Toniolo and Linder, Equation (10-)

Percentage Accurate: 36.2% → 95.9%

Time: 18.7s

Alternatives: 11

Speedup: 28.1×

• 39.9% of points produce a very large (infinite) output. You may want to add a precondition.

Specification

Math

\[\begin{array}{l} \\ \frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) - 1\right)} \end{array} \]

FPCore

(FPCore (t l k)
 :precision binary64
 (/
  2.0
  (*
   (* (* (/ (pow t 3.0) (* l l)) (sin k)) (tan k))
   (- (+ 1.0 (pow (/ k t) 2.0)) 1.0))))

C

double code(double t, double l, double k) {
	return 2.0 / ((((pow(t, 3.0) / (l * l)) * sin(k)) * tan(k)) * ((1.0 + pow((k / t), 2.0)) - 1.0));
}

Fortran

real(8) function code(t, l, k)
    real(8), intent (in) :: t
    real(8), intent (in) :: l
    real(8), intent (in) :: k
    code = 2.0d0 / (((((t ** 3.0d0) / (l * l)) * sin(k)) * tan(k)) * ((1.0d0 + ((k / t) ** 2.0d0)) - 1.0d0))
end function

Java

public static double code(double t, double l, double k) {
	return 2.0 / ((((Math.pow(t, 3.0) / (l * l)) * Math.sin(k)) * Math.tan(k)) * ((1.0 + Math.pow((k / t), 2.0)) - 1.0));
}

Python

def code(t, l, k):
	return 2.0 / ((((math.pow(t, 3.0) / (l * l)) * math.sin(k)) * math.tan(k)) * ((1.0 + math.pow((k / t), 2.0)) - 1.0))

Julia

function code(t, l, k)
	return Float64(2.0 / Float64(Float64(Float64(Float64((t ^ 3.0) / Float64(l * l)) * sin(k)) * tan(k)) * Float64(Float64(1.0 + (Float64(k / t) ^ 2.0)) - 1.0)))
end

MATLAB

function tmp = code(t, l, k)
	tmp = 2.0 / (((((t ^ 3.0) / (l * l)) * sin(k)) * tan(k)) * ((1.0 + ((k / t) ^ 2.0)) - 1.0));
end

Wolfram

code[t_, l_, k_] := N[(2.0 / N[(N[(N[(N[(N[Power[t, 3.0], $MachinePrecision] / N[(l * l), $MachinePrecision]), $MachinePrecision] * N[Sin[k], $MachinePrecision]), $MachinePrecision] * N[Tan[k], $MachinePrecision]), $MachinePrecision] * N[(N[(1.0 + N[Power[N[(k / t), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] - 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Initial Program: 36.2% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) - 1\right)} \end{array} \]

FPCore

(FPCore (t l k)
 :precision binary64
 (/
  2.0
  (*
   (* (* (/ (pow t 3.0) (* l l)) (sin k)) (tan k))
   (- (+ 1.0 (pow (/ k t) 2.0)) 1.0))))

C

double code(double t, double l, double k) {
	return 2.0 / ((((pow(t, 3.0) / (l * l)) * sin(k)) * tan(k)) * ((1.0 + pow((k / t), 2.0)) - 1.0));
}

Fortran

real(8) function code(t, l, k)
    real(8), intent (in) :: t
    real(8), intent (in) :: l
    real(8), intent (in) :: k
    code = 2.0d0 / (((((t ** 3.0d0) / (l * l)) * sin(k)) * tan(k)) * ((1.0d0 + ((k / t) ** 2.0d0)) - 1.0d0))
end function

Java

public static double code(double t, double l, double k) {
	return 2.0 / ((((Math.pow(t, 3.0) / (l * l)) * Math.sin(k)) * Math.tan(k)) * ((1.0 + Math.pow((k / t), 2.0)) - 1.0));
}

Python

def code(t, l, k):
	return 2.0 / ((((math.pow(t, 3.0) / (l * l)) * math.sin(k)) * math.tan(k)) * ((1.0 + math.pow((k / t), 2.0)) - 1.0))

Julia

function code(t, l, k)
	return Float64(2.0 / Float64(Float64(Float64(Float64((t ^ 3.0) / Float64(l * l)) * sin(k)) * tan(k)) * Float64(Float64(1.0 + (Float64(k / t) ^ 2.0)) - 1.0)))
end

MATLAB

function tmp = code(t, l, k)
	tmp = 2.0 / (((((t ^ 3.0) / (l * l)) * sin(k)) * tan(k)) * ((1.0 + ((k / t) ^ 2.0)) - 1.0));
end

Wolfram

code[t_, l_, k_] := N[(2.0 / N[(N[(N[(N[(N[Power[t, 3.0], $MachinePrecision] / N[(l * l), $MachinePrecision]), $MachinePrecision] * N[Sin[k], $MachinePrecision]), $MachinePrecision] * N[Tan[k], $MachinePrecision]), $MachinePrecision] * N[(N[(1.0 + N[Power[N[(k / t), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] - 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Alternative 1: 95.9% accurate, 1.9× speedup

Math

\[\begin{array}{l} k_m = \left|k\right| \\ \begin{array}{l} \mathbf{if}\;k\_m \leq 8.5 \cdot 10^{-5}:\\ \;\;\;\;\ell \cdot \frac{\frac{\frac{2 \cdot \ell}{k\_m \cdot \left(k\_m \cdot t\right)}}{k\_m}}{k\_m}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\frac{2 \cdot \ell}{\frac{k\_m}{\cos k\_m}}}{t} \cdot \frac{\ell}{k\_m}}{0.5 - 0.5 \cdot \cos \left(k\_m \cdot 2\right)}\\ \end{array} \end{array} \]

FPCore

k_m = (fabs.f64 k)
(FPCore (t l k_m)
 :precision binary64
 (if (<= k_m 8.5e-5)
   (* l (/ (/ (/ (* 2.0 l) (* k_m (* k_m t))) k_m) k_m))
   (/
    (* (/ (/ (* 2.0 l) (/ k_m (cos k_m))) t) (/ l k_m))
    (- 0.5 (* 0.5 (cos (* k_m 2.0)))))))

C

k_m = fabs(k);
double code(double t, double l, double k_m) {
	double tmp;
	if (k_m <= 8.5e-5) {
		tmp = l * ((((2.0 * l) / (k_m * (k_m * t))) / k_m) / k_m);
	} else {
		tmp = ((((2.0 * l) / (k_m / cos(k_m))) / t) * (l / k_m)) / (0.5 - (0.5 * cos((k_m * 2.0))));
	}
	return tmp;
}

Fortran

k_m = abs(k)
real(8) function code(t, l, k_m)
    real(8), intent (in) :: t
    real(8), intent (in) :: l
    real(8), intent (in) :: k_m
    real(8) :: tmp
    if (k_m <= 8.5d-5) then
        tmp = l * ((((2.0d0 * l) / (k_m * (k_m * t))) / k_m) / k_m)
    else
        tmp = ((((2.0d0 * l) / (k_m / cos(k_m))) / t) * (l / k_m)) / (0.5d0 - (0.5d0 * cos((k_m * 2.0d0))))
    end if
    code = tmp
end function

Java

k_m = Math.abs(k);
public static double code(double t, double l, double k_m) {
	double tmp;
	if (k_m <= 8.5e-5) {
		tmp = l * ((((2.0 * l) / (k_m * (k_m * t))) / k_m) / k_m);
	} else {
		tmp = ((((2.0 * l) / (k_m / Math.cos(k_m))) / t) * (l / k_m)) / (0.5 - (0.5 * Math.cos((k_m * 2.0))));
	}
	return tmp;
}

Python

k_m = math.fabs(k)
def code(t, l, k_m):
	tmp = 0
	if k_m <= 8.5e-5:
		tmp = l * ((((2.0 * l) / (k_m * (k_m * t))) / k_m) / k_m)
	else:
		tmp = ((((2.0 * l) / (k_m / math.cos(k_m))) / t) * (l / k_m)) / (0.5 - (0.5 * math.cos((k_m * 2.0))))
	return tmp

Julia

k_m = abs(k)
function code(t, l, k_m)
	tmp = 0.0
	if (k_m <= 8.5e-5)
		tmp = Float64(l * Float64(Float64(Float64(Float64(2.0 * l) / Float64(k_m * Float64(k_m * t))) / k_m) / k_m));
	else
		tmp = Float64(Float64(Float64(Float64(Float64(2.0 * l) / Float64(k_m / cos(k_m))) / t) * Float64(l / k_m)) / Float64(0.5 - Float64(0.5 * cos(Float64(k_m * 2.0)))));
	end
	return tmp
end

MATLAB

k_m = abs(k);
function tmp_2 = code(t, l, k_m)
	tmp = 0.0;
	if (k_m <= 8.5e-5)
		tmp = l * ((((2.0 * l) / (k_m * (k_m * t))) / k_m) / k_m);
	else
		tmp = ((((2.0 * l) / (k_m / cos(k_m))) / t) * (l / k_m)) / (0.5 - (0.5 * cos((k_m * 2.0))));
	end
	tmp_2 = tmp;
end

Wolfram

k_m = N[Abs[k], $MachinePrecision]
code[t_, l_, k$95$m_] := If[LessEqual[k$95$m, 8.5e-5], N[(l * N[(N[(N[(N[(2.0 * l), $MachinePrecision] / N[(k$95$m * N[(k$95$m * t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / k$95$m), $MachinePrecision] / k$95$m), $MachinePrecision]), $MachinePrecision], N[(N[(N[(N[(N[(2.0 * l), $MachinePrecision] / N[(k$95$m / N[Cos[k$95$m], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / t), $MachinePrecision] * N[(l / k$95$m), $MachinePrecision]), $MachinePrecision] / N[(0.5 - N[(0.5 * N[Cos[N[(k$95$m * 2.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if k < 8.500000000000001e-5

   1. Initial program 35.0%

      \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) - 1\right)} \]
   2. Add Preprocessing

   3. Taylor expanded in k around 0

      \[\leadsto \mathsf{/.f64}\left(2, \color{blue}{\left(\frac{{k}^{4} \cdot t}{{\ell}^{2}}\right)}\right) \]
   4. Step-by-step derivation

      1. associate-/l* N/A

         \[\leadsto \mathsf{/.f64}\left(2, \left({k}^{4} \cdot \color{blue}{\frac{t}{{\ell}^{2}}}\right)\right) \]

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

         \[\leadsto \mathsf{/.f64}\left(2, \mathsf{*.f64}\left(\left({k}^{4}\right), \color{blue}{\left(\frac{t}{{\ell}^{2}}\right)}\right)\right) \]

      3. metadata-eval N/A

         \[\leadsto \mathsf{/.f64}\left(2, \mathsf{*.f64}\left(\left({k}^{\left(2 \cdot 2\right)}\right), \left(\frac{t}{{\ell}^{2}}\right)\right)\right) \]

      4. pow-sqr N/A

         \[\leadsto \mathsf{/.f64}\left(2, \mathsf{*.f64}\left(\left({k}^{2} \cdot {k}^{2}\right), \left(\frac{\color{blue}{t}}{{\ell}^{2}}\right)\right)\right) \]

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

         \[\leadsto \mathsf{/.f64}\left(2, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\left({k}^{2}\right), \left({k}^{2}\right)\right), \left(\frac{\color{blue}{t}}{{\ell}^{2}}\right)\right)\right) \]

      6. unpow2 N/A

         \[\leadsto \mathsf{/.f64}\left(2, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\left(k \cdot k\right), \left({k}^{2}\right)\right), \left(\frac{t}{{\ell}^{2}}\right)\right)\right) \]

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

         \[\leadsto \mathsf{/.f64}\left(2, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(k, k\right), \left({k}^{2}\right)\right), \left(\frac{t}{{\ell}^{2}}\right)\right)\right) \]

      8. unpow2 N/A

         \[\leadsto \mathsf{/.f64}\left(2, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(k, k\right), \left(k \cdot k\right)\right), \left(\frac{t}{{\ell}^{2}}\right)\right)\right) \]

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

         \[\leadsto \mathsf{/.f64}\left(2, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(k, k\right), \mathsf{*.f64}\left(k, k\right)\right), \left(\frac{t}{{\ell}^{2}}\right)\right)\right) \]

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

          \[\leadsto \mathsf{/.f64}\left(2, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(k, k\right), \mathsf{*.f64}\left(k, k\right)\right), \mathsf{/.f64}\left(t, \color{blue}{\left({\ell}^{2}\right)}\right)\right)\right) \]

      11. unpow2 N/A

          \[\leadsto \mathsf{/.f64}\left(2, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(k, k\right), \mathsf{*.f64}\left(k, k\right)\right), \mathsf{/.f64}\left(t, \left(\ell \cdot \color{blue}{\ell}\right)\right)\right)\right) \]

      12. *-lowering-*.f64 58.1%

          \[\leadsto \mathsf{/.f64}\left(2, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(k, k\right), \mathsf{*.f64}\left(k, k\right)\right), \mathsf{/.f64}\left(t, \mathsf{*.f64}\left(\ell, \color{blue}{\ell}\right)\right)\right)\right) \]

   5. Simplified 58.1%

      \[\leadsto \frac{2}{\color{blue}{\left(\left(k \cdot k\right) \cdot \left(k \cdot k\right)\right) \cdot \frac{t}{\ell \cdot \ell}}} \]
   6. Step-by-step derivation

      1. associate-/r* N/A

         \[\leadsto \frac{\frac{2}{\left(k \cdot k\right) \cdot \left(k \cdot k\right)}}{\color{blue}{\frac{t}{\ell \cdot \ell}}} \]

      2. associate-/r* N/A

         \[\leadsto \frac{\frac{2}{\left(k \cdot k\right) \cdot \left(k \cdot k\right)}}{\frac{\frac{t}{\ell}}{\color{blue}{\ell}}} \]

      3. associate-/r/ N/A

         \[\leadsto \frac{\frac{2}{\left(k \cdot k\right) \cdot \left(k \cdot k\right)}}{\frac{t}{\ell}} \cdot \color{blue}{\ell} \]

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

         \[\leadsto \mathsf{*.f64}\left(\left(\frac{\frac{2}{\left(k \cdot k\right) \cdot \left(k \cdot k\right)}}{\frac{t}{\ell}}\right), \color{blue}{\ell}\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\left(\frac{2}{\left(k \cdot k\right) \cdot \left(k \cdot k\right)}\right), \left(\frac{t}{\ell}\right)\right), \ell\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(2, \left(\left(k \cdot k\right) \cdot \left(k \cdot k\right)\right)\right), \left(\frac{t}{\ell}\right)\right), \ell\right) \]

      7. associate-*l* N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(2, \left(k \cdot \left(k \cdot \left(k \cdot k\right)\right)\right)\right), \left(\frac{t}{\ell}\right)\right), \ell\right) \]

      8. cube-unmult N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(2, \left(k \cdot {k}^{3}\right)\right), \left(\frac{t}{\ell}\right)\right), \ell\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(2, \mathsf{*.f64}\left(k, \left({k}^{3}\right)\right)\right), \left(\frac{t}{\ell}\right)\right), \ell\right) \]

      10. cube-unmult N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(2, \mathsf{*.f64}\left(k, \left(k \cdot \left(k \cdot k\right)\right)\right)\right), \left(\frac{t}{\ell}\right)\right), \ell\right) \]

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

          \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(2, \mathsf{*.f64}\left(k, \mathsf{*.f64}\left(k, \left(k \cdot k\right)\right)\right)\right), \left(\frac{t}{\ell}\right)\right), \ell\right) \]

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

          \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(2, \mathsf{*.f64}\left(k, \mathsf{*.f64}\left(k, \mathsf{*.f64}\left(k, k\right)\right)\right)\right), \left(\frac{t}{\ell}\right)\right), \ell\right) \]

      13. /-lowering-/.f64 69.3%

          \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(2, \mathsf{*.f64}\left(k, \mathsf{*.f64}\left(k, \mathsf{*.f64}\left(k, k\right)\right)\right)\right), \mathsf{/.f64}\left(t, \ell\right)\right), \ell\right) \]

   7. Applied egg-rr 69.3%

      \[\leadsto \color{blue}{\frac{\frac{2}{k \cdot \left(k \cdot \left(k \cdot k\right)\right)}}{\frac{t}{\ell}} \cdot \ell} \]
   8. Step-by-step derivation

      1. div-inv N/A

         \[\leadsto \mathsf{*.f64}\left(\left(\frac{2}{k \cdot \left(k \cdot \left(k \cdot k\right)\right)} \cdot \frac{1}{\frac{t}{\ell}}\right), \ell\right) \]

      2. associate-/r* N/A

         \[\leadsto \mathsf{*.f64}\left(\left(\frac{\frac{2}{k}}{k \cdot \left(k \cdot k\right)} \cdot \frac{1}{\frac{t}{\ell}}\right), \ell\right) \]

      3. clear-num N/A

         \[\leadsto \mathsf{*.f64}\left(\left(\frac{\frac{2}{k}}{k \cdot \left(k \cdot k\right)} \cdot \frac{\ell}{t}\right), \ell\right) \]

      4. associate-*l/ N/A

         \[\leadsto \mathsf{*.f64}\left(\left(\frac{\frac{2}{k} \cdot \frac{\ell}{t}}{k \cdot \left(k \cdot k\right)}\right), \ell\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\left(\frac{2}{k} \cdot \frac{\ell}{t}\right), \left(k \cdot \left(k \cdot k\right)\right)\right), \ell\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\left(\frac{2}{k}\right), \left(\frac{\ell}{t}\right)\right), \left(k \cdot \left(k \cdot k\right)\right)\right), \ell\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(2, k\right), \left(\frac{\ell}{t}\right)\right), \left(k \cdot \left(k \cdot k\right)\right)\right), \ell\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(2, k\right), \mathsf{/.f64}\left(\ell, t\right)\right), \left(k \cdot \left(k \cdot k\right)\right)\right), \ell\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(2, k\right), \mathsf{/.f64}\left(\ell, t\right)\right), \mathsf{*.f64}\left(k, \left(k \cdot k\right)\right)\right), \ell\right) \]

      10. *-lowering-*.f64 72.8%

          \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(2, k\right), \mathsf{/.f64}\left(\ell, t\right)\right), \mathsf{*.f64}\left(k, \mathsf{*.f64}\left(k, k\right)\right)\right), \ell\right) \]

   9. Applied egg-rr 72.8%

      \[\leadsto \color{blue}{\frac{\frac{2}{k} \cdot \frac{\ell}{t}}{k \cdot \left(k \cdot k\right)}} \cdot \ell \]
   10. Step-by-step derivation

       1. associate-/r* N/A

          \[\leadsto \mathsf{*.f64}\left(\left(\frac{\frac{\frac{2}{k} \cdot \frac{\ell}{t}}{k}}{k \cdot k}\right), \ell\right) \]

       2. associate-/r* N/A

          \[\leadsto \mathsf{*.f64}\left(\left(\frac{\frac{\frac{\frac{2}{k} \cdot \frac{\ell}{t}}{k}}{k}}{k}\right), \ell\right) \]

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

          \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\left(\frac{\frac{\frac{2}{k} \cdot \frac{\ell}{t}}{k}}{k}\right), k\right), \ell\right) \]

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

          \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(\left(\frac{\frac{2}{k} \cdot \frac{\ell}{t}}{k}\right), k\right), k\right), \ell\right) \]

       5. frac-times N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(\left(\frac{\frac{2 \cdot \ell}{k \cdot t}}{k}\right), k\right), k\right), \ell\right) \]

       6. associate-/l/ N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(\left(\frac{2 \cdot \ell}{k \cdot \left(k \cdot t\right)}\right), k\right), k\right), \ell\right) \]

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

          \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(\left(2 \cdot \ell\right), \left(k \cdot \left(k \cdot t\right)\right)\right), k\right), k\right), \ell\right) \]

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

          \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(2, \ell\right), \left(k \cdot \left(k \cdot t\right)\right)\right), k\right), k\right), \ell\right) \]

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

          \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(2, \ell\right), \mathsf{*.f64}\left(k, \left(k \cdot t\right)\right)\right), k\right), k\right), \ell\right) \]

       10. *-lowering-*.f64 78.5%

           \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(2, \ell\right), \mathsf{*.f64}\left(k, \mathsf{*.f64}\left(k, t\right)\right)\right), k\right), k\right), \ell\right) \]

   11. Applied egg-rr 78.5%

       \[\leadsto \color{blue}{\frac{\frac{\frac{2 \cdot \ell}{k \cdot \left(k \cdot t\right)}}{k}}{k}} \cdot \ell \]

   if 8.500000000000001e-5 < k

   1. Initial program 21.4%

      \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) - 1\right)} \]
   2. Add Preprocessing

   3. Taylor expanded in t around 0

      \[\leadsto \color{blue}{2 \cdot \frac{{\ell}^{2} \cdot \cos k}{{k}^{2} \cdot \left(t \cdot {\sin k}^{2}\right)}} \]
   4. Step-by-step derivation

      1. associate-*r/ N/A

         \[\leadsto \frac{2 \cdot \left({\ell}^{2} \cdot \cos k\right)}{\color{blue}{{k}^{2} \cdot \left(t \cdot {\sin k}^{2}\right)}} \]

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

         \[\leadsto \mathsf{/.f64}\left(\left(2 \cdot \left({\ell}^{2} \cdot \cos k\right)\right), \color{blue}{\left({k}^{2} \cdot \left(t \cdot {\sin k}^{2}\right)\right)}\right) \]

      3. associate-*r* N/A

         \[\leadsto \mathsf{/.f64}\left(\left(\left(2 \cdot {\ell}^{2}\right) \cdot \cos k\right), \left(\color{blue}{{k}^{2}} \cdot \left(t \cdot {\sin k}^{2}\right)\right)\right) \]

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

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\left(2 \cdot {\ell}^{2}\right), \cos k\right), \left(\color{blue}{{k}^{2}} \cdot \left(t \cdot {\sin k}^{2}\right)\right)\right) \]

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

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(2, \left({\ell}^{2}\right)\right), \cos k\right), \left({\color{blue}{k}}^{2} \cdot \left(t \cdot {\sin k}^{2}\right)\right)\right) \]

      6. unpow2 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(2, \left(\ell \cdot \ell\right)\right), \cos k\right), \left({k}^{2} \cdot \left(t \cdot {\sin k}^{2}\right)\right)\right) \]

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

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(2, \mathsf{*.f64}\left(\ell, \ell\right)\right), \cos k\right), \left({k}^{2} \cdot \left(t \cdot {\sin k}^{2}\right)\right)\right) \]

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

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(2, \mathsf{*.f64}\left(\ell, \ell\right)\right), \mathsf{cos.f64}\left(k\right)\right), \left({k}^{\color{blue}{2}} \cdot \left(t \cdot {\sin k}^{2}\right)\right)\right) \]

      9. unpow2 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(2, \mathsf{*.f64}\left(\ell, \ell\right)\right), \mathsf{cos.f64}\left(k\right)\right), \left(\left(k \cdot k\right) \cdot \left(\color{blue}{t} \cdot {\sin k}^{2}\right)\right)\right) \]

      10. associate-*l* N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(2, \mathsf{*.f64}\left(\ell, \ell\right)\right), \mathsf{cos.f64}\left(k\right)\right), \left(k \cdot \color{blue}{\left(k \cdot \left(t \cdot {\sin k}^{2}\right)\right)}\right)\right) \]

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

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(2, \mathsf{*.f64}\left(\ell, \ell\right)\right), \mathsf{cos.f64}\left(k\right)\right), \mathsf{*.f64}\left(k, \color{blue}{\left(k \cdot \left(t \cdot {\sin k}^{2}\right)\right)}\right)\right) \]

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

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(2, \mathsf{*.f64}\left(\ell, \ell\right)\right), \mathsf{cos.f64}\left(k\right)\right), \mathsf{*.f64}\left(k, \mathsf{*.f64}\left(k, \color{blue}{\left(t \cdot {\sin k}^{2}\right)}\right)\right)\right) \]

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

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(2, \mathsf{*.f64}\left(\ell, \ell\right)\right), \mathsf{cos.f64}\left(k\right)\right), \mathsf{*.f64}\left(k, \mathsf{*.f64}\left(k, \mathsf{*.f64}\left(t, \color{blue}{\left({\sin k}^{2}\right)}\right)\right)\right)\right) \]

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

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(2, \mathsf{*.f64}\left(\ell, \ell\right)\right), \mathsf{cos.f64}\left(k\right)\right), \mathsf{*.f64}\left(k, \mathsf{*.f64}\left(k, \mathsf{*.f64}\left(t, \mathsf{pow.f64}\left(\sin k, \color{blue}{2}\right)\right)\right)\right)\right) \]

      15. sin-lowering-sin.f64 67.6%

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(2, \mathsf{*.f64}\left(\ell, \ell\right)\right), \mathsf{cos.f64}\left(k\right)\right), \mathsf{*.f64}\left(k, \mathsf{*.f64}\left(k, \mathsf{*.f64}\left(t, \mathsf{pow.f64}\left(\mathsf{sin.f64}\left(k\right), 2\right)\right)\right)\right)\right) \]

   5. Simplified 67.6%

      \[\leadsto \color{blue}{\frac{\left(2 \cdot \left(\ell \cdot \ell\right)\right) \cdot \cos k}{k \cdot \left(k \cdot \left(t \cdot {\sin k}^{2}\right)\right)}} \]
   6. Step-by-step derivation

      1. associate-/r* N/A

         \[\leadsto \frac{\frac{\left(2 \cdot \left(\ell \cdot \ell\right)\right) \cdot \cos k}{k}}{\color{blue}{k \cdot \left(t \cdot {\sin k}^{2}\right)}} \]

      2. associate-*r* N/A

         \[\leadsto \frac{\frac{\left(2 \cdot \left(\ell \cdot \ell\right)\right) \cdot \cos k}{k}}{\left(k \cdot t\right) \cdot \color{blue}{{\sin k}^{2}}} \]

      3. associate-/r* N/A

         \[\leadsto \frac{\frac{\frac{\left(2 \cdot \left(\ell \cdot \ell\right)\right) \cdot \cos k}{k}}{k \cdot t}}{\color{blue}{{\sin k}^{2}}} \]

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

         \[\leadsto \mathsf{/.f64}\left(\left(\frac{\frac{\left(2 \cdot \left(\ell \cdot \ell\right)\right) \cdot \cos k}{k}}{k \cdot t}\right), \color{blue}{\left({\sin k}^{2}\right)}\right) \]

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

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\left(\frac{\left(2 \cdot \left(\ell \cdot \ell\right)\right) \cdot \cos k}{k}\right), \left(k \cdot t\right)\right), \left({\color{blue}{\sin k}}^{2}\right)\right) \]

      6. associate-/l* N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\left(\left(2 \cdot \left(\ell \cdot \ell\right)\right) \cdot \frac{\cos k}{k}\right), \left(k \cdot t\right)\right), \left({\sin \color{blue}{k}}^{2}\right)\right) \]

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

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\left(2 \cdot \left(\ell \cdot \ell\right)\right), \left(\frac{\cos k}{k}\right)\right), \left(k \cdot t\right)\right), \left({\sin \color{blue}{k}}^{2}\right)\right) \]

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

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(2, \left(\ell \cdot \ell\right)\right), \left(\frac{\cos k}{k}\right)\right), \left(k \cdot t\right)\right), \left({\sin k}^{2}\right)\right) \]

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

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(2, \mathsf{*.f64}\left(\ell, \ell\right)\right), \left(\frac{\cos k}{k}\right)\right), \left(k \cdot t\right)\right), \left({\sin k}^{2}\right)\right) \]

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

          \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(2, \mathsf{*.f64}\left(\ell, \ell\right)\right), \mathsf{/.f64}\left(\cos k, k\right)\right), \left(k \cdot t\right)\right), \left({\sin k}^{2}\right)\right) \]

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

          \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(2, \mathsf{*.f64}\left(\ell, \ell\right)\right), \mathsf{/.f64}\left(\mathsf{cos.f64}\left(k\right), k\right)\right), \left(k \cdot t\right)\right), \left({\sin k}^{2}\right)\right) \]

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

          \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(2, \mathsf{*.f64}\left(\ell, \ell\right)\right), \mathsf{/.f64}\left(\mathsf{cos.f64}\left(k\right), k\right)\right), \mathsf{*.f64}\left(k, t\right)\right), \left({\sin k}^{2}\right)\right) \]

      13. unpow2 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(2, \mathsf{*.f64}\left(\ell, \ell\right)\right), \mathsf{/.f64}\left(\mathsf{cos.f64}\left(k\right), k\right)\right), \mathsf{*.f64}\left(k, t\right)\right), \left(\sin k \cdot \color{blue}{\sin k}\right)\right) \]

      14. sqr-sin-a N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(2, \mathsf{*.f64}\left(\ell, \ell\right)\right), \mathsf{/.f64}\left(\mathsf{cos.f64}\left(k\right), k\right)\right), \mathsf{*.f64}\left(k, t\right)\right), \left(\frac{1}{2} - \color{blue}{\frac{1}{2} \cdot \cos \left(2 \cdot k\right)}\right)\right) \]

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

          \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(2, \mathsf{*.f64}\left(\ell, \ell\right)\right), \mathsf{/.f64}\left(\mathsf{cos.f64}\left(k\right), k\right)\right), \mathsf{*.f64}\left(k, t\right)\right), \mathsf{\_.f64}\left(\frac{1}{2}, \color{blue}{\left(\frac{1}{2} \cdot \cos \left(2 \cdot k\right)\right)}\right)\right) \]

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

          \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(2, \mathsf{*.f64}\left(\ell, \ell\right)\right), \mathsf{/.f64}\left(\mathsf{cos.f64}\left(k\right), k\right)\right), \mathsf{*.f64}\left(k, t\right)\right), \mathsf{\_.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(\frac{1}{2}, \color{blue}{\cos \left(2 \cdot k\right)}\right)\right)\right) \]

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

          \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(2, \mathsf{*.f64}\left(\ell, \ell\right)\right), \mathsf{/.f64}\left(\mathsf{cos.f64}\left(k\right), k\right)\right), \mathsf{*.f64}\left(k, t\right)\right), \mathsf{\_.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(\frac{1}{2}, \mathsf{cos.f64}\left(\left(2 \cdot k\right)\right)\right)\right)\right) \]

      18. *-lowering-*.f64 71.6%

          \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(2, \mathsf{*.f64}\left(\ell, \ell\right)\right), \mathsf{/.f64}\left(\mathsf{cos.f64}\left(k\right), k\right)\right), \mathsf{*.f64}\left(k, t\right)\right), \mathsf{\_.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(\frac{1}{2}, \mathsf{cos.f64}\left(\mathsf{*.f64}\left(2, k\right)\right)\right)\right)\right) \]

   7. Applied egg-rr 71.6%

      \[\leadsto \color{blue}{\frac{\frac{\left(2 \cdot \left(\ell \cdot \ell\right)\right) \cdot \frac{\cos k}{k}}{k \cdot t}}{0.5 - 0.5 \cdot \cos \left(2 \cdot k\right)}} \]
   8. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\left(\frac{\cos k}{k} \cdot \left(2 \cdot \left(\ell \cdot \ell\right)\right)\right), \mathsf{*.f64}\left(k, t\right)\right), \mathsf{\_.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(\frac{1}{2}, \mathsf{cos.f64}\left(\mathsf{*.f64}\left(2, k\right)\right)\right)\right)\right) \]

      2. associate-*r* N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\left(\frac{\cos k}{k} \cdot \left(\left(2 \cdot \ell\right) \cdot \ell\right)\right), \mathsf{*.f64}\left(k, t\right)\right), \mathsf{\_.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(\frac{1}{2}, \mathsf{cos.f64}\left(\mathsf{*.f64}\left(2, k\right)\right)\right)\right)\right) \]

      3. associate-*r* N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\left(\left(\frac{\cos k}{k} \cdot \left(2 \cdot \ell\right)\right) \cdot \ell\right), \mathsf{*.f64}\left(k, t\right)\right), \mathsf{\_.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(\frac{1}{2}, \mathsf{cos.f64}\left(\mathsf{*.f64}\left(2, k\right)\right)\right)\right)\right) \]

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

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\left(\frac{\cos k}{k} \cdot \left(2 \cdot \ell\right)\right), \ell\right), \mathsf{*.f64}\left(k, t\right)\right), \mathsf{\_.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(\frac{1}{2}, \mathsf{cos.f64}\left(\mathsf{*.f64}\left(2, k\right)\right)\right)\right)\right) \]

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

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\left(\frac{\cos k}{k}\right), \left(2 \cdot \ell\right)\right), \ell\right), \mathsf{*.f64}\left(k, t\right)\right), \mathsf{\_.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(\frac{1}{2}, \mathsf{cos.f64}\left(\mathsf{*.f64}\left(2, k\right)\right)\right)\right)\right) \]

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

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(\cos k, k\right), \left(2 \cdot \ell\right)\right), \ell\right), \mathsf{*.f64}\left(k, t\right)\right), \mathsf{\_.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(\frac{1}{2}, \mathsf{cos.f64}\left(\mathsf{*.f64}\left(2, k\right)\right)\right)\right)\right) \]

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

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{cos.f64}\left(k\right), k\right), \left(2 \cdot \ell\right)\right), \ell\right), \mathsf{*.f64}\left(k, t\right)\right), \mathsf{\_.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(\frac{1}{2}, \mathsf{cos.f64}\left(\mathsf{*.f64}\left(2, k\right)\right)\right)\right)\right) \]

      8. *-lowering-*.f64 81.0%

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{cos.f64}\left(k\right), k\right), \mathsf{*.f64}\left(2, \ell\right)\right), \ell\right), \mathsf{*.f64}\left(k, t\right)\right), \mathsf{\_.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(\frac{1}{2}, \mathsf{cos.f64}\left(\mathsf{*.f64}\left(2, k\right)\right)\right)\right)\right) \]

   9. Applied egg-rr 81.0%

      \[\leadsto \frac{\frac{\color{blue}{\left(\frac{\cos k}{k} \cdot \left(2 \cdot \ell\right)\right) \cdot \ell}}{k \cdot t}}{0.5 - 0.5 \cdot \cos \left(2 \cdot k\right)} \]
   10. Step-by-step derivation

       1. *-commutative N/A

          \[\leadsto \mathsf{/.f64}\left(\left(\frac{\left(\frac{\cos k}{k} \cdot \left(2 \cdot \ell\right)\right) \cdot \ell}{t \cdot k}\right), \mathsf{\_.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(\frac{1}{2}, \mathsf{cos.f64}\left(\mathsf{*.f64}\left(2, k\right)\right)\right)\right)\right) \]

       2. times-frac N/A

          \[\leadsto \mathsf{/.f64}\left(\left(\frac{\frac{\cos k}{k} \cdot \left(2 \cdot \ell\right)}{t} \cdot \frac{\ell}{k}\right), \mathsf{\_.f64}\left(\color{blue}{\frac{1}{2}}, \mathsf{*.f64}\left(\frac{1}{2}, \mathsf{cos.f64}\left(\mathsf{*.f64}\left(2, k\right)\right)\right)\right)\right) \]

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

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\left(\frac{\frac{\cos k}{k} \cdot \left(2 \cdot \ell\right)}{t}\right), \left(\frac{\ell}{k}\right)\right), \mathsf{\_.f64}\left(\color{blue}{\frac{1}{2}}, \mathsf{*.f64}\left(\frac{1}{2}, \mathsf{cos.f64}\left(\mathsf{*.f64}\left(2, k\right)\right)\right)\right)\right) \]

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

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(\left(\frac{\cos k}{k} \cdot \left(2 \cdot \ell\right)\right), t\right), \left(\frac{\ell}{k}\right)\right), \mathsf{\_.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(\frac{1}{2}, \mathsf{cos.f64}\left(\mathsf{*.f64}\left(2, k\right)\right)\right)\right)\right) \]

       5. *-commutative N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(\left(\left(2 \cdot \ell\right) \cdot \frac{\cos k}{k}\right), t\right), \left(\frac{\ell}{k}\right)\right), \mathsf{\_.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(\frac{1}{2}, \mathsf{cos.f64}\left(\mathsf{*.f64}\left(2, k\right)\right)\right)\right)\right) \]

       6. clear-num N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(\left(\left(2 \cdot \ell\right) \cdot \frac{1}{\frac{k}{\cos k}}\right), t\right), \left(\frac{\ell}{k}\right)\right), \mathsf{\_.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(\frac{1}{2}, \mathsf{cos.f64}\left(\mathsf{*.f64}\left(2, k\right)\right)\right)\right)\right) \]

       7. un-div-inv N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(\left(\frac{2 \cdot \ell}{\frac{k}{\cos k}}\right), t\right), \left(\frac{\ell}{k}\right)\right), \mathsf{\_.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(\frac{1}{2}, \mathsf{cos.f64}\left(\mathsf{*.f64}\left(2, k\right)\right)\right)\right)\right) \]

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

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(\left(2 \cdot \ell\right), \left(\frac{k}{\cos k}\right)\right), t\right), \left(\frac{\ell}{k}\right)\right), \mathsf{\_.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(\frac{1}{2}, \mathsf{cos.f64}\left(\mathsf{*.f64}\left(2, k\right)\right)\right)\right)\right) \]

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

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(2, \ell\right), \left(\frac{k}{\cos k}\right)\right), t\right), \left(\frac{\ell}{k}\right)\right), \mathsf{\_.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(\frac{1}{2}, \mathsf{cos.f64}\left(\mathsf{*.f64}\left(2, k\right)\right)\right)\right)\right) \]

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

           \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(2, \ell\right), \mathsf{/.f64}\left(k, \cos k\right)\right), t\right), \left(\frac{\ell}{k}\right)\right), \mathsf{\_.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(\frac{1}{2}, \mathsf{cos.f64}\left(\mathsf{*.f64}\left(2, k\right)\right)\right)\right)\right) \]

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

           \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(2, \ell\right), \mathsf{/.f64}\left(k, \mathsf{cos.f64}\left(k\right)\right)\right), t\right), \left(\frac{\ell}{k}\right)\right), \mathsf{\_.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(\frac{1}{2}, \mathsf{cos.f64}\left(\mathsf{*.f64}\left(2, k\right)\right)\right)\right)\right) \]

       12. /-lowering-/.f64 99.3%

           \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(2, \ell\right), \mathsf{/.f64}\left(k, \mathsf{cos.f64}\left(k\right)\right)\right), t\right), \mathsf{/.f64}\left(\ell, k\right)\right), \mathsf{\_.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(\frac{1}{2}, \mathsf{cos.f64}\left(\mathsf{*.f64}\left(2, k\right)\right)\right)\right)\right) \]

   11. Applied egg-rr 99.3%

       \[\leadsto \frac{\color{blue}{\frac{\frac{2 \cdot \ell}{\frac{k}{\cos k}}}{t} \cdot \frac{\ell}{k}}}{0.5 - 0.5 \cdot \cos \left(2 \cdot k\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 83.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;k \leq 8.5 \cdot 10^{-5}:\\ \;\;\;\;\ell \cdot \frac{\frac{\frac{2 \cdot \ell}{k \cdot \left(k \cdot t\right)}}{k}}{k}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\frac{2 \cdot \ell}{\frac{k}{\cos k}}}{t} \cdot \frac{\ell}{k}}{0.5 - 0.5 \cdot \cos \left(k \cdot 2\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 92.4% accurate, 1.8× speedup

Math

\[\begin{array}{l} k_m = \left|k\right| \\ \begin{array}{l} t_1 := 0.5 + \cos \left(k\_m \cdot 2\right) \cdot -0.5\\ \mathbf{if}\;k\_m \leq 8.5 \cdot 10^{-5}:\\ \;\;\;\;\ell \cdot \frac{\frac{\frac{2 \cdot \ell}{k\_m \cdot \left(k\_m \cdot t\right)}}{k\_m}}{k\_m}\\ \mathbf{elif}\;k\_m \leq 7.2 \cdot 10^{+188}:\\ \;\;\;\;\left(\frac{\cos k\_m}{k\_m} \cdot \frac{2 \cdot \ell}{k\_m}\right) \cdot \frac{\frac{\ell}{t}}{t\_1}\\ \mathbf{else}:\\ \;\;\;\;\frac{\ell}{t\_1} \cdot \frac{\frac{2 \cdot \ell}{\frac{k\_m}{\cos k\_m}}}{k\_m \cdot t}\\ \end{array} \end{array} \]

FPCore

k_m = (fabs.f64 k)
(FPCore (t l k_m)
 :precision binary64
 (let* ((t_1 (+ 0.5 (* (cos (* k_m 2.0)) -0.5))))
   (if (<= k_m 8.5e-5)
     (* l (/ (/ (/ (* 2.0 l) (* k_m (* k_m t))) k_m) k_m))
     (if (<= k_m 7.2e+188)
       (* (* (/ (cos k_m) k_m) (/ (* 2.0 l) k_m)) (/ (/ l t) t_1))
       (* (/ l t_1) (/ (/ (* 2.0 l) (/ k_m (cos k_m))) (* k_m t)))))))

C

k_m = fabs(k);
double code(double t, double l, double k_m) {
	double t_1 = 0.5 + (cos((k_m * 2.0)) * -0.5);
	double tmp;
	if (k_m <= 8.5e-5) {
		tmp = l * ((((2.0 * l) / (k_m * (k_m * t))) / k_m) / k_m);
	} else if (k_m <= 7.2e+188) {
		tmp = ((cos(k_m) / k_m) * ((2.0 * l) / k_m)) * ((l / t) / t_1);
	} else {
		tmp = (l / t_1) * (((2.0 * l) / (k_m / cos(k_m))) / (k_m * t));
	}
	return tmp;
}

Fortran

k_m = abs(k)
real(8) function code(t, l, k_m)
    real(8), intent (in) :: t
    real(8), intent (in) :: l
    real(8), intent (in) :: k_m
    real(8) :: t_1
    real(8) :: tmp
    t_1 = 0.5d0 + (cos((k_m * 2.0d0)) * (-0.5d0))
    if (k_m <= 8.5d-5) then
        tmp = l * ((((2.0d0 * l) / (k_m * (k_m * t))) / k_m) / k_m)
    else if (k_m <= 7.2d+188) then
        tmp = ((cos(k_m) / k_m) * ((2.0d0 * l) / k_m)) * ((l / t) / t_1)
    else
        tmp = (l / t_1) * (((2.0d0 * l) / (k_m / cos(k_m))) / (k_m * t))
    end if
    code = tmp
end function

Java

k_m = Math.abs(k);
public static double code(double t, double l, double k_m) {
	double t_1 = 0.5 + (Math.cos((k_m * 2.0)) * -0.5);
	double tmp;
	if (k_m <= 8.5e-5) {
		tmp = l * ((((2.0 * l) / (k_m * (k_m * t))) / k_m) / k_m);
	} else if (k_m <= 7.2e+188) {
		tmp = ((Math.cos(k_m) / k_m) * ((2.0 * l) / k_m)) * ((l / t) / t_1);
	} else {
		tmp = (l / t_1) * (((2.0 * l) / (k_m / Math.cos(k_m))) / (k_m * t));
	}
	return tmp;
}

Python

k_m = math.fabs(k)
def code(t, l, k_m):
	t_1 = 0.5 + (math.cos((k_m * 2.0)) * -0.5)
	tmp = 0
	if k_m <= 8.5e-5:
		tmp = l * ((((2.0 * l) / (k_m * (k_m * t))) / k_m) / k_m)
	elif k_m <= 7.2e+188:
		tmp = ((math.cos(k_m) / k_m) * ((2.0 * l) / k_m)) * ((l / t) / t_1)
	else:
		tmp = (l / t_1) * (((2.0 * l) / (k_m / math.cos(k_m))) / (k_m * t))
	return tmp

