_, b$95$m_, angle_] := If[LessEqual[angle, 2.2e+208], N[(N[(a$95$m + b$95$m), $MachinePrecision] * N[(N[(b$95$m - a$95$m), $MachinePrecision] * N[(N[(Pi * angle), $MachinePrecision] * 0.011111111111111112), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(angle * N[(N[(0.011111111111111112 * N[(Pi * N[(b$95$m - a$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(N[(2.0 * N[(angle * angle), $MachinePrecision]), $MachinePrecision] * N[(N[(N[(b$95$m - a$95$m), $MachinePrecision] * N[(Pi * N[(Pi * Pi), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * -1.1431184270690443e-7), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(1.0 / N[(a$95$m + b$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if angle < 2.20000000000000014e208

   1. Initial program 63.1%

      \[\left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin \left(\pi \cdot \frac{angle}{180}\right)\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right) \]
   2. Step-by-step derivation

      1. associate-*l* N/A

         \[\leadsto \left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \color{blue}{\left(\sin \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right) \cdot \cos \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right)\right)} \]

      2. *-commutative N/A

         \[\leadsto \left(\sin \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right) \cdot \cos \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right)\right) \cdot \color{blue}{\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right)} \]

      3. associate-*l* N/A

         \[\leadsto \sin \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right) \cdot \color{blue}{\left(\cos \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right) \cdot \left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right)\right)} \]

      4. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\sin \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right), \color{blue}{\left(\cos \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right) \cdot \left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right)\right)}\right) \]

      5. sin-lowering-sin.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sin.f64}\left(\left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right)\right), \left(\color{blue}{\cos \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right)} \cdot \left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right)\right)\right) \]

      6. associate-*r/ N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sin.f64}\left(\left(\frac{\mathsf{PI}\left(\right) \cdot angle}{180}\right)\right), \left(\cos \color{blue}{\left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right)} \cdot \left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right)\right)\right) \]

      7. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sin.f64}\left(\mathsf{/.f64}\left(\left(\mathsf{PI}\left(\right) \cdot angle\right), 180\right)\right), \left(\cos \color{blue}{\left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right)} \cdot \left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right)\right)\right) \]

      8. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sin.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{PI}\left(\right), angle\right), 180\right)\right), \left(\cos \left(\color{blue}{\mathsf{PI}\left(\right)} \cdot \frac{angle}{180}\right) \cdot \left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right)\right)\right) \]

      9. PI-lowering-PI.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sin.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{PI.f64}\left(\right), angle\right), 180\right)\right), \left(\cos \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right) \cdot \left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right)\right)\right) \]

      10. *-commutative N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{sin.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{PI.f64}\left(\right), angle\right), 180\right)\right), \left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \color{blue}{\cos \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right)}\right)\right) \]

      11. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{sin.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{PI.f64}\left(\right), angle\right), 180\right)\right), \mathsf{*.f64}\left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right), \color{blue}{\cos \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right)}\right)\right) \]

   3. Simplified 64.6%

      \[\leadsto \color{blue}{\sin \left(\frac{\pi \cdot angle}{180}\right) \cdot \left(\left(2 \cdot \left(b \cdot b - a \cdot a\right)\right) \cdot \cos \left(\frac{\pi \cdot angle}{180}\right)\right)} \]
   4. Add Preprocessing

   5. Taylor expanded in angle around 0

      \[\leadsto \color{blue}{\frac{1}{90} \cdot \left(angle \cdot \left(\mathsf{PI}\left(\right) \cdot \left({b}^{2} - {a}^{2}\right)\right)\right)} \]
   6. Step-by-step derivation

      1. associate-*r* N/A

         \[\leadsto \frac{1}{90} \cdot \left(\left(angle \cdot \mathsf{PI}\left(\right)\right) \cdot \color{blue}{\left({b}^{2} - {a}^{2}\right)}\right) \]

      2. associate-*r* N/A

         \[\leadsto \left(\frac{1}{90} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right) \cdot \color{blue}{\left({b}^{2} - {a}^{2}\right)} \]

      3. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\left(\frac{1}{90} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right), \color{blue}{\left({b}^{2} - {a}^{2}\right)}\right) \]

      4. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{1}{90}, \left(angle \cdot \mathsf{PI}\left(\right)\right)\right), \left(\color{blue}{{b}^{2}} - {a}^{2}\right)\right) \]

      5. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{1}{90}, \mathsf{*.f64}\left(angle, \mathsf{PI}\left(\right)\right)\right), \left({b}^{\color{blue}{2}} - {a}^{2}\right)\right) \]

      6. PI-lowering-PI.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{1}{90}, \mathsf{*.f64}\left(angle, \mathsf{PI.f64}\left(\right)\right)\right), \left({b}^{2} - {a}^{2}\right)\right) \]

      7. --lowering--.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{1}{90}, \mathsf{*.f64}\left(angle, \mathsf{PI.f64}\left(\right)\right)\right), \mathsf{\_.f64}\left(\left({b}^{2}\right), \color{blue}{\left({a}^{2}\right)}\right)\right) \]

      8. unpow2 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{1}{90}, \mathsf{*.f64}\left(angle, \mathsf{PI.f64}\left(\right)\right)\right), \mathsf{\_.f64}\left(\left(b \cdot b\right), \left({\color{blue}{a}}^{2}\right)\right)\right) \]

      9. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{1}{90}, \mathsf{*.f64}\left(angle, \mathsf{PI.f64}\left(\right)\right)\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(b, b\right), \left({\color{blue}{a}}^{2}\right)\right)\right) \]

      10. unpow2 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{1}{90}, \mathsf{*.f64}\left(angle, \mathsf{PI.f64}\left(\right)\right)\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(b, b\right), \left(a \cdot \color{blue}{a}\right)\right)\right) \]

      11. *-lowering-*.f64 61.7%

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{1}{90}, \mathsf{*.f64}\left(angle, \mathsf{PI.f64}\left(\right)\right)\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(b, b\right), \mathsf{*.f64}\left(a, \color{blue}{a}\right)\right)\right) \]

   7. Simplified 61.7%

      \[\leadsto \color{blue}{\left(0.011111111111111112 \cdot \left(angle \cdot \pi\right)\right) \cdot \left(b \cdot b - a \cdot a\right)} \]
   8. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \left(b \cdot b - a \cdot a\right) \cdot \color{blue}{\left(\frac{1}{90} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)} \]

      2. difference-of-squares N/A

         \[\leadsto \left(\left(b + a\right) \cdot \left(b - a\right)\right) \cdot \left(\color{blue}{\frac{1}{90}} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right) \]

      3. associate-*l* N/A

         \[\leadsto \left(b + a\right) \cdot \color{blue}{\left(\left(b - a\right) \cdot \left(\frac{1}{90} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)\right)} \]

      4. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\left(b + a\right), \color{blue}{\left(\left(b - a\right) \cdot \left(\frac{1}{90} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)\right)}\right) \]

      5. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(b, a\right), \left(\color{blue}{\left(b - a\right)} \cdot \left(\frac{1}{90} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)\right)\right) \]

      6. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(b, a\right), \mathsf{*.f64}\left(\left(b - a\right), \color{blue}{\left(\frac{1}{90} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}\right)\right) \]

      7. --lowering--.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(b, a\right), \mathsf{*.f64}\left(\mathsf{\_.f64}\left(b, a\right), \left(\color{blue}{\frac{1}{90}} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)\right)\right) \]

      8. *-commutative N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(b, a\right), \mathsf{*.f64}\left(\mathsf{\_.f64}\left(b, a\right), \left(\frac{1}{90} \cdot \left(\mathsf{PI}\left(\right) \cdot \color{blue}{angle}\right)\right)\right)\right) \]

      9. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(b, a\right), \mathsf{*.f64}\left(\mathsf{\_.f64}\left(b, a\right), \mathsf{*.f64}\left(\frac{1}{90}, \color{blue}{\left(\mathsf{PI}\left(\right) \cdot angle\right)}\right)\right)\right) \]

      10. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(b, a\right), \mathsf{*.f64}\left(\mathsf{\_.f64}\left(b, a\right), \mathsf{*.f64}\left(\frac{1}{90}, \mathsf{*.f64}\left(\mathsf{PI}\left(\right), \color{blue}{angle}\right)\right)\right)\right) \]

      11. PI-lowering-PI.f64 71.2%

          \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(b, a\right), \mathsf{*.f64}\left(\mathsf{\_.f64}\left(b, a\right), \mathsf{*.f64}\left(\frac{1}{90}, \mathsf{*.f64}\left(\mathsf{PI.f64}\left(\right), angle\right)\right)\right)\right) \]

   9. Applied egg-rr 71.2%

      \[\leadsto \color{blue}{\left(b + a\right) \cdot \left(\left(b - a\right) \cdot \left(0.011111111111111112 \cdot \left(\pi \cdot angle\right)\right)\right)} \]

   if 2.20000000000000014e208 < angle

   1. Initial program 31.1%

      \[\left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin \left(\pi \cdot \frac{angle}{180}\right)\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right) \]
   2. Step-by-step derivation

      1. associate-*l* N/A

         \[\leadsto \left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \color{blue}{\left(\sin \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right) \cdot \cos \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right)\right)} \]

      2. *-commutative N/A

         \[\leadsto \left(\sin \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right) \cdot \cos \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right)\right) \cdot \color{blue}{\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right)} \]

      3. associate-*l* N/A

         \[\leadsto \sin \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right) \cdot \color{blue}{\left(\cos \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right) \cdot \left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right)\right)} \]

      4. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\sin \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right), \color{blue}{\left(\cos \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right) \cdot \left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right)\right)}\right) \]

      5. sin-lowering-sin.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sin.f64}\left(\left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right)\right), \left(\color{blue}{\cos \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right)} \cdot \left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right)\right)\right) \]

      6. associate-*r/ N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sin.f64}\left(\left(\frac{\mathsf{PI}\left(\right) \cdot angle}{180}\right)\right), \left(\cos \color{blue}{\left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right)} \cdot \left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right)\right)\right) \]

      7. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sin.f64}\left(\mathsf{/.f64}\left(\left(\mathsf{PI}\left(\right) \cdot angle\right), 180\right)\right), \left(\cos \color{blue}{\left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right)} \cdot \left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right)\right)\right) \]

      8. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sin.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{PI}\left(\right), angle\right), 180\right)\right), \left(\cos \left(\color{blue}{\mathsf{PI}\left(\right)} \cdot \frac{angle}{180}\right) \cdot \left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right)\right)\right) \]

      9. PI-lowering-PI.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sin.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{PI.f64}\left(\right), angle\right), 180\right)\right), \left(\cos \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right) \cdot \left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right)\right)\right) \]

      10. *-commutative N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{sin.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{PI.f64}\left(\right), angle\right), 180\right)\right), \left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \color{blue}{\cos \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right)}\right)\right) \]

      11. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{sin.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{PI.f64}\left(\right), angle\right), 180\right)\right), \mathsf{*.f64}\left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right), \color{blue}{\cos \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right)}\right)\right) \]

   3. Simplified 42.7%

      \[\leadsto \color{blue}{\sin \left(\frac{\pi \cdot angle}{180}\right) \cdot \left(\left(2 \cdot \left(b \cdot b - a \cdot a\right)\right) \cdot \cos \left(\frac{\pi \cdot angle}{180}\right)\right)} \]
   4. Add Preprocessing
   5. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \left(\left(2 \cdot \left(b \cdot b - a \cdot a\right)\right) \cdot \cos \left(\frac{\mathsf{PI}\left(\right) \cdot angle}{180}\right)\right) \cdot \color{blue}{\sin \left(\frac{\mathsf{PI}\left(\right) \cdot angle}{180}\right)} \]

      2. *-commutative N/A

         \[\leadsto \left(\cos \left(\frac{\mathsf{PI}\left(\right) \cdot angle}{180}\right) \cdot \left(2 \cdot \left(b \cdot b - a \cdot a\right)\right)\right) \cdot \sin \color{blue}{\left(\frac{\mathsf{PI}\left(\right) \cdot angle}{180}\right)} \]

      3. associate-*r* N/A

         \[\leadsto \left(\left(\cos \left(\frac{\mathsf{PI}\left(\right) \cdot angle}{180}\right) \cdot 2\right) \cdot \left(b \cdot b - a \cdot a\right)\right) \cdot \sin \color{blue}{\left(\frac{\mathsf{PI}\left(\right) \cdot angle}{180}\right)} \]

      4. associate-*r/ N/A

         \[\leadsto \left(\left(\cos \left(\frac{\mathsf{PI}\left(\right) \cdot angle}{180}\right) \cdot 2\right) \cdot \left(b \cdot b - a \cdot a\right)\right) \cdot \sin \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right) \]

      5. associate-*r* N/A

         \[\leadsto \left(\cos \left(\frac{\mathsf{PI}\left(\right) \cdot angle}{180}\right) \cdot 2\right) \cdot \color{blue}{\left(\left(b \cdot b - a \cdot a\right) \cdot \sin \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right)\right)} \]

      6. pow2 N/A

         \[\leadsto \left(\cos \left(\frac{\mathsf{PI}\left(\right) \cdot angle}{180}\right) \cdot 2\right) \cdot \left(\left({b}^{2} - a \cdot a\right) \cdot \sin \left(\color{blue}{\mathsf{PI}\left(\right)} \cdot \frac{angle}{180}\right)\right) \]

      7. pow2 N/A

         \[\leadsto \left(\cos \left(\frac{\mathsf{PI}\left(\right) \cdot angle}{180}\right) \cdot 2\right) \cdot \left(\left({b}^{2} - {a}^{2}\right) \cdot \sin \left(\mathsf{PI}\left(\right) \cdot \color{blue}{\frac{angle}{180}}\right)\right) \]

      8. *-commutative N/A

         \[\leadsto \left(\left({b}^{2} - {a}^{2}\right) \cdot \sin \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right)\right) \cdot \color{blue}{\left(\cos \left(\frac{\mathsf{PI}\left(\right) \cdot angle}{180}\right) \cdot 2\right)} \]

      9. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\left(\left({b}^{2} - {a}^{2}\right) \cdot \sin \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right)\right), \color{blue}{\left(\cos \left(\frac{\mathsf{PI}\left(\right) \cdot angle}{180}\right) \cdot 2\right)}\right) \]

   6. Applied egg-rr 36.8%

      \[\leadsto \color{blue}{\left(\left(b + a\right) \cdot \left(\sin \left(\frac{\pi}{\frac{180}{angle}}\right) \cdot \left(b - a\right)\right)\right) \cdot \left(2 \cdot \cos \left(\frac{\pi}{\frac{180}{angle}}\right)\right)} \]
   7. Step-by-step derivation

      1. associate-*l* N/A

         \[\leadsto \left(b + a\right) \cdot \color{blue}{\left(\left(\sin \left(\frac{\mathsf{PI}\left(\right)}{\frac{180}{angle}}\right) \cdot \left(b - a\right)\right) \cdot \left(2 \cdot \cos \left(\frac{\mathsf{PI}\left(\right)}{\frac{180}{angle}}\right)\right)\right)} \]

      2. flip-+ N/A

         \[\leadsto \frac{b \cdot b - a \cdot a}{b - a} \cdot \left(\color{blue}{\left(\sin \left(\frac{\mathsf{PI}\left(\right)}{\frac{180}{angle}}\right) \cdot \left(b - a\right)\right)} \cdot \left(2 \cdot \cos \left(\frac{\mathsf{PI}\left(\right)}{\frac{180}{angle}}\right)\right)\right) \]

      3. associate-*l/ N/A

         \[\leadsto \frac{\left(b \cdot b - a \cdot a\right) \cdot \left(\left(\sin \left(\frac{\mathsf{PI}\left(\right)}{\frac{180}{angle}}\right) \cdot \left(b - a\right)\right) \cdot \left(2 \cdot \cos \left(\frac{\mathsf{PI}\left(\right)}{\frac{180}{angle}}\right)\right)\right)}{\color{blue}{b - a}} \]

      4. flip-- N/A

         \[\leadsto \frac{\left(b \cdot b - a \cdot a\right) \cdot \left(\left(\sin \left(\frac{\mathsf{PI}\left(\right)}{\frac{180}{angle}}\right) \cdot \left(b - a\right)\right) \cdot \left(2 \cdot \cos \left(\frac{\mathsf{PI}\left(\right)}{\frac{180}{angle}}\right)\right)\right)}{\frac{b \cdot b - a \cdot a}{\color{blue}{b + a}}} \]

      5. div-inv N/A

         \[\leadsto \frac{\left(b \cdot b - a \cdot a\right) \cdot \left(\left(\sin \left(\frac{\mathsf{PI}\left(\right)}{\frac{180}{angle}}\right) \cdot \left(b - a\right)\right) \cdot \left(2 \cdot \cos \left(\frac{\mathsf{PI}\left(\right)}{\frac{180}{angle}}\right)\right)\right)}{\left(b \cdot b - a \cdot a\right) \cdot \color{blue}{\frac{1}{b + a}}} \]

      6. times-frac N/A

         \[\leadsto \frac{b \cdot b - a \cdot a}{b \cdot b - a \cdot a} \cdot \color{blue}{\frac{\left(\sin \left(\frac{\mathsf{PI}\left(\right)}{\frac{180}{angle}}\right) \cdot \left(b - a\right)\right) \cdot \left(2 \cdot \cos \left(\frac{\mathsf{PI}\left(\right)}{\frac{180}{angle}}\right)\right)}{\frac{1}{b + a}}} \]

   8. Applied egg-rr 36.8%

      \[\leadsto \color{blue}{1 \cdot \frac{\left(\left(\left(b - a\right) \cdot \sin \left(\frac{\pi}{\frac{180}{angle}}\right)\right) \cdot 2\right) \cdot \cos \left(\frac{\pi}{\frac{180}{angle}}\right)}{\frac{1}{b + a}}} \]
   9. Step-by-step derivation

      1. *-lft-identity N/A

         \[\leadsto \frac{\left(\left(\left(b - a\right) \cdot \sin \left(\frac{\mathsf{PI}\left(\right)}{\frac{180}{angle}}\right)\right) \cdot 2\right) \cdot \cos \left(\frac{\mathsf{PI}\left(\right)}{\frac{180}{angle}}\right)}{\color{blue}{\frac{1}{b + a}}} \]

      2. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\left(\left(\left(\left(b - a\right) \cdot \sin \left(\frac{\mathsf{PI}\left(\right)}{\frac{180}{angle}}\right)\right) \cdot 2\right) \cdot \cos \left(\frac{\mathsf{PI}\left(\right)}{\frac{180}{angle}}\right)\right), \color{blue}{\left(\frac{1}{b + a}\right)}\right) \]

   10. Applied egg-rr 42.7%

       \[\leadsto \color{blue}{\frac{\left(\sin \left(\frac{\pi \cdot angle}{180}\right) \cdot \left(\left(b - a\right) \cdot 2\right)\right) \cdot \cos \left(\frac{\pi \cdot angle}{180}\right)}{\frac{1}{b + a}}} \]

   11. Taylor expanded in angle around 0

       \[\leadsto \mathsf{/.f64}\left(\color{blue}{\left(angle \cdot \left(\frac{1}{90} \cdot \left(\mathsf{PI}\left(\right) \cdot \left(b - a\right)\right) + 2 \cdot \left({angle}^{2} \cdot \left(\frac{-1}{11664000} \cdot \left({\mathsf{PI}\left(\right)}^{3} \cdot \left(b - a\right)\right) + \frac{-1}{34992000} \cdot \left({\mathsf{PI}\left(\right)}^{3} \cdot \left(b - a\right)\right)\right)\right)\right)\right)}, \mathsf{/.f64}\left(1, \mathsf{+.f64}\left(b, a\right)\right)\right) \]
   12. Step-by-step derivation

       1. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(angle, \left(\frac{1}{90} \cdot \left(\mathsf{PI}\left(\right) \cdot \left(b - a\right)\right) + 2 \cdot \left({angle}^{2} \cdot \left(\frac{-1}{11664000} \cdot \left({\mathsf{PI}\left(\right)}^{3} \cdot \left(b - a\right)\right) + \frac{-1}{34992000} \cdot \left({\mathsf{PI}\left(\right)}^{3} \cdot \left(b - a\right)\right)\right)\right)\right)\right), \mathsf{/.f64}\left(\color{blue}{1}, \mathsf{+.f64}\left(b, a\right)\right)\right) \]

       2. +-lowering-+.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(angle, \mathsf{+.f64}\left(\left(\frac{1}{90} \cdot \left(\mathsf{PI}\left(\right) \cdot \left(b - a\right)\right)\right), \left(2 \cdot \left({angle}^{2} \cdot \left(\frac{-1}{11664000} \cdot \left({\mathsf{PI}\left(\right)}^{3} \cdot \left(b - a\right)\right) + \frac{-1}{34992000} \cdot \left({\mathsf{PI}\left(\right)}^{3} \cdot \left(b - a\right)\right)\right)\right)\right)\right)\right), \mathsf{/.f64}\left(1, \mathsf{+.f64}\left(b, a\right)\right)\right) \]

       3. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(angle, \mathsf{+.f64}\left(\mathsf{*.f64}\left(\frac{1}{90}, \left(\mathsf{PI}\left(\right) \cdot \left(b - a\right)\right)\right), \left(2 \cdot \left({angle}^{2} \cdot \left(\frac{-1}{11664000} \cdot \left({\mathsf{PI}\left(\right)}^{3} \cdot \left(b - a\right)\right) + \frac{-1}{34992000} \cdot \left({\mathsf{PI}\left(\right)}^{3} \cdot \left(b - a\right)\right)\right)\right)\right)\right)\right), \mathsf{/.f64}\left(1, \mathsf{+.f64}\left(b, a\right)\right)\right) \]

       4. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(angle, \mathsf{+.f64}\left(\mathsf{*.f64}\left(\frac{1}{90}, \mathsf{*.f64}\left(\mathsf{PI}\left(\right), \left(b - a\right)\right)\right), \left(2 \cdot \left({angle}^{2} \cdot \left(\frac{-1}{11664000} \cdot \left({\mathsf{PI}\left(\right)}^{3} \cdot \left(b - a\right)\right) + \frac{-1}{34992000} \cdot \left({\mathsf{PI}\left(\right)}^{3} \cdot \left(b - a\right)\right)\right)\right)\right)\right)\right), \mathsf{/.f64}\left(1, \mathsf{+.f64}\left(b, a\right)\right)\right) \]

       5. PI-lowering-PI.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(angle, \mathsf{+.f64}\left(\mathsf{*.f64}\left(\frac{1}{90}, \mathsf{*.f64}\left(\mathsf{PI.f64}\left(\right), \left(b - a\right)\right)\right), \left(2 \cdot \left({angle}^{2} \cdot \left(\frac{-1}{11664000} \cdot \left({\mathsf{PI}\left(\right)}^{3} \cdot \left(b - a\right)\right) + \frac{-1}{34992000} \cdot \left({\mathsf{PI}\left(\right)}^{3} \cdot \left(b - a\right)\right)\right)\right)\right)\right)\right), \mathsf{/.f64}\left(1, \mathsf{+.f64}\left(b, a\right)\right)\right) \]

       6. --lowering--.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(angle, \mathsf{+.f64}\left(\mathsf{*.f64}\left(\frac{1}{90}, \mathsf{*.f64}\left(\mathsf{PI.f64}\left(\right), \mathsf{\_.f64}\left(b, a\right)\right)\right), \left(2 \cdot \left({angle}^{2} \cdot \left(\frac{-1}{11664000} \cdot \left({\mathsf{PI}\left(\right)}^{3} \cdot \left(b - a\right)\right) + \frac{-1}{34992000} \cdot \left({\mathsf{PI}\left(\right)}^{3} \cdot \left(b - a\right)\right)\right)\right)\right)\right)\right), \mathsf{/.f64}\left(1, \mathsf{+.f64}\left(b, a\right)\right)\right) \]

       7. associate-*r* N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(angle, \mathsf{+.f64}\left(\mathsf{*.f64}\left(\frac{1}{90}, \mathsf{*.f64}\left(\mathsf{PI.f64}\left(\right), \mathsf{\_.f64}\left(b, a\right)\right)\right), \left(\left(2 \cdot {angle}^{2}\right) \cdot \left(\frac{-1}{11664000} \cdot \left({\mathsf{PI}\left(\right)}^{3} \cdot \left(b - a\right)\right) + \frac{-1}{34992000} \cdot \left({\mathsf{PI}\left(\right)}^{3} \cdot \left(b - a\right)\right)\right)\right)\right)\right), \mathsf{/.f64}\left(1, \mathsf{+.f64}\left(b, a\right)\right)\right) \]

