ab-angle->ABCF B

Percentage Accurate: 54.7% → 67.6%

Time: 23.0s

Alternatives: 21

Speedup: 23.3×

• 32.6% of points produce a very large (infinite) output. You may want to add a precondition.

Specification

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 54.7% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 67.6% accurate, 1.0× speedup

Math

\[\begin{array}{l} a_m = \left|a\right| \\ angle\_m = \left|angle\right| \\ angle\_s = \mathsf{copysign}\left(1, angle\right) \\ \begin{array}{l} t_0 := \frac{\pi}{\frac{180}{angle\_m}}\\ t_1 := 2 \cdot \sin \left(\frac{\left(angle\_m \cdot \left(\pi + 1\right)\right) \cdot \frac{180}{angle\_m} - 180}{\frac{32400}{angle\_m}}\right)\\ angle\_s \cdot \begin{array}{l} \mathbf{if}\;a\_m \leq 1.85 \cdot 10^{-126}:\\ \;\;\;\;\left(b + a\_m\right) \cdot \left(t\_1 \cdot \left(\left(b - a\_m\right) \cdot \cos \left(\sqrt{\pi} \cdot \left(\frac{angle\_m}{180} \cdot \sqrt{\pi}\right)\right)\right)\right)\\ \mathbf{elif}\;a\_m \leq 2 \cdot 10^{+157}:\\ \;\;\;\;\left(b + a\_m\right) \cdot \left(\left(\left(b - a\_m\right) \cdot \cos t\_0\right) \cdot \left(2 \cdot \sin \left(\frac{180 \cdot \left(\frac{angle\_m}{180} \cdot \left(-1 + \pi \cdot \pi\right)\right) - angle\_m \cdot \left(-1 + \pi\right)}{180 \cdot \left(-1 + \pi\right)}\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(b + a\_m\right) \cdot \left(t\_1 \cdot \left(\left(b - a\_m\right) \cdot \cos \left(e^{\log t\_0}\right)\right)\right)\\ \end{array} \end{array} \end{array} \]

FPCore

a_m = (fabs.f64 a)
angle\_m = (fabs.f64 angle)
angle\_s = (copysign.f64 #s(literal 1 binary64) angle)
(FPCore (angle_s a_m b angle_m)
 :precision binary64
 (let* ((t_0 (/ PI (/ 180.0 angle_m)))
        (t_1
         (*
          2.0
          (sin
           (/
            (- (* (* angle_m (+ PI 1.0)) (/ 180.0 angle_m)) 180.0)
            (/ 32400.0 angle_m))))))
   (*
    angle_s
    (if (<= a_m 1.85e-126)
      (*
       (+ b a_m)
       (*
        t_1
        (* (- b a_m) (cos (* (sqrt PI) (* (/ angle_m 180.0) (sqrt PI)))))))
      (if (<= a_m 2e+157)
        (*
         (+ b a_m)
         (*
          (* (- b a_m) (cos t_0))
          (*
           2.0
           (sin
            (/
             (-
              (* 180.0 (* (/ angle_m 180.0) (+ -1.0 (* PI PI))))
              (* angle_m (+ -1.0 PI)))
             (* 180.0 (+ -1.0 PI)))))))
        (* (+ b a_m) (* t_1 (* (- b a_m) (cos (exp (log t_0)))))))))))

C

a_m = fabs(a);
angle\_m = fabs(angle);
angle\_s = copysign(1.0, angle);
double code(double angle_s, double a_m, double b, double angle_m) {
	double t_0 = ((double) M_PI) / (180.0 / angle_m);
	double t_1 = 2.0 * sin(((((angle_m * (((double) M_PI) + 1.0)) * (180.0 / angle_m)) - 180.0) / (32400.0 / angle_m)));
	double tmp;
	if (a_m <= 1.85e-126) {
		tmp = (b + a_m) * (t_1 * ((b - a_m) * cos((sqrt(((double) M_PI)) * ((angle_m / 180.0) * sqrt(((double) M_PI)))))));
	} else if (a_m <= 2e+157) {
		tmp = (b + a_m) * (((b - a_m) * cos(t_0)) * (2.0 * sin((((180.0 * ((angle_m / 180.0) * (-1.0 + (((double) M_PI) * ((double) M_PI))))) - (angle_m * (-1.0 + ((double) M_PI)))) / (180.0 * (-1.0 + ((double) M_PI)))))));
	} else {
		tmp = (b + a_m) * (t_1 * ((b - a_m) * cos(exp(log(t_0)))));
	}
	return angle_s * tmp;
}

Java

a_m = Math.abs(a);
angle\_m = Math.abs(angle);
angle\_s = Math.copySign(1.0, angle);
public static double code(double angle_s, double a_m, double b, double angle_m) {
	double t_0 = Math.PI / (180.0 / angle_m);
	double t_1 = 2.0 * Math.sin(((((angle_m * (Math.PI + 1.0)) * (180.0 / angle_m)) - 180.0) / (32400.0 / angle_m)));
	double tmp;
	if (a_m <= 1.85e-126) {
		tmp = (b + a_m) * (t_1 * ((b - a_m) * Math.cos((Math.sqrt(Math.PI) * ((angle_m / 180.0) * Math.sqrt(Math.PI))))));
	} else if (a_m <= 2e+157) {
		tmp = (b + a_m) * (((b - a_m) * Math.cos(t_0)) * (2.0 * Math.sin((((180.0 * ((angle_m / 180.0) * (-1.0 + (Math.PI * Math.PI)))) - (angle_m * (-1.0 + Math.PI))) / (180.0 * (-1.0 + Math.PI))))));
	} else {
		tmp = (b + a_m) * (t_1 * ((b - a_m) * Math.cos(Math.exp(Math.log(t_0)))));
	}
	return angle_s * tmp;
}

Python

a_m = math.fabs(a)
angle\_m = math.fabs(angle)
angle\_s = math.copysign(1.0, angle)
def code(angle_s, a_m, b, angle_m):
	t_0 = math.pi / (180.0 / angle_m)
	t_1 = 2.0 * math.sin(((((angle_m * (math.pi + 1.0)) * (180.0 / angle_m)) - 180.0) / (32400.0 / angle_m)))
	tmp = 0
	if a_m <= 1.85e-126:
		tmp = (b + a_m) * (t_1 * ((b - a_m) * math.cos((math.sqrt(math.pi) * ((angle_m / 180.0) * math.sqrt(math.pi))))))
	elif a_m <= 2e+157:
		tmp = (b + a_m) * (((b - a_m) * math.cos(t_0)) * (2.0 * math.sin((((180.0 * ((angle_m / 180.0) * (-1.0 + (math.pi * math.pi)))) - (angle_m * (-1.0 + math.pi))) / (180.0 * (-1.0 + math.pi))))))
	else:
		tmp = (b + a_m) * (t_1 * ((b - a_m) * math.cos(math.exp(math.log(t_0)))))
	return angle_s * tmp

Julia

a_m = abs(a)
angle\_m = abs(angle)
angle\_s = copysign(1.0, angle)
function code(angle_s, a_m, b, angle_m)
	t_0 = Float64(pi / Float64(180.0 / angle_m))
	t_1 = Float64(2.0 * sin(Float64(Float64(Float64(Float64(angle_m * Float64(pi + 1.0)) * Float64(180.0 / angle_m)) - 180.0) / Float64(32400.0 / angle_m))))
	tmp = 0.0
	if (a_m <= 1.85e-126)
		tmp = Float64(Float64(b + a_m) * Float64(t_1 * Float64(Float64(b - a_m) * cos(Float64(sqrt(pi) * Float64(Float64(angle_m / 180.0) * sqrt(pi)))))));
	elseif (a_m <= 2e+157)
		tmp = Float64(Float64(b + a_m) * Float64(Float64(Float64(b - a_m) * cos(t_0)) * Float64(2.0 * sin(Float64(Float64(Float64(180.0 * Float64(Float64(angle_m / 180.0) * Float64(-1.0 + Float64(pi * pi)))) - Float64(angle_m * Float64(-1.0 + pi))) / Float64(180.0 * Float64(-1.0 + pi)))))));
	else
		tmp = Float64(Float64(b + a_m) * Float64(t_1 * Float64(Float64(b - a_m) * cos(exp(log(t_0))))));
	end
	return Float64(angle_s * tmp)
end

MATLAB

a_m = abs(a);
angle\_m = abs(angle);
angle\_s = sign(angle) * abs(1.0);
function tmp_2 = code(angle_s, a_m, b, angle_m)
	t_0 = pi / (180.0 / angle_m);
	t_1 = 2.0 * sin(((((angle_m * (pi + 1.0)) * (180.0 / angle_m)) - 180.0) / (32400.0 / angle_m)));
	tmp = 0.0;
	if (a_m <= 1.85e-126)
		tmp = (b + a_m) * (t_1 * ((b - a_m) * cos((sqrt(pi) * ((angle_m / 180.0) * sqrt(pi))))));
	elseif (a_m <= 2e+157)
		tmp = (b + a_m) * (((b - a_m) * cos(t_0)) * (2.0 * sin((((180.0 * ((angle_m / 180.0) * (-1.0 + (pi * pi)))) - (angle_m * (-1.0 + pi))) / (180.0 * (-1.0 + pi))))));
	else
		tmp = (b + a_m) * (t_1 * ((b - a_m) * cos(exp(log(t_0)))));
	end
	tmp_2 = angle_s * tmp;
end

Wolfram

a_m = N[Abs[a], $MachinePrecision]
angle\_m = N[Abs[angle], $MachinePrecision]
angle\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[angle]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[angle$95$s_, a$95$m_, b_, angle$95$m_] := Block[{t$95$0 = N[(Pi / N[(180.0 / angle$95$m), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(2.0 * N[Sin[N[(N[(N[(N[(angle$95$m * N[(Pi + 1.0), $MachinePrecision]), $MachinePrecision] * N[(180.0 / angle$95$m), $MachinePrecision]), $MachinePrecision] - 180.0), $MachinePrecision] / N[(32400.0 / angle$95$m), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]}, N[(angle$95$s * If[LessEqual[a$95$m, 1.85e-126], N[(N[(b + a$95$m), $MachinePrecision] * N[(t$95$1 * N[(N[(b - a$95$m), $MachinePrecision] * N[Cos[N[(N[Sqrt[Pi], $MachinePrecision] * N[(N[(angle$95$m / 180.0), $MachinePrecision] * N[Sqrt[Pi], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[a$95$m, 2e+157], N[(N[(b + a$95$m), $MachinePrecision] * N[(N[(N[(b - a$95$m), $MachinePrecision] * N[Cos[t$95$0], $MachinePrecision]), $MachinePrecision] * N[(2.0 * N[Sin[N[(N[(N[(180.0 * N[(N[(angle$95$m / 180.0), $MachinePrecision] * N[(-1.0 + N[(Pi * Pi), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(angle$95$m * N[(-1.0 + Pi), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(180.0 * N[(-1.0 + Pi), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(b + a$95$m), $MachinePrecision] * N[(t$95$1 * N[(N[(b - a$95$m), $MachinePrecision] * N[Cos[N[Exp[N[Log[t$95$0], $MachinePrecision]], $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if a < 1.85e-126

   1. Initial program 49.6%

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

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

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

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

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

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

   3. Simplified 50.2%

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

      1. associate-*r* N/A

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

      2. *-commutative N/A

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

      3. difference-of-squares N/A

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

      4. associate-*r/ N/A

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

      5. associate-*l* N/A

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

      6. associate-*l* N/A

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

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

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

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

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

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

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

   6. Applied egg-rr 58.6%

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

      1. expm1-log1p-u N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(b, a\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{cos.f64}\left(\mathsf{/.f64}\left(\mathsf{PI.f64}\left(\right), \mathsf{/.f64}\left(180, angle\right)\right)\right), \mathsf{\_.f64}\left(b, a\right)\right), \mathsf{*.f64}\left(2, \mathsf{sin.f64}\left(\left(\frac{\mathsf{expm1}\left(\mathsf{log1p}\left(\mathsf{PI}\left(\right)\right)\right)}{\frac{180}{angle}}\right)\right)\right)\right)\right) \]

      2. expm1-undefine N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(b, a\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{cos.f64}\left(\mathsf{/.f64}\left(\mathsf{PI.f64}\left(\right), \mathsf{/.f64}\left(180, angle\right)\right)\right), \mathsf{\_.f64}\left(b, a\right)\right), \mathsf{*.f64}\left(2, \mathsf{sin.f64}\left(\left(\frac{e^{\mathsf{log1p}\left(\mathsf{PI}\left(\right)\right)} - 1}{\frac{180}{angle}}\right)\right)\right)\right)\right) \]

      3. log1p-undefine N/A

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

      4. +-commutative N/A

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

      5. rem-exp-log N/A

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

      6. sub-div N/A

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

      7. associate-/r/ N/A

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

      8. associate-*l/ N/A

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

      9. frac-sub N/A

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

      10. *-commutative N/A

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

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

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

   8. Applied egg-rr 58.6%

      \[\leadsto \left(b + a\right) \cdot \left(\left(\cos \left(\frac{\pi}{\frac{180}{angle}}\right) \cdot \left(b - a\right)\right) \cdot \left(2 \cdot \sin \color{blue}{\left(\frac{\left(\left(\pi + 1\right) \cdot angle\right) \cdot \frac{180}{angle} - 180}{\frac{32400}{angle}}\right)}\right)\right) \]
   9. Step-by-step derivation

      1. div-inv N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(b, a\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{cos.f64}\left(\left(\mathsf{PI}\left(\right) \cdot \frac{1}{\frac{180}{angle}}\right)\right), \mathsf{\_.f64}\left(b, a\right)\right), \mathsf{*.f64}\left(2, \mathsf{sin.f64}\left(\mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(\mathsf{PI.f64}\left(\right), 1\right), angle\right), \mathsf{/.f64}\left(180, angle\right)\right), 180\right), \mathsf{/.f64}\left(32400, angle\right)\right)\right)\right)\right)\right) \]

      2. add-sqr-sqrt N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(b, a\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{cos.f64}\left(\left(\left(\sqrt{\mathsf{PI}\left(\right)} \cdot \sqrt{\mathsf{PI}\left(\right)}\right) \cdot \frac{1}{\frac{180}{angle}}\right)\right), \mathsf{\_.f64}\left(b, a\right)\right), \mathsf{*.f64}\left(2, \mathsf{sin.f64}\left(\mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(\mathsf{PI.f64}\left(\right), 1\right), angle\right), \mathsf{/.f64}\left(180, angle\right)\right), 180\right), \mathsf{/.f64}\left(32400, angle\right)\right)\right)\right)\right)\right) \]

      3. clear-num N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(b, a\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{cos.f64}\left(\left(\left(\sqrt{\mathsf{PI}\left(\right)} \cdot \sqrt{\mathsf{PI}\left(\right)}\right) \cdot \frac{angle}{180}\right)\right), \mathsf{\_.f64}\left(b, a\right)\right), \mathsf{*.f64}\left(2, \mathsf{sin.f64}\left(\mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(\mathsf{PI.f64}\left(\right), 1\right), angle\right), \mathsf{/.f64}\left(180, angle\right)\right), 180\right), \mathsf{/.f64}\left(32400, angle\right)\right)\right)\right)\right)\right) \]

      4. associate-*l* N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(b, a\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{cos.f64}\left(\left(\sqrt{\mathsf{PI}\left(\right)} \cdot \left(\sqrt{\mathsf{PI}\left(\right)} \cdot \frac{angle}{180}\right)\right)\right), \mathsf{\_.f64}\left(b, a\right)\right), \mathsf{*.f64}\left(2, \mathsf{sin.f64}\left(\mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(\mathsf{PI.f64}\left(\right), 1\right), angle\right), \mathsf{/.f64}\left(180, angle\right)\right), 180\right), \mathsf{/.f64}\left(32400, angle\right)\right)\right)\right)\right)\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(b, a\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{cos.f64}\left(\mathsf{*.f64}\left(\left(\sqrt{\mathsf{PI}\left(\right)}\right), \left(\sqrt{\mathsf{PI}\left(\right)} \cdot \frac{angle}{180}\right)\right)\right), \mathsf{\_.f64}\left(b, a\right)\right), \mathsf{*.f64}\left(2, \mathsf{sin.f64}\left(\mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(\mathsf{PI.f64}\left(\right), 1\right), angle\right), \mathsf{/.f64}\left(180, angle\right)\right), 180\right), \mathsf{/.f64}\left(32400, angle\right)\right)\right)\right)\right)\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(b, a\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{cos.f64}\left(\mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{PI}\left(\right)\right), \left(\sqrt{\mathsf{PI}\left(\right)} \cdot \frac{angle}{180}\right)\right)\right), \mathsf{\_.f64}\left(b, a\right)\right), \mathsf{*.f64}\left(2, \mathsf{sin.f64}\left(\mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(\mathsf{PI.f64}\left(\right), 1\right), angle\right), \mathsf{/.f64}\left(180, angle\right)\right), 180\right), \mathsf{/.f64}\left(32400, angle\right)\right)\right)\right)\right)\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(b, a\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{cos.f64}\left(\mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{PI.f64}\left(\right)\right), \left(\sqrt{\mathsf{PI}\left(\right)} \cdot \frac{angle}{180}\right)\right)\right), \mathsf{\_.f64}\left(b, a\right)\right), \mathsf{*.f64}\left(2, \mathsf{sin.f64}\left(\mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(\mathsf{PI.f64}\left(\right), 1\right), angle\right), \mathsf{/.f64}\left(180, angle\right)\right), 180\right), \mathsf{/.f64}\left(32400, angle\right)\right)\right)\right)\right)\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(b, a\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{cos.f64}\left(\mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{PI.f64}\left(\right)\right), \mathsf{*.f64}\left(\left(\sqrt{\mathsf{PI}\left(\right)}\right), \left(\frac{angle}{180}\right)\right)\right)\right), \mathsf{\_.f64}\left(b, a\right)\right), \mathsf{*.f64}\left(2, \mathsf{sin.f64}\left(\mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(\mathsf{PI.f64}\left(\right), 1\right), angle\right), \mathsf{/.f64}\left(180, angle\right)\right), 180\right), \mathsf{/.f64}\left(32400, angle\right)\right)\right)\right)\right)\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(b, a\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{cos.f64}\left(\mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{PI.f64}\left(\right)\right), \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{PI}\left(\right)\right), \left(\frac{angle}{180}\right)\right)\right)\right), \mathsf{\_.f64}\left(b, a\right)\right), \mathsf{*.f64}\left(2, \mathsf{sin.f64}\left(\mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(\mathsf{PI.f64}\left(\right), 1\right), angle\right), \mathsf{/.f64}\left(180, angle\right)\right), 180\right), \mathsf{/.f64}\left(32400, angle\right)\right)\right)\right)\right)\right) \]

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

          \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(b, a\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{cos.f64}\left(\mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{PI.f64}\left(\right)\right), \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{PI.f64}\left(\right)\right), \left(\frac{angle}{180}\right)\right)\right)\right), \mathsf{\_.f64}\left(b, a\right)\right), \mathsf{*.f64}\left(2, \mathsf{sin.f64}\left(\mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(\mathsf{PI.f64}\left(\right), 1\right), angle\right), \mathsf{/.f64}\left(180, angle\right)\right), 180\right), \mathsf{/.f64}\left(32400, angle\right)\right)\right)\right)\right)\right) \]

      11. /-lowering-/.f64 62.3%

          \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(b, a\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{cos.f64}\left(\mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{PI.f64}\left(\right)\right), \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{PI.f64}\left(\right)\right), \mathsf{/.f64}\left(angle, 180\right)\right)\right)\right), \mathsf{\_.f64}\left(b, a\right)\right), \mathsf{*.f64}\left(2, \mathsf{sin.f64}\left(\mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(\mathsf{PI.f64}\left(\right), 1\right), angle\right), \mathsf{/.f64}\left(180, angle\right)\right), 180\right), \mathsf{/.f64}\left(32400, angle\right)\right)\right)\right)\right)\right) \]

   10. Applied egg-rr 62.3%

       \[\leadsto \left(b + a\right) \cdot \left(\left(\cos \color{blue}{\left(\sqrt{\pi} \cdot \left(\sqrt{\pi} \cdot \frac{angle}{180}\right)\right)} \cdot \left(b - a\right)\right) \cdot \left(2 \cdot \sin \left(\frac{\left(\left(\pi + 1\right) \cdot angle\right) \cdot \frac{180}{angle} - 180}{\frac{32400}{angle}}\right)\right)\right) \]

   if 1.85e-126 < a < 1.99999999999999997e157

   1. Initial program 50.3%

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

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

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

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

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

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

   3. Simplified 49.4%

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

      1. associate-*r* N/A

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

      2. *-commutative N/A

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

      3. difference-of-squares N/A

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

      4. associate-*r/ N/A

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

      5. associate-*l* N/A

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

      6. associate-*l* N/A

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

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

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

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

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

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

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

   6. Applied egg-rr 60.1%

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

      1. expm1-log1p-u N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(b, a\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{cos.f64}\left(\mathsf{/.f64}\left(\mathsf{PI.f64}\left(\right), \mathsf{/.f64}\left(180, angle\right)\right)\right), \mathsf{\_.f64}\left(b, a\right)\right), \mathsf{*.f64}\left(2, \mathsf{sin.f64}\left(\left(\frac{\mathsf{expm1}\left(\mathsf{log1p}\left(\mathsf{PI}\left(\right)\right)\right)}{\frac{180}{angle}}\right)\right)\right)\right)\right) \]

      2. expm1-undefine N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(b, a\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{cos.f64}\left(\mathsf{/.f64}\left(\mathsf{PI.f64}\left(\right), \mathsf{/.f64}\left(180, angle\right)\right)\right), \mathsf{\_.f64}\left(b, a\right)\right), \mathsf{*.f64}\left(2, \mathsf{sin.f64}\left(\left(\frac{e^{\mathsf{log1p}\left(\mathsf{PI}\left(\right)\right)} - 1}{\frac{180}{angle}}\right)\right)\right)\right)\right) \]

      3. log1p-undefine N/A

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

      4. +-commutative N/A

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

      5. rem-exp-log N/A

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

      6. sub-div N/A

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

      7. clear-num N/A

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

      8. div-inv N/A

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

      9. flip-+ N/A

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

      10. clear-num N/A

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

      11. associate-*l/ N/A

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

      12. frac-sub N/A

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

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

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

   8. Applied egg-rr 65.5%

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

   if 1.99999999999999997e157 < a

   1. Initial program 46.6%

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

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

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

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

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

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

   3. Simplified 42.0%

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

      1. associate-*r* N/A

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

      2. *-commutative N/A

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

      3. difference-of-squares N/A

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

      4. associate-*r/ N/A

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

      5. associate-*l* N/A

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

      6. associate-*l* N/A

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

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

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

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

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

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

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

   6. Applied egg-rr 77.1%

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

      1. expm1-log1p-u N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(b, a\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{cos.f64}\left(\mathsf{/.f64}\left(\mathsf{PI.f64}\left(\right), \mathsf{/.f64}\left(180, angle\right)\right)\right), \mathsf{\_.f64}\left(b, a\right)\right), \mathsf{*.f64}\left(2, \mathsf{sin.f64}\left(\left(\frac{\mathsf{expm1}\left(\mathsf{log1p}\left(\mathsf{PI}\left(\right)\right)\right)}{\frac{180}{angle}}\right)\right)\right)\right)\right) \]

      2. expm1-undefine N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(b, a\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{cos.f64}\left(\mathsf{/.f64}\left(\mathsf{PI.f64}\left(\right), \mathsf{/.f64}\left(180, angle\right)\right)\right), \mathsf{\_.f64}\left(b, a\right)\right), \mathsf{*.f64}\left(2, \mathsf{sin.f64}\left(\left(\frac{e^{\mathsf{log1p}\left(\mathsf{PI}\left(\right)\right)} - 1}{\frac{180}{angle}}\right)\right)\right)\right)\right) \]

      3. log1p-undefine N/A

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

      4. +-commutative N/A

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

      5. rem-exp-log N/A

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

      6. sub-div N/A

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

      7. associate-/r/ N/A

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

      8. associate-*l/ N/A

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

      9. frac-sub N/A

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

      10. *-commutative N/A

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

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

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

   8. Applied egg-rr 86.3%

      \[\leadsto \left(b + a\right) \cdot \left(\left(\cos \left(\frac{\pi}{\frac{180}{angle}}\right) \cdot \left(b - a\right)\right) \cdot \left(2 \cdot \sin \color{blue}{\left(\frac{\left(\left(\pi + 1\right) \cdot angle\right) \cdot \frac{180}{angle} - 180}{\frac{32400}{angle}}\right)}\right)\right) \]
   9. Step-by-step derivation

      1. associate-/r/ N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(b, a\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{cos.f64}\left(\left(\frac{\mathsf{PI}\left(\right)}{180} \cdot angle\right)\right), \mathsf{\_.f64}\left(b, a\right)\right), \mathsf{*.f64}\left(2, \mathsf{sin.f64}\left(\mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(\mathsf{PI.f64}\left(\right), 1\right), angle\right), \mathsf{/.f64}\left(180, angle\right)\right), 180\right), \mathsf{/.f64}\left(32400, angle\right)\right)\right)\right)\right)\right) \]

      2. associate-*l/ N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(b, a\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{cos.f64}\left(\left(\frac{\mathsf{PI}\left(\right) \cdot angle}{180}\right)\right), \mathsf{\_.f64}\left(b, a\right)\right), \mathsf{*.f64}\left(2, \mathsf{sin.f64}\left(\mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(\mathsf{PI.f64}\left(\right), 1\right), angle\right), \mathsf{/.f64}\left(180, angle\right)\right), 180\right), \mathsf{/.f64}\left(32400, angle\right)\right)\right)\right)\right)\right) \]

      3. clear-num N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(b, a\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{cos.f64}\left(\left(\frac{1}{\frac{180}{\mathsf{PI}\left(\right) \cdot angle}}\right)\right), \mathsf{\_.f64}\left(b, a\right)\right), \mathsf{*.f64}\left(2, \mathsf{sin.f64}\left(\mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(\mathsf{PI.f64}\left(\right), 1\right), angle\right), \mathsf{/.f64}\left(180, angle\right)\right), 180\right), \mathsf{/.f64}\left(32400, angle\right)\right)\right)\right)\right)\right) \]

      4. rem-exp-log N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(b, a\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{cos.f64}\left(\left(\frac{1}{e^{\log \left(\frac{180}{\mathsf{PI}\left(\right) \cdot angle}\right)}}\right)\right), \mathsf{\_.f64}\left(b, a\right)\right), \mathsf{*.f64}\left(2, \mathsf{sin.f64}\left(\mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(\mathsf{PI.f64}\left(\right), 1\right), angle\right), \mathsf{/.f64}\left(180, angle\right)\right), 180\right), \mathsf{/.f64}\left(32400, angle\right)\right)\right)\right)\right)\right) \]

      5. rec-exp N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(b, a\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{cos.f64}\left(\left(e^{\mathsf{neg}\left(\log \left(\frac{180}{\mathsf{PI}\left(\right) \cdot angle}\right)\right)}\right)\right), \mathsf{\_.f64}\left(b, a\right)\right), \mathsf{*.f64}\left(2, \mathsf{sin.f64}\left(\mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(\mathsf{PI.f64}\left(\right), 1\right), angle\right), \mathsf{/.f64}\left(180, angle\right)\right), 180\right), \mathsf{/.f64}\left(32400, angle\right)\right)\right)\right)\right)\right) \]

      6. associate-/l/ N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(b, a\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{cos.f64}\left(\left(e^{\mathsf{neg}\left(\log \left(\frac{\frac{180}{angle}}{\mathsf{PI}\left(\right)}\right)\right)}\right)\right), \mathsf{\_.f64}\left(b, a\right)\right), \mathsf{*.f64}\left(2, \mathsf{sin.f64}\left(\mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(\mathsf{PI.f64}\left(\right), 1\right), angle\right), \mathsf{/.f64}\left(180, angle\right)\right), 180\right), \mathsf{/.f64}\left(32400, angle\right)\right)\right)\right)\right)\right) \]

      7. neg-log N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(b, a\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{cos.f64}\left(\left(e^{\log \left(\frac{1}{\frac{\frac{180}{angle}}{\mathsf{PI}\left(\right)}}\right)}\right)\right), \mathsf{\_.f64}\left(b, a\right)\right), \mathsf{*.f64}\left(2, \mathsf{sin.f64}\left(\mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(\mathsf{PI.f64}\left(\right), 1\right), angle\right), \mathsf{/.f64}\left(180, angle\right)\right), 180\right), \mathsf{/.f64}\left(32400, angle\right)\right)\right)\right)\right)\right) \]

      8. clear-num N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(b, a\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{cos.f64}\left(\left(e^{\log \left(\frac{\mathsf{PI}\left(\right)}{\frac{180}{angle}}\right)}\right)\right), \mathsf{\_.f64}\left(b, a\right)\right), \mathsf{*.f64}\left(2, \mathsf{sin.f64}\left(\mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(\mathsf{PI.f64}\left(\right), 1\right), angle\right), \mathsf{/.f64}\left(180, angle\right)\right), 180\right), \mathsf{/.f64}\left(32400, angle\right)\right)\right)\right)\right)\right) \]

      9. associate-/r/ N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(b, a\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{cos.f64}\left(\left(e^{\log \left(\frac{\mathsf{PI}\left(\right)}{180} \cdot angle\right)}\right)\right), \mathsf{\_.f64}\left(b, a\right)\right), \mathsf{*.f64}\left(2, \mathsf{sin.f64}\left(\mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(\mathsf{PI.f64}\left(\right), 1\right), angle\right), \mathsf{/.f64}\left(180, angle\right)\right), 180\right), \mathsf{/.f64}\left(32400, angle\right)\right)\right)\right)\right)\right) \]

      10. associate-*l/ N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(b, a\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{cos.f64}\left(\left(e^{\log \left(\frac{\mathsf{PI}\left(\right) \cdot angle}{180}\right)}\right)\right), \mathsf{\_.f64}\left(b, a\right)\right), \mathsf{*.f64}\left(2, \mathsf{sin.f64}\left(\mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(\mathsf{PI.f64}\left(\right), 1\right), angle\right), \mathsf{/.f64}\left(180, angle\right)\right), 180\right), \mathsf{/.f64}\left(32400, angle\right)\right)\right)\right)\right)\right) \]

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

          \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(b, a\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{cos.f64}\left(\mathsf{exp.f64}\left(\log \left(\frac{\mathsf{PI}\left(\right) \cdot angle}{180}\right)\right)\right), \mathsf{\_.f64}\left(b, a\right)\right), \mathsf{*.f64}\left(2, \mathsf{sin.f64}\left(\mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(\mathsf{PI.f64}\left(\right), 1\right), angle\right), \mathsf{/.f64}\left(180, angle\right)\right), 180\right), \mathsf{/.f64}\left(32400, angle\right)\right)\right)\right)\right)\right) \]

      12. associate-*l/ N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(b, a\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{cos.f64}\left(\mathsf{exp.f64}\left(\log \left(\frac{\mathsf{PI}\left(\right)}{180} \cdot angle\right)\right)\right), \mathsf{\_.f64}\left(b, a\right)\right), \mathsf{*.f64}\left(2, \mathsf{sin.f64}\left(\mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(\mathsf{PI.f64}\left(\right), 1\right), angle\right), \mathsf{/.f64}\left(180, angle\right)\right), 180\right), \mathsf{/.f64}\left(32400, angle\right)\right)\right)\right)\right)\right) \]

      13. associate-/r/ N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(b, a\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{cos.f64}\left(\mathsf{exp.f64}\left(\log \left(\frac{\mathsf{PI}\left(\right)}{\frac{180}{angle}}\right)\right)\right), \mathsf{\_.f64}\left(b, a\right)\right), \mathsf{*.f64}\left(2, \mathsf{sin.f64}\left(\mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(\mathsf{PI.f64}\left(\right), 1\right), angle\right), \mathsf{/.f64}\left(180, angle\right)\right), 180\right), \mathsf{/.f64}\left(32400, angle\right)\right)\right)\right)\right)\right) \]

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

          \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(b, a\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{cos.f64}\left(\mathsf{exp.f64}\left(\mathsf{log.f64}\left(\left(\frac{\mathsf{PI}\left(\right)}{\frac{180}{angle}}\right)\right)\right)\right), \mathsf{\_.f64}\left(b, a\right)\right), \mathsf{*.f64}\left(2, \mathsf{sin.f64}\left(\mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(\mathsf{PI.f64}\left(\right), 1\right), angle\right), \mathsf{/.f64}\left(180, angle\right)\right), 180\right), \mathsf{/.f64}\left(32400, angle\right)\right)\right)\right)\right)\right) \]

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

          \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(b, a\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{cos.f64}\left(\mathsf{exp.f64}\left(\mathsf{log.f64}\left(\mathsf{/.f64}\left(\mathsf{PI}\left(\right), \left(\frac{180}{angle}\right)\right)\right)\right)\right), \mathsf{\_.f64}\left(b, a\right)\right), \mathsf{*.f64}\left(2, \mathsf{sin.f64}\left(\mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(\mathsf{PI.f64}\left(\right), 1\right), angle\right), \mathsf{/.f64}\left(180, angle\right)\right), 180\right), \mathsf{/.f64}\left(32400, angle\right)\right)\right)\right)\right)\right) \]

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

          \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(b, a\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{cos.f64}\left(\mathsf{exp.f64}\left(\mathsf{log.f64}\left(\mathsf{/.f64}\left(\mathsf{PI.f64}\left(\right), \left(\frac{180}{angle}\right)\right)\right)\right)\right), \mathsf{\_.f64}\left(b, a\right)\right), \mathsf{*.f64}\left(2, \mathsf{sin.f64}\left(\mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(\mathsf{PI.f64}\left(\right), 1\right), angle\right), \mathsf{/.f64}\left(180, angle\right)\right), 180\right), \mathsf{/.f64}\left(32400, angle\right)\right)\right)\right)\right)\right) \]

      17. /-lowering-/.f64 54.5%

          \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(b, a\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{cos.f64}\left(\mathsf{exp.f64}\left(\mathsf{log.f64}\left(\mathsf{/.f64}\left(\mathsf{PI.f64}\left(\right), \mathsf{/.f64}\left(180, angle\right)\right)\right)\right)\right), \mathsf{\_.f64}\left(b, a\right)\right), \mathsf{*.f64}\left(2, \mathsf{sin.f64}\left(\mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(\mathsf{PI.f64}\left(\right), 1\right), angle\right), \mathsf{/.f64}\left(180, angle\right)\right), 180\right), \mathsf{/.f64}\left(32400, angle\right)\right)\right)\right)\right)\right) \]

   10. Applied egg-rr 54.5%

       \[\leadsto \left(b + a\right) \cdot \left(\left(\cos \color{blue}{\left(e^{\log \left(\frac{\pi}{\frac{180}{angle}}\right)}\right)} \cdot \left(b - a\right)\right) \cdot \left(2 \cdot \sin \left(\frac{\left(\left(\pi + 1\right) \cdot angle\right) \cdot \frac{180}{angle} - 180}{\frac{32400}{angle}}\right)\right)\right) \]
3. Recombined 3 regimes into one program.

4. Final simplification 62.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \leq 1.85 \cdot 10^{-126}:\\ \;\;\;\;\left(b + a\right) \cdot \left(\left(2 \cdot \sin \left(\frac{\left(angle \cdot \left(\pi + 1\right)\right) \cdot \frac{180}{angle} - 180}{\frac{32400}{angle}}\right)\right) \cdot \left(\left(b - a\right) \cdot \cos \left(\sqrt{\pi} \cdot \left(\frac{angle}{180} \cdot \sqrt{\pi}\right)\right)\right)\right)\\ \mathbf{elif}\;a \leq 2 \cdot 10^{+157}:\\ \;\;\;\;\left(b + a\right) \cdot \left(\left(\left(b - a\right) \cdot \cos \left(\frac{\pi}{\frac{180}{angle}}\right)\right) \cdot \left(2 \cdot \sin \left(\frac{180 \cdot \left(\frac{angle}{180} \cdot \left(-1 + \pi \cdot \pi\right)\right) - angle \cdot \left(-1 + \pi\right)}{180 \cdot \left(-1 + \pi\right)}\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(b + a\right) \cdot \left(\left(2 \cdot \sin \left(\frac{\left(angle \cdot \left(\pi + 1\right)\right) \cdot \frac{180}{angle} - 180}{\frac{32400}{angle}}\right)\right) \cdot \left(\left(b - a\right) \cdot \cos \left(e^{\log \left(\frac{\pi}{\frac{180}{angle}}\right)}\right)\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 66.7% accurate, 0.5× speedup

Math

\[\begin{array}{l} a_m = \left|a\right| \\ angle\_m = \left|angle\right| \\ angle\_s = \mathsf{copysign}\left(1, angle\right) \\ angle\_s \cdot \left(\left(b + a\_m\right) \cdot \left(\left(\cos \left(\frac{{\left(e^{-1}\right)}^{\log 180}}{{\left(e^{-1}\right)}^{\log \left(\pi \cdot angle\_m\right)}}\right) \cdot \left(b - a\_m\right)\right) \cdot \left(2 \cdot \sin \left(\frac{\left(angle\_m \cdot \left(\pi + 1\right)\right) \cdot \frac{180}{angle\_m} - 180}{\frac{32400}{angle\_m}}\right)\right)\right)\right) \end{array} \]

FPCore

a_m = (fabs.f64 a)
angle\_m = (fabs.f64 angle)
angle\_s = (copysign.f64 #s(literal 1 binary64) angle)
(FPCore (angle_s a_m b angle_m)
 :precision binary64
 (*
  angle_s
  (*
   (+ b a_m)
   (*
    (*
     (cos
      (/ (pow (exp -1.0) (log 180.0)) (pow (exp -1.0) (log (* PI angle_m)))))
     (- b a_m))
    (*
     2.0
     (sin
      (/
       (- (* (* angle_m (+ PI 1.0)) (/ 180.0 angle_m)) 180.0)
       (/ 32400.0 angle_m))))))))

C

a_m = fabs(a);
angle\_m = fabs(angle);
angle\_s = copysign(1.0, angle);
double code(double angle_s, double a_m, double b, double angle_m) {
	return angle_s * ((b + a_m) * ((cos((pow(exp(-1.0), log(180.0)) / pow(exp(-1.0), log((((double) M_PI) * angle_m))))) * (b - a_m)) * (2.0 * sin(((((angle_m * (((double) M_PI) + 1.0)) * (180.0 / angle_m)) - 180.0) / (32400.0 / angle_m))))));
}

Java

a_m = Math.abs(a);
angle\_m = Math.abs(angle);
angle\_s = Math.copySign(1.0, angle);
public static double code(double angle_s, double a_m, double b, double angle_m) {
	return angle_s * ((b + a_m) * ((Math.cos((Math.pow(Math.exp(-1.0), Math.log(180.0)) / Math.pow(Math.exp(-1.0), Math.log((Math.PI * angle_m))))) * (b - a_m)) * (2.0 * Math.sin(((((angle_m * (Math.PI + 1.0)) * (180.0 / angle_m)) - 180.0) / (32400.0 / angle_m))))));
}

Python

a_m = math.fabs(a)
angle\_m = math.fabs(angle)
angle\_s = math.copysign(1.0, angle)
def code(angle_s, a_m, b, angle_m):
	return angle_s * ((b + a_m) * ((math.cos((math.pow(math.exp(-1.0), math.log(180.0)) / math.pow(math.exp(-1.0), math.log((math.pi * angle_m))))) * (b - a_m)) * (2.0 * math.sin(((((angle_m * (math.pi + 1.0)) * (180.0 / angle_m)) - 180.0) / (32400.0 / angle_m))))))

Julia

a_m = abs(a)
angle\_m = abs(angle)
angle\_s = copysign(1.0, angle)
function code(angle_s, a_m, b, angle_m)
	return Float64(angle_s * Float64(Float64(b + a_m) * Float64(Float64(cos(Float64((exp(-1.0) ^ log(180.0)) / (exp(-1.0) ^ log(Float64(pi * angle_m))))) * Float64(b - a_m)) * Float64(2.0 * sin(Float64(Float64(Float64(Float64(angle_m * Float64(pi + 1.0)) * Float64(180.0 / angle_m)) - 180.0) / Float64(32400.0 / angle_m)))))))
end

MATLAB

a_m = abs(a);
angle\_m = abs(angle);
angle\_s = sign(angle) * abs(1.0);
function tmp = code(angle_s, a_m, b, angle_m)
	tmp = angle_s * ((b + a_m) * ((cos(((exp(-1.0) ^ log(180.0)) / (exp(-1.0) ^ log((pi * angle_m))))) * (b - a_m)) * (2.0 * sin(((((angle_m * (pi + 1.0)) * (180.0 / angle_m)) - 180.0) / (32400.0 / angle_m))))));
end

Wolfram

a_m = N[Abs[a], $MachinePrecision]
angle\_m = N[Abs[angle], $MachinePrecision]
angle\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[angle]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[angle$95$s_, a$95$m_, b_, angle$95$m_] := N[(angle$95$s * N[(N[(b + a$95$m), $MachinePrecision] * N[(N[(N[Cos[N[(N[Power[N[Exp[-1.0], $MachinePrecision], N[Log[180.0], $MachinePrecision]], $MachinePrecision] / N[Power[N[Exp[-1.0], $MachinePrecision], N[Log[N[(Pi * angle$95$m), $MachinePrecision]], $MachinePrecision]], $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * N[(b - a$95$m), $MachinePrecision]), $MachinePrecision] * N[(2.0 * N[Sin[N[(N[(N[(N[(angle$95$m * N[(Pi + 1.0), $MachinePrecision]), $MachinePrecision] * N[(180.0 / angle$95$m), $MachinePrecision]), $MachinePrecision] - 180.0), $MachinePrecision] / N[(32400.0 / angle$95$m), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 49.5%

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

   1. associate-*l* N/A

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

   2. *-commutative N/A

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

   3. associate-*l* N/A

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

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

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

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

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

   6. associate-*r/ N/A

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

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

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

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

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

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

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

   10. *-commutative N/A

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

   11. associate-*l* N/A

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

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

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

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

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

3. Simplified 49.3%

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

   1. associate-*r* N/A

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

   2. *-commutative N/A

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

   3. difference-of-squares N/A

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

   4. associate-*r/ N/A

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

   5. associate-*l* N/A

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

   6. associate-*l* N/A

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

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

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

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

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

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

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

6. Applied egg-rr 60.6%

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

   1. expm1-log1p-u N/A

      \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(b, a\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{cos.f64}\left(\mathsf{/.f64}\left(\mathsf{PI.f64}\left(\right), \mathsf{/.f64}\left(180, angle\right)\right)\right), \mathsf{\_.f64}\left(b, a\right)\right), \mathsf{*.f64}\left(2, \mathsf{sin.f64}\left(\left(\frac{\mathsf{expm1}\left(\mathsf{log1p}\left(\mathsf{PI}\left(\right)\right)\right)}{\frac{180}{angle}}\right)\right)\right)\right)\right) \]

   2. expm1-undefine N/A

      \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(b, a\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{cos.f64}\left(\mathsf{/.f64}\left(\mathsf{PI.f64}\left(\right), \mathsf{/.f64}\left(180, angle\right)\right)\right), \mathsf{\_.f64}\left(b, a\right)\right), \mathsf{*.f64}\left(2, \mathsf{sin.f64}\left(\left(\frac{e^{\mathsf{log1p}\left(\mathsf{PI}\left(\right)\right)} - 1}{\frac{180}{angle}}\right)\right)\right)\right)\right) \]

   3. log1p-undefine N/A

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

   4. +-commutative N/A

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

   5. rem-exp-log N/A

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

   6. sub-div N/A

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

   7. associate-/r/ N/A

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

   8. associate-*l/ N/A

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

   9. frac-sub N/A

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

   10. *-commutative N/A

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

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

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

8. Applied egg-rr 62.5%

   \[\leadsto \left(b + a\right) \cdot \left(\left(\cos \left(\frac{\pi}{\frac{180}{angle}}\right) \cdot \left(b - a\right)\right) \cdot \left(2 \cdot \sin \color{blue}{\left(\frac{\left(\left(\pi + 1\right) \cdot angle\right) \cdot \frac{180}{angle} - 180}{\frac{32400}{angle}}\right)}\right)\right) \]
9. Step-by-step derivation

   1. associate-/r/ N/A

      \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(b, a\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{cos.f64}\left(\left(\frac{\mathsf{PI}\left(\right)}{180} \cdot angle\right)\right), \mathsf{\_.f64}\left(b, a\right)\right), \mathsf{*.f64}\left(2, \mathsf{sin.f64}\left(\mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(\mathsf{PI.f64}\left(\right), 1\right), angle\right), \mathsf{/.f64}\left(180, angle\right)\right), 180\right), \mathsf{/.f64}\left(32400, angle\right)\right)\right)\right)\right)\right) \]

   2. associate-*l/ N/A

      \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(b, a\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{cos.f64}\left(\left(\frac{\mathsf{PI}\left(\right) \cdot angle}{180}\right)\right), \mathsf{\_.f64}\left(b, a\right)\right), \mathsf{*.f64}\left(2, \mathsf{sin.f64}\left(\mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(\mathsf{PI.f64}\left(\right), 1\right), angle\right), \mathsf{/.f64}\left(180, angle\right)\right), 180\right), \mathsf{/.f64}\left(32400, angle\right)\right)\right)\right)\right)\right) \]

   3. clear-num N/A

      \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(b, a\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{cos.f64}\left(\left(\frac{1}{\frac{180}{\mathsf{PI}\left(\right) \cdot angle}}\right)\right), \mathsf{\_.f64}\left(b, a\right)\right), \mathsf{*.f64}\left(2, \mathsf{sin.f64}\left(\mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(\mathsf{PI.f64}\left(\right), 1\right), angle\right), \mathsf{/.f64}\left(180, angle\right)\right), 180\right), \mathsf{/.f64}\left(32400, angle\right)\right)\right)\right)\right)\right) \]

   4. unpow-1 N/A

      \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(b, a\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{cos.f64}\left(\left({\left(\frac{180}{\mathsf{PI}\left(\right) \cdot angle}\right)}^{-1}\right)\right), \mathsf{\_.f64}\left(b, a\right)\right), \mathsf{*.f64}\left(2, \mathsf{sin.f64}\left(\mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(\mathsf{PI.f64}\left(\right), 1\right), angle\right), \mathsf{/.f64}\left(180, angle\right)\right), 180\right), \mathsf{/.f64}\left(32400, angle\right)\right)\right)\right)\right)\right) \]

   5. exp-to-pow N/A

      \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(b, a\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{cos.f64}\left(\left(e^{\log \left(\frac{180}{\mathsf{PI}\left(\right) \cdot angle}\right) \cdot -1}\right)\right), \mathsf{\_.f64}\left(b, a\right)\right), \mathsf{*.f64}\left(2, \mathsf{sin.f64}\left(\mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(\mathsf{PI.f64}\left(\right), 1\right), angle\right), \mathsf{/.f64}\left(180, angle\right)\right), 180\right), \mathsf{/.f64}\left(32400, angle\right)\right)\right)\right)\right)\right) \]

   6. *-commutative N/A

      \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(b, a\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{cos.f64}\left(\left(e^{-1 \cdot \log \left(\frac{180}{\mathsf{PI}\left(\right) \cdot angle}\right)}\right)\right), \mathsf{\_.f64}\left(b, a\right)\right), \mathsf{*.f64}\left(2, \mathsf{sin.f64}\left(\mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(\mathsf{PI.f64}\left(\right), 1\right), angle\right), \mathsf{/.f64}\left(180, angle\right)\right), 180\right), \mathsf{/.f64}\left(32400, angle\right)\right)\right)\right)\right)\right) \]

   7. exp-prod N/A

      \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(b, a\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{cos.f64}\left(\left({\left(e^{-1}\right)}^{\log \left(\frac{180}{\mathsf{PI}\left(\right) \cdot angle}\right)}\right)\right), \mathsf{\_.f64}\left(b, a\right)\right), \mathsf{*.f64}\left(2, \mathsf{sin.f64}\left(\mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(\mathsf{PI.f64}\left(\right), 1\right), angle\right), \mathsf{/.f64}\left(180, angle\right)\right), 180\right), \mathsf{/.f64}\left(32400, angle\right)\right)\right)\right)\right)\right) \]

   8. log-div N/A

      \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(b, a\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{cos.f64}\left(\left({\left(e^{-1}\right)}^{\left(\log 180 - \log \left(\mathsf{PI}\left(\right) \cdot angle\right)\right)}\right)\right), \mathsf{\_.f64}\left(b, a\right)\right), \mathsf{*.f64}\left(2, \mathsf{sin.f64}\left(\mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(\mathsf{PI.f64}\left(\right), 1\right), angle\right), \mathsf{/.f64}\left(180, angle\right)\right), 180\right), \mathsf{/.f64}\left(32400, angle\right)\right)\right)\right)\right)\right) \]

   9. pow-sub N/A

      \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(b, a\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{cos.f64}\left(\left(\frac{{\left(e^{-1}\right)}^{\log 180}}{{\left(e^{-1}\right)}^{\log \left(\mathsf{PI}\left(\right) \cdot angle\right)}}\right)\right), \mathsf{\_.f64}\left(b, a\right)\right), \mathsf{*.f64}\left(2, \mathsf{sin.f64}\left(\mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(\mathsf{PI.f64}\left(\right), 1\right), angle\right), \mathsf{/.f64}\left(180, angle\right)\right), 180\right), \mathsf{/.f64}\left(32400, angle\right)\right)\right)\right)\right)\right) \]

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

       \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(b, a\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{cos.f64}\left(\mathsf{/.f64}\left(\left({\left(e^{-1}\right)}^{\log 180}\right), \left({\left(e^{-1}\right)}^{\log \left(\mathsf{PI}\left(\right) \cdot angle\right)}\right)\right)\right), \mathsf{\_.f64}\left(b, a\right)\right), \mathsf{*.f64}\left(2, \mathsf{sin.f64}\left(\mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(\mathsf{PI.f64}\left(\right), 1\right), angle\right), \mathsf{/.f64}\left(180, angle\right)\right), 180\right), \mathsf{/.f64}\left(32400, angle\right)\right)\right)\right)\right)\right) \]

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

       \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(b, a\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{cos.f64}\left(\mathsf{/.f64}\left(\mathsf{pow.f64}\left(\left(e^{-1}\right), \log 180\right), \left({\left(e^{-1}\right)}^{\log \left(\mathsf{PI}\left(\right) \cdot angle\right)}\right)\right)\right), \mathsf{\_.f64}\left(b, a\right)\right), \mathsf{*.f64}\left(2, \mathsf{sin.f64}\left(\mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(\mathsf{PI.f64}\left(\right), 1\right), angle\right), \mathsf{/.f64}\left(180, angle\right)\right), 180\right), \mathsf{/.f64}\left(32400, angle\right)\right)\right)\right)\right)\right) \]

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

       \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(b, a\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{cos.f64}\left(\mathsf{/.f64}\left(\mathsf{pow.f64}\left(\mathsf{exp.f64}\left(-1\right), \log 180\right), \left({\left(e^{-1}\right)}^{\log \left(\mathsf{PI}\left(\right) \cdot angle\right)}\right)\right)\right), \mathsf{\_.f64}\left(b, a\right)\right), \mathsf{*.f64}\left(2, \mathsf{sin.f64}\left(\mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(\mathsf{PI.f64}\left(\right), 1\right), angle\right), \mathsf{/.f64}\left(180, angle\right)\right), 180\right), \mathsf{/.f64}\left(32400, angle\right)\right)\right)\right)\right)\right) \]

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

       \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(b, a\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{cos.f64}\left(\mathsf{/.f64}\left(\mathsf{pow.f64}\left(\mathsf{exp.f64}\left(-1\right), \mathsf{log.f64}\left(180\right)\right), \left({\left(e^{-1}\right)}^{\log \left(\mathsf{PI}\left(\right) \cdot angle\right)}\right)\right)\right), \mathsf{\_.f64}\left(b, a\right)\right), \mathsf{*.f64}\left(2, \mathsf{sin.f64}\left(\mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(\mathsf{PI.f64}\left(\right), 1\right), angle\right), \mathsf{/.f64}\left(180, angle\right)\right), 180\right), \mathsf{/.f64}\left(32400, angle\right)\right)\right)\right)\right)\right) \]