Julia

k_m = abs(k)
function code(t, l, k_m)
	t_1 = Float64(0.5 + Float64(cos(Float64(k_m * 2.0)) * -0.5))
	tmp = 0.0
	if (k_m <= 8.5e-5)
		tmp = Float64(l * Float64(Float64(Float64(Float64(2.0 * l) / Float64(k_m * Float64(k_m * t))) / k_m) / k_m));
	elseif (k_m <= 7.2e+188)
		tmp = Float64(Float64(Float64(cos(k_m) / k_m) * Float64(Float64(2.0 * l) / k_m)) * Float64(Float64(l / t) / t_1));
	else
		tmp = Float64(Float64(l / t_1) * Float64(Float64(Float64(2.0 * l) / Float64(k_m / cos(k_m))) / Float64(k_m * t)));
	end
	return tmp
end

MATLAB

k_m = abs(k);
function tmp_2 = code(t, l, k_m)
	t_1 = 0.5 + (cos((k_m * 2.0)) * -0.5);
	tmp = 0.0;
	if (k_m <= 8.5e-5)
		tmp = l * ((((2.0 * l) / (k_m * (k_m * t))) / k_m) / k_m);
	elseif (k_m <= 7.2e+188)
		tmp = ((cos(k_m) / k_m) * ((2.0 * l) / k_m)) * ((l / t) / t_1);
	else
		tmp = (l / t_1) * (((2.0 * l) / (k_m / cos(k_m))) / (k_m * t));
	end
	tmp_2 = tmp;
end

Wolfram

k_m = N[Abs[k], $MachinePrecision]
code[t_, l_, k$95$m_] := Block[{t$95$1 = N[(0.5 + N[(N[Cos[N[(k$95$m * 2.0), $MachinePrecision]], $MachinePrecision] * -0.5), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[k$95$m, 8.5e-5], N[(l * N[(N[(N[(N[(2.0 * l), $MachinePrecision] / N[(k$95$m * N[(k$95$m * t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / k$95$m), $MachinePrecision] / k$95$m), $MachinePrecision]), $MachinePrecision], If[LessEqual[k$95$m, 7.2e+188], N[(N[(N[(N[Cos[k$95$m], $MachinePrecision] / k$95$m), $MachinePrecision] * N[(N[(2.0 * l), $MachinePrecision] / k$95$m), $MachinePrecision]), $MachinePrecision] * N[(N[(l / t), $MachinePrecision] / t$95$1), $MachinePrecision]), $MachinePrecision], N[(N[(l / t$95$1), $MachinePrecision] * N[(N[(N[(2.0 * l), $MachinePrecision] / N[(k$95$m / N[Cos[k$95$m], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(k$95$m * t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if k < 8.500000000000001e-5

   1. Initial program 35.0%

      \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) - 1\right)} \]
   2. Add Preprocessing

   3. Taylor expanded in k around 0

      \[\leadsto \mathsf{/.f64}\left(2, \color{blue}{\left(\frac{{k}^{4} \cdot t}{{\ell}^{2}}\right)}\right) \]
   4. Step-by-step derivation

      1. associate-/l* N/A

         \[\leadsto \mathsf{/.f64}\left(2, \left({k}^{4} \cdot \color{blue}{\frac{t}{{\ell}^{2}}}\right)\right) \]

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

         \[\leadsto \mathsf{/.f64}\left(2, \mathsf{*.f64}\left(\left({k}^{4}\right), \color{blue}{\left(\frac{t}{{\ell}^{2}}\right)}\right)\right) \]

      3. metadata-eval N/A

         \[\leadsto \mathsf{/.f64}\left(2, \mathsf{*.f64}\left(\left({k}^{\left(2 \cdot 2\right)}\right), \left(\frac{t}{{\ell}^{2}}\right)\right)\right) \]

      4. pow-sqr N/A

         \[\leadsto \mathsf{/.f64}\left(2, \mathsf{*.f64}\left(\left({k}^{2} \cdot {k}^{2}\right), \left(\frac{\color{blue}{t}}{{\ell}^{2}}\right)\right)\right) \]

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

         \[\leadsto \mathsf{/.f64}\left(2, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\left({k}^{2}\right), \left({k}^{2}\right)\right), \left(\frac{\color{blue}{t}}{{\ell}^{2}}\right)\right)\right) \]

      6. unpow2 N/A

         \[\leadsto \mathsf{/.f64}\left(2, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\left(k \cdot k\right), \left({k}^{2}\right)\right), \left(\frac{t}{{\ell}^{2}}\right)\right)\right) \]

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

         \[\leadsto \mathsf{/.f64}\left(2, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(k, k\right), \left({k}^{2}\right)\right), \left(\frac{t}{{\ell}^{2}}\right)\right)\right) \]

      8. unpow2 N/A

         \[\leadsto \mathsf{/.f64}\left(2, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(k, k\right), \left(k \cdot k\right)\right), \left(\frac{t}{{\ell}^{2}}\right)\right)\right) \]

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

         \[\leadsto \mathsf{/.f64}\left(2, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(k, k\right), \mathsf{*.f64}\left(k, k\right)\right), \left(\frac{t}{{\ell}^{2}}\right)\right)\right) \]

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

          \[\leadsto \mathsf{/.f64}\left(2, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(k, k\right), \mathsf{*.f64}\left(k, k\right)\right), \mathsf{/.f64}\left(t, \color{blue}{\left({\ell}^{2}\right)}\right)\right)\right) \]

      11. unpow2 N/A

          \[\leadsto \mathsf{/.f64}\left(2, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(k, k\right), \mathsf{*.f64}\left(k, k\right)\right), \mathsf{/.f64}\left(t, \left(\ell \cdot \color{blue}{\ell}\right)\right)\right)\right) \]

      12. *-lowering-*.f64 58.1%

          \[\leadsto \mathsf{/.f64}\left(2, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(k, k\right), \mathsf{*.f64}\left(k, k\right)\right), \mathsf{/.f64}\left(t, \mathsf{*.f64}\left(\ell, \color{blue}{\ell}\right)\right)\right)\right) \]

   5. Simplified 58.1%

      \[\leadsto \frac{2}{\color{blue}{\left(\left(k \cdot k\right) \cdot \left(k \cdot k\right)\right) \cdot \frac{t}{\ell \cdot \ell}}} \]
   6. Step-by-step derivation

      1. associate-/r* N/A

         \[\leadsto \frac{\frac{2}{\left(k \cdot k\right) \cdot \left(k \cdot k\right)}}{\color{blue}{\frac{t}{\ell \cdot \ell}}} \]

      2. associate-/r* N/A

         \[\leadsto \frac{\frac{2}{\left(k \cdot k\right) \cdot \left(k \cdot k\right)}}{\frac{\frac{t}{\ell}}{\color{blue}{\ell}}} \]

      3. associate-/r/ N/A

         \[\leadsto \frac{\frac{2}{\left(k \cdot k\right) \cdot \left(k \cdot k\right)}}{\frac{t}{\ell}} \cdot \color{blue}{\ell} \]

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

         \[\leadsto \mathsf{*.f64}\left(\left(\frac{\frac{2}{\left(k \cdot k\right) \cdot \left(k \cdot k\right)}}{\frac{t}{\ell}}\right), \color{blue}{\ell}\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\left(\frac{2}{\left(k \cdot k\right) \cdot \left(k \cdot k\right)}\right), \left(\frac{t}{\ell}\right)\right), \ell\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(2, \left(\left(k \cdot k\right) \cdot \left(k \cdot k\right)\right)\right), \left(\frac{t}{\ell}\right)\right), \ell\right) \]

      7. associate-*l* N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(2, \left(k \cdot \left(k \cdot \left(k \cdot k\right)\right)\right)\right), \left(\frac{t}{\ell}\right)\right), \ell\right) \]

      8. cube-unmult N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(2, \left(k \cdot {k}^{3}\right)\right), \left(\frac{t}{\ell}\right)\right), \ell\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(2, \mathsf{*.f64}\left(k, \left({k}^{3}\right)\right)\right), \left(\frac{t}{\ell}\right)\right), \ell\right) \]

      10. cube-unmult N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(2, \mathsf{*.f64}\left(k, \left(k \cdot \left(k \cdot k\right)\right)\right)\right), \left(\frac{t}{\ell}\right)\right), \ell\right) \]

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

          \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(2, \mathsf{*.f64}\left(k, \mathsf{*.f64}\left(k, \left(k \cdot k\right)\right)\right)\right), \left(\frac{t}{\ell}\right)\right), \ell\right) \]

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

          \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(2, \mathsf{*.f64}\left(k, \mathsf{*.f64}\left(k, \mathsf{*.f64}\left(k, k\right)\right)\right)\right), \left(\frac{t}{\ell}\right)\right), \ell\right) \]

      13. /-lowering-/.f64 69.3%

          \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(2, \mathsf{*.f64}\left(k, \mathsf{*.f64}\left(k, \mathsf{*.f64}\left(k, k\right)\right)\right)\right), \mathsf{/.f64}\left(t, \ell\right)\right), \ell\right) \]

   7. Applied egg-rr 69.3%

      \[\leadsto \color{blue}{\frac{\frac{2}{k \cdot \left(k \cdot \left(k \cdot k\right)\right)}}{\frac{t}{\ell}} \cdot \ell} \]
   8. Step-by-step derivation

      1. div-inv N/A

         \[\leadsto \mathsf{*.f64}\left(\left(\frac{2}{k \cdot \left(k \cdot \left(k \cdot k\right)\right)} \cdot \frac{1}{\frac{t}{\ell}}\right), \ell\right) \]

      2. associate-/r* N/A

         \[\leadsto \mathsf{*.f64}\left(\left(\frac{\frac{2}{k}}{k \cdot \left(k \cdot k\right)} \cdot \frac{1}{\frac{t}{\ell}}\right), \ell\right) \]

      3. clear-num N/A

         \[\leadsto \mathsf{*.f64}\left(\left(\frac{\frac{2}{k}}{k \cdot \left(k \cdot k\right)} \cdot \frac{\ell}{t}\right), \ell\right) \]

      4. associate-*l/ N/A

         \[\leadsto \mathsf{*.f64}\left(\left(\frac{\frac{2}{k} \cdot \frac{\ell}{t}}{k \cdot \left(k \cdot k\right)}\right), \ell\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\left(\frac{2}{k} \cdot \frac{\ell}{t}\right), \left(k \cdot \left(k \cdot k\right)\right)\right), \ell\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\left(\frac{2}{k}\right), \left(\frac{\ell}{t}\right)\right), \left(k \cdot \left(k \cdot k\right)\right)\right), \ell\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(2, k\right), \left(\frac{\ell}{t}\right)\right), \left(k \cdot \left(k \cdot k\right)\right)\right), \ell\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(2, k\right), \mathsf{/.f64}\left(\ell, t\right)\right), \left(k \cdot \left(k \cdot k\right)\right)\right), \ell\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(2, k\right), \mathsf{/.f64}\left(\ell, t\right)\right), \mathsf{*.f64}\left(k, \left(k \cdot k\right)\right)\right), \ell\right) \]

      10. *-lowering-*.f64 72.8%

          \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(2, k\right), \mathsf{/.f64}\left(\ell, t\right)\right), \mathsf{*.f64}\left(k, \mathsf{*.f64}\left(k, k\right)\right)\right), \ell\right) \]

   9. Applied egg-rr 72.8%

      \[\leadsto \color{blue}{\frac{\frac{2}{k} \cdot \frac{\ell}{t}}{k \cdot \left(k \cdot k\right)}} \cdot \ell \]
   10. Step-by-step derivation

       1. associate-/r* N/A

          \[\leadsto \mathsf{*.f64}\left(\left(\frac{\frac{\frac{2}{k} \cdot \frac{\ell}{t}}{k}}{k \cdot k}\right), \ell\right) \]

       2. associate-/r* N/A

          \[\leadsto \mathsf{*.f64}\left(\left(\frac{\frac{\frac{\frac{2}{k} \cdot \frac{\ell}{t}}{k}}{k}}{k}\right), \ell\right) \]

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

          \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\left(\frac{\frac{\frac{2}{k} \cdot \frac{\ell}{t}}{k}}{k}\right), k\right), \ell\right) \]

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

          \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(\left(\frac{\frac{2}{k} \cdot \frac{\ell}{t}}{k}\right), k\right), k\right), \ell\right) \]

       5. frac-times N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(\left(\frac{\frac{2 \cdot \ell}{k \cdot t}}{k}\right), k\right), k\right), \ell\right) \]

       6. associate-/l/ N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(\left(\frac{2 \cdot \ell}{k \cdot \left(k \cdot t\right)}\right), k\right), k\right), \ell\right) \]

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

          \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(\left(2 \cdot \ell\right), \left(k \cdot \left(k \cdot t\right)\right)\right), k\right), k\right), \ell\right) \]

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

          \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(2, \ell\right), \left(k \cdot \left(k \cdot t\right)\right)\right), k\right), k\right), \ell\right) \]

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

          \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(2, \ell\right), \mathsf{*.f64}\left(k, \left(k \cdot t\right)\right)\right), k\right), k\right), \ell\right) \]

       10. *-lowering-*.f64 78.5%

           \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(2, \ell\right), \mathsf{*.f64}\left(k, \mathsf{*.f64}\left(k, t\right)\right)\right), k\right), k\right), \ell\right) \]

   11. Applied egg-rr 78.5%

       \[\leadsto \color{blue}{\frac{\frac{\frac{2 \cdot \ell}{k \cdot \left(k \cdot t\right)}}{k}}{k}} \cdot \ell \]

   if 8.500000000000001e-5 < k < 7.20000000000000041e188

   1. Initial program 16.4%

      \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) - 1\right)} \]
   2. Add Preprocessing

   3. Taylor expanded in t around 0

      \[\leadsto \color{blue}{2 \cdot \frac{{\ell}^{2} \cdot \cos k}{{k}^{2} \cdot \left(t \cdot {\sin k}^{2}\right)}} \]
   4. Step-by-step derivation

      1. associate-*r/ N/A

         \[\leadsto \frac{2 \cdot \left({\ell}^{2} \cdot \cos k\right)}{\color{blue}{{k}^{2} \cdot \left(t \cdot {\sin k}^{2}\right)}} \]

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

         \[\leadsto \mathsf{/.f64}\left(\left(2 \cdot \left({\ell}^{2} \cdot \cos k\right)\right), \color{blue}{\left({k}^{2} \cdot \left(t \cdot {\sin k}^{2}\right)\right)}\right) \]

      3. associate-*r* N/A

         \[\leadsto \mathsf{/.f64}\left(\left(\left(2 \cdot {\ell}^{2}\right) \cdot \cos k\right), \left(\color{blue}{{k}^{2}} \cdot \left(t \cdot {\sin k}^{2}\right)\right)\right) \]

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

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\left(2 \cdot {\ell}^{2}\right), \cos k\right), \left(\color{blue}{{k}^{2}} \cdot \left(t \cdot {\sin k}^{2}\right)\right)\right) \]

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

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(2, \left({\ell}^{2}\right)\right), \cos k\right), \left({\color{blue}{k}}^{2} \cdot \left(t \cdot {\sin k}^{2}\right)\right)\right) \]

      6. unpow2 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(2, \left(\ell \cdot \ell\right)\right), \cos k\right), \left({k}^{2} \cdot \left(t \cdot {\sin k}^{2}\right)\right)\right) \]

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

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(2, \mathsf{*.f64}\left(\ell, \ell\right)\right), \cos k\right), \left({k}^{2} \cdot \left(t \cdot {\sin k}^{2}\right)\right)\right) \]

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

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(2, \mathsf{*.f64}\left(\ell, \ell\right)\right), \mathsf{cos.f64}\left(k\right)\right), \left({k}^{\color{blue}{2}} \cdot \left(t \cdot {\sin k}^{2}\right)\right)\right) \]

      9. unpow2 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(2, \mathsf{*.f64}\left(\ell, \ell\right)\right), \mathsf{cos.f64}\left(k\right)\right), \left(\left(k \cdot k\right) \cdot \left(\color{blue}{t} \cdot {\sin k}^{2}\right)\right)\right) \]

      10. associate-*l* N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(2, \mathsf{*.f64}\left(\ell, \ell\right)\right), \mathsf{cos.f64}\left(k\right)\right), \left(k \cdot \color{blue}{\left(k \cdot \left(t \cdot {\sin k}^{2}\right)\right)}\right)\right) \]

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

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(2, \mathsf{*.f64}\left(\ell, \ell\right)\right), \mathsf{cos.f64}\left(k\right)\right), \mathsf{*.f64}\left(k, \color{blue}{\left(k \cdot \left(t \cdot {\sin k}^{2}\right)\right)}\right)\right) \]

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

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(2, \mathsf{*.f64}\left(\ell, \ell\right)\right), \mathsf{cos.f64}\left(k\right)\right), \mathsf{*.f64}\left(k, \mathsf{*.f64}\left(k, \color{blue}{\left(t \cdot {\sin k}^{2}\right)}\right)\right)\right) \]

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

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(2, \mathsf{*.f64}\left(\ell, \ell\right)\right), \mathsf{cos.f64}\left(k\right)\right), \mathsf{*.f64}\left(k, \mathsf{*.f64}\left(k, \mathsf{*.f64}\left(t, \color{blue}{\left({\sin k}^{2}\right)}\right)\right)\right)\right) \]

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

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(2, \mathsf{*.f64}\left(\ell, \ell\right)\right), \mathsf{cos.f64}\left(k\right)\right), \mathsf{*.f64}\left(k, \mathsf{*.f64}\left(k, \mathsf{*.f64}\left(t, \mathsf{pow.f64}\left(\sin k, \color{blue}{2}\right)\right)\right)\right)\right) \]

      15. sin-lowering-sin.f64 75.4%

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(2, \mathsf{*.f64}\left(\ell, \ell\right)\right), \mathsf{cos.f64}\left(k\right)\right), \mathsf{*.f64}\left(k, \mathsf{*.f64}\left(k, \mathsf{*.f64}\left(t, \mathsf{pow.f64}\left(\mathsf{sin.f64}\left(k\right), 2\right)\right)\right)\right)\right) \]

   5. Simplified 75.4%

      \[\leadsto \color{blue}{\frac{\left(2 \cdot \left(\ell \cdot \ell\right)\right) \cdot \cos k}{k \cdot \left(k \cdot \left(t \cdot {\sin k}^{2}\right)\right)}} \]
   6. Step-by-step derivation

      1. associate-/r* N/A

         \[\leadsto \frac{\frac{\left(2 \cdot \left(\ell \cdot \ell\right)\right) \cdot \cos k}{k}}{\color{blue}{k \cdot \left(t \cdot {\sin k}^{2}\right)}} \]

      2. associate-*r* N/A

         \[\leadsto \frac{\frac{\left(2 \cdot \left(\ell \cdot \ell\right)\right) \cdot \cos k}{k}}{\left(k \cdot t\right) \cdot \color{blue}{{\sin k}^{2}}} \]

      3. associate-/r* N/A

         \[\leadsto \frac{\frac{\frac{\left(2 \cdot \left(\ell \cdot \ell\right)\right) \cdot \cos k}{k}}{k \cdot t}}{\color{blue}{{\sin k}^{2}}} \]

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

         \[\leadsto \mathsf{/.f64}\left(\left(\frac{\frac{\left(2 \cdot \left(\ell \cdot \ell\right)\right) \cdot \cos k}{k}}{k \cdot t}\right), \color{blue}{\left({\sin k}^{2}\right)}\right) \]

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

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\left(\frac{\left(2 \cdot \left(\ell \cdot \ell\right)\right) \cdot \cos k}{k}\right), \left(k \cdot t\right)\right), \left({\color{blue}{\sin k}}^{2}\right)\right) \]

      6. associate-/l* N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\left(\left(2 \cdot \left(\ell \cdot \ell\right)\right) \cdot \frac{\cos k}{k}\right), \left(k \cdot t\right)\right), \left({\sin \color{blue}{k}}^{2}\right)\right) \]

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

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\left(2 \cdot \left(\ell \cdot \ell\right)\right), \left(\frac{\cos k}{k}\right)\right), \left(k \cdot t\right)\right), \left({\sin \color{blue}{k}}^{2}\right)\right) \]

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

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(2, \left(\ell \cdot \ell\right)\right), \left(\frac{\cos k}{k}\right)\right), \left(k \cdot t\right)\right), \left({\sin k}^{2}\right)\right) \]

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

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(2, \mathsf{*.f64}\left(\ell, \ell\right)\right), \left(\frac{\cos k}{k}\right)\right), \left(k \cdot t\right)\right), \left({\sin k}^{2}\right)\right) \]

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

          \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(2, \mathsf{*.f64}\left(\ell, \ell\right)\right), \mathsf{/.f64}\left(\cos k, k\right)\right), \left(k \cdot t\right)\right), \left({\sin k}^{2}\right)\right) \]

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

          \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(2, \mathsf{*.f64}\left(\ell, \ell\right)\right), \mathsf{/.f64}\left(\mathsf{cos.f64}\left(k\right), k\right)\right), \left(k \cdot t\right)\right), \left({\sin k}^{2}\right)\right) \]

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

          \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(2, \mathsf{*.f64}\left(\ell, \ell\right)\right), \mathsf{/.f64}\left(\mathsf{cos.f64}\left(k\right), k\right)\right), \mathsf{*.f64}\left(k, t\right)\right), \left({\sin k}^{2}\right)\right) \]

      13. unpow2 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(2, \mathsf{*.f64}\left(\ell, \ell\right)\right), \mathsf{/.f64}\left(\mathsf{cos.f64}\left(k\right), k\right)\right), \mathsf{*.f64}\left(k, t\right)\right), \left(\sin k \cdot \color{blue}{\sin k}\right)\right) \]

      14. sqr-sin-a N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(2, \mathsf{*.f64}\left(\ell, \ell\right)\right), \mathsf{/.f64}\left(\mathsf{cos.f64}\left(k\right), k\right)\right), \mathsf{*.f64}\left(k, t\right)\right), \left(\frac{1}{2} - \color{blue}{\frac{1}{2} \cdot \cos \left(2 \cdot k\right)}\right)\right) \]

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

          \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(2, \mathsf{*.f64}\left(\ell, \ell\right)\right), \mathsf{/.f64}\left(\mathsf{cos.f64}\left(k\right), k\right)\right), \mathsf{*.f64}\left(k, t\right)\right), \mathsf{\_.f64}\left(\frac{1}{2}, \color{blue}{\left(\frac{1}{2} \cdot \cos \left(2 \cdot k\right)\right)}\right)\right) \]

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

          \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(2, \mathsf{*.f64}\left(\ell, \ell\right)\right), \mathsf{/.f64}\left(\mathsf{cos.f64}\left(k\right), k\right)\right), \mathsf{*.f64}\left(k, t\right)\right), \mathsf{\_.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(\frac{1}{2}, \color{blue}{\cos \left(2 \cdot k\right)}\right)\right)\right) \]

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

          \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(2, \mathsf{*.f64}\left(\ell, \ell\right)\right), \mathsf{/.f64}\left(\mathsf{cos.f64}\left(k\right), k\right)\right), \mathsf{*.f64}\left(k, t\right)\right), \mathsf{\_.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(\frac{1}{2}, \mathsf{cos.f64}\left(\left(2 \cdot k\right)\right)\right)\right)\right) \]

      18. *-lowering-*.f64 76.5%

          \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(2, \mathsf{*.f64}\left(\ell, \ell\right)\right), \mathsf{/.f64}\left(\mathsf{cos.f64}\left(k\right), k\right)\right), \mathsf{*.f64}\left(k, t\right)\right), \mathsf{\_.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(\frac{1}{2}, \mathsf{cos.f64}\left(\mathsf{*.f64}\left(2, k\right)\right)\right)\right)\right) \]

   7. Applied egg-rr 76.5%

      \[\leadsto \color{blue}{\frac{\frac{\left(2 \cdot \left(\ell \cdot \ell\right)\right) \cdot \frac{\cos k}{k}}{k \cdot t}}{0.5 - 0.5 \cdot \cos \left(2 \cdot k\right)}} \]
   8. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\left(\frac{\cos k}{k} \cdot \left(2 \cdot \left(\ell \cdot \ell\right)\right)\right), \mathsf{*.f64}\left(k, t\right)\right), \mathsf{\_.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(\frac{1}{2}, \mathsf{cos.f64}\left(\mathsf{*.f64}\left(2, k\right)\right)\right)\right)\right) \]

      2. associate-*r* N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\left(\frac{\cos k}{k} \cdot \left(\left(2 \cdot \ell\right) \cdot \ell\right)\right), \mathsf{*.f64}\left(k, t\right)\right), \mathsf{\_.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(\frac{1}{2}, \mathsf{cos.f64}\left(\mathsf{*.f64}\left(2, k\right)\right)\right)\right)\right) \]

      3. associate-*r* N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\left(\left(\frac{\cos k}{k} \cdot \left(2 \cdot \ell\right)\right) \cdot \ell\right), \mathsf{*.f64}\left(k, t\right)\right), \mathsf{\_.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(\frac{1}{2}, \mathsf{cos.f64}\left(\mathsf{*.f64}\left(2, k\right)\right)\right)\right)\right) \]

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

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\left(\frac{\cos k}{k} \cdot \left(2 \cdot \ell\right)\right), \ell\right), \mathsf{*.f64}\left(k, t\right)\right), \mathsf{\_.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(\frac{1}{2}, \mathsf{cos.f64}\left(\mathsf{*.f64}\left(2, k\right)\right)\right)\right)\right) \]

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

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\left(\frac{\cos k}{k}\right), \left(2 \cdot \ell\right)\right), \ell\right), \mathsf{*.f64}\left(k, t\right)\right), \mathsf{\_.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(\frac{1}{2}, \mathsf{cos.f64}\left(\mathsf{*.f64}\left(2, k\right)\right)\right)\right)\right) \]

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

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(\cos k, k\right), \left(2 \cdot \ell\right)\right), \ell\right), \mathsf{*.f64}\left(k, t\right)\right), \mathsf{\_.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(\frac{1}{2}, \mathsf{cos.f64}\left(\mathsf{*.f64}\left(2, k\right)\right)\right)\right)\right) \]

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

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{cos.f64}\left(k\right), k\right), \left(2 \cdot \ell\right)\right), \ell\right), \mathsf{*.f64}\left(k, t\right)\right), \mathsf{\_.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(\frac{1}{2}, \mathsf{cos.f64}\left(\mathsf{*.f64}\left(2, k\right)\right)\right)\right)\right) \]

      8. *-lowering-*.f64 76.9%

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{cos.f64}\left(k\right), k\right), \mathsf{*.f64}\left(2, \ell\right)\right), \ell\right), \mathsf{*.f64}\left(k, t\right)\right), \mathsf{\_.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(\frac{1}{2}, \mathsf{cos.f64}\left(\mathsf{*.f64}\left(2, k\right)\right)\right)\right)\right) \]

   9. Applied egg-rr 76.9%

      \[\leadsto \frac{\frac{\color{blue}{\left(\frac{\cos k}{k} \cdot \left(2 \cdot \ell\right)\right) \cdot \ell}}{k \cdot t}}{0.5 - 0.5 \cdot \cos \left(2 \cdot k\right)} \]
   10. Step-by-step derivation

       1. times-frac N/A

          \[\leadsto \frac{\frac{\frac{\cos k}{k} \cdot \left(2 \cdot \ell\right)}{k} \cdot \frac{\ell}{t}}{\color{blue}{\frac{1}{2}} - \frac{1}{2} \cdot \cos \left(2 \cdot k\right)} \]

       2. cancel-sign-sub-inv N/A

          \[\leadsto \frac{\frac{\frac{\cos k}{k} \cdot \left(2 \cdot \ell\right)}{k} \cdot \frac{\ell}{t}}{\frac{1}{2} + \color{blue}{\left(\mathsf{neg}\left(\frac{1}{2}\right)\right) \cdot \cos \left(2 \cdot k\right)}} \]

       3. metadata-eval N/A

          \[\leadsto \frac{\frac{\frac{\cos k}{k} \cdot \left(2 \cdot \ell\right)}{k} \cdot \frac{\ell}{t}}{\frac{1}{2} + \frac{-1}{2} \cdot \cos \color{blue}{\left(2 \cdot k\right)}} \]

       4. *-commutative N/A

          \[\leadsto \frac{\frac{\frac{\cos k}{k} \cdot \left(2 \cdot \ell\right)}{k} \cdot \frac{\ell}{t}}{\frac{1}{2} + \cos \left(2 \cdot k\right) \cdot \color{blue}{\frac{-1}{2}}} \]

       5. associate-/l* N/A

          \[\leadsto \frac{\frac{\cos k}{k} \cdot \left(2 \cdot \ell\right)}{k} \cdot \color{blue}{\frac{\frac{\ell}{t}}{\frac{1}{2} + \cos \left(2 \cdot k\right) \cdot \frac{-1}{2}}} \]

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

          \[\leadsto \mathsf{*.f64}\left(\left(\frac{\frac{\cos k}{k} \cdot \left(2 \cdot \ell\right)}{k}\right), \color{blue}{\left(\frac{\frac{\ell}{t}}{\frac{1}{2} + \cos \left(2 \cdot k\right) \cdot \frac{-1}{2}}\right)}\right) \]

       7. associate-/l* N/A

          \[\leadsto \mathsf{*.f64}\left(\left(\frac{\cos k}{k} \cdot \frac{2 \cdot \ell}{k}\right), \left(\frac{\color{blue}{\frac{\ell}{t}}}{\frac{1}{2} + \cos \left(2 \cdot k\right) \cdot \frac{-1}{2}}\right)\right) \]

       8. associate-*l/ N/A

          \[\leadsto \mathsf{*.f64}\left(\left(\frac{\cos k}{k} \cdot \left(\frac{2}{k} \cdot \ell\right)\right), \left(\frac{\frac{\ell}{\color{blue}{t}}}{\frac{1}{2} + \cos \left(2 \cdot k\right) \cdot \frac{-1}{2}}\right)\right) \]

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

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\left(\frac{\cos k}{k}\right), \left(\frac{2}{k} \cdot \ell\right)\right), \left(\frac{\color{blue}{\frac{\ell}{t}}}{\frac{1}{2} + \cos \left(2 \cdot k\right) \cdot \frac{-1}{2}}\right)\right) \]

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

           \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(\cos k, k\right), \left(\frac{2}{k} \cdot \ell\right)\right), \left(\frac{\frac{\color{blue}{\ell}}{t}}{\frac{1}{2} + \cos \left(2 \cdot k\right) \cdot \frac{-1}{2}}\right)\right) \]

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

           \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{cos.f64}\left(k\right), k\right), \left(\frac{2}{k} \cdot \ell\right)\right), \left(\frac{\frac{\ell}{t}}{\frac{1}{2} + \cos \left(2 \cdot k\right) \cdot \frac{-1}{2}}\right)\right) \]

       12. associate-*l/ N/A

           \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{cos.f64}\left(k\right), k\right), \left(\frac{2 \cdot \ell}{k}\right)\right), \left(\frac{\frac{\ell}{\color{blue}{t}}}{\frac{1}{2} + \cos \left(2 \cdot k\right) \cdot \frac{-1}{2}}\right)\right) \]

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

           \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{cos.f64}\left(k\right), k\right), \mathsf{/.f64}\left(\left(2 \cdot \ell\right), k\right)\right), \left(\frac{\frac{\ell}{\color{blue}{t}}}{\frac{1}{2} + \cos \left(2 \cdot k\right) \cdot \frac{-1}{2}}\right)\right) \]

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

           \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{cos.f64}\left(k\right), k\right), \mathsf{/.f64}\left(\mathsf{*.f64}\left(2, \ell\right), k\right)\right), \left(\frac{\frac{\ell}{t}}{\frac{1}{2} + \cos \left(2 \cdot k\right) \cdot \frac{-1}{2}}\right)\right) \]

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

           \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{cos.f64}\left(k\right), k\right), \mathsf{/.f64}\left(\mathsf{*.f64}\left(2, \ell\right), k\right)\right), \mathsf{/.f64}\left(\left(\frac{\ell}{t}\right), \color{blue}{\left(\frac{1}{2} + \cos \left(2 \cdot k\right) \cdot \frac{-1}{2}\right)}\right)\right) \]

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

           \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{cos.f64}\left(k\right), k\right), \mathsf{/.f64}\left(\mathsf{*.f64}\left(2, \ell\right), k\right)\right), \mathsf{/.f64}\left(\mathsf{/.f64}\left(\ell, t\right), \left(\color{blue}{\frac{1}{2}} + \cos \left(2 \cdot k\right) \cdot \frac{-1}{2}\right)\right)\right) \]

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

           \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{cos.f64}\left(k\right), k\right), \mathsf{/.f64}\left(\mathsf{*.f64}\left(2, \ell\right), k\right)\right), \mathsf{/.f64}\left(\mathsf{/.f64}\left(\ell, t\right), \mathsf{+.f64}\left(\frac{1}{2}, \color{blue}{\left(\cos \left(2 \cdot k\right) \cdot \frac{-1}{2}\right)}\right)\right)\right) \]

   11. Applied egg-rr 92.5%

       \[\leadsto \color{blue}{\left(\frac{\cos k}{k} \cdot \frac{2 \cdot \ell}{k}\right) \cdot \frac{\frac{\ell}{t}}{0.5 + \cos \left(k \cdot 2\right) \cdot -0.5}} \]

   if 7.20000000000000041e188 < k

   1. Initial program 29.4%

      \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) - 1\right)} \]
   2. Add Preprocessing

   3. Taylor expanded in t around 0

      \[\leadsto \color{blue}{2 \cdot \frac{{\ell}^{2} \cdot \cos k}{{k}^{2} \cdot \left(t \cdot {\sin k}^{2}\right)}} \]
   4. Step-by-step derivation

      1. associate-*r/ N/A

         \[\leadsto \frac{2 \cdot \left({\ell}^{2} \cdot \cos k\right)}{\color{blue}{{k}^{2} \cdot \left(t \cdot {\sin k}^{2}\right)}} \]

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

         \[\leadsto \mathsf{/.f64}\left(\left(2 \cdot \left({\ell}^{2} \cdot \cos k\right)\right), \color{blue}{\left({k}^{2} \cdot \left(t \cdot {\sin k}^{2}\right)\right)}\right) \]

      3. associate-*r* N/A

         \[\leadsto \mathsf{/.f64}\left(\left(\left(2 \cdot {\ell}^{2}\right) \cdot \cos k\right), \left(\color{blue}{{k}^{2}} \cdot \left(t \cdot {\sin k}^{2}\right)\right)\right) \]

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

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\left(2 \cdot {\ell}^{2}\right), \cos k\right), \left(\color{blue}{{k}^{2}} \cdot \left(t \cdot {\sin k}^{2}\right)\right)\right) \]

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

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(2, \left({\ell}^{2}\right)\right), \cos k\right), \left({\color{blue}{k}}^{2} \cdot \left(t \cdot {\sin k}^{2}\right)\right)\right) \]

      6. unpow2 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(2, \left(\ell \cdot \ell\right)\right), \cos k\right), \left({k}^{2} \cdot \left(t \cdot {\sin k}^{2}\right)\right)\right) \]

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

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(2, \mathsf{*.f64}\left(\ell, \ell\right)\right), \cos k\right), \left({k}^{2} \cdot \left(t \cdot {\sin k}^{2}\right)\right)\right) \]

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

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(2, \mathsf{*.f64}\left(\ell, \ell\right)\right), \mathsf{cos.f64}\left(k\right)\right), \left({k}^{\color{blue}{2}} \cdot \left(t \cdot {\sin k}^{2}\right)\right)\right) \]

      9. unpow2 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(2, \mathsf{*.f64}\left(\ell, \ell\right)\right), \mathsf{cos.f64}\left(k\right)\right), \left(\left(k \cdot k\right) \cdot \left(\color{blue}{t} \cdot {\sin k}^{2}\right)\right)\right) \]

      10. associate-*l* N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(2, \mathsf{*.f64}\left(\ell, \ell\right)\right), \mathsf{cos.f64}\left(k\right)\right), \left(k \cdot \color{blue}{\left(k \cdot \left(t \cdot {\sin k}^{2}\right)\right)}\right)\right) \]

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

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(2, \mathsf{*.f64}\left(\ell, \ell\right)\right), \mathsf{cos.f64}\left(k\right)\right), \mathsf{*.f64}\left(k, \color{blue}{\left(k \cdot \left(t \cdot {\sin k}^{2}\right)\right)}\right)\right) \]

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

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(2, \mathsf{*.f64}\left(\ell, \ell\right)\right), \mathsf{cos.f64}\left(k\right)\right), \mathsf{*.f64}\left(k, \mathsf{*.f64}\left(k, \color{blue}{\left(t \cdot {\sin k}^{2}\right)}\right)\right)\right) \]

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

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(2, \mathsf{*.f64}\left(\ell, \ell\right)\right), \mathsf{cos.f64}\left(k\right)\right), \mathsf{*.f64}\left(k, \mathsf{*.f64}\left(k, \mathsf{*.f64}\left(t, \color{blue}{\left({\sin k}^{2}\right)}\right)\right)\right)\right) \]

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

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(2, \mathsf{*.f64}\left(\ell, \ell\right)\right), \mathsf{cos.f64}\left(k\right)\right), \mathsf{*.f64}\left(k, \mathsf{*.f64}\left(k, \mathsf{*.f64}\left(t, \mathsf{pow.f64}\left(\sin k, \color{blue}{2}\right)\right)\right)\right)\right) \]

      15. sin-lowering-sin.f64 55.1%

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(2, \mathsf{*.f64}\left(\ell, \ell\right)\right), \mathsf{cos.f64}\left(k\right)\right), \mathsf{*.f64}\left(k, \mathsf{*.f64}\left(k, \mathsf{*.f64}\left(t, \mathsf{pow.f64}\left(\mathsf{sin.f64}\left(k\right), 2\right)\right)\right)\right)\right) \]

   5. Simplified 55.1%

      \[\leadsto \color{blue}{\frac{\left(2 \cdot \left(\ell \cdot \ell\right)\right) \cdot \cos k}{k \cdot \left(k \cdot \left(t \cdot {\sin k}^{2}\right)\right)}} \]
   6. Step-by-step derivation

      1. associate-/r* N/A

         \[\leadsto \frac{\frac{\left(2 \cdot \left(\ell \cdot \ell\right)\right) \cdot \cos k}{k}}{\color{blue}{k \cdot \left(t \cdot {\sin k}^{2}\right)}} \]

      2. associate-*r* N/A

         \[\leadsto \frac{\frac{\left(2 \cdot \left(\ell \cdot \ell\right)\right) \cdot \cos k}{k}}{\left(k \cdot t\right) \cdot \color{blue}{{\sin k}^{2}}} \]

      3. associate-/r* N/A

         \[\leadsto \frac{\frac{\frac{\left(2 \cdot \left(\ell \cdot \ell\right)\right) \cdot \cos k}{k}}{k \cdot t}}{\color{blue}{{\sin k}^{2}}} \]

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

         \[\leadsto \mathsf{/.f64}\left(\left(\frac{\frac{\left(2 \cdot \left(\ell \cdot \ell\right)\right) \cdot \cos k}{k}}{k \cdot t}\right), \color{blue}{\left({\sin k}^{2}\right)}\right) \]

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

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\left(\frac{\left(2 \cdot \left(\ell \cdot \ell\right)\right) \cdot \cos k}{k}\right), \left(k \cdot t\right)\right), \left({\color{blue}{\sin k}}^{2}\right)\right) \]

      6. associate-/l* N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\left(\left(2 \cdot \left(\ell \cdot \ell\right)\right) \cdot \frac{\cos k}{k}\right), \left(k \cdot t\right)\right), \left({\sin \color{blue}{k}}^{2}\right)\right) \]

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

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\left(2 \cdot \left(\ell \cdot \ell\right)\right), \left(\frac{\cos k}{k}\right)\right), \left(k \cdot t\right)\right), \left({\sin \color{blue}{k}}^{2}\right)\right) \]

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

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(2, \left(\ell \cdot \ell\right)\right), \left(\frac{\cos k}{k}\right)\right), \left(k \cdot t\right)\right), \left({\sin k}^{2}\right)\right) \]

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

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(2, \mathsf{*.f64}\left(\ell, \ell\right)\right), \left(\frac{\cos k}{k}\right)\right), \left(k \cdot t\right)\right), \left({\sin k}^{2}\right)\right) \]

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

          \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(2, \mathsf{*.f64}\left(\ell, \ell\right)\right), \mathsf{/.f64}\left(\cos k, k\right)\right), \left(k \cdot t\right)\right), \left({\sin k}^{2}\right)\right) \]

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

          \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(2, \mathsf{*.f64}\left(\ell, \ell\right)\right), \mathsf{/.f64}\left(\mathsf{cos.f64}\left(k\right), k\right)\right), \left(k \cdot t\right)\right), \left({\sin k}^{2}\right)\right) \]

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

          \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(2, \mathsf{*.f64}\left(\ell, \ell\right)\right), \mathsf{/.f64}\left(\mathsf{cos.f64}\left(k\right), k\right)\right), \mathsf{*.f64}\left(k, t\right)\right), \left({\sin k}^{2}\right)\right) \]

      13. unpow2 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(2, \mathsf{*.f64}\left(\ell, \ell\right)\right), \mathsf{/.f64}\left(\mathsf{cos.f64}\left(k\right), k\right)\right), \mathsf{*.f64}\left(k, t\right)\right), \left(\sin k \cdot \color{blue}{\sin k}\right)\right) \]

      14. sqr-sin-a N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(2, \mathsf{*.f64}\left(\ell, \ell\right)\right), \mathsf{/.f64}\left(\mathsf{cos.f64}\left(k\right), k\right)\right), \mathsf{*.f64}\left(k, t\right)\right), \left(\frac{1}{2} - \color{blue}{\frac{1}{2} \cdot \cos \left(2 \cdot k\right)}\right)\right) \]

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

          \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(2, \mathsf{*.f64}\left(\ell, \ell\right)\right), \mathsf{/.f64}\left(\mathsf{cos.f64}\left(k\right), k\right)\right), \mathsf{*.f64}\left(k, t\right)\right), \mathsf{\_.f64}\left(\frac{1}{2}, \color{blue}{\left(\frac{1}{2} \cdot \cos \left(2 \cdot k\right)\right)}\right)\right) \]

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

          \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(2, \mathsf{*.f64}\left(\ell, \ell\right)\right), \mathsf{/.f64}\left(\mathsf{cos.f64}\left(k\right), k\right)\right), \mathsf{*.f64}\left(k, t\right)\right), \mathsf{\_.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(\frac{1}{2}, \color{blue}{\cos \left(2 \cdot k\right)}\right)\right)\right) \]

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

          \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(2, \mathsf{*.f64}\left(\ell, \ell\right)\right), \mathsf{/.f64}\left(\mathsf{cos.f64}\left(k\right), k\right)\right), \mathsf{*.f64}\left(k, t\right)\right), \mathsf{\_.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(\frac{1}{2}, \mathsf{cos.f64}\left(\left(2 \cdot k\right)\right)\right)\right)\right) \]