       8. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(angle, \mathsf{+.f64}\left(\mathsf{*.f64}\left(\frac{1}{90}, \mathsf{*.f64}\left(\mathsf{PI.f64}\left(\right), \mathsf{\_.f64}\left(b, a\right)\right)\right), \mathsf{*.f64}\left(\left(2 \cdot {angle}^{2}\right), \left(\frac{-1}{11664000} \cdot \left({\mathsf{PI}\left(\right)}^{3} \cdot \left(b - a\right)\right) + \frac{-1}{34992000} \cdot \left({\mathsf{PI}\left(\right)}^{3} \cdot \left(b - a\right)\right)\right)\right)\right)\right), \mathsf{/.f64}\left(1, \mathsf{+.f64}\left(b, a\right)\right)\right) \]

       9. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(angle, \mathsf{+.f64}\left(\mathsf{*.f64}\left(\frac{1}{90}, \mathsf{*.f64}\left(\mathsf{PI.f64}\left(\right), \mathsf{\_.f64}\left(b, a\right)\right)\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(2, \left({angle}^{2}\right)\right), \left(\frac{-1}{11664000} \cdot \left({\mathsf{PI}\left(\right)}^{3} \cdot \left(b - a\right)\right) + \frac{-1}{34992000} \cdot \left({\mathsf{PI}\left(\right)}^{3} \cdot \left(b - a\right)\right)\right)\right)\right)\right), \mathsf{/.f64}\left(1, \mathsf{+.f64}\left(b, a\right)\right)\right) \]

       10. unpow2 N/A

           \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(angle, \mathsf{+.f64}\left(\mathsf{*.f64}\left(\frac{1}{90}, \mathsf{*.f64}\left(\mathsf{PI.f64}\left(\right), \mathsf{\_.f64}\left(b, a\right)\right)\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(2, \left(angle \cdot angle\right)\right), \left(\frac{-1}{11664000} \cdot \left({\mathsf{PI}\left(\right)}^{3} \cdot \left(b - a\right)\right) + \frac{-1}{34992000} \cdot \left({\mathsf{PI}\left(\right)}^{3} \cdot \left(b - a\right)\right)\right)\right)\right)\right), \mathsf{/.f64}\left(1, \mathsf{+.f64}\left(b, a\right)\right)\right) \]

       11. *-lowering-*.f64 N/A

           \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(angle, \mathsf{+.f64}\left(\mathsf{*.f64}\left(\frac{1}{90}, \mathsf{*.f64}\left(\mathsf{PI.f64}\left(\right), \mathsf{\_.f64}\left(b, a\right)\right)\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(2, \mathsf{*.f64}\left(angle, angle\right)\right), \left(\frac{-1}{11664000} \cdot \left({\mathsf{PI}\left(\right)}^{3} \cdot \left(b - a\right)\right) + \frac{-1}{34992000} \cdot \left({\mathsf{PI}\left(\right)}^{3} \cdot \left(b - a\right)\right)\right)\right)\right)\right), \mathsf{/.f64}\left(1, \mathsf{+.f64}\left(b, a\right)\right)\right) \]

       12. distribute-rgt-out N/A

           \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(angle, \mathsf{+.f64}\left(\mathsf{*.f64}\left(\frac{1}{90}, \mathsf{*.f64}\left(\mathsf{PI.f64}\left(\right), \mathsf{\_.f64}\left(b, a\right)\right)\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(2, \mathsf{*.f64}\left(angle, angle\right)\right), \left(\left({\mathsf{PI}\left(\right)}^{3} \cdot \left(b - a\right)\right) \cdot \left(\frac{-1}{11664000} + \frac{-1}{34992000}\right)\right)\right)\right)\right), \mathsf{/.f64}\left(1, \mathsf{+.f64}\left(b, a\right)\right)\right) \]

   13. Simplified 25.4%

       \[\leadsto \frac{\color{blue}{angle \cdot \left(0.011111111111111112 \cdot \left(\pi \cdot \left(b - a\right)\right) + \left(2 \cdot \left(angle \cdot angle\right)\right) \cdot \left(\left(\left(b - a\right) \cdot \left(\pi \cdot \left(\pi \cdot \pi\right)\right)\right) \cdot -1.1431184270690443 \cdot 10^{-7}\right)\right)}}{\frac{1}{b + a}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 68.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;angle \leq 2.2 \cdot 10^{+208}:\\ \;\;\;\;\left(a + b\right) \cdot \left(\left(b - a\right) \cdot \left(\left(\pi \cdot angle\right) \cdot 0.011111111111111112\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{angle \cdot \left(0.011111111111111112 \cdot \left(\pi \cdot \left(b - a\right)\right) + \left(2 \cdot \left(angle \cdot angle\right)\right) \cdot \left(\left(\left(b - a\right) \cdot \left(\pi \cdot \left(\pi \cdot \pi\right)\right)\right) \cdot -1.1431184270690443 \cdot 10^{-7}\right)\right)}{\frac{1}{a + b}}\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 62.4% accurate, 12.0× speedup

Math

\[\begin{array}{l} a_m = \left|a\right| \\ b_m = \left|b\right| \\ \begin{array}{l} \mathbf{if}\;angle \leq 2.2 \cdot 10^{+208}:\\ \;\;\;\;\left(a\_m + b\_m\right) \cdot \left(\left(b\_m - a\_m\right) \cdot \left(\left(\pi \cdot angle\right) \cdot 0.011111111111111112\right)\right)\\ \mathbf{elif}\;angle \leq 5.6 \cdot 10^{+227}:\\ \;\;\;\;\left(\left(angle \cdot 0.011111111111111112\right) \cdot \left(\pi \cdot \left(b\_m \cdot b\_m - a\_m \cdot a\_m\right)\right)\right) \cdot \left(1 + \left(-1.54320987654321 \cdot 10^{-5} \cdot \left(angle \cdot angle\right)\right) \cdot \left(\pi \cdot \pi\right)\right)\\ \mathbf{else}:\\ \;\;\;\;-0.011111111111111112 \cdot \left(\pi \cdot \left(angle \cdot \left(a\_m \cdot a\_m\right)\right)\right)\\ \end{array} \end{array} \]

FPCore

a_m = (fabs.f64 a)
b_m = (fabs.f64 b)
(FPCore (a_m b_m angle)
 :precision binary64
 (if (<= angle 2.2e+208)
   (* (+ a_m b_m) (* (- b_m a_m) (* (* PI angle) 0.011111111111111112)))
   (if (<= angle 5.6e+227)
     (*
      (* (* angle 0.011111111111111112) (* PI (- (* b_m b_m) (* a_m a_m))))
      (+ 1.0 (* (* -1.54320987654321e-5 (* angle angle)) (* PI PI))))
     (* -0.011111111111111112 (* PI (* angle (* a_m a_m)))))))

C

a_m = fabs(a);
b_m = fabs(b);
double code(double a_m, double b_m, double angle) {
	double tmp;
	if (angle <= 2.2e+208) {
		tmp = (a_m + b_m) * ((b_m - a_m) * ((((double) M_PI) * angle) * 0.011111111111111112));
	} else if (angle <= 5.6e+227) {
		tmp = ((angle * 0.011111111111111112) * (((double) M_PI) * ((b_m * b_m) - (a_m * a_m)))) * (1.0 + ((-1.54320987654321e-5 * (angle * angle)) * (((double) M_PI) * ((double) M_PI))));
	} else {
		tmp = -0.011111111111111112 * (((double) M_PI) * (angle * (a_m * a_m)));
	}
	return tmp;
}

Java

a_m = Math.abs(a);
b_m = Math.abs(b);
public static double code(double a_m, double b_m, double angle) {
	double tmp;
	if (angle <= 2.2e+208) {
		tmp = (a_m + b_m) * ((b_m - a_m) * ((Math.PI * angle) * 0.011111111111111112));
	} else if (angle <= 5.6e+227) {
		tmp = ((angle * 0.011111111111111112) * (Math.PI * ((b_m * b_m) - (a_m * a_m)))) * (1.0 + ((-1.54320987654321e-5 * (angle * angle)) * (Math.PI * Math.PI)));
	} else {
		tmp = -0.011111111111111112 * (Math.PI * (angle * (a_m * a_m)));
	}
	return tmp;
}

Python

a_m = math.fabs(a)
b_m = math.fabs(b)
def code(a_m, b_m, angle):
	tmp = 0
	if angle <= 2.2e+208:
		tmp = (a_m + b_m) * ((b_m - a_m) * ((math.pi * angle) * 0.011111111111111112))
	elif angle <= 5.6e+227:
		tmp = ((angle * 0.011111111111111112) * (math.pi * ((b_m * b_m) - (a_m * a_m)))) * (1.0 + ((-1.54320987654321e-5 * (angle * angle)) * (math.pi * math.pi)))
	else:
		tmp = -0.011111111111111112 * (math.pi * (angle * (a_m * a_m)))
	return tmp

Julia

a_m = abs(a)
b_m = abs(b)
function code(a_m, b_m, angle)
	tmp = 0.0
	if (angle <= 2.2e+208)
		tmp = Float64(Float64(a_m + b_m) * Float64(Float64(b_m - a_m) * Float64(Float64(pi * angle) * 0.011111111111111112)));
	elseif (angle <= 5.6e+227)
		tmp = Float64(Float64(Float64(angle * 0.011111111111111112) * Float64(pi * Float64(Float64(b_m * b_m) - Float64(a_m * a_m)))) * Float64(1.0 + Float64(Float64(-1.54320987654321e-5 * Float64(angle * angle)) * Float64(pi * pi))));
	else
		tmp = Float64(-0.011111111111111112 * Float64(pi * Float64(angle * Float64(a_m * a_m))));
	end
	return tmp
end

MATLAB

a_m = abs(a);
b_m = abs(b);
function tmp_2 = code(a_m, b_m, angle)
	tmp = 0.0;
	if (angle <= 2.2e+208)
		tmp = (a_m + b_m) * ((b_m - a_m) * ((pi * angle) * 0.011111111111111112));
	elseif (angle <= 5.6e+227)
		tmp = ((angle * 0.011111111111111112) * (pi * ((b_m * b_m) - (a_m * a_m)))) * (1.0 + ((-1.54320987654321e-5 * (angle * angle)) * (pi * pi)));
	else
		tmp = -0.011111111111111112 * (pi * (angle * (a_m * a_m)));
	end
	tmp_2 = tmp;
end

Wolfram

a_m = N[Abs[a], $MachinePrecision]
b_m = N[Abs[b], $MachinePrecision]
code[a$95$m_, b$95$m_, angle_] := If[LessEqual[angle, 2.2e+208], N[(N[(a$95$m + b$95$m), $MachinePrecision] * N[(N[(b$95$m - a$95$m), $MachinePrecision] * N[(N[(Pi * angle), $MachinePrecision] * 0.011111111111111112), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[angle, 5.6e+227], N[(N[(N[(angle * 0.011111111111111112), $MachinePrecision] * N[(Pi * N[(N[(b$95$m * b$95$m), $MachinePrecision] - N[(a$95$m * a$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(1.0 + N[(N[(-1.54320987654321e-5 * N[(angle * angle), $MachinePrecision]), $MachinePrecision] * N[(Pi * Pi), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(-0.011111111111111112 * N[(Pi * N[(angle * N[(a$95$m * a$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if angle < 2.20000000000000014e208

   1. Initial program 63.1%

      \[\left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin \left(\pi \cdot \frac{angle}{180}\right)\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right) \]
   2. Step-by-step derivation

      1. associate-*l* N/A

         \[\leadsto \left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \color{blue}{\left(\sin \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right) \cdot \cos \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right)\right)} \]

      2. *-commutative N/A

         \[\leadsto \left(\sin \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right) \cdot \cos \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right)\right) \cdot \color{blue}{\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right)} \]

      3. associate-*l* N/A

         \[\leadsto \sin \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right) \cdot \color{blue}{\left(\cos \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right) \cdot \left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right)\right)} \]

      4. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\sin \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right), \color{blue}{\left(\cos \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right) \cdot \left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right)\right)}\right) \]

      5. sin-lowering-sin.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sin.f64}\left(\left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right)\right), \left(\color{blue}{\cos \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right)} \cdot \left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right)\right)\right) \]

      6. associate-*r/ N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sin.f64}\left(\left(\frac{\mathsf{PI}\left(\right) \cdot angle}{180}\right)\right), \left(\cos \color{blue}{\left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right)} \cdot \left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right)\right)\right) \]

      7. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sin.f64}\left(\mathsf{/.f64}\left(\left(\mathsf{PI}\left(\right) \cdot angle\right), 180\right)\right), \left(\cos \color{blue}{\left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right)} \cdot \left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right)\right)\right) \]

      8. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sin.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{PI}\left(\right), angle\right), 180\right)\right), \left(\cos \left(\color{blue}{\mathsf{PI}\left(\right)} \cdot \frac{angle}{180}\right) \cdot \left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right)\right)\right) \]

      9. PI-lowering-PI.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sin.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{PI.f64}\left(\right), angle\right), 180\right)\right), \left(\cos \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right) \cdot \left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right)\right)\right) \]

      10. *-commutative N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{sin.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{PI.f64}\left(\right), angle\right), 180\right)\right), \left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \color{blue}{\cos \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right)}\right)\right) \]

      11. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{sin.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{PI.f64}\left(\right), angle\right), 180\right)\right), \mathsf{*.f64}\left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right), \color{blue}{\cos \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right)}\right)\right) \]

   3. Simplified 64.6%

      \[\leadsto \color{blue}{\sin \left(\frac{\pi \cdot angle}{180}\right) \cdot \left(\left(2 \cdot \left(b \cdot b - a \cdot a\right)\right) \cdot \cos \left(\frac{\pi \cdot angle}{180}\right)\right)} \]
   4. Add Preprocessing

   5. Taylor expanded in angle around 0

      \[\leadsto \color{blue}{\frac{1}{90} \cdot \left(angle \cdot \left(\mathsf{PI}\left(\right) \cdot \left({b}^{2} - {a}^{2}\right)\right)\right)} \]
   6. Step-by-step derivation

      1. associate-*r* N/A

         \[\leadsto \frac{1}{90} \cdot \left(\left(angle \cdot \mathsf{PI}\left(\right)\right) \cdot \color{blue}{\left({b}^{2} - {a}^{2}\right)}\right) \]

      2. associate-*r* N/A

         \[\leadsto \left(\frac{1}{90} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right) \cdot \color{blue}{\left({b}^{2} - {a}^{2}\right)} \]

      3. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\left(\frac{1}{90} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right), \color{blue}{\left({b}^{2} - {a}^{2}\right)}\right) \]

      4. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{1}{90}, \left(angle \cdot \mathsf{PI}\left(\right)\right)\right), \left(\color{blue}{{b}^{2}} - {a}^{2}\right)\right) \]

      5. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{1}{90}, \mathsf{*.f64}\left(angle, \mathsf{PI}\left(\right)\right)\right), \left({b}^{\color{blue}{2}} - {a}^{2}\right)\right) \]

      6. PI-lowering-PI.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{1}{90}, \mathsf{*.f64}\left(angle, \mathsf{PI.f64}\left(\right)\right)\right), \left({b}^{2} - {a}^{2}\right)\right) \]

      7. --lowering--.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{1}{90}, \mathsf{*.f64}\left(angle, \mathsf{PI.f64}\left(\right)\right)\right), \mathsf{\_.f64}\left(\left({b}^{2}\right), \color{blue}{\left({a}^{2}\right)}\right)\right) \]

      8. unpow2 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{1}{90}, \mathsf{*.f64}\left(angle, \mathsf{PI.f64}\left(\right)\right)\right), \mathsf{\_.f64}\left(\left(b \cdot b\right), \left({\color{blue}{a}}^{2}\right)\right)\right) \]

      9. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{1}{90}, \mathsf{*.f64}\left(angle, \mathsf{PI.f64}\left(\right)\right)\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(b, b\right), \left({\color{blue}{a}}^{2}\right)\right)\right) \]

      10. unpow2 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{1}{90}, \mathsf{*.f64}\left(angle, \mathsf{PI.f64}\left(\right)\right)\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(b, b\right), \left(a \cdot \color{blue}{a}\right)\right)\right) \]

      11. *-lowering-*.f64 61.7%

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{1}{90}, \mathsf{*.f64}\left(angle, \mathsf{PI.f64}\left(\right)\right)\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(b, b\right), \mathsf{*.f64}\left(a, \color{blue}{a}\right)\right)\right) \]

   7. Simplified 61.7%

      \[\leadsto \color{blue}{\left(0.011111111111111112 \cdot \left(angle \cdot \pi\right)\right) \cdot \left(b \cdot b - a \cdot a\right)} \]
   8. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \left(b \cdot b - a \cdot a\right) \cdot \color{blue}{\left(\frac{1}{90} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)} \]

      2. difference-of-squares N/A

         \[\leadsto \left(\left(b + a\right) \cdot \left(b - a\right)\right) \cdot \left(\color{blue}{\frac{1}{90}} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right) \]

      3. associate-*l* N/A

         \[\leadsto \left(b + a\right) \cdot \color{blue}{\left(\left(b - a\right) \cdot \left(\frac{1}{90} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)\right)} \]

      4. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\left(b + a\right), \color{blue}{\left(\left(b - a\right) \cdot \left(\frac{1}{90} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)\right)}\right) \]

      5. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(b, a\right), \left(\color{blue}{\left(b - a\right)} \cdot \left(\frac{1}{90} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)\right)\right) \]

      6. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(b, a\right), \mathsf{*.f64}\left(\left(b - a\right), \color{blue}{\left(\frac{1}{90} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}\right)\right) \]

      7. --lowering--.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(b, a\right), \mathsf{*.f64}\left(\mathsf{\_.f64}\left(b, a\right), \left(\color{blue}{\frac{1}{90}} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)\right)\right) \]

      8. *-commutative N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(b, a\right), \mathsf{*.f64}\left(\mathsf{\_.f64}\left(b, a\right), \left(\frac{1}{90} \cdot \left(\mathsf{PI}\left(\right) \cdot \color{blue}{angle}\right)\right)\right)\right) \]

      9. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(b, a\right), \mathsf{*.f64}\left(\mathsf{\_.f64}\left(b, a\right), \mathsf{*.f64}\left(\frac{1}{90}, \color{blue}{\left(\mathsf{PI}\left(\right) \cdot angle\right)}\right)\right)\right) \]

      10. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(b, a\right), \mathsf{*.f64}\left(\mathsf{\_.f64}\left(b, a\right), \mathsf{*.f64}\left(\frac{1}{90}, \mathsf{*.f64}\left(\mathsf{PI}\left(\right), \color{blue}{angle}\right)\right)\right)\right) \]

      11. PI-lowering-PI.f64 71.2%

          \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(b, a\right), \mathsf{*.f64}\left(\mathsf{\_.f64}\left(b, a\right), \mathsf{*.f64}\left(\frac{1}{90}, \mathsf{*.f64}\left(\mathsf{PI.f64}\left(\right), angle\right)\right)\right)\right) \]

   9. Applied egg-rr 71.2%

      \[\leadsto \color{blue}{\left(b + a\right) \cdot \left(\left(b - a\right) \cdot \left(0.011111111111111112 \cdot \left(\pi \cdot angle\right)\right)\right)} \]

   if 2.20000000000000014e208 < angle < 5.59999999999999968e227

   1. Initial program 55.3%

      \[\left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin \left(\pi \cdot \frac{angle}{180}\right)\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right) \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. pow2 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\left(2 \cdot \left(b \cdot b - {a}^{2}\right)\right), \mathsf{sin.f64}\left(\mathsf{*.f64}\left(\mathsf{PI.f64}\left(\right), \mathsf{/.f64}\left(angle, 180\right)\right)\right)\right), \mathsf{cos.f64}\left(\mathsf{*.f64}\left(\mathsf{PI.f64}\left(\right), \mathsf{/.f64}\left(angle, 180\right)\right)\right)\right) \]

      2. pow2 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\left(2 \cdot \left(b \cdot b - a \cdot a\right)\right), \mathsf{sin.f64}\left(\mathsf{*.f64}\left(\mathsf{PI.f64}\left(\right), \mathsf{/.f64}\left(angle, 180\right)\right)\right)\right), \mathsf{cos.f64}\left(\mathsf{*.f64}\left(\mathsf{PI.f64}\left(\right), \mathsf{/.f64}\left(angle, 180\right)\right)\right)\right) \]

      3. flip-- N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\left(2 \cdot \frac{\left(b \cdot b\right) \cdot \left(b \cdot b\right) - \left(a \cdot a\right) \cdot \left(a \cdot a\right)}{b \cdot b + a \cdot a}\right), \mathsf{sin.f64}\left(\mathsf{*.f64}\left(\mathsf{PI.f64}\left(\right), \mathsf{/.f64}\left(angle, 180\right)\right)\right)\right), \mathsf{cos.f64}\left(\mathsf{*.f64}\left(\mathsf{PI.f64}\left(\right), \mathsf{/.f64}\left(angle, 180\right)\right)\right)\right) \]

      4. clear-num N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\left(2 \cdot \frac{1}{\frac{b \cdot b + a \cdot a}{\left(b \cdot b\right) \cdot \left(b \cdot b\right) - \left(a \cdot a\right) \cdot \left(a \cdot a\right)}}\right), \mathsf{sin.f64}\left(\mathsf{*.f64}\left(\mathsf{PI.f64}\left(\right), \mathsf{/.f64}\left(angle, 180\right)\right)\right)\right), \mathsf{cos.f64}\left(\mathsf{*.f64}\left(\mathsf{PI.f64}\left(\right), \mathsf{/.f64}\left(angle, 180\right)\right)\right)\right) \]

      5. un-div-inv N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\left(\frac{2}{\frac{b \cdot b + a \cdot a}{\left(b \cdot b\right) \cdot \left(b \cdot b\right) - \left(a \cdot a\right) \cdot \left(a \cdot a\right)}}\right), \mathsf{sin.f64}\left(\mathsf{*.f64}\left(\mathsf{PI.f64}\left(\right), \mathsf{/.f64}\left(angle, 180\right)\right)\right)\right), \mathsf{cos.f64}\left(\mathsf{*.f64}\left(\color{blue}{\mathsf{PI.f64}\left(\right)}, \mathsf{/.f64}\left(angle, 180\right)\right)\right)\right) \]

      6. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(2, \left(\frac{b \cdot b + a \cdot a}{\left(b \cdot b\right) \cdot \left(b \cdot b\right) - \left(a \cdot a\right) \cdot \left(a \cdot a\right)}\right)\right), \mathsf{sin.f64}\left(\mathsf{*.f64}\left(\mathsf{PI.f64}\left(\right), \mathsf{/.f64}\left(angle, 180\right)\right)\right)\right), \mathsf{cos.f64}\left(\mathsf{*.f64}\left(\color{blue}{\mathsf{PI.f64}\left(\right)}, \mathsf{/.f64}\left(angle, 180\right)\right)\right)\right) \]

      7. clear-num N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(2, \left(\frac{1}{\frac{\left(b \cdot b\right) \cdot \left(b \cdot b\right) - \left(a \cdot a\right) \cdot \left(a \cdot a\right)}{b \cdot b + a \cdot a}}\right)\right), \mathsf{sin.f64}\left(\mathsf{*.f64}\left(\mathsf{PI.f64}\left(\right), \mathsf{/.f64}\left(angle, 180\right)\right)\right)\right), \mathsf{cos.f64}\left(\mathsf{*.f64}\left(\mathsf{PI.f64}\left(\right), \mathsf{/.f64}\left(angle, 180\right)\right)\right)\right) \]

      8. flip-- N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(2, \left(\frac{1}{b \cdot b - a \cdot a}\right)\right), \mathsf{sin.f64}\left(\mathsf{*.f64}\left(\mathsf{PI.f64}\left(\right), \mathsf{/.f64}\left(angle, 180\right)\right)\right)\right), \mathsf{cos.f64}\left(\mathsf{*.f64}\left(\mathsf{PI.f64}\left(\right), \mathsf{/.f64}\left(angle, 180\right)\right)\right)\right) \]

      9. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(2, \mathsf{/.f64}\left(1, \left(b \cdot b - a \cdot a\right)\right)\right), \mathsf{sin.f64}\left(\mathsf{*.f64}\left(\mathsf{PI.f64}\left(\right), \mathsf{/.f64}\left(angle, 180\right)\right)\right)\right), \mathsf{cos.f64}\left(\mathsf{*.f64}\left(\mathsf{PI.f64}\left(\right), \mathsf{/.f64}\left(angle, 180\right)\right)\right)\right) \]