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

       \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(b, a\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{cos.f64}\left(\mathsf{/.f64}\left(\mathsf{pow.f64}\left(\mathsf{exp.f64}\left(-1\right), \mathsf{log.f64}\left(180\right)\right), \mathsf{pow.f64}\left(\left(e^{-1}\right), \log \left(\mathsf{PI}\left(\right) \cdot angle\right)\right)\right)\right), \mathsf{\_.f64}\left(b, a\right)\right), \mathsf{*.f64}\left(2, \mathsf{sin.f64}\left(\mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(\mathsf{PI.f64}\left(\right), 1\right), angle\right), \mathsf{/.f64}\left(180, angle\right)\right), 180\right), \mathsf{/.f64}\left(32400, angle\right)\right)\right)\right)\right)\right) \]

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

       \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(b, a\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{cos.f64}\left(\mathsf{/.f64}\left(\mathsf{pow.f64}\left(\mathsf{exp.f64}\left(-1\right), \mathsf{log.f64}\left(180\right)\right), \mathsf{pow.f64}\left(\mathsf{exp.f64}\left(-1\right), \log \left(\mathsf{PI}\left(\right) \cdot angle\right)\right)\right)\right), \mathsf{\_.f64}\left(b, a\right)\right), \mathsf{*.f64}\left(2, \mathsf{sin.f64}\left(\mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(\mathsf{PI.f64}\left(\right), 1\right), angle\right), \mathsf{/.f64}\left(180, angle\right)\right), 180\right), \mathsf{/.f64}\left(32400, angle\right)\right)\right)\right)\right)\right) \]

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

       \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(b, a\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{cos.f64}\left(\mathsf{/.f64}\left(\mathsf{pow.f64}\left(\mathsf{exp.f64}\left(-1\right), \mathsf{log.f64}\left(180\right)\right), \mathsf{pow.f64}\left(\mathsf{exp.f64}\left(-1\right), \mathsf{log.f64}\left(\left(\mathsf{PI}\left(\right) \cdot angle\right)\right)\right)\right)\right), \mathsf{\_.f64}\left(b, a\right)\right), \mathsf{*.f64}\left(2, \mathsf{sin.f64}\left(\mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(\mathsf{PI.f64}\left(\right), 1\right), angle\right), \mathsf{/.f64}\left(180, angle\right)\right), 180\right), \mathsf{/.f64}\left(32400, angle\right)\right)\right)\right)\right)\right) \]

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

       \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(b, a\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{cos.f64}\left(\mathsf{/.f64}\left(\mathsf{pow.f64}\left(\mathsf{exp.f64}\left(-1\right), \mathsf{log.f64}\left(180\right)\right), \mathsf{pow.f64}\left(\mathsf{exp.f64}\left(-1\right), \mathsf{log.f64}\left(\mathsf{*.f64}\left(\mathsf{PI}\left(\right), angle\right)\right)\right)\right)\right), \mathsf{\_.f64}\left(b, a\right)\right), \mathsf{*.f64}\left(2, \mathsf{sin.f64}\left(\mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(\mathsf{PI.f64}\left(\right), 1\right), angle\right), \mathsf{/.f64}\left(180, angle\right)\right), 180\right), \mathsf{/.f64}\left(32400, angle\right)\right)\right)\right)\right)\right) \]

   18. PI-lowering-PI.f64 37.8%

       \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(b, a\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{cos.f64}\left(\mathsf{/.f64}\left(\mathsf{pow.f64}\left(\mathsf{exp.f64}\left(-1\right), \mathsf{log.f64}\left(180\right)\right), \mathsf{pow.f64}\left(\mathsf{exp.f64}\left(-1\right), \mathsf{log.f64}\left(\mathsf{*.f64}\left(\mathsf{PI.f64}\left(\right), angle\right)\right)\right)\right)\right), \mathsf{\_.f64}\left(b, a\right)\right), \mathsf{*.f64}\left(2, \mathsf{sin.f64}\left(\mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(\mathsf{PI.f64}\left(\right), 1\right), angle\right), \mathsf{/.f64}\left(180, angle\right)\right), 180\right), \mathsf{/.f64}\left(32400, angle\right)\right)\right)\right)\right)\right) \]

10. Applied egg-rr 37.8%

    \[\leadsto \left(b + a\right) \cdot \left(\left(\cos \color{blue}{\left(\frac{{\left(e^{-1}\right)}^{\log 180}}{{\left(e^{-1}\right)}^{\log \left(\pi \cdot angle\right)}}\right)} \cdot \left(b - a\right)\right) \cdot \left(2 \cdot \sin \left(\frac{\left(\left(\pi + 1\right) \cdot angle\right) \cdot \frac{180}{angle} - 180}{\frac{32400}{angle}}\right)\right)\right) \]

11. Final simplification 37.8%

    \[\leadsto \left(b + a\right) \cdot \left(\left(\cos \left(\frac{{\left(e^{-1}\right)}^{\log 180}}{{\left(e^{-1}\right)}^{\log \left(\pi \cdot angle\right)}}\right) \cdot \left(b - a\right)\right) \cdot \left(2 \cdot \sin \left(\frac{\left(angle \cdot \left(\pi + 1\right)\right) \cdot \frac{180}{angle} - 180}{\frac{32400}{angle}}\right)\right)\right) \]
12. Add Preprocessing

Alternative 3: 67.7% accurate, 0.5× speedup

Math

\[\begin{array}{l} a_m = \left|a\right| \\ angle\_m = \left|angle\right| \\ angle\_s = \mathsf{copysign}\left(1, angle\right) \\ angle\_s \cdot \left(\left(b + a\_m\right) \cdot \left(\left(\left(b - a\_m\right) \cdot \cos \left(\frac{{\left(e^{-1}\right)}^{\log 180}}{{\left(e^{-1}\right)}^{\log \left(\pi \cdot angle\_m\right)}} + \left(angle\_m \cdot -0.005555555555555556 + \frac{angle\_m}{180}\right)\right)\right) \cdot \left(2 \cdot \sin \left(\frac{\pi}{\frac{180}{angle\_m}}\right)\right)\right)\right) \end{array} \]

FPCore

a_m = (fabs.f64 a)
angle\_m = (fabs.f64 angle)
angle\_s = (copysign.f64 #s(literal 1 binary64) angle)
(FPCore (angle_s a_m b angle_m)
 :precision binary64
 (*
  angle_s
  (*
   (+ b a_m)
   (*
    (*
     (- b a_m)
     (cos
      (+
       (/ (pow (exp -1.0) (log 180.0)) (pow (exp -1.0) (log (* PI angle_m))))
       (+ (* angle_m -0.005555555555555556) (/ angle_m 180.0)))))
    (* 2.0 (sin (/ PI (/ 180.0 angle_m))))))))

C

a_m = fabs(a);
angle\_m = fabs(angle);
angle\_s = copysign(1.0, angle);
double code(double angle_s, double a_m, double b, double angle_m) {
	return angle_s * ((b + a_m) * (((b - a_m) * cos(((pow(exp(-1.0), log(180.0)) / pow(exp(-1.0), log((((double) M_PI) * angle_m)))) + ((angle_m * -0.005555555555555556) + (angle_m / 180.0))))) * (2.0 * sin((((double) M_PI) / (180.0 / angle_m))))));
}

Java

a_m = Math.abs(a);
angle\_m = Math.abs(angle);
angle\_s = Math.copySign(1.0, angle);
public static double code(double angle_s, double a_m, double b, double angle_m) {
	return angle_s * ((b + a_m) * (((b - a_m) * Math.cos(((Math.pow(Math.exp(-1.0), Math.log(180.0)) / Math.pow(Math.exp(-1.0), Math.log((Math.PI * angle_m)))) + ((angle_m * -0.005555555555555556) + (angle_m / 180.0))))) * (2.0 * Math.sin((Math.PI / (180.0 / angle_m))))));
}

Python

a_m = math.fabs(a)
angle\_m = math.fabs(angle)
angle\_s = math.copysign(1.0, angle)
def code(angle_s, a_m, b, angle_m):
	return angle_s * ((b + a_m) * (((b - a_m) * math.cos(((math.pow(math.exp(-1.0), math.log(180.0)) / math.pow(math.exp(-1.0), math.log((math.pi * angle_m)))) + ((angle_m * -0.005555555555555556) + (angle_m / 180.0))))) * (2.0 * math.sin((math.pi / (180.0 / angle_m))))))

Julia

a_m = abs(a)
angle\_m = abs(angle)
angle\_s = copysign(1.0, angle)
function code(angle_s, a_m, b, angle_m)
	return Float64(angle_s * Float64(Float64(b + a_m) * Float64(Float64(Float64(b - a_m) * cos(Float64(Float64((exp(-1.0) ^ log(180.0)) / (exp(-1.0) ^ log(Float64(pi * angle_m)))) + Float64(Float64(angle_m * -0.005555555555555556) + Float64(angle_m / 180.0))))) * Float64(2.0 * sin(Float64(pi / Float64(180.0 / angle_m)))))))
end

MATLAB

a_m = abs(a);
angle\_m = abs(angle);
angle\_s = sign(angle) * abs(1.0);
function tmp = code(angle_s, a_m, b, angle_m)
	tmp = angle_s * ((b + a_m) * (((b - a_m) * cos((((exp(-1.0) ^ log(180.0)) / (exp(-1.0) ^ log((pi * angle_m)))) + ((angle_m * -0.005555555555555556) + (angle_m / 180.0))))) * (2.0 * sin((pi / (180.0 / angle_m))))));
end

Wolfram

a_m = N[Abs[a], $MachinePrecision]
angle\_m = N[Abs[angle], $MachinePrecision]
angle\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[angle]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[angle$95$s_, a$95$m_, b_, angle$95$m_] := N[(angle$95$s * N[(N[(b + a$95$m), $MachinePrecision] * N[(N[(N[(b - a$95$m), $MachinePrecision] * N[Cos[N[(N[(N[Power[N[Exp[-1.0], $MachinePrecision], N[Log[180.0], $MachinePrecision]], $MachinePrecision] / N[Power[N[Exp[-1.0], $MachinePrecision], N[Log[N[(Pi * angle$95$m), $MachinePrecision]], $MachinePrecision]], $MachinePrecision]), $MachinePrecision] + N[(N[(angle$95$m * -0.005555555555555556), $MachinePrecision] + N[(angle$95$m / 180.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[(2.0 * N[Sin[N[(Pi / N[(180.0 / angle$95$m), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 49.5%

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

   1. associate-*l* N/A

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

   2. *-commutative N/A

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

   3. associate-*l* N/A

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

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

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

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

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

   6. associate-*r/ N/A

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

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

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

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

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

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

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

   10. *-commutative N/A

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

   11. associate-*l* N/A

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

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

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

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

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

3. Simplified 49.3%

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

   1. associate-*r* N/A

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

   2. *-commutative N/A

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

   3. difference-of-squares N/A

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

   4. associate-*r/ N/A

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

   5. associate-*l* N/A

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

   6. associate-*l* N/A

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

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

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

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

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

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

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

6. Applied egg-rr 60.6%

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

   1. expm1-log1p-u N/A

      \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(b, a\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{cos.f64}\left(\left(\frac{\mathsf{expm1}\left(\mathsf{log1p}\left(\mathsf{PI}\left(\right)\right)\right)}{\frac{180}{angle}}\right)\right), \mathsf{\_.f64}\left(b, a\right)\right), \mathsf{*.f64}\left(2, \mathsf{sin.f64}\left(\mathsf{/.f64}\left(\mathsf{PI.f64}\left(\right), \mathsf{/.f64}\left(180, angle\right)\right)\right)\right)\right)\right) \]

   2. expm1-undefine N/A

      \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(b, a\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{cos.f64}\left(\left(\frac{e^{\mathsf{log1p}\left(\mathsf{PI}\left(\right)\right)} - 1}{\frac{180}{angle}}\right)\right), \mathsf{\_.f64}\left(b, a\right)\right), \mathsf{*.f64}\left(2, \mathsf{sin.f64}\left(\mathsf{/.f64}\left(\mathsf{PI.f64}\left(\right), \mathsf{/.f64}\left(180, angle\right)\right)\right)\right)\right)\right) \]

   3. log1p-undefine N/A

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

   4. +-commutative N/A

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

   5. rem-exp-log N/A

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

   6. sub-div N/A

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

   7. clear-num N/A

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

   8. associate-/r/ N/A

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

   9. div-inv N/A

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

   10. metadata-eval N/A

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

   11. prod-diff N/A

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

   12. metadata-eval N/A

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

   13. associate-/r/ N/A

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

   14. clear-num N/A

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

   15. fmm-def N/A

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

   16. associate-/r/ N/A

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

8. Applied egg-rr 60.1%

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

   1. associate-/r/ N/A

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

   2. associate-*l/ N/A

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

   3. clear-num N/A

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

   4. unpow-1 N/A

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

   5. exp-to-pow N/A

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

   6. *-commutative N/A

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

   7. exp-prod N/A

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

   8. log-div N/A

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

   9. pow-sub N/A

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

   18. PI-lowering-PI.f64 36.7%

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

10. Applied egg-rr 36.7%

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

11. Final simplification 36.7%

    \[\leadsto \left(b + a\right) \cdot \left(\left(\left(b - a\right) \cdot \cos \left(\frac{{\left(e^{-1}\right)}^{\log 180}}{{\left(e^{-1}\right)}^{\log \left(\pi \cdot angle\right)}} + \left(angle \cdot -0.005555555555555556 + \frac{angle}{180}\right)\right)\right) \cdot \left(2 \cdot \sin \left(\frac{\pi}{\frac{180}{angle}}\right)\right)\right) \]
12. Add Preprocessing

Alternative 4: 67.4% accurate, 1.0× speedup

Math

\[\begin{array}{l} a_m = \left|a\right| \\ angle\_m = \left|angle\right| \\ angle\_s = \mathsf{copysign}\left(1, angle\right) \\ \begin{array}{l} t_0 := 2 \cdot \sin \left(\frac{\left(angle\_m \cdot \left(\pi + 1\right)\right) \cdot \frac{180}{angle\_m} - 180}{\frac{32400}{angle\_m}}\right)\\ angle\_s \cdot \begin{array}{l} \mathbf{if}\;a\_m \leq 8 \cdot 10^{+156}:\\ \;\;\;\;\left(b + a\_m\right) \cdot \left(t\_0 \cdot \left(\left(b - a\_m\right) \cdot \cos \left(\frac{\sqrt{\pi}}{180} \cdot \frac{\sqrt{\pi}}{\frac{1}{angle\_m}}\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(b + a\_m\right) \cdot \left(t\_0 \cdot \left(\left(b - a\_m\right) \cdot \cos \left(e^{\log \left(\frac{\pi}{\frac{180}{angle\_m}}\right)}\right)\right)\right)\\ \end{array} \end{array} \end{array} \]

FPCore

a_m = (fabs.f64 a)
angle\_m = (fabs.f64 angle)
angle\_s = (copysign.f64 #s(literal 1 binary64) angle)
(FPCore (angle_s a_m b angle_m)
 :precision binary64
 (let* ((t_0
         (*
          2.0
          (sin
           (/
            (- (* (* angle_m (+ PI 1.0)) (/ 180.0 angle_m)) 180.0)
            (/ 32400.0 angle_m))))))
   (*
    angle_s
    (if (<= a_m 8e+156)
      (*
       (+ b a_m)
       (*
        t_0
        (*
         (- b a_m)
         (cos (* (/ (sqrt PI) 180.0) (/ (sqrt PI) (/ 1.0 angle_m)))))))
      (*
       (+ b a_m)
       (* t_0 (* (- b a_m) (cos (exp (log (/ PI (/ 180.0 angle_m))))))))))))

C

a_m = fabs(a);
angle\_m = fabs(angle);
angle\_s = copysign(1.0, angle);
double code(double angle_s, double a_m, double b, double angle_m) {
	double t_0 = 2.0 * sin(((((angle_m * (((double) M_PI) + 1.0)) * (180.0 / angle_m)) - 180.0) / (32400.0 / angle_m)));
	double tmp;
	if (a_m <= 8e+156) {
		tmp = (b + a_m) * (t_0 * ((b - a_m) * cos(((sqrt(((double) M_PI)) / 180.0) * (sqrt(((double) M_PI)) / (1.0 / angle_m))))));
	} else {
		tmp = (b + a_m) * (t_0 * ((b - a_m) * cos(exp(log((((double) M_PI) / (180.0 / angle_m)))))));
	}
	return angle_s * tmp;
}

Java

a_m = Math.abs(a);
angle\_m = Math.abs(angle);
angle\_s = Math.copySign(1.0, angle);
public static double code(double angle_s, double a_m, double b, double angle_m) {
	double t_0 = 2.0 * Math.sin(((((angle_m * (Math.PI + 1.0)) * (180.0 / angle_m)) - 180.0) / (32400.0 / angle_m)));
	double tmp;
	if (a_m <= 8e+156) {
		tmp = (b + a_m) * (t_0 * ((b - a_m) * Math.cos(((Math.sqrt(Math.PI) / 180.0) * (Math.sqrt(Math.PI) / (1.0 / angle_m))))));
	} else {
		tmp = (b + a_m) * (t_0 * ((b - a_m) * Math.cos(Math.exp(Math.log((Math.PI / (180.0 / angle_m)))))));
	}
	return angle_s * tmp;
}

Python

a_m = math.fabs(a)
angle\_m = math.fabs(angle)
angle\_s = math.copysign(1.0, angle)
def code(angle_s, a_m, b, angle_m):
	t_0 = 2.0 * math.sin(((((angle_m * (math.pi + 1.0)) * (180.0 / angle_m)) - 180.0) / (32400.0 / angle_m)))
	tmp = 0
	if a_m <= 8e+156:
		tmp = (b + a_m) * (t_0 * ((b - a_m) * math.cos(((math.sqrt(math.pi) / 180.0) * (math.sqrt(math.pi) / (1.0 / angle_m))))))
	else:
		tmp = (b + a_m) * (t_0 * ((b - a_m) * math.cos(math.exp(math.log((math.pi / (180.0 / angle_m)))))))
	return angle_s * tmp

Julia

a_m = abs(a)
angle\_m = abs(angle)
angle\_s = copysign(1.0, angle)
function code(angle_s, a_m, b, angle_m)
	t_0 = Float64(2.0 * sin(Float64(Float64(Float64(Float64(angle_m * Float64(pi + 1.0)) * Float64(180.0 / angle_m)) - 180.0) / Float64(32400.0 / angle_m))))
	tmp = 0.0
	if (a_m <= 8e+156)
		tmp = Float64(Float64(b + a_m) * Float64(t_0 * Float64(Float64(b - a_m) * cos(Float64(Float64(sqrt(pi) / 180.0) * Float64(sqrt(pi) / Float64(1.0 / angle_m)))))));
	else
		tmp = Float64(Float64(b + a_m) * Float64(t_0 * Float64(Float64(b - a_m) * cos(exp(log(Float64(pi / Float64(180.0 / angle_m))))))));
	end
	return Float64(angle_s * tmp)
end

MATLAB

a_m = abs(a);
angle\_m = abs(angle);
angle\_s = sign(angle) * abs(1.0);
function tmp_2 = code(angle_s, a_m, b, angle_m)
	t_0 = 2.0 * sin(((((angle_m * (pi + 1.0)) * (180.0 / angle_m)) - 180.0) / (32400.0 / angle_m)));
	tmp = 0.0;
	if (a_m <= 8e+156)
		tmp = (b + a_m) * (t_0 * ((b - a_m) * cos(((sqrt(pi) / 180.0) * (sqrt(pi) / (1.0 / angle_m))))));
	else
		tmp = (b + a_m) * (t_0 * ((b - a_m) * cos(exp(log((pi / (180.0 / angle_m)))))));
	end
	tmp_2 = angle_s * tmp;
end

Wolfram

a_m = N[Abs[a], $MachinePrecision]
angle\_m = N[Abs[angle], $MachinePrecision]
angle\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[angle]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[angle$95$s_, a$95$m_, b_, angle$95$m_] := Block[{t$95$0 = N[(2.0 * N[Sin[N[(N[(N[(N[(angle$95$m * N[(Pi + 1.0), $MachinePrecision]), $MachinePrecision] * N[(180.0 / angle$95$m), $MachinePrecision]), $MachinePrecision] - 180.0), $MachinePrecision] / N[(32400.0 / angle$95$m), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]}, N[(angle$95$s * If[LessEqual[a$95$m, 8e+156], N[(N[(b + a$95$m), $MachinePrecision] * N[(t$95$0 * N[(N[(b - a$95$m), $MachinePrecision] * N[Cos[N[(N[(N[Sqrt[Pi], $MachinePrecision] / 180.0), $MachinePrecision] * N[(N[Sqrt[Pi], $MachinePrecision] / N[(1.0 / angle$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(b + a$95$m), $MachinePrecision] * N[(t$95$0 * N[(N[(b - a$95$m), $MachinePrecision] * N[Cos[N[Exp[N[Log[N[(Pi / N[(180.0 / angle$95$m), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]], $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if a < 7.9999999999999999e156

   1. Initial program 49.8%

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

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

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

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

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

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

   3. Simplified 50.0%

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

      1. associate-*r* N/A

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

      2. *-commutative N/A

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

      3. difference-of-squares N/A

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

      4. associate-*r/ N/A

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

      5. associate-*l* N/A

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

      6. associate-*l* N/A

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

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

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

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

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

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

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

   6. Applied egg-rr 59.0%

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

      1. expm1-log1p-u N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(b, a\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{cos.f64}\left(\mathsf{/.f64}\left(\mathsf{PI.f64}\left(\right), \mathsf{/.f64}\left(180, angle\right)\right)\right), \mathsf{\_.f64}\left(b, a\right)\right), \mathsf{*.f64}\left(2, \mathsf{sin.f64}\left(\left(\frac{\mathsf{expm1}\left(\mathsf{log1p}\left(\mathsf{PI}\left(\right)\right)\right)}{\frac{180}{angle}}\right)\right)\right)\right)\right) \]

      2. expm1-undefine N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(b, a\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{cos.f64}\left(\mathsf{/.f64}\left(\mathsf{PI.f64}\left(\right), \mathsf{/.f64}\left(180, angle\right)\right)\right), \mathsf{\_.f64}\left(b, a\right)\right), \mathsf{*.f64}\left(2, \mathsf{sin.f64}\left(\left(\frac{e^{\mathsf{log1p}\left(\mathsf{PI}\left(\right)\right)} - 1}{\frac{180}{angle}}\right)\right)\right)\right)\right) \]

      3. log1p-undefine N/A

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

      4. +-commutative N/A

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

      5. rem-exp-log N/A

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

      6. sub-div N/A

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

      7. associate-/r/ N/A

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

      8. associate-*l/ N/A

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

      9. frac-sub N/A

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

      10. *-commutative N/A

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

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

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

   8. Applied egg-rr 60.2%

      \[\leadsto \left(b + a\right) \cdot \left(\left(\cos \left(\frac{\pi}{\frac{180}{angle}}\right) \cdot \left(b - a\right)\right) \cdot \left(2 \cdot \sin \color{blue}{\left(\frac{\left(\left(\pi + 1\right) \cdot angle\right) \cdot \frac{180}{angle} - 180}{\frac{32400}{angle}}\right)}\right)\right) \]
   9. Step-by-step derivation

      1. add-sqr-sqrt N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(b, a\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{cos.f64}\left(\left(\frac{\sqrt{\mathsf{PI}\left(\right)} \cdot \sqrt{\mathsf{PI}\left(\right)}}{\frac{180}{angle}}\right)\right), \mathsf{\_.f64}\left(b, a\right)\right), \mathsf{*.f64}\left(2, \mathsf{sin.f64}\left(\mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(\mathsf{PI.f64}\left(\right), 1\right), angle\right), \mathsf{/.f64}\left(180, angle\right)\right), 180\right), \mathsf{/.f64}\left(32400, angle\right)\right)\right)\right)\right)\right) \]

      2. div-inv N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(b, a\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{cos.f64}\left(\left(\frac{\sqrt{\mathsf{PI}\left(\right)} \cdot \sqrt{\mathsf{PI}\left(\right)}}{180 \cdot \frac{1}{angle}}\right)\right), \mathsf{\_.f64}\left(b, a\right)\right), \mathsf{*.f64}\left(2, \mathsf{sin.f64}\left(\mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(\mathsf{PI.f64}\left(\right), 1\right), angle\right), \mathsf{/.f64}\left(180, angle\right)\right), 180\right), \mathsf{/.f64}\left(32400, angle\right)\right)\right)\right)\right)\right) \]

      3. times-frac N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(b, a\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{cos.f64}\left(\left(\frac{\sqrt{\mathsf{PI}\left(\right)}}{180} \cdot \frac{\sqrt{\mathsf{PI}\left(\right)}}{\frac{1}{angle}}\right)\right), \mathsf{\_.f64}\left(b, a\right)\right), \mathsf{*.f64}\left(2, \mathsf{sin.f64}\left(\mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(\mathsf{PI.f64}\left(\right), 1\right), angle\right), \mathsf{/.f64}\left(180, angle\right)\right), 180\right), \mathsf{/.f64}\left(32400, angle\right)\right)\right)\right)\right)\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(b, a\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{cos.f64}\left(\mathsf{*.f64}\left(\left(\frac{\sqrt{\mathsf{PI}\left(\right)}}{180}\right), \left(\frac{\sqrt{\mathsf{PI}\left(\right)}}{\frac{1}{angle}}\right)\right)\right), \mathsf{\_.f64}\left(b, a\right)\right), \mathsf{*.f64}\left(2, \mathsf{sin.f64}\left(\mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(\mathsf{PI.f64}\left(\right), 1\right), angle\right), \mathsf{/.f64}\left(180, angle\right)\right), 180\right), \mathsf{/.f64}\left(32400, angle\right)\right)\right)\right)\right)\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(b, a\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{cos.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(\left(\sqrt{\mathsf{PI}\left(\right)}\right), 180\right), \left(\frac{\sqrt{\mathsf{PI}\left(\right)}}{\frac{1}{angle}}\right)\right)\right), \mathsf{\_.f64}\left(b, a\right)\right), \mathsf{*.f64}\left(2, \mathsf{sin.f64}\left(\mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(\mathsf{PI.f64}\left(\right), 1\right), angle\right), \mathsf{/.f64}\left(180, angle\right)\right), 180\right), \mathsf{/.f64}\left(32400, angle\right)\right)\right)\right)\right)\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(b, a\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{cos.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{PI}\left(\right)\right), 180\right), \left(\frac{\sqrt{\mathsf{PI}\left(\right)}}{\frac{1}{angle}}\right)\right)\right), \mathsf{\_.f64}\left(b, a\right)\right), \mathsf{*.f64}\left(2, \mathsf{sin.f64}\left(\mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(\mathsf{PI.f64}\left(\right), 1\right), angle\right), \mathsf{/.f64}\left(180, angle\right)\right), 180\right), \mathsf{/.f64}\left(32400, angle\right)\right)\right)\right)\right)\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(b, a\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{cos.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{PI.f64}\left(\right)\right), 180\right), \left(\frac{\sqrt{\mathsf{PI}\left(\right)}}{\frac{1}{angle}}\right)\right)\right), \mathsf{\_.f64}\left(b, a\right)\right), \mathsf{*.f64}\left(2, \mathsf{sin.f64}\left(\mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(\mathsf{PI.f64}\left(\right), 1\right), angle\right), \mathsf{/.f64}\left(180, angle\right)\right), 180\right), \mathsf{/.f64}\left(32400, angle\right)\right)\right)\right)\right)\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(b, a\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{cos.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{PI.f64}\left(\right)\right), 180\right), \mathsf{/.f64}\left(\left(\sqrt{\mathsf{PI}\left(\right)}\right), \left(\frac{1}{angle}\right)\right)\right)\right), \mathsf{\_.f64}\left(b, a\right)\right), \mathsf{*.f64}\left(2, \mathsf{sin.f64}\left(\mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(\mathsf{PI.f64}\left(\right), 1\right), angle\right), \mathsf{/.f64}\left(180, angle\right)\right), 180\right), \mathsf{/.f64}\left(32400, angle\right)\right)\right)\right)\right)\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(b, a\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{cos.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{PI.f64}\left(\right)\right), 180\right), \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{PI}\left(\right)\right), \left(\frac{1}{angle}\right)\right)\right)\right), \mathsf{\_.f64}\left(b, a\right)\right), \mathsf{*.f64}\left(2, \mathsf{sin.f64}\left(\mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(\mathsf{PI.f64}\left(\right), 1\right), angle\right), \mathsf{/.f64}\left(180, angle\right)\right), 180\right), \mathsf{/.f64}\left(32400, angle\right)\right)\right)\right)\right)\right) \]

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

          \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(b, a\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{cos.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{PI.f64}\left(\right)\right), 180\right), \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{PI.f64}\left(\right)\right), \left(\frac{1}{angle}\right)\right)\right)\right), \mathsf{\_.f64}\left(b, a\right)\right), \mathsf{*.f64}\left(2, \mathsf{sin.f64}\left(\mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(\mathsf{PI.f64}\left(\right), 1\right), angle\right), \mathsf{/.f64}\left(180, angle\right)\right), 180\right), \mathsf{/.f64}\left(32400, angle\right)\right)\right)\right)\right)\right) \]

      11. /-lowering-/.f64 60.9%

          \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(b, a\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{cos.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{PI.f64}\left(\right)\right), 180\right), \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{PI.f64}\left(\right)\right), \mathsf{/.f64}\left(1, angle\right)\right)\right)\right), \mathsf{\_.f64}\left(b, a\right)\right), \mathsf{*.f64}\left(2, \mathsf{sin.f64}\left(\mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(\mathsf{PI.f64}\left(\right), 1\right), angle\right), \mathsf{/.f64}\left(180, angle\right)\right), 180\right), \mathsf{/.f64}\left(32400, angle\right)\right)\right)\right)\right)\right) \]

   10. Applied egg-rr 60.9%

       \[\leadsto \left(b + a\right) \cdot \left(\left(\cos \color{blue}{\left(\frac{\sqrt{\pi}}{180} \cdot \frac{\sqrt{\pi}}{\frac{1}{angle}}\right)} \cdot \left(b - a\right)\right) \cdot \left(2 \cdot \sin \left(\frac{\left(\left(\pi + 1\right) \cdot angle\right) \cdot \frac{180}{angle} - 180}{\frac{32400}{angle}}\right)\right)\right) \]

   if 7.9999999999999999e156 < a

   1. Initial program 46.6%

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

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

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

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

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

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

   3. Simplified 42.0%

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

      1. associate-*r* N/A

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

      2. *-commutative N/A

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

      3. difference-of-squares N/A

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

      4. associate-*r/ N/A

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

      5. associate-*l* N/A

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

      6. associate-*l* N/A

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

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

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

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

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

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

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

   6. Applied egg-rr 77.1%

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

      1. expm1-log1p-u N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(b, a\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{cos.f64}\left(\mathsf{/.f64}\left(\mathsf{PI.f64}\left(\right), \mathsf{/.f64}\left(180, angle\right)\right)\right), \mathsf{\_.f64}\left(b, a\right)\right), \mathsf{*.f64}\left(2, \mathsf{sin.f64}\left(\left(\frac{\mathsf{expm1}\left(\mathsf{log1p}\left(\mathsf{PI}\left(\right)\right)\right)}{\frac{180}{angle}}\right)\right)\right)\right)\right) \]

      2. expm1-undefine N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(b, a\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{cos.f64}\left(\mathsf{/.f64}\left(\mathsf{PI.f64}\left(\right), \mathsf{/.f64}\left(180, angle\right)\right)\right), \mathsf{\_.f64}\left(b, a\right)\right), \mathsf{*.f64}\left(2, \mathsf{sin.f64}\left(\left(\frac{e^{\mathsf{log1p}\left(\mathsf{PI}\left(\right)\right)} - 1}{\frac{180}{angle}}\right)\right)\right)\right)\right) \]

      3. log1p-undefine N/A

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

      4. +-commutative N/A

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

      5. rem-exp-log N/A

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

      6. sub-div N/A

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

      7. associate-/r/ N/A

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

      8. associate-*l/ N/A

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

      9. frac-sub N/A

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

      10. *-commutative N/A

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

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

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

   8. Applied egg-rr 86.3%

      \[\leadsto \left(b + a\right) \cdot \left(\left(\cos \left(\frac{\pi}{\frac{180}{angle}}\right) \cdot \left(b - a\right)\right) \cdot \left(2 \cdot \sin \color{blue}{\left(\frac{\left(\left(\pi + 1\right) \cdot angle\right) \cdot \frac{180}{angle} - 180}{\frac{32400}{angle}}\right)}\right)\right) \]
   9. Step-by-step derivation

      1. associate-/r/ N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(b, a\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{cos.f64}\left(\left(\frac{\mathsf{PI}\left(\right)}{180} \cdot angle\right)\right), \mathsf{\_.f64}\left(b, a\right)\right), \mathsf{*.f64}\left(2, \mathsf{sin.f64}\left(\mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(\mathsf{PI.f64}\left(\right), 1\right), angle\right), \mathsf{/.f64}\left(180, angle\right)\right), 180\right), \mathsf{/.f64}\left(32400, angle\right)\right)\right)\right)\right)\right) \]

      2. associate-*l/ N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(b, a\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{cos.f64}\left(\left(\frac{\mathsf{PI}\left(\right) \cdot angle}{180}\right)\right), \mathsf{\_.f64}\left(b, a\right)\right), \mathsf{*.f64}\left(2, \mathsf{sin.f64}\left(\mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(\mathsf{PI.f64}\left(\right), 1\right), angle\right), \mathsf{/.f64}\left(180, angle\right)\right), 180\right), \mathsf{/.f64}\left(32400, angle\right)\right)\right)\right)\right)\right) \]

      3. clear-num N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(b, a\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{cos.f64}\left(\left(\frac{1}{\frac{180}{\mathsf{PI}\left(\right) \cdot angle}}\right)\right), \mathsf{\_.f64}\left(b, a\right)\right), \mathsf{*.f64}\left(2, \mathsf{sin.f64}\left(\mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(\mathsf{PI.f64}\left(\right), 1\right), angle\right), \mathsf{/.f64}\left(180, angle\right)\right), 180\right), \mathsf{/.f64}\left(32400, angle\right)\right)\right)\right)\right)\right) \]

      4. rem-exp-log N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(b, a\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{cos.f64}\left(\left(\frac{1}{e^{\log \left(\frac{180}{\mathsf{PI}\left(\right) \cdot angle}\right)}}\right)\right), \mathsf{\_.f64}\left(b, a\right)\right), \mathsf{*.f64}\left(2, \mathsf{sin.f64}\left(\mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(\mathsf{PI.f64}\left(\right), 1\right), angle\right), \mathsf{/.f64}\left(180, angle\right)\right), 180\right), \mathsf{/.f64}\left(32400, angle\right)\right)\right)\right)\right)\right) \]

      5. rec-exp N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(b, a\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{cos.f64}\left(\left(e^{\mathsf{neg}\left(\log \left(\frac{180}{\mathsf{PI}\left(\right) \cdot angle}\right)\right)}\right)\right), \mathsf{\_.f64}\left(b, a\right)\right), \mathsf{*.f64}\left(2, \mathsf{sin.f64}\left(\mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(\mathsf{PI.f64}\left(\right), 1\right), angle\right), \mathsf{/.f64}\left(180, angle\right)\right), 180\right), \mathsf{/.f64}\left(32400, angle\right)\right)\right)\right)\right)\right) \]

      6. associate-/l/ N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(b, a\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{cos.f64}\left(\left(e^{\mathsf{neg}\left(\log \left(\frac{\frac{180}{angle}}{\mathsf{PI}\left(\right)}\right)\right)}\right)\right), \mathsf{\_.f64}\left(b, a\right)\right), \mathsf{*.f64}\left(2, \mathsf{sin.f64}\left(\mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(\mathsf{PI.f64}\left(\right), 1\right), angle\right), \mathsf{/.f64}\left(180, angle\right)\right), 180\right), \mathsf{/.f64}\left(32400, angle\right)\right)\right)\right)\right)\right) \]

      7. neg-log N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(b, a\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{cos.f64}\left(\left(e^{\log \left(\frac{1}{\frac{\frac{180}{angle}}{\mathsf{PI}\left(\right)}}\right)}\right)\right), \mathsf{\_.f64}\left(b, a\right)\right), \mathsf{*.f64}\left(2, \mathsf{sin.f64}\left(\mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(\mathsf{PI.f64}\left(\right), 1\right), angle\right), \mathsf{/.f64}\left(180, angle\right)\right), 180\right), \mathsf{/.f64}\left(32400, angle\right)\right)\right)\right)\right)\right) \]

      8. clear-num N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(b, a\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{cos.f64}\left(\left(e^{\log \left(\frac{\mathsf{PI}\left(\right)}{\frac{180}{angle}}\right)}\right)\right), \mathsf{\_.f64}\left(b, a\right)\right), \mathsf{*.f64}\left(2, \mathsf{sin.f64}\left(\mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(\mathsf{PI.f64}\left(\right), 1\right), angle\right), \mathsf{/.f64}\left(180, angle\right)\right), 180\right), \mathsf{/.f64}\left(32400, angle\right)\right)\right)\right)\right)\right) \]

      9. associate-/r/ N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(b, a\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{cos.f64}\left(\left(e^{\log \left(\frac{\mathsf{PI}\left(\right)}{180} \cdot angle\right)}\right)\right), \mathsf{\_.f64}\left(b, a\right)\right), \mathsf{*.f64}\left(2, \mathsf{sin.f64}\left(\mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(\mathsf{PI.f64}\left(\right), 1\right), angle\right), \mathsf{/.f64}\left(180, angle\right)\right), 180\right), \mathsf{/.f64}\left(32400, angle\right)\right)\right)\right)\right)\right) \]

      10. associate-*l/ N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(b, a\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{cos.f64}\left(\left(e^{\log \left(\frac{\mathsf{PI}\left(\right) \cdot angle}{180}\right)}\right)\right), \mathsf{\_.f64}\left(b, a\right)\right), \mathsf{*.f64}\left(2, \mathsf{sin.f64}\left(\mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(\mathsf{PI.f64}\left(\right), 1\right), angle\right), \mathsf{/.f64}\left(180, angle\right)\right), 180\right), \mathsf{/.f64}\left(32400, angle\right)\right)\right)\right)\right)\right) \]

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

          \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(b, a\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{cos.f64}\left(\mathsf{exp.f64}\left(\log \left(\frac{\mathsf{PI}\left(\right) \cdot angle}{180}\right)\right)\right), \mathsf{\_.f64}\left(b, a\right)\right), \mathsf{*.f64}\left(2, \mathsf{sin.f64}\left(\mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(\mathsf{PI.f64}\left(\right), 1\right), angle\right), \mathsf{/.f64}\left(180, angle\right)\right), 180\right), \mathsf{/.f64}\left(32400, angle\right)\right)\right)\right)\right)\right) \]

      12. associate-*l/ N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(b, a\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{cos.f64}\left(\mathsf{exp.f64}\left(\log \left(\frac{\mathsf{PI}\left(\right)}{180} \cdot angle\right)\right)\right), \mathsf{\_.f64}\left(b, a\right)\right), \mathsf{*.f64}\left(2, \mathsf{sin.f64}\left(\mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(\mathsf{PI.f64}\left(\right), 1\right), angle\right), \mathsf{/.f64}\left(180, angle\right)\right), 180\right), \mathsf{/.f64}\left(32400, angle\right)\right)\right)\right)\right)\right) \]

      13. associate-/r/ N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(b, a\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{cos.f64}\left(\mathsf{exp.f64}\left(\log \left(\frac{\mathsf{PI}\left(\right)}{\frac{180}{angle}}\right)\right)\right), \mathsf{\_.f64}\left(b, a\right)\right), \mathsf{*.f64}\left(2, \mathsf{sin.f64}\left(\mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(\mathsf{PI.f64}\left(\right), 1\right), angle\right), \mathsf{/.f64}\left(180, angle\right)\right), 180\right), \mathsf{/.f64}\left(32400, angle\right)\right)\right)\right)\right)\right) \]

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

          \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(b, a\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{cos.f64}\left(\mathsf{exp.f64}\left(\mathsf{log.f64}\left(\left(\frac{\mathsf{PI}\left(\right)}{\frac{180}{angle}}\right)\right)\right)\right), \mathsf{\_.f64}\left(b, a\right)\right), \mathsf{*.f64}\left(2, \mathsf{sin.f64}\left(\mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(\mathsf{PI.f64}\left(\right), 1\right), angle\right), \mathsf{/.f64}\left(180, angle\right)\right), 180\right), \mathsf{/.f64}\left(32400, angle\right)\right)\right)\right)\right)\right) \]

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

          \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(b, a\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{cos.f64}\left(\mathsf{exp.f64}\left(\mathsf{log.f64}\left(\mathsf{/.f64}\left(\mathsf{PI}\left(\right), \left(\frac{180}{angle}\right)\right)\right)\right)\right), \mathsf{\_.f64}\left(b, a\right)\right), \mathsf{*.f64}\left(2, \mathsf{sin.f64}\left(\mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(\mathsf{PI.f64}\left(\right), 1\right), angle\right), \mathsf{/.f64}\left(180, angle\right)\right), 180\right), \mathsf{/.f64}\left(32400, angle\right)\right)\right)\right)\right)\right) \]

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

          \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(b, a\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{cos.f64}\left(\mathsf{exp.f64}\left(\mathsf{log.f64}\left(\mathsf{/.f64}\left(\mathsf{PI.f64}\left(\right), \left(\frac{180}{angle}\right)\right)\right)\right)\right), \mathsf{\_.f64}\left(b, a\right)\right), \mathsf{*.f64}\left(2, \mathsf{sin.f64}\left(\mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(\mathsf{PI.f64}\left(\right), 1\right), angle\right), \mathsf{/.f64}\left(180, angle\right)\right), 180\right), \mathsf{/.f64}\left(32400, angle\right)\right)\right)\right)\right)\right) \]

      17. /-lowering-/.f64 54.5%

          \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(b, a\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{cos.f64}\left(\mathsf{exp.f64}\left(\mathsf{log.f64}\left(\mathsf{/.f64}\left(\mathsf{PI.f64}\left(\right), \mathsf{/.f64}\left(180, angle\right)\right)\right)\right)\right), \mathsf{\_.f64}\left(b, a\right)\right), \mathsf{*.f64}\left(2, \mathsf{sin.f64}\left(\mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(\mathsf{PI.f64}\left(\right), 1\right), angle\right), \mathsf{/.f64}\left(180, angle\right)\right), 180\right), \mathsf{/.f64}\left(32400, angle\right)\right)\right)\right)\right)\right) \]

   10. Applied egg-rr 54.5%

       \[\leadsto \left(b + a\right) \cdot \left(\left(\cos \color{blue}{\left(e^{\log \left(\frac{\pi}{\frac{180}{angle}}\right)}\right)} \cdot \left(b - a\right)\right) \cdot \left(2 \cdot \sin \left(\frac{\left(\left(\pi + 1\right) \cdot angle\right) \cdot \frac{180}{angle} - 180}{\frac{32400}{angle}}\right)\right)\right) \]
3. Recombined 2 regimes into one program.

4. Final simplification 60.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \leq 8 \cdot 10^{+156}:\\ \;\;\;\;\left(b + a\right) \cdot \left(\left(2 \cdot \sin \left(\frac{\left(angle \cdot \left(\pi + 1\right)\right) \cdot \frac{180}{angle} - 180}{\frac{32400}{angle}}\right)\right) \cdot \left(\left(b - a\right) \cdot \cos \left(\frac{\sqrt{\pi}}{180} \cdot \frac{\sqrt{\pi}}{\frac{1}{angle}}\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(b + a\right) \cdot \left(\left(2 \cdot \sin \left(\frac{\left(angle \cdot \left(\pi + 1\right)\right) \cdot \frac{180}{angle} - 180}{\frac{32400}{angle}}\right)\right) \cdot \left(\left(b - a\right) \cdot \cos \left(e^{\log \left(\frac{\pi}{\frac{180}{angle}}\right)}\right)\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 67.4% accurate, 1.0× speedup

Math

\[\begin{array}{l} a_m = \left|a\right| \\ angle\_m = \left|angle\right| \\ angle\_s = \mathsf{copysign}\left(1, angle\right) \\ angle\_s \cdot \left(\left(b + a\_m\right) \cdot \left(\left(2 \cdot \sin \left(\frac{\left(angle\_m \cdot \left(\pi + 1\right)\right) \cdot \frac{180}{angle\_m} - 180}{\frac{32400}{angle\_m}}\right)\right) \cdot \left(\left(b - a\_m\right) \cdot \cos \left(e^{\log \left(\frac{\pi}{\frac{180}{angle\_m}}\right)}\right)\right)\right)\right) \end{array} \]

FPCore

a_m = (fabs.f64 a)
angle\_m = (fabs.f64 angle)
angle\_s = (copysign.f64 #s(literal 1 binary64) angle)
(FPCore (angle_s a_m b angle_m)
 :precision binary64
 (*
  angle_s
  (*
   (+ b a_m)
   (*
    (*
     2.0
     (sin
      (/
       (- (* (* angle_m (+ PI 1.0)) (/ 180.0 angle_m)) 180.0)
       (/ 32400.0 angle_m))))
    (* (- b a_m) (cos (exp (log (/ PI (/ 180.0 angle_m))))))))))

C

a_m = fabs(a);
angle\_m = fabs(angle);
angle\_s = copysign(1.0, angle);
double code(double angle_s, double a_m, double b, double angle_m) {
	return angle_s * ((b + a_m) * ((2.0 * sin(((((angle_m * (((double) M_PI) + 1.0)) * (180.0 / angle_m)) - 180.0) / (32400.0 / angle_m)))) * ((b - a_m) * cos(exp(log((((double) M_PI) / (180.0 / angle_m))))))));
}

Java

a_m = Math.abs(a);
angle\_m = Math.abs(angle);
angle\_s = Math.copySign(1.0, angle);
public static double code(double angle_s, double a_m, double b, double angle_m) {
	return angle_s * ((b + a_m) * ((2.0 * Math.sin(((((angle_m * (Math.PI + 1.0)) * (180.0 / angle_m)) - 180.0) / (32400.0 / angle_m)))) * ((b - a_m) * Math.cos(Math.exp(Math.log((Math.PI / (180.0 / angle_m))))))));
}

Python

a_m = math.fabs(a)
angle\_m = math.fabs(angle)
angle\_s = math.copysign(1.0, angle)
def code(angle_s, a_m, b, angle_m):
	return angle_s * ((b + a_m) * ((2.0 * math.sin(((((angle_m * (math.pi + 1.0)) * (180.0 / angle_m)) - 180.0) / (32400.0 / angle_m)))) * ((b - a_m) * math.cos(math.exp(math.log((math.pi / (180.0 / angle_m))))))))

Julia

a_m = abs(a)
angle\_m = abs(angle)
angle\_s = copysign(1.0, angle)
function code(angle_s, a_m, b, angle_m)
	return Float64(angle_s * Float64(Float64(b + a_m) * Float64(Float64(2.0 * sin(Float64(Float64(Float64(Float64(angle_m * Float64(pi + 1.0)) * Float64(180.0 / angle_m)) - 180.0) / Float64(32400.0 / angle_m)))) * Float64(Float64(b - a_m) * cos(exp(log(Float64(pi / Float64(180.0 / angle_m)))))))))
end

MATLAB

a_m = abs(a);
angle\_m = abs(angle);
angle\_s = sign(angle) * abs(1.0);
function tmp = code(angle_s, a_m, b, angle_m)
	tmp = angle_s * ((b + a_m) * ((2.0 * sin(((((angle_m * (pi + 1.0)) * (180.0 / angle_m)) - 180.0) / (32400.0 / angle_m)))) * ((b - a_m) * cos(exp(log((pi / (180.0 / angle_m))))))));
end

Wolfram

a_m = N[Abs[a], $MachinePrecision]
angle\_m = N[Abs[angle], $MachinePrecision]
angle\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[angle]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[angle$95$s_, a$95$m_, b_, angle$95$m_] := N[(angle$95$s * N[(N[(b + a$95$m), $MachinePrecision] * N[(N[(2.0 * N[Sin[N[(N[(N[(N[(angle$95$m * N[(Pi + 1.0), $MachinePrecision]), $MachinePrecision] * N[(180.0 / angle$95$m), $MachinePrecision]), $MachinePrecision] - 180.0), $MachinePrecision] / N[(32400.0 / angle$95$m), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[(N[(b - a$95$m), $MachinePrecision] * N[Cos[N[Exp[N[Log[N[(Pi / N[(180.0 / angle$95$m), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]], $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 49.5%

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

   1. associate-*l* N/A

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

   2. *-commutative N/A

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

   3. associate-*l* N/A

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

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

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

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

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

   6. associate-*r/ N/A

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

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

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

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

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

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

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

   10. *-commutative N/A

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

   11. associate-*l* N/A

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

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

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

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

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

3. Simplified 49.3%

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

   1. associate-*r* N/A

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

   2. *-commutative N/A

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

   3. difference-of-squares N/A

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

   4. associate-*r/ N/A

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

   5. associate-*l* N/A

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

   6. associate-*l* N/A

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

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

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

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

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

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

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

6. Applied egg-rr 60.6%

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

   1. expm1-log1p-u N/A

      \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(b, a\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{cos.f64}\left(\mathsf{/.f64}\left(\mathsf{PI.f64}\left(\right), \mathsf{/.f64}\left(180, angle\right)\right)\right), \mathsf{\_.f64}\left(b, a\right)\right), \mathsf{*.f64}\left(2, \mathsf{sin.f64}\left(\left(\frac{\mathsf{expm1}\left(\mathsf{log1p}\left(\mathsf{PI}\left(\right)\right)\right)}{\frac{180}{angle}}\right)\right)\right)\right)\right) \]

   2. expm1-undefine N/A

      \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(b, a\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{cos.f64}\left(\mathsf{/.f64}\left(\mathsf{PI.f64}\left(\right), \mathsf{/.f64}\left(180, angle\right)\right)\right), \mathsf{\_.f64}\left(b, a\right)\right), \mathsf{*.f64}\left(2, \mathsf{sin.f64}\left(\left(\frac{e^{\mathsf{log1p}\left(\mathsf{PI}\left(\right)\right)} - 1}{\frac{180}{angle}}\right)\right)\right)\right)\right) \]

   3. log1p-undefine N/A

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

   4. +-commutative N/A

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

   5. rem-exp-log N/A

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

   6. sub-div N/A

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

   7. associate-/r/ N/A

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

   8. associate-*l/ N/A

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

   9. frac-sub N/A

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

   10. *-commutative N/A

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

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

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

8. Applied egg-rr 62.5%

   \[\leadsto \left(b + a\right) \cdot \left(\left(\cos \left(\frac{\pi}{\frac{180}{angle}}\right) \cdot \left(b - a\right)\right) \cdot \left(2 \cdot \sin \color{blue}{\left(\frac{\left(\left(\pi + 1\right) \cdot angle\right) \cdot \frac{180}{angle} - 180}{\frac{32400}{angle}}\right)}\right)\right) \]
9. Step-by-step derivation

   1. associate-/r/ N/A

      \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(b, a\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{cos.f64}\left(\left(\frac{\mathsf{PI}\left(\right)}{180} \cdot angle\right)\right), \mathsf{\_.f64}\left(b, a\right)\right), \mathsf{*.f64}\left(2, \mathsf{sin.f64}\left(\mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(\mathsf{PI.f64}\left(\right), 1\right), angle\right), \mathsf{/.f64}\left(180, angle\right)\right), 180\right), \mathsf{/.f64}\left(32400, angle\right)\right)\right)\right)\right)\right) \]

   2. associate-*l/ N/A

      \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(b, a\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{cos.f64}\left(\left(\frac{\mathsf{PI}\left(\right) \cdot angle}{180}\right)\right), \mathsf{\_.f64}\left(b, a\right)\right), \mathsf{*.f64}\left(2, \mathsf{sin.f64}\left(\mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(\mathsf{PI.f64}\left(\right), 1\right), angle\right), \mathsf{/.f64}\left(180, angle\right)\right), 180\right), \mathsf{/.f64}\left(32400, angle\right)\right)\right)\right)\right)\right) \]

   3. clear-num N/A

      \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(b, a\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{cos.f64}\left(\left(\frac{1}{\frac{180}{\mathsf{PI}\left(\right) \cdot angle}}\right)\right), \mathsf{\_.f64}\left(b, a\right)\right), \mathsf{*.f64}\left(2, \mathsf{sin.f64}\left(\mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(\mathsf{PI.f64}\left(\right), 1\right), angle\right), \mathsf{/.f64}\left(180, angle\right)\right), 180\right), \mathsf{/.f64}\left(32400, angle\right)\right)\right)\right)\right)\right) \]