      18. *-lowering-*.f64 63.5%

          \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(2, \mathsf{*.f64}\left(\ell, \ell\right)\right), \mathsf{/.f64}\left(\mathsf{cos.f64}\left(k\right), k\right)\right), \mathsf{*.f64}\left(k, t\right)\right), \mathsf{\_.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(\frac{1}{2}, \mathsf{cos.f64}\left(\mathsf{*.f64}\left(2, k\right)\right)\right)\right)\right) \]

   7. Applied egg-rr 63.5%

      \[\leadsto \color{blue}{\frac{\frac{\left(2 \cdot \left(\ell \cdot \ell\right)\right) \cdot \frac{\cos k}{k}}{k \cdot t}}{0.5 - 0.5 \cdot \cos \left(2 \cdot k\right)}} \]
   8. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\left(\frac{\cos k}{k} \cdot \left(2 \cdot \left(\ell \cdot \ell\right)\right)\right), \mathsf{*.f64}\left(k, t\right)\right), \mathsf{\_.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(\frac{1}{2}, \mathsf{cos.f64}\left(\mathsf{*.f64}\left(2, k\right)\right)\right)\right)\right) \]

      2. associate-*r* N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\left(\frac{\cos k}{k} \cdot \left(\left(2 \cdot \ell\right) \cdot \ell\right)\right), \mathsf{*.f64}\left(k, t\right)\right), \mathsf{\_.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(\frac{1}{2}, \mathsf{cos.f64}\left(\mathsf{*.f64}\left(2, k\right)\right)\right)\right)\right) \]

      3. associate-*r* N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\left(\left(\frac{\cos k}{k} \cdot \left(2 \cdot \ell\right)\right) \cdot \ell\right), \mathsf{*.f64}\left(k, t\right)\right), \mathsf{\_.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(\frac{1}{2}, \mathsf{cos.f64}\left(\mathsf{*.f64}\left(2, k\right)\right)\right)\right)\right) \]

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

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\left(\frac{\cos k}{k} \cdot \left(2 \cdot \ell\right)\right), \ell\right), \mathsf{*.f64}\left(k, t\right)\right), \mathsf{\_.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(\frac{1}{2}, \mathsf{cos.f64}\left(\mathsf{*.f64}\left(2, k\right)\right)\right)\right)\right) \]

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

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\left(\frac{\cos k}{k}\right), \left(2 \cdot \ell\right)\right), \ell\right), \mathsf{*.f64}\left(k, t\right)\right), \mathsf{\_.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(\frac{1}{2}, \mathsf{cos.f64}\left(\mathsf{*.f64}\left(2, k\right)\right)\right)\right)\right) \]

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

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(\cos k, k\right), \left(2 \cdot \ell\right)\right), \ell\right), \mathsf{*.f64}\left(k, t\right)\right), \mathsf{\_.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(\frac{1}{2}, \mathsf{cos.f64}\left(\mathsf{*.f64}\left(2, k\right)\right)\right)\right)\right) \]

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

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{cos.f64}\left(k\right), k\right), \left(2 \cdot \ell\right)\right), \ell\right), \mathsf{*.f64}\left(k, t\right)\right), \mathsf{\_.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(\frac{1}{2}, \mathsf{cos.f64}\left(\mathsf{*.f64}\left(2, k\right)\right)\right)\right)\right) \]

      8. *-lowering-*.f64 87.7%

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{cos.f64}\left(k\right), k\right), \mathsf{*.f64}\left(2, \ell\right)\right), \ell\right), \mathsf{*.f64}\left(k, t\right)\right), \mathsf{\_.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(\frac{1}{2}, \mathsf{cos.f64}\left(\mathsf{*.f64}\left(2, k\right)\right)\right)\right)\right) \]

   9. Applied egg-rr 87.7%

      \[\leadsto \frac{\frac{\color{blue}{\left(\frac{\cos k}{k} \cdot \left(2 \cdot \ell\right)\right) \cdot \ell}}{k \cdot t}}{0.5 - 0.5 \cdot \cos \left(2 \cdot k\right)} \]
   10. Step-by-step derivation

       1. cancel-sign-sub-inv N/A

          \[\leadsto \frac{\frac{\left(\frac{\cos k}{k} \cdot \left(2 \cdot \ell\right)\right) \cdot \ell}{k \cdot t}}{\frac{1}{2} + \color{blue}{\left(\mathsf{neg}\left(\frac{1}{2}\right)\right) \cdot \cos \left(2 \cdot k\right)}} \]

       2. metadata-eval N/A

          \[\leadsto \frac{\frac{\left(\frac{\cos k}{k} \cdot \left(2 \cdot \ell\right)\right) \cdot \ell}{k \cdot t}}{\frac{1}{2} + \frac{-1}{2} \cdot \cos \color{blue}{\left(2 \cdot k\right)}} \]

       3. *-commutative N/A

          \[\leadsto \frac{\frac{\left(\frac{\cos k}{k} \cdot \left(2 \cdot \ell\right)\right) \cdot \ell}{k \cdot t}}{\frac{1}{2} + \cos \left(2 \cdot k\right) \cdot \color{blue}{\frac{-1}{2}}} \]

       4. associate-/l/ N/A

          \[\leadsto \frac{\left(\frac{\cos k}{k} \cdot \left(2 \cdot \ell\right)\right) \cdot \ell}{\color{blue}{\left(\frac{1}{2} + \cos \left(2 \cdot k\right) \cdot \frac{-1}{2}\right) \cdot \left(k \cdot t\right)}} \]

       5. *-commutative N/A

          \[\leadsto \frac{\ell \cdot \left(\frac{\cos k}{k} \cdot \left(2 \cdot \ell\right)\right)}{\color{blue}{\left(\frac{1}{2} + \cos \left(2 \cdot k\right) \cdot \frac{-1}{2}\right)} \cdot \left(k \cdot t\right)} \]

       6. times-frac N/A

          \[\leadsto \frac{\ell}{\frac{1}{2} + \cos \left(2 \cdot k\right) \cdot \frac{-1}{2}} \cdot \color{blue}{\frac{\frac{\cos k}{k} \cdot \left(2 \cdot \ell\right)}{k \cdot t}} \]

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

          \[\leadsto \mathsf{*.f64}\left(\left(\frac{\ell}{\frac{1}{2} + \cos \left(2 \cdot k\right) \cdot \frac{-1}{2}}\right), \color{blue}{\left(\frac{\frac{\cos k}{k} \cdot \left(2 \cdot \ell\right)}{k \cdot t}\right)}\right) \]

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

          \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\ell, \left(\frac{1}{2} + \cos \left(2 \cdot k\right) \cdot \frac{-1}{2}\right)\right), \left(\frac{\color{blue}{\frac{\cos k}{k} \cdot \left(2 \cdot \ell\right)}}{k \cdot t}\right)\right) \]

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

          \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\ell, \mathsf{+.f64}\left(\frac{1}{2}, \left(\cos \left(2 \cdot k\right) \cdot \frac{-1}{2}\right)\right)\right), \left(\frac{\frac{\cos k}{k} \cdot \color{blue}{\left(2 \cdot \ell\right)}}{k \cdot t}\right)\right) \]

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

           \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\ell, \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(\cos \left(2 \cdot k\right), \frac{-1}{2}\right)\right)\right), \left(\frac{\frac{\cos k}{k} \cdot \left(2 \cdot \color{blue}{\ell}\right)}{k \cdot t}\right)\right) \]

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

           \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\ell, \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(\mathsf{cos.f64}\left(\left(2 \cdot k\right)\right), \frac{-1}{2}\right)\right)\right), \left(\frac{\frac{\cos k}{k} \cdot \left(2 \cdot \ell\right)}{k \cdot t}\right)\right) \]

       12. *-commutative N/A

           \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\ell, \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(\mathsf{cos.f64}\left(\left(k \cdot 2\right)\right), \frac{-1}{2}\right)\right)\right), \left(\frac{\frac{\cos k}{k} \cdot \left(2 \cdot \ell\right)}{k \cdot t}\right)\right) \]

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

           \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\ell, \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(\mathsf{cos.f64}\left(\mathsf{*.f64}\left(k, 2\right)\right), \frac{-1}{2}\right)\right)\right), \left(\frac{\frac{\cos k}{k} \cdot \left(2 \cdot \ell\right)}{k \cdot t}\right)\right) \]

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

           \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\ell, \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(\mathsf{cos.f64}\left(\mathsf{*.f64}\left(k, 2\right)\right), \frac{-1}{2}\right)\right)\right), \mathsf{/.f64}\left(\left(\frac{\cos k}{k} \cdot \left(2 \cdot \ell\right)\right), \color{blue}{\left(k \cdot t\right)}\right)\right) \]

   11. Applied egg-rr 88.0%

       \[\leadsto \color{blue}{\frac{\ell}{0.5 + \cos \left(k \cdot 2\right) \cdot -0.5} \cdot \frac{\frac{2 \cdot \ell}{\frac{k}{\cos k}}}{k \cdot t}} \]
3. Recombined 3 regimes into one program.

4. Final simplification 81.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;k \leq 8.5 \cdot 10^{-5}:\\ \;\;\;\;\ell \cdot \frac{\frac{\frac{2 \cdot \ell}{k \cdot \left(k \cdot t\right)}}{k}}{k}\\ \mathbf{elif}\;k \leq 7.2 \cdot 10^{+188}:\\ \;\;\;\;\left(\frac{\cos k}{k} \cdot \frac{2 \cdot \ell}{k}\right) \cdot \frac{\frac{\ell}{t}}{0.5 + \cos \left(k \cdot 2\right) \cdot -0.5}\\ \mathbf{else}:\\ \;\;\;\;\frac{\ell}{0.5 + \cos \left(k \cdot 2\right) \cdot -0.5} \cdot \frac{\frac{2 \cdot \ell}{\frac{k}{\cos k}}}{k \cdot t}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 89.7% accurate, 1.8× speedup

Math

\[\begin{array}{l} k_m = \left|k\right| \\ \begin{array}{l} t_1 := k\_m \cdot \left(k\_m \cdot t\right)\\ t_2 := \cos \left(k\_m \cdot 2\right)\\ \mathbf{if}\;k\_m \leq 8.5 \cdot 10^{-5}:\\ \;\;\;\;\ell \cdot \frac{\frac{\frac{2 \cdot \ell}{t\_1}}{k\_m}}{k\_m}\\ \mathbf{elif}\;k\_m \leq 1.12 \cdot 10^{+154}:\\ \;\;\;\;\left(2 \cdot \ell\right) \cdot \left(\frac{\ell}{t} \cdot \frac{\cos k\_m}{\left(k\_m \cdot k\_m\right) \cdot \left(0.5 + t\_2 \cdot -0.5\right)}\right)\\ \mathbf{else}:\\ \;\;\;\;\left(2 \cdot \ell\right) \cdot \left(\ell \cdot \frac{\cos k\_m}{\left(0.5 - 0.5 \cdot t\_2\right) \cdot t\_1}\right)\\ \end{array} \end{array} \]

FPCore

k_m = (fabs.f64 k)
(FPCore (t l k_m)
 :precision binary64
 (let* ((t_1 (* k_m (* k_m t))) (t_2 (cos (* k_m 2.0))))
   (if (<= k_m 8.5e-5)
     (* l (/ (/ (/ (* 2.0 l) t_1) k_m) k_m))
     (if (<= k_m 1.12e+154)
       (*
        (* 2.0 l)
        (* (/ l t) (/ (cos k_m) (* (* k_m k_m) (+ 0.5 (* t_2 -0.5))))))
       (* (* 2.0 l) (* l (/ (cos k_m) (* (- 0.5 (* 0.5 t_2)) t_1))))))))

C

k_m = fabs(k);
double code(double t, double l, double k_m) {
	double t_1 = k_m * (k_m * t);
	double t_2 = cos((k_m * 2.0));
	double tmp;
	if (k_m <= 8.5e-5) {
		tmp = l * ((((2.0 * l) / t_1) / k_m) / k_m);
	} else if (k_m <= 1.12e+154) {
		tmp = (2.0 * l) * ((l / t) * (cos(k_m) / ((k_m * k_m) * (0.5 + (t_2 * -0.5)))));
	} else {
		tmp = (2.0 * l) * (l * (cos(k_m) / ((0.5 - (0.5 * t_2)) * t_1)));
	}
	return tmp;
}

Fortran

k_m = abs(k)
real(8) function code(t, l, k_m)
    real(8), intent (in) :: t
    real(8), intent (in) :: l
    real(8), intent (in) :: k_m
    real(8) :: t_1
    real(8) :: t_2
    real(8) :: tmp
    t_1 = k_m * (k_m * t)
    t_2 = cos((k_m * 2.0d0))
    if (k_m <= 8.5d-5) then
        tmp = l * ((((2.0d0 * l) / t_1) / k_m) / k_m)
    else if (k_m <= 1.12d+154) then
        tmp = (2.0d0 * l) * ((l / t) * (cos(k_m) / ((k_m * k_m) * (0.5d0 + (t_2 * (-0.5d0))))))
    else
        tmp = (2.0d0 * l) * (l * (cos(k_m) / ((0.5d0 - (0.5d0 * t_2)) * t_1)))
    end if
    code = tmp
end function

Java

k_m = Math.abs(k);
public static double code(double t, double l, double k_m) {
	double t_1 = k_m * (k_m * t);
	double t_2 = Math.cos((k_m * 2.0));
	double tmp;
	if (k_m <= 8.5e-5) {
		tmp = l * ((((2.0 * l) / t_1) / k_m) / k_m);
	} else if (k_m <= 1.12e+154) {
		tmp = (2.0 * l) * ((l / t) * (Math.cos(k_m) / ((k_m * k_m) * (0.5 + (t_2 * -0.5)))));
	} else {
		tmp = (2.0 * l) * (l * (Math.cos(k_m) / ((0.5 - (0.5 * t_2)) * t_1)));
	}
	return tmp;
}

Python

k_m = math.fabs(k)
def code(t, l, k_m):
	t_1 = k_m * (k_m * t)
	t_2 = math.cos((k_m * 2.0))
	tmp = 0
	if k_m <= 8.5e-5:
		tmp = l * ((((2.0 * l) / t_1) / k_m) / k_m)
	elif k_m <= 1.12e+154:
		tmp = (2.0 * l) * ((l / t) * (math.cos(k_m) / ((k_m * k_m) * (0.5 + (t_2 * -0.5)))))
	else:
		tmp = (2.0 * l) * (l * (math.cos(k_m) / ((0.5 - (0.5 * t_2)) * t_1)))
	return tmp

Julia

k_m = abs(k)
function code(t, l, k_m)
	t_1 = Float64(k_m * Float64(k_m * t))
	t_2 = cos(Float64(k_m * 2.0))
	tmp = 0.0
	if (k_m <= 8.5e-5)
		tmp = Float64(l * Float64(Float64(Float64(Float64(2.0 * l) / t_1) / k_m) / k_m));
	elseif (k_m <= 1.12e+154)
		tmp = Float64(Float64(2.0 * l) * Float64(Float64(l / t) * Float64(cos(k_m) / Float64(Float64(k_m * k_m) * Float64(0.5 + Float64(t_2 * -0.5))))));
	else
		tmp = Float64(Float64(2.0 * l) * Float64(l * Float64(cos(k_m) / Float64(Float64(0.5 - Float64(0.5 * t_2)) * t_1))));
	end
	return tmp
end

MATLAB

k_m = abs(k);
function tmp_2 = code(t, l, k_m)
	t_1 = k_m * (k_m * t);
	t_2 = cos((k_m * 2.0));
	tmp = 0.0;
	if (k_m <= 8.5e-5)
		tmp = l * ((((2.0 * l) / t_1) / k_m) / k_m);
	elseif (k_m <= 1.12e+154)
		tmp = (2.0 * l) * ((l / t) * (cos(k_m) / ((k_m * k_m) * (0.5 + (t_2 * -0.5)))));
	else
		tmp = (2.0 * l) * (l * (cos(k_m) / ((0.5 - (0.5 * t_2)) * t_1)));
	end
	tmp_2 = tmp;
end

Wolfram

k_m = N[Abs[k], $MachinePrecision]
code[t_, l_, k$95$m_] := Block[{t$95$1 = N[(k$95$m * N[(k$95$m * t), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[Cos[N[(k$95$m * 2.0), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[k$95$m, 8.5e-5], N[(l * N[(N[(N[(N[(2.0 * l), $MachinePrecision] / t$95$1), $MachinePrecision] / k$95$m), $MachinePrecision] / k$95$m), $MachinePrecision]), $MachinePrecision], If[LessEqual[k$95$m, 1.12e+154], N[(N[(2.0 * l), $MachinePrecision] * N[(N[(l / t), $MachinePrecision] * N[(N[Cos[k$95$m], $MachinePrecision] / N[(N[(k$95$m * k$95$m), $MachinePrecision] * N[(0.5 + N[(t$95$2 * -0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(2.0 * l), $MachinePrecision] * N[(l * N[(N[Cos[k$95$m], $MachinePrecision] / N[(N[(0.5 - N[(0.5 * t$95$2), $MachinePrecision]), $MachinePrecision] * t$95$1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

2. if k < 8.500000000000001e-5

   1. Initial program 35.0%

      \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) - 1\right)} \]
   2. Add Preprocessing

   3. Taylor expanded in k around 0

      \[\leadsto \mathsf{/.f64}\left(2, \color{blue}{\left(\frac{{k}^{4} \cdot t}{{\ell}^{2}}\right)}\right) \]
   4. Step-by-step derivation

      1. associate-/l* N/A

         \[\leadsto \mathsf{/.f64}\left(2, \left({k}^{4} \cdot \color{blue}{\frac{t}{{\ell}^{2}}}\right)\right) \]

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

         \[\leadsto \mathsf{/.f64}\left(2, \mathsf{*.f64}\left(\left({k}^{4}\right), \color{blue}{\left(\frac{t}{{\ell}^{2}}\right)}\right)\right) \]

      3. metadata-eval N/A

         \[\leadsto \mathsf{/.f64}\left(2, \mathsf{*.f64}\left(\left({k}^{\left(2 \cdot 2\right)}\right), \left(\frac{t}{{\ell}^{2}}\right)\right)\right) \]

      4. pow-sqr N/A

         \[\leadsto \mathsf{/.f64}\left(2, \mathsf{*.f64}\left(\left({k}^{2} \cdot {k}^{2}\right), \left(\frac{\color{blue}{t}}{{\ell}^{2}}\right)\right)\right) \]

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

         \[\leadsto \mathsf{/.f64}\left(2, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\left({k}^{2}\right), \left({k}^{2}\right)\right), \left(\frac{\color{blue}{t}}{{\ell}^{2}}\right)\right)\right) \]

      6. unpow2 N/A

         \[\leadsto \mathsf{/.f64}\left(2, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\left(k \cdot k\right), \left({k}^{2}\right)\right), \left(\frac{t}{{\ell}^{2}}\right)\right)\right) \]

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

         \[\leadsto \mathsf{/.f64}\left(2, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(k, k\right), \left({k}^{2}\right)\right), \left(\frac{t}{{\ell}^{2}}\right)\right)\right) \]

      8. unpow2 N/A

         \[\leadsto \mathsf{/.f64}\left(2, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(k, k\right), \left(k \cdot k\right)\right), \left(\frac{t}{{\ell}^{2}}\right)\right)\right) \]

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

         \[\leadsto \mathsf{/.f64}\left(2, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(k, k\right), \mathsf{*.f64}\left(k, k\right)\right), \left(\frac{t}{{\ell}^{2}}\right)\right)\right) \]

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

          \[\leadsto \mathsf{/.f64}\left(2, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(k, k\right), \mathsf{*.f64}\left(k, k\right)\right), \mathsf{/.f64}\left(t, \color{blue}{\left({\ell}^{2}\right)}\right)\right)\right) \]

      11. unpow2 N/A

          \[\leadsto \mathsf{/.f64}\left(2, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(k, k\right), \mathsf{*.f64}\left(k, k\right)\right), \mathsf{/.f64}\left(t, \left(\ell \cdot \color{blue}{\ell}\right)\right)\right)\right) \]

      12. *-lowering-*.f64 58.1%

          \[\leadsto \mathsf{/.f64}\left(2, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(k, k\right), \mathsf{*.f64}\left(k, k\right)\right), \mathsf{/.f64}\left(t, \mathsf{*.f64}\left(\ell, \color{blue}{\ell}\right)\right)\right)\right) \]

   5. Simplified 58.1%

      \[\leadsto \frac{2}{\color{blue}{\left(\left(k \cdot k\right) \cdot \left(k \cdot k\right)\right) \cdot \frac{t}{\ell \cdot \ell}}} \]
   6. Step-by-step derivation

      1. associate-/r* N/A

         \[\leadsto \frac{\frac{2}{\left(k \cdot k\right) \cdot \left(k \cdot k\right)}}{\color{blue}{\frac{t}{\ell \cdot \ell}}} \]

      2. associate-/r* N/A

         \[\leadsto \frac{\frac{2}{\left(k \cdot k\right) \cdot \left(k \cdot k\right)}}{\frac{\frac{t}{\ell}}{\color{blue}{\ell}}} \]

      3. associate-/r/ N/A

         \[\leadsto \frac{\frac{2}{\left(k \cdot k\right) \cdot \left(k \cdot k\right)}}{\frac{t}{\ell}} \cdot \color{blue}{\ell} \]

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

         \[\leadsto \mathsf{*.f64}\left(\left(\frac{\frac{2}{\left(k \cdot k\right) \cdot \left(k \cdot k\right)}}{\frac{t}{\ell}}\right), \color{blue}{\ell}\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\left(\frac{2}{\left(k \cdot k\right) \cdot \left(k \cdot k\right)}\right), \left(\frac{t}{\ell}\right)\right), \ell\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(2, \left(\left(k \cdot k\right) \cdot \left(k \cdot k\right)\right)\right), \left(\frac{t}{\ell}\right)\right), \ell\right) \]

      7. associate-*l* N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(2, \left(k \cdot \left(k \cdot \left(k \cdot k\right)\right)\right)\right), \left(\frac{t}{\ell}\right)\right), \ell\right) \]

      8. cube-unmult N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(2, \left(k \cdot {k}^{3}\right)\right), \left(\frac{t}{\ell}\right)\right), \ell\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(2, \mathsf{*.f64}\left(k, \left({k}^{3}\right)\right)\right), \left(\frac{t}{\ell}\right)\right), \ell\right) \]

      10. cube-unmult N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(2, \mathsf{*.f64}\left(k, \left(k \cdot \left(k \cdot k\right)\right)\right)\right), \left(\frac{t}{\ell}\right)\right), \ell\right) \]

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

          \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(2, \mathsf{*.f64}\left(k, \mathsf{*.f64}\left(k, \left(k \cdot k\right)\right)\right)\right), \left(\frac{t}{\ell}\right)\right), \ell\right) \]

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

          \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(2, \mathsf{*.f64}\left(k, \mathsf{*.f64}\left(k, \mathsf{*.f64}\left(k, k\right)\right)\right)\right), \left(\frac{t}{\ell}\right)\right), \ell\right) \]

      13. /-lowering-/.f64 69.3%

          \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(2, \mathsf{*.f64}\left(k, \mathsf{*.f64}\left(k, \mathsf{*.f64}\left(k, k\right)\right)\right)\right), \mathsf{/.f64}\left(t, \ell\right)\right), \ell\right) \]

   7. Applied egg-rr 69.3%

      \[\leadsto \color{blue}{\frac{\frac{2}{k \cdot \left(k \cdot \left(k \cdot k\right)\right)}}{\frac{t}{\ell}} \cdot \ell} \]
   8. Step-by-step derivation

      1. div-inv N/A

         \[\leadsto \mathsf{*.f64}\left(\left(\frac{2}{k \cdot \left(k \cdot \left(k \cdot k\right)\right)} \cdot \frac{1}{\frac{t}{\ell}}\right), \ell\right) \]

      2. associate-/r* N/A

         \[\leadsto \mathsf{*.f64}\left(\left(\frac{\frac{2}{k}}{k \cdot \left(k \cdot k\right)} \cdot \frac{1}{\frac{t}{\ell}}\right), \ell\right) \]

      3. clear-num N/A

         \[\leadsto \mathsf{*.f64}\left(\left(\frac{\frac{2}{k}}{k \cdot \left(k \cdot k\right)} \cdot \frac{\ell}{t}\right), \ell\right) \]

      4. associate-*l/ N/A

         \[\leadsto \mathsf{*.f64}\left(\left(\frac{\frac{2}{k} \cdot \frac{\ell}{t}}{k \cdot \left(k \cdot k\right)}\right), \ell\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\left(\frac{2}{k} \cdot \frac{\ell}{t}\right), \left(k \cdot \left(k \cdot k\right)\right)\right), \ell\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\left(\frac{2}{k}\right), \left(\frac{\ell}{t}\right)\right), \left(k \cdot \left(k \cdot k\right)\right)\right), \ell\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(2, k\right), \left(\frac{\ell}{t}\right)\right), \left(k \cdot \left(k \cdot k\right)\right)\right), \ell\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(2, k\right), \mathsf{/.f64}\left(\ell, t\right)\right), \left(k \cdot \left(k \cdot k\right)\right)\right), \ell\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(2, k\right), \mathsf{/.f64}\left(\ell, t\right)\right), \mathsf{*.f64}\left(k, \left(k \cdot k\right)\right)\right), \ell\right) \]

      10. *-lowering-*.f64 72.8%

          \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(2, k\right), \mathsf{/.f64}\left(\ell, t\right)\right), \mathsf{*.f64}\left(k, \mathsf{*.f64}\left(k, k\right)\right)\right), \ell\right) \]

   9. Applied egg-rr 72.8%

      \[\leadsto \color{blue}{\frac{\frac{2}{k} \cdot \frac{\ell}{t}}{k \cdot \left(k \cdot k\right)}} \cdot \ell \]
   10. Step-by-step derivation

       1. associate-/r* N/A

          \[\leadsto \mathsf{*.f64}\left(\left(\frac{\frac{\frac{2}{k} \cdot \frac{\ell}{t}}{k}}{k \cdot k}\right), \ell\right) \]

       2. associate-/r* N/A

          \[\leadsto \mathsf{*.f64}\left(\left(\frac{\frac{\frac{\frac{2}{k} \cdot \frac{\ell}{t}}{k}}{k}}{k}\right), \ell\right) \]

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

          \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\left(\frac{\frac{\frac{2}{k} \cdot \frac{\ell}{t}}{k}}{k}\right), k\right), \ell\right) \]

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

          \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(\left(\frac{\frac{2}{k} \cdot \frac{\ell}{t}}{k}\right), k\right), k\right), \ell\right) \]

       5. frac-times N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(\left(\frac{\frac{2 \cdot \ell}{k \cdot t}}{k}\right), k\right), k\right), \ell\right) \]

       6. associate-/l/ N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(\left(\frac{2 \cdot \ell}{k \cdot \left(k \cdot t\right)}\right), k\right), k\right), \ell\right) \]

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

          \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(\left(2 \cdot \ell\right), \left(k \cdot \left(k \cdot t\right)\right)\right), k\right), k\right), \ell\right) \]

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

          \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(2, \ell\right), \left(k \cdot \left(k \cdot t\right)\right)\right), k\right), k\right), \ell\right) \]

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

          \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(2, \ell\right), \mathsf{*.f64}\left(k, \left(k \cdot t\right)\right)\right), k\right), k\right), \ell\right) \]

       10. *-lowering-*.f64 78.5%

           \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(2, \ell\right), \mathsf{*.f64}\left(k, \mathsf{*.f64}\left(k, t\right)\right)\right), k\right), k\right), \ell\right) \]

   11. Applied egg-rr 78.5%

       \[\leadsto \color{blue}{\frac{\frac{\frac{2 \cdot \ell}{k \cdot \left(k \cdot t\right)}}{k}}{k}} \cdot \ell \]

   if 8.500000000000001e-5 < k < 1.11999999999999994e154

   1. Initial program 12.3%

      \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) - 1\right)} \]
   2. Add Preprocessing

   3. Taylor expanded in t around 0

      \[\leadsto \color{blue}{2 \cdot \frac{{\ell}^{2} \cdot \cos k}{{k}^{2} \cdot \left(t \cdot {\sin k}^{2}\right)}} \]
   4. Step-by-step derivation

      1. associate-*r/ N/A

         \[\leadsto \frac{2 \cdot \left({\ell}^{2} \cdot \cos k\right)}{\color{blue}{{k}^{2} \cdot \left(t \cdot {\sin k}^{2}\right)}} \]

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

         \[\leadsto \mathsf{/.f64}\left(\left(2 \cdot \left({\ell}^{2} \cdot \cos k\right)\right), \color{blue}{\left({k}^{2} \cdot \left(t \cdot {\sin k}^{2}\right)\right)}\right) \]

      3. associate-*r* N/A

         \[\leadsto \mathsf{/.f64}\left(\left(\left(2 \cdot {\ell}^{2}\right) \cdot \cos k\right), \left(\color{blue}{{k}^{2}} \cdot \left(t \cdot {\sin k}^{2}\right)\right)\right) \]

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

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\left(2 \cdot {\ell}^{2}\right), \cos k\right), \left(\color{blue}{{k}^{2}} \cdot \left(t \cdot {\sin k}^{2}\right)\right)\right) \]

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

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(2, \left({\ell}^{2}\right)\right), \cos k\right), \left({\color{blue}{k}}^{2} \cdot \left(t \cdot {\sin k}^{2}\right)\right)\right) \]

      6. unpow2 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(2, \left(\ell \cdot \ell\right)\right), \cos k\right), \left({k}^{2} \cdot \left(t \cdot {\sin k}^{2}\right)\right)\right) \]

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

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(2, \mathsf{*.f64}\left(\ell, \ell\right)\right), \cos k\right), \left({k}^{2} \cdot \left(t \cdot {\sin k}^{2}\right)\right)\right) \]

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

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(2, \mathsf{*.f64}\left(\ell, \ell\right)\right), \mathsf{cos.f64}\left(k\right)\right), \left({k}^{\color{blue}{2}} \cdot \left(t \cdot {\sin k}^{2}\right)\right)\right) \]

      9. unpow2 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(2, \mathsf{*.f64}\left(\ell, \ell\right)\right), \mathsf{cos.f64}\left(k\right)\right), \left(\left(k \cdot k\right) \cdot \left(\color{blue}{t} \cdot {\sin k}^{2}\right)\right)\right) \]

      10. associate-*l* N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(2, \mathsf{*.f64}\left(\ell, \ell\right)\right), \mathsf{cos.f64}\left(k\right)\right), \left(k \cdot \color{blue}{\left(k \cdot \left(t \cdot {\sin k}^{2}\right)\right)}\right)\right) \]

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

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(2, \mathsf{*.f64}\left(\ell, \ell\right)\right), \mathsf{cos.f64}\left(k\right)\right), \mathsf{*.f64}\left(k, \color{blue}{\left(k \cdot \left(t \cdot {\sin k}^{2}\right)\right)}\right)\right) \]

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

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(2, \mathsf{*.f64}\left(\ell, \ell\right)\right), \mathsf{cos.f64}\left(k\right)\right), \mathsf{*.f64}\left(k, \mathsf{*.f64}\left(k, \color{blue}{\left(t \cdot {\sin k}^{2}\right)}\right)\right)\right) \]

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

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(2, \mathsf{*.f64}\left(\ell, \ell\right)\right), \mathsf{cos.f64}\left(k\right)\right), \mathsf{*.f64}\left(k, \mathsf{*.f64}\left(k, \mathsf{*.f64}\left(t, \color{blue}{\left({\sin k}^{2}\right)}\right)\right)\right)\right) \]

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

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(2, \mathsf{*.f64}\left(\ell, \ell\right)\right), \mathsf{cos.f64}\left(k\right)\right), \mathsf{*.f64}\left(k, \mathsf{*.f64}\left(k, \mathsf{*.f64}\left(t, \mathsf{pow.f64}\left(\sin k, \color{blue}{2}\right)\right)\right)\right)\right) \]

      15. sin-lowering-sin.f64 78.9%

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(2, \mathsf{*.f64}\left(\ell, \ell\right)\right), \mathsf{cos.f64}\left(k\right)\right), \mathsf{*.f64}\left(k, \mathsf{*.f64}\left(k, \mathsf{*.f64}\left(t, \mathsf{pow.f64}\left(\mathsf{sin.f64}\left(k\right), 2\right)\right)\right)\right)\right) \]

   5. Simplified 78.9%

      \[\leadsto \color{blue}{\frac{\left(2 \cdot \left(\ell \cdot \ell\right)\right) \cdot \cos k}{k \cdot \left(k \cdot \left(t \cdot {\sin k}^{2}\right)\right)}} \]
   6. Step-by-step derivation

      1. associate-/l* N/A

         \[\leadsto \left(2 \cdot \left(\ell \cdot \ell\right)\right) \cdot \color{blue}{\frac{\cos k}{k \cdot \left(k \cdot \left(t \cdot {\sin k}^{2}\right)\right)}} \]

      2. associate-*r* N/A

         \[\leadsto \left(\left(2 \cdot \ell\right) \cdot \ell\right) \cdot \frac{\color{blue}{\cos k}}{k \cdot \left(k \cdot \left(t \cdot {\sin k}^{2}\right)\right)} \]

      3. associate-*l* N/A

         \[\leadsto \left(2 \cdot \ell\right) \cdot \color{blue}{\left(\ell \cdot \frac{\cos k}{k \cdot \left(k \cdot \left(t \cdot {\sin k}^{2}\right)\right)}\right)} \]

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

         \[\leadsto \mathsf{*.f64}\left(\left(2 \cdot \ell\right), \color{blue}{\left(\ell \cdot \frac{\cos k}{k \cdot \left(k \cdot \left(t \cdot {\sin k}^{2}\right)\right)}\right)}\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(2, \ell\right), \left(\color{blue}{\ell} \cdot \frac{\cos k}{k \cdot \left(k \cdot \left(t \cdot {\sin k}^{2}\right)\right)}\right)\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(2, \ell\right), \mathsf{*.f64}\left(\ell, \color{blue}{\left(\frac{\cos k}{k \cdot \left(k \cdot \left(t \cdot {\sin k}^{2}\right)\right)}\right)}\right)\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(2, \ell\right), \mathsf{*.f64}\left(\ell, \mathsf{/.f64}\left(\cos k, \color{blue}{\left(k \cdot \left(k \cdot \left(t \cdot {\sin k}^{2}\right)\right)\right)}\right)\right)\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(2, \ell\right), \mathsf{*.f64}\left(\ell, \mathsf{/.f64}\left(\mathsf{cos.f64}\left(k\right), \left(\color{blue}{k} \cdot \left(k \cdot \left(t \cdot {\sin k}^{2}\right)\right)\right)\right)\right)\right) \]

      9. associate-*r* N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(2, \ell\right), \mathsf{*.f64}\left(\ell, \mathsf{/.f64}\left(\mathsf{cos.f64}\left(k\right), \left(\left(k \cdot k\right) \cdot \color{blue}{\left(t \cdot {\sin k}^{2}\right)}\right)\right)\right)\right) \]

      10. associate-*r* N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(2, \ell\right), \mathsf{*.f64}\left(\ell, \mathsf{/.f64}\left(\mathsf{cos.f64}\left(k\right), \left(\left(\left(k \cdot k\right) \cdot t\right) \cdot \color{blue}{{\sin k}^{2}}\right)\right)\right)\right) \]

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

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(2, \ell\right), \mathsf{*.f64}\left(\ell, \mathsf{/.f64}\left(\mathsf{cos.f64}\left(k\right), \mathsf{*.f64}\left(\left(\left(k \cdot k\right) \cdot t\right), \color{blue}{\left({\sin k}^{2}\right)}\right)\right)\right)\right) \]

      12. *-commutative N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(2, \ell\right), \mathsf{*.f64}\left(\ell, \mathsf{/.f64}\left(\mathsf{cos.f64}\left(k\right), \mathsf{*.f64}\left(\left(t \cdot \left(k \cdot k\right)\right), \left({\color{blue}{\sin k}}^{2}\right)\right)\right)\right)\right) \]

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

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(2, \ell\right), \mathsf{*.f64}\left(\ell, \mathsf{/.f64}\left(\mathsf{cos.f64}\left(k\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(t, \left(k \cdot k\right)\right), \left({\color{blue}{\sin k}}^{2}\right)\right)\right)\right)\right) \]

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

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(2, \ell\right), \mathsf{*.f64}\left(\ell, \mathsf{/.f64}\left(\mathsf{cos.f64}\left(k\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(t, \mathsf{*.f64}\left(k, k\right)\right), \left({\sin k}^{2}\right)\right)\right)\right)\right) \]

      15. unpow2 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(2, \ell\right), \mathsf{*.f64}\left(\ell, \mathsf{/.f64}\left(\mathsf{cos.f64}\left(k\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(t, \mathsf{*.f64}\left(k, k\right)\right), \left(\sin k \cdot \color{blue}{\sin k}\right)\right)\right)\right)\right) \]

      16. sqr-sin-a N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(2, \ell\right), \mathsf{*.f64}\left(\ell, \mathsf{/.f64}\left(\mathsf{cos.f64}\left(k\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(t, \mathsf{*.f64}\left(k, k\right)\right), \left(\frac{1}{2} - \color{blue}{\frac{1}{2} \cdot \cos \left(2 \cdot k\right)}\right)\right)\right)\right)\right) \]

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

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(2, \ell\right), \mathsf{*.f64}\left(\ell, \mathsf{/.f64}\left(\mathsf{cos.f64}\left(k\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(t, \mathsf{*.f64}\left(k, k\right)\right), \mathsf{\_.f64}\left(\frac{1}{2}, \color{blue}{\left(\frac{1}{2} \cdot \cos \left(2 \cdot k\right)\right)}\right)\right)\right)\right)\right) \]

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

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(2, \ell\right), \mathsf{*.f64}\left(\ell, \mathsf{/.f64}\left(\mathsf{cos.f64}\left(k\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(t, \mathsf{*.f64}\left(k, k\right)\right), \mathsf{\_.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(\frac{1}{2}, \color{blue}{\cos \left(2 \cdot k\right)}\right)\right)\right)\right)\right)\right) \]

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

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(2, \ell\right), \mathsf{*.f64}\left(\ell, \mathsf{/.f64}\left(\mathsf{cos.f64}\left(k\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(t, \mathsf{*.f64}\left(k, k\right)\right), \mathsf{\_.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(\frac{1}{2}, \mathsf{cos.f64}\left(\left(2 \cdot k\right)\right)\right)\right)\right)\right)\right)\right) \]

      20. *-lowering-*.f64 92.5%

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(2, \ell\right), \mathsf{*.f64}\left(\ell, \mathsf{/.f64}\left(\mathsf{cos.f64}\left(k\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(t, \mathsf{*.f64}\left(k, k\right)\right), \mathsf{\_.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(\frac{1}{2}, \mathsf{cos.f64}\left(\mathsf{*.f64}\left(2, k\right)\right)\right)\right)\right)\right)\right)\right) \]

   7. Applied egg-rr 92.5%

      \[\leadsto \color{blue}{\left(2 \cdot \ell\right) \cdot \left(\ell \cdot \frac{\cos k}{\left(t \cdot \left(k \cdot k\right)\right) \cdot \left(0.5 - 0.5 \cdot \cos \left(2 \cdot k\right)\right)}\right)} \]
   8. Step-by-step derivation

      1. associate-*r/ N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(2, \ell\right), \left(\frac{\ell \cdot \cos k}{\color{blue}{\left(t \cdot \left(k \cdot k\right)\right) \cdot \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot k\right)\right)}}\right)\right) \]

      2. associate-*l* N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(2, \ell\right), \left(\frac{\ell \cdot \cos k}{t \cdot \color{blue}{\left(\left(k \cdot k\right) \cdot \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot k\right)\right)\right)}}\right)\right) \]

      3. times-frac N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(2, \ell\right), \left(\frac{\ell}{t} \cdot \color{blue}{\frac{\cos k}{\left(k \cdot k\right) \cdot \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot k\right)\right)}}\right)\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(2, \ell\right), \mathsf{*.f64}\left(\left(\frac{\ell}{t}\right), \color{blue}{\left(\frac{\cos k}{\left(k \cdot k\right) \cdot \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot k\right)\right)}\right)}\right)\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(2, \ell\right), \mathsf{*.f64}\left(\mathsf{/.f64}\left(\ell, t\right), \left(\frac{\color{blue}{\cos k}}{\left(k \cdot k\right) \cdot \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot k\right)\right)}\right)\right)\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(2, \ell\right), \mathsf{*.f64}\left(\mathsf{/.f64}\left(\ell, t\right), \mathsf{/.f64}\left(\cos k, \color{blue}{\left(\left(k \cdot k\right) \cdot \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot k\right)\right)\right)}\right)\right)\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(2, \ell\right), \mathsf{*.f64}\left(\mathsf{/.f64}\left(\ell, t\right), \mathsf{/.f64}\left(\mathsf{cos.f64}\left(k\right), \left(\color{blue}{\left(k \cdot k\right)} \cdot \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot k\right)\right)\right)\right)\right)\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(2, \ell\right), \mathsf{*.f64}\left(\mathsf{/.f64}\left(\ell, t\right), \mathsf{/.f64}\left(\mathsf{cos.f64}\left(k\right), \mathsf{*.f64}\left(\left(k \cdot k\right), \color{blue}{\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot k\right)\right)}\right)\right)\right)\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(2, \ell\right), \mathsf{*.f64}\left(\mathsf{/.f64}\left(\ell, t\right), \mathsf{/.f64}\left(\mathsf{cos.f64}\left(k\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(k, k\right), \left(\color{blue}{\frac{1}{2}} - \frac{1}{2} \cdot \cos \left(2 \cdot k\right)\right)\right)\right)\right)\right) \]

      10. sub-neg N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(2, \ell\right), \mathsf{*.f64}\left(\mathsf{/.f64}\left(\ell, t\right), \mathsf{/.f64}\left(\mathsf{cos.f64}\left(k\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(k, k\right), \left(\frac{1}{2} + \color{blue}{\left(\mathsf{neg}\left(\frac{1}{2} \cdot \cos \left(2 \cdot k\right)\right)\right)}\right)\right)\right)\right)\right) \]

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

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(2, \ell\right), \mathsf{*.f64}\left(\mathsf{/.f64}\left(\ell, t\right), \mathsf{/.f64}\left(\mathsf{cos.f64}\left(k\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(k, k\right), \mathsf{+.f64}\left(\frac{1}{2}, \color{blue}{\left(\mathsf{neg}\left(\frac{1}{2} \cdot \cos \left(2 \cdot k\right)\right)\right)}\right)\right)\right)\right)\right) \]

      12. *-commutative N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(2, \ell\right), \mathsf{*.f64}\left(\mathsf{/.f64}\left(\ell, t\right), \mathsf{/.f64}\left(\mathsf{cos.f64}\left(k\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(k, k\right), \mathsf{+.f64}\left(\frac{1}{2}, \left(\mathsf{neg}\left(\cos \left(2 \cdot k\right) \cdot \frac{1}{2}\right)\right)\right)\right)\right)\right)\right) \]

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

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(2, \ell\right), \mathsf{*.f64}\left(\mathsf{/.f64}\left(\ell, t\right), \mathsf{/.f64}\left(\mathsf{cos.f64}\left(k\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(k, k\right), \mathsf{+.f64}\left(\frac{1}{2}, \left(\cos \left(2 \cdot k\right) \cdot \color{blue}{\left(\mathsf{neg}\left(\frac{1}{2}\right)\right)}\right)\right)\right)\right)\right)\right) \]