      10. --lowering--.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(2, \mathsf{/.f64}\left(1, \mathsf{\_.f64}\left(\left(b \cdot b\right), \left(a \cdot a\right)\right)\right)\right), \mathsf{sin.f64}\left(\mathsf{*.f64}\left(\mathsf{PI.f64}\left(\right), \mathsf{/.f64}\left(angle, 180\right)\right)\right)\right), \mathsf{cos.f64}\left(\mathsf{*.f64}\left(\mathsf{PI.f64}\left(\right), \mathsf{/.f64}\left(angle, 180\right)\right)\right)\right) \]

      11. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(2, \mathsf{/.f64}\left(1, \mathsf{\_.f64}\left(\mathsf{*.f64}\left(b, b\right), \left(a \cdot a\right)\right)\right)\right), \mathsf{sin.f64}\left(\mathsf{*.f64}\left(\mathsf{PI.f64}\left(\right), \mathsf{/.f64}\left(angle, 180\right)\right)\right)\right), \mathsf{cos.f64}\left(\mathsf{*.f64}\left(\mathsf{PI.f64}\left(\right), \mathsf{/.f64}\left(angle, 180\right)\right)\right)\right) \]

      12. *-lowering-*.f64 55.3%

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(2, \mathsf{/.f64}\left(1, \mathsf{\_.f64}\left(\mathsf{*.f64}\left(b, b\right), \mathsf{*.f64}\left(a, a\right)\right)\right)\right), \mathsf{sin.f64}\left(\mathsf{*.f64}\left(\mathsf{PI.f64}\left(\right), \mathsf{/.f64}\left(angle, 180\right)\right)\right)\right), \mathsf{cos.f64}\left(\mathsf{*.f64}\left(\mathsf{PI.f64}\left(\right), \mathsf{/.f64}\left(angle, 180\right)\right)\right)\right) \]

   4. Applied egg-rr 55.3%

      \[\leadsto \left(\color{blue}{\frac{2}{\frac{1}{b \cdot b - a \cdot a}}} \cdot \sin \left(\pi \cdot \frac{angle}{180}\right)\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right) \]

   5. Taylor expanded in angle around 0

      \[\leadsto \mathsf{*.f64}\left(\color{blue}{\left(\frac{1}{90} \cdot \left(angle \cdot \left(\mathsf{PI}\left(\right) \cdot \left({b}^{2} - {a}^{2}\right)\right)\right)\right)}, \mathsf{cos.f64}\left(\mathsf{*.f64}\left(\mathsf{PI.f64}\left(\right), \mathsf{/.f64}\left(angle, 180\right)\right)\right)\right) \]
   6. Step-by-step derivation

      1. associate-*r* N/A

         \[\leadsto \mathsf{*.f64}\left(\left(\left(\frac{1}{90} \cdot angle\right) \cdot \left(\mathsf{PI}\left(\right) \cdot \left({b}^{2} - {a}^{2}\right)\right)\right), \mathsf{cos.f64}\left(\color{blue}{\mathsf{*.f64}\left(\mathsf{PI.f64}\left(\right), \mathsf{/.f64}\left(angle, 180\right)\right)}\right)\right) \]

      2. unpow2 N/A

         \[\leadsto \mathsf{*.f64}\left(\left(\left(\frac{1}{90} \cdot angle\right) \cdot \left(\mathsf{PI}\left(\right) \cdot \left(b \cdot b - {a}^{2}\right)\right)\right), \mathsf{cos.f64}\left(\mathsf{*.f64}\left(\mathsf{PI.f64}\left(\right), \mathsf{/.f64}\left(angle, 180\right)\right)\right)\right) \]

      3. unpow2 N/A

         \[\leadsto \mathsf{*.f64}\left(\left(\left(\frac{1}{90} \cdot angle\right) \cdot \left(\mathsf{PI}\left(\right) \cdot \left(b \cdot b - a \cdot a\right)\right)\right), \mathsf{cos.f64}\left(\mathsf{*.f64}\left(\mathsf{PI.f64}\left(\right), \mathsf{/.f64}\left(angle, 180\right)\right)\right)\right) \]

      4. difference-of-squares N/A

         \[\leadsto \mathsf{*.f64}\left(\left(\left(\frac{1}{90} \cdot angle\right) \cdot \left(\mathsf{PI}\left(\right) \cdot \left(\left(b + a\right) \cdot \left(b - a\right)\right)\right)\right), \mathsf{cos.f64}\left(\mathsf{*.f64}\left(\mathsf{PI.f64}\left(\right), \mathsf{/.f64}\left(angle, \color{blue}{180}\right)\right)\right)\right) \]

      5. +-commutative N/A

         \[\leadsto \mathsf{*.f64}\left(\left(\left(\frac{1}{90} \cdot angle\right) \cdot \left(\mathsf{PI}\left(\right) \cdot \left(\left(a + b\right) \cdot \left(b - a\right)\right)\right)\right), \mathsf{cos.f64}\left(\mathsf{*.f64}\left(\mathsf{PI.f64}\left(\right), \mathsf{/.f64}\left(angle, 180\right)\right)\right)\right) \]

      6. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\left(\frac{1}{90} \cdot angle\right), \left(\mathsf{PI}\left(\right) \cdot \left(\left(a + b\right) \cdot \left(b - a\right)\right)\right)\right), \mathsf{cos.f64}\left(\color{blue}{\mathsf{*.f64}\left(\mathsf{PI.f64}\left(\right), \mathsf{/.f64}\left(angle, 180\right)\right)}\right)\right) \]

      7. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{1}{90}, angle\right), \left(\mathsf{PI}\left(\right) \cdot \left(\left(a + b\right) \cdot \left(b - a\right)\right)\right)\right), \mathsf{cos.f64}\left(\mathsf{*.f64}\left(\color{blue}{\mathsf{PI.f64}\left(\right)}, \mathsf{/.f64}\left(angle, 180\right)\right)\right)\right) \]

      8. +-commutative N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{1}{90}, angle\right), \left(\mathsf{PI}\left(\right) \cdot \left(\left(b + a\right) \cdot \left(b - a\right)\right)\right)\right), \mathsf{cos.f64}\left(\mathsf{*.f64}\left(\mathsf{PI.f64}\left(\right), \mathsf{/.f64}\left(angle, 180\right)\right)\right)\right) \]

      9. difference-of-squares N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{1}{90}, angle\right), \left(\mathsf{PI}\left(\right) \cdot \left(b \cdot b - a \cdot a\right)\right)\right), \mathsf{cos.f64}\left(\mathsf{*.f64}\left(\mathsf{PI.f64}\left(\right), \mathsf{/.f64}\left(angle, \color{blue}{180}\right)\right)\right)\right) \]

      10. unpow2 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{1}{90}, angle\right), \left(\mathsf{PI}\left(\right) \cdot \left({b}^{2} - a \cdot a\right)\right)\right), \mathsf{cos.f64}\left(\mathsf{*.f64}\left(\mathsf{PI.f64}\left(\right), \mathsf{/.f64}\left(angle, 180\right)\right)\right)\right) \]

      11. unpow2 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{1}{90}, angle\right), \left(\mathsf{PI}\left(\right) \cdot \left({b}^{2} - {a}^{2}\right)\right)\right), \mathsf{cos.f64}\left(\mathsf{*.f64}\left(\mathsf{PI.f64}\left(\right), \mathsf{/.f64}\left(angle, 180\right)\right)\right)\right) \]

      12. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{1}{90}, angle\right), \mathsf{*.f64}\left(\mathsf{PI}\left(\right), \left({b}^{2} - {a}^{2}\right)\right)\right), \mathsf{cos.f64}\left(\mathsf{*.f64}\left(\mathsf{PI.f64}\left(\right), \color{blue}{\mathsf{/.f64}\left(angle, 180\right)}\right)\right)\right) \]

      13. PI-lowering-PI.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{1}{90}, angle\right), \mathsf{*.f64}\left(\mathsf{PI.f64}\left(\right), \left({b}^{2} - {a}^{2}\right)\right)\right), \mathsf{cos.f64}\left(\mathsf{*.f64}\left(\mathsf{PI.f64}\left(\right), \mathsf{/.f64}\left(\color{blue}{angle}, 180\right)\right)\right)\right) \]

      14. --lowering--.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{1}{90}, angle\right), \mathsf{*.f64}\left(\mathsf{PI.f64}\left(\right), \mathsf{\_.f64}\left(\left({b}^{2}\right), \left({a}^{2}\right)\right)\right)\right), \mathsf{cos.f64}\left(\mathsf{*.f64}\left(\mathsf{PI.f64}\left(\right), \mathsf{/.f64}\left(angle, \color{blue}{180}\right)\right)\right)\right) \]

      15. unpow2 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{1}{90}, angle\right), \mathsf{*.f64}\left(\mathsf{PI.f64}\left(\right), \mathsf{\_.f64}\left(\left(b \cdot b\right), \left({a}^{2}\right)\right)\right)\right), \mathsf{cos.f64}\left(\mathsf{*.f64}\left(\mathsf{PI.f64}\left(\right), \mathsf{/.f64}\left(angle, 180\right)\right)\right)\right) \]

      16. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{1}{90}, angle\right), \mathsf{*.f64}\left(\mathsf{PI.f64}\left(\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(b, b\right), \left({a}^{2}\right)\right)\right)\right), \mathsf{cos.f64}\left(\mathsf{*.f64}\left(\mathsf{PI.f64}\left(\right), \mathsf{/.f64}\left(angle, 180\right)\right)\right)\right) \]

      17. unpow2 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{1}{90}, angle\right), \mathsf{*.f64}\left(\mathsf{PI.f64}\left(\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(b, b\right), \left(a \cdot a\right)\right)\right)\right), \mathsf{cos.f64}\left(\mathsf{*.f64}\left(\mathsf{PI.f64}\left(\right), \mathsf{/.f64}\left(angle, 180\right)\right)\right)\right) \]

      18. *-lowering-*.f64 52.0%

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{1}{90}, angle\right), \mathsf{*.f64}\left(\mathsf{PI.f64}\left(\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(b, b\right), \mathsf{*.f64}\left(a, a\right)\right)\right)\right), \mathsf{cos.f64}\left(\mathsf{*.f64}\left(\mathsf{PI.f64}\left(\right), \mathsf{/.f64}\left(angle, 180\right)\right)\right)\right) \]

   7. Simplified 52.0%

      \[\leadsto \color{blue}{\left(\left(0.011111111111111112 \cdot angle\right) \cdot \left(\pi \cdot \left(b \cdot b - a \cdot a\right)\right)\right)} \cdot \cos \left(\pi \cdot \frac{angle}{180}\right) \]

   8. Taylor expanded in angle around 0

      \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{1}{90}, angle\right), \mathsf{*.f64}\left(\mathsf{PI.f64}\left(\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(b, b\right), \mathsf{*.f64}\left(a, a\right)\right)\right)\right), \color{blue}{\left(1 + \frac{-1}{64800} \cdot \left({angle}^{2} \cdot {\mathsf{PI}\left(\right)}^{2}\right)\right)}\right) \]
   9. Step-by-step derivation

      1. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{1}{90}, angle\right), \mathsf{*.f64}\left(\mathsf{PI.f64}\left(\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(b, b\right), \mathsf{*.f64}\left(a, a\right)\right)\right)\right), \mathsf{+.f64}\left(1, \color{blue}{\left(\frac{-1}{64800} \cdot \left({angle}^{2} \cdot {\mathsf{PI}\left(\right)}^{2}\right)\right)}\right)\right) \]

      2. associate-*r* N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{1}{90}, angle\right), \mathsf{*.f64}\left(\mathsf{PI.f64}\left(\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(b, b\right), \mathsf{*.f64}\left(a, a\right)\right)\right)\right), \mathsf{+.f64}\left(1, \left(\left(\frac{-1}{64800} \cdot {angle}^{2}\right) \cdot \color{blue}{{\mathsf{PI}\left(\right)}^{2}}\right)\right)\right) \]

      3. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{1}{90}, angle\right), \mathsf{*.f64}\left(\mathsf{PI.f64}\left(\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(b, b\right), \mathsf{*.f64}\left(a, a\right)\right)\right)\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\left(\frac{-1}{64800} \cdot {angle}^{2}\right), \color{blue}{\left({\mathsf{PI}\left(\right)}^{2}\right)}\right)\right)\right) \]

      4. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{1}{90}, angle\right), \mathsf{*.f64}\left(\mathsf{PI.f64}\left(\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(b, b\right), \mathsf{*.f64}\left(a, a\right)\right)\right)\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{-1}{64800}, \left({angle}^{2}\right)\right), \left({\color{blue}{\mathsf{PI}\left(\right)}}^{2}\right)\right)\right)\right) \]

      5. unpow2 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{1}{90}, angle\right), \mathsf{*.f64}\left(\mathsf{PI.f64}\left(\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(b, b\right), \mathsf{*.f64}\left(a, a\right)\right)\right)\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{-1}{64800}, \left(angle \cdot angle\right)\right), \left({\mathsf{PI}\left(\right)}^{2}\right)\right)\right)\right) \]

      6. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{1}{90}, angle\right), \mathsf{*.f64}\left(\mathsf{PI.f64}\left(\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(b, b\right), \mathsf{*.f64}\left(a, a\right)\right)\right)\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{-1}{64800}, \mathsf{*.f64}\left(angle, angle\right)\right), \left({\mathsf{PI}\left(\right)}^{2}\right)\right)\right)\right) \]

      7. unpow2 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{1}{90}, angle\right), \mathsf{*.f64}\left(\mathsf{PI.f64}\left(\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(b, b\right), \mathsf{*.f64}\left(a, a\right)\right)\right)\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{-1}{64800}, \mathsf{*.f64}\left(angle, angle\right)\right), \left(\mathsf{PI}\left(\right) \cdot \color{blue}{\mathsf{PI}\left(\right)}\right)\right)\right)\right) \]

      8. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{1}{90}, angle\right), \mathsf{*.f64}\left(\mathsf{PI.f64}\left(\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(b, b\right), \mathsf{*.f64}\left(a, a\right)\right)\right)\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{-1}{64800}, \mathsf{*.f64}\left(angle, angle\right)\right), \mathsf{*.f64}\left(\mathsf{PI}\left(\right), \color{blue}{\mathsf{PI}\left(\right)}\right)\right)\right)\right) \]

      9. PI-lowering-PI.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{1}{90}, angle\right), \mathsf{*.f64}\left(\mathsf{PI.f64}\left(\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(b, b\right), \mathsf{*.f64}\left(a, a\right)\right)\right)\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{-1}{64800}, \mathsf{*.f64}\left(angle, angle\right)\right), \mathsf{*.f64}\left(\mathsf{PI.f64}\left(\right), \mathsf{PI}\left(\right)\right)\right)\right)\right) \]

      10. PI-lowering-PI.f64 51.3%

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{1}{90}, angle\right), \mathsf{*.f64}\left(\mathsf{PI.f64}\left(\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(b, b\right), \mathsf{*.f64}\left(a, a\right)\right)\right)\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{-1}{64800}, \mathsf{*.f64}\left(angle, angle\right)\right), \mathsf{*.f64}\left(\mathsf{PI.f64}\left(\right), \mathsf{PI.f64}\left(\right)\right)\right)\right)\right) \]

   10. Simplified 51.3%

       \[\leadsto \left(\left(0.011111111111111112 \cdot angle\right) \cdot \left(\pi \cdot \left(b \cdot b - a \cdot a\right)\right)\right) \cdot \color{blue}{\left(1 + \left(-1.54320987654321 \cdot 10^{-5} \cdot \left(angle \cdot angle\right)\right) \cdot \left(\pi \cdot \pi\right)\right)} \]

   if 5.59999999999999968e227 < angle

   1. Initial program 23.7%

      \[\left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin \left(\pi \cdot \frac{angle}{180}\right)\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right) \]
   2. Step-by-step derivation

      1. associate-*l* N/A

         \[\leadsto \left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \color{blue}{\left(\sin \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right) \cdot \cos \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right)\right)} \]

      2. *-commutative N/A

         \[\leadsto \left(\sin \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right) \cdot \cos \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right)\right) \cdot \color{blue}{\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right)} \]

      3. associate-*l* N/A

         \[\leadsto \sin \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right) \cdot \color{blue}{\left(\cos \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right) \cdot \left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right)\right)} \]

      4. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\sin \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right), \color{blue}{\left(\cos \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right) \cdot \left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right)\right)}\right) \]

      5. sin-lowering-sin.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sin.f64}\left(\left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right)\right), \left(\color{blue}{\cos \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right)} \cdot \left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right)\right)\right) \]

      6. associate-*r/ N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sin.f64}\left(\left(\frac{\mathsf{PI}\left(\right) \cdot angle}{180}\right)\right), \left(\cos \color{blue}{\left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right)} \cdot \left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right)\right)\right) \]

      7. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sin.f64}\left(\mathsf{/.f64}\left(\left(\mathsf{PI}\left(\right) \cdot angle\right), 180\right)\right), \left(\cos \color{blue}{\left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right)} \cdot \left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right)\right)\right) \]

      8. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sin.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{PI}\left(\right), angle\right), 180\right)\right), \left(\cos \left(\color{blue}{\mathsf{PI}\left(\right)} \cdot \frac{angle}{180}\right) \cdot \left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right)\right)\right) \]

      9. PI-lowering-PI.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sin.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{PI.f64}\left(\right), angle\right), 180\right)\right), \left(\cos \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right) \cdot \left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right)\right)\right) \]

      10. *-commutative N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{sin.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{PI.f64}\left(\right), angle\right), 180\right)\right), \left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \color{blue}{\cos \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right)}\right)\right) \]

      11. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{sin.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{PI.f64}\left(\right), angle\right), 180\right)\right), \mathsf{*.f64}\left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right), \color{blue}{\cos \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right)}\right)\right) \]

   3. Simplified 38.8%

      \[\leadsto \color{blue}{\sin \left(\frac{\pi \cdot angle}{180}\right) \cdot \left(\left(2 \cdot \left(b \cdot b - a \cdot a\right)\right) \cdot \cos \left(\frac{\pi \cdot angle}{180}\right)\right)} \]
   4. Add Preprocessing

   5. Taylor expanded in angle around 0

      \[\leadsto \color{blue}{\frac{1}{90} \cdot \left(angle \cdot \left(\mathsf{PI}\left(\right) \cdot \left({b}^{2} - {a}^{2}\right)\right)\right)} \]
   6. Step-by-step derivation

      1. associate-*r* N/A

         \[\leadsto \frac{1}{90} \cdot \left(\left(angle \cdot \mathsf{PI}\left(\right)\right) \cdot \color{blue}{\left({b}^{2} - {a}^{2}\right)}\right) \]

      2. associate-*r* N/A

         \[\leadsto \left(\frac{1}{90} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right) \cdot \color{blue}{\left({b}^{2} - {a}^{2}\right)} \]

      3. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\left(\frac{1}{90} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right), \color{blue}{\left({b}^{2} - {a}^{2}\right)}\right) \]

      4. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{1}{90}, \left(angle \cdot \mathsf{PI}\left(\right)\right)\right), \left(\color{blue}{{b}^{2}} - {a}^{2}\right)\right) \]

      5. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{1}{90}, \mathsf{*.f64}\left(angle, \mathsf{PI}\left(\right)\right)\right), \left({b}^{\color{blue}{2}} - {a}^{2}\right)\right) \]

      6. PI-lowering-PI.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{1}{90}, \mathsf{*.f64}\left(angle, \mathsf{PI.f64}\left(\right)\right)\right), \left({b}^{2} - {a}^{2}\right)\right) \]

      7. --lowering--.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{1}{90}, \mathsf{*.f64}\left(angle, \mathsf{PI.f64}\left(\right)\right)\right), \mathsf{\_.f64}\left(\left({b}^{2}\right), \color{blue}{\left({a}^{2}\right)}\right)\right) \]

      8. unpow2 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{1}{90}, \mathsf{*.f64}\left(angle, \mathsf{PI.f64}\left(\right)\right)\right), \mathsf{\_.f64}\left(\left(b \cdot b\right), \left({\color{blue}{a}}^{2}\right)\right)\right) \]

      9. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{1}{90}, \mathsf{*.f64}\left(angle, \mathsf{PI.f64}\left(\right)\right)\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(b, b\right), \left({\color{blue}{a}}^{2}\right)\right)\right) \]

      10. unpow2 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{1}{90}, \mathsf{*.f64}\left(angle, \mathsf{PI.f64}\left(\right)\right)\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(b, b\right), \left(a \cdot \color{blue}{a}\right)\right)\right) \]

      11. *-lowering-*.f64 24.7%

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{1}{90}, \mathsf{*.f64}\left(angle, \mathsf{PI.f64}\left(\right)\right)\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(b, b\right), \mathsf{*.f64}\left(a, \color{blue}{a}\right)\right)\right) \]

   7. Simplified 24.7%

      \[\leadsto \color{blue}{\left(0.011111111111111112 \cdot \left(angle \cdot \pi\right)\right) \cdot \left(b \cdot b - a \cdot a\right)} \]

   8. Taylor expanded in b around 0

      \[\leadsto \color{blue}{\frac{-1}{90} \cdot \left({a}^{2} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)} \]
   9. Step-by-step derivation

      1. *-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 10.1%

         \[\leadsto \mathsf{*.f64}\left(\frac{-1}{90}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(a, a\right), angle\right), \mathsf{PI.f64}\left(\right)\right)\right) \]

   10. Simplified 10.1%

       \[\leadsto \color{blue}{-0.011111111111111112 \cdot \left(\left(\left(a \cdot a\right) \cdot angle\right) \cdot \pi\right)} \]
3. Recombined 3 regimes into one program.