   4. rem-exp-log N/A

      \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(b, a\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{cos.f64}\left(\left(\frac{1}{e^{\log \left(\frac{180}{\mathsf{PI}\left(\right) \cdot angle}\right)}}\right)\right), \mathsf{\_.f64}\left(b, a\right)\right), \mathsf{*.f64}\left(2, \mathsf{sin.f64}\left(\mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(\mathsf{PI.f64}\left(\right), 1\right), angle\right), \mathsf{/.f64}\left(180, angle\right)\right), 180\right), \mathsf{/.f64}\left(32400, angle\right)\right)\right)\right)\right)\right) \]

   5. rec-exp N/A

      \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(b, a\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{cos.f64}\left(\left(e^{\mathsf{neg}\left(\log \left(\frac{180}{\mathsf{PI}\left(\right) \cdot angle}\right)\right)}\right)\right), \mathsf{\_.f64}\left(b, a\right)\right), \mathsf{*.f64}\left(2, \mathsf{sin.f64}\left(\mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(\mathsf{PI.f64}\left(\right), 1\right), angle\right), \mathsf{/.f64}\left(180, angle\right)\right), 180\right), \mathsf{/.f64}\left(32400, angle\right)\right)\right)\right)\right)\right) \]

   6. associate-/l/ N/A

      \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(b, a\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{cos.f64}\left(\left(e^{\mathsf{neg}\left(\log \left(\frac{\frac{180}{angle}}{\mathsf{PI}\left(\right)}\right)\right)}\right)\right), \mathsf{\_.f64}\left(b, a\right)\right), \mathsf{*.f64}\left(2, \mathsf{sin.f64}\left(\mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(\mathsf{PI.f64}\left(\right), 1\right), angle\right), \mathsf{/.f64}\left(180, angle\right)\right), 180\right), \mathsf{/.f64}\left(32400, angle\right)\right)\right)\right)\right)\right) \]

   7. neg-log N/A

      \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(b, a\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{cos.f64}\left(\left(e^{\log \left(\frac{1}{\frac{\frac{180}{angle}}{\mathsf{PI}\left(\right)}}\right)}\right)\right), \mathsf{\_.f64}\left(b, a\right)\right), \mathsf{*.f64}\left(2, \mathsf{sin.f64}\left(\mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(\mathsf{PI.f64}\left(\right), 1\right), angle\right), \mathsf{/.f64}\left(180, angle\right)\right), 180\right), \mathsf{/.f64}\left(32400, angle\right)\right)\right)\right)\right)\right) \]

   8. clear-num N/A

      \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(b, a\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{cos.f64}\left(\left(e^{\log \left(\frac{\mathsf{PI}\left(\right)}{\frac{180}{angle}}\right)}\right)\right), \mathsf{\_.f64}\left(b, a\right)\right), \mathsf{*.f64}\left(2, \mathsf{sin.f64}\left(\mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(\mathsf{PI.f64}\left(\right), 1\right), angle\right), \mathsf{/.f64}\left(180, angle\right)\right), 180\right), \mathsf{/.f64}\left(32400, angle\right)\right)\right)\right)\right)\right) \]

   9. associate-/r/ N/A

      \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(b, a\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{cos.f64}\left(\left(e^{\log \left(\frac{\mathsf{PI}\left(\right)}{180} \cdot angle\right)}\right)\right), \mathsf{\_.f64}\left(b, a\right)\right), \mathsf{*.f64}\left(2, \mathsf{sin.f64}\left(\mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(\mathsf{PI.f64}\left(\right), 1\right), angle\right), \mathsf{/.f64}\left(180, angle\right)\right), 180\right), \mathsf{/.f64}\left(32400, angle\right)\right)\right)\right)\right)\right) \]

   10. associate-*l/ N/A

       \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(b, a\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{cos.f64}\left(\left(e^{\log \left(\frac{\mathsf{PI}\left(\right) \cdot angle}{180}\right)}\right)\right), \mathsf{\_.f64}\left(b, a\right)\right), \mathsf{*.f64}\left(2, \mathsf{sin.f64}\left(\mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(\mathsf{PI.f64}\left(\right), 1\right), angle\right), \mathsf{/.f64}\left(180, angle\right)\right), 180\right), \mathsf{/.f64}\left(32400, angle\right)\right)\right)\right)\right)\right) \]

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

       \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(b, a\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{cos.f64}\left(\mathsf{exp.f64}\left(\log \left(\frac{\mathsf{PI}\left(\right) \cdot angle}{180}\right)\right)\right), \mathsf{\_.f64}\left(b, a\right)\right), \mathsf{*.f64}\left(2, \mathsf{sin.f64}\left(\mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(\mathsf{PI.f64}\left(\right), 1\right), angle\right), \mathsf{/.f64}\left(180, angle\right)\right), 180\right), \mathsf{/.f64}\left(32400, angle\right)\right)\right)\right)\right)\right) \]

   12. associate-*l/ N/A

       \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(b, a\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{cos.f64}\left(\mathsf{exp.f64}\left(\log \left(\frac{\mathsf{PI}\left(\right)}{180} \cdot angle\right)\right)\right), \mathsf{\_.f64}\left(b, a\right)\right), \mathsf{*.f64}\left(2, \mathsf{sin.f64}\left(\mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(\mathsf{PI.f64}\left(\right), 1\right), angle\right), \mathsf{/.f64}\left(180, angle\right)\right), 180\right), \mathsf{/.f64}\left(32400, angle\right)\right)\right)\right)\right)\right) \]

   13. associate-/r/ N/A

       \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(b, a\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{cos.f64}\left(\mathsf{exp.f64}\left(\log \left(\frac{\mathsf{PI}\left(\right)}{\frac{180}{angle}}\right)\right)\right), \mathsf{\_.f64}\left(b, a\right)\right), \mathsf{*.f64}\left(2, \mathsf{sin.f64}\left(\mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(\mathsf{PI.f64}\left(\right), 1\right), angle\right), \mathsf{/.f64}\left(180, angle\right)\right), 180\right), \mathsf{/.f64}\left(32400, angle\right)\right)\right)\right)\right)\right) \]

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

       \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(b, a\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{cos.f64}\left(\mathsf{exp.f64}\left(\mathsf{log.f64}\left(\left(\frac{\mathsf{PI}\left(\right)}{\frac{180}{angle}}\right)\right)\right)\right), \mathsf{\_.f64}\left(b, a\right)\right), \mathsf{*.f64}\left(2, \mathsf{sin.f64}\left(\mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(\mathsf{PI.f64}\left(\right), 1\right), angle\right), \mathsf{/.f64}\left(180, angle\right)\right), 180\right), \mathsf{/.f64}\left(32400, angle\right)\right)\right)\right)\right)\right) \]

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

       \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(b, a\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{cos.f64}\left(\mathsf{exp.f64}\left(\mathsf{log.f64}\left(\mathsf{/.f64}\left(\mathsf{PI}\left(\right), \left(\frac{180}{angle}\right)\right)\right)\right)\right), \mathsf{\_.f64}\left(b, a\right)\right), \mathsf{*.f64}\left(2, \mathsf{sin.f64}\left(\mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(\mathsf{PI.f64}\left(\right), 1\right), angle\right), \mathsf{/.f64}\left(180, angle\right)\right), 180\right), \mathsf{/.f64}\left(32400, angle\right)\right)\right)\right)\right)\right) \]

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

       \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(b, a\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{cos.f64}\left(\mathsf{exp.f64}\left(\mathsf{log.f64}\left(\mathsf{/.f64}\left(\mathsf{PI.f64}\left(\right), \left(\frac{180}{angle}\right)\right)\right)\right)\right), \mathsf{\_.f64}\left(b, a\right)\right), \mathsf{*.f64}\left(2, \mathsf{sin.f64}\left(\mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(\mathsf{PI.f64}\left(\right), 1\right), angle\right), \mathsf{/.f64}\left(180, angle\right)\right), 180\right), \mathsf{/.f64}\left(32400, angle\right)\right)\right)\right)\right)\right) \]

   17. /-lowering-/.f64 35.5%

       \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(b, a\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{cos.f64}\left(\mathsf{exp.f64}\left(\mathsf{log.f64}\left(\mathsf{/.f64}\left(\mathsf{PI.f64}\left(\right), \mathsf{/.f64}\left(180, angle\right)\right)\right)\right)\right), \mathsf{\_.f64}\left(b, a\right)\right), \mathsf{*.f64}\left(2, \mathsf{sin.f64}\left(\mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(\mathsf{PI.f64}\left(\right), 1\right), angle\right), \mathsf{/.f64}\left(180, angle\right)\right), 180\right), \mathsf{/.f64}\left(32400, angle\right)\right)\right)\right)\right)\right) \]

10. Applied egg-rr 35.5%

    \[\leadsto \left(b + a\right) \cdot \left(\left(\cos \color{blue}{\left(e^{\log \left(\frac{\pi}{\frac{180}{angle}}\right)}\right)} \cdot \left(b - a\right)\right) \cdot \left(2 \cdot \sin \left(\frac{\left(\left(\pi + 1\right) \cdot angle\right) \cdot \frac{180}{angle} - 180}{\frac{32400}{angle}}\right)\right)\right) \]

11. Final simplification 35.5%

    \[\leadsto \left(b + a\right) \cdot \left(\left(2 \cdot \sin \left(\frac{\left(angle \cdot \left(\pi + 1\right)\right) \cdot \frac{180}{angle} - 180}{\frac{32400}{angle}}\right)\right) \cdot \left(\left(b - a\right) \cdot \cos \left(e^{\log \left(\frac{\pi}{\frac{180}{angle}}\right)}\right)\right)\right) \]
12. Add Preprocessing

Alternative 6: 67.5% accurate, 1.0× speedup

Math

\[\begin{array}{l} a_m = \left|a\right| \\ angle\_m = \left|angle\right| \\ angle\_s = \mathsf{copysign}\left(1, angle\right) \\ angle\_s \cdot \begin{array}{l} \mathbf{if}\;\frac{angle\_m}{180} \leq 5 \cdot 10^{+188}:\\ \;\;\;\;\left(b + a\_m\right) \cdot \left(\left(2 \cdot \sin \left(\frac{\pi}{\frac{180}{angle\_m}}\right)\right) \cdot \left(\left(b - a\_m\right) \cdot \cos \left(e^{0 - \log \left(\frac{180}{\pi \cdot angle\_m}\right)}\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(b + a\_m\right) \cdot \left(\left(b - a\_m\right) \cdot \left(2 \cdot \sin \left(\frac{\left(angle\_m \cdot \left(\pi + 1\right)\right) \cdot \frac{180}{angle\_m} - 180}{\frac{32400}{angle\_m}}\right)\right)\right)\\ \end{array} \end{array} \]

FPCore

a_m = (fabs.f64 a)
angle\_m = (fabs.f64 angle)
angle\_s = (copysign.f64 #s(literal 1 binary64) angle)
(FPCore (angle_s a_m b angle_m)
 :precision binary64
 (*
  angle_s
  (if (<= (/ angle_m 180.0) 5e+188)
    (*
     (+ b a_m)
     (*
      (* 2.0 (sin (/ PI (/ 180.0 angle_m))))
      (* (- b a_m) (cos (exp (- 0.0 (log (/ 180.0 (* PI angle_m)))))))))
    (*
     (+ b a_m)
     (*
      (- b a_m)
      (*
       2.0
       (sin
        (/
         (- (* (* angle_m (+ PI 1.0)) (/ 180.0 angle_m)) 180.0)
         (/ 32400.0 angle_m)))))))))

C

a_m = fabs(a);
angle\_m = fabs(angle);
angle\_s = copysign(1.0, angle);
double code(double angle_s, double a_m, double b, double angle_m) {
	double tmp;
	if ((angle_m / 180.0) <= 5e+188) {
		tmp = (b + a_m) * ((2.0 * sin((((double) M_PI) / (180.0 / angle_m)))) * ((b - a_m) * cos(exp((0.0 - log((180.0 / (((double) M_PI) * angle_m))))))));
	} else {
		tmp = (b + a_m) * ((b - a_m) * (2.0 * sin(((((angle_m * (((double) M_PI) + 1.0)) * (180.0 / angle_m)) - 180.0) / (32400.0 / angle_m)))));
	}
	return angle_s * tmp;
}

Java

a_m = Math.abs(a);
angle\_m = Math.abs(angle);
angle\_s = Math.copySign(1.0, angle);
public static double code(double angle_s, double a_m, double b, double angle_m) {
	double tmp;
	if ((angle_m / 180.0) <= 5e+188) {
		tmp = (b + a_m) * ((2.0 * Math.sin((Math.PI / (180.0 / angle_m)))) * ((b - a_m) * Math.cos(Math.exp((0.0 - Math.log((180.0 / (Math.PI * angle_m))))))));
	} else {
		tmp = (b + a_m) * ((b - a_m) * (2.0 * Math.sin(((((angle_m * (Math.PI + 1.0)) * (180.0 / angle_m)) - 180.0) / (32400.0 / angle_m)))));
	}
	return angle_s * tmp;
}

Python

a_m = math.fabs(a)
angle\_m = math.fabs(angle)
angle\_s = math.copysign(1.0, angle)
def code(angle_s, a_m, b, angle_m):
	tmp = 0
	if (angle_m / 180.0) <= 5e+188:
		tmp = (b + a_m) * ((2.0 * math.sin((math.pi / (180.0 / angle_m)))) * ((b - a_m) * math.cos(math.exp((0.0 - math.log((180.0 / (math.pi * angle_m))))))))
	else:
		tmp = (b + a_m) * ((b - a_m) * (2.0 * math.sin(((((angle_m * (math.pi + 1.0)) * (180.0 / angle_m)) - 180.0) / (32400.0 / angle_m)))))
	return angle_s * tmp

Julia

a_m = abs(a)
angle\_m = abs(angle)
angle\_s = copysign(1.0, angle)
function code(angle_s, a_m, b, angle_m)
	tmp = 0.0
	if (Float64(angle_m / 180.0) <= 5e+188)
		tmp = Float64(Float64(b + a_m) * Float64(Float64(2.0 * sin(Float64(pi / Float64(180.0 / angle_m)))) * Float64(Float64(b - a_m) * cos(exp(Float64(0.0 - log(Float64(180.0 / Float64(pi * angle_m)))))))));
	else
		tmp = Float64(Float64(b + a_m) * Float64(Float64(b - a_m) * Float64(2.0 * sin(Float64(Float64(Float64(Float64(angle_m * Float64(pi + 1.0)) * Float64(180.0 / angle_m)) - 180.0) / Float64(32400.0 / angle_m))))));
	end
	return Float64(angle_s * tmp)
end

MATLAB

a_m = abs(a);
angle\_m = abs(angle);
angle\_s = sign(angle) * abs(1.0);
function tmp_2 = code(angle_s, a_m, b, angle_m)
	tmp = 0.0;
	if ((angle_m / 180.0) <= 5e+188)
		tmp = (b + a_m) * ((2.0 * sin((pi / (180.0 / angle_m)))) * ((b - a_m) * cos(exp((0.0 - log((180.0 / (pi * angle_m))))))));
	else
		tmp = (b + a_m) * ((b - a_m) * (2.0 * sin(((((angle_m * (pi + 1.0)) * (180.0 / angle_m)) - 180.0) / (32400.0 / angle_m)))));
	end
	tmp_2 = angle_s * tmp;
end

Wolfram

a_m = N[Abs[a], $MachinePrecision]
angle\_m = N[Abs[angle], $MachinePrecision]
angle\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[angle]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[angle$95$s_, a$95$m_, b_, angle$95$m_] := N[(angle$95$s * If[LessEqual[N[(angle$95$m / 180.0), $MachinePrecision], 5e+188], N[(N[(b + a$95$m), $MachinePrecision] * N[(N[(2.0 * N[Sin[N[(Pi / N[(180.0 / angle$95$m), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[(N[(b - a$95$m), $MachinePrecision] * N[Cos[N[Exp[N[(0.0 - N[Log[N[(180.0 / N[(Pi * angle$95$m), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(b + a$95$m), $MachinePrecision] * N[(N[(b - a$95$m), $MachinePrecision] * N[(2.0 * N[Sin[N[(N[(N[(N[(angle$95$m * N[(Pi + 1.0), $MachinePrecision]), $MachinePrecision] * N[(180.0 / angle$95$m), $MachinePrecision]), $MachinePrecision] - 180.0), $MachinePrecision] / N[(32400.0 / angle$95$m), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if (/.f64 angle #s(literal 180 binary64)) < 5.0000000000000001e188

   1. Initial program 53.3%

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

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

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

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

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

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

   3. Simplified 53.5%

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

      1. associate-*r* N/A

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

      2. *-commutative N/A

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

      3. difference-of-squares N/A

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

      4. associate-*r/ N/A

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

      5. associate-*l* N/A

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

      6. associate-*l* N/A

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

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

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

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

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

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

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

   6. Applied egg-rr 64.8%

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

      1. associate-/r/ N/A

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

      2. associate-*l/ N/A

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

      3. clear-num N/A

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

      4. inv-pow N/A

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

      5. metadata-eval N/A

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

      6. pow-to-exp N/A

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

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

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

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

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

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

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

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

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

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

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

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

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

      13. metadata-eval 37.2%

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

   8. Applied egg-rr 37.2%

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

   if 5.0000000000000001e188 < (/.f64 angle #s(literal 180 binary64))

   1. Initial program 17.2%

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

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

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

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

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

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

   3. Simplified 13.5%

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

      1. associate-*r* N/A

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

      2. *-commutative N/A

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

      3. difference-of-squares N/A

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

      4. associate-*r/ N/A

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

      5. associate-*l* N/A

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

      6. associate-*l* N/A

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

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

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

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

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

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

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

   6. Applied egg-rr 25.1%

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

      1. expm1-log1p-u N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(b, a\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{cos.f64}\left(\mathsf{/.f64}\left(\mathsf{PI.f64}\left(\right), \mathsf{/.f64}\left(180, angle\right)\right)\right), \mathsf{\_.f64}\left(b, a\right)\right), \mathsf{*.f64}\left(2, \mathsf{sin.f64}\left(\left(\frac{\mathsf{expm1}\left(\mathsf{log1p}\left(\mathsf{PI}\left(\right)\right)\right)}{\frac{180}{angle}}\right)\right)\right)\right)\right) \]

      2. expm1-undefine N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(b, a\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{cos.f64}\left(\mathsf{/.f64}\left(\mathsf{PI.f64}\left(\right), \mathsf{/.f64}\left(180, angle\right)\right)\right), \mathsf{\_.f64}\left(b, a\right)\right), \mathsf{*.f64}\left(2, \mathsf{sin.f64}\left(\left(\frac{e^{\mathsf{log1p}\left(\mathsf{PI}\left(\right)\right)} - 1}{\frac{180}{angle}}\right)\right)\right)\right)\right) \]

      3. log1p-undefine N/A

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

      4. +-commutative N/A

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

      5. rem-exp-log N/A

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

      6. sub-div N/A

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

      7. associate-/r/ N/A

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

      8. associate-*l/ N/A

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

      9. frac-sub N/A

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

      10. *-commutative N/A

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

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

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

   8. Applied egg-rr 23.7%

      \[\leadsto \left(b + a\right) \cdot \left(\left(\cos \left(\frac{\pi}{\frac{180}{angle}}\right) \cdot \left(b - a\right)\right) \cdot \left(2 \cdot \sin \color{blue}{\left(\frac{\left(\left(\pi + 1\right) \cdot angle\right) \cdot \frac{180}{angle} - 180}{\frac{32400}{angle}}\right)}\right)\right) \]

   9. Taylor expanded in angle around 0

      \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(b, a\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(\color{blue}{1}, \mathsf{\_.f64}\left(b, a\right)\right), \mathsf{*.f64}\left(2, \mathsf{sin.f64}\left(\mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(\mathsf{PI.f64}\left(\right), 1\right), angle\right), \mathsf{/.f64}\left(180, angle\right)\right), 180\right), \mathsf{/.f64}\left(32400, angle\right)\right)\right)\right)\right)\right) \]
   10. Step-by-step derivation

   11. Simplified 35.0%

       \[\leadsto \left(b + a\right) \cdot \left(\left(\color{blue}{1} \cdot \left(b - a\right)\right) \cdot \left(2 \cdot \sin \left(\frac{\left(\left(\pi + 1\right) \cdot angle\right) \cdot \frac{180}{angle} - 180}{\frac{32400}{angle}}\right)\right)\right) \]
3. Recombined 2 regimes into one program.

4. Final simplification 37.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{angle}{180} \leq 5 \cdot 10^{+188}:\\ \;\;\;\;\left(b + a\right) \cdot \left(\left(2 \cdot \sin \left(\frac{\pi}{\frac{180}{angle}}\right)\right) \cdot \left(\left(b - a\right) \cdot \cos \left(e^{0 - \log \left(\frac{180}{\pi \cdot angle}\right)}\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(b + a\right) \cdot \left(\left(b - a\right) \cdot \left(2 \cdot \sin \left(\frac{\left(angle \cdot \left(\pi + 1\right)\right) \cdot \frac{180}{angle} - 180}{\frac{32400}{angle}}\right)\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 67.4% accurate, 1.7× speedup

Math

\[\begin{array}{l} a_m = \left|a\right| \\ angle\_m = \left|angle\right| \\ angle\_s = \mathsf{copysign}\left(1, angle\right) \\ \begin{array}{l} t_0 := \frac{\pi}{\frac{180}{angle\_m}}\\ angle\_s \cdot \begin{array}{l} \mathbf{if}\;\frac{angle\_m}{180} \leq 2 \cdot 10^{+165}:\\ \;\;\;\;\left(b + a\_m\right) \cdot \left(\left(\left(b - a\_m\right) \cdot \cos t\_0\right) \cdot \left(2 \cdot \sin \left(\frac{\frac{1}{angle\_m} \cdot \left(\left(\pi + 1\right) \cdot \left(180 \cdot angle\_m\right)\right) - 180}{\frac{32400}{angle\_m}}\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(b + a\_m\right) \cdot \left(\left(2 \cdot \sin t\_0\right) \cdot \left(\left(b - a\_m\right) \cdot \cos \left(\frac{180 \cdot \left(\frac{angle\_m}{180} \cdot \left(-1 + \pi \cdot \pi\right)\right) - angle\_m \cdot \left(-1 + \pi\right)}{180 \cdot \left(-1 + \pi\right)}\right)\right)\right)\\ \end{array} \end{array} \end{array} \]

FPCore

a_m = (fabs.f64 a)
angle\_m = (fabs.f64 angle)
angle\_s = (copysign.f64 #s(literal 1 binary64) angle)
(FPCore (angle_s a_m b angle_m)
 :precision binary64
 (let* ((t_0 (/ PI (/ 180.0 angle_m))))
   (*
    angle_s
    (if (<= (/ angle_m 180.0) 2e+165)
      (*
       (+ b a_m)
       (*
        (* (- b a_m) (cos t_0))
        (*
         2.0
         (sin
          (/
           (- (* (/ 1.0 angle_m) (* (+ PI 1.0) (* 180.0 angle_m))) 180.0)
           (/ 32400.0 angle_m))))))
      (*
       (+ b a_m)
       (*
        (* 2.0 (sin t_0))
        (*
         (- b a_m)
         (cos
          (/
           (-
            (* 180.0 (* (/ angle_m 180.0) (+ -1.0 (* PI PI))))
            (* angle_m (+ -1.0 PI)))
           (* 180.0 (+ -1.0 PI)))))))))))

C

a_m = fabs(a);
angle\_m = fabs(angle);
angle\_s = copysign(1.0, angle);
double code(double angle_s, double a_m, double b, double angle_m) {
	double t_0 = ((double) M_PI) / (180.0 / angle_m);
	double tmp;
	if ((angle_m / 180.0) <= 2e+165) {
		tmp = (b + a_m) * (((b - a_m) * cos(t_0)) * (2.0 * sin(((((1.0 / angle_m) * ((((double) M_PI) + 1.0) * (180.0 * angle_m))) - 180.0) / (32400.0 / angle_m)))));
	} else {
		tmp = (b + a_m) * ((2.0 * sin(t_0)) * ((b - a_m) * cos((((180.0 * ((angle_m / 180.0) * (-1.0 + (((double) M_PI) * ((double) M_PI))))) - (angle_m * (-1.0 + ((double) M_PI)))) / (180.0 * (-1.0 + ((double) M_PI)))))));
	}
	return angle_s * tmp;
}

Java

a_m = Math.abs(a);
angle\_m = Math.abs(angle);
angle\_s = Math.copySign(1.0, angle);
public static double code(double angle_s, double a_m, double b, double angle_m) {
	double t_0 = Math.PI / (180.0 / angle_m);
	double tmp;
	if ((angle_m / 180.0) <= 2e+165) {
		tmp = (b + a_m) * (((b - a_m) * Math.cos(t_0)) * (2.0 * Math.sin(((((1.0 / angle_m) * ((Math.PI + 1.0) * (180.0 * angle_m))) - 180.0) / (32400.0 / angle_m)))));
	} else {
		tmp = (b + a_m) * ((2.0 * Math.sin(t_0)) * ((b - a_m) * Math.cos((((180.0 * ((angle_m / 180.0) * (-1.0 + (Math.PI * Math.PI)))) - (angle_m * (-1.0 + Math.PI))) / (180.0 * (-1.0 + Math.PI))))));
	}
	return angle_s * tmp;
}

Python

a_m = math.fabs(a)
angle\_m = math.fabs(angle)
angle\_s = math.copysign(1.0, angle)
def code(angle_s, a_m, b, angle_m):
	t_0 = math.pi / (180.0 / angle_m)
	tmp = 0
	if (angle_m / 180.0) <= 2e+165:
		tmp = (b + a_m) * (((b - a_m) * math.cos(t_0)) * (2.0 * math.sin(((((1.0 / angle_m) * ((math.pi + 1.0) * (180.0 * angle_m))) - 180.0) / (32400.0 / angle_m)))))
	else:
		tmp = (b + a_m) * ((2.0 * math.sin(t_0)) * ((b - a_m) * math.cos((((180.0 * ((angle_m / 180.0) * (-1.0 + (math.pi * math.pi)))) - (angle_m * (-1.0 + math.pi))) / (180.0 * (-1.0 + math.pi))))))
	return angle_s * tmp

Julia

a_m = abs(a)
angle\_m = abs(angle)
angle\_s = copysign(1.0, angle)
function code(angle_s, a_m, b, angle_m)
	t_0 = Float64(pi / Float64(180.0 / angle_m))
	tmp = 0.0
	if (Float64(angle_m / 180.0) <= 2e+165)
		tmp = Float64(Float64(b + a_m) * Float64(Float64(Float64(b - a_m) * cos(t_0)) * Float64(2.0 * sin(Float64(Float64(Float64(Float64(1.0 / angle_m) * Float64(Float64(pi + 1.0) * Float64(180.0 * angle_m))) - 180.0) / Float64(32400.0 / angle_m))))));
	else
		tmp = Float64(Float64(b + a_m) * Float64(Float64(2.0 * sin(t_0)) * Float64(Float64(b - a_m) * cos(Float64(Float64(Float64(180.0 * Float64(Float64(angle_m / 180.0) * Float64(-1.0 + Float64(pi * pi)))) - Float64(angle_m * Float64(-1.0 + pi))) / Float64(180.0 * Float64(-1.0 + pi)))))));
	end
	return Float64(angle_s * tmp)
end

MATLAB

a_m = abs(a);
angle\_m = abs(angle);
angle\_s = sign(angle) * abs(1.0);
function tmp_2 = code(angle_s, a_m, b, angle_m)
	t_0 = pi / (180.0 / angle_m);
	tmp = 0.0;
	if ((angle_m / 180.0) <= 2e+165)
		tmp = (b + a_m) * (((b - a_m) * cos(t_0)) * (2.0 * sin(((((1.0 / angle_m) * ((pi + 1.0) * (180.0 * angle_m))) - 180.0) / (32400.0 / angle_m)))));
	else
		tmp = (b + a_m) * ((2.0 * sin(t_0)) * ((b - a_m) * cos((((180.0 * ((angle_m / 180.0) * (-1.0 + (pi * pi)))) - (angle_m * (-1.0 + pi))) / (180.0 * (-1.0 + pi))))));
	end
	tmp_2 = angle_s * tmp;
end

Wolfram

a_m = N[Abs[a], $MachinePrecision]
angle\_m = N[Abs[angle], $MachinePrecision]
angle\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[angle]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[angle$95$s_, a$95$m_, b_, angle$95$m_] := Block[{t$95$0 = N[(Pi / N[(180.0 / angle$95$m), $MachinePrecision]), $MachinePrecision]}, N[(angle$95$s * If[LessEqual[N[(angle$95$m / 180.0), $MachinePrecision], 2e+165], N[(N[(b + a$95$m), $MachinePrecision] * N[(N[(N[(b - a$95$m), $MachinePrecision] * N[Cos[t$95$0], $MachinePrecision]), $MachinePrecision] * N[(2.0 * N[Sin[N[(N[(N[(N[(1.0 / angle$95$m), $MachinePrecision] * N[(N[(Pi + 1.0), $MachinePrecision] * N[(180.0 * angle$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - 180.0), $MachinePrecision] / N[(32400.0 / angle$95$m), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(b + a$95$m), $MachinePrecision] * N[(N[(2.0 * N[Sin[t$95$0], $MachinePrecision]), $MachinePrecision] * N[(N[(b - a$95$m), $MachinePrecision] * N[Cos[N[(N[(N[(180.0 * N[(N[(angle$95$m / 180.0), $MachinePrecision] * N[(-1.0 + N[(Pi * Pi), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(angle$95$m * N[(-1.0 + Pi), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(180.0 * N[(-1.0 + Pi), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 angle #s(literal 180 binary64)) < 1.9999999999999998e165

   1. Initial program 54.7%

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

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

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

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

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

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

   3. Simplified 54.9%

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

      1. associate-*r* N/A

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

      2. *-commutative N/A

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

      3. difference-of-squares N/A

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

      4. associate-*r/ N/A

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

      5. associate-*l* N/A

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

      6. associate-*l* N/A

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

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

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

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

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

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

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

   6. Applied egg-rr 65.9%

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

      1. expm1-log1p-u N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(b, a\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{cos.f64}\left(\mathsf{/.f64}\left(\mathsf{PI.f64}\left(\right), \mathsf{/.f64}\left(180, angle\right)\right)\right), \mathsf{\_.f64}\left(b, a\right)\right), \mathsf{*.f64}\left(2, \mathsf{sin.f64}\left(\left(\frac{\mathsf{expm1}\left(\mathsf{log1p}\left(\mathsf{PI}\left(\right)\right)\right)}{\frac{180}{angle}}\right)\right)\right)\right)\right) \]

      2. expm1-undefine N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(b, a\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{cos.f64}\left(\mathsf{/.f64}\left(\mathsf{PI.f64}\left(\right), \mathsf{/.f64}\left(180, angle\right)\right)\right), \mathsf{\_.f64}\left(b, a\right)\right), \mathsf{*.f64}\left(2, \mathsf{sin.f64}\left(\left(\frac{e^{\mathsf{log1p}\left(\mathsf{PI}\left(\right)\right)} - 1}{\frac{180}{angle}}\right)\right)\right)\right)\right) \]

      3. log1p-undefine N/A

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

      4. +-commutative N/A

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

      5. rem-exp-log N/A

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

      6. sub-div N/A

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

      7. associate-/r/ N/A

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

      8. associate-*l/ N/A

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

      9. frac-sub N/A

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

      10. *-commutative N/A

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

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

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

   8. Applied egg-rr 67.8%

      \[\leadsto \left(b + a\right) \cdot \left(\left(\cos \left(\frac{\pi}{\frac{180}{angle}}\right) \cdot \left(b - a\right)\right) \cdot \left(2 \cdot \sin \color{blue}{\left(\frac{\left(\left(\pi + 1\right) \cdot angle\right) \cdot \frac{180}{angle} - 180}{\frac{32400}{angle}}\right)}\right)\right) \]
   9. Step-by-step derivation

      1. associate-*r/ N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(b, a\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{cos.f64}\left(\mathsf{/.f64}\left(\mathsf{PI.f64}\left(\right), \mathsf{/.f64}\left(180, angle\right)\right)\right), \mathsf{\_.f64}\left(b, a\right)\right), \mathsf{*.f64}\left(2, \mathsf{sin.f64}\left(\mathsf{/.f64}\left(\mathsf{\_.f64}\left(\left(\frac{\left(\left(\mathsf{PI}\left(\right) + 1\right) \cdot angle\right) \cdot 180}{angle}\right), 180\right), \mathsf{/.f64}\left(32400, angle\right)\right)\right)\right)\right)\right) \]

      2. div-inv N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(b, a\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{cos.f64}\left(\mathsf{/.f64}\left(\mathsf{PI.f64}\left(\right), \mathsf{/.f64}\left(180, angle\right)\right)\right), \mathsf{\_.f64}\left(b, a\right)\right), \mathsf{*.f64}\left(2, \mathsf{sin.f64}\left(\mathsf{/.f64}\left(\mathsf{\_.f64}\left(\left(\left(\left(\left(\mathsf{PI}\left(\right) + 1\right) \cdot angle\right) \cdot 180\right) \cdot \frac{1}{angle}\right), 180\right), \mathsf{/.f64}\left(32400, angle\right)\right)\right)\right)\right)\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(b, a\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{cos.f64}\left(\mathsf{/.f64}\left(\mathsf{PI.f64}\left(\right), \mathsf{/.f64}\left(180, angle\right)\right)\right), \mathsf{\_.f64}\left(b, a\right)\right), \mathsf{*.f64}\left(2, \mathsf{sin.f64}\left(\mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(\left(\left(\left(\mathsf{PI}\left(\right) + 1\right) \cdot angle\right) \cdot 180\right), \left(\frac{1}{angle}\right)\right), 180\right), \mathsf{/.f64}\left(32400, angle\right)\right)\right)\right)\right)\right) \]

      4. associate-*l* N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(b, a\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{cos.f64}\left(\mathsf{/.f64}\left(\mathsf{PI.f64}\left(\right), \mathsf{/.f64}\left(180, angle\right)\right)\right), \mathsf{\_.f64}\left(b, a\right)\right), \mathsf{*.f64}\left(2, \mathsf{sin.f64}\left(\mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(\left(\left(\mathsf{PI}\left(\right) + 1\right) \cdot \left(angle \cdot 180\right)\right), \left(\frac{1}{angle}\right)\right), 180\right), \mathsf{/.f64}\left(32400, angle\right)\right)\right)\right)\right)\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(b, a\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{cos.f64}\left(\mathsf{/.f64}\left(\mathsf{PI.f64}\left(\right), \mathsf{/.f64}\left(180, angle\right)\right)\right), \mathsf{\_.f64}\left(b, a\right)\right), \mathsf{*.f64}\left(2, \mathsf{sin.f64}\left(\mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\left(\mathsf{PI}\left(\right) + 1\right), \left(angle \cdot 180\right)\right), \left(\frac{1}{angle}\right)\right), 180\right), \mathsf{/.f64}\left(32400, angle\right)\right)\right)\right)\right)\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(b, a\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{cos.f64}\left(\mathsf{/.f64}\left(\mathsf{PI.f64}\left(\right), \mathsf{/.f64}\left(180, angle\right)\right)\right), \mathsf{\_.f64}\left(b, a\right)\right), \mathsf{*.f64}\left(2, \mathsf{sin.f64}\left(\mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(\mathsf{PI}\left(\right), 1\right), \left(angle \cdot 180\right)\right), \left(\frac{1}{angle}\right)\right), 180\right), \mathsf{/.f64}\left(32400, angle\right)\right)\right)\right)\right)\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(b, a\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{cos.f64}\left(\mathsf{/.f64}\left(\mathsf{PI.f64}\left(\right), \mathsf{/.f64}\left(180, angle\right)\right)\right), \mathsf{\_.f64}\left(b, a\right)\right), \mathsf{*.f64}\left(2, \mathsf{sin.f64}\left(\mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(\mathsf{PI.f64}\left(\right), 1\right), \left(angle \cdot 180\right)\right), \left(\frac{1}{angle}\right)\right), 180\right), \mathsf{/.f64}\left(32400, angle\right)\right)\right)\right)\right)\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(b, a\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{cos.f64}\left(\mathsf{/.f64}\left(\mathsf{PI.f64}\left(\right), \mathsf{/.f64}\left(180, angle\right)\right)\right), \mathsf{\_.f64}\left(b, a\right)\right), \mathsf{*.f64}\left(2, \mathsf{sin.f64}\left(\mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(\mathsf{PI.f64}\left(\right), 1\right), \mathsf{*.f64}\left(angle, 180\right)\right), \left(\frac{1}{angle}\right)\right), 180\right), \mathsf{/.f64}\left(32400, angle\right)\right)\right)\right)\right)\right) \]

      9. /-lowering-/.f64 67.3%

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(b, a\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{cos.f64}\left(\mathsf{/.f64}\left(\mathsf{PI.f64}\left(\right), \mathsf{/.f64}\left(180, angle\right)\right)\right), \mathsf{\_.f64}\left(b, a\right)\right), \mathsf{*.f64}\left(2, \mathsf{sin.f64}\left(\mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(\mathsf{PI.f64}\left(\right), 1\right), \mathsf{*.f64}\left(angle, 180\right)\right), \mathsf{/.f64}\left(1, angle\right)\right), 180\right), \mathsf{/.f64}\left(32400, angle\right)\right)\right)\right)\right)\right) \]

   10. Applied egg-rr 67.3%

       \[\leadsto \left(b + a\right) \cdot \left(\left(\cos \left(\frac{\pi}{\frac{180}{angle}}\right) \cdot \left(b - a\right)\right) \cdot \left(2 \cdot \sin \left(\frac{\color{blue}{\left(\left(\pi + 1\right) \cdot \left(angle \cdot 180\right)\right) \cdot \frac{1}{angle}} - 180}{\frac{32400}{angle}}\right)\right)\right) \]

   if 1.9999999999999998e165 < (/.f64 angle #s(literal 180 binary64))

   1. Initial program 14.7%

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

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

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

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

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

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

   3. Simplified 11.6%

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

      1. associate-*r* N/A

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

      2. *-commutative N/A

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

      3. difference-of-squares N/A

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

      4. associate-*r/ N/A

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

      5. associate-*l* N/A

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

      6. associate-*l* N/A

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

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

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

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

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

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

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

   6. Applied egg-rr 24.7%

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

      1. expm1-log1p-u N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(b, a\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{cos.f64}\left(\left(\frac{\mathsf{expm1}\left(\mathsf{log1p}\left(\mathsf{PI}\left(\right)\right)\right)}{\frac{180}{angle}}\right)\right), \mathsf{\_.f64}\left(b, a\right)\right), \mathsf{*.f64}\left(2, \mathsf{sin.f64}\left(\mathsf{/.f64}\left(\mathsf{PI.f64}\left(\right), \mathsf{/.f64}\left(180, angle\right)\right)\right)\right)\right)\right) \]

      2. expm1-undefine N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(b, a\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{cos.f64}\left(\left(\frac{e^{\mathsf{log1p}\left(\mathsf{PI}\left(\right)\right)} - 1}{\frac{180}{angle}}\right)\right), \mathsf{\_.f64}\left(b, a\right)\right), \mathsf{*.f64}\left(2, \mathsf{sin.f64}\left(\mathsf{/.f64}\left(\mathsf{PI.f64}\left(\right), \mathsf{/.f64}\left(180, angle\right)\right)\right)\right)\right)\right) \]

      3. log1p-undefine N/A

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

      4. +-commutative N/A

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

      5. rem-exp-log N/A

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

      6. sub-div N/A

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

      7. clear-num N/A

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

      8. div-inv N/A

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

      9. flip-+ N/A

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

      10. clear-num N/A

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

      11. associate-*l/ N/A

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

      12. frac-sub N/A

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

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

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

   8. Applied egg-rr 42.4%

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

4. Final simplification 64.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{angle}{180} \leq 2 \cdot 10^{+165}:\\ \;\;\;\;\left(b + a\right) \cdot \left(\left(\left(b - a\right) \cdot \cos \left(\frac{\pi}{\frac{180}{angle}}\right)\right) \cdot \left(2 \cdot \sin \left(\frac{\frac{1}{angle} \cdot \left(\left(\pi + 1\right) \cdot \left(180 \cdot angle\right)\right) - 180}{\frac{32400}{angle}}\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(b + a\right) \cdot \left(\left(2 \cdot \sin \left(\frac{\pi}{\frac{180}{angle}}\right)\right) \cdot \left(\left(b - a\right) \cdot \cos \left(\frac{180 \cdot \left(\frac{angle}{180} \cdot \left(-1 + \pi \cdot \pi\right)\right) - angle \cdot \left(-1 + \pi\right)}{180 \cdot \left(-1 + \pi\right)}\right)\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 66.9% accurate, 1.8× speedup

Math

\[\begin{array}{l} a_m = \left|a\right| \\ angle\_m = \left|angle\right| \\ angle\_s = \mathsf{copysign}\left(1, angle\right) \\ \begin{array}{l} t_0 := angle\_m \cdot \left(\pi + 1\right)\\ angle\_s \cdot \begin{array}{l} \mathbf{if}\;\frac{angle\_m}{180} \leq 2 \cdot 10^{+196}:\\ \;\;\;\;\left(b + a\_m\right) \cdot \left(\cos \left(\frac{\pi}{\frac{180}{angle\_m}}\right) \cdot \left(\left(\left(b - a\_m\right) \cdot 2\right) \cdot \sin \left(\frac{\frac{t\_0}{\frac{angle\_m}{180}} + -180}{\frac{32400}{angle\_m}}\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(b + a\_m\right) \cdot \left(\left(b - a\_m\right) \cdot \left(2 \cdot \sin \left(\frac{t\_0 \cdot \frac{180}{angle\_m} - 180}{\frac{32400}{angle\_m}}\right)\right)\right)\\ \end{array} \end{array} \end{array} \]

FPCore

a_m = (fabs.f64 a)
angle\_m = (fabs.f64 angle)
angle\_s = (copysign.f64 #s(literal 1 binary64) angle)
(FPCore (angle_s a_m b angle_m)
 :precision binary64
 (let* ((t_0 (* angle_m (+ PI 1.0))))
   (*
    angle_s
    (if (<= (/ angle_m 180.0) 2e+196)
      (*
       (+ b a_m)
       (*
        (cos (/ PI (/ 180.0 angle_m)))
        (*
         (* (- b a_m) 2.0)
         (sin (/ (+ (/ t_0 (/ angle_m 180.0)) -180.0) (/ 32400.0 angle_m))))))
      (*
       (+ b a_m)
       (*
        (- b a_m)
        (*
         2.0
         (sin
          (/ (- (* t_0 (/ 180.0 angle_m)) 180.0) (/ 32400.0 angle_m))))))))))

C

a_m = fabs(a);
angle\_m = fabs(angle);
angle\_s = copysign(1.0, angle);
double code(double angle_s, double a_m, double b, double angle_m) {
	double t_0 = angle_m * (((double) M_PI) + 1.0);
	double tmp;
	if ((angle_m / 180.0) <= 2e+196) {
		tmp = (b + a_m) * (cos((((double) M_PI) / (180.0 / angle_m))) * (((b - a_m) * 2.0) * sin((((t_0 / (angle_m / 180.0)) + -180.0) / (32400.0 / angle_m)))));
	} else {
		tmp = (b + a_m) * ((b - a_m) * (2.0 * sin((((t_0 * (180.0 / angle_m)) - 180.0) / (32400.0 / angle_m)))));
	}
	return angle_s * tmp;
}

Java

a_m = Math.abs(a);
angle\_m = Math.abs(angle);
angle\_s = Math.copySign(1.0, angle);
public static double code(double angle_s, double a_m, double b, double angle_m) {
	double t_0 = angle_m * (Math.PI + 1.0);
	double tmp;
	if ((angle_m / 180.0) <= 2e+196) {
		tmp = (b + a_m) * (Math.cos((Math.PI / (180.0 / angle_m))) * (((b - a_m) * 2.0) * Math.sin((((t_0 / (angle_m / 180.0)) + -180.0) / (32400.0 / angle_m)))));
	} else {
		tmp = (b + a_m) * ((b - a_m) * (2.0 * Math.sin((((t_0 * (180.0 / angle_m)) - 180.0) / (32400.0 / angle_m)))));
	}
	return angle_s * tmp;
}

Python

a_m = math.fabs(a)
angle\_m = math.fabs(angle)
angle\_s = math.copysign(1.0, angle)
def code(angle_s, a_m, b, angle_m):
	t_0 = angle_m * (math.pi + 1.0)
	tmp = 0
	if (angle_m / 180.0) <= 2e+196:
		tmp = (b + a_m) * (math.cos((math.pi / (180.0 / angle_m))) * (((b - a_m) * 2.0) * math.sin((((t_0 / (angle_m / 180.0)) + -180.0) / (32400.0 / angle_m)))))
	else:
		tmp = (b + a_m) * ((b - a_m) * (2.0 * math.sin((((t_0 * (180.0 / angle_m)) - 180.0) / (32400.0 / angle_m)))))
	return angle_s * tmp

Julia

a_m = abs(a)
angle\_m = abs(angle)
angle\_s = copysign(1.0, angle)
function code(angle_s, a_m, b, angle_m)
	t_0 = Float64(angle_m * Float64(pi + 1.0))
	tmp = 0.0
	if (Float64(angle_m / 180.0) <= 2e+196)
		tmp = Float64(Float64(b + a_m) * Float64(cos(Float64(pi / Float64(180.0 / angle_m))) * Float64(Float64(Float64(b - a_m) * 2.0) * sin(Float64(Float64(Float64(t_0 / Float64(angle_m / 180.0)) + -180.0) / Float64(32400.0 / angle_m))))));
	else
		tmp = Float64(Float64(b + a_m) * Float64(Float64(b - a_m) * Float64(2.0 * sin(Float64(Float64(Float64(t_0 * Float64(180.0 / angle_m)) - 180.0) / Float64(32400.0 / angle_m))))));
	end
	return Float64(angle_s * tmp)
end

MATLAB

a_m = abs(a);
angle\_m = abs(angle);
angle\_s = sign(angle) * abs(1.0);
function tmp_2 = code(angle_s, a_m, b, angle_m)
	t_0 = angle_m * (pi + 1.0);
	tmp = 0.0;
	if ((angle_m / 180.0) <= 2e+196)
		tmp = (b + a_m) * (cos((pi / (180.0 / angle_m))) * (((b - a_m) * 2.0) * sin((((t_0 / (angle_m / 180.0)) + -180.0) / (32400.0 / angle_m)))));
	else
		tmp = (b + a_m) * ((b - a_m) * (2.0 * sin((((t_0 * (180.0 / angle_m)) - 180.0) / (32400.0 / angle_m)))));
	end
	tmp_2 = angle_s * tmp;
end

Wolfram

a_m = N[Abs[a], $MachinePrecision]
angle\_m = N[Abs[angle], $MachinePrecision]
angle\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[angle]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[angle$95$s_, a$95$m_, b_, angle$95$m_] := Block[{t$95$0 = N[(angle$95$m * N[(Pi + 1.0), $MachinePrecision]), $MachinePrecision]}, N[(angle$95$s * If[LessEqual[N[(angle$95$m / 180.0), $MachinePrecision], 2e+196], N[(N[(b + a$95$m), $MachinePrecision] * N[(N[Cos[N[(Pi / N[(180.0 / angle$95$m), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * N[(N[(N[(b - a$95$m), $MachinePrecision] * 2.0), $MachinePrecision] * N[Sin[N[(N[(N[(t$95$0 / N[(angle$95$m / 180.0), $MachinePrecision]), $MachinePrecision] + -180.0), $MachinePrecision] / N[(32400.0 / angle$95$m), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(b + a$95$m), $MachinePrecision] * N[(N[(b - a$95$m), $MachinePrecision] * N[(2.0 * N[Sin[N[(N[(N[(t$95$0 * N[(180.0 / angle$95$m), $MachinePrecision]), $MachinePrecision] - 180.0), $MachinePrecision] / N[(32400.0 / angle$95$m), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 angle #s(literal 180 binary64)) < 1.9999999999999999e196

   1. Initial program 52.9%

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

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

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

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

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

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

   3. Simplified 53.1%

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

      1. associate-*r* N/A

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

      2. *-commutative N/A

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

      3. difference-of-squares N/A

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

      4. associate-*r/ N/A

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

      5. associate-*l* N/A

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

      6. associate-*l* N/A

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

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

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

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

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

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

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

   6. Applied egg-rr 64.3%

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

      1. expm1-log1p-u N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(b, a\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{cos.f64}\left(\mathsf{/.f64}\left(\mathsf{PI.f64}\left(\right), \mathsf{/.f64}\left(180, angle\right)\right)\right), \mathsf{\_.f64}\left(b, a\right)\right), \mathsf{*.f64}\left(2, \mathsf{sin.f64}\left(\left(\frac{\mathsf{expm1}\left(\mathsf{log1p}\left(\mathsf{PI}\left(\right)\right)\right)}{\frac{180}{angle}}\right)\right)\right)\right)\right) \]

      2. expm1-undefine N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(b, a\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{cos.f64}\left(\mathsf{/.f64}\left(\mathsf{PI.f64}\left(\right), \mathsf{/.f64}\left(180, angle\right)\right)\right), \mathsf{\_.f64}\left(b, a\right)\right), \mathsf{*.f64}\left(2, \mathsf{sin.f64}\left(\left(\frac{e^{\mathsf{log1p}\left(\mathsf{PI}\left(\right)\right)} - 1}{\frac{180}{angle}}\right)\right)\right)\right)\right) \]

      3. log1p-undefine N/A

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

      4. +-commutative N/A

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

      5. rem-exp-log N/A

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

      6. sub-div N/A

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

      7. associate-/r/ N/A

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

      8. associate-*l/ N/A

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

      9. frac-sub N/A

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

      10. *-commutative N/A

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

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

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

   8. Applied egg-rr 67.0%

      \[\leadsto \left(b + a\right) \cdot \left(\left(\cos \left(\frac{\pi}{\frac{180}{angle}}\right) \cdot \left(b - a\right)\right) \cdot \left(2 \cdot \sin \color{blue}{\left(\frac{\left(\left(\pi + 1\right) \cdot angle\right) \cdot \frac{180}{angle} - 180}{\frac{32400}{angle}}\right)}\right)\right) \]
   9. Step-by-step derivation

      1. associate-*l* N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(b, a\right), \left(\cos \left(\frac{\mathsf{PI}\left(\right)}{\frac{180}{angle}}\right) \cdot \color{blue}{\left(\left(b - a\right) \cdot \left(2 \cdot \sin \left(\frac{\left(\left(\mathsf{PI}\left(\right) + 1\right) \cdot angle\right) \cdot \frac{180}{angle} - 180}{\frac{32400}{angle}}\right)\right)\right)}\right)\right) \]

      2. *-commutative N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(b, a\right), \left(\left(\left(b - a\right) \cdot \left(2 \cdot \sin \left(\frac{\left(\left(\mathsf{PI}\left(\right) + 1\right) \cdot angle\right) \cdot \frac{180}{angle} - 180}{\frac{32400}{angle}}\right)\right)\right) \cdot \color{blue}{\cos \left(\frac{\mathsf{PI}\left(\right)}{\frac{180}{angle}}\right)}\right)\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(b, a\right), \mathsf{*.f64}\left(\left(\left(b - a\right) \cdot \left(2 \cdot \sin \left(\frac{\left(\left(\mathsf{PI}\left(\right) + 1\right) \cdot angle\right) \cdot \frac{180}{angle} - 180}{\frac{32400}{angle}}\right)\right)\right), \color{blue}{\cos \left(\frac{\mathsf{PI}\left(\right)}{\frac{180}{angle}}\right)}\right)\right) \]

   10. Applied egg-rr 66.8%

       \[\leadsto \left(b + a\right) \cdot \color{blue}{\left(\left(\left(\left(b - a\right) \cdot 2\right) \cdot \sin \left(\frac{\frac{\left(\pi + 1\right) \cdot angle}{\frac{angle}{180}} + -180}{\frac{32400}{angle}}\right)\right) \cdot \cos \left(\frac{\pi}{\frac{180}{angle}}\right)\right)} \]

   if 1.9999999999999999e196 < (/.f64 angle #s(literal 180 binary64))

   1. Initial program 17.9%

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

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

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

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

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

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

   3. Simplified 13.9%

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

      1. associate-*r* N/A

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

      2. *-commutative N/A

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

      3. difference-of-squares N/A

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

      4. associate-*r/ N/A

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

      5. associate-*l* N/A

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

      6. associate-*l* N/A

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

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

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

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

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

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

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

   6. Applied egg-rr 26.5%

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

      1. expm1-log1p-u N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(b, a\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{cos.f64}\left(\mathsf{/.f64}\left(\mathsf{PI.f64}\left(\right), \mathsf{/.f64}\left(180, angle\right)\right)\right), \mathsf{\_.f64}\left(b, a\right)\right), \mathsf{*.f64}\left(2, \mathsf{sin.f64}\left(\left(\frac{\mathsf{expm1}\left(\mathsf{log1p}\left(\mathsf{PI}\left(\right)\right)\right)}{\frac{180}{angle}}\right)\right)\right)\right)\right) \]

      2. expm1-undefine N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(b, a\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{cos.f64}\left(\mathsf{/.f64}\left(\mathsf{PI.f64}\left(\right), \mathsf{/.f64}\left(180, angle\right)\right)\right), \mathsf{\_.f64}\left(b, a\right)\right), \mathsf{*.f64}\left(2, \mathsf{sin.f64}\left(\left(\frac{e^{\mathsf{log1p}\left(\mathsf{PI}\left(\right)\right)} - 1}{\frac{180}{angle}}\right)\right)\right)\right)\right) \]

      3. log1p-undefine N/A

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

      4. +-commutative N/A

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

      5. rem-exp-log N/A

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

      6. sub-div N/A

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

      7. associate-/r/ N/A

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

      8. associate-*l/ N/A

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

      9. frac-sub N/A

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

      10. *-commutative N/A

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

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

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

   8. Applied egg-rr 21.0%

      \[\leadsto \left(b + a\right) \cdot \left(\left(\cos \left(\frac{\pi}{\frac{180}{angle}}\right) \cdot \left(b - a\right)\right) \cdot \left(2 \cdot \sin \color{blue}{\left(\frac{\left(\left(\pi + 1\right) \cdot angle\right) \cdot \frac{180}{angle} - 180}{\frac{32400}{angle}}\right)}\right)\right) \]

   9. Taylor expanded in angle around 0

      \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(b, a\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(\color{blue}{1}, \mathsf{\_.f64}\left(b, a\right)\right), \mathsf{*.f64}\left(2, \mathsf{sin.f64}\left(\mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(\mathsf{PI.f64}\left(\right), 1\right), angle\right), \mathsf{/.f64}\left(180, angle\right)\right), 180\right), \mathsf{/.f64}\left(32400, angle\right)\right)\right)\right)\right)\right) \]
   10. Step-by-step derivation

   11. Simplified 33.7%

       \[\leadsto \left(b + a\right) \cdot \left(\left(\color{blue}{1} \cdot \left(b - a\right)\right) \cdot \left(2 \cdot \sin \left(\frac{\left(\left(\pi + 1\right) \cdot angle\right) \cdot \frac{180}{angle} - 180}{\frac{32400}{angle}}\right)\right)\right) \]
3. Recombined 2 regimes into one program.