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

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(2, \ell\right), \mathsf{*.f64}\left(\mathsf{/.f64}\left(\ell, t\right), \mathsf{/.f64}\left(\mathsf{cos.f64}\left(k\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(k, k\right), \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(\cos \left(2 \cdot k\right), \color{blue}{\left(\mathsf{neg}\left(\frac{1}{2}\right)\right)}\right)\right)\right)\right)\right)\right) \]

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

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(2, \ell\right), \mathsf{*.f64}\left(\mathsf{/.f64}\left(\ell, t\right), \mathsf{/.f64}\left(\mathsf{cos.f64}\left(k\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(k, k\right), \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(\mathsf{cos.f64}\left(\left(2 \cdot k\right)\right), \left(\mathsf{neg}\left(\color{blue}{\frac{1}{2}}\right)\right)\right)\right)\right)\right)\right)\right) \]

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

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(2, \ell\right), \mathsf{*.f64}\left(\mathsf{/.f64}\left(\ell, t\right), \mathsf{/.f64}\left(\mathsf{cos.f64}\left(k\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(k, k\right), \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(\mathsf{cos.f64}\left(\mathsf{*.f64}\left(2, k\right)\right), \left(\mathsf{neg}\left(\frac{1}{2}\right)\right)\right)\right)\right)\right)\right)\right) \]

      17. metadata-eval 95.4%

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(2, \ell\right), \mathsf{*.f64}\left(\mathsf{/.f64}\left(\ell, t\right), \mathsf{/.f64}\left(\mathsf{cos.f64}\left(k\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(k, k\right), \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(\mathsf{cos.f64}\left(\mathsf{*.f64}\left(2, k\right)\right), \frac{-1}{2}\right)\right)\right)\right)\right)\right) \]

   9. Applied egg-rr 95.4%

      \[\leadsto \left(2 \cdot \ell\right) \cdot \color{blue}{\left(\frac{\ell}{t} \cdot \frac{\cos k}{\left(k \cdot k\right) \cdot \left(0.5 + \cos \left(2 \cdot k\right) \cdot -0.5\right)}\right)} \]

   if 1.11999999999999994e154 < k

   1. Initial program 28.2%

      \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) - 1\right)} \]
   2. Add Preprocessing

   3. Taylor expanded in t around 0

      \[\leadsto \color{blue}{2 \cdot \frac{{\ell}^{2} \cdot \cos k}{{k}^{2} \cdot \left(t \cdot {\sin k}^{2}\right)}} \]
   4. Step-by-step derivation

      1. associate-*r/ N/A

         \[\leadsto \frac{2 \cdot \left({\ell}^{2} \cdot \cos k\right)}{\color{blue}{{k}^{2} \cdot \left(t \cdot {\sin k}^{2}\right)}} \]

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

         \[\leadsto \mathsf{/.f64}\left(\left(2 \cdot \left({\ell}^{2} \cdot \cos k\right)\right), \color{blue}{\left({k}^{2} \cdot \left(t \cdot {\sin k}^{2}\right)\right)}\right) \]

      3. associate-*r* N/A

         \[\leadsto \mathsf{/.f64}\left(\left(\left(2 \cdot {\ell}^{2}\right) \cdot \cos k\right), \left(\color{blue}{{k}^{2}} \cdot \left(t \cdot {\sin k}^{2}\right)\right)\right) \]

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

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\left(2 \cdot {\ell}^{2}\right), \cos k\right), \left(\color{blue}{{k}^{2}} \cdot \left(t \cdot {\sin k}^{2}\right)\right)\right) \]

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

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(2, \left({\ell}^{2}\right)\right), \cos k\right), \left({\color{blue}{k}}^{2} \cdot \left(t \cdot {\sin k}^{2}\right)\right)\right) \]

      6. unpow2 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(2, \left(\ell \cdot \ell\right)\right), \cos k\right), \left({k}^{2} \cdot \left(t \cdot {\sin k}^{2}\right)\right)\right) \]

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

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(2, \mathsf{*.f64}\left(\ell, \ell\right)\right), \cos k\right), \left({k}^{2} \cdot \left(t \cdot {\sin k}^{2}\right)\right)\right) \]

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

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(2, \mathsf{*.f64}\left(\ell, \ell\right)\right), \mathsf{cos.f64}\left(k\right)\right), \left({k}^{\color{blue}{2}} \cdot \left(t \cdot {\sin k}^{2}\right)\right)\right) \]

      9. unpow2 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(2, \mathsf{*.f64}\left(\ell, \ell\right)\right), \mathsf{cos.f64}\left(k\right)\right), \left(\left(k \cdot k\right) \cdot \left(\color{blue}{t} \cdot {\sin k}^{2}\right)\right)\right) \]

      10. associate-*l* N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(2, \mathsf{*.f64}\left(\ell, \ell\right)\right), \mathsf{cos.f64}\left(k\right)\right), \left(k \cdot \color{blue}{\left(k \cdot \left(t \cdot {\sin k}^{2}\right)\right)}\right)\right) \]

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

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(2, \mathsf{*.f64}\left(\ell, \ell\right)\right), \mathsf{cos.f64}\left(k\right)\right), \mathsf{*.f64}\left(k, \color{blue}{\left(k \cdot \left(t \cdot {\sin k}^{2}\right)\right)}\right)\right) \]

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

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(2, \mathsf{*.f64}\left(\ell, \ell\right)\right), \mathsf{cos.f64}\left(k\right)\right), \mathsf{*.f64}\left(k, \mathsf{*.f64}\left(k, \color{blue}{\left(t \cdot {\sin k}^{2}\right)}\right)\right)\right) \]

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

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(2, \mathsf{*.f64}\left(\ell, \ell\right)\right), \mathsf{cos.f64}\left(k\right)\right), \mathsf{*.f64}\left(k, \mathsf{*.f64}\left(k, \mathsf{*.f64}\left(t, \color{blue}{\left({\sin k}^{2}\right)}\right)\right)\right)\right) \]

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

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(2, \mathsf{*.f64}\left(\ell, \ell\right)\right), \mathsf{cos.f64}\left(k\right)\right), \mathsf{*.f64}\left(k, \mathsf{*.f64}\left(k, \mathsf{*.f64}\left(t, \mathsf{pow.f64}\left(\sin k, \color{blue}{2}\right)\right)\right)\right)\right) \]

      15. sin-lowering-sin.f64 59.2%

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(2, \mathsf{*.f64}\left(\ell, \ell\right)\right), \mathsf{cos.f64}\left(k\right)\right), \mathsf{*.f64}\left(k, \mathsf{*.f64}\left(k, \mathsf{*.f64}\left(t, \mathsf{pow.f64}\left(\mathsf{sin.f64}\left(k\right), 2\right)\right)\right)\right)\right) \]

   5. Simplified 59.2%

      \[\leadsto \color{blue}{\frac{\left(2 \cdot \left(\ell \cdot \ell\right)\right) \cdot \cos k}{k \cdot \left(k \cdot \left(t \cdot {\sin k}^{2}\right)\right)}} \]
   6. Step-by-step derivation

      1. associate-/l* N/A

         \[\leadsto \left(2 \cdot \left(\ell \cdot \ell\right)\right) \cdot \color{blue}{\frac{\cos k}{k \cdot \left(k \cdot \left(t \cdot {\sin k}^{2}\right)\right)}} \]

      2. associate-*r* N/A

         \[\leadsto \left(\left(2 \cdot \ell\right) \cdot \ell\right) \cdot \frac{\color{blue}{\cos k}}{k \cdot \left(k \cdot \left(t \cdot {\sin k}^{2}\right)\right)} \]

      3. associate-*l* N/A

         \[\leadsto \left(2 \cdot \ell\right) \cdot \color{blue}{\left(\ell \cdot \frac{\cos k}{k \cdot \left(k \cdot \left(t \cdot {\sin k}^{2}\right)\right)}\right)} \]

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

         \[\leadsto \mathsf{*.f64}\left(\left(2 \cdot \ell\right), \color{blue}{\left(\ell \cdot \frac{\cos k}{k \cdot \left(k \cdot \left(t \cdot {\sin k}^{2}\right)\right)}\right)}\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(2, \ell\right), \left(\color{blue}{\ell} \cdot \frac{\cos k}{k \cdot \left(k \cdot \left(t \cdot {\sin k}^{2}\right)\right)}\right)\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(2, \ell\right), \mathsf{*.f64}\left(\ell, \color{blue}{\left(\frac{\cos k}{k \cdot \left(k \cdot \left(t \cdot {\sin k}^{2}\right)\right)}\right)}\right)\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(2, \ell\right), \mathsf{*.f64}\left(\ell, \mathsf{/.f64}\left(\cos k, \color{blue}{\left(k \cdot \left(k \cdot \left(t \cdot {\sin k}^{2}\right)\right)\right)}\right)\right)\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(2, \ell\right), \mathsf{*.f64}\left(\ell, \mathsf{/.f64}\left(\mathsf{cos.f64}\left(k\right), \left(\color{blue}{k} \cdot \left(k \cdot \left(t \cdot {\sin k}^{2}\right)\right)\right)\right)\right)\right) \]

      9. associate-*r* N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(2, \ell\right), \mathsf{*.f64}\left(\ell, \mathsf{/.f64}\left(\mathsf{cos.f64}\left(k\right), \left(\left(k \cdot k\right) \cdot \color{blue}{\left(t \cdot {\sin k}^{2}\right)}\right)\right)\right)\right) \]

      10. associate-*r* N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(2, \ell\right), \mathsf{*.f64}\left(\ell, \mathsf{/.f64}\left(\mathsf{cos.f64}\left(k\right), \left(\left(\left(k \cdot k\right) \cdot t\right) \cdot \color{blue}{{\sin k}^{2}}\right)\right)\right)\right) \]

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

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(2, \ell\right), \mathsf{*.f64}\left(\ell, \mathsf{/.f64}\left(\mathsf{cos.f64}\left(k\right), \mathsf{*.f64}\left(\left(\left(k \cdot k\right) \cdot t\right), \color{blue}{\left({\sin k}^{2}\right)}\right)\right)\right)\right) \]

      12. *-commutative N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(2, \ell\right), \mathsf{*.f64}\left(\ell, \mathsf{/.f64}\left(\mathsf{cos.f64}\left(k\right), \mathsf{*.f64}\left(\left(t \cdot \left(k \cdot k\right)\right), \left({\color{blue}{\sin k}}^{2}\right)\right)\right)\right)\right) \]

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

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(2, \ell\right), \mathsf{*.f64}\left(\ell, \mathsf{/.f64}\left(\mathsf{cos.f64}\left(k\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(t, \left(k \cdot k\right)\right), \left({\color{blue}{\sin k}}^{2}\right)\right)\right)\right)\right) \]

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

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(2, \ell\right), \mathsf{*.f64}\left(\ell, \mathsf{/.f64}\left(\mathsf{cos.f64}\left(k\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(t, \mathsf{*.f64}\left(k, k\right)\right), \left({\sin k}^{2}\right)\right)\right)\right)\right) \]

      15. unpow2 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(2, \ell\right), \mathsf{*.f64}\left(\ell, \mathsf{/.f64}\left(\mathsf{cos.f64}\left(k\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(t, \mathsf{*.f64}\left(k, k\right)\right), \left(\sin k \cdot \color{blue}{\sin k}\right)\right)\right)\right)\right) \]

      16. sqr-sin-a N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(2, \ell\right), \mathsf{*.f64}\left(\ell, \mathsf{/.f64}\left(\mathsf{cos.f64}\left(k\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(t, \mathsf{*.f64}\left(k, k\right)\right), \left(\frac{1}{2} - \color{blue}{\frac{1}{2} \cdot \cos \left(2 \cdot k\right)}\right)\right)\right)\right)\right) \]

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

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(2, \ell\right), \mathsf{*.f64}\left(\ell, \mathsf{/.f64}\left(\mathsf{cos.f64}\left(k\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(t, \mathsf{*.f64}\left(k, k\right)\right), \mathsf{\_.f64}\left(\frac{1}{2}, \color{blue}{\left(\frac{1}{2} \cdot \cos \left(2 \cdot k\right)\right)}\right)\right)\right)\right)\right) \]

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

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(2, \ell\right), \mathsf{*.f64}\left(\ell, \mathsf{/.f64}\left(\mathsf{cos.f64}\left(k\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(t, \mathsf{*.f64}\left(k, k\right)\right), \mathsf{\_.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(\frac{1}{2}, \color{blue}{\cos \left(2 \cdot k\right)}\right)\right)\right)\right)\right)\right) \]

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

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(2, \ell\right), \mathsf{*.f64}\left(\ell, \mathsf{/.f64}\left(\mathsf{cos.f64}\left(k\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(t, \mathsf{*.f64}\left(k, k\right)\right), \mathsf{\_.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(\frac{1}{2}, \mathsf{cos.f64}\left(\left(2 \cdot k\right)\right)\right)\right)\right)\right)\right)\right) \]

      20. *-lowering-*.f64 51.9%

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(2, \ell\right), \mathsf{*.f64}\left(\ell, \mathsf{/.f64}\left(\mathsf{cos.f64}\left(k\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(t, \mathsf{*.f64}\left(k, k\right)\right), \mathsf{\_.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(\frac{1}{2}, \mathsf{cos.f64}\left(\mathsf{*.f64}\left(2, k\right)\right)\right)\right)\right)\right)\right)\right) \]

   7. Applied egg-rr 51.9%

      \[\leadsto \color{blue}{\left(2 \cdot \ell\right) \cdot \left(\ell \cdot \frac{\cos k}{\left(t \cdot \left(k \cdot k\right)\right) \cdot \left(0.5 - 0.5 \cdot \cos \left(2 \cdot k\right)\right)}\right)} \]
   8. Step-by-step derivation

      1. associate-*r* N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(2, \ell\right), \mathsf{*.f64}\left(\ell, \mathsf{/.f64}\left(\mathsf{cos.f64}\left(k\right), \mathsf{*.f64}\left(\left(\left(t \cdot k\right) \cdot k\right), \mathsf{\_.f64}\left(\color{blue}{\frac{1}{2}}, \mathsf{*.f64}\left(\frac{1}{2}, \mathsf{cos.f64}\left(\mathsf{*.f64}\left(2, k\right)\right)\right)\right)\right)\right)\right)\right) \]

      2. *-commutative N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(2, \ell\right), \mathsf{*.f64}\left(\ell, \mathsf{/.f64}\left(\mathsf{cos.f64}\left(k\right), \mathsf{*.f64}\left(\left(\left(k \cdot t\right) \cdot k\right), \mathsf{\_.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(\frac{1}{2}, \mathsf{cos.f64}\left(\mathsf{*.f64}\left(2, k\right)\right)\right)\right)\right)\right)\right)\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(2, \ell\right), \mathsf{*.f64}\left(\ell, \mathsf{/.f64}\left(\mathsf{cos.f64}\left(k\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(\left(k \cdot t\right), k\right), \mathsf{\_.f64}\left(\color{blue}{\frac{1}{2}}, \mathsf{*.f64}\left(\frac{1}{2}, \mathsf{cos.f64}\left(\mathsf{*.f64}\left(2, k\right)\right)\right)\right)\right)\right)\right)\right) \]

      4. *-lowering-*.f64 65.2%

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(2, \ell\right), \mathsf{*.f64}\left(\ell, \mathsf{/.f64}\left(\mathsf{cos.f64}\left(k\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(k, t\right), k\right), \mathsf{\_.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(\frac{1}{2}, \mathsf{cos.f64}\left(\mathsf{*.f64}\left(2, k\right)\right)\right)\right)\right)\right)\right)\right) \]

   9. Applied egg-rr 65.2%

      \[\leadsto \left(2 \cdot \ell\right) \cdot \left(\ell \cdot \frac{\cos k}{\color{blue}{\left(\left(k \cdot t\right) \cdot k\right)} \cdot \left(0.5 - 0.5 \cdot \cos \left(2 \cdot k\right)\right)}\right) \]
3. Recombined 3 regimes into one program.

4. Final simplification 78.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;k \leq 8.5 \cdot 10^{-5}:\\ \;\;\;\;\ell \cdot \frac{\frac{\frac{2 \cdot \ell}{k \cdot \left(k \cdot t\right)}}{k}}{k}\\ \mathbf{elif}\;k \leq 1.12 \cdot 10^{+154}:\\ \;\;\;\;\left(2 \cdot \ell\right) \cdot \left(\frac{\ell}{t} \cdot \frac{\cos k}{\left(k \cdot k\right) \cdot \left(0.5 + \cos \left(k \cdot 2\right) \cdot -0.5\right)}\right)\\ \mathbf{else}:\\ \;\;\;\;\left(2 \cdot \ell\right) \cdot \left(\ell \cdot \frac{\cos k}{\left(0.5 - 0.5 \cdot \cos \left(k \cdot 2\right)\right) \cdot \left(k \cdot \left(k \cdot t\right)\right)}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 90.5% accurate, 1.9× speedup

Math

\[\begin{array}{l} k_m = \left|k\right| \\ \begin{array}{l} \mathbf{if}\;k\_m \leq 8.5 \cdot 10^{-5}:\\ \;\;\;\;\ell \cdot \frac{\frac{\frac{2 \cdot \ell}{k\_m \cdot \left(k\_m \cdot t\right)}}{k\_m}}{k\_m}\\ \mathbf{else}:\\ \;\;\;\;\left(\frac{\cos k\_m}{k\_m} \cdot \frac{2 \cdot \ell}{k\_m}\right) \cdot \frac{\frac{\ell}{t}}{0.5 + \cos \left(k\_m \cdot 2\right) \cdot -0.5}\\ \end{array} \end{array} \]

FPCore

k_m = (fabs.f64 k)
(FPCore (t l k_m)
 :precision binary64
 (if (<= k_m 8.5e-5)
   (* l (/ (/ (/ (* 2.0 l) (* k_m (* k_m t))) k_m) k_m))
   (*
    (* (/ (cos k_m) k_m) (/ (* 2.0 l) k_m))
    (/ (/ l t) (+ 0.5 (* (cos (* k_m 2.0)) -0.5))))))

C

k_m = fabs(k);
double code(double t, double l, double k_m) {
	double tmp;
	if (k_m <= 8.5e-5) {
		tmp = l * ((((2.0 * l) / (k_m * (k_m * t))) / k_m) / k_m);
	} else {
		tmp = ((cos(k_m) / k_m) * ((2.0 * l) / k_m)) * ((l / t) / (0.5 + (cos((k_m * 2.0)) * -0.5)));
	}
	return tmp;
}

Fortran

k_m = abs(k)
real(8) function code(t, l, k_m)
    real(8), intent (in) :: t
    real(8), intent (in) :: l
    real(8), intent (in) :: k_m
    real(8) :: tmp
    if (k_m <= 8.5d-5) then
        tmp = l * ((((2.0d0 * l) / (k_m * (k_m * t))) / k_m) / k_m)
    else
        tmp = ((cos(k_m) / k_m) * ((2.0d0 * l) / k_m)) * ((l / t) / (0.5d0 + (cos((k_m * 2.0d0)) * (-0.5d0))))
    end if
    code = tmp
end function

Java

k_m = Math.abs(k);
public static double code(double t, double l, double k_m) {
	double tmp;
	if (k_m <= 8.5e-5) {
		tmp = l * ((((2.0 * l) / (k_m * (k_m * t))) / k_m) / k_m);
	} else {
		tmp = ((Math.cos(k_m) / k_m) * ((2.0 * l) / k_m)) * ((l / t) / (0.5 + (Math.cos((k_m * 2.0)) * -0.5)));
	}
	return tmp;
}

Python

k_m = math.fabs(k)
def code(t, l, k_m):
	tmp = 0
	if k_m <= 8.5e-5:
		tmp = l * ((((2.0 * l) / (k_m * (k_m * t))) / k_m) / k_m)
	else:
		tmp = ((math.cos(k_m) / k_m) * ((2.0 * l) / k_m)) * ((l / t) / (0.5 + (math.cos((k_m * 2.0)) * -0.5)))
	return tmp

Julia

k_m = abs(k)
function code(t, l, k_m)
	tmp = 0.0
	if (k_m <= 8.5e-5)
		tmp = Float64(l * Float64(Float64(Float64(Float64(2.0 * l) / Float64(k_m * Float64(k_m * t))) / k_m) / k_m));
	else
		tmp = Float64(Float64(Float64(cos(k_m) / k_m) * Float64(Float64(2.0 * l) / k_m)) * Float64(Float64(l / t) / Float64(0.5 + Float64(cos(Float64(k_m * 2.0)) * -0.5))));
	end
	return tmp
end

MATLAB

k_m = abs(k);
function tmp_2 = code(t, l, k_m)
	tmp = 0.0;
	if (k_m <= 8.5e-5)
		tmp = l * ((((2.0 * l) / (k_m * (k_m * t))) / k_m) / k_m);
	else
		tmp = ((cos(k_m) / k_m) * ((2.0 * l) / k_m)) * ((l / t) / (0.5 + (cos((k_m * 2.0)) * -0.5)));
	end
	tmp_2 = tmp;
end

Wolfram

k_m = N[Abs[k], $MachinePrecision]
code[t_, l_, k$95$m_] := If[LessEqual[k$95$m, 8.5e-5], N[(l * N[(N[(N[(N[(2.0 * l), $MachinePrecision] / N[(k$95$m * N[(k$95$m * t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / k$95$m), $MachinePrecision] / k$95$m), $MachinePrecision]), $MachinePrecision], N[(N[(N[(N[Cos[k$95$m], $MachinePrecision] / k$95$m), $MachinePrecision] * N[(N[(2.0 * l), $MachinePrecision] / k$95$m), $MachinePrecision]), $MachinePrecision] * N[(N[(l / t), $MachinePrecision] / N[(0.5 + N[(N[Cos[N[(k$95$m * 2.0), $MachinePrecision]], $MachinePrecision] * -0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if k < 8.500000000000001e-5

   1. Initial program 35.0%

      \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) - 1\right)} \]
   2. Add Preprocessing

   3. Taylor expanded in k around 0

      \[\leadsto \mathsf{/.f64}\left(2, \color{blue}{\left(\frac{{k}^{4} \cdot t}{{\ell}^{2}}\right)}\right) \]
   4. Step-by-step derivation

      1. associate-/l* N/A

         \[\leadsto \mathsf{/.f64}\left(2, \left({k}^{4} \cdot \color{blue}{\frac{t}{{\ell}^{2}}}\right)\right) \]

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

         \[\leadsto \mathsf{/.f64}\left(2, \mathsf{*.f64}\left(\left({k}^{4}\right), \color{blue}{\left(\frac{t}{{\ell}^{2}}\right)}\right)\right) \]

      3. metadata-eval N/A

         \[\leadsto \mathsf{/.f64}\left(2, \mathsf{*.f64}\left(\left({k}^{\left(2 \cdot 2\right)}\right), \left(\frac{t}{{\ell}^{2}}\right)\right)\right) \]

      4. pow-sqr N/A

         \[\leadsto \mathsf{/.f64}\left(2, \mathsf{*.f64}\left(\left({k}^{2} \cdot {k}^{2}\right), \left(\frac{\color{blue}{t}}{{\ell}^{2}}\right)\right)\right) \]

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

         \[\leadsto \mathsf{/.f64}\left(2, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\left({k}^{2}\right), \left({k}^{2}\right)\right), \left(\frac{\color{blue}{t}}{{\ell}^{2}}\right)\right)\right) \]

      6. unpow2 N/A

         \[\leadsto \mathsf{/.f64}\left(2, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\left(k \cdot k\right), \left({k}^{2}\right)\right), \left(\frac{t}{{\ell}^{2}}\right)\right)\right) \]

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

         \[\leadsto \mathsf{/.f64}\left(2, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(k, k\right), \left({k}^{2}\right)\right), \left(\frac{t}{{\ell}^{2}}\right)\right)\right) \]

      8. unpow2 N/A

         \[\leadsto \mathsf{/.f64}\left(2, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(k, k\right), \left(k \cdot k\right)\right), \left(\frac{t}{{\ell}^{2}}\right)\right)\right) \]

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

         \[\leadsto \mathsf{/.f64}\left(2, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(k, k\right), \mathsf{*.f64}\left(k, k\right)\right), \left(\frac{t}{{\ell}^{2}}\right)\right)\right) \]

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

          \[\leadsto \mathsf{/.f64}\left(2, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(k, k\right), \mathsf{*.f64}\left(k, k\right)\right), \mathsf{/.f64}\left(t, \color{blue}{\left({\ell}^{2}\right)}\right)\right)\right) \]

      11. unpow2 N/A

          \[\leadsto \mathsf{/.f64}\left(2, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(k, k\right), \mathsf{*.f64}\left(k, k\right)\right), \mathsf{/.f64}\left(t, \left(\ell \cdot \color{blue}{\ell}\right)\right)\right)\right) \]

      12. *-lowering-*.f64 58.1%

          \[\leadsto \mathsf{/.f64}\left(2, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(k, k\right), \mathsf{*.f64}\left(k, k\right)\right), \mathsf{/.f64}\left(t, \mathsf{*.f64}\left(\ell, \color{blue}{\ell}\right)\right)\right)\right) \]

   5. Simplified 58.1%

      \[\leadsto \frac{2}{\color{blue}{\left(\left(k \cdot k\right) \cdot \left(k \cdot k\right)\right) \cdot \frac{t}{\ell \cdot \ell}}} \]
   6. Step-by-step derivation

      1. associate-/r* N/A

         \[\leadsto \frac{\frac{2}{\left(k \cdot k\right) \cdot \left(k \cdot k\right)}}{\color{blue}{\frac{t}{\ell \cdot \ell}}} \]

      2. associate-/r* N/A

         \[\leadsto \frac{\frac{2}{\left(k \cdot k\right) \cdot \left(k \cdot k\right)}}{\frac{\frac{t}{\ell}}{\color{blue}{\ell}}} \]

      3. associate-/r/ N/A

         \[\leadsto \frac{\frac{2}{\left(k \cdot k\right) \cdot \left(k \cdot k\right)}}{\frac{t}{\ell}} \cdot \color{blue}{\ell} \]

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

         \[\leadsto \mathsf{*.f64}\left(\left(\frac{\frac{2}{\left(k \cdot k\right) \cdot \left(k \cdot k\right)}}{\frac{t}{\ell}}\right), \color{blue}{\ell}\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\left(\frac{2}{\left(k \cdot k\right) \cdot \left(k \cdot k\right)}\right), \left(\frac{t}{\ell}\right)\right), \ell\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(2, \left(\left(k \cdot k\right) \cdot \left(k \cdot k\right)\right)\right), \left(\frac{t}{\ell}\right)\right), \ell\right) \]

      7. associate-*l* N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(2, \left(k \cdot \left(k \cdot \left(k \cdot k\right)\right)\right)\right), \left(\frac{t}{\ell}\right)\right), \ell\right) \]

      8. cube-unmult N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(2, \left(k \cdot {k}^{3}\right)\right), \left(\frac{t}{\ell}\right)\right), \ell\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(2, \mathsf{*.f64}\left(k, \left({k}^{3}\right)\right)\right), \left(\frac{t}{\ell}\right)\right), \ell\right) \]

      10. cube-unmult N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(2, \mathsf{*.f64}\left(k, \left(k \cdot \left(k \cdot k\right)\right)\right)\right), \left(\frac{t}{\ell}\right)\right), \ell\right) \]

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

          \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(2, \mathsf{*.f64}\left(k, \mathsf{*.f64}\left(k, \left(k \cdot k\right)\right)\right)\right), \left(\frac{t}{\ell}\right)\right), \ell\right) \]

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

          \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(2, \mathsf{*.f64}\left(k, \mathsf{*.f64}\left(k, \mathsf{*.f64}\left(k, k\right)\right)\right)\right), \left(\frac{t}{\ell}\right)\right), \ell\right) \]

      13. /-lowering-/.f64 69.3%

          \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(2, \mathsf{*.f64}\left(k, \mathsf{*.f64}\left(k, \mathsf{*.f64}\left(k, k\right)\right)\right)\right), \mathsf{/.f64}\left(t, \ell\right)\right), \ell\right) \]

   7. Applied egg-rr 69.3%

      \[\leadsto \color{blue}{\frac{\frac{2}{k \cdot \left(k \cdot \left(k \cdot k\right)\right)}}{\frac{t}{\ell}} \cdot \ell} \]
   8. Step-by-step derivation

      1. div-inv N/A

         \[\leadsto \mathsf{*.f64}\left(\left(\frac{2}{k \cdot \left(k \cdot \left(k \cdot k\right)\right)} \cdot \frac{1}{\frac{t}{\ell}}\right), \ell\right) \]

      2. associate-/r* N/A

         \[\leadsto \mathsf{*.f64}\left(\left(\frac{\frac{2}{k}}{k \cdot \left(k \cdot k\right)} \cdot \frac{1}{\frac{t}{\ell}}\right), \ell\right) \]

      3. clear-num N/A

         \[\leadsto \mathsf{*.f64}\left(\left(\frac{\frac{2}{k}}{k \cdot \left(k \cdot k\right)} \cdot \frac{\ell}{t}\right), \ell\right) \]

      4. associate-*l/ N/A

         \[\leadsto \mathsf{*.f64}\left(\left(\frac{\frac{2}{k} \cdot \frac{\ell}{t}}{k \cdot \left(k \cdot k\right)}\right), \ell\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\left(\frac{2}{k} \cdot \frac{\ell}{t}\right), \left(k \cdot \left(k \cdot k\right)\right)\right), \ell\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\left(\frac{2}{k}\right), \left(\frac{\ell}{t}\right)\right), \left(k \cdot \left(k \cdot k\right)\right)\right), \ell\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(2, k\right), \left(\frac{\ell}{t}\right)\right), \left(k \cdot \left(k \cdot k\right)\right)\right), \ell\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(2, k\right), \mathsf{/.f64}\left(\ell, t\right)\right), \left(k \cdot \left(k \cdot k\right)\right)\right), \ell\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(2, k\right), \mathsf{/.f64}\left(\ell, t\right)\right), \mathsf{*.f64}\left(k, \left(k \cdot k\right)\right)\right), \ell\right) \]

      10. *-lowering-*.f64 72.8%

          \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(2, k\right), \mathsf{/.f64}\left(\ell, t\right)\right), \mathsf{*.f64}\left(k, \mathsf{*.f64}\left(k, k\right)\right)\right), \ell\right) \]

   9. Applied egg-rr 72.8%

      \[\leadsto \color{blue}{\frac{\frac{2}{k} \cdot \frac{\ell}{t}}{k \cdot \left(k \cdot k\right)}} \cdot \ell \]
   10. Step-by-step derivation

       1. associate-/r* N/A

          \[\leadsto \mathsf{*.f64}\left(\left(\frac{\frac{\frac{2}{k} \cdot \frac{\ell}{t}}{k}}{k \cdot k}\right), \ell\right) \]

       2. associate-/r* N/A

          \[\leadsto \mathsf{*.f64}\left(\left(\frac{\frac{\frac{\frac{2}{k} \cdot \frac{\ell}{t}}{k}}{k}}{k}\right), \ell\right) \]

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

          \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\left(\frac{\frac{\frac{2}{k} \cdot \frac{\ell}{t}}{k}}{k}\right), k\right), \ell\right) \]

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

          \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(\left(\frac{\frac{2}{k} \cdot \frac{\ell}{t}}{k}\right), k\right), k\right), \ell\right) \]

       5. frac-times N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(\left(\frac{\frac{2 \cdot \ell}{k \cdot t}}{k}\right), k\right), k\right), \ell\right) \]

       6. associate-/l/ N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(\left(\frac{2 \cdot \ell}{k \cdot \left(k \cdot t\right)}\right), k\right), k\right), \ell\right) \]

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

          \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(\left(2 \cdot \ell\right), \left(k \cdot \left(k \cdot t\right)\right)\right), k\right), k\right), \ell\right) \]

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

          \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(2, \ell\right), \left(k \cdot \left(k \cdot t\right)\right)\right), k\right), k\right), \ell\right) \]

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

          \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(2, \ell\right), \mathsf{*.f64}\left(k, \left(k \cdot t\right)\right)\right), k\right), k\right), \ell\right) \]

       10. *-lowering-*.f64 78.5%

           \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(2, \ell\right), \mathsf{*.f64}\left(k, \mathsf{*.f64}\left(k, t\right)\right)\right), k\right), k\right), \ell\right) \]

   11. Applied egg-rr 78.5%

       \[\leadsto \color{blue}{\frac{\frac{\frac{2 \cdot \ell}{k \cdot \left(k \cdot t\right)}}{k}}{k}} \cdot \ell \]

   if 8.500000000000001e-5 < k

   1. Initial program 21.4%

      \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) - 1\right)} \]
   2. Add Preprocessing

   3. Taylor expanded in t around 0

      \[\leadsto \color{blue}{2 \cdot \frac{{\ell}^{2} \cdot \cos k}{{k}^{2} \cdot \left(t \cdot {\sin k}^{2}\right)}} \]
   4. Step-by-step derivation

      1. associate-*r/ N/A

         \[\leadsto \frac{2 \cdot \left({\ell}^{2} \cdot \cos k\right)}{\color{blue}{{k}^{2} \cdot \left(t \cdot {\sin k}^{2}\right)}} \]

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

         \[\leadsto \mathsf{/.f64}\left(\left(2 \cdot \left({\ell}^{2} \cdot \cos k\right)\right), \color{blue}{\left({k}^{2} \cdot \left(t \cdot {\sin k}^{2}\right)\right)}\right) \]

      3. associate-*r* N/A

         \[\leadsto \mathsf{/.f64}\left(\left(\left(2 \cdot {\ell}^{2}\right) \cdot \cos k\right), \left(\color{blue}{{k}^{2}} \cdot \left(t \cdot {\sin k}^{2}\right)\right)\right) \]

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

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\left(2 \cdot {\ell}^{2}\right), \cos k\right), \left(\color{blue}{{k}^{2}} \cdot \left(t \cdot {\sin k}^{2}\right)\right)\right) \]

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

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(2, \left({\ell}^{2}\right)\right), \cos k\right), \left({\color{blue}{k}}^{2} \cdot \left(t \cdot {\sin k}^{2}\right)\right)\right) \]

      6. unpow2 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(2, \left(\ell \cdot \ell\right)\right), \cos k\right), \left({k}^{2} \cdot \left(t \cdot {\sin k}^{2}\right)\right)\right) \]

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

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(2, \mathsf{*.f64}\left(\ell, \ell\right)\right), \cos k\right), \left({k}^{2} \cdot \left(t \cdot {\sin k}^{2}\right)\right)\right) \]

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

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(2, \mathsf{*.f64}\left(\ell, \ell\right)\right), \mathsf{cos.f64}\left(k\right)\right), \left({k}^{\color{blue}{2}} \cdot \left(t \cdot {\sin k}^{2}\right)\right)\right) \]

      9. unpow2 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(2, \mathsf{*.f64}\left(\ell, \ell\right)\right), \mathsf{cos.f64}\left(k\right)\right), \left(\left(k \cdot k\right) \cdot \left(\color{blue}{t} \cdot {\sin k}^{2}\right)\right)\right) \]

      10. associate-*l* N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(2, \mathsf{*.f64}\left(\ell, \ell\right)\right), \mathsf{cos.f64}\left(k\right)\right), \left(k \cdot \color{blue}{\left(k \cdot \left(t \cdot {\sin k}^{2}\right)\right)}\right)\right) \]

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

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(2, \mathsf{*.f64}\left(\ell, \ell\right)\right), \mathsf{cos.f64}\left(k\right)\right), \mathsf{*.f64}\left(k, \color{blue}{\left(k \cdot \left(t \cdot {\sin k}^{2}\right)\right)}\right)\right) \]

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

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(2, \mathsf{*.f64}\left(\ell, \ell\right)\right), \mathsf{cos.f64}\left(k\right)\right), \mathsf{*.f64}\left(k, \mathsf{*.f64}\left(k, \color{blue}{\left(t \cdot {\sin k}^{2}\right)}\right)\right)\right) \]

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

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(2, \mathsf{*.f64}\left(\ell, \ell\right)\right), \mathsf{cos.f64}\left(k\right)\right), \mathsf{*.f64}\left(k, \mathsf{*.f64}\left(k, \mathsf{*.f64}\left(t, \color{blue}{\left({\sin k}^{2}\right)}\right)\right)\right)\right) \]

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

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(2, \mathsf{*.f64}\left(\ell, \ell\right)\right), \mathsf{cos.f64}\left(k\right)\right), \mathsf{*.f64}\left(k, \mathsf{*.f64}\left(k, \mathsf{*.f64}\left(t, \mathsf{pow.f64}\left(\sin k, \color{blue}{2}\right)\right)\right)\right)\right) \]

      15. sin-lowering-sin.f64 67.6%

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(2, \mathsf{*.f64}\left(\ell, \ell\right)\right), \mathsf{cos.f64}\left(k\right)\right), \mathsf{*.f64}\left(k, \mathsf{*.f64}\left(k, \mathsf{*.f64}\left(t, \mathsf{pow.f64}\left(\mathsf{sin.f64}\left(k\right), 2\right)\right)\right)\right)\right) \]

   5. Simplified 67.6%

      \[\leadsto \color{blue}{\frac{\left(2 \cdot \left(\ell \cdot \ell\right)\right) \cdot \cos k}{k \cdot \left(k \cdot \left(t \cdot {\sin k}^{2}\right)\right)}} \]
   6. Step-by-step derivation

      1. associate-/r* N/A

         \[\leadsto \frac{\frac{\left(2 \cdot \left(\ell \cdot \ell\right)\right) \cdot \cos k}{k}}{\color{blue}{k \cdot \left(t \cdot {\sin k}^{2}\right)}} \]

      2. associate-*r* N/A

         \[\leadsto \frac{\frac{\left(2 \cdot \left(\ell \cdot \ell\right)\right) \cdot \cos k}{k}}{\left(k \cdot t\right) \cdot \color{blue}{{\sin k}^{2}}} \]

      3. associate-/r* N/A

         \[\leadsto \frac{\frac{\frac{\left(2 \cdot \left(\ell \cdot \ell\right)\right) \cdot \cos k}{k}}{k \cdot t}}{\color{blue}{{\sin k}^{2}}} \]

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

         \[\leadsto \mathsf{/.f64}\left(\left(\frac{\frac{\left(2 \cdot \left(\ell \cdot \ell\right)\right) \cdot \cos k}{k}}{k \cdot t}\right), \color{blue}{\left({\sin k}^{2}\right)}\right) \]

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

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\left(\frac{\left(2 \cdot \left(\ell \cdot \ell\right)\right) \cdot \cos k}{k}\right), \left(k \cdot t\right)\right), \left({\color{blue}{\sin k}}^{2}\right)\right) \]

      6. associate-/l* N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\left(\left(2 \cdot \left(\ell \cdot \ell\right)\right) \cdot \frac{\cos k}{k}\right), \left(k \cdot t\right)\right), \left({\sin \color{blue}{k}}^{2}\right)\right) \]

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

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\left(2 \cdot \left(\ell \cdot \ell\right)\right), \left(\frac{\cos k}{k}\right)\right), \left(k \cdot t\right)\right), \left({\sin \color{blue}{k}}^{2}\right)\right) \]

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

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(2, \left(\ell \cdot \ell\right)\right), \left(\frac{\cos k}{k}\right)\right), \left(k \cdot t\right)\right), \left({\sin k}^{2}\right)\right) \]

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

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(2, \mathsf{*.f64}\left(\ell, \ell\right)\right), \left(\frac{\cos k}{k}\right)\right), \left(k \cdot t\right)\right), \left({\sin k}^{2}\right)\right) \]

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

          \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(2, \mathsf{*.f64}\left(\ell, \ell\right)\right), \mathsf{/.f64}\left(\cos k, k\right)\right), \left(k \cdot t\right)\right), \left({\sin k}^{2}\right)\right) \]

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

          \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(2, \mathsf{*.f64}\left(\ell, \ell\right)\right), \mathsf{/.f64}\left(\mathsf{cos.f64}\left(k\right), k\right)\right), \left(k \cdot t\right)\right), \left({\sin k}^{2}\right)\right) \]

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

          \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(2, \mathsf{*.f64}\left(\ell, \ell\right)\right), \mathsf{/.f64}\left(\mathsf{cos.f64}\left(k\right), k\right)\right), \mathsf{*.f64}\left(k, t\right)\right), \left({\sin k}^{2}\right)\right) \]

      13. unpow2 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(2, \mathsf{*.f64}\left(\ell, \ell\right)\right), \mathsf{/.f64}\left(\mathsf{cos.f64}\left(k\right), k\right)\right), \mathsf{*.f64}\left(k, t\right)\right), \left(\sin k \cdot \color{blue}{\sin k}\right)\right) \]

      14. sqr-sin-a N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(2, \mathsf{*.f64}\left(\ell, \ell\right)\right), \mathsf{/.f64}\left(\mathsf{cos.f64}\left(k\right), k\right)\right), \mathsf{*.f64}\left(k, t\right)\right), \left(\frac{1}{2} - \color{blue}{\frac{1}{2} \cdot \cos \left(2 \cdot k\right)}\right)\right) \]

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

          \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(2, \mathsf{*.f64}\left(\ell, \ell\right)\right), \mathsf{/.f64}\left(\mathsf{cos.f64}\left(k\right), k\right)\right), \mathsf{*.f64}\left(k, t\right)\right), \mathsf{\_.f64}\left(\frac{1}{2}, \color{blue}{\left(\frac{1}{2} \cdot \cos \left(2 \cdot k\right)\right)}\right)\right) \]

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

          \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(2, \mathsf{*.f64}\left(\ell, \ell\right)\right), \mathsf{/.f64}\left(\mathsf{cos.f64}\left(k\right), k\right)\right), \mathsf{*.f64}\left(k, t\right)\right), \mathsf{\_.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(\frac{1}{2}, \color{blue}{\cos \left(2 \cdot k\right)}\right)\right)\right) \]

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

          \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(2, \mathsf{*.f64}\left(\ell, \ell\right)\right), \mathsf{/.f64}\left(\mathsf{cos.f64}\left(k\right), k\right)\right), \mathsf{*.f64}\left(k, t\right)\right), \mathsf{\_.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(\frac{1}{2}, \mathsf{cos.f64}\left(\left(2 \cdot k\right)\right)\right)\right)\right) \]

      18. *-lowering-*.f64 71.6%

          \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(2, \mathsf{*.f64}\left(\ell, \ell\right)\right), \mathsf{/.f64}\left(\mathsf{cos.f64}\left(k\right), k\right)\right), \mathsf{*.f64}\left(k, t\right)\right), \mathsf{\_.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(\frac{1}{2}, \mathsf{cos.f64}\left(\mathsf{*.f64}\left(2, k\right)\right)\right)\right)\right) \]

   7. Applied egg-rr 71.6%

      \[\leadsto \color{blue}{\frac{\frac{\left(2 \cdot \left(\ell \cdot \ell\right)\right) \cdot \frac{\cos k}{k}}{k \cdot t}}{0.5 - 0.5 \cdot \cos \left(2 \cdot k\right)}} \]
   8. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\left(\frac{\cos k}{k} \cdot \left(2 \cdot \left(\ell \cdot \ell\right)\right)\right), \mathsf{*.f64}\left(k, t\right)\right), \mathsf{\_.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(\frac{1}{2}, \mathsf{cos.f64}\left(\mathsf{*.f64}\left(2, k\right)\right)\right)\right)\right) \]