4. Final simplification 67.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;angle \leq 2.2 \cdot 10^{+208}:\\ \;\;\;\;\left(a + b\right) \cdot \left(\left(b - a\right) \cdot \left(\left(\pi \cdot angle\right) \cdot 0.011111111111111112\right)\right)\\ \mathbf{elif}\;angle \leq 5.6 \cdot 10^{+227}:\\ \;\;\;\;\left(\left(angle \cdot 0.011111111111111112\right) \cdot \left(\pi \cdot \left(b \cdot b - a \cdot a\right)\right)\right) \cdot \left(1 + \left(-1.54320987654321 \cdot 10^{-5} \cdot \left(angle \cdot angle\right)\right) \cdot \left(\pi \cdot \pi\right)\right)\\ \mathbf{else}:\\ \;\;\;\;-0.011111111111111112 \cdot \left(\pi \cdot \left(angle \cdot \left(a \cdot a\right)\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 12: 62.4% accurate, 23.3× speedup

Math

\[\begin{array}{l} a_m = \left|a\right| \\ b_m = \left|b\right| \\ \begin{array}{l} \mathbf{if}\;angle \leq 1.36 \cdot 10^{+228}:\\ \;\;\;\;\left(a\_m + b\_m\right) \cdot \left(\left(b\_m - a\_m\right) \cdot \left(\left(\pi \cdot angle\right) \cdot 0.011111111111111112\right)\right)\\ \mathbf{else}:\\ \;\;\;\;-0.011111111111111112 \cdot \left(\pi \cdot \left(angle \cdot \left(a\_m \cdot a\_m\right)\right)\right)\\ \end{array} \end{array} \]

FPCore

a_m = (fabs.f64 a)
b_m = (fabs.f64 b)
(FPCore (a_m b_m angle)
 :precision binary64
 (if (<= angle 1.36e+228)
   (* (+ a_m b_m) (* (- b_m a_m) (* (* PI angle) 0.011111111111111112)))
   (* -0.011111111111111112 (* PI (* angle (* a_m a_m))))))

C

a_m = fabs(a);
b_m = fabs(b);
double code(double a_m, double b_m, double angle) {
	double tmp;
	if (angle <= 1.36e+228) {
		tmp = (a_m + b_m) * ((b_m - a_m) * ((((double) M_PI) * angle) * 0.011111111111111112));
	} else {
		tmp = -0.011111111111111112 * (((double) M_PI) * (angle * (a_m * a_m)));
	}
	return tmp;
}

Java

a_m = Math.abs(a);
b_m = Math.abs(b);
public static double code(double a_m, double b_m, double angle) {
	double tmp;
	if (angle <= 1.36e+228) {
		tmp = (a_m + b_m) * ((b_m - a_m) * ((Math.PI * angle) * 0.011111111111111112));
	} else {
		tmp = -0.011111111111111112 * (Math.PI * (angle * (a_m * a_m)));
	}
	return tmp;
}

Python

a_m = math.fabs(a)
b_m = math.fabs(b)
def code(a_m, b_m, angle):
	tmp = 0
	if angle <= 1.36e+228:
		tmp = (a_m + b_m) * ((b_m - a_m) * ((math.pi * angle) * 0.011111111111111112))
	else:
		tmp = -0.011111111111111112 * (math.pi * (angle * (a_m * a_m)))
	return tmp

Julia

a_m = abs(a)
b_m = abs(b)
function code(a_m, b_m, angle)
	tmp = 0.0
	if (angle <= 1.36e+228)
		tmp = Float64(Float64(a_m + b_m) * Float64(Float64(b_m - a_m) * Float64(Float64(pi * angle) * 0.011111111111111112)));
	else
		tmp = Float64(-0.011111111111111112 * Float64(pi * Float64(angle * Float64(a_m * a_m))));
	end
	return tmp
end

MATLAB

a_m = abs(a);
b_m = abs(b);
function tmp_2 = code(a_m, b_m, angle)
	tmp = 0.0;
	if (angle <= 1.36e+228)
		tmp = (a_m + b_m) * ((b_m - a_m) * ((pi * angle) * 0.011111111111111112));
	else
		tmp = -0.011111111111111112 * (pi * (angle * (a_m * a_m)));
	end
	tmp_2 = tmp;
end

Wolfram

a_m = N[Abs[a], $MachinePrecision]
b_m = N[Abs[b], $MachinePrecision]
code[a$95$m_, b$95$m_, angle_] := If[LessEqual[angle, 1.36e+228], N[(N[(a$95$m + b$95$m), $MachinePrecision] * N[(N[(b$95$m - a$95$m), $MachinePrecision] * N[(N[(Pi * angle), $MachinePrecision] * 0.011111111111111112), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(-0.011111111111111112 * N[(Pi * N[(angle * N[(a$95$m * a$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if angle < 1.35999999999999996e228

   1. Initial program 62.7%

      \[\left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin \left(\pi \cdot \frac{angle}{180}\right)\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right) \]
   2. Step-by-step derivation

      1. associate-*l* N/A

         \[\leadsto \left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \color{blue}{\left(\sin \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right) \cdot \cos \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right)\right)} \]

      2. *-commutative N/A

         \[\leadsto \left(\sin \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right) \cdot \cos \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right)\right) \cdot \color{blue}{\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right)} \]

      3. associate-*l* N/A

         \[\leadsto \sin \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right) \cdot \color{blue}{\left(\cos \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right) \cdot \left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right)\right)} \]

      4. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\sin \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right), \color{blue}{\left(\cos \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right) \cdot \left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right)\right)}\right) \]

      5. sin-lowering-sin.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sin.f64}\left(\left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right)\right), \left(\color{blue}{\cos \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right)} \cdot \left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right)\right)\right) \]

      6. associate-*r/ N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sin.f64}\left(\left(\frac{\mathsf{PI}\left(\right) \cdot angle}{180}\right)\right), \left(\cos \color{blue}{\left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right)} \cdot \left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right)\right)\right) \]

      7. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sin.f64}\left(\mathsf{/.f64}\left(\left(\mathsf{PI}\left(\right) \cdot angle\right), 180\right)\right), \left(\cos \color{blue}{\left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right)} \cdot \left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right)\right)\right) \]

      8. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sin.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{PI}\left(\right), angle\right), 180\right)\right), \left(\cos \left(\color{blue}{\mathsf{PI}\left(\right)} \cdot \frac{angle}{180}\right) \cdot \left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right)\right)\right) \]

      9. PI-lowering-PI.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sin.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{PI.f64}\left(\right), angle\right), 180\right)\right), \left(\cos \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right) \cdot \left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right)\right)\right) \]

      10. *-commutative N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{sin.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{PI.f64}\left(\right), angle\right), 180\right)\right), \left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \color{blue}{\cos \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right)}\right)\right) \]

      11. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{sin.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{PI.f64}\left(\right), angle\right), 180\right)\right), \mathsf{*.f64}\left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right), \color{blue}{\cos \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right)}\right)\right) \]

   3. Simplified 64.6%

      \[\leadsto \color{blue}{\sin \left(\frac{\pi \cdot angle}{180}\right) \cdot \left(\left(2 \cdot \left(b \cdot b - a \cdot a\right)\right) \cdot \cos \left(\frac{\pi \cdot angle}{180}\right)\right)} \]
   4. Add Preprocessing

   5. Taylor expanded in angle around 0

      \[\leadsto \color{blue}{\frac{1}{90} \cdot \left(angle \cdot \left(\mathsf{PI}\left(\right) \cdot \left({b}^{2} - {a}^{2}\right)\right)\right)} \]
   6. Step-by-step derivation

      1. associate-*r* N/A

         \[\leadsto \frac{1}{90} \cdot \left(\left(angle \cdot \mathsf{PI}\left(\right)\right) \cdot \color{blue}{\left({b}^{2} - {a}^{2}\right)}\right) \]

      2. associate-*r* N/A

         \[\leadsto \left(\frac{1}{90} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right) \cdot \color{blue}{\left({b}^{2} - {a}^{2}\right)} \]

      3. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\left(\frac{1}{90} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right), \color{blue}{\left({b}^{2} - {a}^{2}\right)}\right) \]

      4. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{1}{90}, \left(angle \cdot \mathsf{PI}\left(\right)\right)\right), \left(\color{blue}{{b}^{2}} - {a}^{2}\right)\right) \]

      5. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{1}{90}, \mathsf{*.f64}\left(angle, \mathsf{PI}\left(\right)\right)\right), \left({b}^{\color{blue}{2}} - {a}^{2}\right)\right) \]

      6. PI-lowering-PI.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{1}{90}, \mathsf{*.f64}\left(angle, \mathsf{PI.f64}\left(\right)\right)\right), \left({b}^{2} - {a}^{2}\right)\right) \]

      7. --lowering--.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{1}{90}, \mathsf{*.f64}\left(angle, \mathsf{PI.f64}\left(\right)\right)\right), \mathsf{\_.f64}\left(\left({b}^{2}\right), \color{blue}{\left({a}^{2}\right)}\right)\right) \]

      8. unpow2 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{1}{90}, \mathsf{*.f64}\left(angle, \mathsf{PI.f64}\left(\right)\right)\right), \mathsf{\_.f64}\left(\left(b \cdot b\right), \left({\color{blue}{a}}^{2}\right)\right)\right) \]

      9. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{1}{90}, \mathsf{*.f64}\left(angle, \mathsf{PI.f64}\left(\right)\right)\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(b, b\right), \left({\color{blue}{a}}^{2}\right)\right)\right) \]

      10. unpow2 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{1}{90}, \mathsf{*.f64}\left(angle, \mathsf{PI.f64}\left(\right)\right)\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(b, b\right), \left(a \cdot \color{blue}{a}\right)\right)\right) \]

      11. *-lowering-*.f64 60.8%

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{1}{90}, \mathsf{*.f64}\left(angle, \mathsf{PI.f64}\left(\right)\right)\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(b, b\right), \mathsf{*.f64}\left(a, \color{blue}{a}\right)\right)\right) \]

   7. Simplified 60.8%

      \[\leadsto \color{blue}{\left(0.011111111111111112 \cdot \left(angle \cdot \pi\right)\right) \cdot \left(b \cdot b - a \cdot a\right)} \]
   8. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \left(b \cdot b - a \cdot a\right) \cdot \color{blue}{\left(\frac{1}{90} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)} \]

      2. difference-of-squares N/A

         \[\leadsto \left(\left(b + a\right) \cdot \left(b - a\right)\right) \cdot \left(\color{blue}{\frac{1}{90}} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right) \]

      3. associate-*l* N/A

         \[\leadsto \left(b + a\right) \cdot \color{blue}{\left(\left(b - a\right) \cdot \left(\frac{1}{90} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)\right)} \]

      4. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\left(b + a\right), \color{blue}{\left(\left(b - a\right) \cdot \left(\frac{1}{90} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)\right)}\right) \]

      5. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(b, a\right), \left(\color{blue}{\left(b - a\right)} \cdot \left(\frac{1}{90} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)\right)\right) \]

      6. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(b, a\right), \mathsf{*.f64}\left(\left(b - a\right), \color{blue}{\left(\frac{1}{90} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}\right)\right) \]

      7. --lowering--.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(b, a\right), \mathsf{*.f64}\left(\mathsf{\_.f64}\left(b, a\right), \left(\color{blue}{\frac{1}{90}} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)\right)\right) \]

      8. *-commutative N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(b, a\right), \mathsf{*.f64}\left(\mathsf{\_.f64}\left(b, a\right), \left(\frac{1}{90} \cdot \left(\mathsf{PI}\left(\right) \cdot \color{blue}{angle}\right)\right)\right)\right) \]

      9. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(b, a\right), \mathsf{*.f64}\left(\mathsf{\_.f64}\left(b, a\right), \mathsf{*.f64}\left(\frac{1}{90}, \color{blue}{\left(\mathsf{PI}\left(\right) \cdot angle\right)}\right)\right)\right) \]

      10. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(b, a\right), \mathsf{*.f64}\left(\mathsf{\_.f64}\left(b, a\right), \mathsf{*.f64}\left(\frac{1}{90}, \mathsf{*.f64}\left(\mathsf{PI}\left(\right), \color{blue}{angle}\right)\right)\right)\right) \]

      11. PI-lowering-PI.f64 70.2%

          \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(b, a\right), \mathsf{*.f64}\left(\mathsf{\_.f64}\left(b, a\right), \mathsf{*.f64}\left(\frac{1}{90}, \mathsf{*.f64}\left(\mathsf{PI.f64}\left(\right), angle\right)\right)\right)\right) \]

   9. Applied egg-rr 70.2%

      \[\leadsto \color{blue}{\left(b + a\right) \cdot \left(\left(b - a\right) \cdot \left(0.011111111111111112 \cdot \left(\pi \cdot angle\right)\right)\right)} \]

   if 1.35999999999999996e228 < angle

   1. Initial program 25.6%

      \[\left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin \left(\pi \cdot \frac{angle}{180}\right)\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right) \]
   2. Step-by-step derivation

      1. associate-*l* N/A

         \[\leadsto \left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \color{blue}{\left(\sin \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right) \cdot \cos \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right)\right)} \]

      2. *-commutative N/A

         \[\leadsto \left(\sin \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right) \cdot \cos \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right)\right) \cdot \color{blue}{\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right)} \]

      3. associate-*l* N/A

         \[\leadsto \sin \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right) \cdot \color{blue}{\left(\cos \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right) \cdot \left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right)\right)} \]

      4. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\sin \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right), \color{blue}{\left(\cos \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right) \cdot \left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right)\right)}\right) \]

      5. sin-lowering-sin.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sin.f64}\left(\left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right)\right), \left(\color{blue}{\cos \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right)} \cdot \left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right)\right)\right) \]

      6. associate-*r/ N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sin.f64}\left(\left(\frac{\mathsf{PI}\left(\right) \cdot angle}{180}\right)\right), \left(\cos \color{blue}{\left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right)} \cdot \left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right)\right)\right) \]

      7. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sin.f64}\left(\mathsf{/.f64}\left(\left(\mathsf{PI}\left(\right) \cdot angle\right), 180\right)\right), \left(\cos \color{blue}{\left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right)} \cdot \left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right)\right)\right) \]

      8. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sin.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{PI}\left(\right), angle\right), 180\right)\right), \left(\cos \left(\color{blue}{\mathsf{PI}\left(\right)} \cdot \frac{angle}{180}\right) \cdot \left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right)\right)\right) \]

      9. PI-lowering-PI.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sin.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{PI.f64}\left(\right), angle\right), 180\right)\right), \left(\cos \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right) \cdot \left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right)\right)\right) \]

      10. *-commutative N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{sin.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{PI.f64}\left(\right), angle\right), 180\right)\right), \left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \color{blue}{\cos \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right)}\right)\right) \]

      11. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{sin.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{PI.f64}\left(\right), angle\right), 180\right)\right), \mathsf{*.f64}\left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right), \color{blue}{\cos \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right)}\right)\right) \]

   3. Simplified 33.7%

      \[\leadsto \color{blue}{\sin \left(\frac{\pi \cdot angle}{180}\right) \cdot \left(\left(2 \cdot \left(b \cdot b - a \cdot a\right)\right) \cdot \cos \left(\frac{\pi \cdot angle}{180}\right)\right)} \]
   4. Add Preprocessing

   5. Taylor expanded in angle around 0

      \[\leadsto \color{blue}{\frac{1}{90} \cdot \left(angle \cdot \left(\mathsf{PI}\left(\right) \cdot \left({b}^{2} - {a}^{2}\right)\right)\right)} \]
   6. Step-by-step derivation

      1. associate-*r* N/A

         \[\leadsto \frac{1}{90} \cdot \left(\left(angle \cdot \mathsf{PI}\left(\right)\right) \cdot \color{blue}{\left({b}^{2} - {a}^{2}\right)}\right) \]

      2. associate-*r* N/A

         \[\leadsto \left(\frac{1}{90} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right) \cdot \color{blue}{\left({b}^{2} - {a}^{2}\right)} \]

      3. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\left(\frac{1}{90} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right), \color{blue}{\left({b}^{2} - {a}^{2}\right)}\right) \]

      4. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{1}{90}, \left(angle \cdot \mathsf{PI}\left(\right)\right)\right), \left(\color{blue}{{b}^{2}} - {a}^{2}\right)\right) \]

      5. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{1}{90}, \mathsf{*.f64}\left(angle, \mathsf{PI}\left(\right)\right)\right), \left({b}^{\color{blue}{2}} - {a}^{2}\right)\right) \]

      6. PI-lowering-PI.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{1}{90}, \mathsf{*.f64}\left(angle, \mathsf{PI.f64}\left(\right)\right)\right), \left({b}^{2} - {a}^{2}\right)\right) \]

      7. --lowering--.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{1}{90}, \mathsf{*.f64}\left(angle, \mathsf{PI.f64}\left(\right)\right)\right), \mathsf{\_.f64}\left(\left({b}^{2}\right), \color{blue}{\left({a}^{2}\right)}\right)\right) \]

      8. unpow2 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{1}{90}, \mathsf{*.f64}\left(angle, \mathsf{PI.f64}\left(\right)\right)\right), \mathsf{\_.f64}\left(\left(b \cdot b\right), \left({\color{blue}{a}}^{2}\right)\right)\right) \]

      9. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{1}{90}, \mathsf{*.f64}\left(angle, \mathsf{PI.f64}\left(\right)\right)\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(b, b\right), \left({\color{blue}{a}}^{2}\right)\right)\right) \]

      10. unpow2 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{1}{90}, \mathsf{*.f64}\left(angle, \mathsf{PI.f64}\left(\right)\right)\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(b, b\right), \left(a \cdot \color{blue}{a}\right)\right)\right) \]

      11. *-lowering-*.f64 18.5%

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{1}{90}, \mathsf{*.f64}\left(angle, \mathsf{PI.f64}\left(\right)\right)\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(b, b\right), \mathsf{*.f64}\left(a, \color{blue}{a}\right)\right)\right) \]

   7. Simplified 18.5%

      \[\leadsto \color{blue}{\left(0.011111111111111112 \cdot \left(angle \cdot \pi\right)\right) \cdot \left(b \cdot b - a \cdot a\right)} \]

   8. Taylor expanded in b around 0

      \[\leadsto \color{blue}{\frac{-1}{90} \cdot \left({a}^{2} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)} \]
   9. Step-by-step derivation

      1. *-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 10.9%

         \[\leadsto \mathsf{*.f64}\left(\frac{-1}{90}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(a, a\right), angle\right), \mathsf{PI.f64}\left(\right)\right)\right) \]

   10. Simplified 10.9%

       \[\leadsto \color{blue}{-0.011111111111111112 \cdot \left(\left(\left(a \cdot a\right) \cdot angle\right) \cdot \pi\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 67.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;angle \leq 1.36 \cdot 10^{+228}:\\ \;\;\;\;\left(a + b\right) \cdot \left(\left(b - a\right) \cdot \left(\left(\pi \cdot angle\right) \cdot 0.011111111111111112\right)\right)\\ \mathbf{else}:\\ \;\;\;\;-0.011111111111111112 \cdot \left(\pi \cdot \left(angle \cdot \left(a \cdot a\right)\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 13: 57.5% accurate, 23.3× speedup

Math

\[\begin{array}{l} a_m = \left|a\right| \\ b_m = \left|b\right| \\ \begin{array}{l} \mathbf{if}\;a\_m \leq 6 \cdot 10^{+146}:\\ \;\;\;\;\left(b\_m \cdot b\_m - a\_m \cdot a\_m\right) \cdot \left(angle \cdot \left(\pi \cdot 0.011111111111111112\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(a\_m \cdot \pi\right) \cdot \left(-0.011111111111111112 \cdot \left(a\_m \cdot angle\right)\right)\\ \end{array} \end{array} \]

FPCore

a_m = (fabs.f64 a)
b_m = (fabs.f64 b)
(FPCore (a_m b_m angle)
 :precision binary64
 (if (<= a_m 6e+146)
   (* (- (* b_m b_m) (* a_m a_m)) (* angle (* PI 0.011111111111111112)))
   (* (* a_m PI) (* -0.011111111111111112 (* a_m angle)))))

C

a_m = fabs(a);
b_m = fabs(b);
double code(double a_m, double b_m, double angle) {
	double tmp;
	if (a_m <= 6e+146) {
		tmp = ((b_m * b_m) - (a_m * a_m)) * (angle * (((double) M_PI) * 0.011111111111111112));
	} else {
		tmp = (a_m * ((double) M_PI)) * (-0.011111111111111112 * (a_m * angle));
	}
	return tmp;
}

Java

a_m = Math.abs(a);
b_m = Math.abs(b);
public static double code(double a_m, double b_m, double angle) {
	double tmp;
	if (a_m <= 6e+146) {
		tmp = ((b_m * b_m) - (a_m * a_m)) * (angle * (Math.PI * 0.011111111111111112));
	} else {
		tmp = (a_m * Math.PI) * (-0.011111111111111112 * (a_m * angle));
	}
	return tmp;
}

Python

a_m = math.fabs(a)
b_m = math.fabs(b)
def code(a_m, b_m, angle):
	tmp = 0
	if a_m <= 6e+146:
		tmp = ((b_m * b_m) - (a_m * a_m)) * (angle * (math.pi * 0.011111111111111112))
	else:
		tmp = (a_m * math.pi) * (-0.011111111111111112 * (a_m * angle))
	return tmp

Julia

a_m = abs(a)
b_m = abs(b)
function code(a_m, b_m, angle)
	tmp = 0.0
	if (a_m <= 6e+146)
		tmp = Float64(Float64(Float64(b_m * b_m) - Float64(a_m * a_m)) * Float64(angle * Float64(pi * 0.011111111111111112)));
	else
		tmp = Float64(Float64(a_m * pi) * Float64(-0.011111111111111112 * Float64(a_m * angle)));
	end
	return tmp
end

MATLAB

a_m = abs(a);
b_m = abs(b);
function tmp_2 = code(a_m, b_m, angle)
	tmp = 0.0;
	if (a_m <= 6e+146)
		tmp = ((b_m * b_m) - (a_m * a_m)) * (angle * (pi * 0.011111111111111112));
	else
		tmp = (a_m * pi) * (-0.011111111111111112 * (a_m * angle));
	end
	tmp_2 = tmp;
end

Wolfram

a_m = N[Abs[a], $MachinePrecision]
b_m = N[Abs[b], $MachinePrecision]
code[a$95$m_, b$95$m_, angle_] := If[LessEqual[a$95$m, 6e+146], N[(N[(N[(b$95$m * b$95$m), $MachinePrecision] - N[(a$95$m * a$95$m), $MachinePrecision]), $MachinePrecision] * N[(angle * N[(Pi * 0.011111111111111112), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(a$95$m * Pi), $MachinePrecision] * N[(-0.011111111111111112 * N[(a$95$m * angle), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if a < 6.00000000000000005e146

   1. Initial program 59.9%

      \[\left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin \left(\pi \cdot \frac{angle}{180}\right)\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right) \]
   2. Step-by-step derivation

      1. associate-*l* N/A

         \[\leadsto \left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \color{blue}{\left(\sin \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right) \cdot \cos \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right)\right)} \]

      2. *-commutative N/A

         \[\leadsto \left(\sin \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right) \cdot \cos \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right)\right) \cdot \color{blue}{\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right)} \]

      3. associate-*l* N/A

         \[\leadsto \sin \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right) \cdot \color{blue}{\left(\cos \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right) \cdot \left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right)\right)} \]

      4. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\sin \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right), \color{blue}{\left(\cos \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right) \cdot \left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right)\right)}\right) \]

      5. sin-lowering-sin.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sin.f64}\left(\left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right)\right), \left(\color{blue}{\cos \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right)} \cdot \left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right)\right)\right) \]

      6. associate-*r/ N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sin.f64}\left(\left(\frac{\mathsf{PI}\left(\right) \cdot angle}{180}\right)\right), \left(\cos \color{blue}{\left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right)} \cdot \left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right)\right)\right) \]

      7. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sin.f64}\left(\mathsf{/.f64}\left(\left(\mathsf{PI}\left(\right) \cdot angle\right), 180\right)\right), \left(\cos \color{blue}{\left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right)} \cdot \left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right)\right)\right) \]

      8. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sin.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{PI}\left(\right), angle\right), 180\right)\right), \left(\cos \left(\color{blue}{\mathsf{PI}\left(\right)} \cdot \frac{angle}{180}\right) \cdot \left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right)\right)\right) \]

      9. PI-lowering-PI.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sin.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{PI.f64}\left(\right), angle\right), 180\right)\right), \left(\cos \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right) \cdot \left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right)\right)\right) \]

      10. *-commutative N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{sin.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{PI.f64}\left(\right), angle\right), 180\right)\right), \left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \color{blue}{\cos \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right)}\right)\right) \]

      11. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{sin.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{PI.f64}\left(\right), angle\right), 180\right)\right), \mathsf{*.f64}\left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right), \color{blue}{\cos \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right)}\right)\right) \]

   3. Simplified 61.9%

      \[\leadsto \color{blue}{\sin \left(\frac{\pi \cdot angle}{180}\right) \cdot \left(\left(2 \cdot \left(b \cdot b - a \cdot a\right)\right) \cdot \cos \left(\frac{\pi \cdot angle}{180}\right)\right)} \]
   4. Add Preprocessing

   5. Taylor expanded in angle around 0

      \[\leadsto \color{blue}{\frac{1}{90} \cdot \left(angle \cdot \left(\mathsf{PI}\left(\right) \cdot \left({b}^{2} - {a}^{2}\right)\right)\right)} \]
   6. Step-by-step derivation

      1. associate-*r* N/A

         \[\leadsto \frac{1}{90} \cdot \left(\left(angle \cdot \mathsf{PI}\left(\right)\right) \cdot \color{blue}{\left({b}^{2} - {a}^{2}\right)}\right) \]

      2. associate-*r* N/A

         \[\leadsto \left(\frac{1}{90} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right) \cdot \color{blue}{\left({b}^{2} - {a}^{2}\right)} \]

      3. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\left(\frac{1}{90} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right), \color{blue}{\left({b}^{2} - {a}^{2}\right)}\right) \]

      4. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{1}{90}, \left(angle \cdot \mathsf{PI}\left(\right)\right)\right), \left(\color{blue}{{b}^{2}} - {a}^{2}\right)\right) \]

      5. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{1}{90}, \mathsf{*.f64}\left(angle, \mathsf{PI}\left(\right)\right)\right), \left({b}^{\color{blue}{2}} - {a}^{2}\right)\right) \]

      6. PI-lowering-PI.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{1}{90}, \mathsf{*.f64}\left(angle, \mathsf{PI.f64}\left(\right)\right)\right), \left({b}^{2} - {a}^{2}\right)\right) \]

      7. --lowering--.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{1}{90}, \mathsf{*.f64}\left(angle, \mathsf{PI.f64}\left(\right)\right)\right), \mathsf{\_.f64}\left(\left({b}^{2}\right), \color{blue}{\left({a}^{2}\right)}\right)\right) \]

      8. unpow2 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{1}{90}, \mathsf{*.f64}\left(angle, \mathsf{PI.f64}\left(\right)\right)\right), \mathsf{\_.f64}\left(\left(b \cdot b\right), \left({\color{blue}{a}}^{2}\right)\right)\right) \]

      9. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{1}{90}, \mathsf{*.f64}\left(angle, \mathsf{PI.f64}\left(\right)\right)\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(b, b\right), \left({\color{blue}{a}}^{2}\right)\right)\right) \]

      10. unpow2 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{1}{90}, \mathsf{*.f64}\left(angle, \mathsf{PI.f64}\left(\right)\right)\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(b, b\right), \left(a \cdot \color{blue}{a}\right)\right)\right) \]

      11. *-lowering-*.f64 58.8%

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{1}{90}, \mathsf{*.f64}\left(angle, \mathsf{PI.f64}\left(\right)\right)\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(b, b\right), \mathsf{*.f64}\left(a, \color{blue}{a}\right)\right)\right) \]