4. Final simplification 63.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{angle}{180} \leq 2 \cdot 10^{+196}:\\ \;\;\;\;\left(b + a\right) \cdot \left(\cos \left(\frac{\pi}{\frac{180}{angle}}\right) \cdot \left(\left(\left(b - a\right) \cdot 2\right) \cdot \sin \left(\frac{\frac{angle \cdot \left(\pi + 1\right)}{\frac{angle}{180}} + -180}{\frac{32400}{angle}}\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(b + a\right) \cdot \left(\left(b - a\right) \cdot \left(2 \cdot \sin \left(\frac{\left(angle \cdot \left(\pi + 1\right)\right) \cdot \frac{180}{angle} - 180}{\frac{32400}{angle}}\right)\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 67.6% accurate, 1.8× speedup

Math

\[\begin{array}{l} a_m = \left|a\right| \\ angle\_m = \left|angle\right| \\ angle\_s = \mathsf{copysign}\left(1, angle\right) \\ angle\_s \cdot \begin{array}{l} \mathbf{if}\;\frac{angle\_m}{180} \leq 2 \cdot 10^{+192}:\\ \;\;\;\;\left(b + a\_m\right) \cdot \left(\left(\left(b - a\_m\right) \cdot \cos \left(\left(angle\_m \cdot -0.005555555555555556 + \frac{angle\_m}{180}\right) + \frac{\pi}{\frac{180}{angle\_m}}\right)\right) \cdot \left(2 \cdot \sin \left(0.005555555555555556 \cdot \frac{\pi}{\frac{1}{angle\_m}}\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(b + a\_m\right) \cdot \left(\left(b - a\_m\right) \cdot \left(2 \cdot \sin \left(\frac{\left(angle\_m \cdot \left(\pi + 1\right)\right) \cdot \frac{180}{angle\_m} - 180}{\frac{32400}{angle\_m}}\right)\right)\right)\\ \end{array} \end{array} \]

FPCore

a_m = (fabs.f64 a)
angle\_m = (fabs.f64 angle)
angle\_s = (copysign.f64 #s(literal 1 binary64) angle)
(FPCore (angle_s a_m b angle_m)
 :precision binary64
 (*
  angle_s
  (if (<= (/ angle_m 180.0) 2e+192)
    (*
     (+ b a_m)
     (*
      (*
       (- b a_m)
       (cos
        (+
         (+ (* angle_m -0.005555555555555556) (/ angle_m 180.0))
         (/ PI (/ 180.0 angle_m)))))
      (* 2.0 (sin (* 0.005555555555555556 (/ PI (/ 1.0 angle_m)))))))
    (*
     (+ b a_m)
     (*
      (- b a_m)
      (*
       2.0
       (sin
        (/
         (- (* (* angle_m (+ PI 1.0)) (/ 180.0 angle_m)) 180.0)
         (/ 32400.0 angle_m)))))))))

C

a_m = fabs(a);
angle\_m = fabs(angle);
angle\_s = copysign(1.0, angle);
double code(double angle_s, double a_m, double b, double angle_m) {
	double tmp;
	if ((angle_m / 180.0) <= 2e+192) {
		tmp = (b + a_m) * (((b - a_m) * cos((((angle_m * -0.005555555555555556) + (angle_m / 180.0)) + (((double) M_PI) / (180.0 / angle_m))))) * (2.0 * sin((0.005555555555555556 * (((double) M_PI) / (1.0 / angle_m))))));
	} else {
		tmp = (b + a_m) * ((b - a_m) * (2.0 * sin(((((angle_m * (((double) M_PI) + 1.0)) * (180.0 / angle_m)) - 180.0) / (32400.0 / angle_m)))));
	}
	return angle_s * tmp;
}

Java

a_m = Math.abs(a);
angle\_m = Math.abs(angle);
angle\_s = Math.copySign(1.0, angle);
public static double code(double angle_s, double a_m, double b, double angle_m) {
	double tmp;
	if ((angle_m / 180.0) <= 2e+192) {
		tmp = (b + a_m) * (((b - a_m) * Math.cos((((angle_m * -0.005555555555555556) + (angle_m / 180.0)) + (Math.PI / (180.0 / angle_m))))) * (2.0 * Math.sin((0.005555555555555556 * (Math.PI / (1.0 / angle_m))))));
	} else {
		tmp = (b + a_m) * ((b - a_m) * (2.0 * Math.sin(((((angle_m * (Math.PI + 1.0)) * (180.0 / angle_m)) - 180.0) / (32400.0 / angle_m)))));
	}
	return angle_s * tmp;
}

Python

a_m = math.fabs(a)
angle\_m = math.fabs(angle)
angle\_s = math.copysign(1.0, angle)
def code(angle_s, a_m, b, angle_m):
	tmp = 0
	if (angle_m / 180.0) <= 2e+192:
		tmp = (b + a_m) * (((b - a_m) * math.cos((((angle_m * -0.005555555555555556) + (angle_m / 180.0)) + (math.pi / (180.0 / angle_m))))) * (2.0 * math.sin((0.005555555555555556 * (math.pi / (1.0 / angle_m))))))
	else:
		tmp = (b + a_m) * ((b - a_m) * (2.0 * math.sin(((((angle_m * (math.pi + 1.0)) * (180.0 / angle_m)) - 180.0) / (32400.0 / angle_m)))))
	return angle_s * tmp

Julia

a_m = abs(a)
angle\_m = abs(angle)
angle\_s = copysign(1.0, angle)
function code(angle_s, a_m, b, angle_m)
	tmp = 0.0
	if (Float64(angle_m / 180.0) <= 2e+192)
		tmp = Float64(Float64(b + a_m) * Float64(Float64(Float64(b - a_m) * cos(Float64(Float64(Float64(angle_m * -0.005555555555555556) + Float64(angle_m / 180.0)) + Float64(pi / Float64(180.0 / angle_m))))) * Float64(2.0 * sin(Float64(0.005555555555555556 * Float64(pi / Float64(1.0 / angle_m)))))));
	else
		tmp = Float64(Float64(b + a_m) * Float64(Float64(b - a_m) * Float64(2.0 * sin(Float64(Float64(Float64(Float64(angle_m * Float64(pi + 1.0)) * Float64(180.0 / angle_m)) - 180.0) / Float64(32400.0 / angle_m))))));
	end
	return Float64(angle_s * tmp)
end

MATLAB

a_m = abs(a);
angle\_m = abs(angle);
angle\_s = sign(angle) * abs(1.0);
function tmp_2 = code(angle_s, a_m, b, angle_m)
	tmp = 0.0;
	if ((angle_m / 180.0) <= 2e+192)
		tmp = (b + a_m) * (((b - a_m) * cos((((angle_m * -0.005555555555555556) + (angle_m / 180.0)) + (pi / (180.0 / angle_m))))) * (2.0 * sin((0.005555555555555556 * (pi / (1.0 / angle_m))))));
	else
		tmp = (b + a_m) * ((b - a_m) * (2.0 * sin(((((angle_m * (pi + 1.0)) * (180.0 / angle_m)) - 180.0) / (32400.0 / angle_m)))));
	end
	tmp_2 = angle_s * tmp;
end

Wolfram

a_m = N[Abs[a], $MachinePrecision]
angle\_m = N[Abs[angle], $MachinePrecision]
angle\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[angle]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[angle$95$s_, a$95$m_, b_, angle$95$m_] := N[(angle$95$s * If[LessEqual[N[(angle$95$m / 180.0), $MachinePrecision], 2e+192], N[(N[(b + a$95$m), $MachinePrecision] * N[(N[(N[(b - a$95$m), $MachinePrecision] * N[Cos[N[(N[(N[(angle$95$m * -0.005555555555555556), $MachinePrecision] + N[(angle$95$m / 180.0), $MachinePrecision]), $MachinePrecision] + N[(Pi / N[(180.0 / angle$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[(2.0 * N[Sin[N[(0.005555555555555556 * N[(Pi / N[(1.0 / angle$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(b + a$95$m), $MachinePrecision] * N[(N[(b - a$95$m), $MachinePrecision] * N[(2.0 * N[Sin[N[(N[(N[(N[(angle$95$m * N[(Pi + 1.0), $MachinePrecision]), $MachinePrecision] * N[(180.0 / angle$95$m), $MachinePrecision]), $MachinePrecision] - 180.0), $MachinePrecision] / N[(32400.0 / angle$95$m), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if (/.f64 angle #s(literal 180 binary64)) < 2.00000000000000008e192

   1. Initial program 53.3%

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

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

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

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

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

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

   3. Simplified 53.5%

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

      1. associate-*r* N/A

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

      2. *-commutative N/A

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

      3. difference-of-squares N/A

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

      4. associate-*r/ N/A

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

      5. associate-*l* N/A

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

      6. associate-*l* N/A

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

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

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

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

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

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

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

   6. Applied egg-rr 64.8%

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

      1. expm1-log1p-u N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(b, a\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{cos.f64}\left(\left(\frac{\mathsf{expm1}\left(\mathsf{log1p}\left(\mathsf{PI}\left(\right)\right)\right)}{\frac{180}{angle}}\right)\right), \mathsf{\_.f64}\left(b, a\right)\right), \mathsf{*.f64}\left(2, \mathsf{sin.f64}\left(\mathsf{/.f64}\left(\mathsf{PI.f64}\left(\right), \mathsf{/.f64}\left(180, angle\right)\right)\right)\right)\right)\right) \]

      2. expm1-undefine N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(b, a\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{cos.f64}\left(\left(\frac{e^{\mathsf{log1p}\left(\mathsf{PI}\left(\right)\right)} - 1}{\frac{180}{angle}}\right)\right), \mathsf{\_.f64}\left(b, a\right)\right), \mathsf{*.f64}\left(2, \mathsf{sin.f64}\left(\mathsf{/.f64}\left(\mathsf{PI.f64}\left(\right), \mathsf{/.f64}\left(180, angle\right)\right)\right)\right)\right)\right) \]

      3. log1p-undefine N/A

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

      4. +-commutative N/A

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

      5. rem-exp-log N/A

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

      6. sub-div N/A

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

      7. clear-num N/A

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

      8. associate-/r/ N/A

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

      9. div-inv N/A

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

      10. metadata-eval N/A

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

      11. prod-diff N/A

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

      12. metadata-eval N/A

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

      13. associate-/r/ N/A

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

      14. clear-num N/A

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

      15. fmm-def N/A

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

      16. associate-/r/ N/A

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

   8. Applied egg-rr 64.6%

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

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

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

      2. div-inv N/A

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

      3. times-frac N/A

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

      4. metadata-eval N/A

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

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

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

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

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

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

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

      8. /-lowering-/.f64 66.7%

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

   10. Applied egg-rr 66.7%

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

   if 2.00000000000000008e192 < (/.f64 angle #s(literal 180 binary64))

   1. Initial program 17.2%

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

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

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

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

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

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

   3. Simplified 13.5%

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

      1. associate-*r* N/A

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

      2. *-commutative N/A

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

      3. difference-of-squares N/A

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

      4. associate-*r/ N/A

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

      5. associate-*l* N/A

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

      6. associate-*l* N/A

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

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

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

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

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

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

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

   6. Applied egg-rr 25.1%

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

      1. expm1-log1p-u N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(b, a\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{cos.f64}\left(\mathsf{/.f64}\left(\mathsf{PI.f64}\left(\right), \mathsf{/.f64}\left(180, angle\right)\right)\right), \mathsf{\_.f64}\left(b, a\right)\right), \mathsf{*.f64}\left(2, \mathsf{sin.f64}\left(\left(\frac{\mathsf{expm1}\left(\mathsf{log1p}\left(\mathsf{PI}\left(\right)\right)\right)}{\frac{180}{angle}}\right)\right)\right)\right)\right) \]

      2. expm1-undefine N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(b, a\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{cos.f64}\left(\mathsf{/.f64}\left(\mathsf{PI.f64}\left(\right), \mathsf{/.f64}\left(180, angle\right)\right)\right), \mathsf{\_.f64}\left(b, a\right)\right), \mathsf{*.f64}\left(2, \mathsf{sin.f64}\left(\left(\frac{e^{\mathsf{log1p}\left(\mathsf{PI}\left(\right)\right)} - 1}{\frac{180}{angle}}\right)\right)\right)\right)\right) \]

      3. log1p-undefine N/A

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

      4. +-commutative N/A

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

      5. rem-exp-log N/A

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

      6. sub-div N/A

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

      7. associate-/r/ N/A

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

      8. associate-*l/ N/A

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

      9. frac-sub N/A

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

      10. *-commutative N/A

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

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

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

   8. Applied egg-rr 23.7%

      \[\leadsto \left(b + a\right) \cdot \left(\left(\cos \left(\frac{\pi}{\frac{180}{angle}}\right) \cdot \left(b - a\right)\right) \cdot \left(2 \cdot \sin \color{blue}{\left(\frac{\left(\left(\pi + 1\right) \cdot angle\right) \cdot \frac{180}{angle} - 180}{\frac{32400}{angle}}\right)}\right)\right) \]

   9. Taylor expanded in angle around 0

      \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(b, a\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(\color{blue}{1}, \mathsf{\_.f64}\left(b, a\right)\right), \mathsf{*.f64}\left(2, \mathsf{sin.f64}\left(\mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(\mathsf{PI.f64}\left(\right), 1\right), angle\right), \mathsf{/.f64}\left(180, angle\right)\right), 180\right), \mathsf{/.f64}\left(32400, angle\right)\right)\right)\right)\right)\right) \]
   10. Step-by-step derivation

   11. Simplified 35.0%

       \[\leadsto \left(b + a\right) \cdot \left(\left(\color{blue}{1} \cdot \left(b - a\right)\right) \cdot \left(2 \cdot \sin \left(\frac{\left(\left(\pi + 1\right) \cdot angle\right) \cdot \frac{180}{angle} - 180}{\frac{32400}{angle}}\right)\right)\right) \]
3. Recombined 2 regimes into one program.

4. Final simplification 63.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{angle}{180} \leq 2 \cdot 10^{+192}:\\ \;\;\;\;\left(b + a\right) \cdot \left(\left(\left(b - a\right) \cdot \cos \left(\left(angle \cdot -0.005555555555555556 + \frac{angle}{180}\right) + \frac{\pi}{\frac{180}{angle}}\right)\right) \cdot \left(2 \cdot \sin \left(0.005555555555555556 \cdot \frac{\pi}{\frac{1}{angle}}\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(b + a\right) \cdot \left(\left(b - a\right) \cdot \left(2 \cdot \sin \left(\frac{\left(angle \cdot \left(\pi + 1\right)\right) \cdot \frac{180}{angle} - 180}{\frac{32400}{angle}}\right)\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 67.7% accurate, 1.8× speedup

Math

\[\begin{array}{l} a_m = \left|a\right| \\ angle\_m = \left|angle\right| \\ angle\_s = \mathsf{copysign}\left(1, angle\right) \\ angle\_s \cdot \begin{array}{l} \mathbf{if}\;\frac{angle\_m}{180} \leq 2 \cdot 10^{+192}:\\ \;\;\;\;\left(b + a\_m\right) \cdot \left(\left(\left(b - a\_m\right) \cdot \cos \left(\frac{\pi}{\frac{180}{angle\_m}}\right)\right) \cdot \left(2 \cdot \sin \left(\left(\pi \cdot angle\_m\right) \cdot 0.005555555555555556\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(b + a\_m\right) \cdot \left(\left(b - a\_m\right) \cdot \left(2 \cdot \sin \left(\frac{\left(angle\_m \cdot \left(\pi + 1\right)\right) \cdot \frac{180}{angle\_m} - 180}{\frac{32400}{angle\_m}}\right)\right)\right)\\ \end{array} \end{array} \]

FPCore

a_m = (fabs.f64 a)
angle\_m = (fabs.f64 angle)
angle\_s = (copysign.f64 #s(literal 1 binary64) angle)
(FPCore (angle_s a_m b angle_m)
 :precision binary64
 (*
  angle_s
  (if (<= (/ angle_m 180.0) 2e+192)
    (*
     (+ b a_m)
     (*
      (* (- b a_m) (cos (/ PI (/ 180.0 angle_m))))
      (* 2.0 (sin (* (* PI angle_m) 0.005555555555555556)))))
    (*
     (+ b a_m)
     (*
      (- b a_m)
      (*
       2.0
       (sin
        (/
         (- (* (* angle_m (+ PI 1.0)) (/ 180.0 angle_m)) 180.0)
         (/ 32400.0 angle_m)))))))))

C

a_m = fabs(a);
angle\_m = fabs(angle);
angle\_s = copysign(1.0, angle);
double code(double angle_s, double a_m, double b, double angle_m) {
	double tmp;
	if ((angle_m / 180.0) <= 2e+192) {
		tmp = (b + a_m) * (((b - a_m) * cos((((double) M_PI) / (180.0 / angle_m)))) * (2.0 * sin(((((double) M_PI) * angle_m) * 0.005555555555555556))));
	} else {
		tmp = (b + a_m) * ((b - a_m) * (2.0 * sin(((((angle_m * (((double) M_PI) + 1.0)) * (180.0 / angle_m)) - 180.0) / (32400.0 / angle_m)))));
	}
	return angle_s * tmp;
}

Java

a_m = Math.abs(a);
angle\_m = Math.abs(angle);
angle\_s = Math.copySign(1.0, angle);
public static double code(double angle_s, double a_m, double b, double angle_m) {
	double tmp;
	if ((angle_m / 180.0) <= 2e+192) {
		tmp = (b + a_m) * (((b - a_m) * Math.cos((Math.PI / (180.0 / angle_m)))) * (2.0 * Math.sin(((Math.PI * angle_m) * 0.005555555555555556))));
	} else {
		tmp = (b + a_m) * ((b - a_m) * (2.0 * Math.sin(((((angle_m * (Math.PI + 1.0)) * (180.0 / angle_m)) - 180.0) / (32400.0 / angle_m)))));
	}
	return angle_s * tmp;
}

Python

a_m = math.fabs(a)
angle\_m = math.fabs(angle)
angle\_s = math.copysign(1.0, angle)
def code(angle_s, a_m, b, angle_m):
	tmp = 0
	if (angle_m / 180.0) <= 2e+192:
		tmp = (b + a_m) * (((b - a_m) * math.cos((math.pi / (180.0 / angle_m)))) * (2.0 * math.sin(((math.pi * angle_m) * 0.005555555555555556))))
	else:
		tmp = (b + a_m) * ((b - a_m) * (2.0 * math.sin(((((angle_m * (math.pi + 1.0)) * (180.0 / angle_m)) - 180.0) / (32400.0 / angle_m)))))
	return angle_s * tmp

Julia

a_m = abs(a)
angle\_m = abs(angle)
angle\_s = copysign(1.0, angle)
function code(angle_s, a_m, b, angle_m)
	tmp = 0.0
	if (Float64(angle_m / 180.0) <= 2e+192)
		tmp = Float64(Float64(b + a_m) * Float64(Float64(Float64(b - a_m) * cos(Float64(pi / Float64(180.0 / angle_m)))) * Float64(2.0 * sin(Float64(Float64(pi * angle_m) * 0.005555555555555556)))));
	else
		tmp = Float64(Float64(b + a_m) * Float64(Float64(b - a_m) * Float64(2.0 * sin(Float64(Float64(Float64(Float64(angle_m * Float64(pi + 1.0)) * Float64(180.0 / angle_m)) - 180.0) / Float64(32400.0 / angle_m))))));
	end
	return Float64(angle_s * tmp)
end

MATLAB

a_m = abs(a);
angle\_m = abs(angle);
angle\_s = sign(angle) * abs(1.0);
function tmp_2 = code(angle_s, a_m, b, angle_m)
	tmp = 0.0;
	if ((angle_m / 180.0) <= 2e+192)
		tmp = (b + a_m) * (((b - a_m) * cos((pi / (180.0 / angle_m)))) * (2.0 * sin(((pi * angle_m) * 0.005555555555555556))));
	else
		tmp = (b + a_m) * ((b - a_m) * (2.0 * sin(((((angle_m * (pi + 1.0)) * (180.0 / angle_m)) - 180.0) / (32400.0 / angle_m)))));
	end
	tmp_2 = angle_s * tmp;
end

Wolfram

a_m = N[Abs[a], $MachinePrecision]
angle\_m = N[Abs[angle], $MachinePrecision]
angle\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[angle]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[angle$95$s_, a$95$m_, b_, angle$95$m_] := N[(angle$95$s * If[LessEqual[N[(angle$95$m / 180.0), $MachinePrecision], 2e+192], N[(N[(b + a$95$m), $MachinePrecision] * N[(N[(N[(b - a$95$m), $MachinePrecision] * N[Cos[N[(Pi / N[(180.0 / angle$95$m), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[(2.0 * N[Sin[N[(N[(Pi * angle$95$m), $MachinePrecision] * 0.005555555555555556), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(b + a$95$m), $MachinePrecision] * N[(N[(b - a$95$m), $MachinePrecision] * N[(2.0 * N[Sin[N[(N[(N[(N[(angle$95$m * N[(Pi + 1.0), $MachinePrecision]), $MachinePrecision] * N[(180.0 / angle$95$m), $MachinePrecision]), $MachinePrecision] - 180.0), $MachinePrecision] / N[(32400.0 / angle$95$m), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if (/.f64 angle #s(literal 180 binary64)) < 2.00000000000000008e192

   1. Initial program 53.3%

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

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

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

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

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

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

   3. Simplified 53.5%

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

      1. associate-*r* N/A

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

      2. *-commutative N/A

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

      3. difference-of-squares N/A

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

      4. associate-*r/ N/A

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

      5. associate-*l* N/A

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

      6. associate-*l* N/A

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

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

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

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

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

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

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

   6. Applied egg-rr 64.8%

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

      1. associate-/r/ N/A

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

      2. associate-*l/ N/A

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

      3. div-inv N/A

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

      4. metadata-eval N/A

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

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

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

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

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

      7. PI-lowering-PI.f64 66.1%

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

   8. Applied egg-rr 66.1%

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

   if 2.00000000000000008e192 < (/.f64 angle #s(literal 180 binary64))

   1. Initial program 17.2%

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

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

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

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

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

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

   3. Simplified 13.5%

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

      1. associate-*r* N/A

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

      2. *-commutative N/A

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

      3. difference-of-squares N/A

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

      4. associate-*r/ N/A

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

      5. associate-*l* N/A

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

      6. associate-*l* N/A

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

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

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

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

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

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

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

   6. Applied egg-rr 25.1%

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

      1. expm1-log1p-u N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(b, a\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{cos.f64}\left(\mathsf{/.f64}\left(\mathsf{PI.f64}\left(\right), \mathsf{/.f64}\left(180, angle\right)\right)\right), \mathsf{\_.f64}\left(b, a\right)\right), \mathsf{*.f64}\left(2, \mathsf{sin.f64}\left(\left(\frac{\mathsf{expm1}\left(\mathsf{log1p}\left(\mathsf{PI}\left(\right)\right)\right)}{\frac{180}{angle}}\right)\right)\right)\right)\right) \]

      2. expm1-undefine N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(b, a\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{cos.f64}\left(\mathsf{/.f64}\left(\mathsf{PI.f64}\left(\right), \mathsf{/.f64}\left(180, angle\right)\right)\right), \mathsf{\_.f64}\left(b, a\right)\right), \mathsf{*.f64}\left(2, \mathsf{sin.f64}\left(\left(\frac{e^{\mathsf{log1p}\left(\mathsf{PI}\left(\right)\right)} - 1}{\frac{180}{angle}}\right)\right)\right)\right)\right) \]

      3. log1p-undefine N/A

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

      4. +-commutative N/A

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

      5. rem-exp-log N/A

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

      6. sub-div N/A

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

      7. associate-/r/ N/A

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

      8. associate-*l/ N/A

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

      9. frac-sub N/A

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

      10. *-commutative N/A

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

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

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

   8. Applied egg-rr 23.7%

      \[\leadsto \left(b + a\right) \cdot \left(\left(\cos \left(\frac{\pi}{\frac{180}{angle}}\right) \cdot \left(b - a\right)\right) \cdot \left(2 \cdot \sin \color{blue}{\left(\frac{\left(\left(\pi + 1\right) \cdot angle\right) \cdot \frac{180}{angle} - 180}{\frac{32400}{angle}}\right)}\right)\right) \]

   9. Taylor expanded in angle around 0

      \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(b, a\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(\color{blue}{1}, \mathsf{\_.f64}\left(b, a\right)\right), \mathsf{*.f64}\left(2, \mathsf{sin.f64}\left(\mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(\mathsf{PI.f64}\left(\right), 1\right), angle\right), \mathsf{/.f64}\left(180, angle\right)\right), 180\right), \mathsf{/.f64}\left(32400, angle\right)\right)\right)\right)\right)\right) \]
   10. Step-by-step derivation

   11. Simplified 35.0%

       \[\leadsto \left(b + a\right) \cdot \left(\left(\color{blue}{1} \cdot \left(b - a\right)\right) \cdot \left(2 \cdot \sin \left(\frac{\left(\left(\pi + 1\right) \cdot angle\right) \cdot \frac{180}{angle} - 180}{\frac{32400}{angle}}\right)\right)\right) \]
3. Recombined 2 regimes into one program.

4. Final simplification 62.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{angle}{180} \leq 2 \cdot 10^{+192}:\\ \;\;\;\;\left(b + a\right) \cdot \left(\left(\left(b - a\right) \cdot \cos \left(\frac{\pi}{\frac{180}{angle}}\right)\right) \cdot \left(2 \cdot \sin \left(\left(\pi \cdot angle\right) \cdot 0.005555555555555556\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(b + a\right) \cdot \left(\left(b - a\right) \cdot \left(2 \cdot \sin \left(\frac{\left(angle \cdot \left(\pi + 1\right)\right) \cdot \frac{180}{angle} - 180}{\frac{32400}{angle}}\right)\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 67.6% accurate, 3.2× speedup

Math

\[\begin{array}{l} a_m = \left|a\right| \\ angle\_m = \left|angle\right| \\ angle\_s = \mathsf{copysign}\left(1, angle\right) \\ angle\_s \cdot \begin{array}{l} \mathbf{if}\;\frac{angle\_m}{180} \leq 10^{+139}:\\ \;\;\;\;\left(b - a\_m\right) \cdot \left(\left(b + a\_m\right) \cdot \sin \left(\left(\pi \cdot angle\_m\right) \cdot 0.011111111111111112\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(b + a\_m\right) \cdot \left(\left(b - a\_m\right) \cdot \left(2 \cdot \sin \left(\frac{\left(angle\_m \cdot \left(\pi + 1\right)\right) \cdot \frac{180}{angle\_m} - 180}{\frac{32400}{angle\_m}}\right)\right)\right)\\ \end{array} \end{array} \]

FPCore

a_m = (fabs.f64 a)
angle\_m = (fabs.f64 angle)
angle\_s = (copysign.f64 #s(literal 1 binary64) angle)
(FPCore (angle_s a_m b angle_m)
 :precision binary64
 (*
  angle_s
  (if (<= (/ angle_m 180.0) 1e+139)
    (* (- b a_m) (* (+ b a_m) (sin (* (* PI angle_m) 0.011111111111111112))))
    (*
     (+ b a_m)
     (*
      (- b a_m)
      (*
       2.0
       (sin
        (/
         (- (* (* angle_m (+ PI 1.0)) (/ 180.0 angle_m)) 180.0)
         (/ 32400.0 angle_m)))))))))

C

a_m = fabs(a);
angle\_m = fabs(angle);
angle\_s = copysign(1.0, angle);
double code(double angle_s, double a_m, double b, double angle_m) {
	double tmp;
	if ((angle_m / 180.0) <= 1e+139) {
		tmp = (b - a_m) * ((b + a_m) * sin(((((double) M_PI) * angle_m) * 0.011111111111111112)));
	} else {
		tmp = (b + a_m) * ((b - a_m) * (2.0 * sin(((((angle_m * (((double) M_PI) + 1.0)) * (180.0 / angle_m)) - 180.0) / (32400.0 / angle_m)))));
	}
	return angle_s * tmp;
}

Java

a_m = Math.abs(a);
angle\_m = Math.abs(angle);
angle\_s = Math.copySign(1.0, angle);
public static double code(double angle_s, double a_m, double b, double angle_m) {
	double tmp;
	if ((angle_m / 180.0) <= 1e+139) {
		tmp = (b - a_m) * ((b + a_m) * Math.sin(((Math.PI * angle_m) * 0.011111111111111112)));
	} else {
		tmp = (b + a_m) * ((b - a_m) * (2.0 * Math.sin(((((angle_m * (Math.PI + 1.0)) * (180.0 / angle_m)) - 180.0) / (32400.0 / angle_m)))));
	}
	return angle_s * tmp;
}

Python

a_m = math.fabs(a)
angle\_m = math.fabs(angle)
angle\_s = math.copysign(1.0, angle)
def code(angle_s, a_m, b, angle_m):
	tmp = 0
	if (angle_m / 180.0) <= 1e+139:
		tmp = (b - a_m) * ((b + a_m) * math.sin(((math.pi * angle_m) * 0.011111111111111112)))
	else:
		tmp = (b + a_m) * ((b - a_m) * (2.0 * math.sin(((((angle_m * (math.pi + 1.0)) * (180.0 / angle_m)) - 180.0) / (32400.0 / angle_m)))))
	return angle_s * tmp

Julia

a_m = abs(a)
angle\_m = abs(angle)
angle\_s = copysign(1.0, angle)
function code(angle_s, a_m, b, angle_m)
	tmp = 0.0
	if (Float64(angle_m / 180.0) <= 1e+139)
		tmp = Float64(Float64(b - a_m) * Float64(Float64(b + a_m) * sin(Float64(Float64(pi * angle_m) * 0.011111111111111112))));
	else
		tmp = Float64(Float64(b + a_m) * Float64(Float64(b - a_m) * Float64(2.0 * sin(Float64(Float64(Float64(Float64(angle_m * Float64(pi + 1.0)) * Float64(180.0 / angle_m)) - 180.0) / Float64(32400.0 / angle_m))))));
	end
	return Float64(angle_s * tmp)
end

MATLAB

a_m = abs(a);
angle\_m = abs(angle);
angle\_s = sign(angle) * abs(1.0);
function tmp_2 = code(angle_s, a_m, b, angle_m)
	tmp = 0.0;
	if ((angle_m / 180.0) <= 1e+139)
		tmp = (b - a_m) * ((b + a_m) * sin(((pi * angle_m) * 0.011111111111111112)));
	else
		tmp = (b + a_m) * ((b - a_m) * (2.0 * sin(((((angle_m * (pi + 1.0)) * (180.0 / angle_m)) - 180.0) / (32400.0 / angle_m)))));
	end
	tmp_2 = angle_s * tmp;
end

Wolfram

a_m = N[Abs[a], $MachinePrecision]
angle\_m = N[Abs[angle], $MachinePrecision]
angle\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[angle]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[angle$95$s_, a$95$m_, b_, angle$95$m_] := N[(angle$95$s * If[LessEqual[N[(angle$95$m / 180.0), $MachinePrecision], 1e+139], N[(N[(b - a$95$m), $MachinePrecision] * N[(N[(b + a$95$m), $MachinePrecision] * N[Sin[N[(N[(Pi * angle$95$m), $MachinePrecision] * 0.011111111111111112), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(b + a$95$m), $MachinePrecision] * N[(N[(b - a$95$m), $MachinePrecision] * N[(2.0 * N[Sin[N[(N[(N[(N[(angle$95$m * N[(Pi + 1.0), $MachinePrecision]), $MachinePrecision] * N[(180.0 / angle$95$m), $MachinePrecision]), $MachinePrecision] - 180.0), $MachinePrecision] / N[(32400.0 / angle$95$m), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if (/.f64 angle #s(literal 180 binary64)) < 1.00000000000000003e139

   1. Initial program 55.3%

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

      1. associate-*l* 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. associate-*l* N/A

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

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

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

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

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

   3. Simplified 55.7%

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

      1. associate-*r* N/A

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

      2. *-commutative N/A

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

      3. associate-*r/ N/A

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

      4. associate-*l* N/A

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

      5. *-commutative N/A

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

      6. difference-of-squares N/A

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

      7. *-commutative N/A

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

      8. associate-*l* N/A

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

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

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

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

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

   6. Applied egg-rr 68.0%

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

   if 1.00000000000000003e139 < (/.f64 angle #s(literal 180 binary64))

   1. Initial program 16.0%

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

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

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

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

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

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

   3. Simplified 12.8%

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

      1. associate-*r* N/A

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

      2. *-commutative N/A

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

      3. difference-of-squares N/A

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

      4. associate-*r/ N/A

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

      5. associate-*l* N/A

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

      6. associate-*l* N/A

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

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

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

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

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

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

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

   6. Applied egg-rr 24.7%

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

      1. expm1-log1p-u N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(b, a\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{cos.f64}\left(\mathsf{/.f64}\left(\mathsf{PI.f64}\left(\right), \mathsf{/.f64}\left(180, angle\right)\right)\right), \mathsf{\_.f64}\left(b, a\right)\right), \mathsf{*.f64}\left(2, \mathsf{sin.f64}\left(\left(\frac{\mathsf{expm1}\left(\mathsf{log1p}\left(\mathsf{PI}\left(\right)\right)\right)}{\frac{180}{angle}}\right)\right)\right)\right)\right) \]

      2. expm1-undefine N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(b, a\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{cos.f64}\left(\mathsf{/.f64}\left(\mathsf{PI.f64}\left(\right), \mathsf{/.f64}\left(180, angle\right)\right)\right), \mathsf{\_.f64}\left(b, a\right)\right), \mathsf{*.f64}\left(2, \mathsf{sin.f64}\left(\left(\frac{e^{\mathsf{log1p}\left(\mathsf{PI}\left(\right)\right)} - 1}{\frac{180}{angle}}\right)\right)\right)\right)\right) \]

      3. log1p-undefine N/A

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

      4. +-commutative N/A

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

      5. rem-exp-log N/A

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

      6. sub-div N/A

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

      7. associate-/r/ N/A

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

      8. associate-*l/ N/A

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

      9. frac-sub N/A

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

      10. *-commutative N/A

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

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

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

   8. Applied egg-rr 28.9%

      \[\leadsto \left(b + a\right) \cdot \left(\left(\cos \left(\frac{\pi}{\frac{180}{angle}}\right) \cdot \left(b - a\right)\right) \cdot \left(2 \cdot \sin \color{blue}{\left(\frac{\left(\left(\pi + 1\right) \cdot angle\right) \cdot \frac{180}{angle} - 180}{\frac{32400}{angle}}\right)}\right)\right) \]

   9. Taylor expanded in angle around 0

      \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(b, a\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(\color{blue}{1}, \mathsf{\_.f64}\left(b, a\right)\right), \mathsf{*.f64}\left(2, \mathsf{sin.f64}\left(\mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(\mathsf{PI.f64}\left(\right), 1\right), angle\right), \mathsf{/.f64}\left(180, angle\right)\right), 180\right), \mathsf{/.f64}\left(32400, angle\right)\right)\right)\right)\right)\right) \]
   10. Step-by-step derivation

   11. Simplified 34.6%

       \[\leadsto \left(b + a\right) \cdot \left(\left(\color{blue}{1} \cdot \left(b - a\right)\right) \cdot \left(2 \cdot \sin \left(\frac{\left(\left(\pi + 1\right) \cdot angle\right) \cdot \frac{180}{angle} - 180}{\frac{32400}{angle}}\right)\right)\right) \]
3. Recombined 2 regimes into one program.

4. Final simplification 63.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{angle}{180} \leq 10^{+139}:\\ \;\;\;\;\left(b - a\right) \cdot \left(\left(b + a\right) \cdot \sin \left(\left(\pi \cdot angle\right) \cdot 0.011111111111111112\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(b + a\right) \cdot \left(\left(b - a\right) \cdot \left(2 \cdot \sin \left(\frac{\left(angle \cdot \left(\pi + 1\right)\right) \cdot \frac{180}{angle} - 180}{\frac{32400}{angle}}\right)\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 12: 67.9% accurate, 3.5× speedup

Math

\[\begin{array}{l} a_m = \left|a\right| \\ angle\_m = \left|angle\right| \\ angle\_s = \mathsf{copysign}\left(1, angle\right) \\ angle\_s \cdot \begin{array}{l} \mathbf{if}\;angle\_m \leq 1.95 \cdot 10^{+141}:\\ \;\;\;\;\left(b - a\_m\right) \cdot \left(\left(b + a\_m\right) \cdot \sin \left(\left(\pi \cdot angle\_m\right) \cdot 0.011111111111111112\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(b + a\_m\right) \cdot \left(\left(b - a\_m\right) \cdot \left(2 \cdot \sin \left(\frac{\pi}{\frac{180}{angle\_m}}\right)\right)\right)\\ \end{array} \end{array} \]

FPCore

a_m = (fabs.f64 a)
angle\_m = (fabs.f64 angle)
angle\_s = (copysign.f64 #s(literal 1 binary64) angle)
(FPCore (angle_s a_m b angle_m)
 :precision binary64
 (*
  angle_s
  (if (<= angle_m 1.95e+141)
    (* (- b a_m) (* (+ b a_m) (sin (* (* PI angle_m) 0.011111111111111112))))
    (* (+ b a_m) (* (- b a_m) (* 2.0 (sin (/ PI (/ 180.0 angle_m)))))))))

C

a_m = fabs(a);
angle\_m = fabs(angle);
angle\_s = copysign(1.0, angle);
double code(double angle_s, double a_m, double b, double angle_m) {
	double tmp;
	if (angle_m <= 1.95e+141) {
		tmp = (b - a_m) * ((b + a_m) * sin(((((double) M_PI) * angle_m) * 0.011111111111111112)));
	} else {
		tmp = (b + a_m) * ((b - a_m) * (2.0 * sin((((double) M_PI) / (180.0 / angle_m)))));
	}
	return angle_s * tmp;
}

Java

a_m = Math.abs(a);
angle\_m = Math.abs(angle);
angle\_s = Math.copySign(1.0, angle);
public static double code(double angle_s, double a_m, double b, double angle_m) {
	double tmp;
	if (angle_m <= 1.95e+141) {
		tmp = (b - a_m) * ((b + a_m) * Math.sin(((Math.PI * angle_m) * 0.011111111111111112)));
	} else {
		tmp = (b + a_m) * ((b - a_m) * (2.0 * Math.sin((Math.PI / (180.0 / angle_m)))));
	}
	return angle_s * tmp;
}

Python

a_m = math.fabs(a)
angle\_m = math.fabs(angle)
angle\_s = math.copysign(1.0, angle)
def code(angle_s, a_m, b, angle_m):
	tmp = 0
	if angle_m <= 1.95e+141:
		tmp = (b - a_m) * ((b + a_m) * math.sin(((math.pi * angle_m) * 0.011111111111111112)))
	else:
		tmp = (b + a_m) * ((b - a_m) * (2.0 * math.sin((math.pi / (180.0 / angle_m)))))
	return angle_s * tmp

Julia

a_m = abs(a)
angle\_m = abs(angle)
angle\_s = copysign(1.0, angle)
function code(angle_s, a_m, b, angle_m)
	tmp = 0.0
	if (angle_m <= 1.95e+141)
		tmp = Float64(Float64(b - a_m) * Float64(Float64(b + a_m) * sin(Float64(Float64(pi * angle_m) * 0.011111111111111112))));
	else
		tmp = Float64(Float64(b + a_m) * Float64(Float64(b - a_m) * Float64(2.0 * sin(Float64(pi / Float64(180.0 / angle_m))))));
	end
	return Float64(angle_s * tmp)
end

MATLAB

a_m = abs(a);
angle\_m = abs(angle);
angle\_s = sign(angle) * abs(1.0);
function tmp_2 = code(angle_s, a_m, b, angle_m)
	tmp = 0.0;
	if (angle_m <= 1.95e+141)
		tmp = (b - a_m) * ((b + a_m) * sin(((pi * angle_m) * 0.011111111111111112)));
	else
		tmp = (b + a_m) * ((b - a_m) * (2.0 * sin((pi / (180.0 / angle_m)))));
	end
	tmp_2 = angle_s * tmp;
end

Wolfram

a_m = N[Abs[a], $MachinePrecision]
angle\_m = N[Abs[angle], $MachinePrecision]
angle\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[angle]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[angle$95$s_, a$95$m_, b_, angle$95$m_] := N[(angle$95$s * If[LessEqual[angle$95$m, 1.95e+141], N[(N[(b - a$95$m), $MachinePrecision] * N[(N[(b + a$95$m), $MachinePrecision] * N[Sin[N[(N[(Pi * angle$95$m), $MachinePrecision] * 0.011111111111111112), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(b + a$95$m), $MachinePrecision] * N[(N[(b - a$95$m), $MachinePrecision] * N[(2.0 * N[Sin[N[(Pi / N[(180.0 / angle$95$m), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if angle < 1.94999999999999996e141

   1. Initial program 55.3%

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

      1. associate-*l* 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. associate-*l* N/A

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

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

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

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

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

   3. Simplified 55.7%

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

      1. associate-*r* N/A

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

      2. *-commutative N/A

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

      3. associate-*r/ N/A

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

      4. associate-*l* N/A

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

      5. *-commutative N/A

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

      6. difference-of-squares N/A

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

      7. *-commutative N/A

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

      8. associate-*l* N/A

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

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

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

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

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

   6. Applied egg-rr 68.0%

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

   if 1.94999999999999996e141 < angle

   1. Initial program 16.0%

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

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

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

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

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

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

   3. Simplified 12.8%

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

      1. associate-*r* N/A

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

      2. *-commutative N/A

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

      3. difference-of-squares N/A

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

      4. associate-*r/ N/A

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

      5. associate-*l* N/A

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

      6. associate-*l* N/A

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

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

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

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

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

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

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

   6. Applied egg-rr 24.7%

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

   7. Taylor expanded in angle around 0

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

      1. --lowering--.f64 31.1%

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

   9. Simplified 31.1%

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

Alternative 13: 67.8% accurate, 3.6× speedup

Math

\[\begin{array}{l} a_m = \left|a\right| \\ angle\_m = \left|angle\right| \\ angle\_s = \mathsf{copysign}\left(1, angle\right) \\ angle\_s \cdot \begin{array}{l} \mathbf{if}\;b \leq 3.1 \cdot 10^{+131}:\\ \;\;\;\;\left(b - a\_m\right) \cdot \left(\left(b + a\_m\right) \cdot \sin \left(\left(\pi \cdot angle\_m\right) \cdot 0.011111111111111112\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(b + a\_m\right) \cdot \left(6.17283950617284 \cdot 10^{-5} \cdot \left(\left(angle\_m \cdot \left(b - a\_m\right)\right) \cdot \left(180 + \left(-180 + 180 \cdot \pi\right)\right)\right)\right)\\ \end{array} \end{array} \]

FPCore

a_m = (fabs.f64 a)
angle\_m = (fabs.f64 angle)
angle\_s = (copysign.f64 #s(literal 1 binary64) angle)
(FPCore (angle_s a_m b angle_m)
 :precision binary64
 (*
  angle_s
  (if (<= b 3.1e+131)
    (* (- b a_m) (* (+ b a_m) (sin (* (* PI angle_m) 0.011111111111111112))))
    (*
     (+ b a_m)
     (*
      6.17283950617284e-5
      (* (* angle_m (- b a_m)) (+ 180.0 (+ -180.0 (* 180.0 PI)))))))))

C

a_m = fabs(a);
angle\_m = fabs(angle);
angle\_s = copysign(1.0, angle);
double code(double angle_s, double a_m, double b, double angle_m) {
	double tmp;
	if (b <= 3.1e+131) {
		tmp = (b - a_m) * ((b + a_m) * sin(((((double) M_PI) * angle_m) * 0.011111111111111112)));
	} else {
		tmp = (b + a_m) * (6.17283950617284e-5 * ((angle_m * (b - a_m)) * (180.0 + (-180.0 + (180.0 * ((double) M_PI))))));
	}
	return angle_s * tmp;
}

Java

a_m = Math.abs(a);
angle\_m = Math.abs(angle);
angle\_s = Math.copySign(1.0, angle);
public static double code(double angle_s, double a_m, double b, double angle_m) {
	double tmp;
	if (b <= 3.1e+131) {
		tmp = (b - a_m) * ((b + a_m) * Math.sin(((Math.PI * angle_m) * 0.011111111111111112)));
	} else {
		tmp = (b + a_m) * (6.17283950617284e-5 * ((angle_m * (b - a_m)) * (180.0 + (-180.0 + (180.0 * Math.PI)))));
	}
	return angle_s * tmp;
}

Python

a_m = math.fabs(a)
angle\_m = math.fabs(angle)
angle\_s = math.copysign(1.0, angle)
def code(angle_s, a_m, b, angle_m):
	tmp = 0
	if b <= 3.1e+131:
		tmp = (b - a_m) * ((b + a_m) * math.sin(((math.pi * angle_m) * 0.011111111111111112)))
	else:
		tmp = (b + a_m) * (6.17283950617284e-5 * ((angle_m * (b - a_m)) * (180.0 + (-180.0 + (180.0 * math.pi)))))
	return angle_s * tmp

Julia

a_m = abs(a)
angle\_m = abs(angle)
angle\_s = copysign(1.0, angle)
function code(angle_s, a_m, b, angle_m)
	tmp = 0.0
	if (b <= 3.1e+131)
		tmp = Float64(Float64(b - a_m) * Float64(Float64(b + a_m) * sin(Float64(Float64(pi * angle_m) * 0.011111111111111112))));
	else
		tmp = Float64(Float64(b + a_m) * Float64(6.17283950617284e-5 * Float64(Float64(angle_m * Float64(b - a_m)) * Float64(180.0 + Float64(-180.0 + Float64(180.0 * pi))))));
	end
	return Float64(angle_s * tmp)
end

MATLAB

a_m = abs(a);
angle\_m = abs(angle);
angle\_s = sign(angle) * abs(1.0);
function tmp_2 = code(angle_s, a_m, b, angle_m)
	tmp = 0.0;
	if (b <= 3.1e+131)
		tmp = (b - a_m) * ((b + a_m) * sin(((pi * angle_m) * 0.011111111111111112)));
	else
		tmp = (b + a_m) * (6.17283950617284e-5 * ((angle_m * (b - a_m)) * (180.0 + (-180.0 + (180.0 * pi)))));
	end
	tmp_2 = angle_s * tmp;
end

Wolfram

a_m = N[Abs[a], $MachinePrecision]
angle\_m = N[Abs[angle], $MachinePrecision]
angle\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[angle]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[angle$95$s_, a$95$m_, b_, angle$95$m_] := N[(angle$95$s * If[LessEqual[b, 3.1e+131], N[(N[(b - a$95$m), $MachinePrecision] * N[(N[(b + a$95$m), $MachinePrecision] * N[Sin[N[(N[(Pi * angle$95$m), $MachinePrecision] * 0.011111111111111112), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(b + a$95$m), $MachinePrecision] * N[(6.17283950617284e-5 * N[(N[(angle$95$m * N[(b - a$95$m), $MachinePrecision]), $MachinePrecision] * N[(180.0 + N[(-180.0 + N[(180.0 * Pi), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if b < 3.10000000000000016e131

   1. Initial program 51.5%

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

      1. associate-*l* N/A

         \[\leadsto \left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \color{blue}{\left(\sin \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right) \cdot \cos \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right)\right)} \]

      2. *-commutative N/A

         \[\leadsto \left(\sin \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right) \cdot \cos \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right)\right) \cdot \color{blue}{\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right)} \]

      3. associate-*l* N/A

         \[\leadsto \sin \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right) \cdot \color{blue}{\left(\cos \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right) \cdot \left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right)\right)} \]

      4. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\sin \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right), \color{blue}{\left(\cos \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right) \cdot \left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right)\right)}\right) \]

      5. sin-lowering-sin.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sin.f64}\left(\left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right)\right), \left(\color{blue}{\cos \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right)} \cdot \left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right)\right)\right) \]

      6. associate-*r/ N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sin.f64}\left(\left(\frac{\mathsf{PI}\left(\right) \cdot angle}{180}\right)\right), \left(\cos \color{blue}{\left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right)} \cdot \left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right)\right)\right) \]

      7. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sin.f64}\left(\mathsf{/.f64}\left(\left(\mathsf{PI}\left(\right) \cdot angle\right), 180\right)\right), \left(\cos \color{blue}{\left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right)} \cdot \left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right)\right)\right) \]

      8. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sin.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{PI}\left(\right), angle\right), 180\right)\right), \left(\cos \left(\color{blue}{\mathsf{PI}\left(\right)} \cdot \frac{angle}{180}\right) \cdot \left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right)\right)\right) \]

      9. PI-lowering-PI.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sin.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{PI.f64}\left(\right), angle\right), 180\right)\right), \left(\cos \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right) \cdot \left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right)\right)\right) \]

      10. *-commutative N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{sin.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{PI.f64}\left(\right), angle\right), 180\right)\right), \left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \color{blue}{\cos \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right)}\right)\right) \]

      11. associate-*l* N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{sin.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{PI.f64}\left(\right), angle\right), 180\right)\right), \left(2 \cdot \color{blue}{\left(\left({b}^{2} - {a}^{2}\right) \cdot \cos \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right)\right)}\right)\right) \]

      12. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{sin.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{PI.f64}\left(\right), angle\right), 180\right)\right), \mathsf{*.f64}\left(2, \color{blue}{\left(\left({b}^{2} - {a}^{2}\right) \cdot \cos \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right)\right)}\right)\right) \]

      13. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{sin.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{PI.f64}\left(\right), angle\right), 180\right)\right), \mathsf{*.f64}\left(2, \mathsf{*.f64}\left(\left({b}^{2} - {a}^{2}\right), \color{blue}{\cos \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right)}\right)\right)\right) \]

   3. Simplified 51.2%

      \[\leadsto \color{blue}{\sin \left(\frac{\pi \cdot angle}{180}\right) \cdot \left(2 \cdot \left(\left(b \cdot b - a \cdot a\right) \cdot \cos \left(\frac{\pi \cdot angle}{180}\right)\right)\right)} \]
   4. Add Preprocessing
   5. Step-by-step derivation