      2. associate-*r* N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\left(\frac{\cos k}{k} \cdot \left(\left(2 \cdot \ell\right) \cdot \ell\right)\right), \mathsf{*.f64}\left(k, t\right)\right), \mathsf{\_.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(\frac{1}{2}, \mathsf{cos.f64}\left(\mathsf{*.f64}\left(2, k\right)\right)\right)\right)\right) \]

      3. associate-*r* N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\left(\left(\frac{\cos k}{k} \cdot \left(2 \cdot \ell\right)\right) \cdot \ell\right), \mathsf{*.f64}\left(k, t\right)\right), \mathsf{\_.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(\frac{1}{2}, \mathsf{cos.f64}\left(\mathsf{*.f64}\left(2, k\right)\right)\right)\right)\right) \]

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

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\left(\frac{\cos k}{k} \cdot \left(2 \cdot \ell\right)\right), \ell\right), \mathsf{*.f64}\left(k, t\right)\right), \mathsf{\_.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(\frac{1}{2}, \mathsf{cos.f64}\left(\mathsf{*.f64}\left(2, k\right)\right)\right)\right)\right) \]

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

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\left(\frac{\cos k}{k}\right), \left(2 \cdot \ell\right)\right), \ell\right), \mathsf{*.f64}\left(k, t\right)\right), \mathsf{\_.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(\frac{1}{2}, \mathsf{cos.f64}\left(\mathsf{*.f64}\left(2, k\right)\right)\right)\right)\right) \]

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

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(\cos k, k\right), \left(2 \cdot \ell\right)\right), \ell\right), \mathsf{*.f64}\left(k, t\right)\right), \mathsf{\_.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(\frac{1}{2}, \mathsf{cos.f64}\left(\mathsf{*.f64}\left(2, k\right)\right)\right)\right)\right) \]

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

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{cos.f64}\left(k\right), k\right), \left(2 \cdot \ell\right)\right), \ell\right), \mathsf{*.f64}\left(k, t\right)\right), \mathsf{\_.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(\frac{1}{2}, \mathsf{cos.f64}\left(\mathsf{*.f64}\left(2, k\right)\right)\right)\right)\right) \]

      8. *-lowering-*.f64 81.0%

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{cos.f64}\left(k\right), k\right), \mathsf{*.f64}\left(2, \ell\right)\right), \ell\right), \mathsf{*.f64}\left(k, t\right)\right), \mathsf{\_.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(\frac{1}{2}, \mathsf{cos.f64}\left(\mathsf{*.f64}\left(2, k\right)\right)\right)\right)\right) \]

   9. Applied egg-rr 81.0%

      \[\leadsto \frac{\frac{\color{blue}{\left(\frac{\cos k}{k} \cdot \left(2 \cdot \ell\right)\right) \cdot \ell}}{k \cdot t}}{0.5 - 0.5 \cdot \cos \left(2 \cdot k\right)} \]
   10. Step-by-step derivation

       1. times-frac N/A

          \[\leadsto \frac{\frac{\frac{\cos k}{k} \cdot \left(2 \cdot \ell\right)}{k} \cdot \frac{\ell}{t}}{\color{blue}{\frac{1}{2}} - \frac{1}{2} \cdot \cos \left(2 \cdot k\right)} \]

       2. cancel-sign-sub-inv N/A

          \[\leadsto \frac{\frac{\frac{\cos k}{k} \cdot \left(2 \cdot \ell\right)}{k} \cdot \frac{\ell}{t}}{\frac{1}{2} + \color{blue}{\left(\mathsf{neg}\left(\frac{1}{2}\right)\right) \cdot \cos \left(2 \cdot k\right)}} \]

       3. metadata-eval N/A

          \[\leadsto \frac{\frac{\frac{\cos k}{k} \cdot \left(2 \cdot \ell\right)}{k} \cdot \frac{\ell}{t}}{\frac{1}{2} + \frac{-1}{2} \cdot \cos \color{blue}{\left(2 \cdot k\right)}} \]

       4. *-commutative N/A

          \[\leadsto \frac{\frac{\frac{\cos k}{k} \cdot \left(2 \cdot \ell\right)}{k} \cdot \frac{\ell}{t}}{\frac{1}{2} + \cos \left(2 \cdot k\right) \cdot \color{blue}{\frac{-1}{2}}} \]

       5. associate-/l* N/A

          \[\leadsto \frac{\frac{\cos k}{k} \cdot \left(2 \cdot \ell\right)}{k} \cdot \color{blue}{\frac{\frac{\ell}{t}}{\frac{1}{2} + \cos \left(2 \cdot k\right) \cdot \frac{-1}{2}}} \]

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

          \[\leadsto \mathsf{*.f64}\left(\left(\frac{\frac{\cos k}{k} \cdot \left(2 \cdot \ell\right)}{k}\right), \color{blue}{\left(\frac{\frac{\ell}{t}}{\frac{1}{2} + \cos \left(2 \cdot k\right) \cdot \frac{-1}{2}}\right)}\right) \]

       7. associate-/l* N/A

          \[\leadsto \mathsf{*.f64}\left(\left(\frac{\cos k}{k} \cdot \frac{2 \cdot \ell}{k}\right), \left(\frac{\color{blue}{\frac{\ell}{t}}}{\frac{1}{2} + \cos \left(2 \cdot k\right) \cdot \frac{-1}{2}}\right)\right) \]

       8. associate-*l/ N/A

          \[\leadsto \mathsf{*.f64}\left(\left(\frac{\cos k}{k} \cdot \left(\frac{2}{k} \cdot \ell\right)\right), \left(\frac{\frac{\ell}{\color{blue}{t}}}{\frac{1}{2} + \cos \left(2 \cdot k\right) \cdot \frac{-1}{2}}\right)\right) \]

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

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\left(\frac{\cos k}{k}\right), \left(\frac{2}{k} \cdot \ell\right)\right), \left(\frac{\color{blue}{\frac{\ell}{t}}}{\frac{1}{2} + \cos \left(2 \cdot k\right) \cdot \frac{-1}{2}}\right)\right) \]

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

           \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(\cos k, k\right), \left(\frac{2}{k} \cdot \ell\right)\right), \left(\frac{\frac{\color{blue}{\ell}}{t}}{\frac{1}{2} + \cos \left(2 \cdot k\right) \cdot \frac{-1}{2}}\right)\right) \]

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

           \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{cos.f64}\left(k\right), k\right), \left(\frac{2}{k} \cdot \ell\right)\right), \left(\frac{\frac{\ell}{t}}{\frac{1}{2} + \cos \left(2 \cdot k\right) \cdot \frac{-1}{2}}\right)\right) \]

       12. associate-*l/ N/A

           \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{cos.f64}\left(k\right), k\right), \left(\frac{2 \cdot \ell}{k}\right)\right), \left(\frac{\frac{\ell}{\color{blue}{t}}}{\frac{1}{2} + \cos \left(2 \cdot k\right) \cdot \frac{-1}{2}}\right)\right) \]

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

           \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{cos.f64}\left(k\right), k\right), \mathsf{/.f64}\left(\left(2 \cdot \ell\right), k\right)\right), \left(\frac{\frac{\ell}{\color{blue}{t}}}{\frac{1}{2} + \cos \left(2 \cdot k\right) \cdot \frac{-1}{2}}\right)\right) \]

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

           \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{cos.f64}\left(k\right), k\right), \mathsf{/.f64}\left(\mathsf{*.f64}\left(2, \ell\right), k\right)\right), \left(\frac{\frac{\ell}{t}}{\frac{1}{2} + \cos \left(2 \cdot k\right) \cdot \frac{-1}{2}}\right)\right) \]

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

           \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{cos.f64}\left(k\right), k\right), \mathsf{/.f64}\left(\mathsf{*.f64}\left(2, \ell\right), k\right)\right), \mathsf{/.f64}\left(\left(\frac{\ell}{t}\right), \color{blue}{\left(\frac{1}{2} + \cos \left(2 \cdot k\right) \cdot \frac{-1}{2}\right)}\right)\right) \]

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

           \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{cos.f64}\left(k\right), k\right), \mathsf{/.f64}\left(\mathsf{*.f64}\left(2, \ell\right), k\right)\right), \mathsf{/.f64}\left(\mathsf{/.f64}\left(\ell, t\right), \left(\color{blue}{\frac{1}{2}} + \cos \left(2 \cdot k\right) \cdot \frac{-1}{2}\right)\right)\right) \]

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

           \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{cos.f64}\left(k\right), k\right), \mathsf{/.f64}\left(\mathsf{*.f64}\left(2, \ell\right), k\right)\right), \mathsf{/.f64}\left(\mathsf{/.f64}\left(\ell, t\right), \mathsf{+.f64}\left(\frac{1}{2}, \color{blue}{\left(\cos \left(2 \cdot k\right) \cdot \frac{-1}{2}\right)}\right)\right)\right) \]

   11. Applied egg-rr 85.6%

       \[\leadsto \color{blue}{\left(\frac{\cos k}{k} \cdot \frac{2 \cdot \ell}{k}\right) \cdot \frac{\frac{\ell}{t}}{0.5 + \cos \left(k \cdot 2\right) \cdot -0.5}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 80.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;k \leq 8.5 \cdot 10^{-5}:\\ \;\;\;\;\ell \cdot \frac{\frac{\frac{2 \cdot \ell}{k \cdot \left(k \cdot t\right)}}{k}}{k}\\ \mathbf{else}:\\ \;\;\;\;\left(\frac{\cos k}{k} \cdot \frac{2 \cdot \ell}{k}\right) \cdot \frac{\frac{\ell}{t}}{0.5 + \cos \left(k \cdot 2\right) \cdot -0.5}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 88.0% accurate, 1.9× speedup

Math

\[\begin{array}{l} k_m = \left|k\right| \\ \begin{array}{l} t_1 := k\_m \cdot \left(k\_m \cdot t\right)\\ \mathbf{if}\;k\_m \leq 8.5 \cdot 10^{-5}:\\ \;\;\;\;\ell \cdot \frac{\frac{\frac{2 \cdot \ell}{t\_1}}{k\_m}}{k\_m}\\ \mathbf{else}:\\ \;\;\;\;\left(2 \cdot \ell\right) \cdot \left(\ell \cdot \frac{\cos k\_m}{\left(0.5 - 0.5 \cdot \cos \left(k\_m \cdot 2\right)\right) \cdot t\_1}\right)\\ \end{array} \end{array} \]

FPCore

k_m = (fabs.f64 k)
(FPCore (t l k_m)
 :precision binary64
 (let* ((t_1 (* k_m (* k_m t))))
   (if (<= k_m 8.5e-5)
     (* l (/ (/ (/ (* 2.0 l) t_1) k_m) k_m))
     (*
      (* 2.0 l)
      (* l (/ (cos k_m) (* (- 0.5 (* 0.5 (cos (* k_m 2.0)))) t_1)))))))

C

k_m = fabs(k);
double code(double t, double l, double k_m) {
	double t_1 = k_m * (k_m * t);
	double tmp;
	if (k_m <= 8.5e-5) {
		tmp = l * ((((2.0 * l) / t_1) / k_m) / k_m);
	} else {
		tmp = (2.0 * l) * (l * (cos(k_m) / ((0.5 - (0.5 * cos((k_m * 2.0)))) * t_1)));
	}
	return tmp;
}

Fortran

k_m = abs(k)
real(8) function code(t, l, k_m)
    real(8), intent (in) :: t
    real(8), intent (in) :: l
    real(8), intent (in) :: k_m
    real(8) :: t_1
    real(8) :: tmp
    t_1 = k_m * (k_m * t)
    if (k_m <= 8.5d-5) then
        tmp = l * ((((2.0d0 * l) / t_1) / k_m) / k_m)
    else
        tmp = (2.0d0 * l) * (l * (cos(k_m) / ((0.5d0 - (0.5d0 * cos((k_m * 2.0d0)))) * t_1)))
    end if
    code = tmp
end function

Java

k_m = Math.abs(k);
public static double code(double t, double l, double k_m) {
	double t_1 = k_m * (k_m * t);
	double tmp;
	if (k_m <= 8.5e-5) {
		tmp = l * ((((2.0 * l) / t_1) / k_m) / k_m);
	} else {
		tmp = (2.0 * l) * (l * (Math.cos(k_m) / ((0.5 - (0.5 * Math.cos((k_m * 2.0)))) * t_1)));
	}
	return tmp;
}

Python

k_m = math.fabs(k)
def code(t, l, k_m):
	t_1 = k_m * (k_m * t)
	tmp = 0
	if k_m <= 8.5e-5:
		tmp = l * ((((2.0 * l) / t_1) / k_m) / k_m)
	else:
		tmp = (2.0 * l) * (l * (math.cos(k_m) / ((0.5 - (0.5 * math.cos((k_m * 2.0)))) * t_1)))
	return tmp

Julia

k_m = abs(k)
function code(t, l, k_m)
	t_1 = Float64(k_m * Float64(k_m * t))
	tmp = 0.0
	if (k_m <= 8.5e-5)
		tmp = Float64(l * Float64(Float64(Float64(Float64(2.0 * l) / t_1) / k_m) / k_m));
	else
		tmp = Float64(Float64(2.0 * l) * Float64(l * Float64(cos(k_m) / Float64(Float64(0.5 - Float64(0.5 * cos(Float64(k_m * 2.0)))) * t_1))));
	end
	return tmp
end

MATLAB

k_m = abs(k);
function tmp_2 = code(t, l, k_m)
	t_1 = k_m * (k_m * t);
	tmp = 0.0;
	if (k_m <= 8.5e-5)
		tmp = l * ((((2.0 * l) / t_1) / k_m) / k_m);
	else
		tmp = (2.0 * l) * (l * (cos(k_m) / ((0.5 - (0.5 * cos((k_m * 2.0)))) * t_1)));
	end
	tmp_2 = tmp;
end

Wolfram

k_m = N[Abs[k], $MachinePrecision]
code[t_, l_, k$95$m_] := Block[{t$95$1 = N[(k$95$m * N[(k$95$m * t), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[k$95$m, 8.5e-5], N[(l * N[(N[(N[(N[(2.0 * l), $MachinePrecision] / t$95$1), $MachinePrecision] / k$95$m), $MachinePrecision] / k$95$m), $MachinePrecision]), $MachinePrecision], N[(N[(2.0 * l), $MachinePrecision] * N[(l * N[(N[Cos[k$95$m], $MachinePrecision] / N[(N[(0.5 - N[(0.5 * N[Cos[N[(k$95$m * 2.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * t$95$1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if k < 8.500000000000001e-5

   1. Initial program 35.0%

      \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) - 1\right)} \]
   2. Add Preprocessing

   3. Taylor expanded in k around 0

      \[\leadsto \mathsf{/.f64}\left(2, \color{blue}{\left(\frac{{k}^{4} \cdot t}{{\ell}^{2}}\right)}\right) \]
   4. Step-by-step derivation

      1. associate-/l* N/A

         \[\leadsto \mathsf{/.f64}\left(2, \left({k}^{4} \cdot \color{blue}{\frac{t}{{\ell}^{2}}}\right)\right) \]

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

         \[\leadsto \mathsf{/.f64}\left(2, \mathsf{*.f64}\left(\left({k}^{4}\right), \color{blue}{\left(\frac{t}{{\ell}^{2}}\right)}\right)\right) \]

      3. metadata-eval N/A

         \[\leadsto \mathsf{/.f64}\left(2, \mathsf{*.f64}\left(\left({k}^{\left(2 \cdot 2\right)}\right), \left(\frac{t}{{\ell}^{2}}\right)\right)\right) \]

      4. pow-sqr N/A

         \[\leadsto \mathsf{/.f64}\left(2, \mathsf{*.f64}\left(\left({k}^{2} \cdot {k}^{2}\right), \left(\frac{\color{blue}{t}}{{\ell}^{2}}\right)\right)\right) \]

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

         \[\leadsto \mathsf{/.f64}\left(2, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\left({k}^{2}\right), \left({k}^{2}\right)\right), \left(\frac{\color{blue}{t}}{{\ell}^{2}}\right)\right)\right) \]

      6. unpow2 N/A

         \[\leadsto \mathsf{/.f64}\left(2, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\left(k \cdot k\right), \left({k}^{2}\right)\right), \left(\frac{t}{{\ell}^{2}}\right)\right)\right) \]

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

         \[\leadsto \mathsf{/.f64}\left(2, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(k, k\right), \left({k}^{2}\right)\right), \left(\frac{t}{{\ell}^{2}}\right)\right)\right) \]

      8. unpow2 N/A

         \[\leadsto \mathsf{/.f64}\left(2, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(k, k\right), \left(k \cdot k\right)\right), \left(\frac{t}{{\ell}^{2}}\right)\right)\right) \]

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

         \[\leadsto \mathsf{/.f64}\left(2, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(k, k\right), \mathsf{*.f64}\left(k, k\right)\right), \left(\frac{t}{{\ell}^{2}}\right)\right)\right) \]

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

          \[\leadsto \mathsf{/.f64}\left(2, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(k, k\right), \mathsf{*.f64}\left(k, k\right)\right), \mathsf{/.f64}\left(t, \color{blue}{\left({\ell}^{2}\right)}\right)\right)\right) \]

      11. unpow2 N/A

          \[\leadsto \mathsf{/.f64}\left(2, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(k, k\right), \mathsf{*.f64}\left(k, k\right)\right), \mathsf{/.f64}\left(t, \left(\ell \cdot \color{blue}{\ell}\right)\right)\right)\right) \]

      12. *-lowering-*.f64 58.1%

          \[\leadsto \mathsf{/.f64}\left(2, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(k, k\right), \mathsf{*.f64}\left(k, k\right)\right), \mathsf{/.f64}\left(t, \mathsf{*.f64}\left(\ell, \color{blue}{\ell}\right)\right)\right)\right) \]

   5. Simplified 58.1%

      \[\leadsto \frac{2}{\color{blue}{\left(\left(k \cdot k\right) \cdot \left(k \cdot k\right)\right) \cdot \frac{t}{\ell \cdot \ell}}} \]
   6. Step-by-step derivation

      1. associate-/r* N/A

         \[\leadsto \frac{\frac{2}{\left(k \cdot k\right) \cdot \left(k \cdot k\right)}}{\color{blue}{\frac{t}{\ell \cdot \ell}}} \]

      2. associate-/r* N/A

         \[\leadsto \frac{\frac{2}{\left(k \cdot k\right) \cdot \left(k \cdot k\right)}}{\frac{\frac{t}{\ell}}{\color{blue}{\ell}}} \]

      3. associate-/r/ N/A

         \[\leadsto \frac{\frac{2}{\left(k \cdot k\right) \cdot \left(k \cdot k\right)}}{\frac{t}{\ell}} \cdot \color{blue}{\ell} \]

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

         \[\leadsto \mathsf{*.f64}\left(\left(\frac{\frac{2}{\left(k \cdot k\right) \cdot \left(k \cdot k\right)}}{\frac{t}{\ell}}\right), \color{blue}{\ell}\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\left(\frac{2}{\left(k \cdot k\right) \cdot \left(k \cdot k\right)}\right), \left(\frac{t}{\ell}\right)\right), \ell\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(2, \left(\left(k \cdot k\right) \cdot \left(k \cdot k\right)\right)\right), \left(\frac{t}{\ell}\right)\right), \ell\right) \]

      7. associate-*l* N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(2, \left(k \cdot \left(k \cdot \left(k \cdot k\right)\right)\right)\right), \left(\frac{t}{\ell}\right)\right), \ell\right) \]

      8. cube-unmult N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(2, \left(k \cdot {k}^{3}\right)\right), \left(\frac{t}{\ell}\right)\right), \ell\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(2, \mathsf{*.f64}\left(k, \left({k}^{3}\right)\right)\right), \left(\frac{t}{\ell}\right)\right), \ell\right) \]

      10. cube-unmult N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(2, \mathsf{*.f64}\left(k, \left(k \cdot \left(k \cdot k\right)\right)\right)\right), \left(\frac{t}{\ell}\right)\right), \ell\right) \]

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

          \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(2, \mathsf{*.f64}\left(k, \mathsf{*.f64}\left(k, \left(k \cdot k\right)\right)\right)\right), \left(\frac{t}{\ell}\right)\right), \ell\right) \]

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

          \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(2, \mathsf{*.f64}\left(k, \mathsf{*.f64}\left(k, \mathsf{*.f64}\left(k, k\right)\right)\right)\right), \left(\frac{t}{\ell}\right)\right), \ell\right) \]

      13. /-lowering-/.f64 69.3%

          \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(2, \mathsf{*.f64}\left(k, \mathsf{*.f64}\left(k, \mathsf{*.f64}\left(k, k\right)\right)\right)\right), \mathsf{/.f64}\left(t, \ell\right)\right), \ell\right) \]

   7. Applied egg-rr 69.3%

      \[\leadsto \color{blue}{\frac{\frac{2}{k \cdot \left(k \cdot \left(k \cdot k\right)\right)}}{\frac{t}{\ell}} \cdot \ell} \]
   8. Step-by-step derivation

      1. div-inv N/A

         \[\leadsto \mathsf{*.f64}\left(\left(\frac{2}{k \cdot \left(k \cdot \left(k \cdot k\right)\right)} \cdot \frac{1}{\frac{t}{\ell}}\right), \ell\right) \]

      2. associate-/r* N/A

         \[\leadsto \mathsf{*.f64}\left(\left(\frac{\frac{2}{k}}{k \cdot \left(k \cdot k\right)} \cdot \frac{1}{\frac{t}{\ell}}\right), \ell\right) \]

      3. clear-num N/A

         \[\leadsto \mathsf{*.f64}\left(\left(\frac{\frac{2}{k}}{k \cdot \left(k \cdot k\right)} \cdot \frac{\ell}{t}\right), \ell\right) \]

      4. associate-*l/ N/A

         \[\leadsto \mathsf{*.f64}\left(\left(\frac{\frac{2}{k} \cdot \frac{\ell}{t}}{k \cdot \left(k \cdot k\right)}\right), \ell\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\left(\frac{2}{k} \cdot \frac{\ell}{t}\right), \left(k \cdot \left(k \cdot k\right)\right)\right), \ell\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\left(\frac{2}{k}\right), \left(\frac{\ell}{t}\right)\right), \left(k \cdot \left(k \cdot k\right)\right)\right), \ell\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(2, k\right), \left(\frac{\ell}{t}\right)\right), \left(k \cdot \left(k \cdot k\right)\right)\right), \ell\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(2, k\right), \mathsf{/.f64}\left(\ell, t\right)\right), \left(k \cdot \left(k \cdot k\right)\right)\right), \ell\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(2, k\right), \mathsf{/.f64}\left(\ell, t\right)\right), \mathsf{*.f64}\left(k, \left(k \cdot k\right)\right)\right), \ell\right) \]

      10. *-lowering-*.f64 72.8%

          \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(2, k\right), \mathsf{/.f64}\left(\ell, t\right)\right), \mathsf{*.f64}\left(k, \mathsf{*.f64}\left(k, k\right)\right)\right), \ell\right) \]

   9. Applied egg-rr 72.8%

      \[\leadsto \color{blue}{\frac{\frac{2}{k} \cdot \frac{\ell}{t}}{k \cdot \left(k \cdot k\right)}} \cdot \ell \]
   10. Step-by-step derivation

       1. associate-/r* N/A

          \[\leadsto \mathsf{*.f64}\left(\left(\frac{\frac{\frac{2}{k} \cdot \frac{\ell}{t}}{k}}{k \cdot k}\right), \ell\right) \]

       2. associate-/r* N/A

          \[\leadsto \mathsf{*.f64}\left(\left(\frac{\frac{\frac{\frac{2}{k} \cdot \frac{\ell}{t}}{k}}{k}}{k}\right), \ell\right) \]

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

          \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\left(\frac{\frac{\frac{2}{k} \cdot \frac{\ell}{t}}{k}}{k}\right), k\right), \ell\right) \]

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

          \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(\left(\frac{\frac{2}{k} \cdot \frac{\ell}{t}}{k}\right), k\right), k\right), \ell\right) \]

       5. frac-times N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(\left(\frac{\frac{2 \cdot \ell}{k \cdot t}}{k}\right), k\right), k\right), \ell\right) \]

       6. associate-/l/ N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(\left(\frac{2 \cdot \ell}{k \cdot \left(k \cdot t\right)}\right), k\right), k\right), \ell\right) \]

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

          \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(\left(2 \cdot \ell\right), \left(k \cdot \left(k \cdot t\right)\right)\right), k\right), k\right), \ell\right) \]

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

          \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(2, \ell\right), \left(k \cdot \left(k \cdot t\right)\right)\right), k\right), k\right), \ell\right) \]

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

          \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(2, \ell\right), \mathsf{*.f64}\left(k, \left(k \cdot t\right)\right)\right), k\right), k\right), \ell\right) \]

       10. *-lowering-*.f64 78.5%

           \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(2, \ell\right), \mathsf{*.f64}\left(k, \mathsf{*.f64}\left(k, t\right)\right)\right), k\right), k\right), \ell\right) \]

   11. Applied egg-rr 78.5%

       \[\leadsto \color{blue}{\frac{\frac{\frac{2 \cdot \ell}{k \cdot \left(k \cdot t\right)}}{k}}{k}} \cdot \ell \]

   if 8.500000000000001e-5 < k

   1. Initial program 21.4%

      \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) - 1\right)} \]
   2. Add Preprocessing

   3. Taylor expanded in t around 0

      \[\leadsto \color{blue}{2 \cdot \frac{{\ell}^{2} \cdot \cos k}{{k}^{2} \cdot \left(t \cdot {\sin k}^{2}\right)}} \]
   4. Step-by-step derivation

      1. associate-*r/ N/A

         \[\leadsto \frac{2 \cdot \left({\ell}^{2} \cdot \cos k\right)}{\color{blue}{{k}^{2} \cdot \left(t \cdot {\sin k}^{2}\right)}} \]

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

         \[\leadsto \mathsf{/.f64}\left(\left(2 \cdot \left({\ell}^{2} \cdot \cos k\right)\right), \color{blue}{\left({k}^{2} \cdot \left(t \cdot {\sin k}^{2}\right)\right)}\right) \]

      3. associate-*r* N/A

         \[\leadsto \mathsf{/.f64}\left(\left(\left(2 \cdot {\ell}^{2}\right) \cdot \cos k\right), \left(\color{blue}{{k}^{2}} \cdot \left(t \cdot {\sin k}^{2}\right)\right)\right) \]

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

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\left(2 \cdot {\ell}^{2}\right), \cos k\right), \left(\color{blue}{{k}^{2}} \cdot \left(t \cdot {\sin k}^{2}\right)\right)\right) \]

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

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(2, \left({\ell}^{2}\right)\right), \cos k\right), \left({\color{blue}{k}}^{2} \cdot \left(t \cdot {\sin k}^{2}\right)\right)\right) \]

      6. unpow2 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(2, \left(\ell \cdot \ell\right)\right), \cos k\right), \left({k}^{2} \cdot \left(t \cdot {\sin k}^{2}\right)\right)\right) \]

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

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(2, \mathsf{*.f64}\left(\ell, \ell\right)\right), \cos k\right), \left({k}^{2} \cdot \left(t \cdot {\sin k}^{2}\right)\right)\right) \]

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

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(2, \mathsf{*.f64}\left(\ell, \ell\right)\right), \mathsf{cos.f64}\left(k\right)\right), \left({k}^{\color{blue}{2}} \cdot \left(t \cdot {\sin k}^{2}\right)\right)\right) \]

      9. unpow2 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(2, \mathsf{*.f64}\left(\ell, \ell\right)\right), \mathsf{cos.f64}\left(k\right)\right), \left(\left(k \cdot k\right) \cdot \left(\color{blue}{t} \cdot {\sin k}^{2}\right)\right)\right) \]

      10. associate-*l* N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(2, \mathsf{*.f64}\left(\ell, \ell\right)\right), \mathsf{cos.f64}\left(k\right)\right), \left(k \cdot \color{blue}{\left(k \cdot \left(t \cdot {\sin k}^{2}\right)\right)}\right)\right) \]

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

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(2, \mathsf{*.f64}\left(\ell, \ell\right)\right), \mathsf{cos.f64}\left(k\right)\right), \mathsf{*.f64}\left(k, \color{blue}{\left(k \cdot \left(t \cdot {\sin k}^{2}\right)\right)}\right)\right) \]

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

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(2, \mathsf{*.f64}\left(\ell, \ell\right)\right), \mathsf{cos.f64}\left(k\right)\right), \mathsf{*.f64}\left(k, \mathsf{*.f64}\left(k, \color{blue}{\left(t \cdot {\sin k}^{2}\right)}\right)\right)\right) \]

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

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(2, \mathsf{*.f64}\left(\ell, \ell\right)\right), \mathsf{cos.f64}\left(k\right)\right), \mathsf{*.f64}\left(k, \mathsf{*.f64}\left(k, \mathsf{*.f64}\left(t, \color{blue}{\left({\sin k}^{2}\right)}\right)\right)\right)\right) \]

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

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(2, \mathsf{*.f64}\left(\ell, \ell\right)\right), \mathsf{cos.f64}\left(k\right)\right), \mathsf{*.f64}\left(k, \mathsf{*.f64}\left(k, \mathsf{*.f64}\left(t, \mathsf{pow.f64}\left(\sin k, \color{blue}{2}\right)\right)\right)\right)\right) \]

      15. sin-lowering-sin.f64 67.6%

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(2, \mathsf{*.f64}\left(\ell, \ell\right)\right), \mathsf{cos.f64}\left(k\right)\right), \mathsf{*.f64}\left(k, \mathsf{*.f64}\left(k, \mathsf{*.f64}\left(t, \mathsf{pow.f64}\left(\mathsf{sin.f64}\left(k\right), 2\right)\right)\right)\right)\right) \]

   5. Simplified 67.6%

      \[\leadsto \color{blue}{\frac{\left(2 \cdot \left(\ell \cdot \ell\right)\right) \cdot \cos k}{k \cdot \left(k \cdot \left(t \cdot {\sin k}^{2}\right)\right)}} \]
   6. Step-by-step derivation

      1. associate-/l* N/A

         \[\leadsto \left(2 \cdot \left(\ell \cdot \ell\right)\right) \cdot \color{blue}{\frac{\cos k}{k \cdot \left(k \cdot \left(t \cdot {\sin k}^{2}\right)\right)}} \]

      2. associate-*r* N/A

         \[\leadsto \left(\left(2 \cdot \ell\right) \cdot \ell\right) \cdot \frac{\color{blue}{\cos k}}{k \cdot \left(k \cdot \left(t \cdot {\sin k}^{2}\right)\right)} \]

      3. associate-*l* N/A

         \[\leadsto \left(2 \cdot \ell\right) \cdot \color{blue}{\left(\ell \cdot \frac{\cos k}{k \cdot \left(k \cdot \left(t \cdot {\sin k}^{2}\right)\right)}\right)} \]

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

         \[\leadsto \mathsf{*.f64}\left(\left(2 \cdot \ell\right), \color{blue}{\left(\ell \cdot \frac{\cos k}{k \cdot \left(k \cdot \left(t \cdot {\sin k}^{2}\right)\right)}\right)}\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(2, \ell\right), \left(\color{blue}{\ell} \cdot \frac{\cos k}{k \cdot \left(k \cdot \left(t \cdot {\sin k}^{2}\right)\right)}\right)\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(2, \ell\right), \mathsf{*.f64}\left(\ell, \color{blue}{\left(\frac{\cos k}{k \cdot \left(k \cdot \left(t \cdot {\sin k}^{2}\right)\right)}\right)}\right)\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(2, \ell\right), \mathsf{*.f64}\left(\ell, \mathsf{/.f64}\left(\cos k, \color{blue}{\left(k \cdot \left(k \cdot \left(t \cdot {\sin k}^{2}\right)\right)\right)}\right)\right)\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(2, \ell\right), \mathsf{*.f64}\left(\ell, \mathsf{/.f64}\left(\mathsf{cos.f64}\left(k\right), \left(\color{blue}{k} \cdot \left(k \cdot \left(t \cdot {\sin k}^{2}\right)\right)\right)\right)\right)\right) \]

      9. associate-*r* N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(2, \ell\right), \mathsf{*.f64}\left(\ell, \mathsf{/.f64}\left(\mathsf{cos.f64}\left(k\right), \left(\left(k \cdot k\right) \cdot \color{blue}{\left(t \cdot {\sin k}^{2}\right)}\right)\right)\right)\right) \]

      10. associate-*r* N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(2, \ell\right), \mathsf{*.f64}\left(\ell, \mathsf{/.f64}\left(\mathsf{cos.f64}\left(k\right), \left(\left(\left(k \cdot k\right) \cdot t\right) \cdot \color{blue}{{\sin k}^{2}}\right)\right)\right)\right) \]

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

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(2, \ell\right), \mathsf{*.f64}\left(\ell, \mathsf{/.f64}\left(\mathsf{cos.f64}\left(k\right), \mathsf{*.f64}\left(\left(\left(k \cdot k\right) \cdot t\right), \color{blue}{\left({\sin k}^{2}\right)}\right)\right)\right)\right) \]

      12. *-commutative N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(2, \ell\right), \mathsf{*.f64}\left(\ell, \mathsf{/.f64}\left(\mathsf{cos.f64}\left(k\right), \mathsf{*.f64}\left(\left(t \cdot \left(k \cdot k\right)\right), \left({\color{blue}{\sin k}}^{2}\right)\right)\right)\right)\right) \]

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

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(2, \ell\right), \mathsf{*.f64}\left(\ell, \mathsf{/.f64}\left(\mathsf{cos.f64}\left(k\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(t, \left(k \cdot k\right)\right), \left({\color{blue}{\sin k}}^{2}\right)\right)\right)\right)\right) \]

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

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(2, \ell\right), \mathsf{*.f64}\left(\ell, \mathsf{/.f64}\left(\mathsf{cos.f64}\left(k\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(t, \mathsf{*.f64}\left(k, k\right)\right), \left({\sin k}^{2}\right)\right)\right)\right)\right) \]

      15. unpow2 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(2, \ell\right), \mathsf{*.f64}\left(\ell, \mathsf{/.f64}\left(\mathsf{cos.f64}\left(k\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(t, \mathsf{*.f64}\left(k, k\right)\right), \left(\sin k \cdot \color{blue}{\sin k}\right)\right)\right)\right)\right) \]

      16. sqr-sin-a N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(2, \ell\right), \mathsf{*.f64}\left(\ell, \mathsf{/.f64}\left(\mathsf{cos.f64}\left(k\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(t, \mathsf{*.f64}\left(k, k\right)\right), \left(\frac{1}{2} - \color{blue}{\frac{1}{2} \cdot \cos \left(2 \cdot k\right)}\right)\right)\right)\right)\right) \]

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

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(2, \ell\right), \mathsf{*.f64}\left(\ell, \mathsf{/.f64}\left(\mathsf{cos.f64}\left(k\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(t, \mathsf{*.f64}\left(k, k\right)\right), \mathsf{\_.f64}\left(\frac{1}{2}, \color{blue}{\left(\frac{1}{2} \cdot \cos \left(2 \cdot k\right)\right)}\right)\right)\right)\right)\right) \]

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

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(2, \ell\right), \mathsf{*.f64}\left(\ell, \mathsf{/.f64}\left(\mathsf{cos.f64}\left(k\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(t, \mathsf{*.f64}\left(k, k\right)\right), \mathsf{\_.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(\frac{1}{2}, \color{blue}{\cos \left(2 \cdot k\right)}\right)\right)\right)\right)\right)\right) \]

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

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(2, \ell\right), \mathsf{*.f64}\left(\ell, \mathsf{/.f64}\left(\mathsf{cos.f64}\left(k\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(t, \mathsf{*.f64}\left(k, k\right)\right), \mathsf{\_.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(\frac{1}{2}, \mathsf{cos.f64}\left(\left(2 \cdot k\right)\right)\right)\right)\right)\right)\right)\right) \]

      20. *-lowering-*.f64 69.3%

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(2, \ell\right), \mathsf{*.f64}\left(\ell, \mathsf{/.f64}\left(\mathsf{cos.f64}\left(k\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(t, \mathsf{*.f64}\left(k, k\right)\right), \mathsf{\_.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(\frac{1}{2}, \mathsf{cos.f64}\left(\mathsf{*.f64}\left(2, k\right)\right)\right)\right)\right)\right)\right)\right) \]

   7. Applied egg-rr 69.3%

      \[\leadsto \color{blue}{\left(2 \cdot \ell\right) \cdot \left(\ell \cdot \frac{\cos k}{\left(t \cdot \left(k \cdot k\right)\right) \cdot \left(0.5 - 0.5 \cdot \cos \left(2 \cdot k\right)\right)}\right)} \]
   8. Step-by-step derivation

      1. associate-*r* N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(2, \ell\right), \mathsf{*.f64}\left(\ell, \mathsf{/.f64}\left(\mathsf{cos.f64}\left(k\right), \mathsf{*.f64}\left(\left(\left(t \cdot k\right) \cdot k\right), \mathsf{\_.f64}\left(\color{blue}{\frac{1}{2}}, \mathsf{*.f64}\left(\frac{1}{2}, \mathsf{cos.f64}\left(\mathsf{*.f64}\left(2, k\right)\right)\right)\right)\right)\right)\right)\right) \]

      2. *-commutative N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(2, \ell\right), \mathsf{*.f64}\left(\ell, \mathsf{/.f64}\left(\mathsf{cos.f64}\left(k\right), \mathsf{*.f64}\left(\left(\left(k \cdot t\right) \cdot k\right), \mathsf{\_.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(\frac{1}{2}, \mathsf{cos.f64}\left(\mathsf{*.f64}\left(2, k\right)\right)\right)\right)\right)\right)\right)\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(2, \ell\right), \mathsf{*.f64}\left(\ell, \mathsf{/.f64}\left(\mathsf{cos.f64}\left(k\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(\left(k \cdot t\right), k\right), \mathsf{\_.f64}\left(\color{blue}{\frac{1}{2}}, \mathsf{*.f64}\left(\frac{1}{2}, \mathsf{cos.f64}\left(\mathsf{*.f64}\left(2, k\right)\right)\right)\right)\right)\right)\right)\right) \]

      4. *-lowering-*.f64 76.9%

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(2, \ell\right), \mathsf{*.f64}\left(\ell, \mathsf{/.f64}\left(\mathsf{cos.f64}\left(k\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(k, t\right), k\right), \mathsf{\_.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(\frac{1}{2}, \mathsf{cos.f64}\left(\mathsf{*.f64}\left(2, k\right)\right)\right)\right)\right)\right)\right)\right) \]

   9. Applied egg-rr 76.9%

      \[\leadsto \left(2 \cdot \ell\right) \cdot \left(\ell \cdot \frac{\cos k}{\color{blue}{\left(\left(k \cdot t\right) \cdot k\right)} \cdot \left(0.5 - 0.5 \cdot \cos \left(2 \cdot k\right)\right)}\right) \]
3. Recombined 2 regimes into one program.