   7. Simplified 58.8%

      \[\leadsto \color{blue}{\left(0.011111111111111112 \cdot \left(angle \cdot \pi\right)\right) \cdot \left(b \cdot b - a \cdot a\right)} \]
   8. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \mathsf{*.f64}\left(\left(\frac{1}{90} \cdot \left(\mathsf{PI}\left(\right) \cdot angle\right)\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(b, \color{blue}{b}\right), \mathsf{*.f64}\left(a, a\right)\right)\right) \]

      2. associate-*r* N/A

         \[\leadsto \mathsf{*.f64}\left(\left(\left(\frac{1}{90} \cdot \mathsf{PI}\left(\right)\right) \cdot angle\right), \mathsf{\_.f64}\left(\color{blue}{\mathsf{*.f64}\left(b, b\right)}, \mathsf{*.f64}\left(a, a\right)\right)\right) \]

      3. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\left(\frac{1}{90} \cdot \mathsf{PI}\left(\right)\right), angle\right), \mathsf{\_.f64}\left(\color{blue}{\mathsf{*.f64}\left(b, b\right)}, \mathsf{*.f64}\left(a, a\right)\right)\right) \]

      4. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{1}{90}, \mathsf{PI}\left(\right)\right), angle\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\color{blue}{b}, b\right), \mathsf{*.f64}\left(a, a\right)\right)\right) \]

      5. PI-lowering-PI.f64 58.9%

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{1}{90}, \mathsf{PI.f64}\left(\right)\right), angle\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(b, b\right), \mathsf{*.f64}\left(a, a\right)\right)\right) \]

   9. Applied egg-rr 58.9%

      \[\leadsto \color{blue}{\left(\left(0.011111111111111112 \cdot \pi\right) \cdot angle\right)} \cdot \left(b \cdot b - a \cdot a\right) \]

   if 6.00000000000000005e146 < a

   1. Initial program 69.4%

      \[\left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin \left(\pi \cdot \frac{angle}{180}\right)\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right) \]
   2. Step-by-step derivation

      1. associate-*l* N/A

         \[\leadsto \left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \color{blue}{\left(\sin \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right) \cdot \cos \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right)\right)} \]

      2. *-commutative N/A

         \[\leadsto \left(\sin \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right) \cdot \cos \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right)\right) \cdot \color{blue}{\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right)} \]

      3. associate-*l* N/A

         \[\leadsto \sin \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right) \cdot \color{blue}{\left(\cos \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right) \cdot \left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right)\right)} \]

      4. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\sin \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right), \color{blue}{\left(\cos \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right) \cdot \left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right)\right)}\right) \]

      5. sin-lowering-sin.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sin.f64}\left(\left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right)\right), \left(\color{blue}{\cos \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right)} \cdot \left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right)\right)\right) \]

      6. associate-*r/ N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sin.f64}\left(\left(\frac{\mathsf{PI}\left(\right) \cdot angle}{180}\right)\right), \left(\cos \color{blue}{\left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right)} \cdot \left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right)\right)\right) \]

      7. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sin.f64}\left(\mathsf{/.f64}\left(\left(\mathsf{PI}\left(\right) \cdot angle\right), 180\right)\right), \left(\cos \color{blue}{\left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right)} \cdot \left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right)\right)\right) \]

      8. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sin.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{PI}\left(\right), angle\right), 180\right)\right), \left(\cos \left(\color{blue}{\mathsf{PI}\left(\right)} \cdot \frac{angle}{180}\right) \cdot \left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right)\right)\right) \]

      9. PI-lowering-PI.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sin.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{PI.f64}\left(\right), angle\right), 180\right)\right), \left(\cos \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right) \cdot \left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right)\right)\right) \]

      10. *-commutative N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{sin.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{PI.f64}\left(\right), angle\right), 180\right)\right), \left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \color{blue}{\cos \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right)}\right)\right) \]

      11. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{sin.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{PI.f64}\left(\right), angle\right), 180\right)\right), \mathsf{*.f64}\left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right), \color{blue}{\cos \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right)}\right)\right) \]

   3. Simplified 72.8%

      \[\leadsto \color{blue}{\sin \left(\frac{\pi \cdot angle}{180}\right) \cdot \left(\left(2 \cdot \left(b \cdot b - a \cdot a\right)\right) \cdot \cos \left(\frac{\pi \cdot angle}{180}\right)\right)} \]
   4. Add Preprocessing

   5. Taylor expanded in angle around 0

      \[\leadsto \color{blue}{\frac{1}{90} \cdot \left(angle \cdot \left(\mathsf{PI}\left(\right) \cdot \left({b}^{2} - {a}^{2}\right)\right)\right)} \]
   6. Step-by-step derivation

      1. associate-*r* N/A

         \[\leadsto \frac{1}{90} \cdot \left(\left(angle \cdot \mathsf{PI}\left(\right)\right) \cdot \color{blue}{\left({b}^{2} - {a}^{2}\right)}\right) \]

      2. associate-*r* N/A

         \[\leadsto \left(\frac{1}{90} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right) \cdot \color{blue}{\left({b}^{2} - {a}^{2}\right)} \]

      3. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\left(\frac{1}{90} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right), \color{blue}{\left({b}^{2} - {a}^{2}\right)}\right) \]

      4. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{1}{90}, \left(angle \cdot \mathsf{PI}\left(\right)\right)\right), \left(\color{blue}{{b}^{2}} - {a}^{2}\right)\right) \]

      5. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{1}{90}, \mathsf{*.f64}\left(angle, \mathsf{PI}\left(\right)\right)\right), \left({b}^{\color{blue}{2}} - {a}^{2}\right)\right) \]

      6. PI-lowering-PI.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{1}{90}, \mathsf{*.f64}\left(angle, \mathsf{PI.f64}\left(\right)\right)\right), \left({b}^{2} - {a}^{2}\right)\right) \]

      7. --lowering--.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{1}{90}, \mathsf{*.f64}\left(angle, \mathsf{PI.f64}\left(\right)\right)\right), \mathsf{\_.f64}\left(\left({b}^{2}\right), \color{blue}{\left({a}^{2}\right)}\right)\right) \]

      8. unpow2 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{1}{90}, \mathsf{*.f64}\left(angle, \mathsf{PI.f64}\left(\right)\right)\right), \mathsf{\_.f64}\left(\left(b \cdot b\right), \left({\color{blue}{a}}^{2}\right)\right)\right) \]

      9. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{1}{90}, \mathsf{*.f64}\left(angle, \mathsf{PI.f64}\left(\right)\right)\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(b, b\right), \left({\color{blue}{a}}^{2}\right)\right)\right) \]

      10. unpow2 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{1}{90}, \mathsf{*.f64}\left(angle, \mathsf{PI.f64}\left(\right)\right)\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(b, b\right), \left(a \cdot \color{blue}{a}\right)\right)\right) \]

      11. *-lowering-*.f64 59.0%

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{1}{90}, \mathsf{*.f64}\left(angle, \mathsf{PI.f64}\left(\right)\right)\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(b, b\right), \mathsf{*.f64}\left(a, \color{blue}{a}\right)\right)\right) \]

   7. Simplified 59.0%

      \[\leadsto \color{blue}{\left(0.011111111111111112 \cdot \left(angle \cdot \pi\right)\right) \cdot \left(b \cdot b - a \cdot a\right)} \]

   8. Taylor expanded in b around 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 72.8%

         \[\leadsto \mathsf{*.f64}\left(\frac{-1}{90}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(a, a\right), angle\right), \mathsf{PI.f64}\left(\right)\right)\right) \]

   10. Simplified 72.8%

       \[\leadsto \color{blue}{-0.011111111111111112 \cdot \left(\left(\left(a \cdot a\right) \cdot angle\right) \cdot \pi\right)} \]
   11. Step-by-step derivation

       1. *-commutative N/A

          \[\leadsto \mathsf{*.f64}\left(\frac{-1}{90}, \left(\mathsf{PI}\left(\right) \cdot \color{blue}{\left(\left(a \cdot a\right) \cdot angle\right)}\right)\right) \]

       2. associate-*l* N/A

          \[\leadsto \mathsf{*.f64}\left(\frac{-1}{90}, \left(\mathsf{PI}\left(\right) \cdot \left(a \cdot \color{blue}{\left(a \cdot angle\right)}\right)\right)\right) \]

       3. associate-*r* N/A

          \[\leadsto \mathsf{*.f64}\left(\frac{-1}{90}, \left(\left(\mathsf{PI}\left(\right) \cdot a\right) \cdot \color{blue}{\left(a \cdot angle\right)}\right)\right) \]

       4. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\frac{-1}{90}, \mathsf{*.f64}\left(\left(\mathsf{PI}\left(\right) \cdot a\right), \color{blue}{\left(a \cdot angle\right)}\right)\right) \]

       5. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\frac{-1}{90}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{PI}\left(\right), a\right), \left(\color{blue}{a} \cdot angle\right)\right)\right) \]

       6. PI-lowering-PI.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\frac{-1}{90}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{PI.f64}\left(\right), a\right), \left(a \cdot angle\right)\right)\right) \]

       7. *-lowering-*.f64 79.3%

          \[\leadsto \mathsf{*.f64}\left(\frac{-1}{90}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{PI.f64}\left(\right), a\right), \mathsf{*.f64}\left(a, \color{blue}{angle}\right)\right)\right) \]

   12. Applied egg-rr 79.3%

       \[\leadsto -0.011111111111111112 \cdot \color{blue}{\left(\left(\pi \cdot a\right) \cdot \left(a \cdot angle\right)\right)} \]
   13. Step-by-step derivation

       1. *-commutative N/A

          \[\leadsto \left(\left(\mathsf{PI}\left(\right) \cdot a\right) \cdot \left(a \cdot angle\right)\right) \cdot \color{blue}{\frac{-1}{90}} \]

       2. associate-*l* N/A

          \[\leadsto \left(\mathsf{PI}\left(\right) \cdot a\right) \cdot \color{blue}{\left(\left(a \cdot angle\right) \cdot \frac{-1}{90}\right)} \]

       3. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\left(\mathsf{PI}\left(\right) \cdot a\right), \color{blue}{\left(\left(a \cdot angle\right) \cdot \frac{-1}{90}\right)}\right) \]

       4. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{PI}\left(\right), a\right), \left(\color{blue}{\left(a \cdot angle\right)} \cdot \frac{-1}{90}\right)\right) \]

       5. PI-lowering-PI.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{PI.f64}\left(\right), a\right), \left(\left(\color{blue}{a} \cdot angle\right) \cdot \frac{-1}{90}\right)\right) \]

       6. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{PI.f64}\left(\right), a\right), \mathsf{*.f64}\left(\left(a \cdot angle\right), \color{blue}{\frac{-1}{90}}\right)\right) \]

       7. *-commutative N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{PI.f64}\left(\right), a\right), \mathsf{*.f64}\left(\left(angle \cdot a\right), \frac{-1}{90}\right)\right) \]

       8. *-lowering-*.f64 79.3%

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{PI.f64}\left(\right), a\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(angle, a\right), \frac{-1}{90}\right)\right) \]

   14. Applied egg-rr 79.3%

       \[\leadsto \color{blue}{\left(\pi \cdot a\right) \cdot \left(\left(angle \cdot a\right) \cdot -0.011111111111111112\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 61.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \leq 6 \cdot 10^{+146}:\\ \;\;\;\;\left(b \cdot b - a \cdot a\right) \cdot \left(angle \cdot \left(\pi \cdot 0.011111111111111112\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(a \cdot \pi\right) \cdot \left(-0.011111111111111112 \cdot \left(a \cdot angle\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 14: 57.5% accurate, 23.3× speedup

Math

\[\begin{array}{l} a_m = \left|a\right| \\ b_m = \left|b\right| \\ \begin{array}{l} \mathbf{if}\;a\_m \leq 7.5 \cdot 10^{+153}:\\ \;\;\;\;\left(angle \cdot 0.011111111111111112\right) \cdot \left(\pi \cdot \left(b\_m \cdot b\_m - a\_m \cdot a\_m\right)\right)\\ \mathbf{else}:\\ \;\;\;\;-0.011111111111111112 \cdot \left(a\_m \cdot \left(a\_m \cdot \left(\pi \cdot angle\right)\right)\right)\\ \end{array} \end{array} \]

FPCore

a_m = (fabs.f64 a)
b_m = (fabs.f64 b)
(FPCore (a_m b_m angle)
 :precision binary64
 (if (<= a_m 7.5e+153)
   (* (* angle 0.011111111111111112) (* PI (- (* b_m b_m) (* a_m a_m))))
   (* -0.011111111111111112 (* a_m (* a_m (* PI angle))))))

C

a_m = fabs(a);
b_m = fabs(b);
double code(double a_m, double b_m, double angle) {
	double tmp;
	if (a_m <= 7.5e+153) {
		tmp = (angle * 0.011111111111111112) * (((double) M_PI) * ((b_m * b_m) - (a_m * a_m)));
	} else {
		tmp = -0.011111111111111112 * (a_m * (a_m * (((double) M_PI) * angle)));
	}
	return tmp;
}

Java

a_m = Math.abs(a);
b_m = Math.abs(b);
public static double code(double a_m, double b_m, double angle) {
	double tmp;
	if (a_m <= 7.5e+153) {
		tmp = (angle * 0.011111111111111112) * (Math.PI * ((b_m * b_m) - (a_m * a_m)));
	} else {
		tmp = -0.011111111111111112 * (a_m * (a_m * (Math.PI * angle)));
	}
	return tmp;
}

Python

a_m = math.fabs(a)
b_m = math.fabs(b)
def code(a_m, b_m, angle):
	tmp = 0
	if a_m <= 7.5e+153:
		tmp = (angle * 0.011111111111111112) * (math.pi * ((b_m * b_m) - (a_m * a_m)))
	else:
		tmp = -0.011111111111111112 * (a_m * (a_m * (math.pi * angle)))
	return tmp

Julia

a_m = abs(a)
b_m = abs(b)
function code(a_m, b_m, angle)
	tmp = 0.0
	if (a_m <= 7.5e+153)
		tmp = Float64(Float64(angle * 0.011111111111111112) * Float64(pi * Float64(Float64(b_m * b_m) - Float64(a_m * a_m))));
	else
		tmp = Float64(-0.011111111111111112 * Float64(a_m * Float64(a_m * Float64(pi * angle))));
	end
	return tmp
end

MATLAB

a_m = abs(a);
b_m = abs(b);
function tmp_2 = code(a_m, b_m, angle)
	tmp = 0.0;
	if (a_m <= 7.5e+153)
		tmp = (angle * 0.011111111111111112) * (pi * ((b_m * b_m) - (a_m * a_m)));
	else
		tmp = -0.011111111111111112 * (a_m * (a_m * (pi * angle)));
	end
	tmp_2 = tmp;
end

Wolfram

a_m = N[Abs[a], $MachinePrecision]
b_m = N[Abs[b], $MachinePrecision]
code[a$95$m_, b$95$m_, angle_] := If[LessEqual[a$95$m, 7.5e+153], N[(N[(angle * 0.011111111111111112), $MachinePrecision] * N[(Pi * N[(N[(b$95$m * b$95$m), $MachinePrecision] - N[(a$95$m * a$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(-0.011111111111111112 * N[(a$95$m * N[(a$95$m * N[(Pi * angle), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if a < 7.50000000000000065e153

   1. Initial program 60.1%

      \[\left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin \left(\pi \cdot \frac{angle}{180}\right)\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right) \]
   2. Step-by-step derivation

      1. associate-*l* N/A

         \[\leadsto \left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \color{blue}{\left(\sin \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right) \cdot \cos \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right)\right)} \]

      2. *-commutative N/A

         \[\leadsto \left(\sin \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right) \cdot \cos \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right)\right) \cdot \color{blue}{\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right)} \]

      3. associate-*l* N/A

         \[\leadsto \sin \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right) \cdot \color{blue}{\left(\cos \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right) \cdot \left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right)\right)} \]

      4. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\sin \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right), \color{blue}{\left(\cos \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right) \cdot \left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right)\right)}\right) \]

      5. sin-lowering-sin.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sin.f64}\left(\left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right)\right), \left(\color{blue}{\cos \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right)} \cdot \left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right)\right)\right) \]

      6. associate-*r/ N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sin.f64}\left(\left(\frac{\mathsf{PI}\left(\right) \cdot angle}{180}\right)\right), \left(\cos \color{blue}{\left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right)} \cdot \left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right)\right)\right) \]

      7. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sin.f64}\left(\mathsf{/.f64}\left(\left(\mathsf{PI}\left(\right) \cdot angle\right), 180\right)\right), \left(\cos \color{blue}{\left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right)} \cdot \left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right)\right)\right) \]

      8. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sin.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{PI}\left(\right), angle\right), 180\right)\right), \left(\cos \left(\color{blue}{\mathsf{PI}\left(\right)} \cdot \frac{angle}{180}\right) \cdot \left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right)\right)\right) \]

      9. PI-lowering-PI.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sin.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{PI.f64}\left(\right), angle\right), 180\right)\right), \left(\cos \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right) \cdot \left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right)\right)\right) \]

      10. *-commutative N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{sin.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{PI.f64}\left(\right), angle\right), 180\right)\right), \left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \color{blue}{\cos \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right)}\right)\right) \]

      11. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{sin.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{PI.f64}\left(\right), angle\right), 180\right)\right), \mathsf{*.f64}\left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right), \color{blue}{\cos \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right)}\right)\right) \]

   3. Simplified 62.0%

      \[\leadsto \color{blue}{\sin \left(\frac{\pi \cdot angle}{180}\right) \cdot \left(\left(2 \cdot \left(b \cdot b - a \cdot a\right)\right) \cdot \cos \left(\frac{\pi \cdot angle}{180}\right)\right)} \]
   4. Add Preprocessing

   5. Taylor expanded in angle around 0

      \[\leadsto \color{blue}{\frac{1}{90} \cdot \left(angle \cdot \left(\mathsf{PI}\left(\right) \cdot \left({b}^{2} - {a}^{2}\right)\right)\right)} \]
   6. Step-by-step derivation

      1. associate-*r* N/A

         \[\leadsto \frac{1}{90} \cdot \left(\left(angle \cdot \mathsf{PI}\left(\right)\right) \cdot \color{blue}{\left({b}^{2} - {a}^{2}\right)}\right) \]

      2. associate-*r* N/A

         \[\leadsto \left(\frac{1}{90} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right) \cdot \color{blue}{\left({b}^{2} - {a}^{2}\right)} \]

      3. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\left(\frac{1}{90} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right), \color{blue}{\left({b}^{2} - {a}^{2}\right)}\right) \]

      4. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{1}{90}, \left(angle \cdot \mathsf{PI}\left(\right)\right)\right), \left(\color{blue}{{b}^{2}} - {a}^{2}\right)\right) \]

      5. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{1}{90}, \mathsf{*.f64}\left(angle, \mathsf{PI}\left(\right)\right)\right), \left({b}^{\color{blue}{2}} - {a}^{2}\right)\right) \]

      6. PI-lowering-PI.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{1}{90}, \mathsf{*.f64}\left(angle, \mathsf{PI.f64}\left(\right)\right)\right), \left({b}^{2} - {a}^{2}\right)\right) \]

      7. --lowering--.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{1}{90}, \mathsf{*.f64}\left(angle, \mathsf{PI.f64}\left(\right)\right)\right), \mathsf{\_.f64}\left(\left({b}^{2}\right), \color{blue}{\left({a}^{2}\right)}\right)\right) \]

      8. unpow2 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{1}{90}, \mathsf{*.f64}\left(angle, \mathsf{PI.f64}\left(\right)\right)\right), \mathsf{\_.f64}\left(\left(b \cdot b\right), \left({\color{blue}{a}}^{2}\right)\right)\right) \]

      9. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{1}{90}, \mathsf{*.f64}\left(angle, \mathsf{PI.f64}\left(\right)\right)\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(b, b\right), \left({\color{blue}{a}}^{2}\right)\right)\right) \]

      10. unpow2 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{1}{90}, \mathsf{*.f64}\left(angle, \mathsf{PI.f64}\left(\right)\right)\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(b, b\right), \left(a \cdot \color{blue}{a}\right)\right)\right) \]

      11. *-lowering-*.f64 59.0%

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{1}{90}, \mathsf{*.f64}\left(angle, \mathsf{PI.f64}\left(\right)\right)\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(b, b\right), \mathsf{*.f64}\left(a, \color{blue}{a}\right)\right)\right) \]

   7. Simplified 59.0%

      \[\leadsto \color{blue}{\left(0.011111111111111112 \cdot \left(angle \cdot \pi\right)\right) \cdot \left(b \cdot b - a \cdot a\right)} \]
   8. Step-by-step derivation

      1. associate-*r* N/A

         \[\leadsto \left(\left(\frac{1}{90} \cdot angle\right) \cdot \mathsf{PI}\left(\right)\right) \cdot \left(\color{blue}{b \cdot b} - a \cdot a\right) \]

      2. associate-*l* N/A

         \[\leadsto \left(\frac{1}{90} \cdot angle\right) \cdot \color{blue}{\left(\mathsf{PI}\left(\right) \cdot \left(b \cdot b - a \cdot a\right)\right)} \]

      3. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\left(\frac{1}{90} \cdot angle\right), \color{blue}{\left(\mathsf{PI}\left(\right) \cdot \left(b \cdot b - a \cdot a\right)\right)}\right) \]

      4. *-commutative N/A

         \[\leadsto \mathsf{*.f64}\left(\left(angle \cdot \frac{1}{90}\right), \left(\color{blue}{\mathsf{PI}\left(\right)} \cdot \left(b \cdot b - a \cdot a\right)\right)\right) \]

      5. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(angle, \frac{1}{90}\right), \left(\color{blue}{\mathsf{PI}\left(\right)} \cdot \left(b \cdot b - a \cdot a\right)\right)\right) \]

      6. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(angle, \frac{1}{90}\right), \mathsf{*.f64}\left(\mathsf{PI}\left(\right), \color{blue}{\left(b \cdot b - a \cdot a\right)}\right)\right) \]

      7. PI-lowering-PI.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(angle, \frac{1}{90}\right), \mathsf{*.f64}\left(\mathsf{PI.f64}\left(\right), \left(\color{blue}{b \cdot b} - a \cdot a\right)\right)\right) \]

      8. --lowering--.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(angle, \frac{1}{90}\right), \mathsf{*.f64}\left(\mathsf{PI.f64}\left(\right), \mathsf{\_.f64}\left(\left(b \cdot b\right), \color{blue}{\left(a \cdot a\right)}\right)\right)\right) \]

      9. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(angle, \frac{1}{90}\right), \mathsf{*.f64}\left(\mathsf{PI.f64}\left(\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(b, b\right), \left(\color{blue}{a} \cdot a\right)\right)\right)\right) \]

      10. *-lowering-*.f64 59.0%

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(angle, \frac{1}{90}\right), \mathsf{*.f64}\left(\mathsf{PI.f64}\left(\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(b, b\right), \mathsf{*.f64}\left(a, \color{blue}{a}\right)\right)\right)\right) \]

   9. Applied egg-rr 59.0%

      \[\leadsto \color{blue}{\left(angle \cdot 0.011111111111111112\right) \cdot \left(\pi \cdot \left(b \cdot b - a \cdot a\right)\right)} \]

   if 7.50000000000000065e153 < a

   1. Initial program 68.3%

      \[\left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin \left(\pi \cdot \frac{angle}{180}\right)\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right) \]
   2. Step-by-step derivation

      1. associate-*l* N/A

         \[\leadsto \left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \color{blue}{\left(\sin \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right) \cdot \cos \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right)\right)} \]

      2. *-commutative N/A

         \[\leadsto \left(\sin \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right) \cdot \cos \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right)\right) \cdot \color{blue}{\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right)} \]

      3. associate-*l* N/A

         \[\leadsto \sin \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right) \cdot \color{blue}{\left(\cos \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right) \cdot \left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right)\right)} \]

      4. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\sin \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right), \color{blue}{\left(\cos \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right) \cdot \left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right)\right)}\right) \]

      5. sin-lowering-sin.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sin.f64}\left(\left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right)\right), \left(\color{blue}{\cos \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right)} \cdot \left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right)\right)\right) \]

      6. associate-*r/ N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sin.f64}\left(\left(\frac{\mathsf{PI}\left(\right) \cdot angle}{180}\right)\right), \left(\cos \color{blue}{\left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right)} \cdot \left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right)\right)\right) \]

      7. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sin.f64}\left(\mathsf{/.f64}\left(\left(\mathsf{PI}\left(\right) \cdot angle\right), 180\right)\right), \left(\cos \color{blue}{\left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right)} \cdot \left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right)\right)\right) \]