      1. associate-*r* N/A

         \[\leadsto \left(\sin \left(\frac{\mathsf{PI}\left(\right) \cdot angle}{180}\right) \cdot 2\right) \cdot \color{blue}{\left(\left(b \cdot b - a \cdot a\right) \cdot \cos \left(\frac{\mathsf{PI}\left(\right) \cdot angle}{180}\right)\right)} \]

      2. *-commutative N/A

         \[\leadsto \left(\left(b \cdot b - a \cdot a\right) \cdot \cos \left(\frac{\mathsf{PI}\left(\right) \cdot angle}{180}\right)\right) \cdot \color{blue}{\left(\sin \left(\frac{\mathsf{PI}\left(\right) \cdot angle}{180}\right) \cdot 2\right)} \]

      3. associate-*r/ N/A

         \[\leadsto \left(\left(b \cdot b - a \cdot a\right) \cdot \cos \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right)\right) \cdot \left(\sin \left(\frac{\mathsf{PI}\left(\right) \cdot angle}{180}\right) \cdot 2\right) \]

      4. associate-*l* N/A

         \[\leadsto \left(b \cdot b - a \cdot a\right) \cdot \color{blue}{\left(\cos \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right) \cdot \left(\sin \left(\frac{\mathsf{PI}\left(\right) \cdot angle}{180}\right) \cdot 2\right)\right)} \]

      5. *-commutative N/A

         \[\leadsto \left(b \cdot b - a \cdot a\right) \cdot \left(\left(\sin \left(\frac{\mathsf{PI}\left(\right) \cdot angle}{180}\right) \cdot 2\right) \cdot \color{blue}{\cos \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right)}\right) \]

      6. difference-of-squares N/A

         \[\leadsto \left(\left(b + a\right) \cdot \left(b - a\right)\right) \cdot \left(\color{blue}{\left(\sin \left(\frac{\mathsf{PI}\left(\right) \cdot angle}{180}\right) \cdot 2\right)} \cdot \cos \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right)\right) \]

      7. *-commutative N/A

         \[\leadsto \left(\left(b - a\right) \cdot \left(b + a\right)\right) \cdot \left(\color{blue}{\left(\sin \left(\frac{\mathsf{PI}\left(\right) \cdot angle}{180}\right) \cdot 2\right)} \cdot \cos \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right)\right) \]

      8. associate-*l* N/A

         \[\leadsto \left(b - a\right) \cdot \color{blue}{\left(\left(b + a\right) \cdot \left(\left(\sin \left(\frac{\mathsf{PI}\left(\right) \cdot angle}{180}\right) \cdot 2\right) \cdot \cos \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right)\right)\right)} \]

      9. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\left(b - a\right), \color{blue}{\left(\left(b + a\right) \cdot \left(\left(\sin \left(\frac{\mathsf{PI}\left(\right) \cdot angle}{180}\right) \cdot 2\right) \cdot \cos \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right)\right)\right)}\right) \]

      10. --lowering--.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{\_.f64}\left(b, a\right), \left(\color{blue}{\left(b + a\right)} \cdot \left(\left(\sin \left(\frac{\mathsf{PI}\left(\right) \cdot angle}{180}\right) \cdot 2\right) \cdot \cos \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right)\right)\right)\right) \]

   6. Applied egg-rr 58.3%

      \[\leadsto \color{blue}{\left(b - a\right) \cdot \left(\left(b + a\right) \cdot \sin \left(\left(\pi \cdot angle\right) \cdot 0.011111111111111112\right)\right)} \]

   if 3.10000000000000016e131 < b

   1. Initial program 38.7%

      \[\left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin \left(\pi \cdot \frac{angle}{180}\right)\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right) \]
   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. associate-*l* N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{sin.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{PI.f64}\left(\right), angle\right), 180\right)\right), \left(2 \cdot \color{blue}{\left(\left({b}^{2} - {a}^{2}\right) \cdot \cos \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right)\right)}\right)\right) \]

      12. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{sin.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{PI.f64}\left(\right), angle\right), 180\right)\right), \mathsf{*.f64}\left(2, \color{blue}{\left(\left({b}^{2} - {a}^{2}\right) \cdot \cos \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right)\right)}\right)\right) \]

      13. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{sin.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{PI.f64}\left(\right), angle\right), 180\right)\right), \mathsf{*.f64}\left(2, \mathsf{*.f64}\left(\left({b}^{2} - {a}^{2}\right), \color{blue}{\cos \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right)}\right)\right)\right) \]

   3. Simplified 39.1%

      \[\leadsto \color{blue}{\sin \left(\frac{\pi \cdot angle}{180}\right) \cdot \left(2 \cdot \left(\left(b \cdot b - a \cdot a\right) \cdot \cos \left(\frac{\pi \cdot angle}{180}\right)\right)\right)} \]
   4. Add Preprocessing
   5. Step-by-step derivation

      1. associate-*r* N/A

         \[\leadsto \left(\sin \left(\frac{\mathsf{PI}\left(\right) \cdot angle}{180}\right) \cdot 2\right) \cdot \color{blue}{\left(\left(b \cdot b - a \cdot a\right) \cdot \cos \left(\frac{\mathsf{PI}\left(\right) \cdot angle}{180}\right)\right)} \]

      2. *-commutative N/A

         \[\leadsto \left(\left(b \cdot b - a \cdot a\right) \cdot \cos \left(\frac{\mathsf{PI}\left(\right) \cdot angle}{180}\right)\right) \cdot \color{blue}{\left(\sin \left(\frac{\mathsf{PI}\left(\right) \cdot angle}{180}\right) \cdot 2\right)} \]

      3. difference-of-squares N/A

         \[\leadsto \left(\left(\left(b + a\right) \cdot \left(b - a\right)\right) \cdot \cos \left(\frac{\mathsf{PI}\left(\right) \cdot angle}{180}\right)\right) \cdot \left(\sin \color{blue}{\left(\frac{\mathsf{PI}\left(\right) \cdot angle}{180}\right)} \cdot 2\right) \]

      4. associate-*r/ N/A

         \[\leadsto \left(\left(\left(b + a\right) \cdot \left(b - a\right)\right) \cdot \cos \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right)\right) \cdot \left(\sin \left(\frac{\mathsf{PI}\left(\right) \cdot angle}{180}\right) \cdot 2\right) \]

      5. associate-*l* N/A

         \[\leadsto \left(\left(b + a\right) \cdot \left(\left(b - a\right) \cdot \cos \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right)\right)\right) \cdot \left(\color{blue}{\sin \left(\frac{\mathsf{PI}\left(\right) \cdot angle}{180}\right)} \cdot 2\right) \]

      6. associate-*l* N/A

         \[\leadsto \left(b + a\right) \cdot \color{blue}{\left(\left(\left(b - a\right) \cdot \cos \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right)\right) \cdot \left(\sin \left(\frac{\mathsf{PI}\left(\right) \cdot angle}{180}\right) \cdot 2\right)\right)} \]

      7. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\left(b + a\right), \color{blue}{\left(\left(\left(b - a\right) \cdot \cos \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right)\right) \cdot \left(\sin \left(\frac{\mathsf{PI}\left(\right) \cdot angle}{180}\right) \cdot 2\right)\right)}\right) \]

      8. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(b, a\right), \left(\color{blue}{\left(\left(b - a\right) \cdot \cos \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right)\right)} \cdot \left(\sin \left(\frac{\mathsf{PI}\left(\right) \cdot angle}{180}\right) \cdot 2\right)\right)\right) \]

      9. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(b, a\right), \mathsf{*.f64}\left(\left(\left(b - a\right) \cdot \cos \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right)\right), \color{blue}{\left(\sin \left(\frac{\mathsf{PI}\left(\right) \cdot angle}{180}\right) \cdot 2\right)}\right)\right) \]

   6. Applied egg-rr 72.7%

      \[\leadsto \color{blue}{\left(b + a\right) \cdot \left(\left(\cos \left(\frac{\pi}{\frac{180}{angle}}\right) \cdot \left(b - a\right)\right) \cdot \left(2 \cdot \sin \left(\frac{\pi}{\frac{180}{angle}}\right)\right)\right)} \]
   7. Step-by-step derivation

      1. expm1-log1p-u N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(b, a\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{cos.f64}\left(\mathsf{/.f64}\left(\mathsf{PI.f64}\left(\right), \mathsf{/.f64}\left(180, angle\right)\right)\right), \mathsf{\_.f64}\left(b, a\right)\right), \mathsf{*.f64}\left(2, \mathsf{sin.f64}\left(\left(\frac{\mathsf{expm1}\left(\mathsf{log1p}\left(\mathsf{PI}\left(\right)\right)\right)}{\frac{180}{angle}}\right)\right)\right)\right)\right) \]

      2. expm1-undefine N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(b, a\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{cos.f64}\left(\mathsf{/.f64}\left(\mathsf{PI.f64}\left(\right), \mathsf{/.f64}\left(180, angle\right)\right)\right), \mathsf{\_.f64}\left(b, a\right)\right), \mathsf{*.f64}\left(2, \mathsf{sin.f64}\left(\left(\frac{e^{\mathsf{log1p}\left(\mathsf{PI}\left(\right)\right)} - 1}{\frac{180}{angle}}\right)\right)\right)\right)\right) \]

      3. log1p-undefine N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(b, a\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{cos.f64}\left(\mathsf{/.f64}\left(\mathsf{PI.f64}\left(\right), \mathsf{/.f64}\left(180, angle\right)\right)\right), \mathsf{\_.f64}\left(b, a\right)\right), \mathsf{*.f64}\left(2, \mathsf{sin.f64}\left(\left(\frac{e^{\log \left(1 + \mathsf{PI}\left(\right)\right)} - 1}{\frac{180}{angle}}\right)\right)\right)\right)\right) \]

      4. +-commutative N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(b, a\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{cos.f64}\left(\mathsf{/.f64}\left(\mathsf{PI.f64}\left(\right), \mathsf{/.f64}\left(180, angle\right)\right)\right), \mathsf{\_.f64}\left(b, a\right)\right), \mathsf{*.f64}\left(2, \mathsf{sin.f64}\left(\left(\frac{e^{\log \left(\mathsf{PI}\left(\right) + 1\right)} - 1}{\frac{180}{angle}}\right)\right)\right)\right)\right) \]

      5. rem-exp-log N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(b, a\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{cos.f64}\left(\mathsf{/.f64}\left(\mathsf{PI.f64}\left(\right), \mathsf{/.f64}\left(180, angle\right)\right)\right), \mathsf{\_.f64}\left(b, a\right)\right), \mathsf{*.f64}\left(2, \mathsf{sin.f64}\left(\left(\frac{\left(\mathsf{PI}\left(\right) + 1\right) - 1}{\frac{180}{angle}}\right)\right)\right)\right)\right) \]

      6. sub-div N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(b, a\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{cos.f64}\left(\mathsf{/.f64}\left(\mathsf{PI.f64}\left(\right), \mathsf{/.f64}\left(180, angle\right)\right)\right), \mathsf{\_.f64}\left(b, a\right)\right), \mathsf{*.f64}\left(2, \mathsf{sin.f64}\left(\left(\frac{\mathsf{PI}\left(\right) + 1}{\frac{180}{angle}} - \frac{1}{\frac{180}{angle}}\right)\right)\right)\right)\right) \]

      7. associate-/r/ N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(b, a\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{cos.f64}\left(\mathsf{/.f64}\left(\mathsf{PI.f64}\left(\right), \mathsf{/.f64}\left(180, angle\right)\right)\right), \mathsf{\_.f64}\left(b, a\right)\right), \mathsf{*.f64}\left(2, \mathsf{sin.f64}\left(\left(\frac{\mathsf{PI}\left(\right) + 1}{180} \cdot angle - \frac{1}{\frac{180}{angle}}\right)\right)\right)\right)\right) \]

      8. associate-*l/ N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(b, a\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{cos.f64}\left(\mathsf{/.f64}\left(\mathsf{PI.f64}\left(\right), \mathsf{/.f64}\left(180, angle\right)\right)\right), \mathsf{\_.f64}\left(b, a\right)\right), \mathsf{*.f64}\left(2, \mathsf{sin.f64}\left(\left(\frac{\left(\mathsf{PI}\left(\right) + 1\right) \cdot angle}{180} - \frac{1}{\frac{180}{angle}}\right)\right)\right)\right)\right) \]

      9. frac-sub N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(b, a\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{cos.f64}\left(\mathsf{/.f64}\left(\mathsf{PI.f64}\left(\right), \mathsf{/.f64}\left(180, angle\right)\right)\right), \mathsf{\_.f64}\left(b, a\right)\right), \mathsf{*.f64}\left(2, \mathsf{sin.f64}\left(\left(\frac{\left(\left(\mathsf{PI}\left(\right) + 1\right) \cdot angle\right) \cdot \frac{180}{angle} - 180 \cdot 1}{180 \cdot \frac{180}{angle}}\right)\right)\right)\right)\right) \]

      10. *-commutative N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(b, a\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{cos.f64}\left(\mathsf{/.f64}\left(\mathsf{PI.f64}\left(\right), \mathsf{/.f64}\left(180, angle\right)\right)\right), \mathsf{\_.f64}\left(b, a\right)\right), \mathsf{*.f64}\left(2, \mathsf{sin.f64}\left(\left(\frac{\left(\left(\mathsf{PI}\left(\right) + 1\right) \cdot angle\right) \cdot \frac{180}{angle} - 180 \cdot 1}{\frac{180}{angle} \cdot 180}\right)\right)\right)\right)\right) \]

      11. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(b, a\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{cos.f64}\left(\mathsf{/.f64}\left(\mathsf{PI.f64}\left(\right), \mathsf{/.f64}\left(180, angle\right)\right)\right), \mathsf{\_.f64}\left(b, a\right)\right), \mathsf{*.f64}\left(2, \mathsf{sin.f64}\left(\mathsf{/.f64}\left(\left(\left(\left(\mathsf{PI}\left(\right) + 1\right) \cdot angle\right) \cdot \frac{180}{angle} - 180 \cdot 1\right), \left(\frac{180}{angle} \cdot 180\right)\right)\right)\right)\right)\right) \]

   8. Applied egg-rr 77.8%

      \[\leadsto \left(b + a\right) \cdot \left(\left(\cos \left(\frac{\pi}{\frac{180}{angle}}\right) \cdot \left(b - a\right)\right) \cdot \left(2 \cdot \sin \color{blue}{\left(\frac{\left(\left(\pi + 1\right) \cdot angle\right) \cdot \frac{180}{angle} - 180}{\frac{32400}{angle}}\right)}\right)\right) \]

   9. Taylor expanded in angle around 0

      \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(b, a\right), \color{blue}{\left(\frac{1}{16200} \cdot \left(angle \cdot \left(\left(b - a\right) \cdot \left(180 \cdot \left(1 + \mathsf{PI}\left(\right)\right) - 180\right)\right)\right)\right)}\right) \]
   10. Step-by-step derivation

       1. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(b, a\right), \mathsf{*.f64}\left(\frac{1}{16200}, \color{blue}{\left(angle \cdot \left(\left(b - a\right) \cdot \left(180 \cdot \left(1 + \mathsf{PI}\left(\right)\right) - 180\right)\right)\right)}\right)\right) \]

       2. associate-*r* N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(b, a\right), \mathsf{*.f64}\left(\frac{1}{16200}, \left(\left(angle \cdot \left(b - a\right)\right) \cdot \color{blue}{\left(180 \cdot \left(1 + \mathsf{PI}\left(\right)\right) - 180\right)}\right)\right)\right) \]

       3. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(b, a\right), \mathsf{*.f64}\left(\frac{1}{16200}, \mathsf{*.f64}\left(\left(angle \cdot \left(b - a\right)\right), \color{blue}{\left(180 \cdot \left(1 + \mathsf{PI}\left(\right)\right) - 180\right)}\right)\right)\right) \]

       4. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(b, a\right), \mathsf{*.f64}\left(\frac{1}{16200}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(angle, \left(b - a\right)\right), \left(\color{blue}{180 \cdot \left(1 + \mathsf{PI}\left(\right)\right)} - 180\right)\right)\right)\right) \]

       5. --lowering--.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(b, a\right), \mathsf{*.f64}\left(\frac{1}{16200}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(angle, \mathsf{\_.f64}\left(b, a\right)\right), \left(180 \cdot \color{blue}{\left(1 + \mathsf{PI}\left(\right)\right)} - 180\right)\right)\right)\right) \]

       6. distribute-lft-in N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(b, a\right), \mathsf{*.f64}\left(\frac{1}{16200}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(angle, \mathsf{\_.f64}\left(b, a\right)\right), \left(\left(180 \cdot 1 + 180 \cdot \mathsf{PI}\left(\right)\right) - 180\right)\right)\right)\right) \]

       7. metadata-eval N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(b, a\right), \mathsf{*.f64}\left(\frac{1}{16200}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(angle, \mathsf{\_.f64}\left(b, a\right)\right), \left(\left(180 + 180 \cdot \mathsf{PI}\left(\right)\right) - 180\right)\right)\right)\right) \]

       8. associate--l+ N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(b, a\right), \mathsf{*.f64}\left(\frac{1}{16200}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(angle, \mathsf{\_.f64}\left(b, a\right)\right), \left(180 + \color{blue}{\left(180 \cdot \mathsf{PI}\left(\right) - 180\right)}\right)\right)\right)\right) \]

       9. +-lowering-+.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(b, a\right), \mathsf{*.f64}\left(\frac{1}{16200}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(angle, \mathsf{\_.f64}\left(b, a\right)\right), \mathsf{+.f64}\left(180, \color{blue}{\left(180 \cdot \mathsf{PI}\left(\right) - 180\right)}\right)\right)\right)\right) \]

       10. sub-neg N/A

           \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(b, a\right), \mathsf{*.f64}\left(\frac{1}{16200}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(angle, \mathsf{\_.f64}\left(b, a\right)\right), \mathsf{+.f64}\left(180, \left(180 \cdot \mathsf{PI}\left(\right) + \color{blue}{\left(\mathsf{neg}\left(180\right)\right)}\right)\right)\right)\right)\right) \]

       11. +-lowering-+.f64 N/A

           \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(b, a\right), \mathsf{*.f64}\left(\frac{1}{16200}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(angle, \mathsf{\_.f64}\left(b, a\right)\right), \mathsf{+.f64}\left(180, \mathsf{+.f64}\left(\left(180 \cdot \mathsf{PI}\left(\right)\right), \color{blue}{\left(\mathsf{neg}\left(180\right)\right)}\right)\right)\right)\right)\right) \]

       12. *-lowering-*.f64 N/A

           \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(b, a\right), \mathsf{*.f64}\left(\frac{1}{16200}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(angle, \mathsf{\_.f64}\left(b, a\right)\right), \mathsf{+.f64}\left(180, \mathsf{+.f64}\left(\mathsf{*.f64}\left(180, \mathsf{PI}\left(\right)\right), \left(\mathsf{neg}\left(\color{blue}{180}\right)\right)\right)\right)\right)\right)\right) \]

       13. PI-lowering-PI.f64 N/A

           \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(b, a\right), \mathsf{*.f64}\left(\frac{1}{16200}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(angle, \mathsf{\_.f64}\left(b, a\right)\right), \mathsf{+.f64}\left(180, \mathsf{+.f64}\left(\mathsf{*.f64}\left(180, \mathsf{PI.f64}\left(\right)\right), \left(\mathsf{neg}\left(180\right)\right)\right)\right)\right)\right)\right) \]

       14. metadata-eval 67.9%

           \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(b, a\right), \mathsf{*.f64}\left(\frac{1}{16200}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(angle, \mathsf{\_.f64}\left(b, a\right)\right), \mathsf{+.f64}\left(180, \mathsf{+.f64}\left(\mathsf{*.f64}\left(180, \mathsf{PI.f64}\left(\right)\right), -180\right)\right)\right)\right)\right) \]

   11. Simplified 67.9%

       \[\leadsto \left(b + a\right) \cdot \color{blue}{\left(6.17283950617284 \cdot 10^{-5} \cdot \left(\left(angle \cdot \left(b - a\right)\right) \cdot \left(180 + \left(180 \cdot \pi + -180\right)\right)\right)\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 59.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq 3.1 \cdot 10^{+131}:\\ \;\;\;\;\left(b - a\right) \cdot \left(\left(b + a\right) \cdot \sin \left(\left(\pi \cdot angle\right) \cdot 0.011111111111111112\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(b + a\right) \cdot \left(6.17283950617284 \cdot 10^{-5} \cdot \left(\left(angle \cdot \left(b - a\right)\right) \cdot \left(180 + \left(-180 + 180 \cdot \pi\right)\right)\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 14: 64.2% accurate, 10.2× speedup

Math

\[\begin{array}{l} a_m = \left|a\right| \\ angle\_m = \left|angle\right| \\ angle\_s = \mathsf{copysign}\left(1, angle\right) \\ \begin{array}{l} t_0 := \pi \cdot \left(b - a\_m\right)\\ angle\_s \cdot \begin{array}{l} \mathbf{if}\;angle\_m \leq 1.8 \cdot 10^{+82}:\\ \;\;\;\;\left(b + a\_m\right) \cdot \left(6.17283950617284 \cdot 10^{-5} \cdot \left(\left(angle\_m \cdot \left(b - a\_m\right)\right) \cdot \left(180 + \left(-180 + 180 \cdot \pi\right)\right)\right)\right)\\ \mathbf{elif}\;angle\_m \leq 2.25 \cdot 10^{+148}:\\ \;\;\;\;\left(b + a\_m\right) \cdot \left(angle\_m \cdot \left(0.011111111111111112 \cdot t\_0 + \left(2 \cdot \left(angle\_m \cdot angle\_m\right)\right) \cdot \left(\left(b - a\_m\right) \cdot \left(\left(\pi \cdot \left(\pi \cdot \pi\right)\right) \cdot -1.1431184270690443 \cdot 10^{-7}\right)\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\left(b + a\_m\right) \cdot t\_0\right) \cdot \left(angle\_m \cdot 0.011111111111111112\right)\\ \end{array} \end{array} \end{array} \]

FPCore

a_m = (fabs.f64 a)
angle\_m = (fabs.f64 angle)
angle\_s = (copysign.f64 #s(literal 1 binary64) angle)
(FPCore (angle_s a_m b angle_m)
 :precision binary64
 (let* ((t_0 (* PI (- b a_m))))
   (*
    angle_s
    (if (<= angle_m 1.8e+82)
      (*
       (+ b a_m)
       (*
        6.17283950617284e-5
        (* (* angle_m (- b a_m)) (+ 180.0 (+ -180.0 (* 180.0 PI))))))
      (if (<= angle_m 2.25e+148)
        (*
         (+ b a_m)
         (*
          angle_m
          (+
           (* 0.011111111111111112 t_0)
           (*
            (* 2.0 (* angle_m angle_m))
            (* (- b a_m) (* (* PI (* PI PI)) -1.1431184270690443e-7))))))
        (* (* (+ b a_m) t_0) (* angle_m 0.011111111111111112)))))))

C

a_m = fabs(a);
angle\_m = fabs(angle);
angle\_s = copysign(1.0, angle);
double code(double angle_s, double a_m, double b, double angle_m) {
	double t_0 = ((double) M_PI) * (b - a_m);
	double tmp;
	if (angle_m <= 1.8e+82) {
		tmp = (b + a_m) * (6.17283950617284e-5 * ((angle_m * (b - a_m)) * (180.0 + (-180.0 + (180.0 * ((double) M_PI))))));
	} else if (angle_m <= 2.25e+148) {
		tmp = (b + a_m) * (angle_m * ((0.011111111111111112 * t_0) + ((2.0 * (angle_m * angle_m)) * ((b - a_m) * ((((double) M_PI) * (((double) M_PI) * ((double) M_PI))) * -1.1431184270690443e-7)))));
	} else {
		tmp = ((b + a_m) * t_0) * (angle_m * 0.011111111111111112);
	}
	return angle_s * tmp;
}

Java

a_m = Math.abs(a);
angle\_m = Math.abs(angle);
angle\_s = Math.copySign(1.0, angle);
public static double code(double angle_s, double a_m, double b, double angle_m) {
	double t_0 = Math.PI * (b - a_m);
	double tmp;
	if (angle_m <= 1.8e+82) {
		tmp = (b + a_m) * (6.17283950617284e-5 * ((angle_m * (b - a_m)) * (180.0 + (-180.0 + (180.0 * Math.PI)))));
	} else if (angle_m <= 2.25e+148) {
		tmp = (b + a_m) * (angle_m * ((0.011111111111111112 * t_0) + ((2.0 * (angle_m * angle_m)) * ((b - a_m) * ((Math.PI * (Math.PI * Math.PI)) * -1.1431184270690443e-7)))));
	} else {
		tmp = ((b + a_m) * t_0) * (angle_m * 0.011111111111111112);
	}
	return angle_s * tmp;
}

Python

a_m = math.fabs(a)
angle\_m = math.fabs(angle)
angle\_s = math.copysign(1.0, angle)
def code(angle_s, a_m, b, angle_m):
	t_0 = math.pi * (b - a_m)
	tmp = 0
	if angle_m <= 1.8e+82:
		tmp = (b + a_m) * (6.17283950617284e-5 * ((angle_m * (b - a_m)) * (180.0 + (-180.0 + (180.0 * math.pi)))))
	elif angle_m <= 2.25e+148:
		tmp = (b + a_m) * (angle_m * ((0.011111111111111112 * t_0) + ((2.0 * (angle_m * angle_m)) * ((b - a_m) * ((math.pi * (math.pi * math.pi)) * -1.1431184270690443e-7)))))
	else:
		tmp = ((b + a_m) * t_0) * (angle_m * 0.011111111111111112)
	return angle_s * tmp

Julia

a_m = abs(a)
angle\_m = abs(angle)
angle\_s = copysign(1.0, angle)
function code(angle_s, a_m, b, angle_m)
	t_0 = Float64(pi * Float64(b - a_m))
	tmp = 0.0
	if (angle_m <= 1.8e+82)
		tmp = Float64(Float64(b + a_m) * Float64(6.17283950617284e-5 * Float64(Float64(angle_m * Float64(b - a_m)) * Float64(180.0 + Float64(-180.0 + Float64(180.0 * pi))))));
	elseif (angle_m <= 2.25e+148)
		tmp = Float64(Float64(b + a_m) * Float64(angle_m * Float64(Float64(0.011111111111111112 * t_0) + Float64(Float64(2.0 * Float64(angle_m * angle_m)) * Float64(Float64(b - a_m) * Float64(Float64(pi * Float64(pi * pi)) * -1.1431184270690443e-7))))));
	else
		tmp = Float64(Float64(Float64(b + a_m) * t_0) * Float64(angle_m * 0.011111111111111112));
	end
	return Float64(angle_s * tmp)
end

MATLAB

a_m = abs(a);
angle\_m = abs(angle);
angle\_s = sign(angle) * abs(1.0);
function tmp_2 = code(angle_s, a_m, b, angle_m)
	t_0 = pi * (b - a_m);
	tmp = 0.0;
	if (angle_m <= 1.8e+82)
		tmp = (b + a_m) * (6.17283950617284e-5 * ((angle_m * (b - a_m)) * (180.0 + (-180.0 + (180.0 * pi)))));
	elseif (angle_m <= 2.25e+148)
		tmp = (b + a_m) * (angle_m * ((0.011111111111111112 * t_0) + ((2.0 * (angle_m * angle_m)) * ((b - a_m) * ((pi * (pi * pi)) * -1.1431184270690443e-7)))));
	else
		tmp = ((b + a_m) * t_0) * (angle_m * 0.011111111111111112);
	end
	tmp_2 = angle_s * tmp;
end

Wolfram

a_m = N[Abs[a], $MachinePrecision]
angle\_m = N[Abs[angle], $MachinePrecision]
angle\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[angle]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[angle$95$s_, a$95$m_, b_, angle$95$m_] := Block[{t$95$0 = N[(Pi * N[(b - a$95$m), $MachinePrecision]), $MachinePrecision]}, N[(angle$95$s * If[LessEqual[angle$95$m, 1.8e+82], N[(N[(b + a$95$m), $MachinePrecision] * N[(6.17283950617284e-5 * N[(N[(angle$95$m * N[(b - a$95$m), $MachinePrecision]), $MachinePrecision] * N[(180.0 + N[(-180.0 + N[(180.0 * Pi), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[angle$95$m, 2.25e+148], N[(N[(b + a$95$m), $MachinePrecision] * N[(angle$95$m * N[(N[(0.011111111111111112 * t$95$0), $MachinePrecision] + N[(N[(2.0 * N[(angle$95$m * angle$95$m), $MachinePrecision]), $MachinePrecision] * N[(N[(b - a$95$m), $MachinePrecision] * N[(N[(Pi * N[(Pi * Pi), $MachinePrecision]), $MachinePrecision] * -1.1431184270690443e-7), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(b + a$95$m), $MachinePrecision] * t$95$0), $MachinePrecision] * N[(angle$95$m * 0.011111111111111112), $MachinePrecision]), $MachinePrecision]]]), $MachinePrecision]]

Derivation

1. Split input into 3 regimes

2. if angle < 1.80000000000000007e82

   1. Initial program 57.2%

      \[\left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin \left(\pi \cdot \frac{angle}{180}\right)\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right) \]
   2. Step-by-step derivation

      1. associate-*l* N/A

         \[\leadsto \left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \color{blue}{\left(\sin \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right) \cdot \cos \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right)\right)} \]

      2. *-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. associate-*l* N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{sin.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{PI.f64}\left(\right), angle\right), 180\right)\right), \left(2 \cdot \color{blue}{\left(\left({b}^{2} - {a}^{2}\right) \cdot \cos \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right)\right)}\right)\right) \]

      12. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{sin.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{PI.f64}\left(\right), angle\right), 180\right)\right), \mathsf{*.f64}\left(2, \color{blue}{\left(\left({b}^{2} - {a}^{2}\right) \cdot \cos \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right)\right)}\right)\right) \]

      13. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{sin.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{PI.f64}\left(\right), angle\right), 180\right)\right), \mathsf{*.f64}\left(2, \mathsf{*.f64}\left(\left({b}^{2} - {a}^{2}\right), \color{blue}{\cos \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right)}\right)\right)\right) \]

   3. Simplified 57.0%

      \[\leadsto \color{blue}{\sin \left(\frac{\pi \cdot angle}{180}\right) \cdot \left(2 \cdot \left(\left(b \cdot b - a \cdot a\right) \cdot \cos \left(\frac{\pi \cdot angle}{180}\right)\right)\right)} \]
   4. Add Preprocessing
   5. Step-by-step derivation

      1. associate-*r* N/A

         \[\leadsto \left(\sin \left(\frac{\mathsf{PI}\left(\right) \cdot angle}{180}\right) \cdot 2\right) \cdot \color{blue}{\left(\left(b \cdot b - a \cdot a\right) \cdot \cos \left(\frac{\mathsf{PI}\left(\right) \cdot angle}{180}\right)\right)} \]

      2. *-commutative N/A

         \[\leadsto \left(\left(b \cdot b - a \cdot a\right) \cdot \cos \left(\frac{\mathsf{PI}\left(\right) \cdot angle}{180}\right)\right) \cdot \color{blue}{\left(\sin \left(\frac{\mathsf{PI}\left(\right) \cdot angle}{180}\right) \cdot 2\right)} \]

      3. difference-of-squares N/A

         \[\leadsto \left(\left(\left(b + a\right) \cdot \left(b - a\right)\right) \cdot \cos \left(\frac{\mathsf{PI}\left(\right) \cdot angle}{180}\right)\right) \cdot \left(\sin \color{blue}{\left(\frac{\mathsf{PI}\left(\right) \cdot angle}{180}\right)} \cdot 2\right) \]

      4. associate-*r/ N/A

         \[\leadsto \left(\left(\left(b + a\right) \cdot \left(b - a\right)\right) \cdot \cos \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right)\right) \cdot \left(\sin \left(\frac{\mathsf{PI}\left(\right) \cdot angle}{180}\right) \cdot 2\right) \]

      5. associate-*l* N/A

         \[\leadsto \left(\left(b + a\right) \cdot \left(\left(b - a\right) \cdot \cos \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right)\right)\right) \cdot \left(\color{blue}{\sin \left(\frac{\mathsf{PI}\left(\right) \cdot angle}{180}\right)} \cdot 2\right) \]

      6. associate-*l* N/A

         \[\leadsto \left(b + a\right) \cdot \color{blue}{\left(\left(\left(b - a\right) \cdot \cos \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right)\right) \cdot \left(\sin \left(\frac{\mathsf{PI}\left(\right) \cdot angle}{180}\right) \cdot 2\right)\right)} \]

      7. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\left(b + a\right), \color{blue}{\left(\left(\left(b - a\right) \cdot \cos \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right)\right) \cdot \left(\sin \left(\frac{\mathsf{PI}\left(\right) \cdot angle}{180}\right) \cdot 2\right)\right)}\right) \]

      8. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(b, a\right), \left(\color{blue}{\left(\left(b - a\right) \cdot \cos \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right)\right)} \cdot \left(\sin \left(\frac{\mathsf{PI}\left(\right) \cdot angle}{180}\right) \cdot 2\right)\right)\right) \]

      9. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(b, a\right), \mathsf{*.f64}\left(\left(\left(b - a\right) \cdot \cos \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right)\right), \color{blue}{\left(\sin \left(\frac{\mathsf{PI}\left(\right) \cdot angle}{180}\right) \cdot 2\right)}\right)\right) \]

   6. Applied egg-rr 69.0%

      \[\leadsto \color{blue}{\left(b + a\right) \cdot \left(\left(\cos \left(\frac{\pi}{\frac{180}{angle}}\right) \cdot \left(b - a\right)\right) \cdot \left(2 \cdot \sin \left(\frac{\pi}{\frac{180}{angle}}\right)\right)\right)} \]
   7. Step-by-step derivation

      1. expm1-log1p-u N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(b, a\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{cos.f64}\left(\mathsf{/.f64}\left(\mathsf{PI.f64}\left(\right), \mathsf{/.f64}\left(180, angle\right)\right)\right), \mathsf{\_.f64}\left(b, a\right)\right), \mathsf{*.f64}\left(2, \mathsf{sin.f64}\left(\left(\frac{\mathsf{expm1}\left(\mathsf{log1p}\left(\mathsf{PI}\left(\right)\right)\right)}{\frac{180}{angle}}\right)\right)\right)\right)\right) \]

      2. expm1-undefine N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(b, a\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{cos.f64}\left(\mathsf{/.f64}\left(\mathsf{PI.f64}\left(\right), \mathsf{/.f64}\left(180, angle\right)\right)\right), \mathsf{\_.f64}\left(b, a\right)\right), \mathsf{*.f64}\left(2, \mathsf{sin.f64}\left(\left(\frac{e^{\mathsf{log1p}\left(\mathsf{PI}\left(\right)\right)} - 1}{\frac{180}{angle}}\right)\right)\right)\right)\right) \]

      3. log1p-undefine N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(b, a\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{cos.f64}\left(\mathsf{/.f64}\left(\mathsf{PI.f64}\left(\right), \mathsf{/.f64}\left(180, angle\right)\right)\right), \mathsf{\_.f64}\left(b, a\right)\right), \mathsf{*.f64}\left(2, \mathsf{sin.f64}\left(\left(\frac{e^{\log \left(1 + \mathsf{PI}\left(\right)\right)} - 1}{\frac{180}{angle}}\right)\right)\right)\right)\right) \]

      4. +-commutative N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(b, a\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{cos.f64}\left(\mathsf{/.f64}\left(\mathsf{PI.f64}\left(\right), \mathsf{/.f64}\left(180, angle\right)\right)\right), \mathsf{\_.f64}\left(b, a\right)\right), \mathsf{*.f64}\left(2, \mathsf{sin.f64}\left(\left(\frac{e^{\log \left(\mathsf{PI}\left(\right) + 1\right)} - 1}{\frac{180}{angle}}\right)\right)\right)\right)\right) \]

      5. rem-exp-log N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(b, a\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{cos.f64}\left(\mathsf{/.f64}\left(\mathsf{PI.f64}\left(\right), \mathsf{/.f64}\left(180, angle\right)\right)\right), \mathsf{\_.f64}\left(b, a\right)\right), \mathsf{*.f64}\left(2, \mathsf{sin.f64}\left(\left(\frac{\left(\mathsf{PI}\left(\right) + 1\right) - 1}{\frac{180}{angle}}\right)\right)\right)\right)\right) \]

      6. sub-div N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(b, a\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{cos.f64}\left(\mathsf{/.f64}\left(\mathsf{PI.f64}\left(\right), \mathsf{/.f64}\left(180, angle\right)\right)\right), \mathsf{\_.f64}\left(b, a\right)\right), \mathsf{*.f64}\left(2, \mathsf{sin.f64}\left(\left(\frac{\mathsf{PI}\left(\right) + 1}{\frac{180}{angle}} - \frac{1}{\frac{180}{angle}}\right)\right)\right)\right)\right) \]

      7. associate-/r/ N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(b, a\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{cos.f64}\left(\mathsf{/.f64}\left(\mathsf{PI.f64}\left(\right), \mathsf{/.f64}\left(180, angle\right)\right)\right), \mathsf{\_.f64}\left(b, a\right)\right), \mathsf{*.f64}\left(2, \mathsf{sin.f64}\left(\left(\frac{\mathsf{PI}\left(\right) + 1}{180} \cdot angle - \frac{1}{\frac{180}{angle}}\right)\right)\right)\right)\right) \]

      8. associate-*l/ N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(b, a\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{cos.f64}\left(\mathsf{/.f64}\left(\mathsf{PI.f64}\left(\right), \mathsf{/.f64}\left(180, angle\right)\right)\right), \mathsf{\_.f64}\left(b, a\right)\right), \mathsf{*.f64}\left(2, \mathsf{sin.f64}\left(\left(\frac{\left(\mathsf{PI}\left(\right) + 1\right) \cdot angle}{180} - \frac{1}{\frac{180}{angle}}\right)\right)\right)\right)\right) \]

      9. frac-sub N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(b, a\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{cos.f64}\left(\mathsf{/.f64}\left(\mathsf{PI.f64}\left(\right), \mathsf{/.f64}\left(180, angle\right)\right)\right), \mathsf{\_.f64}\left(b, a\right)\right), \mathsf{*.f64}\left(2, \mathsf{sin.f64}\left(\left(\frac{\left(\left(\mathsf{PI}\left(\right) + 1\right) \cdot angle\right) \cdot \frac{180}{angle} - 180 \cdot 1}{180 \cdot \frac{180}{angle}}\right)\right)\right)\right)\right) \]

      10. *-commutative N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(b, a\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{cos.f64}\left(\mathsf{/.f64}\left(\mathsf{PI.f64}\left(\right), \mathsf{/.f64}\left(180, angle\right)\right)\right), \mathsf{\_.f64}\left(b, a\right)\right), \mathsf{*.f64}\left(2, \mathsf{sin.f64}\left(\left(\frac{\left(\left(\mathsf{PI}\left(\right) + 1\right) \cdot angle\right) \cdot \frac{180}{angle} - 180 \cdot 1}{\frac{180}{angle} \cdot 180}\right)\right)\right)\right)\right) \]

      11. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(b, a\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{cos.f64}\left(\mathsf{/.f64}\left(\mathsf{PI.f64}\left(\right), \mathsf{/.f64}\left(180, angle\right)\right)\right), \mathsf{\_.f64}\left(b, a\right)\right), \mathsf{*.f64}\left(2, \mathsf{sin.f64}\left(\mathsf{/.f64}\left(\left(\left(\left(\mathsf{PI}\left(\right) + 1\right) \cdot angle\right) \cdot \frac{180}{angle} - 180 \cdot 1\right), \left(\frac{180}{angle} \cdot 180\right)\right)\right)\right)\right)\right) \]

   8. Applied egg-rr 70.5%

      \[\leadsto \left(b + a\right) \cdot \left(\left(\cos \left(\frac{\pi}{\frac{180}{angle}}\right) \cdot \left(b - a\right)\right) \cdot \left(2 \cdot \sin \color{blue}{\left(\frac{\left(\left(\pi + 1\right) \cdot angle\right) \cdot \frac{180}{angle} - 180}{\frac{32400}{angle}}\right)}\right)\right) \]

   9. Taylor expanded in angle around 0

      \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(b, a\right), \color{blue}{\left(\frac{1}{16200} \cdot \left(angle \cdot \left(\left(b - a\right) \cdot \left(180 \cdot \left(1 + \mathsf{PI}\left(\right)\right) - 180\right)\right)\right)\right)}\right) \]
   10. Step-by-step derivation

       1. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(b, a\right), \mathsf{*.f64}\left(\frac{1}{16200}, \color{blue}{\left(angle \cdot \left(\left(b - a\right) \cdot \left(180 \cdot \left(1 + \mathsf{PI}\left(\right)\right) - 180\right)\right)\right)}\right)\right) \]

       2. associate-*r* N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(b, a\right), \mathsf{*.f64}\left(\frac{1}{16200}, \left(\left(angle \cdot \left(b - a\right)\right) \cdot \color{blue}{\left(180 \cdot \left(1 + \mathsf{PI}\left(\right)\right) - 180\right)}\right)\right)\right) \]

       3. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(b, a\right), \mathsf{*.f64}\left(\frac{1}{16200}, \mathsf{*.f64}\left(\left(angle \cdot \left(b - a\right)\right), \color{blue}{\left(180 \cdot \left(1 + \mathsf{PI}\left(\right)\right) - 180\right)}\right)\right)\right) \]

       4. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(b, a\right), \mathsf{*.f64}\left(\frac{1}{16200}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(angle, \left(b - a\right)\right), \left(\color{blue}{180 \cdot \left(1 + \mathsf{PI}\left(\right)\right)} - 180\right)\right)\right)\right) \]

       5. --lowering--.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(b, a\right), \mathsf{*.f64}\left(\frac{1}{16200}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(angle, \mathsf{\_.f64}\left(b, a\right)\right), \left(180 \cdot \color{blue}{\left(1 + \mathsf{PI}\left(\right)\right)} - 180\right)\right)\right)\right) \]

       6. distribute-lft-in N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(b, a\right), \mathsf{*.f64}\left(\frac{1}{16200}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(angle, \mathsf{\_.f64}\left(b, a\right)\right), \left(\left(180 \cdot 1 + 180 \cdot \mathsf{PI}\left(\right)\right) - 180\right)\right)\right)\right) \]

       7. metadata-eval N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(b, a\right), \mathsf{*.f64}\left(\frac{1}{16200}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(angle, \mathsf{\_.f64}\left(b, a\right)\right), \left(\left(180 + 180 \cdot \mathsf{PI}\left(\right)\right) - 180\right)\right)\right)\right) \]

       8. associate--l+ N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(b, a\right), \mathsf{*.f64}\left(\frac{1}{16200}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(angle, \mathsf{\_.f64}\left(b, a\right)\right), \left(180 + \color{blue}{\left(180 \cdot \mathsf{PI}\left(\right) - 180\right)}\right)\right)\right)\right) \]

       9. +-lowering-+.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(b, a\right), \mathsf{*.f64}\left(\frac{1}{16200}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(angle, \mathsf{\_.f64}\left(b, a\right)\right), \mathsf{+.f64}\left(180, \color{blue}{\left(180 \cdot \mathsf{PI}\left(\right) - 180\right)}\right)\right)\right)\right) \]

       10. sub-neg N/A

           \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(b, a\right), \mathsf{*.f64}\left(\frac{1}{16200}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(angle, \mathsf{\_.f64}\left(b, a\right)\right), \mathsf{+.f64}\left(180, \left(180 \cdot \mathsf{PI}\left(\right) + \color{blue}{\left(\mathsf{neg}\left(180\right)\right)}\right)\right)\right)\right)\right) \]

       11. +-lowering-+.f64 N/A

           \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(b, a\right), \mathsf{*.f64}\left(\frac{1}{16200}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(angle, \mathsf{\_.f64}\left(b, a\right)\right), \mathsf{+.f64}\left(180, \mathsf{+.f64}\left(\left(180 \cdot \mathsf{PI}\left(\right)\right), \color{blue}{\left(\mathsf{neg}\left(180\right)\right)}\right)\right)\right)\right)\right) \]

       12. *-lowering-*.f64 N/A

           \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(b, a\right), \mathsf{*.f64}\left(\frac{1}{16200}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(angle, \mathsf{\_.f64}\left(b, a\right)\right), \mathsf{+.f64}\left(180, \mathsf{+.f64}\left(\mathsf{*.f64}\left(180, \mathsf{PI}\left(\right)\right), \left(\mathsf{neg}\left(\color{blue}{180}\right)\right)\right)\right)\right)\right)\right) \]

       13. PI-lowering-PI.f64 N/A

           \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(b, a\right), \mathsf{*.f64}\left(\frac{1}{16200}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(angle, \mathsf{\_.f64}\left(b, a\right)\right), \mathsf{+.f64}\left(180, \mathsf{+.f64}\left(\mathsf{*.f64}\left(180, \mathsf{PI.f64}\left(\right)\right), \left(\mathsf{neg}\left(180\right)\right)\right)\right)\right)\right)\right) \]

       14. metadata-eval 64.9%

           \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(b, a\right), \mathsf{*.f64}\left(\frac{1}{16200}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(angle, \mathsf{\_.f64}\left(b, a\right)\right), \mathsf{+.f64}\left(180, \mathsf{+.f64}\left(\mathsf{*.f64}\left(180, \mathsf{PI.f64}\left(\right)\right), -180\right)\right)\right)\right)\right) \]

   11. Simplified 64.9%

       \[\leadsto \left(b + a\right) \cdot \color{blue}{\left(6.17283950617284 \cdot 10^{-5} \cdot \left(\left(angle \cdot \left(b - a\right)\right) \cdot \left(180 + \left(180 \cdot \pi + -180\right)\right)\right)\right)} \]

   if 1.80000000000000007e82 < angle < 2.24999999999999997e148

   1. Initial program 26.6%

      \[\left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin \left(\pi \cdot \frac{angle}{180}\right)\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right) \]
   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. associate-*l* N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{sin.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{PI.f64}\left(\right), angle\right), 180\right)\right), \left(2 \cdot \color{blue}{\left(\left({b}^{2} - {a}^{2}\right) \cdot \cos \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right)\right)}\right)\right) \]

      12. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{sin.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{PI.f64}\left(\right), angle\right), 180\right)\right), \mathsf{*.f64}\left(2, \color{blue}{\left(\left({b}^{2} - {a}^{2}\right) \cdot \cos \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right)\right)}\right)\right) \]

      13. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{sin.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{PI.f64}\left(\right), angle\right), 180\right)\right), \mathsf{*.f64}\left(2, \mathsf{*.f64}\left(\left({b}^{2} - {a}^{2}\right), \color{blue}{\cos \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right)}\right)\right)\right) \]

   3. Simplified 33.6%

      \[\leadsto \color{blue}{\sin \left(\frac{\pi \cdot angle}{180}\right) \cdot \left(2 \cdot \left(\left(b \cdot b - a \cdot a\right) \cdot \cos \left(\frac{\pi \cdot angle}{180}\right)\right)\right)} \]
   4. Add Preprocessing
   5. Step-by-step derivation

      1. associate-*r* N/A

         \[\leadsto \left(\sin \left(\frac{\mathsf{PI}\left(\right) \cdot angle}{180}\right) \cdot 2\right) \cdot \color{blue}{\left(\left(b \cdot b - a \cdot a\right) \cdot \cos \left(\frac{\mathsf{PI}\left(\right) \cdot angle}{180}\right)\right)} \]

      2. *-commutative N/A

         \[\leadsto \left(\left(b \cdot b - a \cdot a\right) \cdot \cos \left(\frac{\mathsf{PI}\left(\right) \cdot angle}{180}\right)\right) \cdot \color{blue}{\left(\sin \left(\frac{\mathsf{PI}\left(\right) \cdot angle}{180}\right) \cdot 2\right)} \]

      3. difference-of-squares N/A

         \[\leadsto \left(\left(\left(b + a\right) \cdot \left(b - a\right)\right) \cdot \cos \left(\frac{\mathsf{PI}\left(\right) \cdot angle}{180}\right)\right) \cdot \left(\sin \color{blue}{\left(\frac{\mathsf{PI}\left(\right) \cdot angle}{180}\right)} \cdot 2\right) \]

      4. associate-*r/ N/A

         \[\leadsto \left(\left(\left(b + a\right) \cdot \left(b - a\right)\right) \cdot \cos \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right)\right) \cdot \left(\sin \left(\frac{\mathsf{PI}\left(\right) \cdot angle}{180}\right) \cdot 2\right) \]

      5. associate-*l* N/A

         \[\leadsto \left(\left(b + a\right) \cdot \left(\left(b - a\right) \cdot \cos \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right)\right)\right) \cdot \left(\color{blue}{\sin \left(\frac{\mathsf{PI}\left(\right) \cdot angle}{180}\right)} \cdot 2\right) \]

      6. associate-*l* N/A

         \[\leadsto \left(b + a\right) \cdot \color{blue}{\left(\left(\left(b - a\right) \cdot \cos \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right)\right) \cdot \left(\sin \left(\frac{\mathsf{PI}\left(\right) \cdot angle}{180}\right) \cdot 2\right)\right)} \]

      7. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\left(b + a\right), \color{blue}{\left(\left(\left(b - a\right) \cdot \cos \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right)\right) \cdot \left(\sin \left(\frac{\mathsf{PI}\left(\right) \cdot angle}{180}\right) \cdot 2\right)\right)}\right) \]

      8. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(b, a\right), \left(\color{blue}{\left(\left(b - a\right) \cdot \cos \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right)\right)} \cdot \left(\sin \left(\frac{\mathsf{PI}\left(\right) \cdot angle}{180}\right) \cdot 2\right)\right)\right) \]

      9. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(b, a\right), \mathsf{*.f64}\left(\left(\left(b - a\right) \cdot \cos \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right)\right), \color{blue}{\left(\sin \left(\frac{\mathsf{PI}\left(\right) \cdot angle}{180}\right) \cdot 2\right)}\right)\right) \]

   6. Applied egg-rr 33.6%

      \[\leadsto \color{blue}{\left(b + a\right) \cdot \left(\left(\cos \left(\frac{\pi}{\frac{180}{angle}}\right) \cdot \left(b - a\right)\right) \cdot \left(2 \cdot \sin \left(\frac{\pi}{\frac{180}{angle}}\right)\right)\right)} \]

   7. Taylor expanded in angle around 0

      \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(b, a\right), \color{blue}{\left(angle \cdot \left(\frac{1}{90} \cdot \left(\mathsf{PI}\left(\right) \cdot \left(b - a\right)\right) + 2 \cdot \left({angle}^{2} \cdot \left(\frac{-1}{11664000} \cdot \left({\mathsf{PI}\left(\right)}^{3} \cdot \left(b - a\right)\right) + \frac{-1}{34992000} \cdot \left({\mathsf{PI}\left(\right)}^{3} \cdot \left(b - a\right)\right)\right)\right)\right)\right)}\right) \]
   8. Step-by-step derivation

      1. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(b, a\right), \mathsf{*.f64}\left(angle, \color{blue}{\left(\frac{1}{90} \cdot \left(\mathsf{PI}\left(\right) \cdot \left(b - a\right)\right) + 2 \cdot \left({angle}^{2} \cdot \left(\frac{-1}{11664000} \cdot \left({\mathsf{PI}\left(\right)}^{3} \cdot \left(b - a\right)\right) + \frac{-1}{34992000} \cdot \left({\mathsf{PI}\left(\right)}^{3} \cdot \left(b - a\right)\right)\right)\right)\right)}\right)\right) \]

      2. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(b, a\right), \mathsf{*.f64}\left(angle, \mathsf{+.f64}\left(\left(\frac{1}{90} \cdot \left(\mathsf{PI}\left(\right) \cdot \left(b - a\right)\right)\right), \color{blue}{\left(2 \cdot \left({angle}^{2} \cdot \left(\frac{-1}{11664000} \cdot \left({\mathsf{PI}\left(\right)}^{3} \cdot \left(b - a\right)\right) + \frac{-1}{34992000} \cdot \left({\mathsf{PI}\left(\right)}^{3} \cdot \left(b - a\right)\right)\right)\right)\right)}\right)\right)\right) \]

      3. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(b, a\right), \mathsf{*.f64}\left(angle, \mathsf{+.f64}\left(\mathsf{*.f64}\left(\frac{1}{90}, \left(\mathsf{PI}\left(\right) \cdot \left(b - a\right)\right)\right), \left(\color{blue}{2} \cdot \left({angle}^{2} \cdot \left(\frac{-1}{11664000} \cdot \left({\mathsf{PI}\left(\right)}^{3} \cdot \left(b - a\right)\right) + \frac{-1}{34992000} \cdot \left({\mathsf{PI}\left(\right)}^{3} \cdot \left(b - a\right)\right)\right)\right)\right)\right)\right)\right) \]