4. Final simplification 78.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;k \leq 8.5 \cdot 10^{-5}:\\ \;\;\;\;\ell \cdot \frac{\frac{\frac{2 \cdot \ell}{k \cdot \left(k \cdot t\right)}}{k}}{k}\\ \mathbf{else}:\\ \;\;\;\;\left(2 \cdot \ell\right) \cdot \left(\ell \cdot \frac{\cos k}{\left(0.5 - 0.5 \cdot \cos \left(k \cdot 2\right)\right) \cdot \left(k \cdot \left(k \cdot t\right)\right)}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 84.7% accurate, 1.9× speedup

Math

\[\begin{array}{l} k_m = \left|k\right| \\ \begin{array}{l} \mathbf{if}\;k\_m \leq 8.5 \cdot 10^{-5}:\\ \;\;\;\;\ell \cdot \frac{\frac{\frac{2 \cdot \ell}{k\_m \cdot \left(k\_m \cdot t\right)}}{k\_m}}{k\_m}\\ \mathbf{else}:\\ \;\;\;\;\left(2 \cdot \ell\right) \cdot \left(\ell \cdot \frac{\cos k\_m}{\left(0.5 - 0.5 \cdot \cos \left(k\_m \cdot 2\right)\right) \cdot \left(t \cdot \left(k\_m \cdot k\_m\right)\right)}\right)\\ \end{array} \end{array} \]

FPCore

k_m = (fabs.f64 k)
(FPCore (t l k_m)
 :precision binary64
 (if (<= k_m 8.5e-5)
   (* l (/ (/ (/ (* 2.0 l) (* k_m (* k_m t))) k_m) k_m))
   (*
    (* 2.0 l)
    (*
     l
     (/ (cos k_m) (* (- 0.5 (* 0.5 (cos (* k_m 2.0)))) (* t (* k_m k_m))))))))

C

k_m = fabs(k);
double code(double t, double l, double k_m) {
	double tmp;
	if (k_m <= 8.5e-5) {
		tmp = l * ((((2.0 * l) / (k_m * (k_m * t))) / k_m) / k_m);
	} else {
		tmp = (2.0 * l) * (l * (cos(k_m) / ((0.5 - (0.5 * cos((k_m * 2.0)))) * (t * (k_m * k_m)))));
	}
	return tmp;
}

Fortran

k_m = abs(k)
real(8) function code(t, l, k_m)
    real(8), intent (in) :: t
    real(8), intent (in) :: l
    real(8), intent (in) :: k_m
    real(8) :: tmp
    if (k_m <= 8.5d-5) then
        tmp = l * ((((2.0d0 * l) / (k_m * (k_m * t))) / k_m) / k_m)
    else
        tmp = (2.0d0 * l) * (l * (cos(k_m) / ((0.5d0 - (0.5d0 * cos((k_m * 2.0d0)))) * (t * (k_m * k_m)))))
    end if
    code = tmp
end function

Java

k_m = Math.abs(k);
public static double code(double t, double l, double k_m) {
	double tmp;
	if (k_m <= 8.5e-5) {
		tmp = l * ((((2.0 * l) / (k_m * (k_m * t))) / k_m) / k_m);
	} else {
		tmp = (2.0 * l) * (l * (Math.cos(k_m) / ((0.5 - (0.5 * Math.cos((k_m * 2.0)))) * (t * (k_m * k_m)))));
	}
	return tmp;
}

Python

k_m = math.fabs(k)
def code(t, l, k_m):
	tmp = 0
	if k_m <= 8.5e-5:
		tmp = l * ((((2.0 * l) / (k_m * (k_m * t))) / k_m) / k_m)
	else:
		tmp = (2.0 * l) * (l * (math.cos(k_m) / ((0.5 - (0.5 * math.cos((k_m * 2.0)))) * (t * (k_m * k_m)))))
	return tmp

Julia

k_m = abs(k)
function code(t, l, k_m)
	tmp = 0.0
	if (k_m <= 8.5e-5)
		tmp = Float64(l * Float64(Float64(Float64(Float64(2.0 * l) / Float64(k_m * Float64(k_m * t))) / k_m) / k_m));
	else
		tmp = Float64(Float64(2.0 * l) * Float64(l * Float64(cos(k_m) / Float64(Float64(0.5 - Float64(0.5 * cos(Float64(k_m * 2.0)))) * Float64(t * Float64(k_m * k_m))))));
	end
	return tmp
end

MATLAB

k_m = abs(k);
function tmp_2 = code(t, l, k_m)
	tmp = 0.0;
	if (k_m <= 8.5e-5)
		tmp = l * ((((2.0 * l) / (k_m * (k_m * t))) / k_m) / k_m);
	else
		tmp = (2.0 * l) * (l * (cos(k_m) / ((0.5 - (0.5 * cos((k_m * 2.0)))) * (t * (k_m * k_m)))));
	end
	tmp_2 = tmp;
end

Wolfram

k_m = N[Abs[k], $MachinePrecision]
code[t_, l_, k$95$m_] := If[LessEqual[k$95$m, 8.5e-5], N[(l * N[(N[(N[(N[(2.0 * l), $MachinePrecision] / N[(k$95$m * N[(k$95$m * t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / k$95$m), $MachinePrecision] / k$95$m), $MachinePrecision]), $MachinePrecision], N[(N[(2.0 * l), $MachinePrecision] * N[(l * N[(N[Cos[k$95$m], $MachinePrecision] / N[(N[(0.5 - N[(0.5 * N[Cos[N[(k$95$m * 2.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(t * N[(k$95$m * k$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if k < 8.500000000000001e-5

   1. Initial program 35.0%

      \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) - 1\right)} \]
   2. Add Preprocessing

   3. Taylor expanded in k around 0

      \[\leadsto \mathsf{/.f64}\left(2, \color{blue}{\left(\frac{{k}^{4} \cdot t}{{\ell}^{2}}\right)}\right) \]
   4. Step-by-step derivation

      1. associate-/l* N/A

         \[\leadsto \mathsf{/.f64}\left(2, \left({k}^{4} \cdot \color{blue}{\frac{t}{{\ell}^{2}}}\right)\right) \]

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

         \[\leadsto \mathsf{/.f64}\left(2, \mathsf{*.f64}\left(\left({k}^{4}\right), \color{blue}{\left(\frac{t}{{\ell}^{2}}\right)}\right)\right) \]

      3. metadata-eval N/A

         \[\leadsto \mathsf{/.f64}\left(2, \mathsf{*.f64}\left(\left({k}^{\left(2 \cdot 2\right)}\right), \left(\frac{t}{{\ell}^{2}}\right)\right)\right) \]

      4. pow-sqr N/A

         \[\leadsto \mathsf{/.f64}\left(2, \mathsf{*.f64}\left(\left({k}^{2} \cdot {k}^{2}\right), \left(\frac{\color{blue}{t}}{{\ell}^{2}}\right)\right)\right) \]

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

         \[\leadsto \mathsf{/.f64}\left(2, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\left({k}^{2}\right), \left({k}^{2}\right)\right), \left(\frac{\color{blue}{t}}{{\ell}^{2}}\right)\right)\right) \]

      6. unpow2 N/A

         \[\leadsto \mathsf{/.f64}\left(2, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\left(k \cdot k\right), \left({k}^{2}\right)\right), \left(\frac{t}{{\ell}^{2}}\right)\right)\right) \]

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

         \[\leadsto \mathsf{/.f64}\left(2, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(k, k\right), \left({k}^{2}\right)\right), \left(\frac{t}{{\ell}^{2}}\right)\right)\right) \]

      8. unpow2 N/A

         \[\leadsto \mathsf{/.f64}\left(2, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(k, k\right), \left(k \cdot k\right)\right), \left(\frac{t}{{\ell}^{2}}\right)\right)\right) \]

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

         \[\leadsto \mathsf{/.f64}\left(2, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(k, k\right), \mathsf{*.f64}\left(k, k\right)\right), \left(\frac{t}{{\ell}^{2}}\right)\right)\right) \]

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

          \[\leadsto \mathsf{/.f64}\left(2, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(k, k\right), \mathsf{*.f64}\left(k, k\right)\right), \mathsf{/.f64}\left(t, \color{blue}{\left({\ell}^{2}\right)}\right)\right)\right) \]

      11. unpow2 N/A

          \[\leadsto \mathsf{/.f64}\left(2, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(k, k\right), \mathsf{*.f64}\left(k, k\right)\right), \mathsf{/.f64}\left(t, \left(\ell \cdot \color{blue}{\ell}\right)\right)\right)\right) \]

      12. *-lowering-*.f64 58.1%

          \[\leadsto \mathsf{/.f64}\left(2, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(k, k\right), \mathsf{*.f64}\left(k, k\right)\right), \mathsf{/.f64}\left(t, \mathsf{*.f64}\left(\ell, \color{blue}{\ell}\right)\right)\right)\right) \]

   5. Simplified 58.1%

      \[\leadsto \frac{2}{\color{blue}{\left(\left(k \cdot k\right) \cdot \left(k \cdot k\right)\right) \cdot \frac{t}{\ell \cdot \ell}}} \]
   6. Step-by-step derivation

      1. associate-/r* N/A

         \[\leadsto \frac{\frac{2}{\left(k \cdot k\right) \cdot \left(k \cdot k\right)}}{\color{blue}{\frac{t}{\ell \cdot \ell}}} \]

      2. associate-/r* N/A

         \[\leadsto \frac{\frac{2}{\left(k \cdot k\right) \cdot \left(k \cdot k\right)}}{\frac{\frac{t}{\ell}}{\color{blue}{\ell}}} \]

      3. associate-/r/ N/A

         \[\leadsto \frac{\frac{2}{\left(k \cdot k\right) \cdot \left(k \cdot k\right)}}{\frac{t}{\ell}} \cdot \color{blue}{\ell} \]

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

         \[\leadsto \mathsf{*.f64}\left(\left(\frac{\frac{2}{\left(k \cdot k\right) \cdot \left(k \cdot k\right)}}{\frac{t}{\ell}}\right), \color{blue}{\ell}\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\left(\frac{2}{\left(k \cdot k\right) \cdot \left(k \cdot k\right)}\right), \left(\frac{t}{\ell}\right)\right), \ell\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(2, \left(\left(k \cdot k\right) \cdot \left(k \cdot k\right)\right)\right), \left(\frac{t}{\ell}\right)\right), \ell\right) \]

      7. associate-*l* N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(2, \left(k \cdot \left(k \cdot \left(k \cdot k\right)\right)\right)\right), \left(\frac{t}{\ell}\right)\right), \ell\right) \]

      8. cube-unmult N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(2, \left(k \cdot {k}^{3}\right)\right), \left(\frac{t}{\ell}\right)\right), \ell\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(2, \mathsf{*.f64}\left(k, \left({k}^{3}\right)\right)\right), \left(\frac{t}{\ell}\right)\right), \ell\right) \]

      10. cube-unmult N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(2, \mathsf{*.f64}\left(k, \left(k \cdot \left(k \cdot k\right)\right)\right)\right), \left(\frac{t}{\ell}\right)\right), \ell\right) \]

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

          \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(2, \mathsf{*.f64}\left(k, \mathsf{*.f64}\left(k, \left(k \cdot k\right)\right)\right)\right), \left(\frac{t}{\ell}\right)\right), \ell\right) \]

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

          \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(2, \mathsf{*.f64}\left(k, \mathsf{*.f64}\left(k, \mathsf{*.f64}\left(k, k\right)\right)\right)\right), \left(\frac{t}{\ell}\right)\right), \ell\right) \]

      13. /-lowering-/.f64 69.3%

          \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(2, \mathsf{*.f64}\left(k, \mathsf{*.f64}\left(k, \mathsf{*.f64}\left(k, k\right)\right)\right)\right), \mathsf{/.f64}\left(t, \ell\right)\right), \ell\right) \]

   7. Applied egg-rr 69.3%

      \[\leadsto \color{blue}{\frac{\frac{2}{k \cdot \left(k \cdot \left(k \cdot k\right)\right)}}{\frac{t}{\ell}} \cdot \ell} \]
   8. Step-by-step derivation

      1. div-inv N/A

         \[\leadsto \mathsf{*.f64}\left(\left(\frac{2}{k \cdot \left(k \cdot \left(k \cdot k\right)\right)} \cdot \frac{1}{\frac{t}{\ell}}\right), \ell\right) \]

      2. associate-/r* N/A

         \[\leadsto \mathsf{*.f64}\left(\left(\frac{\frac{2}{k}}{k \cdot \left(k \cdot k\right)} \cdot \frac{1}{\frac{t}{\ell}}\right), \ell\right) \]

      3. clear-num N/A

         \[\leadsto \mathsf{*.f64}\left(\left(\frac{\frac{2}{k}}{k \cdot \left(k \cdot k\right)} \cdot \frac{\ell}{t}\right), \ell\right) \]

      4. associate-*l/ N/A

         \[\leadsto \mathsf{*.f64}\left(\left(\frac{\frac{2}{k} \cdot \frac{\ell}{t}}{k \cdot \left(k \cdot k\right)}\right), \ell\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\left(\frac{2}{k} \cdot \frac{\ell}{t}\right), \left(k \cdot \left(k \cdot k\right)\right)\right), \ell\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\left(\frac{2}{k}\right), \left(\frac{\ell}{t}\right)\right), \left(k \cdot \left(k \cdot k\right)\right)\right), \ell\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(2, k\right), \left(\frac{\ell}{t}\right)\right), \left(k \cdot \left(k \cdot k\right)\right)\right), \ell\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(2, k\right), \mathsf{/.f64}\left(\ell, t\right)\right), \left(k \cdot \left(k \cdot k\right)\right)\right), \ell\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(2, k\right), \mathsf{/.f64}\left(\ell, t\right)\right), \mathsf{*.f64}\left(k, \left(k \cdot k\right)\right)\right), \ell\right) \]

      10. *-lowering-*.f64 72.8%

          \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(2, k\right), \mathsf{/.f64}\left(\ell, t\right)\right), \mathsf{*.f64}\left(k, \mathsf{*.f64}\left(k, k\right)\right)\right), \ell\right) \]

   9. Applied egg-rr 72.8%

      \[\leadsto \color{blue}{\frac{\frac{2}{k} \cdot \frac{\ell}{t}}{k \cdot \left(k \cdot k\right)}} \cdot \ell \]
   10. Step-by-step derivation

       1. associate-/r* N/A

          \[\leadsto \mathsf{*.f64}\left(\left(\frac{\frac{\frac{2}{k} \cdot \frac{\ell}{t}}{k}}{k \cdot k}\right), \ell\right) \]

       2. associate-/r* N/A

          \[\leadsto \mathsf{*.f64}\left(\left(\frac{\frac{\frac{\frac{2}{k} \cdot \frac{\ell}{t}}{k}}{k}}{k}\right), \ell\right) \]

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

          \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\left(\frac{\frac{\frac{2}{k} \cdot \frac{\ell}{t}}{k}}{k}\right), k\right), \ell\right) \]

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

          \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(\left(\frac{\frac{2}{k} \cdot \frac{\ell}{t}}{k}\right), k\right), k\right), \ell\right) \]

       5. frac-times N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(\left(\frac{\frac{2 \cdot \ell}{k \cdot t}}{k}\right), k\right), k\right), \ell\right) \]

       6. associate-/l/ N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(\left(\frac{2 \cdot \ell}{k \cdot \left(k \cdot t\right)}\right), k\right), k\right), \ell\right) \]

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

          \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(\left(2 \cdot \ell\right), \left(k \cdot \left(k \cdot t\right)\right)\right), k\right), k\right), \ell\right) \]

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

          \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(2, \ell\right), \left(k \cdot \left(k \cdot t\right)\right)\right), k\right), k\right), \ell\right) \]

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

          \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(2, \ell\right), \mathsf{*.f64}\left(k, \left(k \cdot t\right)\right)\right), k\right), k\right), \ell\right) \]

       10. *-lowering-*.f64 78.5%

           \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(2, \ell\right), \mathsf{*.f64}\left(k, \mathsf{*.f64}\left(k, t\right)\right)\right), k\right), k\right), \ell\right) \]

   11. Applied egg-rr 78.5%

       \[\leadsto \color{blue}{\frac{\frac{\frac{2 \cdot \ell}{k \cdot \left(k \cdot t\right)}}{k}}{k}} \cdot \ell \]

   if 8.500000000000001e-5 < k

   1. Initial program 21.4%

      \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) - 1\right)} \]
   2. Add Preprocessing

   3. Taylor expanded in t around 0

      \[\leadsto \color{blue}{2 \cdot \frac{{\ell}^{2} \cdot \cos k}{{k}^{2} \cdot \left(t \cdot {\sin k}^{2}\right)}} \]
   4. Step-by-step derivation

      1. associate-*r/ N/A

         \[\leadsto \frac{2 \cdot \left({\ell}^{2} \cdot \cos k\right)}{\color{blue}{{k}^{2} \cdot \left(t \cdot {\sin k}^{2}\right)}} \]

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

         \[\leadsto \mathsf{/.f64}\left(\left(2 \cdot \left({\ell}^{2} \cdot \cos k\right)\right), \color{blue}{\left({k}^{2} \cdot \left(t \cdot {\sin k}^{2}\right)\right)}\right) \]

      3. associate-*r* N/A

         \[\leadsto \mathsf{/.f64}\left(\left(\left(2 \cdot {\ell}^{2}\right) \cdot \cos k\right), \left(\color{blue}{{k}^{2}} \cdot \left(t \cdot {\sin k}^{2}\right)\right)\right) \]

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

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\left(2 \cdot {\ell}^{2}\right), \cos k\right), \left(\color{blue}{{k}^{2}} \cdot \left(t \cdot {\sin k}^{2}\right)\right)\right) \]

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

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(2, \left({\ell}^{2}\right)\right), \cos k\right), \left({\color{blue}{k}}^{2} \cdot \left(t \cdot {\sin k}^{2}\right)\right)\right) \]

      6. unpow2 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(2, \left(\ell \cdot \ell\right)\right), \cos k\right), \left({k}^{2} \cdot \left(t \cdot {\sin k}^{2}\right)\right)\right) \]

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

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(2, \mathsf{*.f64}\left(\ell, \ell\right)\right), \cos k\right), \left({k}^{2} \cdot \left(t \cdot {\sin k}^{2}\right)\right)\right) \]

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

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(2, \mathsf{*.f64}\left(\ell, \ell\right)\right), \mathsf{cos.f64}\left(k\right)\right), \left({k}^{\color{blue}{2}} \cdot \left(t \cdot {\sin k}^{2}\right)\right)\right) \]

      9. unpow2 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(2, \mathsf{*.f64}\left(\ell, \ell\right)\right), \mathsf{cos.f64}\left(k\right)\right), \left(\left(k \cdot k\right) \cdot \left(\color{blue}{t} \cdot {\sin k}^{2}\right)\right)\right) \]

      10. associate-*l* N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(2, \mathsf{*.f64}\left(\ell, \ell\right)\right), \mathsf{cos.f64}\left(k\right)\right), \left(k \cdot \color{blue}{\left(k \cdot \left(t \cdot {\sin k}^{2}\right)\right)}\right)\right) \]

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

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(2, \mathsf{*.f64}\left(\ell, \ell\right)\right), \mathsf{cos.f64}\left(k\right)\right), \mathsf{*.f64}\left(k, \color{blue}{\left(k \cdot \left(t \cdot {\sin k}^{2}\right)\right)}\right)\right) \]

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

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(2, \mathsf{*.f64}\left(\ell, \ell\right)\right), \mathsf{cos.f64}\left(k\right)\right), \mathsf{*.f64}\left(k, \mathsf{*.f64}\left(k, \color{blue}{\left(t \cdot {\sin k}^{2}\right)}\right)\right)\right) \]

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

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(2, \mathsf{*.f64}\left(\ell, \ell\right)\right), \mathsf{cos.f64}\left(k\right)\right), \mathsf{*.f64}\left(k, \mathsf{*.f64}\left(k, \mathsf{*.f64}\left(t, \color{blue}{\left({\sin k}^{2}\right)}\right)\right)\right)\right) \]

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

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(2, \mathsf{*.f64}\left(\ell, \ell\right)\right), \mathsf{cos.f64}\left(k\right)\right), \mathsf{*.f64}\left(k, \mathsf{*.f64}\left(k, \mathsf{*.f64}\left(t, \mathsf{pow.f64}\left(\sin k, \color{blue}{2}\right)\right)\right)\right)\right) \]

      15. sin-lowering-sin.f64 67.6%

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(2, \mathsf{*.f64}\left(\ell, \ell\right)\right), \mathsf{cos.f64}\left(k\right)\right), \mathsf{*.f64}\left(k, \mathsf{*.f64}\left(k, \mathsf{*.f64}\left(t, \mathsf{pow.f64}\left(\mathsf{sin.f64}\left(k\right), 2\right)\right)\right)\right)\right) \]

   5. Simplified 67.6%

      \[\leadsto \color{blue}{\frac{\left(2 \cdot \left(\ell \cdot \ell\right)\right) \cdot \cos k}{k \cdot \left(k \cdot \left(t \cdot {\sin k}^{2}\right)\right)}} \]
   6. Step-by-step derivation

      1. associate-/l* N/A

         \[\leadsto \left(2 \cdot \left(\ell \cdot \ell\right)\right) \cdot \color{blue}{\frac{\cos k}{k \cdot \left(k \cdot \left(t \cdot {\sin k}^{2}\right)\right)}} \]

      2. associate-*r* N/A

         \[\leadsto \left(\left(2 \cdot \ell\right) \cdot \ell\right) \cdot \frac{\color{blue}{\cos k}}{k \cdot \left(k \cdot \left(t \cdot {\sin k}^{2}\right)\right)} \]

      3. associate-*l* N/A

         \[\leadsto \left(2 \cdot \ell\right) \cdot \color{blue}{\left(\ell \cdot \frac{\cos k}{k \cdot \left(k \cdot \left(t \cdot {\sin k}^{2}\right)\right)}\right)} \]

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

         \[\leadsto \mathsf{*.f64}\left(\left(2 \cdot \ell\right), \color{blue}{\left(\ell \cdot \frac{\cos k}{k \cdot \left(k \cdot \left(t \cdot {\sin k}^{2}\right)\right)}\right)}\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(2, \ell\right), \left(\color{blue}{\ell} \cdot \frac{\cos k}{k \cdot \left(k \cdot \left(t \cdot {\sin k}^{2}\right)\right)}\right)\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(2, \ell\right), \mathsf{*.f64}\left(\ell, \color{blue}{\left(\frac{\cos k}{k \cdot \left(k \cdot \left(t \cdot {\sin k}^{2}\right)\right)}\right)}\right)\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(2, \ell\right), \mathsf{*.f64}\left(\ell, \mathsf{/.f64}\left(\cos k, \color{blue}{\left(k \cdot \left(k \cdot \left(t \cdot {\sin k}^{2}\right)\right)\right)}\right)\right)\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(2, \ell\right), \mathsf{*.f64}\left(\ell, \mathsf{/.f64}\left(\mathsf{cos.f64}\left(k\right), \left(\color{blue}{k} \cdot \left(k \cdot \left(t \cdot {\sin k}^{2}\right)\right)\right)\right)\right)\right) \]

      9. associate-*r* N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(2, \ell\right), \mathsf{*.f64}\left(\ell, \mathsf{/.f64}\left(\mathsf{cos.f64}\left(k\right), \left(\left(k \cdot k\right) \cdot \color{blue}{\left(t \cdot {\sin k}^{2}\right)}\right)\right)\right)\right) \]

      10. associate-*r* N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(2, \ell\right), \mathsf{*.f64}\left(\ell, \mathsf{/.f64}\left(\mathsf{cos.f64}\left(k\right), \left(\left(\left(k \cdot k\right) \cdot t\right) \cdot \color{blue}{{\sin k}^{2}}\right)\right)\right)\right) \]

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

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(2, \ell\right), \mathsf{*.f64}\left(\ell, \mathsf{/.f64}\left(\mathsf{cos.f64}\left(k\right), \mathsf{*.f64}\left(\left(\left(k \cdot k\right) \cdot t\right), \color{blue}{\left({\sin k}^{2}\right)}\right)\right)\right)\right) \]

      12. *-commutative N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(2, \ell\right), \mathsf{*.f64}\left(\ell, \mathsf{/.f64}\left(\mathsf{cos.f64}\left(k\right), \mathsf{*.f64}\left(\left(t \cdot \left(k \cdot k\right)\right), \left({\color{blue}{\sin k}}^{2}\right)\right)\right)\right)\right) \]

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

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(2, \ell\right), \mathsf{*.f64}\left(\ell, \mathsf{/.f64}\left(\mathsf{cos.f64}\left(k\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(t, \left(k \cdot k\right)\right), \left({\color{blue}{\sin k}}^{2}\right)\right)\right)\right)\right) \]

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

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(2, \ell\right), \mathsf{*.f64}\left(\ell, \mathsf{/.f64}\left(\mathsf{cos.f64}\left(k\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(t, \mathsf{*.f64}\left(k, k\right)\right), \left({\sin k}^{2}\right)\right)\right)\right)\right) \]

      15. unpow2 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(2, \ell\right), \mathsf{*.f64}\left(\ell, \mathsf{/.f64}\left(\mathsf{cos.f64}\left(k\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(t, \mathsf{*.f64}\left(k, k\right)\right), \left(\sin k \cdot \color{blue}{\sin k}\right)\right)\right)\right)\right) \]

      16. sqr-sin-a N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(2, \ell\right), \mathsf{*.f64}\left(\ell, \mathsf{/.f64}\left(\mathsf{cos.f64}\left(k\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(t, \mathsf{*.f64}\left(k, k\right)\right), \left(\frac{1}{2} - \color{blue}{\frac{1}{2} \cdot \cos \left(2 \cdot k\right)}\right)\right)\right)\right)\right) \]

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

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(2, \ell\right), \mathsf{*.f64}\left(\ell, \mathsf{/.f64}\left(\mathsf{cos.f64}\left(k\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(t, \mathsf{*.f64}\left(k, k\right)\right), \mathsf{\_.f64}\left(\frac{1}{2}, \color{blue}{\left(\frac{1}{2} \cdot \cos \left(2 \cdot k\right)\right)}\right)\right)\right)\right)\right) \]

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

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(2, \ell\right), \mathsf{*.f64}\left(\ell, \mathsf{/.f64}\left(\mathsf{cos.f64}\left(k\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(t, \mathsf{*.f64}\left(k, k\right)\right), \mathsf{\_.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(\frac{1}{2}, \color{blue}{\cos \left(2 \cdot k\right)}\right)\right)\right)\right)\right)\right) \]

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

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(2, \ell\right), \mathsf{*.f64}\left(\ell, \mathsf{/.f64}\left(\mathsf{cos.f64}\left(k\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(t, \mathsf{*.f64}\left(k, k\right)\right), \mathsf{\_.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(\frac{1}{2}, \mathsf{cos.f64}\left(\left(2 \cdot k\right)\right)\right)\right)\right)\right)\right)\right) \]

      20. *-lowering-*.f64 69.3%

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(2, \ell\right), \mathsf{*.f64}\left(\ell, \mathsf{/.f64}\left(\mathsf{cos.f64}\left(k\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(t, \mathsf{*.f64}\left(k, k\right)\right), \mathsf{\_.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(\frac{1}{2}, \mathsf{cos.f64}\left(\mathsf{*.f64}\left(2, k\right)\right)\right)\right)\right)\right)\right)\right) \]

   7. Applied egg-rr 69.3%

      \[\leadsto \color{blue}{\left(2 \cdot \ell\right) \cdot \left(\ell \cdot \frac{\cos k}{\left(t \cdot \left(k \cdot k\right)\right) \cdot \left(0.5 - 0.5 \cdot \cos \left(2 \cdot k\right)\right)}\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 76.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;k \leq 8.5 \cdot 10^{-5}:\\ \;\;\;\;\ell \cdot \frac{\frac{\frac{2 \cdot \ell}{k \cdot \left(k \cdot t\right)}}{k}}{k}\\ \mathbf{else}:\\ \;\;\;\;\left(2 \cdot \ell\right) \cdot \left(\ell \cdot \frac{\cos k}{\left(0.5 - 0.5 \cdot \cos \left(k \cdot 2\right)\right) \cdot \left(t \cdot \left(k \cdot k\right)\right)}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 71.1% accurate, 19.1× speedup

Math

\[\begin{array}{l} k_m = \left|k\right| \\ \begin{array}{l} \mathbf{if}\;t \leq 3.05 \cdot 10^{-144}:\\ \;\;\;\;\frac{\left(\ell \cdot \frac{\ell}{t}\right) \cdot \left(-0.3333333333333333 + \frac{2}{k\_m \cdot k\_m}\right)}{k\_m \cdot k\_m}\\ \mathbf{else}:\\ \;\;\;\;\ell \cdot \frac{\frac{\frac{2 \cdot \ell}{k\_m \cdot \left(k\_m \cdot t\right)}}{k\_m}}{k\_m}\\ \end{array} \end{array} \]

FPCore

k_m = (fabs.f64 k)
(FPCore (t l k_m)
 :precision binary64
 (if (<= t 3.05e-144)
   (/
    (* (* l (/ l t)) (+ -0.3333333333333333 (/ 2.0 (* k_m k_m))))
    (* k_m k_m))
   (* l (/ (/ (/ (* 2.0 l) (* k_m (* k_m t))) k_m) k_m))))

C

k_m = fabs(k);
double code(double t, double l, double k_m) {
	double tmp;
	if (t <= 3.05e-144) {
		tmp = ((l * (l / t)) * (-0.3333333333333333 + (2.0 / (k_m * k_m)))) / (k_m * k_m);
	} else {
		tmp = l * ((((2.0 * l) / (k_m * (k_m * t))) / k_m) / k_m);
	}
	return tmp;
}

Fortran

k_m = abs(k)
real(8) function code(t, l, k_m)
    real(8), intent (in) :: t
    real(8), intent (in) :: l
    real(8), intent (in) :: k_m
    real(8) :: tmp
    if (t <= 3.05d-144) then
        tmp = ((l * (l / t)) * ((-0.3333333333333333d0) + (2.0d0 / (k_m * k_m)))) / (k_m * k_m)
    else
        tmp = l * ((((2.0d0 * l) / (k_m * (k_m * t))) / k_m) / k_m)
    end if
    code = tmp
end function

Java

k_m = Math.abs(k);
public static double code(double t, double l, double k_m) {
	double tmp;
	if (t <= 3.05e-144) {
		tmp = ((l * (l / t)) * (-0.3333333333333333 + (2.0 / (k_m * k_m)))) / (k_m * k_m);
	} else {
		tmp = l * ((((2.0 * l) / (k_m * (k_m * t))) / k_m) / k_m);
	}
	return tmp;
}

Python

k_m = math.fabs(k)
def code(t, l, k_m):
	tmp = 0
	if t <= 3.05e-144:
		tmp = ((l * (l / t)) * (-0.3333333333333333 + (2.0 / (k_m * k_m)))) / (k_m * k_m)
	else:
		tmp = l * ((((2.0 * l) / (k_m * (k_m * t))) / k_m) / k_m)
	return tmp

Julia

k_m = abs(k)
function code(t, l, k_m)
	tmp = 0.0
	if (t <= 3.05e-144)
		tmp = Float64(Float64(Float64(l * Float64(l / t)) * Float64(-0.3333333333333333 + Float64(2.0 / Float64(k_m * k_m)))) / Float64(k_m * k_m));
	else
		tmp = Float64(l * Float64(Float64(Float64(Float64(2.0 * l) / Float64(k_m * Float64(k_m * t))) / k_m) / k_m));
	end
	return tmp
end

MATLAB

k_m = abs(k);
function tmp_2 = code(t, l, k_m)
	tmp = 0.0;
	if (t <= 3.05e-144)
		tmp = ((l * (l / t)) * (-0.3333333333333333 + (2.0 / (k_m * k_m)))) / (k_m * k_m);
	else
		tmp = l * ((((2.0 * l) / (k_m * (k_m * t))) / k_m) / k_m);
	end
	tmp_2 = tmp;
end

Wolfram

k_m = N[Abs[k], $MachinePrecision]
code[t_, l_, k$95$m_] := If[LessEqual[t, 3.05e-144], N[(N[(N[(l * N[(l / t), $MachinePrecision]), $MachinePrecision] * N[(-0.3333333333333333 + N[(2.0 / N[(k$95$m * k$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(k$95$m * k$95$m), $MachinePrecision]), $MachinePrecision], N[(l * N[(N[(N[(N[(2.0 * l), $MachinePrecision] / N[(k$95$m * N[(k$95$m * t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / k$95$m), $MachinePrecision] / k$95$m), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if t < 3.05e-144

   1. Initial program 29.6%

      \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) - 1\right)} \]
   2. Add Preprocessing

   3. Taylor expanded in t around 0

      \[\leadsto \color{blue}{2 \cdot \frac{{\ell}^{2} \cdot \cos k}{{k}^{2} \cdot \left(t \cdot {\sin k}^{2}\right)}} \]
   4. Step-by-step derivation

      1. associate-*r/ N/A

         \[\leadsto \frac{2 \cdot \left({\ell}^{2} \cdot \cos k\right)}{\color{blue}{{k}^{2} \cdot \left(t \cdot {\sin k}^{2}\right)}} \]

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

         \[\leadsto \mathsf{/.f64}\left(\left(2 \cdot \left({\ell}^{2} \cdot \cos k\right)\right), \color{blue}{\left({k}^{2} \cdot \left(t \cdot {\sin k}^{2}\right)\right)}\right) \]

      3. associate-*r* N/A

         \[\leadsto \mathsf{/.f64}\left(\left(\left(2 \cdot {\ell}^{2}\right) \cdot \cos k\right), \left(\color{blue}{{k}^{2}} \cdot \left(t \cdot {\sin k}^{2}\right)\right)\right) \]

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

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\left(2 \cdot {\ell}^{2}\right), \cos k\right), \left(\color{blue}{{k}^{2}} \cdot \left(t \cdot {\sin k}^{2}\right)\right)\right) \]

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

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(2, \left({\ell}^{2}\right)\right), \cos k\right), \left({\color{blue}{k}}^{2} \cdot \left(t \cdot {\sin k}^{2}\right)\right)\right) \]

      6. unpow2 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(2, \left(\ell \cdot \ell\right)\right), \cos k\right), \left({k}^{2} \cdot \left(t \cdot {\sin k}^{2}\right)\right)\right) \]

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

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(2, \mathsf{*.f64}\left(\ell, \ell\right)\right), \cos k\right), \left({k}^{2} \cdot \left(t \cdot {\sin k}^{2}\right)\right)\right) \]

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

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(2, \mathsf{*.f64}\left(\ell, \ell\right)\right), \mathsf{cos.f64}\left(k\right)\right), \left({k}^{\color{blue}{2}} \cdot \left(t \cdot {\sin k}^{2}\right)\right)\right) \]

      9. unpow2 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(2, \mathsf{*.f64}\left(\ell, \ell\right)\right), \mathsf{cos.f64}\left(k\right)\right), \left(\left(k \cdot k\right) \cdot \left(\color{blue}{t} \cdot {\sin k}^{2}\right)\right)\right) \]

      10. associate-*l* N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(2, \mathsf{*.f64}\left(\ell, \ell\right)\right), \mathsf{cos.f64}\left(k\right)\right), \left(k \cdot \color{blue}{\left(k \cdot \left(t \cdot {\sin k}^{2}\right)\right)}\right)\right) \]

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

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(2, \mathsf{*.f64}\left(\ell, \ell\right)\right), \mathsf{cos.f64}\left(k\right)\right), \mathsf{*.f64}\left(k, \color{blue}{\left(k \cdot \left(t \cdot {\sin k}^{2}\right)\right)}\right)\right) \]

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

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(2, \mathsf{*.f64}\left(\ell, \ell\right)\right), \mathsf{cos.f64}\left(k\right)\right), \mathsf{*.f64}\left(k, \mathsf{*.f64}\left(k, \color{blue}{\left(t \cdot {\sin k}^{2}\right)}\right)\right)\right) \]

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

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(2, \mathsf{*.f64}\left(\ell, \ell\right)\right), \mathsf{cos.f64}\left(k\right)\right), \mathsf{*.f64}\left(k, \mathsf{*.f64}\left(k, \mathsf{*.f64}\left(t, \color{blue}{\left({\sin k}^{2}\right)}\right)\right)\right)\right) \]

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

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(2, \mathsf{*.f64}\left(\ell, \ell\right)\right), \mathsf{cos.f64}\left(k\right)\right), \mathsf{*.f64}\left(k, \mathsf{*.f64}\left(k, \mathsf{*.f64}\left(t, \mathsf{pow.f64}\left(\sin k, \color{blue}{2}\right)\right)\right)\right)\right) \]

      15. sin-lowering-sin.f64 69.6%

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(2, \mathsf{*.f64}\left(\ell, \ell\right)\right), \mathsf{cos.f64}\left(k\right)\right), \mathsf{*.f64}\left(k, \mathsf{*.f64}\left(k, \mathsf{*.f64}\left(t, \mathsf{pow.f64}\left(\mathsf{sin.f64}\left(k\right), 2\right)\right)\right)\right)\right) \]

   5. Simplified 69.6%

      \[\leadsto \color{blue}{\frac{\left(2 \cdot \left(\ell \cdot \ell\right)\right) \cdot \cos k}{k \cdot \left(k \cdot \left(t \cdot {\sin k}^{2}\right)\right)}} \]
   6. Step-by-step derivation

      1. associate-/l* N/A

         \[\leadsto \left(2 \cdot \left(\ell \cdot \ell\right)\right) \cdot \color{blue}{\frac{\cos k}{k \cdot \left(k \cdot \left(t \cdot {\sin k}^{2}\right)\right)}} \]

      2. associate-*r* N/A

         \[\leadsto \left(\left(2 \cdot \ell\right) \cdot \ell\right) \cdot \frac{\color{blue}{\cos k}}{k \cdot \left(k \cdot \left(t \cdot {\sin k}^{2}\right)\right)} \]

      3. associate-*l* N/A

         \[\leadsto \left(2 \cdot \ell\right) \cdot \color{blue}{\left(\ell \cdot \frac{\cos k}{k \cdot \left(k \cdot \left(t \cdot {\sin k}^{2}\right)\right)}\right)} \]

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

         \[\leadsto \mathsf{*.f64}\left(\left(2 \cdot \ell\right), \color{blue}{\left(\ell \cdot \frac{\cos k}{k \cdot \left(k \cdot \left(t \cdot {\sin k}^{2}\right)\right)}\right)}\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(2, \ell\right), \left(\color{blue}{\ell} \cdot \frac{\cos k}{k \cdot \left(k \cdot \left(t \cdot {\sin k}^{2}\right)\right)}\right)\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(2, \ell\right), \mathsf{*.f64}\left(\ell, \color{blue}{\left(\frac{\cos k}{k \cdot \left(k \cdot \left(t \cdot {\sin k}^{2}\right)\right)}\right)}\right)\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(2, \ell\right), \mathsf{*.f64}\left(\ell, \mathsf{/.f64}\left(\cos k, \color{blue}{\left(k \cdot \left(k \cdot \left(t \cdot {\sin k}^{2}\right)\right)\right)}\right)\right)\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(2, \ell\right), \mathsf{*.f64}\left(\ell, \mathsf{/.f64}\left(\mathsf{cos.f64}\left(k\right), \left(\color{blue}{k} \cdot \left(k \cdot \left(t \cdot {\sin k}^{2}\right)\right)\right)\right)\right)\right) \]

      9. associate-*r* N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(2, \ell\right), \mathsf{*.f64}\left(\ell, \mathsf{/.f64}\left(\mathsf{cos.f64}\left(k\right), \left(\left(k \cdot k\right) \cdot \color{blue}{\left(t \cdot {\sin k}^{2}\right)}\right)\right)\right)\right) \]

      10. associate-*r* N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(2, \ell\right), \mathsf{*.f64}\left(\ell, \mathsf{/.f64}\left(\mathsf{cos.f64}\left(k\right), \left(\left(\left(k \cdot k\right) \cdot t\right) \cdot \color{blue}{{\sin k}^{2}}\right)\right)\right)\right) \]

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

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(2, \ell\right), \mathsf{*.f64}\left(\ell, \mathsf{/.f64}\left(\mathsf{cos.f64}\left(k\right), \mathsf{*.f64}\left(\left(\left(k \cdot k\right) \cdot t\right), \color{blue}{\left({\sin k}^{2}\right)}\right)\right)\right)\right) \]

      12. *-commutative N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(2, \ell\right), \mathsf{*.f64}\left(\ell, \mathsf{/.f64}\left(\mathsf{cos.f64}\left(k\right), \mathsf{*.f64}\left(\left(t \cdot \left(k \cdot k\right)\right), \left({\color{blue}{\sin k}}^{2}\right)\right)\right)\right)\right) \]

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

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(2, \ell\right), \mathsf{*.f64}\left(\ell, \mathsf{/.f64}\left(\mathsf{cos.f64}\left(k\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(t, \left(k \cdot k\right)\right), \left({\color{blue}{\sin k}}^{2}\right)\right)\right)\right)\right) \]

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

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(2, \ell\right), \mathsf{*.f64}\left(\ell, \mathsf{/.f64}\left(\mathsf{cos.f64}\left(k\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(t, \mathsf{*.f64}\left(k, k\right)\right), \left({\sin k}^{2}\right)\right)\right)\right)\right) \]

      15. unpow2 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(2, \ell\right), \mathsf{*.f64}\left(\ell, \mathsf{/.f64}\left(\mathsf{cos.f64}\left(k\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(t, \mathsf{*.f64}\left(k, k\right)\right), \left(\sin k \cdot \color{blue}{\sin k}\right)\right)\right)\right)\right) \]

      16. sqr-sin-a N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(2, \ell\right), \mathsf{*.f64}\left(\ell, \mathsf{/.f64}\left(\mathsf{cos.f64}\left(k\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(t, \mathsf{*.f64}\left(k, k\right)\right), \left(\frac{1}{2} - \color{blue}{\frac{1}{2} \cdot \cos \left(2 \cdot k\right)}\right)\right)\right)\right)\right) \]

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

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(2, \ell\right), \mathsf{*.f64}\left(\ell, \mathsf{/.f64}\left(\mathsf{cos.f64}\left(k\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(t, \mathsf{*.f64}\left(k, k\right)\right), \mathsf{\_.f64}\left(\frac{1}{2}, \color{blue}{\left(\frac{1}{2} \cdot \cos \left(2 \cdot k\right)\right)}\right)\right)\right)\right)\right) \]

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

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(2, \ell\right), \mathsf{*.f64}\left(\ell, \mathsf{/.f64}\left(\mathsf{cos.f64}\left(k\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(t, \mathsf{*.f64}\left(k, k\right)\right), \mathsf{\_.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(\frac{1}{2}, \color{blue}{\cos \left(2 \cdot k\right)}\right)\right)\right)\right)\right)\right) \]

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

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(2, \ell\right), \mathsf{*.f64}\left(\ell, \mathsf{/.f64}\left(\mathsf{cos.f64}\left(k\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(t, \mathsf{*.f64}\left(k, k\right)\right), \mathsf{\_.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(\frac{1}{2}, \mathsf{cos.f64}\left(\left(2 \cdot k\right)\right)\right)\right)\right)\right)\right)\right) \]

      20. *-lowering-*.f64 72.2%

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(2, \ell\right), \mathsf{*.f64}\left(\ell, \mathsf{/.f64}\left(\mathsf{cos.f64}\left(k\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(t, \mathsf{*.f64}\left(k, k\right)\right), \mathsf{\_.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(\frac{1}{2}, \mathsf{cos.f64}\left(\mathsf{*.f64}\left(2, k\right)\right)\right)\right)\right)\right)\right)\right) \]

   7. Applied egg-rr 72.2%

      \[\leadsto \color{blue}{\left(2 \cdot \ell\right) \cdot \left(\ell \cdot \frac{\cos k}{\left(t \cdot \left(k \cdot k\right)\right) \cdot \left(0.5 - 0.5 \cdot \cos \left(2 \cdot k\right)\right)}\right)} \]

   8. Taylor expanded in k around 0

      \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(2, \ell\right), \color{blue}{\left(\frac{{k}^{2} \cdot \left(\frac{-1}{2} \cdot \frac{\ell}{t} - \frac{-1}{3} \cdot \frac{\ell}{t}\right) + \frac{\ell}{t}}{{k}^{4}}\right)}\right) \]
   9. Step-by-step derivation

      1. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(2, \ell\right), \mathsf{/.f64}\left(\left({k}^{2} \cdot \left(\frac{-1}{2} \cdot \frac{\ell}{t} - \frac{-1}{3} \cdot \frac{\ell}{t}\right) + \frac{\ell}{t}\right), \color{blue}{\left({k}^{4}\right)}\right)\right) \]

      2. +-commutative N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(2, \ell\right), \mathsf{/.f64}\left(\left(\frac{\ell}{t} + {k}^{2} \cdot \left(\frac{-1}{2} \cdot \frac{\ell}{t} - \frac{-1}{3} \cdot \frac{\ell}{t}\right)\right), \left({\color{blue}{k}}^{4}\right)\right)\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(2, \ell\right), \mathsf{/.f64}\left(\mathsf{+.f64}\left(\left(\frac{\ell}{t}\right), \left({k}^{2} \cdot \left(\frac{-1}{2} \cdot \frac{\ell}{t} - \frac{-1}{3} \cdot \frac{\ell}{t}\right)\right)\right), \left({\color{blue}{k}}^{4}\right)\right)\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(2, \ell\right), \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{/.f64}\left(\ell, t\right), \left({k}^{2} \cdot \left(\frac{-1}{2} \cdot \frac{\ell}{t} - \frac{-1}{3} \cdot \frac{\ell}{t}\right)\right)\right), \left({k}^{4}\right)\right)\right) \]

      5. unpow2 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(2, \ell\right), \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{/.f64}\left(\ell, t\right), \left(\left(k \cdot k\right) \cdot \left(\frac{-1}{2} \cdot \frac{\ell}{t} - \frac{-1}{3} \cdot \frac{\ell}{t}\right)\right)\right), \left({k}^{4}\right)\right)\right) \]

      6. associate-*l* N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(2, \ell\right), \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{/.f64}\left(\ell, t\right), \left(k \cdot \left(k \cdot \left(\frac{-1}{2} \cdot \frac{\ell}{t} - \frac{-1}{3} \cdot \frac{\ell}{t}\right)\right)\right)\right), \left({k}^{4}\right)\right)\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(2, \ell\right), \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{/.f64}\left(\ell, t\right), \mathsf{*.f64}\left(k, \left(k \cdot \left(\frac{-1}{2} \cdot \frac{\ell}{t} - \frac{-1}{3} \cdot \frac{\ell}{t}\right)\right)\right)\right), \left({k}^{4}\right)\right)\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(2, \ell\right), \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{/.f64}\left(\ell, t\right), \mathsf{*.f64}\left(k, \mathsf{*.f64}\left(k, \left(\frac{-1}{2} \cdot \frac{\ell}{t} - \frac{-1}{3} \cdot \frac{\ell}{t}\right)\right)\right)\right), \left({k}^{4}\right)\right)\right) \]

      9. distribute-rgt-out-- N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(2, \ell\right), \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{/.f64}\left(\ell, t\right), \mathsf{*.f64}\left(k, \mathsf{*.f64}\left(k, \left(\frac{\ell}{t} \cdot \left(\frac{-1}{2} - \frac{-1}{3}\right)\right)\right)\right)\right), \left({k}^{4}\right)\right)\right) \]

      10. metadata-eval N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(2, \ell\right), \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{/.f64}\left(\ell, t\right), \mathsf{*.f64}\left(k, \mathsf{*.f64}\left(k, \left(\frac{\ell}{t} \cdot \frac{-1}{6}\right)\right)\right)\right), \left({k}^{4}\right)\right)\right) \]