      8. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sin.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{PI}\left(\right), angle\right), 180\right)\right), \left(\cos \left(\color{blue}{\mathsf{PI}\left(\right)} \cdot \frac{angle}{180}\right) \cdot \left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right)\right)\right) \]

      9. PI-lowering-PI.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sin.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{PI.f64}\left(\right), angle\right), 180\right)\right), \left(\cos \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right) \cdot \left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right)\right)\right) \]

      10. *-commutative N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{sin.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{PI.f64}\left(\right), angle\right), 180\right)\right), \left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \color{blue}{\cos \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right)}\right)\right) \]

      11. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{sin.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{PI.f64}\left(\right), angle\right), 180\right)\right), \mathsf{*.f64}\left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right), \color{blue}{\cos \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right)}\right)\right) \]

   3. Simplified 71.9%

      \[\leadsto \color{blue}{\sin \left(\frac{\pi \cdot angle}{180}\right) \cdot \left(\left(2 \cdot \left(b \cdot b - a \cdot a\right)\right) \cdot \cos \left(\frac{\pi \cdot angle}{180}\right)\right)} \]
   4. Add Preprocessing

   5. Taylor expanded in angle around 0

      \[\leadsto \color{blue}{\frac{1}{90} \cdot \left(angle \cdot \left(\mathsf{PI}\left(\right) \cdot \left({b}^{2} - {a}^{2}\right)\right)\right)} \]
   6. Step-by-step derivation

      1. associate-*r* N/A

         \[\leadsto \frac{1}{90} \cdot \left(\left(angle \cdot \mathsf{PI}\left(\right)\right) \cdot \color{blue}{\left({b}^{2} - {a}^{2}\right)}\right) \]

      2. associate-*r* N/A

         \[\leadsto \left(\frac{1}{90} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right) \cdot \color{blue}{\left({b}^{2} - {a}^{2}\right)} \]

      3. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\left(\frac{1}{90} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right), \color{blue}{\left({b}^{2} - {a}^{2}\right)}\right) \]

      4. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{1}{90}, \left(angle \cdot \mathsf{PI}\left(\right)\right)\right), \left(\color{blue}{{b}^{2}} - {a}^{2}\right)\right) \]

      5. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{1}{90}, \mathsf{*.f64}\left(angle, \mathsf{PI}\left(\right)\right)\right), \left({b}^{\color{blue}{2}} - {a}^{2}\right)\right) \]

      6. PI-lowering-PI.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{1}{90}, \mathsf{*.f64}\left(angle, \mathsf{PI.f64}\left(\right)\right)\right), \left({b}^{2} - {a}^{2}\right)\right) \]

      7. --lowering--.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{1}{90}, \mathsf{*.f64}\left(angle, \mathsf{PI.f64}\left(\right)\right)\right), \mathsf{\_.f64}\left(\left({b}^{2}\right), \color{blue}{\left({a}^{2}\right)}\right)\right) \]

      8. unpow2 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{1}{90}, \mathsf{*.f64}\left(angle, \mathsf{PI.f64}\left(\right)\right)\right), \mathsf{\_.f64}\left(\left(b \cdot b\right), \left({\color{blue}{a}}^{2}\right)\right)\right) \]

      9. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{1}{90}, \mathsf{*.f64}\left(angle, \mathsf{PI.f64}\left(\right)\right)\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(b, b\right), \left({\color{blue}{a}}^{2}\right)\right)\right) \]

      10. unpow2 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{1}{90}, \mathsf{*.f64}\left(angle, \mathsf{PI.f64}\left(\right)\right)\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(b, b\right), \left(a \cdot \color{blue}{a}\right)\right)\right) \]

      11. *-lowering-*.f64 57.6%

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{1}{90}, \mathsf{*.f64}\left(angle, \mathsf{PI.f64}\left(\right)\right)\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(b, b\right), \mathsf{*.f64}\left(a, \color{blue}{a}\right)\right)\right) \]

   7. Simplified 57.6%

      \[\leadsto \color{blue}{\left(0.011111111111111112 \cdot \left(angle \cdot \pi\right)\right) \cdot \left(b \cdot b - a \cdot a\right)} \]

   8. Taylor expanded in b around 0

      \[\leadsto \color{blue}{\frac{-1}{90} \cdot \left({a}^{2} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)} \]
   9. Step-by-step derivation

      1. *-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 71.9%

         \[\leadsto \mathsf{*.f64}\left(\frac{-1}{90}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(a, a\right), angle\right), \mathsf{PI.f64}\left(\right)\right)\right) \]

   10. Simplified 71.9%

       \[\leadsto \color{blue}{-0.011111111111111112 \cdot \left(\left(\left(a \cdot a\right) \cdot angle\right) \cdot \pi\right)} \]
   11. Step-by-step derivation

       1. *-commutative N/A

          \[\leadsto \left(\left(\left(a \cdot a\right) \cdot angle\right) \cdot \mathsf{PI}\left(\right)\right) \cdot \color{blue}{\frac{-1}{90}} \]

       2. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\left(\left(\left(a \cdot a\right) \cdot angle\right) \cdot \mathsf{PI}\left(\right)\right), \color{blue}{\frac{-1}{90}}\right) \]

       3. associate-*l* N/A

          \[\leadsto \mathsf{*.f64}\left(\left(\left(a \cdot a\right) \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right), \frac{-1}{90}\right) \]

       4. *-commutative N/A

          \[\leadsto \mathsf{*.f64}\left(\left(\left(a \cdot a\right) \cdot \left(\mathsf{PI}\left(\right) \cdot angle\right)\right), \frac{-1}{90}\right) \]

       5. associate-*l* N/A

          \[\leadsto \mathsf{*.f64}\left(\left(a \cdot \left(a \cdot \left(\mathsf{PI}\left(\right) \cdot angle\right)\right)\right), \frac{-1}{90}\right) \]

       6. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(a, \left(a \cdot \left(\mathsf{PI}\left(\right) \cdot angle\right)\right)\right), \frac{-1}{90}\right) \]

       7. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(a, \mathsf{*.f64}\left(a, \left(\mathsf{PI}\left(\right) \cdot angle\right)\right)\right), \frac{-1}{90}\right) \]

       8. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(a, \mathsf{*.f64}\left(a, \mathsf{*.f64}\left(\mathsf{PI}\left(\right), angle\right)\right)\right), \frac{-1}{90}\right) \]

       9. PI-lowering-PI.f64 78.6%

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(a, \mathsf{*.f64}\left(a, \mathsf{*.f64}\left(\mathsf{PI.f64}\left(\right), angle\right)\right)\right), \frac{-1}{90}\right) \]

   12. Applied egg-rr 78.6%

       \[\leadsto \color{blue}{\left(a \cdot \left(a \cdot \left(\pi \cdot angle\right)\right)\right) \cdot -0.011111111111111112} \]
3. Recombined 2 regimes into one program.

4. Final simplification 61.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \leq 7.5 \cdot 10^{+153}:\\ \;\;\;\;\left(angle \cdot 0.011111111111111112\right) \cdot \left(\pi \cdot \left(b \cdot b - a \cdot a\right)\right)\\ \mathbf{else}:\\ \;\;\;\;-0.011111111111111112 \cdot \left(a \cdot \left(a \cdot \left(\pi \cdot angle\right)\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 15: 57.5% accurate, 23.3× speedup

Math

\[\begin{array}{l} a_m = \left|a\right| \\ b_m = \left|b\right| \\ \begin{array}{l} \mathbf{if}\;a\_m \leq 4.4 \cdot 10^{+146}:\\ \;\;\;\;\left(\left(\pi \cdot angle\right) \cdot 0.011111111111111112\right) \cdot \left(b\_m \cdot b\_m - a\_m \cdot a\_m\right)\\ \mathbf{else}:\\ \;\;\;\;\left(a\_m \cdot \pi\right) \cdot \left(-0.011111111111111112 \cdot \left(a\_m \cdot angle\right)\right)\\ \end{array} \end{array} \]

FPCore

a_m = (fabs.f64 a)
b_m = (fabs.f64 b)
(FPCore (a_m b_m angle)
 :precision binary64
 (if (<= a_m 4.4e+146)
   (* (* (* PI angle) 0.011111111111111112) (- (* b_m b_m) (* a_m a_m)))
   (* (* a_m PI) (* -0.011111111111111112 (* a_m angle)))))

C

a_m = fabs(a);
b_m = fabs(b);
double code(double a_m, double b_m, double angle) {
	double tmp;
	if (a_m <= 4.4e+146) {
		tmp = ((((double) M_PI) * angle) * 0.011111111111111112) * ((b_m * b_m) - (a_m * a_m));
	} else {
		tmp = (a_m * ((double) M_PI)) * (-0.011111111111111112 * (a_m * angle));
	}
	return tmp;
}

Java

a_m = Math.abs(a);
b_m = Math.abs(b);
public static double code(double a_m, double b_m, double angle) {
	double tmp;
	if (a_m <= 4.4e+146) {
		tmp = ((Math.PI * angle) * 0.011111111111111112) * ((b_m * b_m) - (a_m * a_m));
	} else {
		tmp = (a_m * Math.PI) * (-0.011111111111111112 * (a_m * angle));
	}
	return tmp;
}

Python

a_m = math.fabs(a)
b_m = math.fabs(b)
def code(a_m, b_m, angle):
	tmp = 0
	if a_m <= 4.4e+146:
		tmp = ((math.pi * angle) * 0.011111111111111112) * ((b_m * b_m) - (a_m * a_m))
	else:
		tmp = (a_m * math.pi) * (-0.011111111111111112 * (a_m * angle))
	return tmp

Julia

a_m = abs(a)
b_m = abs(b)
function code(a_m, b_m, angle)
	tmp = 0.0
	if (a_m <= 4.4e+146)
		tmp = Float64(Float64(Float64(pi * angle) * 0.011111111111111112) * Float64(Float64(b_m * b_m) - Float64(a_m * a_m)));
	else
		tmp = Float64(Float64(a_m * pi) * Float64(-0.011111111111111112 * Float64(a_m * angle)));
	end
	return tmp
end

MATLAB

a_m = abs(a);
b_m = abs(b);
function tmp_2 = code(a_m, b_m, angle)
	tmp = 0.0;
	if (a_m <= 4.4e+146)
		tmp = ((pi * angle) * 0.011111111111111112) * ((b_m * b_m) - (a_m * a_m));
	else
		tmp = (a_m * pi) * (-0.011111111111111112 * (a_m * angle));
	end
	tmp_2 = tmp;
end

Wolfram

a_m = N[Abs[a], $MachinePrecision]
b_m = N[Abs[b], $MachinePrecision]
code[a$95$m_, b$95$m_, angle_] := If[LessEqual[a$95$m, 4.4e+146], N[(N[(N[(Pi * angle), $MachinePrecision] * 0.011111111111111112), $MachinePrecision] * N[(N[(b$95$m * b$95$m), $MachinePrecision] - N[(a$95$m * a$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(a$95$m * Pi), $MachinePrecision] * N[(-0.011111111111111112 * N[(a$95$m * angle), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if a < 4.3999999999999996e146

   1. Initial program 59.9%

      \[\left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin \left(\pi \cdot \frac{angle}{180}\right)\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right) \]
   2. Step-by-step derivation

      1. associate-*l* N/A

         \[\leadsto \left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \color{blue}{\left(\sin \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right) \cdot \cos \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right)\right)} \]

      2. *-commutative N/A

         \[\leadsto \left(\sin \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right) \cdot \cos \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right)\right) \cdot \color{blue}{\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right)} \]

      3. associate-*l* N/A

         \[\leadsto \sin \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right) \cdot \color{blue}{\left(\cos \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right) \cdot \left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right)\right)} \]

      4. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\sin \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right), \color{blue}{\left(\cos \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right) \cdot \left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right)\right)}\right) \]

      5. sin-lowering-sin.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sin.f64}\left(\left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right)\right), \left(\color{blue}{\cos \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right)} \cdot \left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right)\right)\right) \]

      6. associate-*r/ N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sin.f64}\left(\left(\frac{\mathsf{PI}\left(\right) \cdot angle}{180}\right)\right), \left(\cos \color{blue}{\left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right)} \cdot \left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right)\right)\right) \]

      7. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sin.f64}\left(\mathsf{/.f64}\left(\left(\mathsf{PI}\left(\right) \cdot angle\right), 180\right)\right), \left(\cos \color{blue}{\left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right)} \cdot \left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right)\right)\right) \]

      8. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sin.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{PI}\left(\right), angle\right), 180\right)\right), \left(\cos \left(\color{blue}{\mathsf{PI}\left(\right)} \cdot \frac{angle}{180}\right) \cdot \left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right)\right)\right) \]

      9. PI-lowering-PI.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sin.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{PI.f64}\left(\right), angle\right), 180\right)\right), \left(\cos \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right) \cdot \left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right)\right)\right) \]

      10. *-commutative N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{sin.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{PI.f64}\left(\right), angle\right), 180\right)\right), \left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \color{blue}{\cos \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right)}\right)\right) \]

      11. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{sin.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{PI.f64}\left(\right), angle\right), 180\right)\right), \mathsf{*.f64}\left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right), \color{blue}{\cos \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right)}\right)\right) \]

   3. Simplified 61.9%

      \[\leadsto \color{blue}{\sin \left(\frac{\pi \cdot angle}{180}\right) \cdot \left(\left(2 \cdot \left(b \cdot b - a \cdot a\right)\right) \cdot \cos \left(\frac{\pi \cdot angle}{180}\right)\right)} \]
   4. Add Preprocessing

   5. Taylor expanded in angle around 0

      \[\leadsto \color{blue}{\frac{1}{90} \cdot \left(angle \cdot \left(\mathsf{PI}\left(\right) \cdot \left({b}^{2} - {a}^{2}\right)\right)\right)} \]
   6. Step-by-step derivation

      1. associate-*r* N/A

         \[\leadsto \frac{1}{90} \cdot \left(\left(angle \cdot \mathsf{PI}\left(\right)\right) \cdot \color{blue}{\left({b}^{2} - {a}^{2}\right)}\right) \]

      2. associate-*r* N/A

         \[\leadsto \left(\frac{1}{90} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right) \cdot \color{blue}{\left({b}^{2} - {a}^{2}\right)} \]

      3. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\left(\frac{1}{90} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right), \color{blue}{\left({b}^{2} - {a}^{2}\right)}\right) \]

      4. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{1}{90}, \left(angle \cdot \mathsf{PI}\left(\right)\right)\right), \left(\color{blue}{{b}^{2}} - {a}^{2}\right)\right) \]

      5. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{1}{90}, \mathsf{*.f64}\left(angle, \mathsf{PI}\left(\right)\right)\right), \left({b}^{\color{blue}{2}} - {a}^{2}\right)\right) \]

      6. PI-lowering-PI.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{1}{90}, \mathsf{*.f64}\left(angle, \mathsf{PI.f64}\left(\right)\right)\right), \left({b}^{2} - {a}^{2}\right)\right) \]

      7. --lowering--.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{1}{90}, \mathsf{*.f64}\left(angle, \mathsf{PI.f64}\left(\right)\right)\right), \mathsf{\_.f64}\left(\left({b}^{2}\right), \color{blue}{\left({a}^{2}\right)}\right)\right) \]

      8. unpow2 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{1}{90}, \mathsf{*.f64}\left(angle, \mathsf{PI.f64}\left(\right)\right)\right), \mathsf{\_.f64}\left(\left(b \cdot b\right), \left({\color{blue}{a}}^{2}\right)\right)\right) \]

      9. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{1}{90}, \mathsf{*.f64}\left(angle, \mathsf{PI.f64}\left(\right)\right)\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(b, b\right), \left({\color{blue}{a}}^{2}\right)\right)\right) \]

      10. unpow2 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{1}{90}, \mathsf{*.f64}\left(angle, \mathsf{PI.f64}\left(\right)\right)\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(b, b\right), \left(a \cdot \color{blue}{a}\right)\right)\right) \]

      11. *-lowering-*.f64 58.8%

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{1}{90}, \mathsf{*.f64}\left(angle, \mathsf{PI.f64}\left(\right)\right)\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(b, b\right), \mathsf{*.f64}\left(a, \color{blue}{a}\right)\right)\right) \]

   7. Simplified 58.8%

      \[\leadsto \color{blue}{\left(0.011111111111111112 \cdot \left(angle \cdot \pi\right)\right) \cdot \left(b \cdot b - a \cdot a\right)} \]

   if 4.3999999999999996e146 < a

   1. Initial program 69.4%

      \[\left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin \left(\pi \cdot \frac{angle}{180}\right)\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right) \]
   2. Step-by-step derivation

      1. associate-*l* N/A

         \[\leadsto \left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \color{blue}{\left(\sin \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right) \cdot \cos \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right)\right)} \]

      2. *-commutative N/A

         \[\leadsto \left(\sin \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right) \cdot \cos \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right)\right) \cdot \color{blue}{\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right)} \]

      3. associate-*l* N/A

         \[\leadsto \sin \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right) \cdot \color{blue}{\left(\cos \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right) \cdot \left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right)\right)} \]

      4. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\sin \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right), \color{blue}{\left(\cos \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right) \cdot \left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right)\right)}\right) \]

      5. sin-lowering-sin.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sin.f64}\left(\left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right)\right), \left(\color{blue}{\cos \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right)} \cdot \left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right)\right)\right) \]

      6. associate-*r/ N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sin.f64}\left(\left(\frac{\mathsf{PI}\left(\right) \cdot angle}{180}\right)\right), \left(\cos \color{blue}{\left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right)} \cdot \left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right)\right)\right) \]

      7. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sin.f64}\left(\mathsf{/.f64}\left(\left(\mathsf{PI}\left(\right) \cdot angle\right), 180\right)\right), \left(\cos \color{blue}{\left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right)} \cdot \left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right)\right)\right) \]

      8. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sin.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{PI}\left(\right), angle\right), 180\right)\right), \left(\cos \left(\color{blue}{\mathsf{PI}\left(\right)} \cdot \frac{angle}{180}\right) \cdot \left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right)\right)\right) \]

      9. PI-lowering-PI.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sin.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{PI.f64}\left(\right), angle\right), 180\right)\right), \left(\cos \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right) \cdot \left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right)\right)\right) \]

      10. *-commutative N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{sin.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{PI.f64}\left(\right), angle\right), 180\right)\right), \left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \color{blue}{\cos \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right)}\right)\right) \]

      11. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{sin.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{PI.f64}\left(\right), angle\right), 180\right)\right), \mathsf{*.f64}\left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right), \color{blue}{\cos \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right)}\right)\right) \]

   3. Simplified 72.8%

      \[\leadsto \color{blue}{\sin \left(\frac{\pi \cdot angle}{180}\right) \cdot \left(\left(2 \cdot \left(b \cdot b - a \cdot a\right)\right) \cdot \cos \left(\frac{\pi \cdot angle}{180}\right)\right)} \]
   4. Add Preprocessing

   5. Taylor expanded in angle around 0

      \[\leadsto \color{blue}{\frac{1}{90} \cdot \left(angle \cdot \left(\mathsf{PI}\left(\right) \cdot \left({b}^{2} - {a}^{2}\right)\right)\right)} \]
   6. Step-by-step derivation

      1. associate-*r* N/A

         \[\leadsto \frac{1}{90} \cdot \left(\left(angle \cdot \mathsf{PI}\left(\right)\right) \cdot \color{blue}{\left({b}^{2} - {a}^{2}\right)}\right) \]

      2. associate-*r* N/A

         \[\leadsto \left(\frac{1}{90} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right) \cdot \color{blue}{\left({b}^{2} - {a}^{2}\right)} \]

      3. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\left(\frac{1}{90} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right), \color{blue}{\left({b}^{2} - {a}^{2}\right)}\right) \]

      4. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{1}{90}, \left(angle \cdot \mathsf{PI}\left(\right)\right)\right), \left(\color{blue}{{b}^{2}} - {a}^{2}\right)\right) \]

      5. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{1}{90}, \mathsf{*.f64}\left(angle, \mathsf{PI}\left(\right)\right)\right), \left({b}^{\color{blue}{2}} - {a}^{2}\right)\right) \]

      6. PI-lowering-PI.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{1}{90}, \mathsf{*.f64}\left(angle, \mathsf{PI.f64}\left(\right)\right)\right), \left({b}^{2} - {a}^{2}\right)\right) \]

      7. --lowering--.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{1}{90}, \mathsf{*.f64}\left(angle, \mathsf{PI.f64}\left(\right)\right)\right), \mathsf{\_.f64}\left(\left({b}^{2}\right), \color{blue}{\left({a}^{2}\right)}\right)\right) \]

      8. unpow2 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{1}{90}, \mathsf{*.f64}\left(angle, \mathsf{PI.f64}\left(\right)\right)\right), \mathsf{\_.f64}\left(\left(b \cdot b\right), \left({\color{blue}{a}}^{2}\right)\right)\right) \]

      9. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{1}{90}, \mathsf{*.f64}\left(angle, \mathsf{PI.f64}\left(\right)\right)\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(b, b\right), \left({\color{blue}{a}}^{2}\right)\right)\right) \]

      10. unpow2 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{1}{90}, \mathsf{*.f64}\left(angle, \mathsf{PI.f64}\left(\right)\right)\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(b, b\right), \left(a \cdot \color{blue}{a}\right)\right)\right) \]

      11. *-lowering-*.f64 59.0%

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{1}{90}, \mathsf{*.f64}\left(angle, \mathsf{PI.f64}\left(\right)\right)\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(b, b\right), \mathsf{*.f64}\left(a, \color{blue}{a}\right)\right)\right) \]

   7. Simplified 59.0%

      \[\leadsto \color{blue}{\left(0.011111111111111112 \cdot \left(angle \cdot \pi\right)\right) \cdot \left(b \cdot b - a \cdot a\right)} \]

   8. Taylor expanded in b around 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 72.8%

         \[\leadsto \mathsf{*.f64}\left(\frac{-1}{90}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(a, a\right), angle\right), \mathsf{PI.f64}\left(\right)\right)\right) \]

   10. Simplified 72.8%

       \[\leadsto \color{blue}{-0.011111111111111112 \cdot \left(\left(\left(a \cdot a\right) \cdot angle\right) \cdot \pi\right)} \]
   11. Step-by-step derivation

       1. *-commutative N/A

          \[\leadsto \mathsf{*.f64}\left(\frac{-1}{90}, \left(\mathsf{PI}\left(\right) \cdot \color{blue}{\left(\left(a \cdot a\right) \cdot angle\right)}\right)\right) \]

       2. associate-*l* N/A

          \[\leadsto \mathsf{*.f64}\left(\frac{-1}{90}, \left(\mathsf{PI}\left(\right) \cdot \left(a \cdot \color{blue}{\left(a \cdot angle\right)}\right)\right)\right) \]

       3. associate-*r* N/A

          \[\leadsto \mathsf{*.f64}\left(\frac{-1}{90}, \left(\left(\mathsf{PI}\left(\right) \cdot a\right) \cdot \color{blue}{\left(a \cdot angle\right)}\right)\right) \]

       4. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\frac{-1}{90}, \mathsf{*.f64}\left(\left(\mathsf{PI}\left(\right) \cdot a\right), \color{blue}{\left(a \cdot angle\right)}\right)\right) \]

       5. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\frac{-1}{90}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{PI}\left(\right), a\right), \left(\color{blue}{a} \cdot angle\right)\right)\right) \]

       6. PI-lowering-PI.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\frac{-1}{90}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{PI.f64}\left(\right), a\right), \left(a \cdot angle\right)\right)\right) \]

       7. *-lowering-*.f64 79.3%

          \[\leadsto \mathsf{*.f64}\left(\frac{-1}{90}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{PI.f64}\left(\right), a\right), \mathsf{*.f64}\left(a, \color{blue}{angle}\right)\right)\right) \]

   12. Applied egg-rr 79.3%

       \[\leadsto -0.011111111111111112 \cdot \color{blue}{\left(\left(\pi \cdot a\right) \cdot \left(a \cdot angle\right)\right)} \]
   13. Step-by-step derivation

       1. *-commutative N/A

          \[\leadsto \left(\left(\mathsf{PI}\left(\right) \cdot a\right) \cdot \left(a \cdot angle\right)\right) \cdot \color{blue}{\frac{-1}{90}} \]

       2. associate-*l* N/A

          \[\leadsto \left(\mathsf{PI}\left(\right) \cdot a\right) \cdot \color{blue}{\left(\left(a \cdot angle\right) \cdot \frac{-1}{90}\right)} \]