      4. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(b, a\right), \mathsf{*.f64}\left(angle, \mathsf{+.f64}\left(\mathsf{*.f64}\left(\frac{1}{90}, \mathsf{*.f64}\left(\mathsf{PI}\left(\right), \left(b - a\right)\right)\right), \left(2 \cdot \left({angle}^{2} \cdot \left(\frac{-1}{11664000} \cdot \left({\mathsf{PI}\left(\right)}^{3} \cdot \left(b - a\right)\right) + \frac{-1}{34992000} \cdot \left({\mathsf{PI}\left(\right)}^{3} \cdot \left(b - a\right)\right)\right)\right)\right)\right)\right)\right) \]

      5. PI-lowering-PI.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(b, a\right), \mathsf{*.f64}\left(angle, \mathsf{+.f64}\left(\mathsf{*.f64}\left(\frac{1}{90}, \mathsf{*.f64}\left(\mathsf{PI.f64}\left(\right), \left(b - a\right)\right)\right), \left(2 \cdot \left({angle}^{2} \cdot \left(\frac{-1}{11664000} \cdot \left({\mathsf{PI}\left(\right)}^{3} \cdot \left(b - a\right)\right) + \frac{-1}{34992000} \cdot \left({\mathsf{PI}\left(\right)}^{3} \cdot \left(b - a\right)\right)\right)\right)\right)\right)\right)\right) \]

      6. --lowering--.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(b, a\right), \mathsf{*.f64}\left(angle, \mathsf{+.f64}\left(\mathsf{*.f64}\left(\frac{1}{90}, \mathsf{*.f64}\left(\mathsf{PI.f64}\left(\right), \mathsf{\_.f64}\left(b, a\right)\right)\right), \left(2 \cdot \left({angle}^{2} \cdot \left(\frac{-1}{11664000} \cdot \left({\mathsf{PI}\left(\right)}^{3} \cdot \left(b - a\right)\right) + \frac{-1}{34992000} \cdot \left({\mathsf{PI}\left(\right)}^{3} \cdot \left(b - a\right)\right)\right)\right)\right)\right)\right)\right) \]

      7. associate-*r* N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(b, a\right), \mathsf{*.f64}\left(angle, \mathsf{+.f64}\left(\mathsf{*.f64}\left(\frac{1}{90}, \mathsf{*.f64}\left(\mathsf{PI.f64}\left(\right), \mathsf{\_.f64}\left(b, a\right)\right)\right), \left(\left(2 \cdot {angle}^{2}\right) \cdot \color{blue}{\left(\frac{-1}{11664000} \cdot \left({\mathsf{PI}\left(\right)}^{3} \cdot \left(b - a\right)\right) + \frac{-1}{34992000} \cdot \left({\mathsf{PI}\left(\right)}^{3} \cdot \left(b - a\right)\right)\right)}\right)\right)\right)\right) \]

      8. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(b, a\right), \mathsf{*.f64}\left(angle, \mathsf{+.f64}\left(\mathsf{*.f64}\left(\frac{1}{90}, \mathsf{*.f64}\left(\mathsf{PI.f64}\left(\right), \mathsf{\_.f64}\left(b, a\right)\right)\right), \mathsf{*.f64}\left(\left(2 \cdot {angle}^{2}\right), \color{blue}{\left(\frac{-1}{11664000} \cdot \left({\mathsf{PI}\left(\right)}^{3} \cdot \left(b - a\right)\right) + \frac{-1}{34992000} \cdot \left({\mathsf{PI}\left(\right)}^{3} \cdot \left(b - a\right)\right)\right)}\right)\right)\right)\right) \]

      9. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(b, a\right), \mathsf{*.f64}\left(angle, \mathsf{+.f64}\left(\mathsf{*.f64}\left(\frac{1}{90}, \mathsf{*.f64}\left(\mathsf{PI.f64}\left(\right), \mathsf{\_.f64}\left(b, a\right)\right)\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(2, \left({angle}^{2}\right)\right), \left(\color{blue}{\frac{-1}{11664000} \cdot \left({\mathsf{PI}\left(\right)}^{3} \cdot \left(b - a\right)\right)} + \frac{-1}{34992000} \cdot \left({\mathsf{PI}\left(\right)}^{3} \cdot \left(b - a\right)\right)\right)\right)\right)\right)\right) \]

      10. unpow2 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(b, a\right), \mathsf{*.f64}\left(angle, \mathsf{+.f64}\left(\mathsf{*.f64}\left(\frac{1}{90}, \mathsf{*.f64}\left(\mathsf{PI.f64}\left(\right), \mathsf{\_.f64}\left(b, a\right)\right)\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(2, \left(angle \cdot angle\right)\right), \left(\frac{-1}{11664000} \cdot \color{blue}{\left({\mathsf{PI}\left(\right)}^{3} \cdot \left(b - a\right)\right)} + \frac{-1}{34992000} \cdot \left({\mathsf{PI}\left(\right)}^{3} \cdot \left(b - a\right)\right)\right)\right)\right)\right)\right) \]

      11. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(b, a\right), \mathsf{*.f64}\left(angle, \mathsf{+.f64}\left(\mathsf{*.f64}\left(\frac{1}{90}, \mathsf{*.f64}\left(\mathsf{PI.f64}\left(\right), \mathsf{\_.f64}\left(b, a\right)\right)\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(2, \mathsf{*.f64}\left(angle, angle\right)\right), \left(\frac{-1}{11664000} \cdot \color{blue}{\left({\mathsf{PI}\left(\right)}^{3} \cdot \left(b - a\right)\right)} + \frac{-1}{34992000} \cdot \left({\mathsf{PI}\left(\right)}^{3} \cdot \left(b - a\right)\right)\right)\right)\right)\right)\right) \]

   9. Simplified 28.2%

      \[\leadsto \left(b + a\right) \cdot \color{blue}{\left(angle \cdot \left(0.011111111111111112 \cdot \left(\pi \cdot \left(b - a\right)\right) + \left(2 \cdot \left(angle \cdot angle\right)\right) \cdot \left(\left(b - a\right) \cdot \left(\left(\pi \cdot \left(\pi \cdot \pi\right)\right) \cdot -1.1431184270690443 \cdot 10^{-7}\right)\right)\right)\right)} \]

   if 2.24999999999999997e148 < angle

   1. Initial program 16.4%

      \[\left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin \left(\pi \cdot \frac{angle}{180}\right)\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right) \]
   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. associate-*l* N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{sin.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{PI.f64}\left(\right), angle\right), 180\right)\right), \left(2 \cdot \color{blue}{\left(\left({b}^{2} - {a}^{2}\right) \cdot \cos \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right)\right)}\right)\right) \]

      12. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{sin.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{PI.f64}\left(\right), angle\right), 180\right)\right), \mathsf{*.f64}\left(2, \color{blue}{\left(\left({b}^{2} - {a}^{2}\right) \cdot \cos \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right)\right)}\right)\right) \]

      13. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{sin.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{PI.f64}\left(\right), angle\right), 180\right)\right), \mathsf{*.f64}\left(2, \mathsf{*.f64}\left(\left({b}^{2} - {a}^{2}\right), \color{blue}{\cos \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right)}\right)\right)\right) \]

   3. Simplified 13.1%

      \[\leadsto \color{blue}{\sin \left(\frac{\pi \cdot angle}{180}\right) \cdot \left(2 \cdot \left(\left(b \cdot b - a \cdot a\right) \cdot \cos \left(\frac{\pi \cdot angle}{180}\right)\right)\right)} \]
   4. Add Preprocessing

   5. Taylor expanded in angle around 0

      \[\leadsto \color{blue}{\frac{1}{90} \cdot \left(angle \cdot \left(\mathsf{PI}\left(\right) \cdot \left({b}^{2} - {a}^{2}\right)\right)\right)} \]
   6. Step-by-step derivation

      1. associate-*r* N/A

         \[\leadsto \left(\frac{1}{90} \cdot angle\right) \cdot \color{blue}{\left(\mathsf{PI}\left(\right) \cdot \left({b}^{2} - {a}^{2}\right)\right)} \]

      2. *-commutative N/A

         \[\leadsto \left(\mathsf{PI}\left(\right) \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \color{blue}{\left(\frac{1}{90} \cdot angle\right)} \]

      3. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\left(\mathsf{PI}\left(\right) \cdot \left({b}^{2} - {a}^{2}\right)\right), \color{blue}{\left(\frac{1}{90} \cdot angle\right)}\right) \]

      4. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{PI}\left(\right), \left({b}^{2} - {a}^{2}\right)\right), \left(\color{blue}{\frac{1}{90}} \cdot angle\right)\right) \]

      5. PI-lowering-PI.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{PI.f64}\left(\right), \left({b}^{2} - {a}^{2}\right)\right), \left(\frac{1}{90} \cdot angle\right)\right) \]

      6. --lowering--.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{PI.f64}\left(\right), \mathsf{\_.f64}\left(\left({b}^{2}\right), \left({a}^{2}\right)\right)\right), \left(\frac{1}{90} \cdot angle\right)\right) \]

      7. unpow2 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{PI.f64}\left(\right), \mathsf{\_.f64}\left(\left(b \cdot b\right), \left({a}^{2}\right)\right)\right), \left(\frac{1}{90} \cdot angle\right)\right) \]

      8. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{PI.f64}\left(\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(b, b\right), \left({a}^{2}\right)\right)\right), \left(\frac{1}{90} \cdot angle\right)\right) \]

      9. unpow2 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{PI.f64}\left(\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(b, b\right), \left(a \cdot a\right)\right)\right), \left(\frac{1}{90} \cdot angle\right)\right) \]

      10. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{PI.f64}\left(\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(b, b\right), \mathsf{*.f64}\left(a, a\right)\right)\right), \left(\frac{1}{90} \cdot angle\right)\right) \]

      11. *-lowering-*.f64 25.9%

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{PI.f64}\left(\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(b, b\right), \mathsf{*.f64}\left(a, a\right)\right)\right), \mathsf{*.f64}\left(\frac{1}{90}, \color{blue}{angle}\right)\right) \]

   7. Simplified 25.9%

      \[\leadsto \color{blue}{\left(\pi \cdot \left(b \cdot b - a \cdot a\right)\right) \cdot \left(0.011111111111111112 \cdot angle\right)} \]
   8. Step-by-step derivation

      1. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\left(\mathsf{PI}\left(\right) \cdot \left(b \cdot b - a \cdot a\right)\right), \color{blue}{\left(\frac{1}{90} \cdot angle\right)}\right) \]

      2. *-commutative N/A

         \[\leadsto \mathsf{*.f64}\left(\left(\left(b \cdot b - a \cdot a\right) \cdot \mathsf{PI}\left(\right)\right), \left(\color{blue}{\frac{1}{90}} \cdot angle\right)\right) \]

      3. difference-of-squares N/A

         \[\leadsto \mathsf{*.f64}\left(\left(\left(\left(b + a\right) \cdot \left(b - a\right)\right) \cdot \mathsf{PI}\left(\right)\right), \left(\frac{1}{90} \cdot angle\right)\right) \]

      4. associate-*l* N/A

         \[\leadsto \mathsf{*.f64}\left(\left(\left(b + a\right) \cdot \left(\left(b - a\right) \cdot \mathsf{PI}\left(\right)\right)\right), \left(\color{blue}{\frac{1}{90}} \cdot angle\right)\right) \]

      5. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\left(b + a\right), \left(\left(b - a\right) \cdot \mathsf{PI}\left(\right)\right)\right), \left(\color{blue}{\frac{1}{90}} \cdot angle\right)\right) \]

      6. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(b, a\right), \left(\left(b - a\right) \cdot \mathsf{PI}\left(\right)\right)\right), \left(\frac{1}{90} \cdot angle\right)\right) \]

      7. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(b, a\right), \mathsf{*.f64}\left(\left(b - a\right), \mathsf{PI}\left(\right)\right)\right), \left(\frac{1}{90} \cdot angle\right)\right) \]

      8. --lowering--.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(b, a\right), \mathsf{*.f64}\left(\mathsf{\_.f64}\left(b, a\right), \mathsf{PI}\left(\right)\right)\right), \left(\frac{1}{90} \cdot angle\right)\right) \]

      9. PI-lowering-PI.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(b, a\right), \mathsf{*.f64}\left(\mathsf{\_.f64}\left(b, a\right), \mathsf{PI.f64}\left(\right)\right)\right), \left(\frac{1}{90} \cdot angle\right)\right) \]

      10. *-commutative N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(b, a\right), \mathsf{*.f64}\left(\mathsf{\_.f64}\left(b, a\right), \mathsf{PI.f64}\left(\right)\right)\right), \left(angle \cdot \color{blue}{\frac{1}{90}}\right)\right) \]

      11. *-lowering-*.f64 25.9%

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(b, a\right), \mathsf{*.f64}\left(\mathsf{\_.f64}\left(b, a\right), \mathsf{PI.f64}\left(\right)\right)\right), \mathsf{*.f64}\left(angle, \color{blue}{\frac{1}{90}}\right)\right) \]

   9. Applied egg-rr 25.9%

      \[\leadsto \color{blue}{\left(\left(b + a\right) \cdot \left(\left(b - a\right) \cdot \pi\right)\right) \cdot \left(angle \cdot 0.011111111111111112\right)} \]
3. Recombined 3 regimes into one program.

4. Final simplification 57.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;angle \leq 1.8 \cdot 10^{+82}:\\ \;\;\;\;\left(b + a\right) \cdot \left(6.17283950617284 \cdot 10^{-5} \cdot \left(\left(angle \cdot \left(b - a\right)\right) \cdot \left(180 + \left(-180 + 180 \cdot \pi\right)\right)\right)\right)\\ \mathbf{elif}\;angle \leq 2.25 \cdot 10^{+148}:\\ \;\;\;\;\left(b + a\right) \cdot \left(angle \cdot \left(0.011111111111111112 \cdot \left(\pi \cdot \left(b - a\right)\right) + \left(2 \cdot \left(angle \cdot angle\right)\right) \cdot \left(\left(b - a\right) \cdot \left(\left(\pi \cdot \left(\pi \cdot \pi\right)\right) \cdot -1.1431184270690443 \cdot 10^{-7}\right)\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\left(b + a\right) \cdot \left(\pi \cdot \left(b - a\right)\right)\right) \cdot \left(angle \cdot 0.011111111111111112\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 15: 63.6% accurate, 17.4× speedup

Math

\[\begin{array}{l} a_m = \left|a\right| \\ angle\_m = \left|angle\right| \\ angle\_s = \mathsf{copysign}\left(1, angle\right) \\ angle\_s \cdot \begin{array}{l} \mathbf{if}\;angle\_m \leq 5.8 \cdot 10^{+78}:\\ \;\;\;\;\left(b + a\_m\right) \cdot \left(6.17283950617284 \cdot 10^{-5} \cdot \left(\left(angle\_m \cdot \left(b - a\_m\right)\right) \cdot \left(180 + \left(-180 + 180 \cdot \pi\right)\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\left(b \cdot b\right) \cdot \left(\pi - \pi \cdot \frac{\frac{a\_m \cdot a\_m}{b}}{b}\right)\right) \cdot \left(angle\_m \cdot 0.011111111111111112\right)\\ \end{array} \end{array} \]

FPCore

a_m = (fabs.f64 a)
angle\_m = (fabs.f64 angle)
angle\_s = (copysign.f64 #s(literal 1 binary64) angle)
(FPCore (angle_s a_m b angle_m)
 :precision binary64
 (*
  angle_s
  (if (<= angle_m 5.8e+78)
    (*
     (+ b a_m)
     (*
      6.17283950617284e-5
      (* (* angle_m (- b a_m)) (+ 180.0 (+ -180.0 (* 180.0 PI))))))
    (*
     (* (* b b) (- PI (* PI (/ (/ (* a_m a_m) b) b))))
     (* angle_m 0.011111111111111112)))))

C

a_m = fabs(a);
angle\_m = fabs(angle);
angle\_s = copysign(1.0, angle);
double code(double angle_s, double a_m, double b, double angle_m) {
	double tmp;
	if (angle_m <= 5.8e+78) {
		tmp = (b + a_m) * (6.17283950617284e-5 * ((angle_m * (b - a_m)) * (180.0 + (-180.0 + (180.0 * ((double) M_PI))))));
	} else {
		tmp = ((b * b) * (((double) M_PI) - (((double) M_PI) * (((a_m * a_m) / b) / b)))) * (angle_m * 0.011111111111111112);
	}
	return angle_s * tmp;
}

Java

a_m = Math.abs(a);
angle\_m = Math.abs(angle);
angle\_s = Math.copySign(1.0, angle);
public static double code(double angle_s, double a_m, double b, double angle_m) {
	double tmp;
	if (angle_m <= 5.8e+78) {
		tmp = (b + a_m) * (6.17283950617284e-5 * ((angle_m * (b - a_m)) * (180.0 + (-180.0 + (180.0 * Math.PI)))));
	} else {
		tmp = ((b * b) * (Math.PI - (Math.PI * (((a_m * a_m) / b) / b)))) * (angle_m * 0.011111111111111112);
	}
	return angle_s * tmp;
}

Python

a_m = math.fabs(a)
angle\_m = math.fabs(angle)
angle\_s = math.copysign(1.0, angle)
def code(angle_s, a_m, b, angle_m):
	tmp = 0
	if angle_m <= 5.8e+78:
		tmp = (b + a_m) * (6.17283950617284e-5 * ((angle_m * (b - a_m)) * (180.0 + (-180.0 + (180.0 * math.pi)))))
	else:
		tmp = ((b * b) * (math.pi - (math.pi * (((a_m * a_m) / b) / b)))) * (angle_m * 0.011111111111111112)
	return angle_s * tmp

Julia

a_m = abs(a)
angle\_m = abs(angle)
angle\_s = copysign(1.0, angle)
function code(angle_s, a_m, b, angle_m)
	tmp = 0.0
	if (angle_m <= 5.8e+78)
		tmp = Float64(Float64(b + a_m) * Float64(6.17283950617284e-5 * Float64(Float64(angle_m * Float64(b - a_m)) * Float64(180.0 + Float64(-180.0 + Float64(180.0 * pi))))));
	else
		tmp = Float64(Float64(Float64(b * b) * Float64(pi - Float64(pi * Float64(Float64(Float64(a_m * a_m) / b) / b)))) * Float64(angle_m * 0.011111111111111112));
	end
	return Float64(angle_s * tmp)
end

MATLAB

a_m = abs(a);
angle\_m = abs(angle);
angle\_s = sign(angle) * abs(1.0);
function tmp_2 = code(angle_s, a_m, b, angle_m)
	tmp = 0.0;
	if (angle_m <= 5.8e+78)
		tmp = (b + a_m) * (6.17283950617284e-5 * ((angle_m * (b - a_m)) * (180.0 + (-180.0 + (180.0 * pi)))));
	else
		tmp = ((b * b) * (pi - (pi * (((a_m * a_m) / b) / b)))) * (angle_m * 0.011111111111111112);
	end
	tmp_2 = angle_s * tmp;
end

Wolfram

a_m = N[Abs[a], $MachinePrecision]
angle\_m = N[Abs[angle], $MachinePrecision]
angle\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[angle]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[angle$95$s_, a$95$m_, b_, angle$95$m_] := N[(angle$95$s * If[LessEqual[angle$95$m, 5.8e+78], N[(N[(b + a$95$m), $MachinePrecision] * N[(6.17283950617284e-5 * N[(N[(angle$95$m * N[(b - a$95$m), $MachinePrecision]), $MachinePrecision] * N[(180.0 + N[(-180.0 + N[(180.0 * Pi), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(b * b), $MachinePrecision] * N[(Pi - N[(Pi * N[(N[(N[(a$95$m * a$95$m), $MachinePrecision] / b), $MachinePrecision] / b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(angle$95$m * 0.011111111111111112), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if angle < 5.80000000000000034e78

   1. Initial program 57.7%

      \[\left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin \left(\pi \cdot \frac{angle}{180}\right)\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right) \]
   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. associate-*l* N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{sin.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{PI.f64}\left(\right), angle\right), 180\right)\right), \left(2 \cdot \color{blue}{\left(\left({b}^{2} - {a}^{2}\right) \cdot \cos \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right)\right)}\right)\right) \]

      12. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{sin.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{PI.f64}\left(\right), angle\right), 180\right)\right), \mathsf{*.f64}\left(2, \color{blue}{\left(\left({b}^{2} - {a}^{2}\right) \cdot \cos \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right)\right)}\right)\right) \]

      13. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{sin.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{PI.f64}\left(\right), angle\right), 180\right)\right), \mathsf{*.f64}\left(2, \mathsf{*.f64}\left(\left({b}^{2} - {a}^{2}\right), \color{blue}{\cos \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right)}\right)\right)\right) \]

   3. Simplified 57.5%

      \[\leadsto \color{blue}{\sin \left(\frac{\pi \cdot angle}{180}\right) \cdot \left(2 \cdot \left(\left(b \cdot b - a \cdot a\right) \cdot \cos \left(\frac{\pi \cdot angle}{180}\right)\right)\right)} \]
   4. Add Preprocessing
   5. Step-by-step derivation

      1. associate-*r* N/A

         \[\leadsto \left(\sin \left(\frac{\mathsf{PI}\left(\right) \cdot angle}{180}\right) \cdot 2\right) \cdot \color{blue}{\left(\left(b \cdot b - a \cdot a\right) \cdot \cos \left(\frac{\mathsf{PI}\left(\right) \cdot angle}{180}\right)\right)} \]

      2. *-commutative N/A

         \[\leadsto \left(\left(b \cdot b - a \cdot a\right) \cdot \cos \left(\frac{\mathsf{PI}\left(\right) \cdot angle}{180}\right)\right) \cdot \color{blue}{\left(\sin \left(\frac{\mathsf{PI}\left(\right) \cdot angle}{180}\right) \cdot 2\right)} \]

      3. difference-of-squares N/A

         \[\leadsto \left(\left(\left(b + a\right) \cdot \left(b - a\right)\right) \cdot \cos \left(\frac{\mathsf{PI}\left(\right) \cdot angle}{180}\right)\right) \cdot \left(\sin \color{blue}{\left(\frac{\mathsf{PI}\left(\right) \cdot angle}{180}\right)} \cdot 2\right) \]

      4. associate-*r/ N/A

         \[\leadsto \left(\left(\left(b + a\right) \cdot \left(b - a\right)\right) \cdot \cos \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right)\right) \cdot \left(\sin \left(\frac{\mathsf{PI}\left(\right) \cdot angle}{180}\right) \cdot 2\right) \]

      5. associate-*l* N/A

         \[\leadsto \left(\left(b + a\right) \cdot \left(\left(b - a\right) \cdot \cos \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right)\right)\right) \cdot \left(\color{blue}{\sin \left(\frac{\mathsf{PI}\left(\right) \cdot angle}{180}\right)} \cdot 2\right) \]

      6. associate-*l* N/A

         \[\leadsto \left(b + a\right) \cdot \color{blue}{\left(\left(\left(b - a\right) \cdot \cos \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right)\right) \cdot \left(\sin \left(\frac{\mathsf{PI}\left(\right) \cdot angle}{180}\right) \cdot 2\right)\right)} \]

      7. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\left(b + a\right), \color{blue}{\left(\left(\left(b - a\right) \cdot \cos \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right)\right) \cdot \left(\sin \left(\frac{\mathsf{PI}\left(\right) \cdot angle}{180}\right) \cdot 2\right)\right)}\right) \]

      8. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(b, a\right), \left(\color{blue}{\left(\left(b - a\right) \cdot \cos \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right)\right)} \cdot \left(\sin \left(\frac{\mathsf{PI}\left(\right) \cdot angle}{180}\right) \cdot 2\right)\right)\right) \]

      9. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(b, a\right), \mathsf{*.f64}\left(\left(\left(b - a\right) \cdot \cos \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right)\right), \color{blue}{\left(\sin \left(\frac{\mathsf{PI}\left(\right) \cdot angle}{180}\right) \cdot 2\right)}\right)\right) \]

   6. Applied egg-rr 69.5%

      \[\leadsto \color{blue}{\left(b + a\right) \cdot \left(\left(\cos \left(\frac{\pi}{\frac{180}{angle}}\right) \cdot \left(b - a\right)\right) \cdot \left(2 \cdot \sin \left(\frac{\pi}{\frac{180}{angle}}\right)\right)\right)} \]
   7. Step-by-step derivation

      1. expm1-log1p-u N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(b, a\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{cos.f64}\left(\mathsf{/.f64}\left(\mathsf{PI.f64}\left(\right), \mathsf{/.f64}\left(180, angle\right)\right)\right), \mathsf{\_.f64}\left(b, a\right)\right), \mathsf{*.f64}\left(2, \mathsf{sin.f64}\left(\left(\frac{\mathsf{expm1}\left(\mathsf{log1p}\left(\mathsf{PI}\left(\right)\right)\right)}{\frac{180}{angle}}\right)\right)\right)\right)\right) \]

      2. expm1-undefine N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(b, a\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{cos.f64}\left(\mathsf{/.f64}\left(\mathsf{PI.f64}\left(\right), \mathsf{/.f64}\left(180, angle\right)\right)\right), \mathsf{\_.f64}\left(b, a\right)\right), \mathsf{*.f64}\left(2, \mathsf{sin.f64}\left(\left(\frac{e^{\mathsf{log1p}\left(\mathsf{PI}\left(\right)\right)} - 1}{\frac{180}{angle}}\right)\right)\right)\right)\right) \]

      3. log1p-undefine N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(b, a\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{cos.f64}\left(\mathsf{/.f64}\left(\mathsf{PI.f64}\left(\right), \mathsf{/.f64}\left(180, angle\right)\right)\right), \mathsf{\_.f64}\left(b, a\right)\right), \mathsf{*.f64}\left(2, \mathsf{sin.f64}\left(\left(\frac{e^{\log \left(1 + \mathsf{PI}\left(\right)\right)} - 1}{\frac{180}{angle}}\right)\right)\right)\right)\right) \]

      4. +-commutative N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(b, a\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{cos.f64}\left(\mathsf{/.f64}\left(\mathsf{PI.f64}\left(\right), \mathsf{/.f64}\left(180, angle\right)\right)\right), \mathsf{\_.f64}\left(b, a\right)\right), \mathsf{*.f64}\left(2, \mathsf{sin.f64}\left(\left(\frac{e^{\log \left(\mathsf{PI}\left(\right) + 1\right)} - 1}{\frac{180}{angle}}\right)\right)\right)\right)\right) \]

      5. rem-exp-log N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(b, a\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{cos.f64}\left(\mathsf{/.f64}\left(\mathsf{PI.f64}\left(\right), \mathsf{/.f64}\left(180, angle\right)\right)\right), \mathsf{\_.f64}\left(b, a\right)\right), \mathsf{*.f64}\left(2, \mathsf{sin.f64}\left(\left(\frac{\left(\mathsf{PI}\left(\right) + 1\right) - 1}{\frac{180}{angle}}\right)\right)\right)\right)\right) \]

      6. sub-div N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(b, a\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{cos.f64}\left(\mathsf{/.f64}\left(\mathsf{PI.f64}\left(\right), \mathsf{/.f64}\left(180, angle\right)\right)\right), \mathsf{\_.f64}\left(b, a\right)\right), \mathsf{*.f64}\left(2, \mathsf{sin.f64}\left(\left(\frac{\mathsf{PI}\left(\right) + 1}{\frac{180}{angle}} - \frac{1}{\frac{180}{angle}}\right)\right)\right)\right)\right) \]

      7. associate-/r/ N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(b, a\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{cos.f64}\left(\mathsf{/.f64}\left(\mathsf{PI.f64}\left(\right), \mathsf{/.f64}\left(180, angle\right)\right)\right), \mathsf{\_.f64}\left(b, a\right)\right), \mathsf{*.f64}\left(2, \mathsf{sin.f64}\left(\left(\frac{\mathsf{PI}\left(\right) + 1}{180} \cdot angle - \frac{1}{\frac{180}{angle}}\right)\right)\right)\right)\right) \]

      8. associate-*l/ N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(b, a\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{cos.f64}\left(\mathsf{/.f64}\left(\mathsf{PI.f64}\left(\right), \mathsf{/.f64}\left(180, angle\right)\right)\right), \mathsf{\_.f64}\left(b, a\right)\right), \mathsf{*.f64}\left(2, \mathsf{sin.f64}\left(\left(\frac{\left(\mathsf{PI}\left(\right) + 1\right) \cdot angle}{180} - \frac{1}{\frac{180}{angle}}\right)\right)\right)\right)\right) \]

      9. frac-sub N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(b, a\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{cos.f64}\left(\mathsf{/.f64}\left(\mathsf{PI.f64}\left(\right), \mathsf{/.f64}\left(180, angle\right)\right)\right), \mathsf{\_.f64}\left(b, a\right)\right), \mathsf{*.f64}\left(2, \mathsf{sin.f64}\left(\left(\frac{\left(\left(\mathsf{PI}\left(\right) + 1\right) \cdot angle\right) \cdot \frac{180}{angle} - 180 \cdot 1}{180 \cdot \frac{180}{angle}}\right)\right)\right)\right)\right) \]

      10. *-commutative N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(b, a\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{cos.f64}\left(\mathsf{/.f64}\left(\mathsf{PI.f64}\left(\right), \mathsf{/.f64}\left(180, angle\right)\right)\right), \mathsf{\_.f64}\left(b, a\right)\right), \mathsf{*.f64}\left(2, \mathsf{sin.f64}\left(\left(\frac{\left(\left(\mathsf{PI}\left(\right) + 1\right) \cdot angle\right) \cdot \frac{180}{angle} - 180 \cdot 1}{\frac{180}{angle} \cdot 180}\right)\right)\right)\right)\right) \]

      11. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(b, a\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{cos.f64}\left(\mathsf{/.f64}\left(\mathsf{PI.f64}\left(\right), \mathsf{/.f64}\left(180, angle\right)\right)\right), \mathsf{\_.f64}\left(b, a\right)\right), \mathsf{*.f64}\left(2, \mathsf{sin.f64}\left(\mathsf{/.f64}\left(\left(\left(\left(\mathsf{PI}\left(\right) + 1\right) \cdot angle\right) \cdot \frac{180}{angle} - 180 \cdot 1\right), \left(\frac{180}{angle} \cdot 180\right)\right)\right)\right)\right)\right) \]

   8. Applied egg-rr 71.1%

      \[\leadsto \left(b + a\right) \cdot \left(\left(\cos \left(\frac{\pi}{\frac{180}{angle}}\right) \cdot \left(b - a\right)\right) \cdot \left(2 \cdot \sin \color{blue}{\left(\frac{\left(\left(\pi + 1\right) \cdot angle\right) \cdot \frac{180}{angle} - 180}{\frac{32400}{angle}}\right)}\right)\right) \]

   9. Taylor expanded in angle around 0

      \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(b, a\right), \color{blue}{\left(\frac{1}{16200} \cdot \left(angle \cdot \left(\left(b - a\right) \cdot \left(180 \cdot \left(1 + \mathsf{PI}\left(\right)\right) - 180\right)\right)\right)\right)}\right) \]
   10. Step-by-step derivation

       1. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(b, a\right), \mathsf{*.f64}\left(\frac{1}{16200}, \color{blue}{\left(angle \cdot \left(\left(b - a\right) \cdot \left(180 \cdot \left(1 + \mathsf{PI}\left(\right)\right) - 180\right)\right)\right)}\right)\right) \]

       2. associate-*r* N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(b, a\right), \mathsf{*.f64}\left(\frac{1}{16200}, \left(\left(angle \cdot \left(b - a\right)\right) \cdot \color{blue}{\left(180 \cdot \left(1 + \mathsf{PI}\left(\right)\right) - 180\right)}\right)\right)\right) \]

       3. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(b, a\right), \mathsf{*.f64}\left(\frac{1}{16200}, \mathsf{*.f64}\left(\left(angle \cdot \left(b - a\right)\right), \color{blue}{\left(180 \cdot \left(1 + \mathsf{PI}\left(\right)\right) - 180\right)}\right)\right)\right) \]

       4. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(b, a\right), \mathsf{*.f64}\left(\frac{1}{16200}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(angle, \left(b - a\right)\right), \left(\color{blue}{180 \cdot \left(1 + \mathsf{PI}\left(\right)\right)} - 180\right)\right)\right)\right) \]

       5. --lowering--.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(b, a\right), \mathsf{*.f64}\left(\frac{1}{16200}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(angle, \mathsf{\_.f64}\left(b, a\right)\right), \left(180 \cdot \color{blue}{\left(1 + \mathsf{PI}\left(\right)\right)} - 180\right)\right)\right)\right) \]

       6. distribute-lft-in N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(b, a\right), \mathsf{*.f64}\left(\frac{1}{16200}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(angle, \mathsf{\_.f64}\left(b, a\right)\right), \left(\left(180 \cdot 1 + 180 \cdot \mathsf{PI}\left(\right)\right) - 180\right)\right)\right)\right) \]

       7. metadata-eval N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(b, a\right), \mathsf{*.f64}\left(\frac{1}{16200}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(angle, \mathsf{\_.f64}\left(b, a\right)\right), \left(\left(180 + 180 \cdot \mathsf{PI}\left(\right)\right) - 180\right)\right)\right)\right) \]

       8. associate--l+ N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(b, a\right), \mathsf{*.f64}\left(\frac{1}{16200}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(angle, \mathsf{\_.f64}\left(b, a\right)\right), \left(180 + \color{blue}{\left(180 \cdot \mathsf{PI}\left(\right) - 180\right)}\right)\right)\right)\right) \]

       9. +-lowering-+.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(b, a\right), \mathsf{*.f64}\left(\frac{1}{16200}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(angle, \mathsf{\_.f64}\left(b, a\right)\right), \mathsf{+.f64}\left(180, \color{blue}{\left(180 \cdot \mathsf{PI}\left(\right) - 180\right)}\right)\right)\right)\right) \]

       10. sub-neg N/A

           \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(b, a\right), \mathsf{*.f64}\left(\frac{1}{16200}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(angle, \mathsf{\_.f64}\left(b, a\right)\right), \mathsf{+.f64}\left(180, \left(180 \cdot \mathsf{PI}\left(\right) + \color{blue}{\left(\mathsf{neg}\left(180\right)\right)}\right)\right)\right)\right)\right) \]

       11. +-lowering-+.f64 N/A

           \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(b, a\right), \mathsf{*.f64}\left(\frac{1}{16200}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(angle, \mathsf{\_.f64}\left(b, a\right)\right), \mathsf{+.f64}\left(180, \mathsf{+.f64}\left(\left(180 \cdot \mathsf{PI}\left(\right)\right), \color{blue}{\left(\mathsf{neg}\left(180\right)\right)}\right)\right)\right)\right)\right) \]

       12. *-lowering-*.f64 N/A

           \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(b, a\right), \mathsf{*.f64}\left(\frac{1}{16200}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(angle, \mathsf{\_.f64}\left(b, a\right)\right), \mathsf{+.f64}\left(180, \mathsf{+.f64}\left(\mathsf{*.f64}\left(180, \mathsf{PI}\left(\right)\right), \left(\mathsf{neg}\left(\color{blue}{180}\right)\right)\right)\right)\right)\right)\right) \]

       13. PI-lowering-PI.f64 N/A

           \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(b, a\right), \mathsf{*.f64}\left(\frac{1}{16200}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(angle, \mathsf{\_.f64}\left(b, a\right)\right), \mathsf{+.f64}\left(180, \mathsf{+.f64}\left(\mathsf{*.f64}\left(180, \mathsf{PI.f64}\left(\right)\right), \left(\mathsf{neg}\left(180\right)\right)\right)\right)\right)\right)\right) \]

       14. metadata-eval 65.5%

           \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(b, a\right), \mathsf{*.f64}\left(\frac{1}{16200}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(angle, \mathsf{\_.f64}\left(b, a\right)\right), \mathsf{+.f64}\left(180, \mathsf{+.f64}\left(\mathsf{*.f64}\left(180, \mathsf{PI.f64}\left(\right)\right), -180\right)\right)\right)\right)\right) \]

   11. Simplified 65.5%

       \[\leadsto \left(b + a\right) \cdot \color{blue}{\left(6.17283950617284 \cdot 10^{-5} \cdot \left(\left(angle \cdot \left(b - a\right)\right) \cdot \left(180 + \left(180 \cdot \pi + -180\right)\right)\right)\right)} \]

   if 5.80000000000000034e78 < angle

   1. Initial program 19.0%

      \[\left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin \left(\pi \cdot \frac{angle}{180}\right)\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right) \]
   2. Step-by-step derivation

      1. associate-*l* 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. associate-*l* N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{sin.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{PI.f64}\left(\right), angle\right), 180\right)\right), \left(2 \cdot \color{blue}{\left(\left({b}^{2} - {a}^{2}\right) \cdot \cos \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right)\right)}\right)\right) \]

      12. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{sin.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{PI.f64}\left(\right), angle\right), 180\right)\right), \mathsf{*.f64}\left(2, \color{blue}{\left(\left({b}^{2} - {a}^{2}\right) \cdot \cos \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right)\right)}\right)\right) \]

      13. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{sin.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{PI.f64}\left(\right), angle\right), 180\right)\right), \mathsf{*.f64}\left(2, \mathsf{*.f64}\left(\left({b}^{2} - {a}^{2}\right), \color{blue}{\cos \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right)}\right)\right)\right) \]

   3. Simplified 18.7%

      \[\leadsto \color{blue}{\sin \left(\frac{\pi \cdot angle}{180}\right) \cdot \left(2 \cdot \left(\left(b \cdot b - a \cdot a\right) \cdot \cos \left(\frac{\pi \cdot angle}{180}\right)\right)\right)} \]
   4. Add Preprocessing

   5. Taylor expanded in angle around 0

      \[\leadsto \color{blue}{\frac{1}{90} \cdot \left(angle \cdot \left(\mathsf{PI}\left(\right) \cdot \left({b}^{2} - {a}^{2}\right)\right)\right)} \]
   6. Step-by-step derivation

      1. associate-*r* N/A

         \[\leadsto \left(\frac{1}{90} \cdot angle\right) \cdot \color{blue}{\left(\mathsf{PI}\left(\right) \cdot \left({b}^{2} - {a}^{2}\right)\right)} \]

      2. *-commutative N/A

         \[\leadsto \left(\mathsf{PI}\left(\right) \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \color{blue}{\left(\frac{1}{90} \cdot angle\right)} \]

      3. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\left(\mathsf{PI}\left(\right) \cdot \left({b}^{2} - {a}^{2}\right)\right), \color{blue}{\left(\frac{1}{90} \cdot angle\right)}\right) \]

      4. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{PI}\left(\right), \left({b}^{2} - {a}^{2}\right)\right), \left(\color{blue}{\frac{1}{90}} \cdot angle\right)\right) \]

      5. PI-lowering-PI.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{PI.f64}\left(\right), \left({b}^{2} - {a}^{2}\right)\right), \left(\frac{1}{90} \cdot angle\right)\right) \]

      6. --lowering--.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{PI.f64}\left(\right), \mathsf{\_.f64}\left(\left({b}^{2}\right), \left({a}^{2}\right)\right)\right), \left(\frac{1}{90} \cdot angle\right)\right) \]

      7. unpow2 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{PI.f64}\left(\right), \mathsf{\_.f64}\left(\left(b \cdot b\right), \left({a}^{2}\right)\right)\right), \left(\frac{1}{90} \cdot angle\right)\right) \]

      8. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{PI.f64}\left(\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(b, b\right), \left({a}^{2}\right)\right)\right), \left(\frac{1}{90} \cdot angle\right)\right) \]

      9. unpow2 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{PI.f64}\left(\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(b, b\right), \left(a \cdot a\right)\right)\right), \left(\frac{1}{90} \cdot angle\right)\right) \]

      10. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{PI.f64}\left(\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(b, b\right), \mathsf{*.f64}\left(a, a\right)\right)\right), \left(\frac{1}{90} \cdot angle\right)\right) \]

      11. *-lowering-*.f64 22.2%

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{PI.f64}\left(\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(b, b\right), \mathsf{*.f64}\left(a, a\right)\right)\right), \mathsf{*.f64}\left(\frac{1}{90}, \color{blue}{angle}\right)\right) \]

   7. Simplified 22.2%

      \[\leadsto \color{blue}{\left(\pi \cdot \left(b \cdot b - a \cdot a\right)\right) \cdot \left(0.011111111111111112 \cdot angle\right)} \]

   8. Taylor expanded in b around inf

      \[\leadsto \mathsf{*.f64}\left(\color{blue}{\left({b}^{2} \cdot \left(\mathsf{PI}\left(\right) + -1 \cdot \frac{{a}^{2} \cdot \mathsf{PI}\left(\right)}{{b}^{2}}\right)\right)}, \mathsf{*.f64}\left(\frac{1}{90}, angle\right)\right) \]
   9. Step-by-step derivation

      1. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\left({b}^{2}\right), \left(\mathsf{PI}\left(\right) + -1 \cdot \frac{{a}^{2} \cdot \mathsf{PI}\left(\right)}{{b}^{2}}\right)\right), \mathsf{*.f64}\left(\color{blue}{\frac{1}{90}}, angle\right)\right) \]

      2. unpow2 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\left(b \cdot b\right), \left(\mathsf{PI}\left(\right) + -1 \cdot \frac{{a}^{2} \cdot \mathsf{PI}\left(\right)}{{b}^{2}}\right)\right), \mathsf{*.f64}\left(\frac{1}{90}, angle\right)\right) \]

      3. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(b, b\right), \left(\mathsf{PI}\left(\right) + -1 \cdot \frac{{a}^{2} \cdot \mathsf{PI}\left(\right)}{{b}^{2}}\right)\right), \mathsf{*.f64}\left(\frac{1}{90}, angle\right)\right) \]

      4. mul-1-neg N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(b, b\right), \left(\mathsf{PI}\left(\right) + \left(\mathsf{neg}\left(\frac{{a}^{2} \cdot \mathsf{PI}\left(\right)}{{b}^{2}}\right)\right)\right)\right), \mathsf{*.f64}\left(\frac{1}{90}, angle\right)\right) \]

      5. unsub-neg N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(b, b\right), \left(\mathsf{PI}\left(\right) - \frac{{a}^{2} \cdot \mathsf{PI}\left(\right)}{{b}^{2}}\right)\right), \mathsf{*.f64}\left(\frac{1}{90}, angle\right)\right) \]

      6. --lowering--.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(b, b\right), \mathsf{\_.f64}\left(\mathsf{PI}\left(\right), \left(\frac{{a}^{2} \cdot \mathsf{PI}\left(\right)}{{b}^{2}}\right)\right)\right), \mathsf{*.f64}\left(\frac{1}{90}, angle\right)\right) \]

      7. PI-lowering-PI.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(b, b\right), \mathsf{\_.f64}\left(\mathsf{PI.f64}\left(\right), \left(\frac{{a}^{2} \cdot \mathsf{PI}\left(\right)}{{b}^{2}}\right)\right)\right), \mathsf{*.f64}\left(\frac{1}{90}, angle\right)\right) \]

      8. *-commutative N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(b, b\right), \mathsf{\_.f64}\left(\mathsf{PI.f64}\left(\right), \left(\frac{\mathsf{PI}\left(\right) \cdot {a}^{2}}{{b}^{2}}\right)\right)\right), \mathsf{*.f64}\left(\frac{1}{90}, angle\right)\right) \]

      9. associate-/l* N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(b, b\right), \mathsf{\_.f64}\left(\mathsf{PI.f64}\left(\right), \left(\mathsf{PI}\left(\right) \cdot \frac{{a}^{2}}{{b}^{2}}\right)\right)\right), \mathsf{*.f64}\left(\frac{1}{90}, angle\right)\right) \]

      10. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(b, b\right), \mathsf{\_.f64}\left(\mathsf{PI.f64}\left(\right), \mathsf{*.f64}\left(\mathsf{PI}\left(\right), \left(\frac{{a}^{2}}{{b}^{2}}\right)\right)\right)\right), \mathsf{*.f64}\left(\frac{1}{90}, angle\right)\right) \]

      11. PI-lowering-PI.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(b, b\right), \mathsf{\_.f64}\left(\mathsf{PI.f64}\left(\right), \mathsf{*.f64}\left(\mathsf{PI.f64}\left(\right), \left(\frac{{a}^{2}}{{b}^{2}}\right)\right)\right)\right), \mathsf{*.f64}\left(\frac{1}{90}, angle\right)\right) \]

      12. unpow2 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(b, b\right), \mathsf{\_.f64}\left(\mathsf{PI.f64}\left(\right), \mathsf{*.f64}\left(\mathsf{PI.f64}\left(\right), \left(\frac{{a}^{2}}{b \cdot b}\right)\right)\right)\right), \mathsf{*.f64}\left(\frac{1}{90}, angle\right)\right) \]

      13. associate-/r* N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(b, b\right), \mathsf{\_.f64}\left(\mathsf{PI.f64}\left(\right), \mathsf{*.f64}\left(\mathsf{PI.f64}\left(\right), \left(\frac{\frac{{a}^{2}}{b}}{b}\right)\right)\right)\right), \mathsf{*.f64}\left(\frac{1}{90}, angle\right)\right) \]

      14. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(b, b\right), \mathsf{\_.f64}\left(\mathsf{PI.f64}\left(\right), \mathsf{*.f64}\left(\mathsf{PI.f64}\left(\right), \mathsf{/.f64}\left(\left(\frac{{a}^{2}}{b}\right), b\right)\right)\right)\right), \mathsf{*.f64}\left(\frac{1}{90}, angle\right)\right) \]

      15. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(b, b\right), \mathsf{\_.f64}\left(\mathsf{PI.f64}\left(\right), \mathsf{*.f64}\left(\mathsf{PI.f64}\left(\right), \mathsf{/.f64}\left(\mathsf{/.f64}\left(\left({a}^{2}\right), b\right), b\right)\right)\right)\right), \mathsf{*.f64}\left(\frac{1}{90}, angle\right)\right) \]

      16. unpow2 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(b, b\right), \mathsf{\_.f64}\left(\mathsf{PI.f64}\left(\right), \mathsf{*.f64}\left(\mathsf{PI.f64}\left(\right), \mathsf{/.f64}\left(\mathsf{/.f64}\left(\left(a \cdot a\right), b\right), b\right)\right)\right)\right), \mathsf{*.f64}\left(\frac{1}{90}, angle\right)\right) \]

      17. *-lowering-*.f64 24.2%

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(b, b\right), \mathsf{\_.f64}\left(\mathsf{PI.f64}\left(\right), \mathsf{*.f64}\left(\mathsf{PI.f64}\left(\right), \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(a, a\right), b\right), b\right)\right)\right)\right), \mathsf{*.f64}\left(\frac{1}{90}, angle\right)\right) \]

   10. Simplified 24.2%

       \[\leadsto \color{blue}{\left(\left(b \cdot b\right) \cdot \left(\pi - \pi \cdot \frac{\frac{a \cdot a}{b}}{b}\right)\right)} \cdot \left(0.011111111111111112 \cdot angle\right) \]
3. Recombined 2 regimes into one program.