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

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(2, \ell\right), \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{/.f64}\left(\ell, t\right), \mathsf{*.f64}\left(k, \mathsf{*.f64}\left(k, \mathsf{*.f64}\left(\left(\frac{\ell}{t}\right), \frac{-1}{6}\right)\right)\right)\right), \left({k}^{4}\right)\right)\right) \]

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

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(2, \ell\right), \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{/.f64}\left(\ell, t\right), \mathsf{*.f64}\left(k, \mathsf{*.f64}\left(k, \mathsf{*.f64}\left(\mathsf{/.f64}\left(\ell, t\right), \frac{-1}{6}\right)\right)\right)\right), \left({k}^{4}\right)\right)\right) \]

      13. pow-lowering-pow.f64 43.9%

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(2, \ell\right), \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{/.f64}\left(\ell, t\right), \mathsf{*.f64}\left(k, \mathsf{*.f64}\left(k, \mathsf{*.f64}\left(\mathsf{/.f64}\left(\ell, t\right), \frac{-1}{6}\right)\right)\right)\right), \mathsf{pow.f64}\left(k, \color{blue}{4}\right)\right)\right) \]

   10. Simplified 43.9%

       \[\leadsto \left(2 \cdot \ell\right) \cdot \color{blue}{\frac{\frac{\ell}{t} + k \cdot \left(k \cdot \left(\frac{\ell}{t} \cdot -0.16666666666666666\right)\right)}{{k}^{4}}} \]

   11. Taylor expanded in k around inf

       \[\leadsto \color{blue}{\frac{\frac{-1}{3} \cdot \frac{{\ell}^{2}}{t} + 2 \cdot \frac{{\ell}^{2}}{{k}^{2} \cdot t}}{{k}^{2}}} \]
   12. Step-by-step derivation

       1. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\left(\frac{-1}{3} \cdot \frac{{\ell}^{2}}{t} + 2 \cdot \frac{{\ell}^{2}}{{k}^{2} \cdot t}\right), \color{blue}{\left({k}^{2}\right)}\right) \]

       2. associate-*r/ N/A

          \[\leadsto \mathsf{/.f64}\left(\left(\frac{-1}{3} \cdot \frac{{\ell}^{2}}{t} + \frac{2 \cdot {\ell}^{2}}{{k}^{2} \cdot t}\right), \left({k}^{2}\right)\right) \]

       3. times-frac N/A

          \[\leadsto \mathsf{/.f64}\left(\left(\frac{-1}{3} \cdot \frac{{\ell}^{2}}{t} + \frac{2}{{k}^{2}} \cdot \frac{{\ell}^{2}}{t}\right), \left({k}^{2}\right)\right) \]

       4. distribute-rgt-out N/A

          \[\leadsto \mathsf{/.f64}\left(\left(\frac{{\ell}^{2}}{t} \cdot \left(\frac{-1}{3} + \frac{2}{{k}^{2}}\right)\right), \left({\color{blue}{k}}^{2}\right)\right) \]

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

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\left(\frac{{\ell}^{2}}{t}\right), \left(\frac{-1}{3} + \frac{2}{{k}^{2}}\right)\right), \left({\color{blue}{k}}^{2}\right)\right) \]

       6. unpow2 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\left(\frac{\ell \cdot \ell}{t}\right), \left(\frac{-1}{3} + \frac{2}{{k}^{2}}\right)\right), \left({k}^{2}\right)\right) \]

       7. associate-/l* N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\left(\ell \cdot \frac{\ell}{t}\right), \left(\frac{-1}{3} + \frac{2}{{k}^{2}}\right)\right), \left({k}^{2}\right)\right) \]

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

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\ell, \left(\frac{\ell}{t}\right)\right), \left(\frac{-1}{3} + \frac{2}{{k}^{2}}\right)\right), \left({k}^{2}\right)\right) \]

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

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\ell, \mathsf{/.f64}\left(\ell, t\right)\right), \left(\frac{-1}{3} + \frac{2}{{k}^{2}}\right)\right), \left({k}^{2}\right)\right) \]

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

           \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\ell, \mathsf{/.f64}\left(\ell, t\right)\right), \mathsf{+.f64}\left(\frac{-1}{3}, \left(\frac{2}{{k}^{2}}\right)\right)\right), \left({k}^{2}\right)\right) \]

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

           \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\ell, \mathsf{/.f64}\left(\ell, t\right)\right), \mathsf{+.f64}\left(\frac{-1}{3}, \mathsf{/.f64}\left(2, \left({k}^{2}\right)\right)\right)\right), \left({k}^{2}\right)\right) \]

       12. unpow2 N/A

           \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\ell, \mathsf{/.f64}\left(\ell, t\right)\right), \mathsf{+.f64}\left(\frac{-1}{3}, \mathsf{/.f64}\left(2, \left(k \cdot k\right)\right)\right)\right), \left({k}^{2}\right)\right) \]

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

           \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\ell, \mathsf{/.f64}\left(\ell, t\right)\right), \mathsf{+.f64}\left(\frac{-1}{3}, \mathsf{/.f64}\left(2, \mathsf{*.f64}\left(k, k\right)\right)\right)\right), \left({k}^{2}\right)\right) \]

       14. unpow2 N/A

           \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\ell, \mathsf{/.f64}\left(\ell, t\right)\right), \mathsf{+.f64}\left(\frac{-1}{3}, \mathsf{/.f64}\left(2, \mathsf{*.f64}\left(k, k\right)\right)\right)\right), \left(k \cdot \color{blue}{k}\right)\right) \]

       15. *-lowering-*.f64 65.4%

           \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\ell, \mathsf{/.f64}\left(\ell, t\right)\right), \mathsf{+.f64}\left(\frac{-1}{3}, \mathsf{/.f64}\left(2, \mathsf{*.f64}\left(k, k\right)\right)\right)\right), \mathsf{*.f64}\left(k, \color{blue}{k}\right)\right) \]

   13. Simplified 65.4%

       \[\leadsto \color{blue}{\frac{\left(\ell \cdot \frac{\ell}{t}\right) \cdot \left(-0.3333333333333333 + \frac{2}{k \cdot k}\right)}{k \cdot k}} \]

   if 3.05e-144 < t

   1. Initial program 36.5%

      \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) - 1\right)} \]
   2. Add Preprocessing

   3. Taylor expanded in k around 0

      \[\leadsto \mathsf{/.f64}\left(2, \color{blue}{\left(\frac{{k}^{4} \cdot t}{{\ell}^{2}}\right)}\right) \]
   4. Step-by-step derivation

      1. associate-/l* N/A

         \[\leadsto \mathsf{/.f64}\left(2, \left({k}^{4} \cdot \color{blue}{\frac{t}{{\ell}^{2}}}\right)\right) \]

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

         \[\leadsto \mathsf{/.f64}\left(2, \mathsf{*.f64}\left(\left({k}^{4}\right), \color{blue}{\left(\frac{t}{{\ell}^{2}}\right)}\right)\right) \]

      3. metadata-eval N/A

         \[\leadsto \mathsf{/.f64}\left(2, \mathsf{*.f64}\left(\left({k}^{\left(2 \cdot 2\right)}\right), \left(\frac{t}{{\ell}^{2}}\right)\right)\right) \]

      4. pow-sqr N/A

         \[\leadsto \mathsf{/.f64}\left(2, \mathsf{*.f64}\left(\left({k}^{2} \cdot {k}^{2}\right), \left(\frac{\color{blue}{t}}{{\ell}^{2}}\right)\right)\right) \]

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

         \[\leadsto \mathsf{/.f64}\left(2, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\left({k}^{2}\right), \left({k}^{2}\right)\right), \left(\frac{\color{blue}{t}}{{\ell}^{2}}\right)\right)\right) \]

      6. unpow2 N/A

         \[\leadsto \mathsf{/.f64}\left(2, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\left(k \cdot k\right), \left({k}^{2}\right)\right), \left(\frac{t}{{\ell}^{2}}\right)\right)\right) \]

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

         \[\leadsto \mathsf{/.f64}\left(2, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(k, k\right), \left({k}^{2}\right)\right), \left(\frac{t}{{\ell}^{2}}\right)\right)\right) \]

      8. unpow2 N/A

         \[\leadsto \mathsf{/.f64}\left(2, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(k, k\right), \left(k \cdot k\right)\right), \left(\frac{t}{{\ell}^{2}}\right)\right)\right) \]

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

         \[\leadsto \mathsf{/.f64}\left(2, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(k, k\right), \mathsf{*.f64}\left(k, k\right)\right), \left(\frac{t}{{\ell}^{2}}\right)\right)\right) \]

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

          \[\leadsto \mathsf{/.f64}\left(2, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(k, k\right), \mathsf{*.f64}\left(k, k\right)\right), \mathsf{/.f64}\left(t, \color{blue}{\left({\ell}^{2}\right)}\right)\right)\right) \]

      11. unpow2 N/A

          \[\leadsto \mathsf{/.f64}\left(2, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(k, k\right), \mathsf{*.f64}\left(k, k\right)\right), \mathsf{/.f64}\left(t, \left(\ell \cdot \color{blue}{\ell}\right)\right)\right)\right) \]

      12. *-lowering-*.f64 61.8%

          \[\leadsto \mathsf{/.f64}\left(2, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(k, k\right), \mathsf{*.f64}\left(k, k\right)\right), \mathsf{/.f64}\left(t, \mathsf{*.f64}\left(\ell, \color{blue}{\ell}\right)\right)\right)\right) \]

   5. Simplified 61.8%

      \[\leadsto \frac{2}{\color{blue}{\left(\left(k \cdot k\right) \cdot \left(k \cdot k\right)\right) \cdot \frac{t}{\ell \cdot \ell}}} \]
   6. Step-by-step derivation

      1. associate-/r* N/A

         \[\leadsto \frac{\frac{2}{\left(k \cdot k\right) \cdot \left(k \cdot k\right)}}{\color{blue}{\frac{t}{\ell \cdot \ell}}} \]

      2. associate-/r* N/A

         \[\leadsto \frac{\frac{2}{\left(k \cdot k\right) \cdot \left(k \cdot k\right)}}{\frac{\frac{t}{\ell}}{\color{blue}{\ell}}} \]

      3. associate-/r/ N/A

         \[\leadsto \frac{\frac{2}{\left(k \cdot k\right) \cdot \left(k \cdot k\right)}}{\frac{t}{\ell}} \cdot \color{blue}{\ell} \]

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

         \[\leadsto \mathsf{*.f64}\left(\left(\frac{\frac{2}{\left(k \cdot k\right) \cdot \left(k \cdot k\right)}}{\frac{t}{\ell}}\right), \color{blue}{\ell}\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\left(\frac{2}{\left(k \cdot k\right) \cdot \left(k \cdot k\right)}\right), \left(\frac{t}{\ell}\right)\right), \ell\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(2, \left(\left(k \cdot k\right) \cdot \left(k \cdot k\right)\right)\right), \left(\frac{t}{\ell}\right)\right), \ell\right) \]

      7. associate-*l* N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(2, \left(k \cdot \left(k \cdot \left(k \cdot k\right)\right)\right)\right), \left(\frac{t}{\ell}\right)\right), \ell\right) \]

      8. cube-unmult N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(2, \left(k \cdot {k}^{3}\right)\right), \left(\frac{t}{\ell}\right)\right), \ell\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(2, \mathsf{*.f64}\left(k, \left({k}^{3}\right)\right)\right), \left(\frac{t}{\ell}\right)\right), \ell\right) \]

      10. cube-unmult N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(2, \mathsf{*.f64}\left(k, \left(k \cdot \left(k \cdot k\right)\right)\right)\right), \left(\frac{t}{\ell}\right)\right), \ell\right) \]

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

          \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(2, \mathsf{*.f64}\left(k, \mathsf{*.f64}\left(k, \left(k \cdot k\right)\right)\right)\right), \left(\frac{t}{\ell}\right)\right), \ell\right) \]

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

          \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(2, \mathsf{*.f64}\left(k, \mathsf{*.f64}\left(k, \mathsf{*.f64}\left(k, k\right)\right)\right)\right), \left(\frac{t}{\ell}\right)\right), \ell\right) \]

      13. /-lowering-/.f64 72.6%

          \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(2, \mathsf{*.f64}\left(k, \mathsf{*.f64}\left(k, \mathsf{*.f64}\left(k, k\right)\right)\right)\right), \mathsf{/.f64}\left(t, \ell\right)\right), \ell\right) \]

   7. Applied egg-rr 72.6%

      \[\leadsto \color{blue}{\frac{\frac{2}{k \cdot \left(k \cdot \left(k \cdot k\right)\right)}}{\frac{t}{\ell}} \cdot \ell} \]
   8. Step-by-step derivation

      1. div-inv N/A

         \[\leadsto \mathsf{*.f64}\left(\left(\frac{2}{k \cdot \left(k \cdot \left(k \cdot k\right)\right)} \cdot \frac{1}{\frac{t}{\ell}}\right), \ell\right) \]

      2. associate-/r* N/A

         \[\leadsto \mathsf{*.f64}\left(\left(\frac{\frac{2}{k}}{k \cdot \left(k \cdot k\right)} \cdot \frac{1}{\frac{t}{\ell}}\right), \ell\right) \]

      3. clear-num N/A

         \[\leadsto \mathsf{*.f64}\left(\left(\frac{\frac{2}{k}}{k \cdot \left(k \cdot k\right)} \cdot \frac{\ell}{t}\right), \ell\right) \]

      4. associate-*l/ N/A

         \[\leadsto \mathsf{*.f64}\left(\left(\frac{\frac{2}{k} \cdot \frac{\ell}{t}}{k \cdot \left(k \cdot k\right)}\right), \ell\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\left(\frac{2}{k} \cdot \frac{\ell}{t}\right), \left(k \cdot \left(k \cdot k\right)\right)\right), \ell\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\left(\frac{2}{k}\right), \left(\frac{\ell}{t}\right)\right), \left(k \cdot \left(k \cdot k\right)\right)\right), \ell\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(2, k\right), \left(\frac{\ell}{t}\right)\right), \left(k \cdot \left(k \cdot k\right)\right)\right), \ell\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(2, k\right), \mathsf{/.f64}\left(\ell, t\right)\right), \left(k \cdot \left(k \cdot k\right)\right)\right), \ell\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(2, k\right), \mathsf{/.f64}\left(\ell, t\right)\right), \mathsf{*.f64}\left(k, \left(k \cdot k\right)\right)\right), \ell\right) \]

      10. *-lowering-*.f64 75.1%

          \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(2, k\right), \mathsf{/.f64}\left(\ell, t\right)\right), \mathsf{*.f64}\left(k, \mathsf{*.f64}\left(k, k\right)\right)\right), \ell\right) \]

   9. Applied egg-rr 75.1%

      \[\leadsto \color{blue}{\frac{\frac{2}{k} \cdot \frac{\ell}{t}}{k \cdot \left(k \cdot k\right)}} \cdot \ell \]
   10. Step-by-step derivation

       1. associate-/r* N/A

          \[\leadsto \mathsf{*.f64}\left(\left(\frac{\frac{\frac{2}{k} \cdot \frac{\ell}{t}}{k}}{k \cdot k}\right), \ell\right) \]

       2. associate-/r* N/A

          \[\leadsto \mathsf{*.f64}\left(\left(\frac{\frac{\frac{\frac{2}{k} \cdot \frac{\ell}{t}}{k}}{k}}{k}\right), \ell\right) \]

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

          \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\left(\frac{\frac{\frac{2}{k} \cdot \frac{\ell}{t}}{k}}{k}\right), k\right), \ell\right) \]

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

          \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(\left(\frac{\frac{2}{k} \cdot \frac{\ell}{t}}{k}\right), k\right), k\right), \ell\right) \]

       5. frac-times N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(\left(\frac{\frac{2 \cdot \ell}{k \cdot t}}{k}\right), k\right), k\right), \ell\right) \]

       6. associate-/l/ N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(\left(\frac{2 \cdot \ell}{k \cdot \left(k \cdot t\right)}\right), k\right), k\right), \ell\right) \]

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

          \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(\left(2 \cdot \ell\right), \left(k \cdot \left(k \cdot t\right)\right)\right), k\right), k\right), \ell\right) \]

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

          \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(2, \ell\right), \left(k \cdot \left(k \cdot t\right)\right)\right), k\right), k\right), \ell\right) \]

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

          \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(2, \ell\right), \mathsf{*.f64}\left(k, \left(k \cdot t\right)\right)\right), k\right), k\right), \ell\right) \]

       10. *-lowering-*.f64 84.4%

           \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(2, \ell\right), \mathsf{*.f64}\left(k, \mathsf{*.f64}\left(k, t\right)\right)\right), k\right), k\right), \ell\right) \]

   11. Applied egg-rr 84.4%

       \[\leadsto \color{blue}{\frac{\frac{\frac{2 \cdot \ell}{k \cdot \left(k \cdot t\right)}}{k}}{k}} \cdot \ell \]
3. Recombined 2 regimes into one program.

4. Final simplification 71.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq 3.05 \cdot 10^{-144}:\\ \;\;\;\;\frac{\left(\ell \cdot \frac{\ell}{t}\right) \cdot \left(-0.3333333333333333 + \frac{2}{k \cdot k}\right)}{k \cdot k}\\ \mathbf{else}:\\ \;\;\;\;\ell \cdot \frac{\frac{\frac{2 \cdot \ell}{k \cdot \left(k \cdot t\right)}}{k}}{k}\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 74.3% accurate, 21.0× speedup

Math

\[\begin{array}{l} k_m = \left|k\right| \\ \begin{array}{l} \mathbf{if}\;t \leq 2.8 \cdot 10^{+19}:\\ \;\;\;\;\frac{2}{k\_m \cdot k\_m} \cdot \left(\ell \cdot \frac{\frac{\ell}{k\_m \cdot t}}{k\_m}\right)\\ \mathbf{else}:\\ \;\;\;\;\ell \cdot \frac{\frac{\frac{2 \cdot \ell}{k\_m \cdot \left(k\_m \cdot t\right)}}{k\_m}}{k\_m}\\ \end{array} \end{array} \]

FPCore

k_m = (fabs.f64 k)
(FPCore (t l k_m)
 :precision binary64
 (if (<= t 2.8e+19)
   (* (/ 2.0 (* k_m k_m)) (* l (/ (/ l (* k_m t)) k_m)))
   (* l (/ (/ (/ (* 2.0 l) (* k_m (* k_m t))) k_m) k_m))))

C

k_m = fabs(k);
double code(double t, double l, double k_m) {
	double tmp;
	if (t <= 2.8e+19) {
		tmp = (2.0 / (k_m * k_m)) * (l * ((l / (k_m * t)) / k_m));
	} else {
		tmp = l * ((((2.0 * l) / (k_m * (k_m * t))) / k_m) / k_m);
	}
	return tmp;
}

Fortran

k_m = abs(k)
real(8) function code(t, l, k_m)
    real(8), intent (in) :: t
    real(8), intent (in) :: l
    real(8), intent (in) :: k_m
    real(8) :: tmp
    if (t <= 2.8d+19) then
        tmp = (2.0d0 / (k_m * k_m)) * (l * ((l / (k_m * t)) / k_m))
    else
        tmp = l * ((((2.0d0 * l) / (k_m * (k_m * t))) / k_m) / k_m)
    end if
    code = tmp
end function

Java

k_m = Math.abs(k);
public static double code(double t, double l, double k_m) {
	double tmp;
	if (t <= 2.8e+19) {
		tmp = (2.0 / (k_m * k_m)) * (l * ((l / (k_m * t)) / k_m));
	} else {
		tmp = l * ((((2.0 * l) / (k_m * (k_m * t))) / k_m) / k_m);
	}
	return tmp;
}

Python

k_m = math.fabs(k)
def code(t, l, k_m):
	tmp = 0
	if t <= 2.8e+19:
		tmp = (2.0 / (k_m * k_m)) * (l * ((l / (k_m * t)) / k_m))
	else:
		tmp = l * ((((2.0 * l) / (k_m * (k_m * t))) / k_m) / k_m)
	return tmp

Julia

k_m = abs(k)
function code(t, l, k_m)
	tmp = 0.0
	if (t <= 2.8e+19)
		tmp = Float64(Float64(2.0 / Float64(k_m * k_m)) * Float64(l * Float64(Float64(l / Float64(k_m * t)) / k_m)));
	else
		tmp = Float64(l * Float64(Float64(Float64(Float64(2.0 * l) / Float64(k_m * Float64(k_m * t))) / k_m) / k_m));
	end
	return tmp
end

MATLAB

k_m = abs(k);
function tmp_2 = code(t, l, k_m)
	tmp = 0.0;
	if (t <= 2.8e+19)
		tmp = (2.0 / (k_m * k_m)) * (l * ((l / (k_m * t)) / k_m));
	else
		tmp = l * ((((2.0 * l) / (k_m * (k_m * t))) / k_m) / k_m);
	end
	tmp_2 = tmp;
end

Wolfram

k_m = N[Abs[k], $MachinePrecision]
code[t_, l_, k$95$m_] := If[LessEqual[t, 2.8e+19], N[(N[(2.0 / N[(k$95$m * k$95$m), $MachinePrecision]), $MachinePrecision] * N[(l * N[(N[(l / N[(k$95$m * t), $MachinePrecision]), $MachinePrecision] / k$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(l * N[(N[(N[(N[(2.0 * l), $MachinePrecision] / N[(k$95$m * N[(k$95$m * t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / k$95$m), $MachinePrecision] / k$95$m), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if t < 2.8e19

   1. Initial program 32.7%

      \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) - 1\right)} \]
   2. Add Preprocessing

   3. Taylor expanded in k around 0

      \[\leadsto \mathsf{/.f64}\left(2, \color{blue}{\left(\frac{{k}^{4} \cdot t}{{\ell}^{2}}\right)}\right) \]
   4. Step-by-step derivation

      1. associate-/l* N/A

         \[\leadsto \mathsf{/.f64}\left(2, \left({k}^{4} \cdot \color{blue}{\frac{t}{{\ell}^{2}}}\right)\right) \]

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

         \[\leadsto \mathsf{/.f64}\left(2, \mathsf{*.f64}\left(\left({k}^{4}\right), \color{blue}{\left(\frac{t}{{\ell}^{2}}\right)}\right)\right) \]

      3. metadata-eval N/A

         \[\leadsto \mathsf{/.f64}\left(2, \mathsf{*.f64}\left(\left({k}^{\left(2 \cdot 2\right)}\right), \left(\frac{t}{{\ell}^{2}}\right)\right)\right) \]

      4. pow-sqr N/A

         \[\leadsto \mathsf{/.f64}\left(2, \mathsf{*.f64}\left(\left({k}^{2} \cdot {k}^{2}\right), \left(\frac{\color{blue}{t}}{{\ell}^{2}}\right)\right)\right) \]

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

         \[\leadsto \mathsf{/.f64}\left(2, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\left({k}^{2}\right), \left({k}^{2}\right)\right), \left(\frac{\color{blue}{t}}{{\ell}^{2}}\right)\right)\right) \]

      6. unpow2 N/A

         \[\leadsto \mathsf{/.f64}\left(2, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\left(k \cdot k\right), \left({k}^{2}\right)\right), \left(\frac{t}{{\ell}^{2}}\right)\right)\right) \]

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

         \[\leadsto \mathsf{/.f64}\left(2, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(k, k\right), \left({k}^{2}\right)\right), \left(\frac{t}{{\ell}^{2}}\right)\right)\right) \]

      8. unpow2 N/A

         \[\leadsto \mathsf{/.f64}\left(2, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(k, k\right), \left(k \cdot k\right)\right), \left(\frac{t}{{\ell}^{2}}\right)\right)\right) \]

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

         \[\leadsto \mathsf{/.f64}\left(2, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(k, k\right), \mathsf{*.f64}\left(k, k\right)\right), \left(\frac{t}{{\ell}^{2}}\right)\right)\right) \]

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

          \[\leadsto \mathsf{/.f64}\left(2, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(k, k\right), \mathsf{*.f64}\left(k, k\right)\right), \mathsf{/.f64}\left(t, \color{blue}{\left({\ell}^{2}\right)}\right)\right)\right) \]

      11. unpow2 N/A

          \[\leadsto \mathsf{/.f64}\left(2, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(k, k\right), \mathsf{*.f64}\left(k, k\right)\right), \mathsf{/.f64}\left(t, \left(\ell \cdot \color{blue}{\ell}\right)\right)\right)\right) \]

      12. *-lowering-*.f64 54.2%

          \[\leadsto \mathsf{/.f64}\left(2, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(k, k\right), \mathsf{*.f64}\left(k, k\right)\right), \mathsf{/.f64}\left(t, \mathsf{*.f64}\left(\ell, \color{blue}{\ell}\right)\right)\right)\right) \]

   5. Simplified 54.2%

      \[\leadsto \frac{2}{\color{blue}{\left(\left(k \cdot k\right) \cdot \left(k \cdot k\right)\right) \cdot \frac{t}{\ell \cdot \ell}}} \]
   6. Step-by-step derivation

      1. associate-/r* N/A

         \[\leadsto \frac{\frac{2}{\left(k \cdot k\right) \cdot \left(k \cdot k\right)}}{\color{blue}{\frac{t}{\ell \cdot \ell}}} \]

      2. associate-/r* N/A

         \[\leadsto \frac{\frac{2}{\left(k \cdot k\right) \cdot \left(k \cdot k\right)}}{\frac{\frac{t}{\ell}}{\color{blue}{\ell}}} \]

      3. associate-/r/ N/A

         \[\leadsto \frac{\frac{2}{\left(k \cdot k\right) \cdot \left(k \cdot k\right)}}{\frac{t}{\ell}} \cdot \color{blue}{\ell} \]

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

         \[\leadsto \mathsf{*.f64}\left(\left(\frac{\frac{2}{\left(k \cdot k\right) \cdot \left(k \cdot k\right)}}{\frac{t}{\ell}}\right), \color{blue}{\ell}\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\left(\frac{2}{\left(k \cdot k\right) \cdot \left(k \cdot k\right)}\right), \left(\frac{t}{\ell}\right)\right), \ell\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(2, \left(\left(k \cdot k\right) \cdot \left(k \cdot k\right)\right)\right), \left(\frac{t}{\ell}\right)\right), \ell\right) \]

      7. associate-*l* N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(2, \left(k \cdot \left(k \cdot \left(k \cdot k\right)\right)\right)\right), \left(\frac{t}{\ell}\right)\right), \ell\right) \]

      8. cube-unmult N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(2, \left(k \cdot {k}^{3}\right)\right), \left(\frac{t}{\ell}\right)\right), \ell\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(2, \mathsf{*.f64}\left(k, \left({k}^{3}\right)\right)\right), \left(\frac{t}{\ell}\right)\right), \ell\right) \]

      10. cube-unmult N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(2, \mathsf{*.f64}\left(k, \left(k \cdot \left(k \cdot k\right)\right)\right)\right), \left(\frac{t}{\ell}\right)\right), \ell\right) \]

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

          \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(2, \mathsf{*.f64}\left(k, \mathsf{*.f64}\left(k, \left(k \cdot k\right)\right)\right)\right), \left(\frac{t}{\ell}\right)\right), \ell\right) \]

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

          \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(2, \mathsf{*.f64}\left(k, \mathsf{*.f64}\left(k, \mathsf{*.f64}\left(k, k\right)\right)\right)\right), \left(\frac{t}{\ell}\right)\right), \ell\right) \]

      13. /-lowering-/.f64 63.3%

          \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(2, \mathsf{*.f64}\left(k, \mathsf{*.f64}\left(k, \mathsf{*.f64}\left(k, k\right)\right)\right)\right), \mathsf{/.f64}\left(t, \ell\right)\right), \ell\right) \]

   7. Applied egg-rr 63.3%

      \[\leadsto \color{blue}{\frac{\frac{2}{k \cdot \left(k \cdot \left(k \cdot k\right)\right)}}{\frac{t}{\ell}} \cdot \ell} \]
   8. Step-by-step derivation

      1. div-inv N/A

         \[\leadsto \mathsf{*.f64}\left(\left(\frac{2}{k \cdot \left(k \cdot \left(k \cdot k\right)\right)} \cdot \frac{1}{\frac{t}{\ell}}\right), \ell\right) \]

      2. associate-/r* N/A

         \[\leadsto \mathsf{*.f64}\left(\left(\frac{\frac{2}{k}}{k \cdot \left(k \cdot k\right)} \cdot \frac{1}{\frac{t}{\ell}}\right), \ell\right) \]

      3. clear-num N/A

         \[\leadsto \mathsf{*.f64}\left(\left(\frac{\frac{2}{k}}{k \cdot \left(k \cdot k\right)} \cdot \frac{\ell}{t}\right), \ell\right) \]

      4. associate-*l/ N/A

         \[\leadsto \mathsf{*.f64}\left(\left(\frac{\frac{2}{k} \cdot \frac{\ell}{t}}{k \cdot \left(k \cdot k\right)}\right), \ell\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\left(\frac{2}{k} \cdot \frac{\ell}{t}\right), \left(k \cdot \left(k \cdot k\right)\right)\right), \ell\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\left(\frac{2}{k}\right), \left(\frac{\ell}{t}\right)\right), \left(k \cdot \left(k \cdot k\right)\right)\right), \ell\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(2, k\right), \left(\frac{\ell}{t}\right)\right), \left(k \cdot \left(k \cdot k\right)\right)\right), \ell\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(2, k\right), \mathsf{/.f64}\left(\ell, t\right)\right), \left(k \cdot \left(k \cdot k\right)\right)\right), \ell\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(2, k\right), \mathsf{/.f64}\left(\ell, t\right)\right), \mathsf{*.f64}\left(k, \left(k \cdot k\right)\right)\right), \ell\right) \]

      10. *-lowering-*.f64 65.6%

          \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(2, k\right), \mathsf{/.f64}\left(\ell, t\right)\right), \mathsf{*.f64}\left(k, \mathsf{*.f64}\left(k, k\right)\right)\right), \ell\right) \]

   9. Applied egg-rr 65.6%

      \[\leadsto \color{blue}{\frac{\frac{2}{k} \cdot \frac{\ell}{t}}{k \cdot \left(k \cdot k\right)}} \cdot \ell \]
   10. Step-by-step derivation

       1. times-frac N/A

          \[\leadsto \left(\frac{\frac{2}{k}}{k} \cdot \frac{\frac{\ell}{t}}{k \cdot k}\right) \cdot \ell \]

       2. associate-*l* N/A

          \[\leadsto \frac{\frac{2}{k}}{k} \cdot \color{blue}{\left(\frac{\frac{\ell}{t}}{k \cdot k} \cdot \ell\right)} \]

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

          \[\leadsto \mathsf{*.f64}\left(\left(\frac{\frac{2}{k}}{k}\right), \color{blue}{\left(\frac{\frac{\ell}{t}}{k \cdot k} \cdot \ell\right)}\right) \]

       4. associate-/l/ N/A

          \[\leadsto \mathsf{*.f64}\left(\left(\frac{2}{k \cdot k}\right), \left(\color{blue}{\frac{\frac{\ell}{t}}{k \cdot k}} \cdot \ell\right)\right) \]

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

          \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(2, \left(k \cdot k\right)\right), \left(\color{blue}{\frac{\frac{\ell}{t}}{k \cdot k}} \cdot \ell\right)\right) \]

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

          \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(2, \mathsf{*.f64}\left(k, k\right)\right), \left(\frac{\frac{\ell}{t}}{\color{blue}{k \cdot k}} \cdot \ell\right)\right) \]

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

          \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(2, \mathsf{*.f64}\left(k, k\right)\right), \mathsf{*.f64}\left(\left(\frac{\frac{\ell}{t}}{k \cdot k}\right), \color{blue}{\ell}\right)\right) \]

       8. associate-/r* N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(2, \mathsf{*.f64}\left(k, k\right)\right), \mathsf{*.f64}\left(\left(\frac{\frac{\frac{\ell}{t}}{k}}{k}\right), \ell\right)\right) \]

       9. associate-/r* N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(2, \mathsf{*.f64}\left(k, k\right)\right), \mathsf{*.f64}\left(\left(\frac{\frac{\ell}{t \cdot k}}{k}\right), \ell\right)\right) \]

       10. *-commutative N/A

           \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(2, \mathsf{*.f64}\left(k, k\right)\right), \mathsf{*.f64}\left(\left(\frac{\frac{\ell}{k \cdot t}}{k}\right), \ell\right)\right) \]

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

           \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(2, \mathsf{*.f64}\left(k, k\right)\right), \mathsf{*.f64}\left(\mathsf{/.f64}\left(\left(\frac{\ell}{k \cdot t}\right), k\right), \ell\right)\right) \]

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

           \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(2, \mathsf{*.f64}\left(k, k\right)\right), \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(\ell, \left(k \cdot t\right)\right), k\right), \ell\right)\right) \]

       13. *-lowering-*.f64 69.9%

           \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(2, \mathsf{*.f64}\left(k, k\right)\right), \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(\ell, \mathsf{*.f64}\left(k, t\right)\right), k\right), \ell\right)\right) \]

   11. Applied egg-rr 69.9%

       \[\leadsto \color{blue}{\frac{2}{k \cdot k} \cdot \left(\frac{\frac{\ell}{k \cdot t}}{k} \cdot \ell\right)} \]

   if 2.8e19 < t

   1. Initial program 27.1%

      \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) - 1\right)} \]
   2. Add Preprocessing

   3. Taylor expanded in k around 0

      \[\leadsto \mathsf{/.f64}\left(2, \color{blue}{\left(\frac{{k}^{4} \cdot t}{{\ell}^{2}}\right)}\right) \]
   4. Step-by-step derivation

      1. associate-/l* N/A

         \[\leadsto \mathsf{/.f64}\left(2, \left({k}^{4} \cdot \color{blue}{\frac{t}{{\ell}^{2}}}\right)\right) \]

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

         \[\leadsto \mathsf{/.f64}\left(2, \mathsf{*.f64}\left(\left({k}^{4}\right), \color{blue}{\left(\frac{t}{{\ell}^{2}}\right)}\right)\right) \]

      3. metadata-eval N/A

         \[\leadsto \mathsf{/.f64}\left(2, \mathsf{*.f64}\left(\left({k}^{\left(2 \cdot 2\right)}\right), \left(\frac{t}{{\ell}^{2}}\right)\right)\right) \]

      4. pow-sqr N/A

         \[\leadsto \mathsf{/.f64}\left(2, \mathsf{*.f64}\left(\left({k}^{2} \cdot {k}^{2}\right), \left(\frac{\color{blue}{t}}{{\ell}^{2}}\right)\right)\right) \]

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

         \[\leadsto \mathsf{/.f64}\left(2, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\left({k}^{2}\right), \left({k}^{2}\right)\right), \left(\frac{\color{blue}{t}}{{\ell}^{2}}\right)\right)\right) \]

      6. unpow2 N/A

         \[\leadsto \mathsf{/.f64}\left(2, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\left(k \cdot k\right), \left({k}^{2}\right)\right), \left(\frac{t}{{\ell}^{2}}\right)\right)\right) \]

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

         \[\leadsto \mathsf{/.f64}\left(2, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(k, k\right), \left({k}^{2}\right)\right), \left(\frac{t}{{\ell}^{2}}\right)\right)\right) \]

      8. unpow2 N/A

         \[\leadsto \mathsf{/.f64}\left(2, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(k, k\right), \left(k \cdot k\right)\right), \left(\frac{t}{{\ell}^{2}}\right)\right)\right) \]

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

         \[\leadsto \mathsf{/.f64}\left(2, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(k, k\right), \mathsf{*.f64}\left(k, k\right)\right), \left(\frac{t}{{\ell}^{2}}\right)\right)\right) \]

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

          \[\leadsto \mathsf{/.f64}\left(2, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(k, k\right), \mathsf{*.f64}\left(k, k\right)\right), \mathsf{/.f64}\left(t, \color{blue}{\left({\ell}^{2}\right)}\right)\right)\right) \]

      11. unpow2 N/A

          \[\leadsto \mathsf{/.f64}\left(2, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(k, k\right), \mathsf{*.f64}\left(k, k\right)\right), \mathsf{/.f64}\left(t, \left(\ell \cdot \color{blue}{\ell}\right)\right)\right)\right) \]

      12. *-lowering-*.f64 56.6%

          \[\leadsto \mathsf{/.f64}\left(2, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(k, k\right), \mathsf{*.f64}\left(k, k\right)\right), \mathsf{/.f64}\left(t, \mathsf{*.f64}\left(\ell, \color{blue}{\ell}\right)\right)\right)\right) \]

   5. Simplified 56.6%

      \[\leadsto \frac{2}{\color{blue}{\left(\left(k \cdot k\right) \cdot \left(k \cdot k\right)\right) \cdot \frac{t}{\ell \cdot \ell}}} \]
   6. Step-by-step derivation

      1. associate-/r* N/A

         \[\leadsto \frac{\frac{2}{\left(k \cdot k\right) \cdot \left(k \cdot k\right)}}{\color{blue}{\frac{t}{\ell \cdot \ell}}} \]

      2. associate-/r* N/A

         \[\leadsto \frac{\frac{2}{\left(k \cdot k\right) \cdot \left(k \cdot k\right)}}{\frac{\frac{t}{\ell}}{\color{blue}{\ell}}} \]

      3. associate-/r/ N/A

         \[\leadsto \frac{\frac{2}{\left(k \cdot k\right) \cdot \left(k \cdot k\right)}}{\frac{t}{\ell}} \cdot \color{blue}{\ell} \]

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

         \[\leadsto \mathsf{*.f64}\left(\left(\frac{\frac{2}{\left(k \cdot k\right) \cdot \left(k \cdot k\right)}}{\frac{t}{\ell}}\right), \color{blue}{\ell}\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\left(\frac{2}{\left(k \cdot k\right) \cdot \left(k \cdot k\right)}\right), \left(\frac{t}{\ell}\right)\right), \ell\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(2, \left(\left(k \cdot k\right) \cdot \left(k \cdot k\right)\right)\right), \left(\frac{t}{\ell}\right)\right), \ell\right) \]

      7. associate-*l* N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(2, \left(k \cdot \left(k \cdot \left(k \cdot k\right)\right)\right)\right), \left(\frac{t}{\ell}\right)\right), \ell\right) \]

      8. cube-unmult N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(2, \left(k \cdot {k}^{3}\right)\right), \left(\frac{t}{\ell}\right)\right), \ell\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(2, \mathsf{*.f64}\left(k, \left({k}^{3}\right)\right)\right), \left(\frac{t}{\ell}\right)\right), \ell\right) \]

      10. cube-unmult N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(2, \mathsf{*.f64}\left(k, \left(k \cdot \left(k \cdot k\right)\right)\right)\right), \left(\frac{t}{\ell}\right)\right), \ell\right) \]

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

          \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(2, \mathsf{*.f64}\left(k, \mathsf{*.f64}\left(k, \left(k \cdot k\right)\right)\right)\right), \left(\frac{t}{\ell}\right)\right), \ell\right) \]

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

          \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(2, \mathsf{*.f64}\left(k, \mathsf{*.f64}\left(k, \mathsf{*.f64}\left(k, k\right)\right)\right)\right), \left(\frac{t}{\ell}\right)\right), \ell\right) \]

      13. /-lowering-/.f64 65.2%

          \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(2, \mathsf{*.f64}\left(k, \mathsf{*.f64}\left(k, \mathsf{*.f64}\left(k, k\right)\right)\right)\right), \mathsf{/.f64}\left(t, \ell\right)\right), \ell\right) \]

   7. Applied egg-rr 65.2%

      \[\leadsto \color{blue}{\frac{\frac{2}{k \cdot \left(k \cdot \left(k \cdot k\right)\right)}}{\frac{t}{\ell}} \cdot \ell} \]
   8. Step-by-step derivation

      1. div-inv N/A

         \[\leadsto \mathsf{*.f64}\left(\left(\frac{2}{k \cdot \left(k \cdot \left(k \cdot k\right)\right)} \cdot \frac{1}{\frac{t}{\ell}}\right), \ell\right) \]

      2. associate-/r* N/A

         \[\leadsto \mathsf{*.f64}\left(\left(\frac{\frac{2}{k}}{k \cdot \left(k \cdot k\right)} \cdot \frac{1}{\frac{t}{\ell}}\right), \ell\right) \]

      3. clear-num N/A

         \[\leadsto \mathsf{*.f64}\left(\left(\frac{\frac{2}{k}}{k \cdot \left(k \cdot k\right)} \cdot \frac{\ell}{t}\right), \ell\right) \]

      4. associate-*l/ N/A

         \[\leadsto \mathsf{*.f64}\left(\left(\frac{\frac{2}{k} \cdot \frac{\ell}{t}}{k \cdot \left(k \cdot k\right)}\right), \ell\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\left(\frac{2}{k} \cdot \frac{\ell}{t}\right), \left(k \cdot \left(k \cdot k\right)\right)\right), \ell\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\left(\frac{2}{k}\right), \left(\frac{\ell}{t}\right)\right), \left(k \cdot \left(k \cdot k\right)\right)\right), \ell\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(2, k\right), \left(\frac{\ell}{t}\right)\right), \left(k \cdot \left(k \cdot k\right)\right)\right), \ell\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(2, k\right), \mathsf{/.f64}\left(\ell, t\right)\right), \left(k \cdot \left(k \cdot k\right)\right)\right), \ell\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(2, k\right), \mathsf{/.f64}\left(\ell, t\right)\right), \mathsf{*.f64}\left(k, \left(k \cdot k\right)\right)\right), \ell\right) \]

      10. *-lowering-*.f64 69.2%

          \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(2, k\right), \mathsf{/.f64}\left(\ell, t\right)\right), \mathsf{*.f64}\left(k, \mathsf{*.f64}\left(k, k\right)\right)\right), \ell\right) \]

   9. Applied egg-rr 69.2%

      \[\leadsto \color{blue}{\frac{\frac{2}{k} \cdot \frac{\ell}{t}}{k \cdot \left(k \cdot k\right)}} \cdot \ell \]
   10. Step-by-step derivation

       1. associate-/r* N/A

          \[\leadsto \mathsf{*.f64}\left(\left(\frac{\frac{\frac{2}{k} \cdot \frac{\ell}{t}}{k}}{k \cdot k}\right), \ell\right) \]

       2. associate-/r* N/A

          \[\leadsto \mathsf{*.f64}\left(\left(\frac{\frac{\frac{\frac{2}{k} \cdot \frac{\ell}{t}}{k}}{k}}{k}\right), \ell\right) \]

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

          \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\left(\frac{\frac{\frac{2}{k} \cdot \frac{\ell}{t}}{k}}{k}\right), k\right), \ell\right) \]

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

          \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(\left(\frac{\frac{2}{k} \cdot \frac{\ell}{t}}{k}\right), k\right), k\right), \ell\right) \]

       5. frac-times N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(\left(\frac{\frac{2 \cdot \ell}{k \cdot t}}{k}\right), k\right), k\right), \ell\right) \]

       6. associate-/l/ N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(\left(\frac{2 \cdot \ell}{k \cdot \left(k \cdot t\right)}\right), k\right), k\right), \ell\right) \]

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

          \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(\left(2 \cdot \ell\right), \left(k \cdot \left(k \cdot t\right)\right)\right), k\right), k\right), \ell\right) \]

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

          \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(2, \ell\right), \left(k \cdot \left(k \cdot t\right)\right)\right), k\right), k\right), \ell\right) \]

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

          \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(2, \ell\right), \mathsf{*.f64}\left(k, \left(k \cdot t\right)\right)\right), k\right), k\right), \ell\right) \]

       10. *-lowering-*.f64 83.8%

           \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(2, \ell\right), \mathsf{*.f64}\left(k, \mathsf{*.f64}\left(k, t\right)\right)\right), k\right), k\right), \ell\right) \]

   11. Applied egg-rr 83.8%

       \[\leadsto \color{blue}{\frac{\frac{\frac{2 \cdot \ell}{k \cdot \left(k \cdot t\right)}}{k}}{k}} \cdot \ell \]
3. Recombined 2 regimes into one program.