       3. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\left(\mathsf{PI}\left(\right) \cdot a\right), \color{blue}{\left(\left(a \cdot angle\right) \cdot \frac{-1}{90}\right)}\right) \]

       4. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{PI}\left(\right), a\right), \left(\color{blue}{\left(a \cdot angle\right)} \cdot \frac{-1}{90}\right)\right) \]

       5. PI-lowering-PI.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{PI.f64}\left(\right), a\right), \left(\left(\color{blue}{a} \cdot angle\right) \cdot \frac{-1}{90}\right)\right) \]

       6. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{PI.f64}\left(\right), a\right), \mathsf{*.f64}\left(\left(a \cdot angle\right), \color{blue}{\frac{-1}{90}}\right)\right) \]

       7. *-commutative N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{PI.f64}\left(\right), a\right), \mathsf{*.f64}\left(\left(angle \cdot a\right), \frac{-1}{90}\right)\right) \]

       8. *-lowering-*.f64 79.3%

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{PI.f64}\left(\right), a\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(angle, a\right), \frac{-1}{90}\right)\right) \]

   14. Applied egg-rr 79.3%

       \[\leadsto \color{blue}{\left(\pi \cdot a\right) \cdot \left(\left(angle \cdot a\right) \cdot -0.011111111111111112\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 61.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \leq 4.4 \cdot 10^{+146}:\\ \;\;\;\;\left(\left(\pi \cdot angle\right) \cdot 0.011111111111111112\right) \cdot \left(b \cdot b - a \cdot a\right)\\ \mathbf{else}:\\ \;\;\;\;\left(a \cdot \pi\right) \cdot \left(-0.011111111111111112 \cdot \left(a \cdot angle\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 16: 51.2% accurate, 29.9× speedup

Math

\[\begin{array}{l} a_m = \left|a\right| \\ b_m = \left|b\right| \\ \begin{array}{l} \mathbf{if}\;a\_m \leq 1.46 \cdot 10^{-18}:\\ \;\;\;\;0.011111111111111112 \cdot \left(angle \cdot \left(\pi \cdot \left(b\_m \cdot b\_m\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;a\_m \cdot \left(angle \cdot \left(-0.011111111111111112 \cdot \left(a\_m \cdot \pi\right)\right)\right)\\ \end{array} \end{array} \]

FPCore

a_m = (fabs.f64 a)
b_m = (fabs.f64 b)
(FPCore (a_m b_m angle)
 :precision binary64
 (if (<= a_m 1.46e-18)
   (* 0.011111111111111112 (* angle (* PI (* b_m b_m))))
   (* a_m (* angle (* -0.011111111111111112 (* a_m PI))))))

C

a_m = fabs(a);
b_m = fabs(b);
double code(double a_m, double b_m, double angle) {
	double tmp;
	if (a_m <= 1.46e-18) {
		tmp = 0.011111111111111112 * (angle * (((double) M_PI) * (b_m * b_m)));
	} else {
		tmp = a_m * (angle * (-0.011111111111111112 * (a_m * ((double) M_PI))));
	}
	return tmp;
}

Java

a_m = Math.abs(a);
b_m = Math.abs(b);
public static double code(double a_m, double b_m, double angle) {
	double tmp;
	if (a_m <= 1.46e-18) {
		tmp = 0.011111111111111112 * (angle * (Math.PI * (b_m * b_m)));
	} else {
		tmp = a_m * (angle * (-0.011111111111111112 * (a_m * Math.PI)));
	}
	return tmp;
}

Python

a_m = math.fabs(a)
b_m = math.fabs(b)
def code(a_m, b_m, angle):
	tmp = 0
	if a_m <= 1.46e-18:
		tmp = 0.011111111111111112 * (angle * (math.pi * (b_m * b_m)))
	else:
		tmp = a_m * (angle * (-0.011111111111111112 * (a_m * math.pi)))
	return tmp

Julia

a_m = abs(a)
b_m = abs(b)
function code(a_m, b_m, angle)
	tmp = 0.0
	if (a_m <= 1.46e-18)
		tmp = Float64(0.011111111111111112 * Float64(angle * Float64(pi * Float64(b_m * b_m))));
	else
		tmp = Float64(a_m * Float64(angle * Float64(-0.011111111111111112 * Float64(a_m * pi))));
	end
	return tmp
end

MATLAB

a_m = abs(a);
b_m = abs(b);
function tmp_2 = code(a_m, b_m, angle)
	tmp = 0.0;
	if (a_m <= 1.46e-18)
		tmp = 0.011111111111111112 * (angle * (pi * (b_m * b_m)));
	else
		tmp = a_m * (angle * (-0.011111111111111112 * (a_m * pi)));
	end
	tmp_2 = tmp;
end

Wolfram

a_m = N[Abs[a], $MachinePrecision]
b_m = N[Abs[b], $MachinePrecision]
code[a$95$m_, b$95$m_, angle_] := If[LessEqual[a$95$m, 1.46e-18], N[(0.011111111111111112 * N[(angle * N[(Pi * N[(b$95$m * b$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(a$95$m * N[(angle * N[(-0.011111111111111112 * N[(a$95$m * Pi), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if a < 1.4599999999999999e-18

   1. Initial program 58.6%

      \[\left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin \left(\pi \cdot \frac{angle}{180}\right)\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right) \]
   2. Step-by-step derivation

      1. associate-*l* N/A

         \[\leadsto \left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \color{blue}{\left(\sin \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right) \cdot \cos \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right)\right)} \]

      2. *-commutative N/A

         \[\leadsto \left(\sin \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right) \cdot \cos \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right)\right) \cdot \color{blue}{\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right)} \]

      3. associate-*l* N/A

         \[\leadsto \sin \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right) \cdot \color{blue}{\left(\cos \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right) \cdot \left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right)\right)} \]

      4. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\sin \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right), \color{blue}{\left(\cos \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right) \cdot \left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right)\right)}\right) \]

      5. sin-lowering-sin.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sin.f64}\left(\left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right)\right), \left(\color{blue}{\cos \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right)} \cdot \left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right)\right)\right) \]

      6. associate-*r/ N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sin.f64}\left(\left(\frac{\mathsf{PI}\left(\right) \cdot angle}{180}\right)\right), \left(\cos \color{blue}{\left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right)} \cdot \left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right)\right)\right) \]

      7. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sin.f64}\left(\mathsf{/.f64}\left(\left(\mathsf{PI}\left(\right) \cdot angle\right), 180\right)\right), \left(\cos \color{blue}{\left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right)} \cdot \left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right)\right)\right) \]

      8. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sin.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{PI}\left(\right), angle\right), 180\right)\right), \left(\cos \left(\color{blue}{\mathsf{PI}\left(\right)} \cdot \frac{angle}{180}\right) \cdot \left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right)\right)\right) \]

      9. PI-lowering-PI.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sin.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{PI.f64}\left(\right), angle\right), 180\right)\right), \left(\cos \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right) \cdot \left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right)\right)\right) \]

      10. *-commutative N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{sin.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{PI.f64}\left(\right), angle\right), 180\right)\right), \left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \color{blue}{\cos \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right)}\right)\right) \]

      11. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{sin.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{PI.f64}\left(\right), angle\right), 180\right)\right), \mathsf{*.f64}\left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right), \color{blue}{\cos \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right)}\right)\right) \]

   3. Simplified 60.8%

      \[\leadsto \color{blue}{\sin \left(\frac{\pi \cdot angle}{180}\right) \cdot \left(\left(2 \cdot \left(b \cdot b - a \cdot a\right)\right) \cdot \cos \left(\frac{\pi \cdot angle}{180}\right)\right)} \]
   4. Add Preprocessing

   5. Taylor expanded in angle around 0

      \[\leadsto \color{blue}{\frac{1}{90} \cdot \left(angle \cdot \left(\mathsf{PI}\left(\right) \cdot \left({b}^{2} - {a}^{2}\right)\right)\right)} \]
   6. Step-by-step derivation

      1. associate-*r* N/A

         \[\leadsto \frac{1}{90} \cdot \left(\left(angle \cdot \mathsf{PI}\left(\right)\right) \cdot \color{blue}{\left({b}^{2} - {a}^{2}\right)}\right) \]

      2. associate-*r* N/A

         \[\leadsto \left(\frac{1}{90} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right) \cdot \color{blue}{\left({b}^{2} - {a}^{2}\right)} \]

      3. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\left(\frac{1}{90} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right), \color{blue}{\left({b}^{2} - {a}^{2}\right)}\right) \]

      4. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{1}{90}, \left(angle \cdot \mathsf{PI}\left(\right)\right)\right), \left(\color{blue}{{b}^{2}} - {a}^{2}\right)\right) \]

      5. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{1}{90}, \mathsf{*.f64}\left(angle, \mathsf{PI}\left(\right)\right)\right), \left({b}^{\color{blue}{2}} - {a}^{2}\right)\right) \]

      6. PI-lowering-PI.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{1}{90}, \mathsf{*.f64}\left(angle, \mathsf{PI.f64}\left(\right)\right)\right), \left({b}^{2} - {a}^{2}\right)\right) \]

      7. --lowering--.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{1}{90}, \mathsf{*.f64}\left(angle, \mathsf{PI.f64}\left(\right)\right)\right), \mathsf{\_.f64}\left(\left({b}^{2}\right), \color{blue}{\left({a}^{2}\right)}\right)\right) \]

      8. unpow2 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{1}{90}, \mathsf{*.f64}\left(angle, \mathsf{PI.f64}\left(\right)\right)\right), \mathsf{\_.f64}\left(\left(b \cdot b\right), \left({\color{blue}{a}}^{2}\right)\right)\right) \]

      9. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{1}{90}, \mathsf{*.f64}\left(angle, \mathsf{PI.f64}\left(\right)\right)\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(b, b\right), \left({\color{blue}{a}}^{2}\right)\right)\right) \]

      10. unpow2 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{1}{90}, \mathsf{*.f64}\left(angle, \mathsf{PI.f64}\left(\right)\right)\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(b, b\right), \left(a \cdot \color{blue}{a}\right)\right)\right) \]

      11. *-lowering-*.f64 57.9%

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{1}{90}, \mathsf{*.f64}\left(angle, \mathsf{PI.f64}\left(\right)\right)\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(b, b\right), \mathsf{*.f64}\left(a, \color{blue}{a}\right)\right)\right) \]

   7. Simplified 57.9%

      \[\leadsto \color{blue}{\left(0.011111111111111112 \cdot \left(angle \cdot \pi\right)\right) \cdot \left(b \cdot b - a \cdot a\right)} \]

   8. Taylor expanded in b around 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 40.1%

         \[\leadsto \mathsf{*.f64}\left(\frac{1}{90}, \mathsf{*.f64}\left(angle, \mathsf{*.f64}\left(\mathsf{PI.f64}\left(\right), \mathsf{*.f64}\left(b, \color{blue}{b}\right)\right)\right)\right) \]

   10. Simplified 40.1%

       \[\leadsto \color{blue}{0.011111111111111112 \cdot \left(angle \cdot \left(\pi \cdot \left(b \cdot b\right)\right)\right)} \]

   if 1.4599999999999999e-18 < a

   1. Initial program 68.8%

      \[\left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin \left(\pi \cdot \frac{angle}{180}\right)\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right) \]
   2. Step-by-step derivation

      1. associate-*l* N/A

         \[\leadsto \left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \color{blue}{\left(\sin \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right) \cdot \cos \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right)\right)} \]

      2. *-commutative N/A

         \[\leadsto \left(\sin \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right) \cdot \cos \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right)\right) \cdot \color{blue}{\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right)} \]

      3. associate-*l* N/A

         \[\leadsto \sin \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right) \cdot \color{blue}{\left(\cos \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right) \cdot \left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right)\right)} \]

      4. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\sin \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right), \color{blue}{\left(\cos \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right) \cdot \left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right)\right)}\right) \]

      5. sin-lowering-sin.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sin.f64}\left(\left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right)\right), \left(\color{blue}{\cos \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right)} \cdot \left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right)\right)\right) \]

      6. associate-*r/ N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sin.f64}\left(\left(\frac{\mathsf{PI}\left(\right) \cdot angle}{180}\right)\right), \left(\cos \color{blue}{\left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right)} \cdot \left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right)\right)\right) \]

      7. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sin.f64}\left(\mathsf{/.f64}\left(\left(\mathsf{PI}\left(\right) \cdot angle\right), 180\right)\right), \left(\cos \color{blue}{\left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right)} \cdot \left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right)\right)\right) \]

      8. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sin.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{PI}\left(\right), angle\right), 180\right)\right), \left(\cos \left(\color{blue}{\mathsf{PI}\left(\right)} \cdot \frac{angle}{180}\right) \cdot \left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right)\right)\right) \]

      9. PI-lowering-PI.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sin.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{PI.f64}\left(\right), angle\right), 180\right)\right), \left(\cos \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right) \cdot \left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right)\right)\right) \]

      10. *-commutative N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{sin.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{PI.f64}\left(\right), angle\right), 180\right)\right), \left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \color{blue}{\cos \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right)}\right)\right) \]

      11. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{sin.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{PI.f64}\left(\right), angle\right), 180\right)\right), \mathsf{*.f64}\left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right), \color{blue}{\cos \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right)}\right)\right) \]

   3. Simplified 70.9%

      \[\leadsto \color{blue}{\sin \left(\frac{\pi \cdot angle}{180}\right) \cdot \left(\left(2 \cdot \left(b \cdot b - a \cdot a\right)\right) \cdot \cos \left(\frac{\pi \cdot angle}{180}\right)\right)} \]
   4. Add Preprocessing

   5. Taylor expanded in angle around 0

      \[\leadsto \color{blue}{\frac{1}{90} \cdot \left(angle \cdot \left(\mathsf{PI}\left(\right) \cdot \left({b}^{2} - {a}^{2}\right)\right)\right)} \]
   6. Step-by-step derivation

      1. associate-*r* N/A

         \[\leadsto \frac{1}{90} \cdot \left(\left(angle \cdot \mathsf{PI}\left(\right)\right) \cdot \color{blue}{\left({b}^{2} - {a}^{2}\right)}\right) \]

      2. associate-*r* N/A

         \[\leadsto \left(\frac{1}{90} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right) \cdot \color{blue}{\left({b}^{2} - {a}^{2}\right)} \]

      3. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\left(\frac{1}{90} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right), \color{blue}{\left({b}^{2} - {a}^{2}\right)}\right) \]

      4. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{1}{90}, \left(angle \cdot \mathsf{PI}\left(\right)\right)\right), \left(\color{blue}{{b}^{2}} - {a}^{2}\right)\right) \]

      5. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{1}{90}, \mathsf{*.f64}\left(angle, \mathsf{PI}\left(\right)\right)\right), \left({b}^{\color{blue}{2}} - {a}^{2}\right)\right) \]

      6. PI-lowering-PI.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{1}{90}, \mathsf{*.f64}\left(angle, \mathsf{PI.f64}\left(\right)\right)\right), \left({b}^{2} - {a}^{2}\right)\right) \]

      7. --lowering--.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{1}{90}, \mathsf{*.f64}\left(angle, \mathsf{PI.f64}\left(\right)\right)\right), \mathsf{\_.f64}\left(\left({b}^{2}\right), \color{blue}{\left({a}^{2}\right)}\right)\right) \]

      8. unpow2 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{1}{90}, \mathsf{*.f64}\left(angle, \mathsf{PI.f64}\left(\right)\right)\right), \mathsf{\_.f64}\left(\left(b \cdot b\right), \left({\color{blue}{a}}^{2}\right)\right)\right) \]

      9. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{1}{90}, \mathsf{*.f64}\left(angle, \mathsf{PI.f64}\left(\right)\right)\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(b, b\right), \left({\color{blue}{a}}^{2}\right)\right)\right) \]

      10. unpow2 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{1}{90}, \mathsf{*.f64}\left(angle, \mathsf{PI.f64}\left(\right)\right)\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(b, b\right), \left(a \cdot \color{blue}{a}\right)\right)\right) \]

      11. *-lowering-*.f64 61.9%

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{1}{90}, \mathsf{*.f64}\left(angle, \mathsf{PI.f64}\left(\right)\right)\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(b, b\right), \mathsf{*.f64}\left(a, \color{blue}{a}\right)\right)\right) \]

   7. Simplified 61.9%

      \[\leadsto \color{blue}{\left(0.011111111111111112 \cdot \left(angle \cdot \pi\right)\right) \cdot \left(b \cdot b - a \cdot a\right)} \]

   8. Taylor expanded in b around 0

      \[\leadsto \color{blue}{\frac{-1}{90} \cdot \left({a}^{2} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)} \]
   9. Step-by-step derivation

      1. *-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 60.9%

         \[\leadsto \mathsf{*.f64}\left(\frac{-1}{90}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(a, a\right), angle\right), \mathsf{PI.f64}\left(\right)\right)\right) \]

   10. Simplified 60.9%

       \[\leadsto \color{blue}{-0.011111111111111112 \cdot \left(\left(\left(a \cdot a\right) \cdot angle\right) \cdot \pi\right)} \]
   11. Step-by-step derivation

       1. *-commutative N/A

          \[\leadsto \mathsf{*.f64}\left(\frac{-1}{90}, \left(\mathsf{PI}\left(\right) \cdot \color{blue}{\left(\left(a \cdot a\right) \cdot angle\right)}\right)\right) \]

       2. associate-*l* N/A

          \[\leadsto \mathsf{*.f64}\left(\frac{-1}{90}, \left(\mathsf{PI}\left(\right) \cdot \left(a \cdot \color{blue}{\left(a \cdot angle\right)}\right)\right)\right) \]

       3. associate-*r* N/A

          \[\leadsto \mathsf{*.f64}\left(\frac{-1}{90}, \left(\left(\mathsf{PI}\left(\right) \cdot a\right) \cdot \color{blue}{\left(a \cdot angle\right)}\right)\right) \]

       4. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\frac{-1}{90}, \mathsf{*.f64}\left(\left(\mathsf{PI}\left(\right) \cdot a\right), \color{blue}{\left(a \cdot angle\right)}\right)\right) \]

       5. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\frac{-1}{90}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{PI}\left(\right), a\right), \left(\color{blue}{a} \cdot angle\right)\right)\right) \]

       6. PI-lowering-PI.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\frac{-1}{90}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{PI.f64}\left(\right), a\right), \left(a \cdot angle\right)\right)\right) \]

       7. *-lowering-*.f64 64.1%

          \[\leadsto \mathsf{*.f64}\left(\frac{-1}{90}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{PI.f64}\left(\right), a\right), \mathsf{*.f64}\left(a, \color{blue}{angle}\right)\right)\right) \]

   12. Applied egg-rr 64.1%

       \[\leadsto -0.011111111111111112 \cdot \color{blue}{\left(\left(\pi \cdot a\right) \cdot \left(a \cdot angle\right)\right)} \]
   13. Step-by-step derivation

       1. associate-*r* N/A

          \[\leadsto \left(\frac{-1}{90} \cdot \left(\mathsf{PI}\left(\right) \cdot a\right)\right) \cdot \color{blue}{\left(a \cdot angle\right)} \]

       2. *-commutative N/A

          \[\leadsto \left(\frac{-1}{90} \cdot \left(\mathsf{PI}\left(\right) \cdot a\right)\right) \cdot \left(angle \cdot \color{blue}{a}\right) \]

       3. associate-*r* N/A

          \[\leadsto \left(\left(\frac{-1}{90} \cdot \left(\mathsf{PI}\left(\right) \cdot a\right)\right) \cdot angle\right) \cdot \color{blue}{a} \]

       4. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\left(\left(\frac{-1}{90} \cdot \left(\mathsf{PI}\left(\right) \cdot a\right)\right) \cdot angle\right), \color{blue}{a}\right) \]

       5. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\left(\frac{-1}{90} \cdot \left(\mathsf{PI}\left(\right) \cdot a\right)\right), angle\right), a\right) \]

       6. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{-1}{90}, \left(\mathsf{PI}\left(\right) \cdot a\right)\right), angle\right), a\right) \]

       7. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{-1}{90}, \mathsf{*.f64}\left(\mathsf{PI}\left(\right), a\right)\right), angle\right), a\right) \]

       8. PI-lowering-PI.f64 64.1%

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{-1}{90}, \mathsf{*.f64}\left(\mathsf{PI.f64}\left(\right), a\right)\right), angle\right), a\right) \]

   14. Applied egg-rr 64.1%

       \[\leadsto \color{blue}{\left(\left(-0.011111111111111112 \cdot \left(\pi \cdot a\right)\right) \cdot angle\right) \cdot a} \]
3. Recombined 2 regimes into one program.

4. Final simplification 45.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \leq 1.46 \cdot 10^{-18}:\\ \;\;\;\;0.011111111111111112 \cdot \left(angle \cdot \left(\pi \cdot \left(b \cdot b\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;a \cdot \left(angle \cdot \left(-0.011111111111111112 \cdot \left(a \cdot \pi\right)\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 17: 51.2% accurate, 29.9× speedup

Math

\[\begin{array}{l} a_m = \left|a\right| \\ b_m = \left|b\right| \\ \begin{array}{l} \mathbf{if}\;a\_m \leq 6.8 \cdot 10^{-18}:\\ \;\;\;\;0.011111111111111112 \cdot \left(angle \cdot \left(\pi \cdot \left(b\_m \cdot b\_m\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;-0.011111111111111112 \cdot \left(\left(a\_m \cdot \pi\right) \cdot \left(a\_m \cdot angle\right)\right)\\ \end{array} \end{array} \]

FPCore

a_m = (fabs.f64 a)
b_m = (fabs.f64 b)
(FPCore (a_m b_m angle)
 :precision binary64
 (if (<= a_m 6.8e-18)
   (* 0.011111111111111112 (* angle (* PI (* b_m b_m))))
   (* -0.011111111111111112 (* (* a_m PI) (* a_m angle)))))

C

a_m = fabs(a);
b_m = fabs(b);
double code(double a_m, double b_m, double angle) {
	double tmp;
	if (a_m <= 6.8e-18) {
		tmp = 0.011111111111111112 * (angle * (((double) M_PI) * (b_m * b_m)));
	} else {
		tmp = -0.011111111111111112 * ((a_m * ((double) M_PI)) * (a_m * angle));
	}
	return tmp;
}

Java

a_m = Math.abs(a);
b_m = Math.abs(b);
public static double code(double a_m, double b_m, double angle) {
	double tmp;
	if (a_m <= 6.8e-18) {
		tmp = 0.011111111111111112 * (angle * (Math.PI * (b_m * b_m)));
	} else {
		tmp = -0.011111111111111112 * ((a_m * Math.PI) * (a_m * angle));
	}
	return tmp;
}

Python

a_m = math.fabs(a)
b_m = math.fabs(b)
def code(a_m, b_m, angle):
	tmp = 0
	if a_m <= 6.8e-18:
		tmp = 0.011111111111111112 * (angle * (math.pi * (b_m * b_m)))
	else:
		tmp = -0.011111111111111112 * ((a_m * math.pi) * (a_m * angle))
	return tmp

Julia

a_m = abs(a)
b_m = abs(b)
function code(a_m, b_m, angle)
	tmp = 0.0
	if (a_m <= 6.8e-18)
		tmp = Float64(0.011111111111111112 * Float64(angle * Float64(pi * Float64(b_m * b_m))));
	else
		tmp = Float64(-0.011111111111111112 * Float64(Float64(a_m * pi) * Float64(a_m * angle)));
	end
	return tmp
end

MATLAB

a_m = abs(a);
b_m = abs(b);
function tmp_2 = code(a_m, b_m, angle)
	tmp = 0.0;
	if (a_m <= 6.8e-18)
		tmp = 0.011111111111111112 * (angle * (pi * (b_m * b_m)));
	else
		tmp = -0.011111111111111112 * ((a_m * pi) * (a_m * angle));
	end
	tmp_2 = tmp;
end

Wolfram

a_m = N[Abs[a], $MachinePrecision]
b_m = N[Abs[b], $MachinePrecision]
code[a$95$m_, b$95$m_, angle_] := If[LessEqual[a$95$m, 6.8e-18], N[(0.011111111111111112 * N[(angle * N[(Pi * N[(b$95$m * b$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(-0.011111111111111112 * N[(N[(a$95$m * Pi), $MachinePrecision] * N[(a$95$m * angle), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if a < 6.80000000000000002e-18

   1. Initial program 58.6%

      \[\left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin \left(\pi \cdot \frac{angle}{180}\right)\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right) \]
   2. Step-by-step derivation

      1. associate-*l* N/A

         \[\leadsto \left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \color{blue}{\left(\sin \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right) \cdot \cos \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right)\right)} \]

      2. *-commutative N/A

         \[\leadsto \left(\sin \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right) \cdot \cos \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right)\right) \cdot \color{blue}{\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right)} \]