4. Final simplification 56.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;angle \leq 5.8 \cdot 10^{+78}:\\ \;\;\;\;\left(b + a\right) \cdot \left(6.17283950617284 \cdot 10^{-5} \cdot \left(\left(angle \cdot \left(b - a\right)\right) \cdot \left(180 + \left(-180 + 180 \cdot \pi\right)\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\left(b \cdot b\right) \cdot \left(\pi - \pi \cdot \frac{\frac{a \cdot a}{b}}{b}\right)\right) \cdot \left(angle \cdot 0.011111111111111112\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 16: 63.6% accurate, 17.4× speedup

Math

\[\begin{array}{l} a_m = \left|a\right| \\ angle\_m = \left|angle\right| \\ angle\_s = \mathsf{copysign}\left(1, angle\right) \\ angle\_s \cdot \begin{array}{l} \mathbf{if}\;angle\_m \leq 1.12 \cdot 10^{+79}:\\ \;\;\;\;\left(b + a\_m\right) \cdot \left(0.011111111111111112 \cdot \left(angle\_m \cdot \left(\pi \cdot \left(b - a\_m\right)\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\left(b \cdot b\right) \cdot \left(\pi - \pi \cdot \frac{\frac{a\_m \cdot a\_m}{b}}{b}\right)\right) \cdot \left(angle\_m \cdot 0.011111111111111112\right)\\ \end{array} \end{array} \]

FPCore

a_m = (fabs.f64 a)
angle\_m = (fabs.f64 angle)
angle\_s = (copysign.f64 #s(literal 1 binary64) angle)
(FPCore (angle_s a_m b angle_m)
 :precision binary64
 (*
  angle_s
  (if (<= angle_m 1.12e+79)
    (* (+ b a_m) (* 0.011111111111111112 (* angle_m (* PI (- b a_m)))))
    (*
     (* (* b b) (- PI (* PI (/ (/ (* a_m a_m) b) b))))
     (* angle_m 0.011111111111111112)))))

C

a_m = fabs(a);
angle\_m = fabs(angle);
angle\_s = copysign(1.0, angle);
double code(double angle_s, double a_m, double b, double angle_m) {
	double tmp;
	if (angle_m <= 1.12e+79) {
		tmp = (b + a_m) * (0.011111111111111112 * (angle_m * (((double) M_PI) * (b - a_m))));
	} else {
		tmp = ((b * b) * (((double) M_PI) - (((double) M_PI) * (((a_m * a_m) / b) / b)))) * (angle_m * 0.011111111111111112);
	}
	return angle_s * tmp;
}

Java

a_m = Math.abs(a);
angle\_m = Math.abs(angle);
angle\_s = Math.copySign(1.0, angle);
public static double code(double angle_s, double a_m, double b, double angle_m) {
	double tmp;
	if (angle_m <= 1.12e+79) {
		tmp = (b + a_m) * (0.011111111111111112 * (angle_m * (Math.PI * (b - a_m))));
	} else {
		tmp = ((b * b) * (Math.PI - (Math.PI * (((a_m * a_m) / b) / b)))) * (angle_m * 0.011111111111111112);
	}
	return angle_s * tmp;
}

Python

a_m = math.fabs(a)
angle\_m = math.fabs(angle)
angle\_s = math.copysign(1.0, angle)
def code(angle_s, a_m, b, angle_m):
	tmp = 0
	if angle_m <= 1.12e+79:
		tmp = (b + a_m) * (0.011111111111111112 * (angle_m * (math.pi * (b - a_m))))
	else:
		tmp = ((b * b) * (math.pi - (math.pi * (((a_m * a_m) / b) / b)))) * (angle_m * 0.011111111111111112)
	return angle_s * tmp

Julia

a_m = abs(a)
angle\_m = abs(angle)
angle\_s = copysign(1.0, angle)
function code(angle_s, a_m, b, angle_m)
	tmp = 0.0
	if (angle_m <= 1.12e+79)
		tmp = Float64(Float64(b + a_m) * Float64(0.011111111111111112 * Float64(angle_m * Float64(pi * Float64(b - a_m)))));
	else
		tmp = Float64(Float64(Float64(b * b) * Float64(pi - Float64(pi * Float64(Float64(Float64(a_m * a_m) / b) / b)))) * Float64(angle_m * 0.011111111111111112));
	end
	return Float64(angle_s * tmp)
end

MATLAB

a_m = abs(a);
angle\_m = abs(angle);
angle\_s = sign(angle) * abs(1.0);
function tmp_2 = code(angle_s, a_m, b, angle_m)
	tmp = 0.0;
	if (angle_m <= 1.12e+79)
		tmp = (b + a_m) * (0.011111111111111112 * (angle_m * (pi * (b - a_m))));
	else
		tmp = ((b * b) * (pi - (pi * (((a_m * a_m) / b) / b)))) * (angle_m * 0.011111111111111112);
	end
	tmp_2 = angle_s * tmp;
end

Wolfram

a_m = N[Abs[a], $MachinePrecision]
angle\_m = N[Abs[angle], $MachinePrecision]
angle\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[angle]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[angle$95$s_, a$95$m_, b_, angle$95$m_] := N[(angle$95$s * If[LessEqual[angle$95$m, 1.12e+79], N[(N[(b + a$95$m), $MachinePrecision] * N[(0.011111111111111112 * N[(angle$95$m * N[(Pi * N[(b - a$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(b * b), $MachinePrecision] * N[(Pi - N[(Pi * N[(N[(N[(a$95$m * a$95$m), $MachinePrecision] / b), $MachinePrecision] / b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(angle$95$m * 0.011111111111111112), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if angle < 1.12e79

   1. Initial program 57.7%

      \[\left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin \left(\pi \cdot \frac{angle}{180}\right)\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right) \]
   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. associate-*l* N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{sin.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{PI.f64}\left(\right), angle\right), 180\right)\right), \left(2 \cdot \color{blue}{\left(\left({b}^{2} - {a}^{2}\right) \cdot \cos \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right)\right)}\right)\right) \]

      12. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{sin.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{PI.f64}\left(\right), angle\right), 180\right)\right), \mathsf{*.f64}\left(2, \color{blue}{\left(\left({b}^{2} - {a}^{2}\right) \cdot \cos \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right)\right)}\right)\right) \]

      13. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{sin.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{PI.f64}\left(\right), angle\right), 180\right)\right), \mathsf{*.f64}\left(2, \mathsf{*.f64}\left(\left({b}^{2} - {a}^{2}\right), \color{blue}{\cos \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right)}\right)\right)\right) \]

   3. Simplified 57.5%

      \[\leadsto \color{blue}{\sin \left(\frac{\pi \cdot angle}{180}\right) \cdot \left(2 \cdot \left(\left(b \cdot b - a \cdot a\right) \cdot \cos \left(\frac{\pi \cdot angle}{180}\right)\right)\right)} \]
   4. Add Preprocessing
   5. Step-by-step derivation

      1. associate-*r* N/A

         \[\leadsto \left(\sin \left(\frac{\mathsf{PI}\left(\right) \cdot angle}{180}\right) \cdot 2\right) \cdot \color{blue}{\left(\left(b \cdot b - a \cdot a\right) \cdot \cos \left(\frac{\mathsf{PI}\left(\right) \cdot angle}{180}\right)\right)} \]

      2. *-commutative N/A

         \[\leadsto \left(\left(b \cdot b - a \cdot a\right) \cdot \cos \left(\frac{\mathsf{PI}\left(\right) \cdot angle}{180}\right)\right) \cdot \color{blue}{\left(\sin \left(\frac{\mathsf{PI}\left(\right) \cdot angle}{180}\right) \cdot 2\right)} \]

      3. difference-of-squares N/A

         \[\leadsto \left(\left(\left(b + a\right) \cdot \left(b - a\right)\right) \cdot \cos \left(\frac{\mathsf{PI}\left(\right) \cdot angle}{180}\right)\right) \cdot \left(\sin \color{blue}{\left(\frac{\mathsf{PI}\left(\right) \cdot angle}{180}\right)} \cdot 2\right) \]

      4. associate-*r/ N/A

         \[\leadsto \left(\left(\left(b + a\right) \cdot \left(b - a\right)\right) \cdot \cos \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right)\right) \cdot \left(\sin \left(\frac{\mathsf{PI}\left(\right) \cdot angle}{180}\right) \cdot 2\right) \]

      5. associate-*l* N/A

         \[\leadsto \left(\left(b + a\right) \cdot \left(\left(b - a\right) \cdot \cos \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right)\right)\right) \cdot \left(\color{blue}{\sin \left(\frac{\mathsf{PI}\left(\right) \cdot angle}{180}\right)} \cdot 2\right) \]

      6. associate-*l* N/A

         \[\leadsto \left(b + a\right) \cdot \color{blue}{\left(\left(\left(b - a\right) \cdot \cos \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right)\right) \cdot \left(\sin \left(\frac{\mathsf{PI}\left(\right) \cdot angle}{180}\right) \cdot 2\right)\right)} \]

      7. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\left(b + a\right), \color{blue}{\left(\left(\left(b - a\right) \cdot \cos \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right)\right) \cdot \left(\sin \left(\frac{\mathsf{PI}\left(\right) \cdot angle}{180}\right) \cdot 2\right)\right)}\right) \]

      8. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(b, a\right), \left(\color{blue}{\left(\left(b - a\right) \cdot \cos \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right)\right)} \cdot \left(\sin \left(\frac{\mathsf{PI}\left(\right) \cdot angle}{180}\right) \cdot 2\right)\right)\right) \]

      9. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(b, a\right), \mathsf{*.f64}\left(\left(\left(b - a\right) \cdot \cos \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right)\right), \color{blue}{\left(\sin \left(\frac{\mathsf{PI}\left(\right) \cdot angle}{180}\right) \cdot 2\right)}\right)\right) \]

   6. Applied egg-rr 69.5%

      \[\leadsto \color{blue}{\left(b + a\right) \cdot \left(\left(\cos \left(\frac{\pi}{\frac{180}{angle}}\right) \cdot \left(b - a\right)\right) \cdot \left(2 \cdot \sin \left(\frac{\pi}{\frac{180}{angle}}\right)\right)\right)} \]

   7. Taylor expanded in angle around 0

      \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(b, a\right), \color{blue}{\left(\frac{1}{90} \cdot \left(angle \cdot \left(\mathsf{PI}\left(\right) \cdot \left(b - a\right)\right)\right)\right)}\right) \]
   8. Step-by-step derivation

      1. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(b, a\right), \mathsf{*.f64}\left(\frac{1}{90}, \color{blue}{\left(angle \cdot \left(\mathsf{PI}\left(\right) \cdot \left(b - a\right)\right)\right)}\right)\right) \]

      2. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(b, a\right), \mathsf{*.f64}\left(\frac{1}{90}, \mathsf{*.f64}\left(angle, \color{blue}{\left(\mathsf{PI}\left(\right) \cdot \left(b - a\right)\right)}\right)\right)\right) \]

      3. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(b, a\right), \mathsf{*.f64}\left(\frac{1}{90}, \mathsf{*.f64}\left(angle, \mathsf{*.f64}\left(\mathsf{PI}\left(\right), \color{blue}{\left(b - a\right)}\right)\right)\right)\right) \]

      4. PI-lowering-PI.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(b, a\right), \mathsf{*.f64}\left(\frac{1}{90}, \mathsf{*.f64}\left(angle, \mathsf{*.f64}\left(\mathsf{PI.f64}\left(\right), \left(\color{blue}{b} - a\right)\right)\right)\right)\right) \]

      5. --lowering--.f64 65.5%

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(b, a\right), \mathsf{*.f64}\left(\frac{1}{90}, \mathsf{*.f64}\left(angle, \mathsf{*.f64}\left(\mathsf{PI.f64}\left(\right), \mathsf{\_.f64}\left(b, \color{blue}{a}\right)\right)\right)\right)\right) \]

   9. Simplified 65.5%

      \[\leadsto \left(b + a\right) \cdot \color{blue}{\left(0.011111111111111112 \cdot \left(angle \cdot \left(\pi \cdot \left(b - a\right)\right)\right)\right)} \]

   if 1.12e79 < angle

   1. Initial program 19.0%

      \[\left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin \left(\pi \cdot \frac{angle}{180}\right)\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right) \]
   2. Step-by-step derivation

      1. associate-*l* 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. associate-*l* N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{sin.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{PI.f64}\left(\right), angle\right), 180\right)\right), \left(2 \cdot \color{blue}{\left(\left({b}^{2} - {a}^{2}\right) \cdot \cos \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right)\right)}\right)\right) \]

      12. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{sin.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{PI.f64}\left(\right), angle\right), 180\right)\right), \mathsf{*.f64}\left(2, \color{blue}{\left(\left({b}^{2} - {a}^{2}\right) \cdot \cos \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right)\right)}\right)\right) \]

      13. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{sin.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{PI.f64}\left(\right), angle\right), 180\right)\right), \mathsf{*.f64}\left(2, \mathsf{*.f64}\left(\left({b}^{2} - {a}^{2}\right), \color{blue}{\cos \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right)}\right)\right)\right) \]

   3. Simplified 18.7%

      \[\leadsto \color{blue}{\sin \left(\frac{\pi \cdot angle}{180}\right) \cdot \left(2 \cdot \left(\left(b \cdot b - a \cdot a\right) \cdot \cos \left(\frac{\pi \cdot angle}{180}\right)\right)\right)} \]
   4. Add Preprocessing

   5. Taylor expanded in angle around 0

      \[\leadsto \color{blue}{\frac{1}{90} \cdot \left(angle \cdot \left(\mathsf{PI}\left(\right) \cdot \left({b}^{2} - {a}^{2}\right)\right)\right)} \]
   6. Step-by-step derivation

      1. associate-*r* N/A

         \[\leadsto \left(\frac{1}{90} \cdot angle\right) \cdot \color{blue}{\left(\mathsf{PI}\left(\right) \cdot \left({b}^{2} - {a}^{2}\right)\right)} \]

      2. *-commutative N/A

         \[\leadsto \left(\mathsf{PI}\left(\right) \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \color{blue}{\left(\frac{1}{90} \cdot angle\right)} \]

      3. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\left(\mathsf{PI}\left(\right) \cdot \left({b}^{2} - {a}^{2}\right)\right), \color{blue}{\left(\frac{1}{90} \cdot angle\right)}\right) \]

      4. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{PI}\left(\right), \left({b}^{2} - {a}^{2}\right)\right), \left(\color{blue}{\frac{1}{90}} \cdot angle\right)\right) \]

      5. PI-lowering-PI.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{PI.f64}\left(\right), \left({b}^{2} - {a}^{2}\right)\right), \left(\frac{1}{90} \cdot angle\right)\right) \]

      6. --lowering--.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{PI.f64}\left(\right), \mathsf{\_.f64}\left(\left({b}^{2}\right), \left({a}^{2}\right)\right)\right), \left(\frac{1}{90} \cdot angle\right)\right) \]

      7. unpow2 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{PI.f64}\left(\right), \mathsf{\_.f64}\left(\left(b \cdot b\right), \left({a}^{2}\right)\right)\right), \left(\frac{1}{90} \cdot angle\right)\right) \]

      8. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{PI.f64}\left(\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(b, b\right), \left({a}^{2}\right)\right)\right), \left(\frac{1}{90} \cdot angle\right)\right) \]

      9. unpow2 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{PI.f64}\left(\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(b, b\right), \left(a \cdot a\right)\right)\right), \left(\frac{1}{90} \cdot angle\right)\right) \]

      10. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{PI.f64}\left(\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(b, b\right), \mathsf{*.f64}\left(a, a\right)\right)\right), \left(\frac{1}{90} \cdot angle\right)\right) \]

      11. *-lowering-*.f64 22.2%

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{PI.f64}\left(\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(b, b\right), \mathsf{*.f64}\left(a, a\right)\right)\right), \mathsf{*.f64}\left(\frac{1}{90}, \color{blue}{angle}\right)\right) \]

   7. Simplified 22.2%

      \[\leadsto \color{blue}{\left(\pi \cdot \left(b \cdot b - a \cdot a\right)\right) \cdot \left(0.011111111111111112 \cdot angle\right)} \]

   8. Taylor expanded in b around inf

      \[\leadsto \mathsf{*.f64}\left(\color{blue}{\left({b}^{2} \cdot \left(\mathsf{PI}\left(\right) + -1 \cdot \frac{{a}^{2} \cdot \mathsf{PI}\left(\right)}{{b}^{2}}\right)\right)}, \mathsf{*.f64}\left(\frac{1}{90}, angle\right)\right) \]
   9. Step-by-step derivation

      1. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\left({b}^{2}\right), \left(\mathsf{PI}\left(\right) + -1 \cdot \frac{{a}^{2} \cdot \mathsf{PI}\left(\right)}{{b}^{2}}\right)\right), \mathsf{*.f64}\left(\color{blue}{\frac{1}{90}}, angle\right)\right) \]

      2. unpow2 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\left(b \cdot b\right), \left(\mathsf{PI}\left(\right) + -1 \cdot \frac{{a}^{2} \cdot \mathsf{PI}\left(\right)}{{b}^{2}}\right)\right), \mathsf{*.f64}\left(\frac{1}{90}, angle\right)\right) \]

      3. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(b, b\right), \left(\mathsf{PI}\left(\right) + -1 \cdot \frac{{a}^{2} \cdot \mathsf{PI}\left(\right)}{{b}^{2}}\right)\right), \mathsf{*.f64}\left(\frac{1}{90}, angle\right)\right) \]

      4. mul-1-neg N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(b, b\right), \left(\mathsf{PI}\left(\right) + \left(\mathsf{neg}\left(\frac{{a}^{2} \cdot \mathsf{PI}\left(\right)}{{b}^{2}}\right)\right)\right)\right), \mathsf{*.f64}\left(\frac{1}{90}, angle\right)\right) \]

      5. unsub-neg N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(b, b\right), \left(\mathsf{PI}\left(\right) - \frac{{a}^{2} \cdot \mathsf{PI}\left(\right)}{{b}^{2}}\right)\right), \mathsf{*.f64}\left(\frac{1}{90}, angle\right)\right) \]

      6. --lowering--.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(b, b\right), \mathsf{\_.f64}\left(\mathsf{PI}\left(\right), \left(\frac{{a}^{2} \cdot \mathsf{PI}\left(\right)}{{b}^{2}}\right)\right)\right), \mathsf{*.f64}\left(\frac{1}{90}, angle\right)\right) \]

      7. PI-lowering-PI.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(b, b\right), \mathsf{\_.f64}\left(\mathsf{PI.f64}\left(\right), \left(\frac{{a}^{2} \cdot \mathsf{PI}\left(\right)}{{b}^{2}}\right)\right)\right), \mathsf{*.f64}\left(\frac{1}{90}, angle\right)\right) \]

      8. *-commutative N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(b, b\right), \mathsf{\_.f64}\left(\mathsf{PI.f64}\left(\right), \left(\frac{\mathsf{PI}\left(\right) \cdot {a}^{2}}{{b}^{2}}\right)\right)\right), \mathsf{*.f64}\left(\frac{1}{90}, angle\right)\right) \]

      9. associate-/l* N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(b, b\right), \mathsf{\_.f64}\left(\mathsf{PI.f64}\left(\right), \left(\mathsf{PI}\left(\right) \cdot \frac{{a}^{2}}{{b}^{2}}\right)\right)\right), \mathsf{*.f64}\left(\frac{1}{90}, angle\right)\right) \]

      10. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(b, b\right), \mathsf{\_.f64}\left(\mathsf{PI.f64}\left(\right), \mathsf{*.f64}\left(\mathsf{PI}\left(\right), \left(\frac{{a}^{2}}{{b}^{2}}\right)\right)\right)\right), \mathsf{*.f64}\left(\frac{1}{90}, angle\right)\right) \]

      11. PI-lowering-PI.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(b, b\right), \mathsf{\_.f64}\left(\mathsf{PI.f64}\left(\right), \mathsf{*.f64}\left(\mathsf{PI.f64}\left(\right), \left(\frac{{a}^{2}}{{b}^{2}}\right)\right)\right)\right), \mathsf{*.f64}\left(\frac{1}{90}, angle\right)\right) \]

      12. unpow2 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(b, b\right), \mathsf{\_.f64}\left(\mathsf{PI.f64}\left(\right), \mathsf{*.f64}\left(\mathsf{PI.f64}\left(\right), \left(\frac{{a}^{2}}{b \cdot b}\right)\right)\right)\right), \mathsf{*.f64}\left(\frac{1}{90}, angle\right)\right) \]

      13. associate-/r* N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(b, b\right), \mathsf{\_.f64}\left(\mathsf{PI.f64}\left(\right), \mathsf{*.f64}\left(\mathsf{PI.f64}\left(\right), \left(\frac{\frac{{a}^{2}}{b}}{b}\right)\right)\right)\right), \mathsf{*.f64}\left(\frac{1}{90}, angle\right)\right) \]

      14. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(b, b\right), \mathsf{\_.f64}\left(\mathsf{PI.f64}\left(\right), \mathsf{*.f64}\left(\mathsf{PI.f64}\left(\right), \mathsf{/.f64}\left(\left(\frac{{a}^{2}}{b}\right), b\right)\right)\right)\right), \mathsf{*.f64}\left(\frac{1}{90}, angle\right)\right) \]

      15. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(b, b\right), \mathsf{\_.f64}\left(\mathsf{PI.f64}\left(\right), \mathsf{*.f64}\left(\mathsf{PI.f64}\left(\right), \mathsf{/.f64}\left(\mathsf{/.f64}\left(\left({a}^{2}\right), b\right), b\right)\right)\right)\right), \mathsf{*.f64}\left(\frac{1}{90}, angle\right)\right) \]

      16. unpow2 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(b, b\right), \mathsf{\_.f64}\left(\mathsf{PI.f64}\left(\right), \mathsf{*.f64}\left(\mathsf{PI.f64}\left(\right), \mathsf{/.f64}\left(\mathsf{/.f64}\left(\left(a \cdot a\right), b\right), b\right)\right)\right)\right), \mathsf{*.f64}\left(\frac{1}{90}, angle\right)\right) \]

      17. *-lowering-*.f64 24.2%

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(b, b\right), \mathsf{\_.f64}\left(\mathsf{PI.f64}\left(\right), \mathsf{*.f64}\left(\mathsf{PI.f64}\left(\right), \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(a, a\right), b\right), b\right)\right)\right)\right), \mathsf{*.f64}\left(\frac{1}{90}, angle\right)\right) \]

   10. Simplified 24.2%

       \[\leadsto \color{blue}{\left(\left(b \cdot b\right) \cdot \left(\pi - \pi \cdot \frac{\frac{a \cdot a}{b}}{b}\right)\right)} \cdot \left(0.011111111111111112 \cdot angle\right) \]
3. Recombined 2 regimes into one program.

4. Final simplification 56.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;angle \leq 1.12 \cdot 10^{+79}:\\ \;\;\;\;\left(b + a\right) \cdot \left(0.011111111111111112 \cdot \left(angle \cdot \left(\pi \cdot \left(b - a\right)\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\left(b \cdot b\right) \cdot \left(\pi - \pi \cdot \frac{\frac{a \cdot a}{b}}{b}\right)\right) \cdot \left(angle \cdot 0.011111111111111112\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 17: 65.0% accurate, 23.3× speedup

Math

\[\begin{array}{l} a_m = \left|a\right| \\ angle\_m = \left|angle\right| \\ angle\_s = \mathsf{copysign}\left(1, angle\right) \\ \begin{array}{l} t_0 := \pi \cdot \left(b - a\_m\right)\\ angle\_s \cdot \begin{array}{l} \mathbf{if}\;angle\_m \leq 400:\\ \;\;\;\;\left(b + a\_m\right) \cdot \left(0.011111111111111112 \cdot \left(angle\_m \cdot t\_0\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\left(b + a\_m\right) \cdot t\_0\right) \cdot \left(angle\_m \cdot 0.011111111111111112\right)\\ \end{array} \end{array} \end{array} \]

FPCore

a_m = (fabs.f64 a)
angle\_m = (fabs.f64 angle)
angle\_s = (copysign.f64 #s(literal 1 binary64) angle)
(FPCore (angle_s a_m b angle_m)
 :precision binary64
 (let* ((t_0 (* PI (- b a_m))))
   (*
    angle_s
    (if (<= angle_m 400.0)
      (* (+ b a_m) (* 0.011111111111111112 (* angle_m t_0)))
      (* (* (+ b a_m) t_0) (* angle_m 0.011111111111111112))))))

C

a_m = fabs(a);
angle\_m = fabs(angle);
angle\_s = copysign(1.0, angle);
double code(double angle_s, double a_m, double b, double angle_m) {
	double t_0 = ((double) M_PI) * (b - a_m);
	double tmp;
	if (angle_m <= 400.0) {
		tmp = (b + a_m) * (0.011111111111111112 * (angle_m * t_0));
	} else {
		tmp = ((b + a_m) * t_0) * (angle_m * 0.011111111111111112);
	}
	return angle_s * tmp;
}

Java

a_m = Math.abs(a);
angle\_m = Math.abs(angle);
angle\_s = Math.copySign(1.0, angle);
public static double code(double angle_s, double a_m, double b, double angle_m) {
	double t_0 = Math.PI * (b - a_m);
	double tmp;
	if (angle_m <= 400.0) {
		tmp = (b + a_m) * (0.011111111111111112 * (angle_m * t_0));
	} else {
		tmp = ((b + a_m) * t_0) * (angle_m * 0.011111111111111112);
	}
	return angle_s * tmp;
}

Python

a_m = math.fabs(a)
angle\_m = math.fabs(angle)
angle\_s = math.copysign(1.0, angle)
def code(angle_s, a_m, b, angle_m):
	t_0 = math.pi * (b - a_m)
	tmp = 0
	if angle_m <= 400.0:
		tmp = (b + a_m) * (0.011111111111111112 * (angle_m * t_0))
	else:
		tmp = ((b + a_m) * t_0) * (angle_m * 0.011111111111111112)
	return angle_s * tmp

Julia

a_m = abs(a)
angle\_m = abs(angle)
angle\_s = copysign(1.0, angle)
function code(angle_s, a_m, b, angle_m)
	t_0 = Float64(pi * Float64(b - a_m))
	tmp = 0.0
	if (angle_m <= 400.0)
		tmp = Float64(Float64(b + a_m) * Float64(0.011111111111111112 * Float64(angle_m * t_0)));
	else
		tmp = Float64(Float64(Float64(b + a_m) * t_0) * Float64(angle_m * 0.011111111111111112));
	end
	return Float64(angle_s * tmp)
end

MATLAB

a_m = abs(a);
angle\_m = abs(angle);
angle\_s = sign(angle) * abs(1.0);
function tmp_2 = code(angle_s, a_m, b, angle_m)
	t_0 = pi * (b - a_m);
	tmp = 0.0;
	if (angle_m <= 400.0)
		tmp = (b + a_m) * (0.011111111111111112 * (angle_m * t_0));
	else
		tmp = ((b + a_m) * t_0) * (angle_m * 0.011111111111111112);
	end
	tmp_2 = angle_s * tmp;
end

Wolfram

a_m = N[Abs[a], $MachinePrecision]
angle\_m = N[Abs[angle], $MachinePrecision]
angle\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[angle]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[angle$95$s_, a$95$m_, b_, angle$95$m_] := Block[{t$95$0 = N[(Pi * N[(b - a$95$m), $MachinePrecision]), $MachinePrecision]}, N[(angle$95$s * If[LessEqual[angle$95$m, 400.0], N[(N[(b + a$95$m), $MachinePrecision] * N[(0.011111111111111112 * N[(angle$95$m * t$95$0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(b + a$95$m), $MachinePrecision] * t$95$0), $MachinePrecision] * N[(angle$95$m * 0.011111111111111112), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if angle < 400

   1. Initial program 61.5%

      \[\left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin \left(\pi \cdot \frac{angle}{180}\right)\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right) \]
   2. Step-by-step derivation

      1. associate-*l* N/A

         \[\leadsto \left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \color{blue}{\left(\sin \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right) \cdot \cos \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right)\right)} \]

      2. *-commutative N/A

         \[\leadsto \left(\sin \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right) \cdot \cos \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right)\right) \cdot \color{blue}{\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right)} \]

      3. associate-*l* N/A

         \[\leadsto \sin \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right) \cdot \color{blue}{\left(\cos \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right) \cdot \left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right)\right)} \]

      4. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\sin \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right), \color{blue}{\left(\cos \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right) \cdot \left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right)\right)}\right) \]

      5. sin-lowering-sin.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sin.f64}\left(\left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right)\right), \left(\color{blue}{\cos \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right)} \cdot \left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right)\right)\right) \]

      6. associate-*r/ N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sin.f64}\left(\left(\frac{\mathsf{PI}\left(\right) \cdot angle}{180}\right)\right), \left(\cos \color{blue}{\left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right)} \cdot \left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right)\right)\right) \]

      7. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sin.f64}\left(\mathsf{/.f64}\left(\left(\mathsf{PI}\left(\right) \cdot angle\right), 180\right)\right), \left(\cos \color{blue}{\left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right)} \cdot \left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right)\right)\right) \]

      8. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sin.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{PI}\left(\right), angle\right), 180\right)\right), \left(\cos \left(\color{blue}{\mathsf{PI}\left(\right)} \cdot \frac{angle}{180}\right) \cdot \left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right)\right)\right) \]

      9. PI-lowering-PI.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sin.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{PI.f64}\left(\right), angle\right), 180\right)\right), \left(\cos \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right) \cdot \left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right)\right)\right) \]

      10. *-commutative N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{sin.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{PI.f64}\left(\right), angle\right), 180\right)\right), \left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \color{blue}{\cos \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right)}\right)\right) \]

      11. associate-*l* N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{sin.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{PI.f64}\left(\right), angle\right), 180\right)\right), \left(2 \cdot \color{blue}{\left(\left({b}^{2} - {a}^{2}\right) \cdot \cos \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right)\right)}\right)\right) \]

      12. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{sin.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{PI.f64}\left(\right), angle\right), 180\right)\right), \mathsf{*.f64}\left(2, \color{blue}{\left(\left({b}^{2} - {a}^{2}\right) \cdot \cos \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right)\right)}\right)\right) \]

      13. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{sin.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{PI.f64}\left(\right), angle\right), 180\right)\right), \mathsf{*.f64}\left(2, \mathsf{*.f64}\left(\left({b}^{2} - {a}^{2}\right), \color{blue}{\cos \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right)}\right)\right)\right) \]

   3. Simplified 61.1%

      \[\leadsto \color{blue}{\sin \left(\frac{\pi \cdot angle}{180}\right) \cdot \left(2 \cdot \left(\left(b \cdot b - a \cdot a\right) \cdot \cos \left(\frac{\pi \cdot angle}{180}\right)\right)\right)} \]
   4. Add Preprocessing
   5. Step-by-step derivation

      1. associate-*r* N/A

         \[\leadsto \left(\sin \left(\frac{\mathsf{PI}\left(\right) \cdot angle}{180}\right) \cdot 2\right) \cdot \color{blue}{\left(\left(b \cdot b - a \cdot a\right) \cdot \cos \left(\frac{\mathsf{PI}\left(\right) \cdot angle}{180}\right)\right)} \]

      2. *-commutative N/A

         \[\leadsto \left(\left(b \cdot b - a \cdot a\right) \cdot \cos \left(\frac{\mathsf{PI}\left(\right) \cdot angle}{180}\right)\right) \cdot \color{blue}{\left(\sin \left(\frac{\mathsf{PI}\left(\right) \cdot angle}{180}\right) \cdot 2\right)} \]

      3. difference-of-squares N/A

         \[\leadsto \left(\left(\left(b + a\right) \cdot \left(b - a\right)\right) \cdot \cos \left(\frac{\mathsf{PI}\left(\right) \cdot angle}{180}\right)\right) \cdot \left(\sin \color{blue}{\left(\frac{\mathsf{PI}\left(\right) \cdot angle}{180}\right)} \cdot 2\right) \]

      4. associate-*r/ N/A

         \[\leadsto \left(\left(\left(b + a\right) \cdot \left(b - a\right)\right) \cdot \cos \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right)\right) \cdot \left(\sin \left(\frac{\mathsf{PI}\left(\right) \cdot angle}{180}\right) \cdot 2\right) \]

      5. associate-*l* N/A

         \[\leadsto \left(\left(b + a\right) \cdot \left(\left(b - a\right) \cdot \cos \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right)\right)\right) \cdot \left(\color{blue}{\sin \left(\frac{\mathsf{PI}\left(\right) \cdot angle}{180}\right)} \cdot 2\right) \]

      6. associate-*l* N/A

         \[\leadsto \left(b + a\right) \cdot \color{blue}{\left(\left(\left(b - a\right) \cdot \cos \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right)\right) \cdot \left(\sin \left(\frac{\mathsf{PI}\left(\right) \cdot angle}{180}\right) \cdot 2\right)\right)} \]

      7. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\left(b + a\right), \color{blue}{\left(\left(\left(b - a\right) \cdot \cos \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right)\right) \cdot \left(\sin \left(\frac{\mathsf{PI}\left(\right) \cdot angle}{180}\right) \cdot 2\right)\right)}\right) \]

      8. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(b, a\right), \left(\color{blue}{\left(\left(b - a\right) \cdot \cos \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right)\right)} \cdot \left(\sin \left(\frac{\mathsf{PI}\left(\right) \cdot angle}{180}\right) \cdot 2\right)\right)\right) \]

      9. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(b, a\right), \mathsf{*.f64}\left(\left(\left(b - a\right) \cdot \cos \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right)\right), \color{blue}{\left(\sin \left(\frac{\mathsf{PI}\left(\right) \cdot angle}{180}\right) \cdot 2\right)}\right)\right) \]

   6. Applied egg-rr 75.2%

      \[\leadsto \color{blue}{\left(b + a\right) \cdot \left(\left(\cos \left(\frac{\pi}{\frac{180}{angle}}\right) \cdot \left(b - a\right)\right) \cdot \left(2 \cdot \sin \left(\frac{\pi}{\frac{180}{angle}}\right)\right)\right)} \]

   7. Taylor expanded in angle around 0

      \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(b, a\right), \color{blue}{\left(\frac{1}{90} \cdot \left(angle \cdot \left(\mathsf{PI}\left(\right) \cdot \left(b - a\right)\right)\right)\right)}\right) \]
   8. Step-by-step derivation

      1. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(b, a\right), \mathsf{*.f64}\left(\frac{1}{90}, \color{blue}{\left(angle \cdot \left(\mathsf{PI}\left(\right) \cdot \left(b - a\right)\right)\right)}\right)\right) \]

      2. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(b, a\right), \mathsf{*.f64}\left(\frac{1}{90}, \mathsf{*.f64}\left(angle, \color{blue}{\left(\mathsf{PI}\left(\right) \cdot \left(b - a\right)\right)}\right)\right)\right) \]

      3. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(b, a\right), \mathsf{*.f64}\left(\frac{1}{90}, \mathsf{*.f64}\left(angle, \mathsf{*.f64}\left(\mathsf{PI}\left(\right), \color{blue}{\left(b - a\right)}\right)\right)\right)\right) \]

      4. PI-lowering-PI.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(b, a\right), \mathsf{*.f64}\left(\frac{1}{90}, \mathsf{*.f64}\left(angle, \mathsf{*.f64}\left(\mathsf{PI.f64}\left(\right), \left(\color{blue}{b} - a\right)\right)\right)\right)\right) \]

      5. --lowering--.f64 69.9%

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(b, a\right), \mathsf{*.f64}\left(\frac{1}{90}, \mathsf{*.f64}\left(angle, \mathsf{*.f64}\left(\mathsf{PI.f64}\left(\right), \mathsf{\_.f64}\left(b, \color{blue}{a}\right)\right)\right)\right)\right) \]

   9. Simplified 69.9%

      \[\leadsto \left(b + a\right) \cdot \color{blue}{\left(0.011111111111111112 \cdot \left(angle \cdot \left(\pi \cdot \left(b - a\right)\right)\right)\right)} \]

   if 400 < angle

   1. Initial program 22.1%

      \[\left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin \left(\pi \cdot \frac{angle}{180}\right)\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right) \]
   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. associate-*l* N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{sin.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{PI.f64}\left(\right), angle\right), 180\right)\right), \left(2 \cdot \color{blue}{\left(\left({b}^{2} - {a}^{2}\right) \cdot \cos \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right)\right)}\right)\right) \]

      12. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{sin.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{PI.f64}\left(\right), angle\right), 180\right)\right), \mathsf{*.f64}\left(2, \color{blue}{\left(\left({b}^{2} - {a}^{2}\right) \cdot \cos \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right)\right)}\right)\right) \]

      13. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{sin.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{PI.f64}\left(\right), angle\right), 180\right)\right), \mathsf{*.f64}\left(2, \mathsf{*.f64}\left(\left({b}^{2} - {a}^{2}\right), \color{blue}{\cos \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right)}\right)\right)\right) \]

   3. Simplified 22.4%

      \[\leadsto \color{blue}{\sin \left(\frac{\pi \cdot angle}{180}\right) \cdot \left(2 \cdot \left(\left(b \cdot b - a \cdot a\right) \cdot \cos \left(\frac{\pi \cdot angle}{180}\right)\right)\right)} \]
   4. Add Preprocessing

   5. Taylor expanded in angle around 0

      \[\leadsto \color{blue}{\frac{1}{90} \cdot \left(angle \cdot \left(\mathsf{PI}\left(\right) \cdot \left({b}^{2} - {a}^{2}\right)\right)\right)} \]
   6. Step-by-step derivation

      1. associate-*r* N/A

         \[\leadsto \left(\frac{1}{90} \cdot angle\right) \cdot \color{blue}{\left(\mathsf{PI}\left(\right) \cdot \left({b}^{2} - {a}^{2}\right)\right)} \]

      2. *-commutative N/A

         \[\leadsto \left(\mathsf{PI}\left(\right) \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \color{blue}{\left(\frac{1}{90} \cdot angle\right)} \]

      3. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\left(\mathsf{PI}\left(\right) \cdot \left({b}^{2} - {a}^{2}\right)\right), \color{blue}{\left(\frac{1}{90} \cdot angle\right)}\right) \]

      4. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{PI}\left(\right), \left({b}^{2} - {a}^{2}\right)\right), \left(\color{blue}{\frac{1}{90}} \cdot angle\right)\right) \]

      5. PI-lowering-PI.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{PI.f64}\left(\right), \left({b}^{2} - {a}^{2}\right)\right), \left(\frac{1}{90} \cdot angle\right)\right) \]

      6. --lowering--.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{PI.f64}\left(\right), \mathsf{\_.f64}\left(\left({b}^{2}\right), \left({a}^{2}\right)\right)\right), \left(\frac{1}{90} \cdot angle\right)\right) \]

      7. unpow2 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{PI.f64}\left(\right), \mathsf{\_.f64}\left(\left(b \cdot b\right), \left({a}^{2}\right)\right)\right), \left(\frac{1}{90} \cdot angle\right)\right) \]

      8. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{PI.f64}\left(\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(b, b\right), \left({a}^{2}\right)\right)\right), \left(\frac{1}{90} \cdot angle\right)\right) \]

      9. unpow2 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{PI.f64}\left(\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(b, b\right), \left(a \cdot a\right)\right)\right), \left(\frac{1}{90} \cdot angle\right)\right) \]

      10. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{PI.f64}\left(\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(b, b\right), \mathsf{*.f64}\left(a, a\right)\right)\right), \left(\frac{1}{90} \cdot angle\right)\right) \]

      11. *-lowering-*.f64 25.6%

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{PI.f64}\left(\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(b, b\right), \mathsf{*.f64}\left(a, a\right)\right)\right), \mathsf{*.f64}\left(\frac{1}{90}, \color{blue}{angle}\right)\right) \]

   7. Simplified 25.6%

      \[\leadsto \color{blue}{\left(\pi \cdot \left(b \cdot b - a \cdot a\right)\right) \cdot \left(0.011111111111111112 \cdot angle\right)} \]
   8. Step-by-step derivation

      1. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\left(\mathsf{PI}\left(\right) \cdot \left(b \cdot b - a \cdot a\right)\right), \color{blue}{\left(\frac{1}{90} \cdot angle\right)}\right) \]

      2. *-commutative N/A

         \[\leadsto \mathsf{*.f64}\left(\left(\left(b \cdot b - a \cdot a\right) \cdot \mathsf{PI}\left(\right)\right), \left(\color{blue}{\frac{1}{90}} \cdot angle\right)\right) \]

      3. difference-of-squares N/A

         \[\leadsto \mathsf{*.f64}\left(\left(\left(\left(b + a\right) \cdot \left(b - a\right)\right) \cdot \mathsf{PI}\left(\right)\right), \left(\frac{1}{90} \cdot angle\right)\right) \]

      4. associate-*l* N/A

         \[\leadsto \mathsf{*.f64}\left(\left(\left(b + a\right) \cdot \left(\left(b - a\right) \cdot \mathsf{PI}\left(\right)\right)\right), \left(\color{blue}{\frac{1}{90}} \cdot angle\right)\right) \]

      5. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\left(b + a\right), \left(\left(b - a\right) \cdot \mathsf{PI}\left(\right)\right)\right), \left(\color{blue}{\frac{1}{90}} \cdot angle\right)\right) \]

      6. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(b, a\right), \left(\left(b - a\right) \cdot \mathsf{PI}\left(\right)\right)\right), \left(\frac{1}{90} \cdot angle\right)\right) \]

      7. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(b, a\right), \mathsf{*.f64}\left(\left(b - a\right), \mathsf{PI}\left(\right)\right)\right), \left(\frac{1}{90} \cdot angle\right)\right) \]

      8. --lowering--.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(b, a\right), \mathsf{*.f64}\left(\mathsf{\_.f64}\left(b, a\right), \mathsf{PI}\left(\right)\right)\right), \left(\frac{1}{90} \cdot angle\right)\right) \]

      9. PI-lowering-PI.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(b, a\right), \mathsf{*.f64}\left(\mathsf{\_.f64}\left(b, a\right), \mathsf{PI.f64}\left(\right)\right)\right), \left(\frac{1}{90} \cdot angle\right)\right) \]

      10. *-commutative N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(b, a\right), \mathsf{*.f64}\left(\mathsf{\_.f64}\left(b, a\right), \mathsf{PI.f64}\left(\right)\right)\right), \left(angle \cdot \color{blue}{\frac{1}{90}}\right)\right) \]

      11. *-lowering-*.f64 26.9%

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(b, a\right), \mathsf{*.f64}\left(\mathsf{\_.f64}\left(b, a\right), \mathsf{PI.f64}\left(\right)\right)\right), \mathsf{*.f64}\left(angle, \color{blue}{\frac{1}{90}}\right)\right) \]

   9. Applied egg-rr 26.9%

      \[\leadsto \color{blue}{\left(\left(b + a\right) \cdot \left(\left(b - a\right) \cdot \pi\right)\right) \cdot \left(angle \cdot 0.011111111111111112\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 56.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;angle \leq 400:\\ \;\;\;\;\left(b + a\right) \cdot \left(0.011111111111111112 \cdot \left(angle \cdot \left(\pi \cdot \left(b - a\right)\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\left(b + a\right) \cdot \left(\pi \cdot \left(b - a\right)\right)\right) \cdot \left(angle \cdot 0.011111111111111112\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 18: 58.8% accurate, 23.3× speedup

Math

\[\begin{array}{l} a_m = \left|a\right| \\ angle\_m = \left|angle\right| \\ angle\_s = \mathsf{copysign}\left(1, angle\right) \\ angle\_s \cdot \begin{array}{l} \mathbf{if}\;a\_m \leq 5.2 \cdot 10^{+150}:\\ \;\;\;\;angle\_m \cdot \left(\pi \cdot \left(0.011111111111111112 \cdot \left(b \cdot b - a\_m \cdot a\_m\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;-0.011111111111111112 \cdot \left(a\_m \cdot \left(a\_m \cdot \left(\pi \cdot angle\_m\right)\right)\right)\\ \end{array} \end{array} \]

FPCore

a_m = (fabs.f64 a)
angle\_m = (fabs.f64 angle)
angle\_s = (copysign.f64 #s(literal 1 binary64) angle)
(FPCore (angle_s a_m b angle_m)
 :precision binary64
 (*
  angle_s
  (if (<= a_m 5.2e+150)
    (* angle_m (* PI (* 0.011111111111111112 (- (* b b) (* a_m a_m)))))
    (* -0.011111111111111112 (* a_m (* a_m (* PI angle_m)))))))

C

a_m = fabs(a);
angle\_m = fabs(angle);
angle\_s = copysign(1.0, angle);
double code(double angle_s, double a_m, double b, double angle_m) {
	double tmp;
	if (a_m <= 5.2e+150) {
		tmp = angle_m * (((double) M_PI) * (0.011111111111111112 * ((b * b) - (a_m * a_m))));
	} else {
		tmp = -0.011111111111111112 * (a_m * (a_m * (((double) M_PI) * angle_m)));
	}
	return angle_s * tmp;
}

Java

a_m = Math.abs(a);
angle\_m = Math.abs(angle);
angle\_s = Math.copySign(1.0, angle);
public static double code(double angle_s, double a_m, double b, double angle_m) {
	double tmp;
	if (a_m <= 5.2e+150) {
		tmp = angle_m * (Math.PI * (0.011111111111111112 * ((b * b) - (a_m * a_m))));
	} else {
		tmp = -0.011111111111111112 * (a_m * (a_m * (Math.PI * angle_m)));
	}
	return angle_s * tmp;
}

Python

a_m = math.fabs(a)
angle\_m = math.fabs(angle)
angle\_s = math.copysign(1.0, angle)
def code(angle_s, a_m, b, angle_m):
	tmp = 0
	if a_m <= 5.2e+150:
		tmp = angle_m * (math.pi * (0.011111111111111112 * ((b * b) - (a_m * a_m))))
	else:
		tmp = -0.011111111111111112 * (a_m * (a_m * (math.pi * angle_m)))
	return angle_s * tmp

Julia

a_m = abs(a)
angle\_m = abs(angle)
angle\_s = copysign(1.0, angle)
function code(angle_s, a_m, b, angle_m)
	tmp = 0.0
	if (a_m <= 5.2e+150)
		tmp = Float64(angle_m * Float64(pi * Float64(0.011111111111111112 * Float64(Float64(b * b) - Float64(a_m * a_m)))));
	else
		tmp = Float64(-0.011111111111111112 * Float64(a_m * Float64(a_m * Float64(pi * angle_m))));
	end
	return Float64(angle_s * tmp)
end

MATLAB

a_m = abs(a);
angle\_m = abs(angle);
angle\_s = sign(angle) * abs(1.0);
function tmp_2 = code(angle_s, a_m, b, angle_m)
	tmp = 0.0;
	if (a_m <= 5.2e+150)
		tmp = angle_m * (pi * (0.011111111111111112 * ((b * b) - (a_m * a_m))));
	else
		tmp = -0.011111111111111112 * (a_m * (a_m * (pi * angle_m)));
	end
	tmp_2 = angle_s * tmp;
end

Wolfram

a_m = N[Abs[a], $MachinePrecision]
angle\_m = N[Abs[angle], $MachinePrecision]
angle\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[angle]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[angle$95$s_, a$95$m_, b_, angle$95$m_] := N[(angle$95$s * If[LessEqual[a$95$m, 5.2e+150], N[(angle$95$m * N[(Pi * N[(0.011111111111111112 * N[(N[(b * b), $MachinePrecision] - N[(a$95$m * a$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(-0.011111111111111112 * N[(a$95$m * N[(a$95$m * N[(Pi * angle$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if a < 5.20000000000000012e150

   1. Initial program 49.9%

      \[\left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin \left(\pi \cdot \frac{angle}{180}\right)\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right) \]
   2. Step-by-step derivation

      1. associate-*l* N/A

         \[\leadsto \left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \color{blue}{\left(\sin \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right) \cdot \cos \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right)\right)} \]

      2. *-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. associate-*l* N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{sin.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{PI.f64}\left(\right), angle\right), 180\right)\right), \left(2 \cdot \color{blue}{\left(\left({b}^{2} - {a}^{2}\right) \cdot \cos \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right)\right)}\right)\right) \]

      12. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{sin.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{PI.f64}\left(\right), angle\right), 180\right)\right), \mathsf{*.f64}\left(2, \color{blue}{\left(\left({b}^{2} - {a}^{2}\right) \cdot \cos \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right)\right)}\right)\right) \]

      13. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{sin.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{PI.f64}\left(\right), angle\right), 180\right)\right), \mathsf{*.f64}\left(2, \mathsf{*.f64}\left(\left({b}^{2} - {a}^{2}\right), \color{blue}{\cos \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right)}\right)\right)\right) \]

   3. Simplified 50.1%

      \[\leadsto \color{blue}{\sin \left(\frac{\pi \cdot angle}{180}\right) \cdot \left(2 \cdot \left(\left(b \cdot b - a \cdot a\right) \cdot \cos \left(\frac{\pi \cdot angle}{180}\right)\right)\right)} \]
   4. Add Preprocessing

   5. Taylor expanded in angle around 0

      \[\leadsto \color{blue}{\frac{1}{90} \cdot \left(angle \cdot \left(\mathsf{PI}\left(\right) \cdot \left({b}^{2} - {a}^{2}\right)\right)\right)} \]
   6. Step-by-step derivation

      1. associate-*r* N/A

         \[\leadsto \left(\frac{1}{90} \cdot angle\right) \cdot \color{blue}{\left(\mathsf{PI}\left(\right) \cdot \left({b}^{2} - {a}^{2}\right)\right)} \]

      2. *-commutative N/A

         \[\leadsto \left(\mathsf{PI}\left(\right) \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \color{blue}{\left(\frac{1}{90} \cdot angle\right)} \]

      3. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\left(\mathsf{PI}\left(\right) \cdot \left({b}^{2} - {a}^{2}\right)\right), \color{blue}{\left(\frac{1}{90} \cdot angle\right)}\right) \]

      4. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{PI}\left(\right), \left({b}^{2} - {a}^{2}\right)\right), \left(\color{blue}{\frac{1}{90}} \cdot angle\right)\right) \]

      5. PI-lowering-PI.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{PI.f64}\left(\right), \left({b}^{2} - {a}^{2}\right)\right), \left(\frac{1}{90} \cdot angle\right)\right) \]

      6. --lowering--.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{PI.f64}\left(\right), \mathsf{\_.f64}\left(\left({b}^{2}\right), \left({a}^{2}\right)\right)\right), \left(\frac{1}{90} \cdot angle\right)\right) \]

      7. unpow2 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{PI.f64}\left(\right), \mathsf{\_.f64}\left(\left(b \cdot b\right), \left({a}^{2}\right)\right)\right), \left(\frac{1}{90} \cdot angle\right)\right) \]

      8. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{PI.f64}\left(\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(b, b\right), \left({a}^{2}\right)\right)\right), \left(\frac{1}{90} \cdot angle\right)\right) \]

      9. unpow2 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{PI.f64}\left(\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(b, b\right), \left(a \cdot a\right)\right)\right), \left(\frac{1}{90} \cdot angle\right)\right) \]

      10. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{PI.f64}\left(\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(b, b\right), \mathsf{*.f64}\left(a, a\right)\right)\right), \left(\frac{1}{90} \cdot angle\right)\right) \]

      11. *-lowering-*.f64 48.5%

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{PI.f64}\left(\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(b, b\right), \mathsf{*.f64}\left(a, a\right)\right)\right), \mathsf{*.f64}\left(\frac{1}{90}, \color{blue}{angle}\right)\right) \]

   7. Simplified 48.5%

      \[\leadsto \color{blue}{\left(\pi \cdot \left(b \cdot b - a \cdot a\right)\right) \cdot \left(0.011111111111111112 \cdot angle\right)} \]
   8. Step-by-step derivation

      1. associate-*r* N/A

         \[\leadsto \left(\left(\mathsf{PI}\left(\right) \cdot \left(b \cdot b - a \cdot a\right)\right) \cdot \frac{1}{90}\right) \cdot \color{blue}{angle} \]

      2. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\left(\left(\mathsf{PI}\left(\right) \cdot \left(b \cdot b - a \cdot a\right)\right) \cdot \frac{1}{90}\right), \color{blue}{angle}\right) \]

      3. associate-*l* N/A

         \[\leadsto \mathsf{*.f64}\left(\left(\mathsf{PI}\left(\right) \cdot \left(\left(b \cdot b - a \cdot a\right) \cdot \frac{1}{90}\right)\right), angle\right) \]

      4. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{PI}\left(\right), \left(\left(b \cdot b - a \cdot a\right) \cdot \frac{1}{90}\right)\right), angle\right) \]

      5. PI-lowering-PI.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{PI.f64}\left(\right), \left(\left(b \cdot b - a \cdot a\right) \cdot \frac{1}{90}\right)\right), angle\right) \]

      6. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{PI.f64}\left(\right), \mathsf{*.f64}\left(\left(b \cdot b - a \cdot a\right), \frac{1}{90}\right)\right), angle\right) \]

      7. --lowering--.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{PI.f64}\left(\right), \mathsf{*.f64}\left(\mathsf{\_.f64}\left(\left(b \cdot b\right), \left(a \cdot a\right)\right), \frac{1}{90}\right)\right), angle\right) \]

      8. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{PI.f64}\left(\right), \mathsf{*.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(b, b\right), \left(a \cdot a\right)\right), \frac{1}{90}\right)\right), angle\right) \]

      9. *-lowering-*.f64 48.6%

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{PI.f64}\left(\right), \mathsf{*.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(b, b\right), \mathsf{*.f64}\left(a, a\right)\right), \frac{1}{90}\right)\right), angle\right) \]

   9. Applied egg-rr 48.6%

      \[\leadsto \color{blue}{\left(\pi \cdot \left(\left(b \cdot b - a \cdot a\right) \cdot 0.011111111111111112\right)\right) \cdot angle} \]

   if 5.20000000000000012e150 < a

   1. Initial program 45.3%

      \[\left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin \left(\pi \cdot \frac{angle}{180}\right)\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right) \]
   2. Step-by-step derivation

      1. associate-*l* N/A

         \[\leadsto \left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \color{blue}{\left(\sin \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right) \cdot \cos \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right)\right)} \]

      2. *-commutative N/A

         \[\leadsto \left(\sin \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right) \cdot \cos \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right)\right) \cdot \color{blue}{\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right)} \]

      3. associate-*l* N/A

         \[\leadsto \sin \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right) \cdot \color{blue}{\left(\cos \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right) \cdot \left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right)\right)} \]

      4. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\sin \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right), \color{blue}{\left(\cos \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right) \cdot \left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right)\right)}\right) \]

      5. sin-lowering-sin.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sin.f64}\left(\left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right)\right), \left(\color{blue}{\cos \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right)} \cdot \left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right)\right)\right) \]

      6. associate-*r/ N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sin.f64}\left(\left(\frac{\mathsf{PI}\left(\right) \cdot angle}{180}\right)\right), \left(\cos \color{blue}{\left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right)} \cdot \left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right)\right)\right) \]

      7. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sin.f64}\left(\mathsf{/.f64}\left(\left(\mathsf{PI}\left(\right) \cdot angle\right), 180\right)\right), \left(\cos \color{blue}{\left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right)} \cdot \left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right)\right)\right) \]

      8. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sin.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{PI}\left(\right), angle\right), 180\right)\right), \left(\cos \left(\color{blue}{\mathsf{PI}\left(\right)} \cdot \frac{angle}{180}\right) \cdot \left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right)\right)\right) \]

      9. PI-lowering-PI.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sin.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{PI.f64}\left(\right), angle\right), 180\right)\right), \left(\cos \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right) \cdot \left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right)\right)\right) \]

      10. *-commutative N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{sin.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{PI.f64}\left(\right), angle\right), 180\right)\right), \left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \color{blue}{\cos \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right)}\right)\right) \]

      11. associate-*l* N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{sin.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{PI.f64}\left(\right), angle\right), 180\right)\right), \left(2 \cdot \color{blue}{\left(\left({b}^{2} - {a}^{2}\right) \cdot \cos \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right)\right)}\right)\right) \]

      12. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{sin.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{PI.f64}\left(\right), angle\right), 180\right)\right), \mathsf{*.f64}\left(2, \color{blue}{\left(\left({b}^{2} - {a}^{2}\right) \cdot \cos \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right)\right)}\right)\right) \]

      13. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{sin.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{PI.f64}\left(\right), angle\right), 180\right)\right), \mathsf{*.f64}\left(2, \mathsf{*.f64}\left(\left({b}^{2} - {a}^{2}\right), \color{blue}{\cos \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right)}\right)\right)\right) \]

   3. Simplified 41.2%

      \[\leadsto \color{blue}{\sin \left(\frac{\pi \cdot angle}{180}\right) \cdot \left(2 \cdot \left(\left(b \cdot b - a \cdot a\right) \cdot \cos \left(\frac{\pi \cdot angle}{180}\right)\right)\right)} \]
   4. Add Preprocessing

   5. Taylor expanded in angle around 0

      \[\leadsto \color{blue}{\frac{1}{90} \cdot \left(angle \cdot \left(\mathsf{PI}\left(\right) \cdot \left({b}^{2} - {a}^{2}\right)\right)\right)} \]
   6. Step-by-step derivation

      1. associate-*r* N/A

         \[\leadsto \left(\frac{1}{90} \cdot angle\right) \cdot \color{blue}{\left(\mathsf{PI}\left(\right) \cdot \left({b}^{2} - {a}^{2}\right)\right)} \]

      2. *-commutative N/A

         \[\leadsto \left(\mathsf{PI}\left(\right) \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \color{blue}{\left(\frac{1}{90} \cdot angle\right)} \]

      3. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\left(\mathsf{PI}\left(\right) \cdot \left({b}^{2} - {a}^{2}\right)\right), \color{blue}{\left(\frac{1}{90} \cdot angle\right)}\right) \]

      4. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{PI}\left(\right), \left({b}^{2} - {a}^{2}\right)\right), \left(\color{blue}{\frac{1}{90}} \cdot angle\right)\right) \]

      5. PI-lowering-PI.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{PI.f64}\left(\right), \left({b}^{2} - {a}^{2}\right)\right), \left(\frac{1}{90} \cdot angle\right)\right) \]

      6. --lowering--.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{PI.f64}\left(\right), \mathsf{\_.f64}\left(\left({b}^{2}\right), \left({a}^{2}\right)\right)\right), \left(\frac{1}{90} \cdot angle\right)\right) \]

      7. unpow2 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{PI.f64}\left(\right), \mathsf{\_.f64}\left(\left(b \cdot b\right), \left({a}^{2}\right)\right)\right), \left(\frac{1}{90} \cdot angle\right)\right) \]

      8. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{PI.f64}\left(\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(b, b\right), \left({a}^{2}\right)\right)\right), \left(\frac{1}{90} \cdot angle\right)\right) \]

      9. unpow2 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{PI.f64}\left(\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(b, b\right), \left(a \cdot a\right)\right)\right), \left(\frac{1}{90} \cdot angle\right)\right) \]

      10. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{PI.f64}\left(\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(b, b\right), \mathsf{*.f64}\left(a, a\right)\right)\right), \left(\frac{1}{90} \cdot angle\right)\right) \]

      11. *-lowering-*.f64 48.9%

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{PI.f64}\left(\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(b, b\right), \mathsf{*.f64}\left(a, a\right)\right)\right), \mathsf{*.f64}\left(\frac{1}{90}, \color{blue}{angle}\right)\right) \]

   7. Simplified 48.9%

      \[\leadsto \color{blue}{\left(\pi \cdot \left(b \cdot b - a \cdot a\right)\right) \cdot \left(0.011111111111111112 \cdot angle\right)} \]

   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 66.3%

         \[\leadsto \mathsf{*.f64}\left(\frac{-1}{90}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(a, a\right), angle\right), \mathsf{PI.f64}\left(\right)\right)\right) \]

   10. Simplified 66.3%

       \[\leadsto \color{blue}{-0.011111111111111112 \cdot \left(\left(\left(a \cdot a\right) \cdot angle\right) \cdot \pi\right)} \]
   11. Step-by-step derivation

       1. associate-*l* N/A

          \[\leadsto \mathsf{*.f64}\left(\frac{-1}{90}, \left(\left(a \cdot a\right) \cdot \color{blue}{\left(angle \cdot \mathsf{PI}\left(\right)\right)}\right)\right) \]

       2. *-commutative N/A

          \[\leadsto \mathsf{*.f64}\left(\frac{-1}{90}, \left(\left(a \cdot a\right) \cdot \left(\mathsf{PI}\left(\right) \cdot \color{blue}{angle}\right)\right)\right) \]

       3. associate-*l* N/A

          \[\leadsto \mathsf{*.f64}\left(\frac{-1}{90}, \left(a \cdot \color{blue}{\left(a \cdot \left(\mathsf{PI}\left(\right) \cdot angle\right)\right)}\right)\right) \]

       4. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\frac{-1}{90}, \mathsf{*.f64}\left(a, \color{blue}{\left(a \cdot \left(\mathsf{PI}\left(\right) \cdot angle\right)\right)}\right)\right) \]

       5. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\frac{-1}{90}, \mathsf{*.f64}\left(a, \mathsf{*.f64}\left(a, \color{blue}{\left(\mathsf{PI}\left(\right) \cdot angle\right)}\right)\right)\right) \]

       6. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\frac{-1}{90}, \mathsf{*.f64}\left(a, \mathsf{*.f64}\left(a, \mathsf{*.f64}\left(\mathsf{PI}\left(\right), \color{blue}{angle}\right)\right)\right)\right) \]

       7. PI-lowering-PI.f64 82.4%

          \[\leadsto \mathsf{*.f64}\left(\frac{-1}{90}, \mathsf{*.f64}\left(a, \mathsf{*.f64}\left(a, \mathsf{*.f64}\left(\mathsf{PI.f64}\left(\right), angle\right)\right)\right)\right) \]

   12. Applied egg-rr 82.4%

       \[\leadsto -0.011111111111111112 \cdot \color{blue}{\left(a \cdot \left(a \cdot \left(\pi \cdot angle\right)\right)\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 51.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \leq 5.2 \cdot 10^{+150}:\\ \;\;\;\;angle \cdot \left(\pi \cdot \left(0.011111111111111112 \cdot \left(b \cdot b - a \cdot a\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;-0.011111111111111112 \cdot \left(a \cdot \left(a \cdot \left(\pi \cdot angle\right)\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 19: 52.1% accurate, 29.9× speedup

Math

\[\begin{array}{l} a_m = \left|a\right| \\ angle\_m = \left|angle\right| \\ angle\_s = \mathsf{copysign}\left(1, angle\right) \\ angle\_s \cdot \begin{array}{l} \mathbf{if}\;a\_m \leq 8.2 \cdot 10^{+39}:\\ \;\;\;\;\left(angle\_m \cdot 0.011111111111111112\right) \cdot \left(\pi \cdot \left(b \cdot b\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(angle\_m \cdot \left(a\_m \cdot \pi\right)\right) \cdot \left(a\_m \cdot -0.011111111111111112\right)\\ \end{array} \end{array} \]

FPCore

a_m = (fabs.f64 a)
angle\_m = (fabs.f64 angle)
angle\_s = (copysign.f64 #s(literal 1 binary64) angle)
(FPCore (angle_s a_m b angle_m)
 :precision binary64
 (*
  angle_s
  (if (<= a_m 8.2e+39)
    (* (* angle_m 0.011111111111111112) (* PI (* b b)))
    (* (* angle_m (* a_m PI)) (* a_m -0.011111111111111112)))))

C

a_m = fabs(a);
angle\_m = fabs(angle);
angle\_s = copysign(1.0, angle);
double code(double angle_s, double a_m, double b, double angle_m) {
	double tmp;
	if (a_m <= 8.2e+39) {
		tmp = (angle_m * 0.011111111111111112) * (((double) M_PI) * (b * b));
	} else {
		tmp = (angle_m * (a_m * ((double) M_PI))) * (a_m * -0.011111111111111112);
	}
	return angle_s * tmp;
}

Java

a_m = Math.abs(a);
angle\_m = Math.abs(angle);
angle\_s = Math.copySign(1.0, angle);
public static double code(double angle_s, double a_m, double b, double angle_m) {
	double tmp;
	if (a_m <= 8.2e+39) {
		tmp = (angle_m * 0.011111111111111112) * (Math.PI * (b * b));
	} else {
		tmp = (angle_m * (a_m * Math.PI)) * (a_m * -0.011111111111111112);
	}
	return angle_s * tmp;
}

Python

a_m = math.fabs(a)
angle\_m = math.fabs(angle)
angle\_s = math.copysign(1.0, angle)
def code(angle_s, a_m, b, angle_m):
	tmp = 0
	if a_m <= 8.2e+39:
		tmp = (angle_m * 0.011111111111111112) * (math.pi * (b * b))
	else:
		tmp = (angle_m * (a_m * math.pi)) * (a_m * -0.011111111111111112)
	return angle_s * tmp

Julia

a_m = abs(a)
angle\_m = abs(angle)
angle\_s = copysign(1.0, angle)
function code(angle_s, a_m, b, angle_m)
	tmp = 0.0
	if (a_m <= 8.2e+39)
		tmp = Float64(Float64(angle_m * 0.011111111111111112) * Float64(pi * Float64(b * b)));
	else
		tmp = Float64(Float64(angle_m * Float64(a_m * pi)) * Float64(a_m * -0.011111111111111112));
	end
	return Float64(angle_s * tmp)
end

MATLAB

a_m = abs(a);
angle\_m = abs(angle);
angle\_s = sign(angle) * abs(1.0);
function tmp_2 = code(angle_s, a_m, b, angle_m)
	tmp = 0.0;
	if (a_m <= 8.2e+39)
		tmp = (angle_m * 0.011111111111111112) * (pi * (b * b));
	else
		tmp = (angle_m * (a_m * pi)) * (a_m * -0.011111111111111112);
	end
	tmp_2 = angle_s * tmp;
end

Wolfram

a_m = N[Abs[a], $MachinePrecision]
angle\_m = N[Abs[angle], $MachinePrecision]
angle\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[angle]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[angle$95$s_, a$95$m_, b_, angle$95$m_] := N[(angle$95$s * If[LessEqual[a$95$m, 8.2e+39], N[(N[(angle$95$m * 0.011111111111111112), $MachinePrecision] * N[(Pi * N[(b * b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(angle$95$m * N[(a$95$m * Pi), $MachinePrecision]), $MachinePrecision] * N[(a$95$m * -0.011111111111111112), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if a < 8.20000000000000008e39

   1. Initial program 50.4%

      \[\left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin \left(\pi \cdot \frac{angle}{180}\right)\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right) \]
   2. Step-by-step derivation

      1. associate-*l* N/A

         \[\leadsto \left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \color{blue}{\left(\sin \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right) \cdot \cos \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right)\right)} \]

      2. *-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. associate-*l* N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{sin.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{PI.f64}\left(\right), angle\right), 180\right)\right), \left(2 \cdot \color{blue}{\left(\left({b}^{2} - {a}^{2}\right) \cdot \cos \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right)\right)}\right)\right) \]

      12. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{sin.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{PI.f64}\left(\right), angle\right), 180\right)\right), \mathsf{*.f64}\left(2, \color{blue}{\left(\left({b}^{2} - {a}^{2}\right) \cdot \cos \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right)\right)}\right)\right) \]

      13. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{sin.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{PI.f64}\left(\right), angle\right), 180\right)\right), \mathsf{*.f64}\left(2, \mathsf{*.f64}\left(\left({b}^{2} - {a}^{2}\right), \color{blue}{\cos \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right)}\right)\right)\right) \]

   3. Simplified 50.9%

      \[\leadsto \color{blue}{\sin \left(\frac{\pi \cdot angle}{180}\right) \cdot \left(2 \cdot \left(\left(b \cdot b - a \cdot a\right) \cdot \cos \left(\frac{\pi \cdot angle}{180}\right)\right)\right)} \]
   4. Add Preprocessing

   5. Taylor expanded in angle around 0

      \[\leadsto \color{blue}{\frac{1}{90} \cdot \left(angle \cdot \left(\mathsf{PI}\left(\right) \cdot \left({b}^{2} - {a}^{2}\right)\right)\right)} \]
   6. Step-by-step derivation

      1. associate-*r* N/A

         \[\leadsto \left(\frac{1}{90} \cdot angle\right) \cdot \color{blue}{\left(\mathsf{PI}\left(\right) \cdot \left({b}^{2} - {a}^{2}\right)\right)} \]

      2. *-commutative N/A

         \[\leadsto \left(\mathsf{PI}\left(\right) \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \color{blue}{\left(\frac{1}{90} \cdot angle\right)} \]

      3. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\left(\mathsf{PI}\left(\right) \cdot \left({b}^{2} - {a}^{2}\right)\right), \color{blue}{\left(\frac{1}{90} \cdot angle\right)}\right) \]

      4. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{PI}\left(\right), \left({b}^{2} - {a}^{2}\right)\right), \left(\color{blue}{\frac{1}{90}} \cdot angle\right)\right) \]

      5. PI-lowering-PI.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{PI.f64}\left(\right), \left({b}^{2} - {a}^{2}\right)\right), \left(\frac{1}{90} \cdot angle\right)\right) \]

      6. --lowering--.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{PI.f64}\left(\right), \mathsf{\_.f64}\left(\left({b}^{2}\right), \left({a}^{2}\right)\right)\right), \left(\frac{1}{90} \cdot angle\right)\right) \]

      7. unpow2 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{PI.f64}\left(\right), \mathsf{\_.f64}\left(\left(b \cdot b\right), \left({a}^{2}\right)\right)\right), \left(\frac{1}{90} \cdot angle\right)\right) \]

      8. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{PI.f64}\left(\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(b, b\right), \left({a}^{2}\right)\right)\right), \left(\frac{1}{90} \cdot angle\right)\right) \]

      9. unpow2 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{PI.f64}\left(\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(b, b\right), \left(a \cdot a\right)\right)\right), \left(\frac{1}{90} \cdot angle\right)\right) \]

      10. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{PI.f64}\left(\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(b, b\right), \mathsf{*.f64}\left(a, a\right)\right)\right), \left(\frac{1}{90} \cdot angle\right)\right) \]

      11. *-lowering-*.f64 49.3%

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{PI.f64}\left(\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(b, b\right), \mathsf{*.f64}\left(a, a\right)\right)\right), \mathsf{*.f64}\left(\frac{1}{90}, \color{blue}{angle}\right)\right) \]

   7. Simplified 49.3%

      \[\leadsto \color{blue}{\left(\pi \cdot \left(b \cdot b - a \cdot a\right)\right) \cdot \left(0.011111111111111112 \cdot angle\right)} \]

   8. Taylor expanded in b around inf

      \[\leadsto \mathsf{*.f64}\left(\color{blue}{\left({b}^{2} \cdot \mathsf{PI}\left(\right)\right)}, \mathsf{*.f64}\left(\frac{1}{90}, angle\right)\right) \]
   9. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \mathsf{*.f64}\left(\left(\mathsf{PI}\left(\right) \cdot {b}^{2}\right), \mathsf{*.f64}\left(\color{blue}{\frac{1}{90}}, angle\right)\right) \]

      2. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{PI}\left(\right), \left({b}^{2}\right)\right), \mathsf{*.f64}\left(\color{blue}{\frac{1}{90}}, angle\right)\right) \]

      3. PI-lowering-PI.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{PI.f64}\left(\right), \left({b}^{2}\right)\right), \mathsf{*.f64}\left(\frac{1}{90}, angle\right)\right) \]

      4. unpow2 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{PI.f64}\left(\right), \left(b \cdot b\right)\right), \mathsf{*.f64}\left(\frac{1}{90}, angle\right)\right) \]

      5. *-lowering-*.f64 41.0%

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{PI.f64}\left(\right), \mathsf{*.f64}\left(b, b\right)\right), \mathsf{*.f64}\left(\frac{1}{90}, angle\right)\right) \]

   10. Simplified 41.0%

       \[\leadsto \color{blue}{\left(\pi \cdot \left(b \cdot b\right)\right)} \cdot \left(0.011111111111111112 \cdot angle\right) \]

   if 8.20000000000000008e39 < a

   1. Initial program 45.3%

      \[\left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin \left(\pi \cdot \frac{angle}{180}\right)\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right) \]
   2. Step-by-step derivation

      1. associate-*l* N/A

         \[\leadsto \left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \color{blue}{\left(\sin \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right) \cdot \cos \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right)\right)} \]

      2. *-commutative N/A

         \[\leadsto \left(\sin \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right) \cdot \cos \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right)\right) \cdot \color{blue}{\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right)} \]

      3. associate-*l* N/A

         \[\leadsto \sin \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right) \cdot \color{blue}{\left(\cos \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right) \cdot \left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right)\right)} \]

      4. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\sin \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right), \color{blue}{\left(\cos \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right) \cdot \left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right)\right)}\right) \]

      5. sin-lowering-sin.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sin.f64}\left(\left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right)\right), \left(\color{blue}{\cos \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right)} \cdot \left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right)\right)\right) \]

      6. associate-*r/ N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sin.f64}\left(\left(\frac{\mathsf{PI}\left(\right) \cdot angle}{180}\right)\right), \left(\cos \color{blue}{\left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right)} \cdot \left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right)\right)\right) \]

      7. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sin.f64}\left(\mathsf{/.f64}\left(\left(\mathsf{PI}\left(\right) \cdot angle\right), 180\right)\right), \left(\cos \color{blue}{\left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right)} \cdot \left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right)\right)\right) \]

      8. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sin.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{PI}\left(\right), angle\right), 180\right)\right), \left(\cos \left(\color{blue}{\mathsf{PI}\left(\right)} \cdot \frac{angle}{180}\right) \cdot \left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right)\right)\right) \]

      9. PI-lowering-PI.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sin.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{PI.f64}\left(\right), angle\right), 180\right)\right), \left(\cos \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right) \cdot \left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right)\right)\right) \]

      10. *-commutative N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{sin.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{PI.f64}\left(\right), angle\right), 180\right)\right), \left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \color{blue}{\cos \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right)}\right)\right) \]

      11. associate-*l* N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{sin.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{PI.f64}\left(\right), angle\right), 180\right)\right), \left(2 \cdot \color{blue}{\left(\left({b}^{2} - {a}^{2}\right) \cdot \cos \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right)\right)}\right)\right) \]

      12. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{sin.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{PI.f64}\left(\right), angle\right), 180\right)\right), \mathsf{*.f64}\left(2, \color{blue}{\left(\left({b}^{2} - {a}^{2}\right) \cdot \cos \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right)\right)}\right)\right) \]

      13. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{sin.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{PI.f64}\left(\right), angle\right), 180\right)\right), \mathsf{*.f64}\left(2, \mathsf{*.f64}\left(\left({b}^{2} - {a}^{2}\right), \color{blue}{\cos \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right)}\right)\right)\right) \]

   3. Simplified 42.1%

      \[\leadsto \color{blue}{\sin \left(\frac{\pi \cdot angle}{180}\right) \cdot \left(2 \cdot \left(\left(b \cdot b - a \cdot a\right) \cdot \cos \left(\frac{\pi \cdot angle}{180}\right)\right)\right)} \]
   4. Add Preprocessing

   5. Taylor expanded in angle around 0

      \[\leadsto \color{blue}{\frac{1}{90} \cdot \left(angle \cdot \left(\mathsf{PI}\left(\right) \cdot \left({b}^{2} - {a}^{2}\right)\right)\right)} \]
   6. Step-by-step derivation

      1. associate-*r* N/A

         \[\leadsto \left(\frac{1}{90} \cdot angle\right) \cdot \color{blue}{\left(\mathsf{PI}\left(\right) \cdot \left({b}^{2} - {a}^{2}\right)\right)} \]

      2. *-commutative N/A

         \[\leadsto \left(\mathsf{PI}\left(\right) \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \color{blue}{\left(\frac{1}{90} \cdot angle\right)} \]

      3. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\left(\mathsf{PI}\left(\right) \cdot \left({b}^{2} - {a}^{2}\right)\right), \color{blue}{\left(\frac{1}{90} \cdot angle\right)}\right) \]

      4. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{PI}\left(\right), \left({b}^{2} - {a}^{2}\right)\right), \left(\color{blue}{\frac{1}{90}} \cdot angle\right)\right) \]

      5. PI-lowering-PI.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{PI.f64}\left(\right), \left({b}^{2} - {a}^{2}\right)\right), \left(\frac{1}{90} \cdot angle\right)\right) \]

      6. --lowering--.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{PI.f64}\left(\right), \mathsf{\_.f64}\left(\left({b}^{2}\right), \left({a}^{2}\right)\right)\right), \left(\frac{1}{90} \cdot angle\right)\right) \]

      7. unpow2 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{PI.f64}\left(\right), \mathsf{\_.f64}\left(\left(b \cdot b\right), \left({a}^{2}\right)\right)\right), \left(\frac{1}{90} \cdot angle\right)\right) \]

      8. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{PI.f64}\left(\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(b, b\right), \left({a}^{2}\right)\right)\right), \left(\frac{1}{90} \cdot angle\right)\right) \]

      9. unpow2 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{PI.f64}\left(\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(b, b\right), \left(a \cdot a\right)\right)\right), \left(\frac{1}{90} \cdot angle\right)\right) \]

      10. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{PI.f64}\left(\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(b, b\right), \mathsf{*.f64}\left(a, a\right)\right)\right), \left(\frac{1}{90} \cdot angle\right)\right) \]

      11. *-lowering-*.f64 45.1%

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{PI.f64}\left(\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(b, b\right), \mathsf{*.f64}\left(a, a\right)\right)\right), \mathsf{*.f64}\left(\frac{1}{90}, \color{blue}{angle}\right)\right) \]

   7. Simplified 45.1%

      \[\leadsto \color{blue}{\left(\pi \cdot \left(b \cdot b - a \cdot a\right)\right) \cdot \left(0.011111111111111112 \cdot angle\right)} \]

   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 51.5%

         \[\leadsto \mathsf{*.f64}\left(\frac{-1}{90}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(a, a\right), angle\right), \mathsf{PI.f64}\left(\right)\right)\right) \]

   10. Simplified 51.5%

       \[\leadsto \color{blue}{-0.011111111111111112 \cdot \left(\left(\left(a \cdot a\right) \cdot angle\right) \cdot \pi\right)} \]
   11. Step-by-step derivation

       1. associate-*l* N/A

          \[\leadsto \mathsf{*.f64}\left(\frac{-1}{90}, \left(\left(a \cdot a\right) \cdot \color{blue}{\left(angle \cdot \mathsf{PI}\left(\right)\right)}\right)\right) \]

       2. *-commutative N/A

          \[\leadsto \mathsf{*.f64}\left(\frac{-1}{90}, \left(\left(a \cdot a\right) \cdot \left(\mathsf{PI}\left(\right) \cdot \color{blue}{angle}\right)\right)\right) \]

       3. associate-*l* N/A

          \[\leadsto \mathsf{*.f64}\left(\frac{-1}{90}, \left(a \cdot \color{blue}{\left(a \cdot \left(\mathsf{PI}\left(\right) \cdot angle\right)\right)}\right)\right) \]

       4. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\frac{-1}{90}, \mathsf{*.f64}\left(a, \color{blue}{\left(a \cdot \left(\mathsf{PI}\left(\right) \cdot angle\right)\right)}\right)\right) \]

       5. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\frac{-1}{90}, \mathsf{*.f64}\left(a, \mathsf{*.f64}\left(a, \color{blue}{\left(\mathsf{PI}\left(\right) \cdot angle\right)}\right)\right)\right) \]

       6. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\frac{-1}{90}, \mathsf{*.f64}\left(a, \mathsf{*.f64}\left(a, \mathsf{*.f64}\left(\mathsf{PI}\left(\right), \color{blue}{angle}\right)\right)\right)\right) \]

       7. PI-lowering-PI.f64 59.6%

          \[\leadsto \mathsf{*.f64}\left(\frac{-1}{90}, \mathsf{*.f64}\left(a, \mathsf{*.f64}\left(a, \mathsf{*.f64}\left(\mathsf{PI.f64}\left(\right), angle\right)\right)\right)\right) \]

   12. Applied egg-rr 59.6%

       \[\leadsto -0.011111111111111112 \cdot \color{blue}{\left(a \cdot \left(a \cdot \left(\pi \cdot angle\right)\right)\right)} \]
   13. Step-by-step derivation

       1. associate-*r* N/A

          \[\leadsto \left(\frac{-1}{90} \cdot a\right) \cdot \color{blue}{\left(a \cdot \left(\mathsf{PI}\left(\right) \cdot angle\right)\right)} \]

       2. *-commutative N/A

          \[\leadsto \left(a \cdot \left(\mathsf{PI}\left(\right) \cdot angle\right)\right) \cdot \color{blue}{\left(\frac{-1}{90} \cdot a\right)} \]

       3. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\left(a \cdot \left(\mathsf{PI}\left(\right) \cdot angle\right)\right), \color{blue}{\left(\frac{-1}{90} \cdot a\right)}\right) \]

       4. associate-*r* N/A

          \[\leadsto \mathsf{*.f64}\left(\left(\left(a \cdot \mathsf{PI}\left(\right)\right) \cdot angle\right), \left(\color{blue}{\frac{-1}{90}} \cdot a\right)\right) \]

       5. *-commutative N/A

          \[\leadsto \mathsf{*.f64}\left(\left(angle \cdot \left(a \cdot \mathsf{PI}\left(\right)\right)\right), \left(\color{blue}{\frac{-1}{90}} \cdot a\right)\right) \]

       6. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(angle, \left(a \cdot \mathsf{PI}\left(\right)\right)\right), \left(\color{blue}{\frac{-1}{90}} \cdot a\right)\right) \]

       7. *-commutative N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(angle, \left(\mathsf{PI}\left(\right) \cdot a\right)\right), \left(\frac{-1}{90} \cdot a\right)\right) \]

       8. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(angle, \mathsf{*.f64}\left(\mathsf{PI}\left(\right), a\right)\right), \left(\frac{-1}{90} \cdot a\right)\right) \]

       9. PI-lowering-PI.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(angle, \mathsf{*.f64}\left(\mathsf{PI.f64}\left(\right), a\right)\right), \left(\frac{-1}{90} \cdot a\right)\right) \]

       10. *-commutative N/A

           \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(angle, \mathsf{*.f64}\left(\mathsf{PI.f64}\left(\right), a\right)\right), \left(a \cdot \color{blue}{\frac{-1}{90}}\right)\right) \]

       11. *-lowering-*.f64 59.7%

           \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(angle, \mathsf{*.f64}\left(\mathsf{PI.f64}\left(\right), a\right)\right), \mathsf{*.f64}\left(a, \color{blue}{\frac{-1}{90}}\right)\right) \]

   14. Applied egg-rr 59.7%

       \[\leadsto \color{blue}{\left(angle \cdot \left(\pi \cdot a\right)\right) \cdot \left(a \cdot -0.011111111111111112\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 44.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \leq 8.2 \cdot 10^{+39}:\\ \;\;\;\;\left(angle \cdot 0.011111111111111112\right) \cdot \left(\pi \cdot \left(b \cdot b\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(angle \cdot \left(a \cdot \pi\right)\right) \cdot \left(a \cdot -0.011111111111111112\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 20: 39.6% accurate, 46.6× speedup

Math

\[\begin{array}{l} a_m = \left|a\right| \\ angle\_m = \left|angle\right| \\ angle\_s = \mathsf{copysign}\left(1, angle\right) \\ angle\_s \cdot \left(\left(angle\_m \cdot \left(a\_m \cdot \pi\right)\right) \cdot \left(a\_m \cdot -0.011111111111111112\right)\right) \end{array} \]

FPCore

a_m = (fabs.f64 a)
angle\_m = (fabs.f64 angle)
angle\_s = (copysign.f64 #s(literal 1 binary64) angle)
(FPCore (angle_s a_m b angle_m)
 :precision binary64
 (* angle_s (* (* angle_m (* a_m PI)) (* a_m -0.011111111111111112))))

C

a_m = fabs(a);
angle\_m = fabs(angle);
angle\_s = copysign(1.0, angle);
double code(double angle_s, double a_m, double b, double angle_m) {
	return angle_s * ((angle_m * (a_m * ((double) M_PI))) * (a_m * -0.011111111111111112));
}

Java

a_m = Math.abs(a);
angle\_m = Math.abs(angle);
angle\_s = Math.copySign(1.0, angle);
public static double code(double angle_s, double a_m, double b, double angle_m) {
	return angle_s * ((angle_m * (a_m * Math.PI)) * (a_m * -0.011111111111111112));
}

Python

a_m = math.fabs(a)
angle\_m = math.fabs(angle)
angle\_s = math.copysign(1.0, angle)
def code(angle_s, a_m, b, angle_m):
	return angle_s * ((angle_m * (a_m * math.pi)) * (a_m * -0.011111111111111112))

Julia

a_m = abs(a)
angle\_m = abs(angle)
angle\_s = copysign(1.0, angle)
function code(angle_s, a_m, b, angle_m)
	return Float64(angle_s * Float64(Float64(angle_m * Float64(a_m * pi)) * Float64(a_m * -0.011111111111111112)))
end

MATLAB

a_m = abs(a);
angle\_m = abs(angle);
angle\_s = sign(angle) * abs(1.0);
function tmp = code(angle_s, a_m, b, angle_m)
	tmp = angle_s * ((angle_m * (a_m * pi)) * (a_m * -0.011111111111111112));
end

Wolfram

a_m = N[Abs[a], $MachinePrecision]
angle\_m = N[Abs[angle], $MachinePrecision]
angle\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[angle]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[angle$95$s_, a$95$m_, b_, angle$95$m_] := N[(angle$95$s * N[(N[(angle$95$m * N[(a$95$m * Pi), $MachinePrecision]), $MachinePrecision] * N[(a$95$m * -0.011111111111111112), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 49.5%

   \[\left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin \left(\pi \cdot \frac{angle}{180}\right)\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right) \]
2. Step-by-step derivation

   1. associate-*l* N/A

      \[\leadsto \left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \color{blue}{\left(\sin \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right) \cdot \cos \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right)\right)} \]

   2. *-commutative N/A

      \[\leadsto \left(\sin \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right) \cdot \cos \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right)\right) \cdot \color{blue}{\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right)} \]

   3. associate-*l* N/A

      \[\leadsto \sin \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right) \cdot \color{blue}{\left(\cos \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right) \cdot \left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right)\right)} \]

   4. *-lowering-*.f64 N/A

      \[\leadsto \mathsf{*.f64}\left(\sin \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right), \color{blue}{\left(\cos \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right) \cdot \left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right)\right)}\right) \]

   5. sin-lowering-sin.f64 N/A

      \[\leadsto \mathsf{*.f64}\left(\mathsf{sin.f64}\left(\left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right)\right), \left(\color{blue}{\cos \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right)} \cdot \left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right)\right)\right) \]

   6. associate-*r/ N/A

      \[\leadsto \mathsf{*.f64}\left(\mathsf{sin.f64}\left(\left(\frac{\mathsf{PI}\left(\right) \cdot angle}{180}\right)\right), \left(\cos \color{blue}{\left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right)} \cdot \left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right)\right)\right) \]

   7. /-lowering-/.f64 N/A

      \[\leadsto \mathsf{*.f64}\left(\mathsf{sin.f64}\left(\mathsf{/.f64}\left(\left(\mathsf{PI}\left(\right) \cdot angle\right), 180\right)\right), \left(\cos \color{blue}{\left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right)} \cdot \left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right)\right)\right) \]

   8. *-lowering-*.f64 N/A

      \[\leadsto \mathsf{*.f64}\left(\mathsf{sin.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{PI}\left(\right), angle\right), 180\right)\right), \left(\cos \left(\color{blue}{\mathsf{PI}\left(\right)} \cdot \frac{angle}{180}\right) \cdot \left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right)\right)\right) \]

   9. PI-lowering-PI.f64 N/A

      \[\leadsto \mathsf{*.f64}\left(\mathsf{sin.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{PI.f64}\left(\right), angle\right), 180\right)\right), \left(\cos \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right) \cdot \left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right)\right)\right) \]

   10. *-commutative N/A

       \[\leadsto \mathsf{*.f64}\left(\mathsf{sin.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{PI.f64}\left(\right), angle\right), 180\right)\right), \left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \color{blue}{\cos \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right)}\right)\right) \]

   11. associate-*l* N/A

       \[\leadsto \mathsf{*.f64}\left(\mathsf{sin.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{PI.f64}\left(\right), angle\right), 180\right)\right), \left(2 \cdot \color{blue}{\left(\left({b}^{2} - {a}^{2}\right) \cdot \cos \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right)\right)}\right)\right) \]

   12. *-lowering-*.f64 N/A

       \[\leadsto \mathsf{*.f64}\left(\mathsf{sin.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{PI.f64}\left(\right), angle\right), 180\right)\right), \mathsf{*.f64}\left(2, \color{blue}{\left(\left({b}^{2} - {a}^{2}\right) \cdot \cos \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right)\right)}\right)\right) \]

   13. *-lowering-*.f64 N/A

       \[\leadsto \mathsf{*.f64}\left(\mathsf{sin.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{PI.f64}\left(\right), angle\right), 180\right)\right), \mathsf{*.f64}\left(2, \mathsf{*.f64}\left(\left({b}^{2} - {a}^{2}\right), \color{blue}{\cos \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right)}\right)\right)\right) \]

3. Simplified 49.3%

   \[\leadsto \color{blue}{\sin \left(\frac{\pi \cdot angle}{180}\right) \cdot \left(2 \cdot \left(\left(b \cdot b - a \cdot a\right) \cdot \cos \left(\frac{\pi \cdot angle}{180}\right)\right)\right)} \]
4. Add Preprocessing

5. Taylor expanded in angle around 0

   \[\leadsto \color{blue}{\frac{1}{90} \cdot \left(angle \cdot \left(\mathsf{PI}\left(\right) \cdot \left({b}^{2} - {a}^{2}\right)\right)\right)} \]
6. Step-by-step derivation

   1. associate-*r* N/A

      \[\leadsto \left(\frac{1}{90} \cdot angle\right) \cdot \color{blue}{\left(\mathsf{PI}\left(\right) \cdot \left({b}^{2} - {a}^{2}\right)\right)} \]

   2. *-commutative N/A

      \[\leadsto \left(\mathsf{PI}\left(\right) \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \color{blue}{\left(\frac{1}{90} \cdot angle\right)} \]

   3. *-lowering-*.f64 N/A

      \[\leadsto \mathsf{*.f64}\left(\left(\mathsf{PI}\left(\right) \cdot \left({b}^{2} - {a}^{2}\right)\right), \color{blue}{\left(\frac{1}{90} \cdot angle\right)}\right) \]

   4. *-lowering-*.f64 N/A

      \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{PI}\left(\right), \left({b}^{2} - {a}^{2}\right)\right), \left(\color{blue}{\frac{1}{90}} \cdot angle\right)\right) \]

   5. PI-lowering-PI.f64 N/A

      \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{PI.f64}\left(\right), \left({b}^{2} - {a}^{2}\right)\right), \left(\frac{1}{90} \cdot angle\right)\right) \]

   6. --lowering--.f64 N/A

      \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{PI.f64}\left(\right), \mathsf{\_.f64}\left(\left({b}^{2}\right), \left({a}^{2}\right)\right)\right), \left(\frac{1}{90} \cdot angle\right)\right) \]

   7. unpow2 N/A

      \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{PI.f64}\left(\right), \mathsf{\_.f64}\left(\left(b \cdot b\right), \left({a}^{2}\right)\right)\right), \left(\frac{1}{90} \cdot angle\right)\right) \]

   8. *-lowering-*.f64 N/A

      \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{PI.f64}\left(\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(b, b\right), \left({a}^{2}\right)\right)\right), \left(\frac{1}{90} \cdot angle\right)\right) \]

   9. unpow2 N/A

      \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{PI.f64}\left(\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(b, b\right), \left(a \cdot a\right)\right)\right), \left(\frac{1}{90} \cdot angle\right)\right) \]

   10. *-lowering-*.f64 N/A

       \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{PI.f64}\left(\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(b, b\right), \mathsf{*.f64}\left(a, a\right)\right)\right), \left(\frac{1}{90} \cdot angle\right)\right) \]

   11. *-lowering-*.f64 48.5%

       \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{PI.f64}\left(\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(b, b\right), \mathsf{*.f64}\left(a, a\right)\right)\right), \mathsf{*.f64}\left(\frac{1}{90}, \color{blue}{angle}\right)\right) \]

7. Simplified 48.5%

   \[\leadsto \color{blue}{\left(\pi \cdot \left(b \cdot b - a \cdot a\right)\right) \cdot \left(0.011111111111111112 \cdot angle\right)} \]

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 35.4%

      \[\leadsto \mathsf{*.f64}\left(\frac{-1}{90}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(a, a\right), angle\right), \mathsf{PI.f64}\left(\right)\right)\right) \]

10. Simplified 35.4%

    \[\leadsto \color{blue}{-0.011111111111111112 \cdot \left(\left(\left(a \cdot a\right) \cdot angle\right) \cdot \pi\right)} \]
11. Step-by-step derivation

    1. associate-*l* N/A

       \[\leadsto \mathsf{*.f64}\left(\frac{-1}{90}, \left(\left(a \cdot a\right) \cdot \color{blue}{\left(angle \cdot \mathsf{PI}\left(\right)\right)}\right)\right) \]

    2. *-commutative N/A

       \[\leadsto \mathsf{*.f64}\left(\frac{-1}{90}, \left(\left(a \cdot a\right) \cdot \left(\mathsf{PI}\left(\right) \cdot \color{blue}{angle}\right)\right)\right) \]

    3. associate-*l* N/A

       \[\leadsto \mathsf{*.f64}\left(\frac{-1}{90}, \left(a \cdot \color{blue}{\left(a \cdot \left(\mathsf{PI}\left(\right) \cdot angle\right)\right)}\right)\right) \]

    4. *-lowering-*.f64 N/A

       \[\leadsto \mathsf{*.f64}\left(\frac{-1}{90}, \mathsf{*.f64}\left(a, \color{blue}{\left(a \cdot \left(\mathsf{PI}\left(\right) \cdot angle\right)\right)}\right)\right) \]

    5. *-lowering-*.f64 N/A

       \[\leadsto \mathsf{*.f64}\left(\frac{-1}{90}, \mathsf{*.f64}\left(a, \mathsf{*.f64}\left(a, \color{blue}{\left(\mathsf{PI}\left(\right) \cdot angle\right)}\right)\right)\right) \]

    6. *-lowering-*.f64 N/A

       \[\leadsto \mathsf{*.f64}\left(\frac{-1}{90}, \mathsf{*.f64}\left(a, \mathsf{*.f64}\left(a, \mathsf{*.f64}\left(\mathsf{PI}\left(\right), \color{blue}{angle}\right)\right)\right)\right) \]

    7. PI-lowering-PI.f64 36.1%

       \[\leadsto \mathsf{*.f64}\left(\frac{-1}{90}, \mathsf{*.f64}\left(a, \mathsf{*.f64}\left(a, \mathsf{*.f64}\left(\mathsf{PI.f64}\left(\right), angle\right)\right)\right)\right) \]

12. Applied egg-rr 36.1%

    \[\leadsto -0.011111111111111112 \cdot \color{blue}{\left(a \cdot \left(a \cdot \left(\pi \cdot angle\right)\right)\right)} \]
13. Step-by-step derivation

    1. associate-*r* N/A

       \[\leadsto \left(\frac{-1}{90} \cdot a\right) \cdot \color{blue}{\left(a \cdot \left(\mathsf{PI}\left(\right) \cdot angle\right)\right)} \]

    2. *-commutative N/A

       \[\leadsto \left(a \cdot \left(\mathsf{PI}\left(\right) \cdot angle\right)\right) \cdot \color{blue}{\left(\frac{-1}{90} \cdot a\right)} \]

    3. *-lowering-*.f64 N/A

       \[\leadsto \mathsf{*.f64}\left(\left(a \cdot \left(\mathsf{PI}\left(\right) \cdot angle\right)\right), \color{blue}{\left(\frac{-1}{90} \cdot a\right)}\right) \]

    4. associate-*r* N/A

       \[\leadsto \mathsf{*.f64}\left(\left(\left(a \cdot \mathsf{PI}\left(\right)\right) \cdot angle\right), \left(\color{blue}{\frac{-1}{90}} \cdot a\right)\right) \]

    5. *-commutative N/A

       \[\leadsto \mathsf{*.f64}\left(\left(angle \cdot \left(a \cdot \mathsf{PI}\left(\right)\right)\right), \left(\color{blue}{\frac{-1}{90}} \cdot a\right)\right) \]

    6. *-lowering-*.f64 N/A

       \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(angle, \left(a \cdot \mathsf{PI}\left(\right)\right)\right), \left(\color{blue}{\frac{-1}{90}} \cdot a\right)\right) \]

    7. *-commutative N/A

       \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(angle, \left(\mathsf{PI}\left(\right) \cdot a\right)\right), \left(\frac{-1}{90} \cdot a\right)\right) \]

    8. *-lowering-*.f64 N/A

       \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(angle, \mathsf{*.f64}\left(\mathsf{PI}\left(\right), a\right)\right), \left(\frac{-1}{90} \cdot a\right)\right) \]

    9. PI-lowering-PI.f64 N/A

       \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(angle, \mathsf{*.f64}\left(\mathsf{PI.f64}\left(\right), a\right)\right), \left(\frac{-1}{90} \cdot a\right)\right) \]

    10. *-commutative N/A

        \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(angle, \mathsf{*.f64}\left(\mathsf{PI.f64}\left(\right), a\right)\right), \left(a \cdot \color{blue}{\frac{-1}{90}}\right)\right) \]

    11. *-lowering-*.f64 36.1%

        \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(angle, \mathsf{*.f64}\left(\mathsf{PI.f64}\left(\right), a\right)\right), \mathsf{*.f64}\left(a, \color{blue}{\frac{-1}{90}}\right)\right) \]

14. Applied egg-rr 36.1%

    \[\leadsto \color{blue}{\left(angle \cdot \left(\pi \cdot a\right)\right) \cdot \left(a \cdot -0.011111111111111112\right)} \]

15. Final simplification 36.1%

    \[\leadsto \left(angle \cdot \left(a \cdot \pi\right)\right) \cdot \left(a \cdot -0.011111111111111112\right) \]
16. Add Preprocessing

Alternative 21: 39.6% accurate, 46.6× speedup

Math

\[\begin{array}{l} a_m = \left|a\right| \\ angle\_m = \left|angle\right| \\ angle\_s = \mathsf{copysign}\left(1, angle\right) \\ angle\_s \cdot \left(-0.011111111111111112 \cdot \left(a\_m \cdot \left(a\_m \cdot \left(\pi \cdot angle\_m\right)\right)\right)\right) \end{array} \]

FPCore

a_m = (fabs.f64 a)
angle\_m = (fabs.f64 angle)
angle\_s = (copysign.f64 #s(literal 1 binary64) angle)
(FPCore (angle_s a_m b angle_m)
 :precision binary64
 (* angle_s (* -0.011111111111111112 (* a_m (* a_m (* PI angle_m))))))

C

a_m = fabs(a);
angle\_m = fabs(angle);
angle\_s = copysign(1.0, angle);
double code(double angle_s, double a_m, double b, double angle_m) {
	return angle_s * (-0.011111111111111112 * (a_m * (a_m * (((double) M_PI) * angle_m))));
}

Java

a_m = Math.abs(a);
angle\_m = Math.abs(angle);
angle\_s = Math.copySign(1.0, angle);
public static double code(double angle_s, double a_m, double b, double angle_m) {
	return angle_s * (-0.011111111111111112 * (a_m * (a_m * (Math.PI * angle_m))));
}

Python

a_m = math.fabs(a)
angle\_m = math.fabs(angle)
angle\_s = math.copysign(1.0, angle)
def code(angle_s, a_m, b, angle_m):
	return angle_s * (-0.011111111111111112 * (a_m * (a_m * (math.pi * angle_m))))

Julia

a_m = abs(a)
angle\_m = abs(angle)
angle\_s = copysign(1.0, angle)
function code(angle_s, a_m, b, angle_m)
	return Float64(angle_s * Float64(-0.011111111111111112 * Float64(a_m * Float64(a_m * Float64(pi * angle_m)))))
end

MATLAB

a_m = abs(a);
angle\_m = abs(angle);
angle\_s = sign(angle) * abs(1.0);
function tmp = code(angle_s, a_m, b, angle_m)
	tmp = angle_s * (-0.011111111111111112 * (a_m * (a_m * (pi * angle_m))));
end

Wolfram

a_m = N[Abs[a], $MachinePrecision]
angle\_m = N[Abs[angle], $MachinePrecision]
angle\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[angle]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[angle$95$s_, a$95$m_, b_, angle$95$m_] := N[(angle$95$s * N[(-0.011111111111111112 * N[(a$95$m * N[(a$95$m * N[(Pi * angle$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 49.5%

   \[\left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin \left(\pi \cdot \frac{angle}{180}\right)\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right) \]
2. Step-by-step derivation

   1. associate-*l* N/A

      \[\leadsto \left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \color{blue}{\left(\sin \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right) \cdot \cos \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right)\right)} \]

   2. *-commutative N/A

      \[\leadsto \left(\sin \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right) \cdot \cos \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right)\right) \cdot \color{blue}{\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right)} \]

   3. associate-*l* N/A

      \[\leadsto \sin \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right) \cdot \color{blue}{\left(\cos \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right) \cdot \left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right)\right)} \]

   4. *-lowering-*.f64 N/A

      \[\leadsto \mathsf{*.f64}\left(\sin \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right), \color{blue}{\left(\cos \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right) \cdot \left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right)\right)}\right) \]

   5. sin-lowering-sin.f64 N/A

      \[\leadsto \mathsf{*.f64}\left(\mathsf{sin.f64}\left(\left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right)\right), \left(\color{blue}{\cos \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right)} \cdot \left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right)\right)\right) \]

   6. associate-*r/ N/A

      \[\leadsto \mathsf{*.f64}\left(\mathsf{sin.f64}\left(\left(\frac{\mathsf{PI}\left(\right) \cdot angle}{180}\right)\right), \left(\cos \color{blue}{\left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right)} \cdot \left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right)\right)\right) \]

   7. /-lowering-/.f64 N/A

      \[\leadsto \mathsf{*.f64}\left(\mathsf{sin.f64}\left(\mathsf{/.f64}\left(\left(\mathsf{PI}\left(\right) \cdot angle\right), 180\right)\right), \left(\cos \color{blue}{\left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right)} \cdot \left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right)\right)\right) \]

   8. *-lowering-*.f64 N/A

      \[\leadsto \mathsf{*.f64}\left(\mathsf{sin.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{PI}\left(\right), angle\right), 180\right)\right), \left(\cos \left(\color{blue}{\mathsf{PI}\left(\right)} \cdot \frac{angle}{180}\right) \cdot \left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right)\right)\right) \]

   9. PI-lowering-PI.f64 N/A

      \[\leadsto \mathsf{*.f64}\left(\mathsf{sin.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{PI.f64}\left(\right), angle\right), 180\right)\right), \left(\cos \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right) \cdot \left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right)\right)\right) \]

   10. *-commutative N/A

       \[\leadsto \mathsf{*.f64}\left(\mathsf{sin.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{PI.f64}\left(\right), angle\right), 180\right)\right), \left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \color{blue}{\cos \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right)}\right)\right) \]

   11. associate-*l* N/A

       \[\leadsto \mathsf{*.f64}\left(\mathsf{sin.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{PI.f64}\left(\right), angle\right), 180\right)\right), \left(2 \cdot \color{blue}{\left(\left({b}^{2} - {a}^{2}\right) \cdot \cos \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right)\right)}\right)\right) \]

   12. *-lowering-*.f64 N/A

       \[\leadsto \mathsf{*.f64}\left(\mathsf{sin.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{PI.f64}\left(\right), angle\right), 180\right)\right), \mathsf{*.f64}\left(2, \color{blue}{\left(\left({b}^{2} - {a}^{2}\right) \cdot \cos \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right)\right)}\right)\right) \]

   13. *-lowering-*.f64 N/A

       \[\leadsto \mathsf{*.f64}\left(\mathsf{sin.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{PI.f64}\left(\right), angle\right), 180\right)\right), \mathsf{*.f64}\left(2, \mathsf{*.f64}\left(\left({b}^{2} - {a}^{2}\right), \color{blue}{\cos \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right)}\right)\right)\right) \]

3. Simplified 49.3%

   \[\leadsto \color{blue}{\sin \left(\frac{\pi \cdot angle}{180}\right) \cdot \left(2 \cdot \left(\left(b \cdot b - a \cdot a\right) \cdot \cos \left(\frac{\pi \cdot angle}{180}\right)\right)\right)} \]
4. Add Preprocessing

5. Taylor expanded in angle around 0

   \[\leadsto \color{blue}{\frac{1}{90} \cdot \left(angle \cdot \left(\mathsf{PI}\left(\right) \cdot \left({b}^{2} - {a}^{2}\right)\right)\right)} \]
6. Step-by-step derivation

   1. associate-*r* N/A

      \[\leadsto \left(\frac{1}{90} \cdot angle\right) \cdot \color{blue}{\left(\mathsf{PI}\left(\right) \cdot \left({b}^{2} - {a}^{2}\right)\right)} \]

   2. *-commutative N/A

      \[\leadsto \left(\mathsf{PI}\left(\right) \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \color{blue}{\left(\frac{1}{90} \cdot angle\right)} \]

   3. *-lowering-*.f64 N/A

      \[\leadsto \mathsf{*.f64}\left(\left(\mathsf{PI}\left(\right) \cdot \left({b}^{2} - {a}^{2}\right)\right), \color{blue}{\left(\frac{1}{90} \cdot angle\right)}\right) \]

   4. *-lowering-*.f64 N/A

      \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{PI}\left(\right), \left({b}^{2} - {a}^{2}\right)\right), \left(\color{blue}{\frac{1}{90}} \cdot angle\right)\right) \]

   5. PI-lowering-PI.f64 N/A

      \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{PI.f64}\left(\right), \left({b}^{2} - {a}^{2}\right)\right), \left(\frac{1}{90} \cdot angle\right)\right) \]

   6. --lowering--.f64 N/A

      \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{PI.f64}\left(\right), \mathsf{\_.f64}\left(\left({b}^{2}\right), \left({a}^{2}\right)\right)\right), \left(\frac{1}{90} \cdot angle\right)\right) \]

   7. unpow2 N/A

      \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{PI.f64}\left(\right), \mathsf{\_.f64}\left(\left(b \cdot b\right), \left({a}^{2}\right)\right)\right), \left(\frac{1}{90} \cdot angle\right)\right) \]

   8. *-lowering-*.f64 N/A

      \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{PI.f64}\left(\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(b, b\right), \left({a}^{2}\right)\right)\right), \left(\frac{1}{90} \cdot angle\right)\right) \]

   9. unpow2 N/A

      \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{PI.f64}\left(\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(b, b\right), \left(a \cdot a\right)\right)\right), \left(\frac{1}{90} \cdot angle\right)\right) \]

   10. *-lowering-*.f64 N/A

       \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{PI.f64}\left(\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(b, b\right), \mathsf{*.f64}\left(a, a\right)\right)\right), \left(\frac{1}{90} \cdot angle\right)\right) \]

   11. *-lowering-*.f64 48.5%

       \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{PI.f64}\left(\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(b, b\right), \mathsf{*.f64}\left(a, a\right)\right)\right), \mathsf{*.f64}\left(\frac{1}{90}, \color{blue}{angle}\right)\right) \]

7. Simplified 48.5%

   \[\leadsto \color{blue}{\left(\pi \cdot \left(b \cdot b - a \cdot a\right)\right) \cdot \left(0.011111111111111112 \cdot angle\right)} \]

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 35.4%

      \[\leadsto \mathsf{*.f64}\left(\frac{-1}{90}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(a, a\right), angle\right), \mathsf{PI.f64}\left(\right)\right)\right) \]

10. Simplified 35.4%

    \[\leadsto \color{blue}{-0.011111111111111112 \cdot \left(\left(\left(a \cdot a\right) \cdot angle\right) \cdot \pi\right)} \]
11. Step-by-step derivation

    1. associate-*l* N/A

       \[\leadsto \mathsf{*.f64}\left(\frac{-1}{90}, \left(\left(a \cdot a\right) \cdot \color{blue}{\left(angle \cdot \mathsf{PI}\left(\right)\right)}\right)\right) \]

    2. *-commutative N/A

       \[\leadsto \mathsf{*.f64}\left(\frac{-1}{90}, \left(\left(a \cdot a\right) \cdot \left(\mathsf{PI}\left(\right) \cdot \color{blue}{angle}\right)\right)\right) \]

    3. associate-*l* N/A

       \[\leadsto \mathsf{*.f64}\left(\frac{-1}{90}, \left(a \cdot \color{blue}{\left(a \cdot \left(\mathsf{PI}\left(\right) \cdot angle\right)\right)}\right)\right) \]

    4. *-lowering-*.f64 N/A

       \[\leadsto \mathsf{*.f64}\left(\frac{-1}{90}, \mathsf{*.f64}\left(a, \color{blue}{\left(a \cdot \left(\mathsf{PI}\left(\right) \cdot angle\right)\right)}\right)\right) \]

    5. *-lowering-*.f64 N/A

       \[\leadsto \mathsf{*.f64}\left(\frac{-1}{90}, \mathsf{*.f64}\left(a, \mathsf{*.f64}\left(a, \color{blue}{\left(\mathsf{PI}\left(\right) \cdot angle\right)}\right)\right)\right) \]

    6. *-lowering-*.f64 N/A

       \[\leadsto \mathsf{*.f64}\left(\frac{-1}{90}, \mathsf{*.f64}\left(a, \mathsf{*.f64}\left(a, \mathsf{*.f64}\left(\mathsf{PI}\left(\right), \color{blue}{angle}\right)\right)\right)\right) \]

    7. PI-lowering-PI.f64 36.1%

       \[\leadsto \mathsf{*.f64}\left(\frac{-1}{90}, \mathsf{*.f64}\left(a, \mathsf{*.f64}\left(a, \mathsf{*.f64}\left(\mathsf{PI.f64}\left(\right), angle\right)\right)\right)\right) \]

12. Applied egg-rr 36.1%

    \[\leadsto -0.011111111111111112 \cdot \color{blue}{\left(a \cdot \left(a \cdot \left(\pi \cdot angle\right)\right)\right)} \]
13. Add Preprocessing

Reproduce

herbie shell --seed 2024163 
(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)))))