      3. associate-*l* N/A

         \[\leadsto \sin \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right) \cdot \color{blue}{\left(\cos \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right) \cdot \left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right)\right)} \]

      4. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\sin \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right), \color{blue}{\left(\cos \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right) \cdot \left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right)\right)}\right) \]

      5. sin-lowering-sin.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sin.f64}\left(\left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right)\right), \left(\color{blue}{\cos \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right)} \cdot \left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right)\right)\right) \]

      6. associate-*r/ N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sin.f64}\left(\left(\frac{\mathsf{PI}\left(\right) \cdot angle}{180}\right)\right), \left(\cos \color{blue}{\left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right)} \cdot \left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right)\right)\right) \]

      7. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sin.f64}\left(\mathsf{/.f64}\left(\left(\mathsf{PI}\left(\right) \cdot angle\right), 180\right)\right), \left(\cos \color{blue}{\left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right)} \cdot \left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right)\right)\right) \]

      8. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sin.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{PI}\left(\right), angle\right), 180\right)\right), \left(\cos \left(\color{blue}{\mathsf{PI}\left(\right)} \cdot \frac{angle}{180}\right) \cdot \left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right)\right)\right) \]

      9. PI-lowering-PI.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sin.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{PI.f64}\left(\right), angle\right), 180\right)\right), \left(\cos \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right) \cdot \left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right)\right)\right) \]

      10. *-commutative N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{sin.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{PI.f64}\left(\right), angle\right), 180\right)\right), \left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \color{blue}{\cos \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right)}\right)\right) \]

      11. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{sin.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{PI.f64}\left(\right), angle\right), 180\right)\right), \mathsf{*.f64}\left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right), \color{blue}{\cos \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right)}\right)\right) \]

   3. Simplified 60.8%

      \[\leadsto \color{blue}{\sin \left(\frac{\pi \cdot angle}{180}\right) \cdot \left(\left(2 \cdot \left(b \cdot b - a \cdot a\right)\right) \cdot \cos \left(\frac{\pi \cdot angle}{180}\right)\right)} \]
   4. Add Preprocessing

   5. Taylor expanded in angle around 0

      \[\leadsto \color{blue}{\frac{1}{90} \cdot \left(angle \cdot \left(\mathsf{PI}\left(\right) \cdot \left({b}^{2} - {a}^{2}\right)\right)\right)} \]
   6. Step-by-step derivation

      1. associate-*r* N/A

         \[\leadsto \frac{1}{90} \cdot \left(\left(angle \cdot \mathsf{PI}\left(\right)\right) \cdot \color{blue}{\left({b}^{2} - {a}^{2}\right)}\right) \]

      2. associate-*r* N/A

         \[\leadsto \left(\frac{1}{90} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right) \cdot \color{blue}{\left({b}^{2} - {a}^{2}\right)} \]

      3. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\left(\frac{1}{90} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right), \color{blue}{\left({b}^{2} - {a}^{2}\right)}\right) \]

      4. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{1}{90}, \left(angle \cdot \mathsf{PI}\left(\right)\right)\right), \left(\color{blue}{{b}^{2}} - {a}^{2}\right)\right) \]

      5. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{1}{90}, \mathsf{*.f64}\left(angle, \mathsf{PI}\left(\right)\right)\right), \left({b}^{\color{blue}{2}} - {a}^{2}\right)\right) \]

      6. PI-lowering-PI.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{1}{90}, \mathsf{*.f64}\left(angle, \mathsf{PI.f64}\left(\right)\right)\right), \left({b}^{2} - {a}^{2}\right)\right) \]

      7. --lowering--.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{1}{90}, \mathsf{*.f64}\left(angle, \mathsf{PI.f64}\left(\right)\right)\right), \mathsf{\_.f64}\left(\left({b}^{2}\right), \color{blue}{\left({a}^{2}\right)}\right)\right) \]

      8. unpow2 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{1}{90}, \mathsf{*.f64}\left(angle, \mathsf{PI.f64}\left(\right)\right)\right), \mathsf{\_.f64}\left(\left(b \cdot b\right), \left({\color{blue}{a}}^{2}\right)\right)\right) \]

      9. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{1}{90}, \mathsf{*.f64}\left(angle, \mathsf{PI.f64}\left(\right)\right)\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(b, b\right), \left({\color{blue}{a}}^{2}\right)\right)\right) \]

      10. unpow2 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{1}{90}, \mathsf{*.f64}\left(angle, \mathsf{PI.f64}\left(\right)\right)\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(b, b\right), \left(a \cdot \color{blue}{a}\right)\right)\right) \]

      11. *-lowering-*.f64 57.9%

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{1}{90}, \mathsf{*.f64}\left(angle, \mathsf{PI.f64}\left(\right)\right)\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(b, b\right), \mathsf{*.f64}\left(a, \color{blue}{a}\right)\right)\right) \]

   7. Simplified 57.9%

      \[\leadsto \color{blue}{\left(0.011111111111111112 \cdot \left(angle \cdot \pi\right)\right) \cdot \left(b \cdot b - a \cdot a\right)} \]

   8. Taylor expanded in b around 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 40.1%

         \[\leadsto \mathsf{*.f64}\left(\frac{1}{90}, \mathsf{*.f64}\left(angle, \mathsf{*.f64}\left(\mathsf{PI.f64}\left(\right), \mathsf{*.f64}\left(b, \color{blue}{b}\right)\right)\right)\right) \]

   10. Simplified 40.1%

       \[\leadsto \color{blue}{0.011111111111111112 \cdot \left(angle \cdot \left(\pi \cdot \left(b \cdot b\right)\right)\right)} \]

   if 6.80000000000000002e-18 < a

   1. Initial program 68.8%

      \[\left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin \left(\pi \cdot \frac{angle}{180}\right)\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right) \]
   2. Step-by-step derivation

      1. associate-*l* N/A

         \[\leadsto \left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \color{blue}{\left(\sin \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right) \cdot \cos \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right)\right)} \]

      2. *-commutative N/A

         \[\leadsto \left(\sin \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right) \cdot \cos \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right)\right) \cdot \color{blue}{\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right)} \]

      3. associate-*l* N/A

         \[\leadsto \sin \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right) \cdot \color{blue}{\left(\cos \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right) \cdot \left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right)\right)} \]

      4. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\sin \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right), \color{blue}{\left(\cos \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right) \cdot \left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right)\right)}\right) \]

      5. sin-lowering-sin.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sin.f64}\left(\left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right)\right), \left(\color{blue}{\cos \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right)} \cdot \left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right)\right)\right) \]

      6. associate-*r/ N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sin.f64}\left(\left(\frac{\mathsf{PI}\left(\right) \cdot angle}{180}\right)\right), \left(\cos \color{blue}{\left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right)} \cdot \left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right)\right)\right) \]

      7. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sin.f64}\left(\mathsf{/.f64}\left(\left(\mathsf{PI}\left(\right) \cdot angle\right), 180\right)\right), \left(\cos \color{blue}{\left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right)} \cdot \left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right)\right)\right) \]

      8. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sin.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{PI}\left(\right), angle\right), 180\right)\right), \left(\cos \left(\color{blue}{\mathsf{PI}\left(\right)} \cdot \frac{angle}{180}\right) \cdot \left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right)\right)\right) \]

      9. PI-lowering-PI.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sin.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{PI.f64}\left(\right), angle\right), 180\right)\right), \left(\cos \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right) \cdot \left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right)\right)\right) \]

      10. *-commutative N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{sin.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{PI.f64}\left(\right), angle\right), 180\right)\right), \left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \color{blue}{\cos \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right)}\right)\right) \]

      11. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{sin.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{PI.f64}\left(\right), angle\right), 180\right)\right), \mathsf{*.f64}\left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right), \color{blue}{\cos \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right)}\right)\right) \]

   3. Simplified 70.9%

      \[\leadsto \color{blue}{\sin \left(\frac{\pi \cdot angle}{180}\right) \cdot \left(\left(2 \cdot \left(b \cdot b - a \cdot a\right)\right) \cdot \cos \left(\frac{\pi \cdot angle}{180}\right)\right)} \]
   4. Add Preprocessing

   5. Taylor expanded in angle around 0

      \[\leadsto \color{blue}{\frac{1}{90} \cdot \left(angle \cdot \left(\mathsf{PI}\left(\right) \cdot \left({b}^{2} - {a}^{2}\right)\right)\right)} \]
   6. Step-by-step derivation

      1. associate-*r* N/A

         \[\leadsto \frac{1}{90} \cdot \left(\left(angle \cdot \mathsf{PI}\left(\right)\right) \cdot \color{blue}{\left({b}^{2} - {a}^{2}\right)}\right) \]

      2. associate-*r* N/A

         \[\leadsto \left(\frac{1}{90} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right) \cdot \color{blue}{\left({b}^{2} - {a}^{2}\right)} \]

      3. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\left(\frac{1}{90} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right), \color{blue}{\left({b}^{2} - {a}^{2}\right)}\right) \]

      4. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{1}{90}, \left(angle \cdot \mathsf{PI}\left(\right)\right)\right), \left(\color{blue}{{b}^{2}} - {a}^{2}\right)\right) \]

      5. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{1}{90}, \mathsf{*.f64}\left(angle, \mathsf{PI}\left(\right)\right)\right), \left({b}^{\color{blue}{2}} - {a}^{2}\right)\right) \]

      6. PI-lowering-PI.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{1}{90}, \mathsf{*.f64}\left(angle, \mathsf{PI.f64}\left(\right)\right)\right), \left({b}^{2} - {a}^{2}\right)\right) \]

      7. --lowering--.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{1}{90}, \mathsf{*.f64}\left(angle, \mathsf{PI.f64}\left(\right)\right)\right), \mathsf{\_.f64}\left(\left({b}^{2}\right), \color{blue}{\left({a}^{2}\right)}\right)\right) \]

      8. unpow2 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{1}{90}, \mathsf{*.f64}\left(angle, \mathsf{PI.f64}\left(\right)\right)\right), \mathsf{\_.f64}\left(\left(b \cdot b\right), \left({\color{blue}{a}}^{2}\right)\right)\right) \]

      9. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{1}{90}, \mathsf{*.f64}\left(angle, \mathsf{PI.f64}\left(\right)\right)\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(b, b\right), \left({\color{blue}{a}}^{2}\right)\right)\right) \]

      10. unpow2 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{1}{90}, \mathsf{*.f64}\left(angle, \mathsf{PI.f64}\left(\right)\right)\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(b, b\right), \left(a \cdot \color{blue}{a}\right)\right)\right) \]

      11. *-lowering-*.f64 61.9%

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{1}{90}, \mathsf{*.f64}\left(angle, \mathsf{PI.f64}\left(\right)\right)\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(b, b\right), \mathsf{*.f64}\left(a, \color{blue}{a}\right)\right)\right) \]

   7. Simplified 61.9%

      \[\leadsto \color{blue}{\left(0.011111111111111112 \cdot \left(angle \cdot \pi\right)\right) \cdot \left(b \cdot b - a \cdot a\right)} \]

   8. Taylor expanded in b around 0

      \[\leadsto \color{blue}{\frac{-1}{90} \cdot \left({a}^{2} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)} \]
   9. Step-by-step derivation

      1. *-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 60.9%

         \[\leadsto \mathsf{*.f64}\left(\frac{-1}{90}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(a, a\right), angle\right), \mathsf{PI.f64}\left(\right)\right)\right) \]

   10. Simplified 60.9%

       \[\leadsto \color{blue}{-0.011111111111111112 \cdot \left(\left(\left(a \cdot a\right) \cdot angle\right) \cdot \pi\right)} \]
   11. Step-by-step derivation

       1. *-commutative N/A

          \[\leadsto \mathsf{*.f64}\left(\frac{-1}{90}, \left(\mathsf{PI}\left(\right) \cdot \color{blue}{\left(\left(a \cdot a\right) \cdot angle\right)}\right)\right) \]

       2. associate-*l* N/A

          \[\leadsto \mathsf{*.f64}\left(\frac{-1}{90}, \left(\mathsf{PI}\left(\right) \cdot \left(a \cdot \color{blue}{\left(a \cdot angle\right)}\right)\right)\right) \]

       3. associate-*r* N/A

          \[\leadsto \mathsf{*.f64}\left(\frac{-1}{90}, \left(\left(\mathsf{PI}\left(\right) \cdot a\right) \cdot \color{blue}{\left(a \cdot angle\right)}\right)\right) \]

       4. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\frac{-1}{90}, \mathsf{*.f64}\left(\left(\mathsf{PI}\left(\right) \cdot a\right), \color{blue}{\left(a \cdot angle\right)}\right)\right) \]

       5. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\frac{-1}{90}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{PI}\left(\right), a\right), \left(\color{blue}{a} \cdot angle\right)\right)\right) \]

       6. PI-lowering-PI.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\frac{-1}{90}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{PI.f64}\left(\right), a\right), \left(a \cdot angle\right)\right)\right) \]

       7. *-lowering-*.f64 64.1%

          \[\leadsto \mathsf{*.f64}\left(\frac{-1}{90}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{PI.f64}\left(\right), a\right), \mathsf{*.f64}\left(a, \color{blue}{angle}\right)\right)\right) \]

   12. Applied egg-rr 64.1%

       \[\leadsto -0.011111111111111112 \cdot \color{blue}{\left(\left(\pi \cdot a\right) \cdot \left(a \cdot angle\right)\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 45.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \leq 6.8 \cdot 10^{-18}:\\ \;\;\;\;0.011111111111111112 \cdot \left(angle \cdot \left(\pi \cdot \left(b \cdot b\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;-0.011111111111111112 \cdot \left(\left(a \cdot \pi\right) \cdot \left(a \cdot angle\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 18: 38.2% accurate, 46.6× speedup

Math

\[\begin{array}{l} a_m = \left|a\right| \\ b_m = \left|b\right| \\ -0.011111111111111112 \cdot \left(\left(a\_m \cdot \pi\right) \cdot \left(a\_m \cdot angle\right)\right) \end{array} \]

FPCore

a_m = (fabs.f64 a)
b_m = (fabs.f64 b)
(FPCore (a_m b_m angle)
 :precision binary64
 (* -0.011111111111111112 (* (* a_m PI) (* a_m angle))))

C

a_m = fabs(a);
b_m = fabs(b);
double code(double a_m, double b_m, double angle) {
	return -0.011111111111111112 * ((a_m * ((double) M_PI)) * (a_m * angle));
}

Java

a_m = Math.abs(a);
b_m = Math.abs(b);
public static double code(double a_m, double b_m, double angle) {
	return -0.011111111111111112 * ((a_m * Math.PI) * (a_m * angle));
}

Python

a_m = math.fabs(a)
b_m = math.fabs(b)
def code(a_m, b_m, angle):
	return -0.011111111111111112 * ((a_m * math.pi) * (a_m * angle))

Julia

a_m = abs(a)
b_m = abs(b)
function code(a_m, b_m, angle)
	return Float64(-0.011111111111111112 * Float64(Float64(a_m * pi) * Float64(a_m * angle)))
end

MATLAB

a_m = abs(a);
b_m = abs(b);
function tmp = code(a_m, b_m, angle)
	tmp = -0.011111111111111112 * ((a_m * pi) * (a_m * angle));
end

Wolfram

a_m = N[Abs[a], $MachinePrecision]
b_m = N[Abs[b], $MachinePrecision]
code[a$95$m_, b$95$m_, angle_] := N[(-0.011111111111111112 * N[(N[(a$95$m * Pi), $MachinePrecision] * N[(a$95$m * angle), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 61.0%

   \[\left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin \left(\pi \cdot \frac{angle}{180}\right)\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right) \]
2. Step-by-step derivation

   1. associate-*l* N/A

      \[\leadsto \left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \color{blue}{\left(\sin \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right) \cdot \cos \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right)\right)} \]

   2. *-commutative N/A

      \[\leadsto \left(\sin \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right) \cdot \cos \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right)\right) \cdot \color{blue}{\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right)} \]

   3. associate-*l* N/A

      \[\leadsto \sin \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right) \cdot \color{blue}{\left(\cos \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right) \cdot \left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right)\right)} \]

   4. *-lowering-*.f64 N/A

      \[\leadsto \mathsf{*.f64}\left(\sin \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right), \color{blue}{\left(\cos \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right) \cdot \left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right)\right)}\right) \]

   5. sin-lowering-sin.f64 N/A

      \[\leadsto \mathsf{*.f64}\left(\mathsf{sin.f64}\left(\left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right)\right), \left(\color{blue}{\cos \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right)} \cdot \left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right)\right)\right) \]

   6. associate-*r/ N/A

      \[\leadsto \mathsf{*.f64}\left(\mathsf{sin.f64}\left(\left(\frac{\mathsf{PI}\left(\right) \cdot angle}{180}\right)\right), \left(\cos \color{blue}{\left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right)} \cdot \left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right)\right)\right) \]

   7. /-lowering-/.f64 N/A

      \[\leadsto \mathsf{*.f64}\left(\mathsf{sin.f64}\left(\mathsf{/.f64}\left(\left(\mathsf{PI}\left(\right) \cdot angle\right), 180\right)\right), \left(\cos \color{blue}{\left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right)} \cdot \left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right)\right)\right) \]

   8. *-lowering-*.f64 N/A

      \[\leadsto \mathsf{*.f64}\left(\mathsf{sin.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{PI}\left(\right), angle\right), 180\right)\right), \left(\cos \left(\color{blue}{\mathsf{PI}\left(\right)} \cdot \frac{angle}{180}\right) \cdot \left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right)\right)\right) \]

   9. PI-lowering-PI.f64 N/A

      \[\leadsto \mathsf{*.f64}\left(\mathsf{sin.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{PI.f64}\left(\right), angle\right), 180\right)\right), \left(\cos \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right) \cdot \left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right)\right)\right) \]

   10. *-commutative N/A

       \[\leadsto \mathsf{*.f64}\left(\mathsf{sin.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{PI.f64}\left(\right), angle\right), 180\right)\right), \left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \color{blue}{\cos \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right)}\right)\right) \]

   11. *-lowering-*.f64 N/A

       \[\leadsto \mathsf{*.f64}\left(\mathsf{sin.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{PI.f64}\left(\right), angle\right), 180\right)\right), \mathsf{*.f64}\left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right), \color{blue}{\cos \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right)}\right)\right) \]

3. Simplified 63.1%

   \[\leadsto \color{blue}{\sin \left(\frac{\pi \cdot angle}{180}\right) \cdot \left(\left(2 \cdot \left(b \cdot b - a \cdot a\right)\right) \cdot \cos \left(\frac{\pi \cdot angle}{180}\right)\right)} \]
4. Add Preprocessing

5. Taylor expanded in angle around 0

   \[\leadsto \color{blue}{\frac{1}{90} \cdot \left(angle \cdot \left(\mathsf{PI}\left(\right) \cdot \left({b}^{2} - {a}^{2}\right)\right)\right)} \]
6. Step-by-step derivation

   1. associate-*r* N/A

      \[\leadsto \frac{1}{90} \cdot \left(\left(angle \cdot \mathsf{PI}\left(\right)\right) \cdot \color{blue}{\left({b}^{2} - {a}^{2}\right)}\right) \]

   2. associate-*r* N/A

      \[\leadsto \left(\frac{1}{90} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right) \cdot \color{blue}{\left({b}^{2} - {a}^{2}\right)} \]

   3. *-lowering-*.f64 N/A

      \[\leadsto \mathsf{*.f64}\left(\left(\frac{1}{90} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right), \color{blue}{\left({b}^{2} - {a}^{2}\right)}\right) \]

   4. *-lowering-*.f64 N/A

      \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{1}{90}, \left(angle \cdot \mathsf{PI}\left(\right)\right)\right), \left(\color{blue}{{b}^{2}} - {a}^{2}\right)\right) \]

   5. *-lowering-*.f64 N/A

      \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{1}{90}, \mathsf{*.f64}\left(angle, \mathsf{PI}\left(\right)\right)\right), \left({b}^{\color{blue}{2}} - {a}^{2}\right)\right) \]

   6. PI-lowering-PI.f64 N/A

      \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{1}{90}, \mathsf{*.f64}\left(angle, \mathsf{PI.f64}\left(\right)\right)\right), \left({b}^{2} - {a}^{2}\right)\right) \]

   7. --lowering--.f64 N/A

      \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{1}{90}, \mathsf{*.f64}\left(angle, \mathsf{PI.f64}\left(\right)\right)\right), \mathsf{\_.f64}\left(\left({b}^{2}\right), \color{blue}{\left({a}^{2}\right)}\right)\right) \]

   8. unpow2 N/A

      \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{1}{90}, \mathsf{*.f64}\left(angle, \mathsf{PI.f64}\left(\right)\right)\right), \mathsf{\_.f64}\left(\left(b \cdot b\right), \left({\color{blue}{a}}^{2}\right)\right)\right) \]

   9. *-lowering-*.f64 N/A

      \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{1}{90}, \mathsf{*.f64}\left(angle, \mathsf{PI.f64}\left(\right)\right)\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(b, b\right), \left({\color{blue}{a}}^{2}\right)\right)\right) \]

   10. unpow2 N/A

       \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{1}{90}, \mathsf{*.f64}\left(angle, \mathsf{PI.f64}\left(\right)\right)\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(b, b\right), \left(a \cdot \color{blue}{a}\right)\right)\right) \]

   11. *-lowering-*.f64 58.8%

       \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{1}{90}, \mathsf{*.f64}\left(angle, \mathsf{PI.f64}\left(\right)\right)\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(b, b\right), \mathsf{*.f64}\left(a, \color{blue}{a}\right)\right)\right) \]

7. Simplified 58.8%

   \[\leadsto \color{blue}{\left(0.011111111111111112 \cdot \left(angle \cdot \pi\right)\right) \cdot \left(b \cdot b - a \cdot a\right)} \]

8. Taylor expanded in b around 0

   \[\leadsto \color{blue}{\frac{-1}{90} \cdot \left({a}^{2} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)} \]
9. Step-by-step derivation

   1. *-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 41.7%

      \[\leadsto \mathsf{*.f64}\left(\frac{-1}{90}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(a, a\right), angle\right), \mathsf{PI.f64}\left(\right)\right)\right) \]

10. Simplified 41.7%

    \[\leadsto \color{blue}{-0.011111111111111112 \cdot \left(\left(\left(a \cdot a\right) \cdot angle\right) \cdot \pi\right)} \]
11. Step-by-step derivation

    1. *-commutative N/A

       \[\leadsto \mathsf{*.f64}\left(\frac{-1}{90}, \left(\mathsf{PI}\left(\right) \cdot \color{blue}{\left(\left(a \cdot a\right) \cdot angle\right)}\right)\right) \]

    2. associate-*l* N/A

       \[\leadsto \mathsf{*.f64}\left(\frac{-1}{90}, \left(\mathsf{PI}\left(\right) \cdot \left(a \cdot \color{blue}{\left(a \cdot angle\right)}\right)\right)\right) \]

    3. associate-*r* N/A

       \[\leadsto \mathsf{*.f64}\left(\frac{-1}{90}, \left(\left(\mathsf{PI}\left(\right) \cdot a\right) \cdot \color{blue}{\left(a \cdot angle\right)}\right)\right) \]

    4. *-lowering-*.f64 N/A

       \[\leadsto \mathsf{*.f64}\left(\frac{-1}{90}, \mathsf{*.f64}\left(\left(\mathsf{PI}\left(\right) \cdot a\right), \color{blue}{\left(a \cdot angle\right)}\right)\right) \]

    5. *-lowering-*.f64 N/A

       \[\leadsto \mathsf{*.f64}\left(\frac{-1}{90}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{PI}\left(\right), a\right), \left(\color{blue}{a} \cdot angle\right)\right)\right) \]

    6. PI-lowering-PI.f64 N/A

       \[\leadsto \mathsf{*.f64}\left(\frac{-1}{90}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{PI.f64}\left(\right), a\right), \left(a \cdot angle\right)\right)\right) \]

    7. *-lowering-*.f64 44.2%

       \[\leadsto \mathsf{*.f64}\left(\frac{-1}{90}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{PI.f64}\left(\right), a\right), \mathsf{*.f64}\left(a, \color{blue}{angle}\right)\right)\right) \]

12. Applied egg-rr 44.2%

    \[\leadsto -0.011111111111111112 \cdot \color{blue}{\left(\left(\pi \cdot a\right) \cdot \left(a \cdot angle\right)\right)} \]

13. Final simplification 44.2%

    \[\leadsto -0.011111111111111112 \cdot \left(\left(a \cdot \pi\right) \cdot \left(a \cdot angle\right)\right) \]
14. Add Preprocessing

Reproduce

herbie shell --seed 2024138 
(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)))))