4. Final simplification 72.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq 2.8 \cdot 10^{+19}:\\ \;\;\;\;\frac{2}{k \cdot k} \cdot \left(\ell \cdot \frac{\frac{\ell}{k \cdot t}}{k}\right)\\ \mathbf{else}:\\ \;\;\;\;\ell \cdot \frac{\frac{\frac{2 \cdot \ell}{k \cdot \left(k \cdot t\right)}}{k}}{k}\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 73.6% accurate, 28.1× speedup

Math

\[\begin{array}{l} k_m = \left|k\right| \\ \frac{2}{k\_m \cdot k\_m} \cdot \left(\ell \cdot \frac{\frac{\ell}{k\_m \cdot t}}{k\_m}\right) \end{array} \]

FPCore

k_m = (fabs.f64 k)
(FPCore (t l k_m)
 :precision binary64
 (* (/ 2.0 (* k_m k_m)) (* l (/ (/ l (* k_m t)) k_m))))

C

k_m = fabs(k);
double code(double t, double l, double k_m) {
	return (2.0 / (k_m * k_m)) * (l * ((l / (k_m * t)) / k_m));
}

Fortran

k_m = abs(k)
real(8) function code(t, l, k_m)
    real(8), intent (in) :: t
    real(8), intent (in) :: l
    real(8), intent (in) :: k_m
    code = (2.0d0 / (k_m * k_m)) * (l * ((l / (k_m * t)) / k_m))
end function

Java

k_m = Math.abs(k);
public static double code(double t, double l, double k_m) {
	return (2.0 / (k_m * k_m)) * (l * ((l / (k_m * t)) / k_m));
}

Python

k_m = math.fabs(k)
def code(t, l, k_m):
	return (2.0 / (k_m * k_m)) * (l * ((l / (k_m * t)) / k_m))

Julia

k_m = abs(k)
function code(t, l, k_m)
	return Float64(Float64(2.0 / Float64(k_m * k_m)) * Float64(l * Float64(Float64(l / Float64(k_m * t)) / k_m)))
end

MATLAB

k_m = abs(k);
function tmp = code(t, l, k_m)
	tmp = (2.0 / (k_m * k_m)) * (l * ((l / (k_m * t)) / k_m));
end

Wolfram

k_m = N[Abs[k], $MachinePrecision]
code[t_, l_, k$95$m_] := N[(N[(2.0 / N[(k$95$m * k$95$m), $MachinePrecision]), $MachinePrecision] * N[(l * N[(N[(l / N[(k$95$m * t), $MachinePrecision]), $MachinePrecision] / k$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 31.6%

   \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) - 1\right)} \]
2. Add Preprocessing

3. Taylor expanded in k around 0

   \[\leadsto \mathsf{/.f64}\left(2, \color{blue}{\left(\frac{{k}^{4} \cdot t}{{\ell}^{2}}\right)}\right) \]
4. Step-by-step derivation

   1. associate-/l* N/A

      \[\leadsto \mathsf{/.f64}\left(2, \left({k}^{4} \cdot \color{blue}{\frac{t}{{\ell}^{2}}}\right)\right) \]

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

      \[\leadsto \mathsf{/.f64}\left(2, \mathsf{*.f64}\left(\left({k}^{4}\right), \color{blue}{\left(\frac{t}{{\ell}^{2}}\right)}\right)\right) \]

   3. metadata-eval N/A

      \[\leadsto \mathsf{/.f64}\left(2, \mathsf{*.f64}\left(\left({k}^{\left(2 \cdot 2\right)}\right), \left(\frac{t}{{\ell}^{2}}\right)\right)\right) \]

   4. pow-sqr N/A

      \[\leadsto \mathsf{/.f64}\left(2, \mathsf{*.f64}\left(\left({k}^{2} \cdot {k}^{2}\right), \left(\frac{\color{blue}{t}}{{\ell}^{2}}\right)\right)\right) \]

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

      \[\leadsto \mathsf{/.f64}\left(2, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\left({k}^{2}\right), \left({k}^{2}\right)\right), \left(\frac{\color{blue}{t}}{{\ell}^{2}}\right)\right)\right) \]

   6. unpow2 N/A

      \[\leadsto \mathsf{/.f64}\left(2, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\left(k \cdot k\right), \left({k}^{2}\right)\right), \left(\frac{t}{{\ell}^{2}}\right)\right)\right) \]

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

      \[\leadsto \mathsf{/.f64}\left(2, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(k, k\right), \left({k}^{2}\right)\right), \left(\frac{t}{{\ell}^{2}}\right)\right)\right) \]

   8. unpow2 N/A

      \[\leadsto \mathsf{/.f64}\left(2, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(k, k\right), \left(k \cdot k\right)\right), \left(\frac{t}{{\ell}^{2}}\right)\right)\right) \]

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

      \[\leadsto \mathsf{/.f64}\left(2, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(k, k\right), \mathsf{*.f64}\left(k, k\right)\right), \left(\frac{t}{{\ell}^{2}}\right)\right)\right) \]

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

       \[\leadsto \mathsf{/.f64}\left(2, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(k, k\right), \mathsf{*.f64}\left(k, k\right)\right), \mathsf{/.f64}\left(t, \color{blue}{\left({\ell}^{2}\right)}\right)\right)\right) \]

   11. unpow2 N/A

       \[\leadsto \mathsf{/.f64}\left(2, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(k, k\right), \mathsf{*.f64}\left(k, k\right)\right), \mathsf{/.f64}\left(t, \left(\ell \cdot \color{blue}{\ell}\right)\right)\right)\right) \]

   12. *-lowering-*.f64 54.6%

       \[\leadsto \mathsf{/.f64}\left(2, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(k, k\right), \mathsf{*.f64}\left(k, k\right)\right), \mathsf{/.f64}\left(t, \mathsf{*.f64}\left(\ell, \color{blue}{\ell}\right)\right)\right)\right) \]

5. Simplified 54.6%

   \[\leadsto \frac{2}{\color{blue}{\left(\left(k \cdot k\right) \cdot \left(k \cdot k\right)\right) \cdot \frac{t}{\ell \cdot \ell}}} \]
6. Step-by-step derivation

   1. associate-/r* N/A

      \[\leadsto \frac{\frac{2}{\left(k \cdot k\right) \cdot \left(k \cdot k\right)}}{\color{blue}{\frac{t}{\ell \cdot \ell}}} \]

   2. associate-/r* N/A

      \[\leadsto \frac{\frac{2}{\left(k \cdot k\right) \cdot \left(k \cdot k\right)}}{\frac{\frac{t}{\ell}}{\color{blue}{\ell}}} \]

   3. associate-/r/ N/A

      \[\leadsto \frac{\frac{2}{\left(k \cdot k\right) \cdot \left(k \cdot k\right)}}{\frac{t}{\ell}} \cdot \color{blue}{\ell} \]

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

      \[\leadsto \mathsf{*.f64}\left(\left(\frac{\frac{2}{\left(k \cdot k\right) \cdot \left(k \cdot k\right)}}{\frac{t}{\ell}}\right), \color{blue}{\ell}\right) \]

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

      \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\left(\frac{2}{\left(k \cdot k\right) \cdot \left(k \cdot k\right)}\right), \left(\frac{t}{\ell}\right)\right), \ell\right) \]

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

      \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(2, \left(\left(k \cdot k\right) \cdot \left(k \cdot k\right)\right)\right), \left(\frac{t}{\ell}\right)\right), \ell\right) \]

   7. associate-*l* N/A

      \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(2, \left(k \cdot \left(k \cdot \left(k \cdot k\right)\right)\right)\right), \left(\frac{t}{\ell}\right)\right), \ell\right) \]

   8. cube-unmult N/A

      \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(2, \left(k \cdot {k}^{3}\right)\right), \left(\frac{t}{\ell}\right)\right), \ell\right) \]

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

      \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(2, \mathsf{*.f64}\left(k, \left({k}^{3}\right)\right)\right), \left(\frac{t}{\ell}\right)\right), \ell\right) \]

   10. cube-unmult N/A

       \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(2, \mathsf{*.f64}\left(k, \left(k \cdot \left(k \cdot k\right)\right)\right)\right), \left(\frac{t}{\ell}\right)\right), \ell\right) \]

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

       \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(2, \mathsf{*.f64}\left(k, \mathsf{*.f64}\left(k, \left(k \cdot k\right)\right)\right)\right), \left(\frac{t}{\ell}\right)\right), \ell\right) \]

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

       \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(2, \mathsf{*.f64}\left(k, \mathsf{*.f64}\left(k, \mathsf{*.f64}\left(k, k\right)\right)\right)\right), \left(\frac{t}{\ell}\right)\right), \ell\right) \]

   13. /-lowering-/.f64 63.6%

       \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(2, \mathsf{*.f64}\left(k, \mathsf{*.f64}\left(k, \mathsf{*.f64}\left(k, k\right)\right)\right)\right), \mathsf{/.f64}\left(t, \ell\right)\right), \ell\right) \]

7. Applied egg-rr 63.6%

   \[\leadsto \color{blue}{\frac{\frac{2}{k \cdot \left(k \cdot \left(k \cdot k\right)\right)}}{\frac{t}{\ell}} \cdot \ell} \]
8. Step-by-step derivation

   1. div-inv N/A

      \[\leadsto \mathsf{*.f64}\left(\left(\frac{2}{k \cdot \left(k \cdot \left(k \cdot k\right)\right)} \cdot \frac{1}{\frac{t}{\ell}}\right), \ell\right) \]

   2. associate-/r* N/A

      \[\leadsto \mathsf{*.f64}\left(\left(\frac{\frac{2}{k}}{k \cdot \left(k \cdot k\right)} \cdot \frac{1}{\frac{t}{\ell}}\right), \ell\right) \]

   3. clear-num N/A

      \[\leadsto \mathsf{*.f64}\left(\left(\frac{\frac{2}{k}}{k \cdot \left(k \cdot k\right)} \cdot \frac{\ell}{t}\right), \ell\right) \]

   4. associate-*l/ N/A

      \[\leadsto \mathsf{*.f64}\left(\left(\frac{\frac{2}{k} \cdot \frac{\ell}{t}}{k \cdot \left(k \cdot k\right)}\right), \ell\right) \]

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

      \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\left(\frac{2}{k} \cdot \frac{\ell}{t}\right), \left(k \cdot \left(k \cdot k\right)\right)\right), \ell\right) \]

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

      \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\left(\frac{2}{k}\right), \left(\frac{\ell}{t}\right)\right), \left(k \cdot \left(k \cdot k\right)\right)\right), \ell\right) \]

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

      \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(2, k\right), \left(\frac{\ell}{t}\right)\right), \left(k \cdot \left(k \cdot k\right)\right)\right), \ell\right) \]

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

      \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(2, k\right), \mathsf{/.f64}\left(\ell, t\right)\right), \left(k \cdot \left(k \cdot k\right)\right)\right), \ell\right) \]

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

      \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(2, k\right), \mathsf{/.f64}\left(\ell, t\right)\right), \mathsf{*.f64}\left(k, \left(k \cdot k\right)\right)\right), \ell\right) \]

   10. *-lowering-*.f64 66.3%

       \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(2, k\right), \mathsf{/.f64}\left(\ell, t\right)\right), \mathsf{*.f64}\left(k, \mathsf{*.f64}\left(k, k\right)\right)\right), \ell\right) \]

9. Applied egg-rr 66.3%

   \[\leadsto \color{blue}{\frac{\frac{2}{k} \cdot \frac{\ell}{t}}{k \cdot \left(k \cdot k\right)}} \cdot \ell \]
10. Step-by-step derivation

    1. times-frac N/A

       \[\leadsto \left(\frac{\frac{2}{k}}{k} \cdot \frac{\frac{\ell}{t}}{k \cdot k}\right) \cdot \ell \]

    2. associate-*l* N/A

       \[\leadsto \frac{\frac{2}{k}}{k} \cdot \color{blue}{\left(\frac{\frac{\ell}{t}}{k \cdot k} \cdot \ell\right)} \]

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

       \[\leadsto \mathsf{*.f64}\left(\left(\frac{\frac{2}{k}}{k}\right), \color{blue}{\left(\frac{\frac{\ell}{t}}{k \cdot k} \cdot \ell\right)}\right) \]

    4. associate-/l/ N/A

       \[\leadsto \mathsf{*.f64}\left(\left(\frac{2}{k \cdot k}\right), \left(\color{blue}{\frac{\frac{\ell}{t}}{k \cdot k}} \cdot \ell\right)\right) \]

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

       \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(2, \left(k \cdot k\right)\right), \left(\color{blue}{\frac{\frac{\ell}{t}}{k \cdot k}} \cdot \ell\right)\right) \]

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

       \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(2, \mathsf{*.f64}\left(k, k\right)\right), \left(\frac{\frac{\ell}{t}}{\color{blue}{k \cdot k}} \cdot \ell\right)\right) \]

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

       \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(2, \mathsf{*.f64}\left(k, k\right)\right), \mathsf{*.f64}\left(\left(\frac{\frac{\ell}{t}}{k \cdot k}\right), \color{blue}{\ell}\right)\right) \]

    8. associate-/r* N/A

       \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(2, \mathsf{*.f64}\left(k, k\right)\right), \mathsf{*.f64}\left(\left(\frac{\frac{\frac{\ell}{t}}{k}}{k}\right), \ell\right)\right) \]

    9. associate-/r* N/A

       \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(2, \mathsf{*.f64}\left(k, k\right)\right), \mathsf{*.f64}\left(\left(\frac{\frac{\ell}{t \cdot k}}{k}\right), \ell\right)\right) \]

    10. *-commutative N/A

        \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(2, \mathsf{*.f64}\left(k, k\right)\right), \mathsf{*.f64}\left(\left(\frac{\frac{\ell}{k \cdot t}}{k}\right), \ell\right)\right) \]

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

        \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(2, \mathsf{*.f64}\left(k, k\right)\right), \mathsf{*.f64}\left(\mathsf{/.f64}\left(\left(\frac{\ell}{k \cdot t}\right), k\right), \ell\right)\right) \]

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

        \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(2, \mathsf{*.f64}\left(k, k\right)\right), \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(\ell, \left(k \cdot t\right)\right), k\right), \ell\right)\right) \]

    13. *-lowering-*.f64 70.6%

        \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(2, \mathsf{*.f64}\left(k, k\right)\right), \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(\ell, \mathsf{*.f64}\left(k, t\right)\right), k\right), \ell\right)\right) \]

11. Applied egg-rr 70.6%

    \[\leadsto \color{blue}{\frac{2}{k \cdot k} \cdot \left(\frac{\frac{\ell}{k \cdot t}}{k} \cdot \ell\right)} \]

12. Final simplification 70.6%

    \[\leadsto \frac{2}{k \cdot k} \cdot \left(\ell \cdot \frac{\frac{\ell}{k \cdot t}}{k}\right) \]
13. Add Preprocessing

Alternative 10: 30.1% accurate, 32.4× speedup

Math

\[\begin{array}{l} k_m = \left|k\right| \\ \left(2 \cdot \ell\right) \cdot \frac{\frac{\ell \cdot -0.16666666666666666}{t}}{k\_m \cdot k\_m} \end{array} \]

FPCore

k_m = (fabs.f64 k)
(FPCore (t l k_m)
 :precision binary64
 (* (* 2.0 l) (/ (/ (* l -0.16666666666666666) t) (* k_m k_m))))

C

k_m = fabs(k);
double code(double t, double l, double k_m) {
	return (2.0 * l) * (((l * -0.16666666666666666) / t) / (k_m * k_m));
}

Fortran

k_m = abs(k)
real(8) function code(t, l, k_m)
    real(8), intent (in) :: t
    real(8), intent (in) :: l
    real(8), intent (in) :: k_m
    code = (2.0d0 * l) * (((l * (-0.16666666666666666d0)) / t) / (k_m * k_m))
end function

Java

k_m = Math.abs(k);
public static double code(double t, double l, double k_m) {
	return (2.0 * l) * (((l * -0.16666666666666666) / t) / (k_m * k_m));
}

Python

k_m = math.fabs(k)
def code(t, l, k_m):
	return (2.0 * l) * (((l * -0.16666666666666666) / t) / (k_m * k_m))

Julia

k_m = abs(k)
function code(t, l, k_m)
	return Float64(Float64(2.0 * l) * Float64(Float64(Float64(l * -0.16666666666666666) / t) / Float64(k_m * k_m)))
end

MATLAB

k_m = abs(k);
function tmp = code(t, l, k_m)
	tmp = (2.0 * l) * (((l * -0.16666666666666666) / t) / (k_m * k_m));
end

Wolfram

k_m = N[Abs[k], $MachinePrecision]
code[t_, l_, k$95$m_] := N[(N[(2.0 * l), $MachinePrecision] * N[(N[(N[(l * -0.16666666666666666), $MachinePrecision] / t), $MachinePrecision] / N[(k$95$m * k$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 31.6%

   \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) - 1\right)} \]
2. Add Preprocessing

3. Taylor expanded in t around 0

   \[\leadsto \color{blue}{2 \cdot \frac{{\ell}^{2} \cdot \cos k}{{k}^{2} \cdot \left(t \cdot {\sin k}^{2}\right)}} \]
4. Step-by-step derivation

   1. associate-*r/ N/A

      \[\leadsto \frac{2 \cdot \left({\ell}^{2} \cdot \cos k\right)}{\color{blue}{{k}^{2} \cdot \left(t \cdot {\sin k}^{2}\right)}} \]

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

      \[\leadsto \mathsf{/.f64}\left(\left(2 \cdot \left({\ell}^{2} \cdot \cos k\right)\right), \color{blue}{\left({k}^{2} \cdot \left(t \cdot {\sin k}^{2}\right)\right)}\right) \]

   3. associate-*r* N/A

      \[\leadsto \mathsf{/.f64}\left(\left(\left(2 \cdot {\ell}^{2}\right) \cdot \cos k\right), \left(\color{blue}{{k}^{2}} \cdot \left(t \cdot {\sin k}^{2}\right)\right)\right) \]

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

      \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\left(2 \cdot {\ell}^{2}\right), \cos k\right), \left(\color{blue}{{k}^{2}} \cdot \left(t \cdot {\sin k}^{2}\right)\right)\right) \]

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

      \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(2, \left({\ell}^{2}\right)\right), \cos k\right), \left({\color{blue}{k}}^{2} \cdot \left(t \cdot {\sin k}^{2}\right)\right)\right) \]

   6. unpow2 N/A

      \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(2, \left(\ell \cdot \ell\right)\right), \cos k\right), \left({k}^{2} \cdot \left(t \cdot {\sin k}^{2}\right)\right)\right) \]

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

      \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(2, \mathsf{*.f64}\left(\ell, \ell\right)\right), \cos k\right), \left({k}^{2} \cdot \left(t \cdot {\sin k}^{2}\right)\right)\right) \]

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

      \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(2, \mathsf{*.f64}\left(\ell, \ell\right)\right), \mathsf{cos.f64}\left(k\right)\right), \left({k}^{\color{blue}{2}} \cdot \left(t \cdot {\sin k}^{2}\right)\right)\right) \]

   9. unpow2 N/A

      \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(2, \mathsf{*.f64}\left(\ell, \ell\right)\right), \mathsf{cos.f64}\left(k\right)\right), \left(\left(k \cdot k\right) \cdot \left(\color{blue}{t} \cdot {\sin k}^{2}\right)\right)\right) \]

   10. associate-*l* N/A

       \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(2, \mathsf{*.f64}\left(\ell, \ell\right)\right), \mathsf{cos.f64}\left(k\right)\right), \left(k \cdot \color{blue}{\left(k \cdot \left(t \cdot {\sin k}^{2}\right)\right)}\right)\right) \]

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

       \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(2, \mathsf{*.f64}\left(\ell, \ell\right)\right), \mathsf{cos.f64}\left(k\right)\right), \mathsf{*.f64}\left(k, \color{blue}{\left(k \cdot \left(t \cdot {\sin k}^{2}\right)\right)}\right)\right) \]

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

       \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(2, \mathsf{*.f64}\left(\ell, \ell\right)\right), \mathsf{cos.f64}\left(k\right)\right), \mathsf{*.f64}\left(k, \mathsf{*.f64}\left(k, \color{blue}{\left(t \cdot {\sin k}^{2}\right)}\right)\right)\right) \]

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

       \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(2, \mathsf{*.f64}\left(\ell, \ell\right)\right), \mathsf{cos.f64}\left(k\right)\right), \mathsf{*.f64}\left(k, \mathsf{*.f64}\left(k, \mathsf{*.f64}\left(t, \color{blue}{\left({\sin k}^{2}\right)}\right)\right)\right)\right) \]

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

       \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(2, \mathsf{*.f64}\left(\ell, \ell\right)\right), \mathsf{cos.f64}\left(k\right)\right), \mathsf{*.f64}\left(k, \mathsf{*.f64}\left(k, \mathsf{*.f64}\left(t, \mathsf{pow.f64}\left(\sin k, \color{blue}{2}\right)\right)\right)\right)\right) \]

   15. sin-lowering-sin.f64 69.3%

       \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(2, \mathsf{*.f64}\left(\ell, \ell\right)\right), \mathsf{cos.f64}\left(k\right)\right), \mathsf{*.f64}\left(k, \mathsf{*.f64}\left(k, \mathsf{*.f64}\left(t, \mathsf{pow.f64}\left(\mathsf{sin.f64}\left(k\right), 2\right)\right)\right)\right)\right) \]

5. Simplified 69.3%

   \[\leadsto \color{blue}{\frac{\left(2 \cdot \left(\ell \cdot \ell\right)\right) \cdot \cos k}{k \cdot \left(k \cdot \left(t \cdot {\sin k}^{2}\right)\right)}} \]
6. Step-by-step derivation

   1. associate-/l* N/A

      \[\leadsto \left(2 \cdot \left(\ell \cdot \ell\right)\right) \cdot \color{blue}{\frac{\cos k}{k \cdot \left(k \cdot \left(t \cdot {\sin k}^{2}\right)\right)}} \]

   2. associate-*r* N/A

      \[\leadsto \left(\left(2 \cdot \ell\right) \cdot \ell\right) \cdot \frac{\color{blue}{\cos k}}{k \cdot \left(k \cdot \left(t \cdot {\sin k}^{2}\right)\right)} \]

   3. associate-*l* N/A

      \[\leadsto \left(2 \cdot \ell\right) \cdot \color{blue}{\left(\ell \cdot \frac{\cos k}{k \cdot \left(k \cdot \left(t \cdot {\sin k}^{2}\right)\right)}\right)} \]

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

      \[\leadsto \mathsf{*.f64}\left(\left(2 \cdot \ell\right), \color{blue}{\left(\ell \cdot \frac{\cos k}{k \cdot \left(k \cdot \left(t \cdot {\sin k}^{2}\right)\right)}\right)}\right) \]

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

      \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(2, \ell\right), \left(\color{blue}{\ell} \cdot \frac{\cos k}{k \cdot \left(k \cdot \left(t \cdot {\sin k}^{2}\right)\right)}\right)\right) \]

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

      \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(2, \ell\right), \mathsf{*.f64}\left(\ell, \color{blue}{\left(\frac{\cos k}{k \cdot \left(k \cdot \left(t \cdot {\sin k}^{2}\right)\right)}\right)}\right)\right) \]

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

      \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(2, \ell\right), \mathsf{*.f64}\left(\ell, \mathsf{/.f64}\left(\cos k, \color{blue}{\left(k \cdot \left(k \cdot \left(t \cdot {\sin k}^{2}\right)\right)\right)}\right)\right)\right) \]

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

      \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(2, \ell\right), \mathsf{*.f64}\left(\ell, \mathsf{/.f64}\left(\mathsf{cos.f64}\left(k\right), \left(\color{blue}{k} \cdot \left(k \cdot \left(t \cdot {\sin k}^{2}\right)\right)\right)\right)\right)\right) \]

   9. associate-*r* N/A

      \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(2, \ell\right), \mathsf{*.f64}\left(\ell, \mathsf{/.f64}\left(\mathsf{cos.f64}\left(k\right), \left(\left(k \cdot k\right) \cdot \color{blue}{\left(t \cdot {\sin k}^{2}\right)}\right)\right)\right)\right) \]

   10. associate-*r* N/A

       \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(2, \ell\right), \mathsf{*.f64}\left(\ell, \mathsf{/.f64}\left(\mathsf{cos.f64}\left(k\right), \left(\left(\left(k \cdot k\right) \cdot t\right) \cdot \color{blue}{{\sin k}^{2}}\right)\right)\right)\right) \]

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

       \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(2, \ell\right), \mathsf{*.f64}\left(\ell, \mathsf{/.f64}\left(\mathsf{cos.f64}\left(k\right), \mathsf{*.f64}\left(\left(\left(k \cdot k\right) \cdot t\right), \color{blue}{\left({\sin k}^{2}\right)}\right)\right)\right)\right) \]

   12. *-commutative N/A

       \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(2, \ell\right), \mathsf{*.f64}\left(\ell, \mathsf{/.f64}\left(\mathsf{cos.f64}\left(k\right), \mathsf{*.f64}\left(\left(t \cdot \left(k \cdot k\right)\right), \left({\color{blue}{\sin k}}^{2}\right)\right)\right)\right)\right) \]

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

       \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(2, \ell\right), \mathsf{*.f64}\left(\ell, \mathsf{/.f64}\left(\mathsf{cos.f64}\left(k\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(t, \left(k \cdot k\right)\right), \left({\color{blue}{\sin k}}^{2}\right)\right)\right)\right)\right) \]

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

       \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(2, \ell\right), \mathsf{*.f64}\left(\ell, \mathsf{/.f64}\left(\mathsf{cos.f64}\left(k\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(t, \mathsf{*.f64}\left(k, k\right)\right), \left({\sin k}^{2}\right)\right)\right)\right)\right) \]

   15. unpow2 N/A

       \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(2, \ell\right), \mathsf{*.f64}\left(\ell, \mathsf{/.f64}\left(\mathsf{cos.f64}\left(k\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(t, \mathsf{*.f64}\left(k, k\right)\right), \left(\sin k \cdot \color{blue}{\sin k}\right)\right)\right)\right)\right) \]

   16. sqr-sin-a N/A

       \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(2, \ell\right), \mathsf{*.f64}\left(\ell, \mathsf{/.f64}\left(\mathsf{cos.f64}\left(k\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(t, \mathsf{*.f64}\left(k, k\right)\right), \left(\frac{1}{2} - \color{blue}{\frac{1}{2} \cdot \cos \left(2 \cdot k\right)}\right)\right)\right)\right)\right) \]

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

       \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(2, \ell\right), \mathsf{*.f64}\left(\ell, \mathsf{/.f64}\left(\mathsf{cos.f64}\left(k\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(t, \mathsf{*.f64}\left(k, k\right)\right), \mathsf{\_.f64}\left(\frac{1}{2}, \color{blue}{\left(\frac{1}{2} \cdot \cos \left(2 \cdot k\right)\right)}\right)\right)\right)\right)\right) \]

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

       \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(2, \ell\right), \mathsf{*.f64}\left(\ell, \mathsf{/.f64}\left(\mathsf{cos.f64}\left(k\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(t, \mathsf{*.f64}\left(k, k\right)\right), \mathsf{\_.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(\frac{1}{2}, \color{blue}{\cos \left(2 \cdot k\right)}\right)\right)\right)\right)\right)\right) \]

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

       \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(2, \ell\right), \mathsf{*.f64}\left(\ell, \mathsf{/.f64}\left(\mathsf{cos.f64}\left(k\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(t, \mathsf{*.f64}\left(k, k\right)\right), \mathsf{\_.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(\frac{1}{2}, \mathsf{cos.f64}\left(\left(2 \cdot k\right)\right)\right)\right)\right)\right)\right)\right) \]

   20. *-lowering-*.f64 70.1%

       \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(2, \ell\right), \mathsf{*.f64}\left(\ell, \mathsf{/.f64}\left(\mathsf{cos.f64}\left(k\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(t, \mathsf{*.f64}\left(k, k\right)\right), \mathsf{\_.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(\frac{1}{2}, \mathsf{cos.f64}\left(\mathsf{*.f64}\left(2, k\right)\right)\right)\right)\right)\right)\right)\right) \]

7. Applied egg-rr 70.1%

   \[\leadsto \color{blue}{\left(2 \cdot \ell\right) \cdot \left(\ell \cdot \frac{\cos k}{\left(t \cdot \left(k \cdot k\right)\right) \cdot \left(0.5 - 0.5 \cdot \cos \left(2 \cdot k\right)\right)}\right)} \]

8. Taylor expanded in k around 0

   \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(2, \ell\right), \color{blue}{\left(\frac{{k}^{2} \cdot \left(\frac{-1}{2} \cdot \frac{\ell}{t} - \frac{-1}{3} \cdot \frac{\ell}{t}\right) + \frac{\ell}{t}}{{k}^{4}}\right)}\right) \]
9. Step-by-step derivation

   1. /-lowering-/.f64 N/A

      \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(2, \ell\right), \mathsf{/.f64}\left(\left({k}^{2} \cdot \left(\frac{-1}{2} \cdot \frac{\ell}{t} - \frac{-1}{3} \cdot \frac{\ell}{t}\right) + \frac{\ell}{t}\right), \color{blue}{\left({k}^{4}\right)}\right)\right) \]

   2. +-commutative N/A

      \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(2, \ell\right), \mathsf{/.f64}\left(\left(\frac{\ell}{t} + {k}^{2} \cdot \left(\frac{-1}{2} \cdot \frac{\ell}{t} - \frac{-1}{3} \cdot \frac{\ell}{t}\right)\right), \left({\color{blue}{k}}^{4}\right)\right)\right) \]

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

      \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(2, \ell\right), \mathsf{/.f64}\left(\mathsf{+.f64}\left(\left(\frac{\ell}{t}\right), \left({k}^{2} \cdot \left(\frac{-1}{2} \cdot \frac{\ell}{t} - \frac{-1}{3} \cdot \frac{\ell}{t}\right)\right)\right), \left({\color{blue}{k}}^{4}\right)\right)\right) \]

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

      \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(2, \ell\right), \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{/.f64}\left(\ell, t\right), \left({k}^{2} \cdot \left(\frac{-1}{2} \cdot \frac{\ell}{t} - \frac{-1}{3} \cdot \frac{\ell}{t}\right)\right)\right), \left({k}^{4}\right)\right)\right) \]

   5. unpow2 N/A

      \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(2, \ell\right), \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{/.f64}\left(\ell, t\right), \left(\left(k \cdot k\right) \cdot \left(\frac{-1}{2} \cdot \frac{\ell}{t} - \frac{-1}{3} \cdot \frac{\ell}{t}\right)\right)\right), \left({k}^{4}\right)\right)\right) \]

   6. associate-*l* N/A

      \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(2, \ell\right), \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{/.f64}\left(\ell, t\right), \left(k \cdot \left(k \cdot \left(\frac{-1}{2} \cdot \frac{\ell}{t} - \frac{-1}{3} \cdot \frac{\ell}{t}\right)\right)\right)\right), \left({k}^{4}\right)\right)\right) \]

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

      \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(2, \ell\right), \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{/.f64}\left(\ell, t\right), \mathsf{*.f64}\left(k, \left(k \cdot \left(\frac{-1}{2} \cdot \frac{\ell}{t} - \frac{-1}{3} \cdot \frac{\ell}{t}\right)\right)\right)\right), \left({k}^{4}\right)\right)\right) \]

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

      \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(2, \ell\right), \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{/.f64}\left(\ell, t\right), \mathsf{*.f64}\left(k, \mathsf{*.f64}\left(k, \left(\frac{-1}{2} \cdot \frac{\ell}{t} - \frac{-1}{3} \cdot \frac{\ell}{t}\right)\right)\right)\right), \left({k}^{4}\right)\right)\right) \]

   9. distribute-rgt-out-- N/A

      \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(2, \ell\right), \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{/.f64}\left(\ell, t\right), \mathsf{*.f64}\left(k, \mathsf{*.f64}\left(k, \left(\frac{\ell}{t} \cdot \left(\frac{-1}{2} - \frac{-1}{3}\right)\right)\right)\right)\right), \left({k}^{4}\right)\right)\right) \]

   10. metadata-eval N/A

       \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(2, \ell\right), \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{/.f64}\left(\ell, t\right), \mathsf{*.f64}\left(k, \mathsf{*.f64}\left(k, \left(\frac{\ell}{t} \cdot \frac{-1}{6}\right)\right)\right)\right), \left({k}^{4}\right)\right)\right) \]

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

       \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(2, \ell\right), \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{/.f64}\left(\ell, t\right), \mathsf{*.f64}\left(k, \mathsf{*.f64}\left(k, \mathsf{*.f64}\left(\left(\frac{\ell}{t}\right), \frac{-1}{6}\right)\right)\right)\right), \left({k}^{4}\right)\right)\right) \]

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

       \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(2, \ell\right), \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{/.f64}\left(\ell, t\right), \mathsf{*.f64}\left(k, \mathsf{*.f64}\left(k, \mathsf{*.f64}\left(\mathsf{/.f64}\left(\ell, t\right), \frac{-1}{6}\right)\right)\right)\right), \left({k}^{4}\right)\right)\right) \]

   13. pow-lowering-pow.f64 49.2%

       \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(2, \ell\right), \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{/.f64}\left(\ell, t\right), \mathsf{*.f64}\left(k, \mathsf{*.f64}\left(k, \mathsf{*.f64}\left(\mathsf{/.f64}\left(\ell, t\right), \frac{-1}{6}\right)\right)\right)\right), \mathsf{pow.f64}\left(k, \color{blue}{4}\right)\right)\right) \]

10. Simplified 49.2%

    \[\leadsto \left(2 \cdot \ell\right) \cdot \color{blue}{\frac{\frac{\ell}{t} + k \cdot \left(k \cdot \left(\frac{\ell}{t} \cdot -0.16666666666666666\right)\right)}{{k}^{4}}} \]

11. Taylor expanded in k around inf

    \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(2, \ell\right), \color{blue}{\left(\frac{-1}{6} \cdot \frac{\ell}{{k}^{2} \cdot t}\right)}\right) \]
12. Step-by-step derivation

    1. associate-*r/ N/A

       \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(2, \ell\right), \left(\frac{\frac{-1}{6} \cdot \ell}{\color{blue}{{k}^{2} \cdot t}}\right)\right) \]

    2. associate-/l/ N/A

       \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(2, \ell\right), \left(\frac{\frac{\frac{-1}{6} \cdot \ell}{t}}{\color{blue}{{k}^{2}}}\right)\right) \]

    3. associate-*r/ N/A

       \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(2, \ell\right), \left(\frac{\frac{-1}{6} \cdot \frac{\ell}{t}}{{\color{blue}{k}}^{2}}\right)\right) \]

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

       \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(2, \ell\right), \mathsf{/.f64}\left(\left(\frac{-1}{6} \cdot \frac{\ell}{t}\right), \color{blue}{\left({k}^{2}\right)}\right)\right) \]

    5. associate-*r/ N/A

       \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(2, \ell\right), \mathsf{/.f64}\left(\left(\frac{\frac{-1}{6} \cdot \ell}{t}\right), \left({\color{blue}{k}}^{2}\right)\right)\right) \]

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

       \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(2, \ell\right), \mathsf{/.f64}\left(\mathsf{/.f64}\left(\left(\frac{-1}{6} \cdot \ell\right), t\right), \left({\color{blue}{k}}^{2}\right)\right)\right) \]

    7. *-commutative N/A

       \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(2, \ell\right), \mathsf{/.f64}\left(\mathsf{/.f64}\left(\left(\ell \cdot \frac{-1}{6}\right), t\right), \left({k}^{2}\right)\right)\right) \]

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

       \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(2, \ell\right), \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\ell, \frac{-1}{6}\right), t\right), \left({k}^{2}\right)\right)\right) \]

    9. unpow2 N/A

       \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(2, \ell\right), \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\ell, \frac{-1}{6}\right), t\right), \left(k \cdot \color{blue}{k}\right)\right)\right) \]

    10. *-lowering-*.f64 24.2%

        \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(2, \ell\right), \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\ell, \frac{-1}{6}\right), t\right), \mathsf{*.f64}\left(k, \color{blue}{k}\right)\right)\right) \]

13. Simplified 24.2%

    \[\leadsto \left(2 \cdot \ell\right) \cdot \color{blue}{\frac{\frac{\ell \cdot -0.16666666666666666}{t}}{k \cdot k}} \]
14. Add Preprocessing

Alternative 11: 29.8% accurate, 38.3× speedup

Math

\[\begin{array}{l} k_m = \left|k\right| \\ \frac{\left(\ell \cdot \frac{\ell}{t}\right) \cdot -0.3333333333333333}{k\_m \cdot k\_m} \end{array} \]

FPCore

k_m = (fabs.f64 k)
(FPCore (t l k_m)
 :precision binary64
 (/ (* (* l (/ l t)) -0.3333333333333333) (* k_m k_m)))

C

k_m = fabs(k);
double code(double t, double l, double k_m) {
	return ((l * (l / t)) * -0.3333333333333333) / (k_m * k_m);
}

Fortran

k_m = abs(k)
real(8) function code(t, l, k_m)
    real(8), intent (in) :: t
    real(8), intent (in) :: l
    real(8), intent (in) :: k_m
    code = ((l * (l / t)) * (-0.3333333333333333d0)) / (k_m * k_m)
end function

Java

k_m = Math.abs(k);
public static double code(double t, double l, double k_m) {
	return ((l * (l / t)) * -0.3333333333333333) / (k_m * k_m);
}

Python

k_m = math.fabs(k)
def code(t, l, k_m):
	return ((l * (l / t)) * -0.3333333333333333) / (k_m * k_m)

Julia

k_m = abs(k)
function code(t, l, k_m)
	return Float64(Float64(Float64(l * Float64(l / t)) * -0.3333333333333333) / Float64(k_m * k_m))
end

MATLAB

k_m = abs(k);
function tmp = code(t, l, k_m)
	tmp = ((l * (l / t)) * -0.3333333333333333) / (k_m * k_m);
end

Wolfram

k_m = N[Abs[k], $MachinePrecision]
code[t_, l_, k$95$m_] := N[(N[(N[(l * N[(l / t), $MachinePrecision]), $MachinePrecision] * -0.3333333333333333), $MachinePrecision] / N[(k$95$m * k$95$m), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 31.6%

   \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) - 1\right)} \]
2. Add Preprocessing

3. Taylor expanded in t around 0

   \[\leadsto \color{blue}{2 \cdot \frac{{\ell}^{2} \cdot \cos k}{{k}^{2} \cdot \left(t \cdot {\sin k}^{2}\right)}} \]
4. Step-by-step derivation

   1. associate-*r/ N/A

      \[\leadsto \frac{2 \cdot \left({\ell}^{2} \cdot \cos k\right)}{\color{blue}{{k}^{2} \cdot \left(t \cdot {\sin k}^{2}\right)}} \]

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

      \[\leadsto \mathsf{/.f64}\left(\left(2 \cdot \left({\ell}^{2} \cdot \cos k\right)\right), \color{blue}{\left({k}^{2} \cdot \left(t \cdot {\sin k}^{2}\right)\right)}\right) \]

   3. associate-*r* N/A

      \[\leadsto \mathsf{/.f64}\left(\left(\left(2 \cdot {\ell}^{2}\right) \cdot \cos k\right), \left(\color{blue}{{k}^{2}} \cdot \left(t \cdot {\sin k}^{2}\right)\right)\right) \]

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

      \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\left(2 \cdot {\ell}^{2}\right), \cos k\right), \left(\color{blue}{{k}^{2}} \cdot \left(t \cdot {\sin k}^{2}\right)\right)\right) \]

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

      \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(2, \left({\ell}^{2}\right)\right), \cos k\right), \left({\color{blue}{k}}^{2} \cdot \left(t \cdot {\sin k}^{2}\right)\right)\right) \]

   6. unpow2 N/A

      \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(2, \left(\ell \cdot \ell\right)\right), \cos k\right), \left({k}^{2} \cdot \left(t \cdot {\sin k}^{2}\right)\right)\right) \]

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

      \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(2, \mathsf{*.f64}\left(\ell, \ell\right)\right), \cos k\right), \left({k}^{2} \cdot \left(t \cdot {\sin k}^{2}\right)\right)\right) \]

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

      \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(2, \mathsf{*.f64}\left(\ell, \ell\right)\right), \mathsf{cos.f64}\left(k\right)\right), \left({k}^{\color{blue}{2}} \cdot \left(t \cdot {\sin k}^{2}\right)\right)\right) \]

   9. unpow2 N/A

      \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(2, \mathsf{*.f64}\left(\ell, \ell\right)\right), \mathsf{cos.f64}\left(k\right)\right), \left(\left(k \cdot k\right) \cdot \left(\color{blue}{t} \cdot {\sin k}^{2}\right)\right)\right) \]

   10. associate-*l* N/A

       \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(2, \mathsf{*.f64}\left(\ell, \ell\right)\right), \mathsf{cos.f64}\left(k\right)\right), \left(k \cdot \color{blue}{\left(k \cdot \left(t \cdot {\sin k}^{2}\right)\right)}\right)\right) \]

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

       \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(2, \mathsf{*.f64}\left(\ell, \ell\right)\right), \mathsf{cos.f64}\left(k\right)\right), \mathsf{*.f64}\left(k, \color{blue}{\left(k \cdot \left(t \cdot {\sin k}^{2}\right)\right)}\right)\right) \]

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

       \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(2, \mathsf{*.f64}\left(\ell, \ell\right)\right), \mathsf{cos.f64}\left(k\right)\right), \mathsf{*.f64}\left(k, \mathsf{*.f64}\left(k, \color{blue}{\left(t \cdot {\sin k}^{2}\right)}\right)\right)\right) \]

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

       \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(2, \mathsf{*.f64}\left(\ell, \ell\right)\right), \mathsf{cos.f64}\left(k\right)\right), \mathsf{*.f64}\left(k, \mathsf{*.f64}\left(k, \mathsf{*.f64}\left(t, \color{blue}{\left({\sin k}^{2}\right)}\right)\right)\right)\right) \]

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

       \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(2, \mathsf{*.f64}\left(\ell, \ell\right)\right), \mathsf{cos.f64}\left(k\right)\right), \mathsf{*.f64}\left(k, \mathsf{*.f64}\left(k, \mathsf{*.f64}\left(t, \mathsf{pow.f64}\left(\sin k, \color{blue}{2}\right)\right)\right)\right)\right) \]

   15. sin-lowering-sin.f64 69.3%

       \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(2, \mathsf{*.f64}\left(\ell, \ell\right)\right), \mathsf{cos.f64}\left(k\right)\right), \mathsf{*.f64}\left(k, \mathsf{*.f64}\left(k, \mathsf{*.f64}\left(t, \mathsf{pow.f64}\left(\mathsf{sin.f64}\left(k\right), 2\right)\right)\right)\right)\right) \]

5. Simplified 69.3%

   \[\leadsto \color{blue}{\frac{\left(2 \cdot \left(\ell \cdot \ell\right)\right) \cdot \cos k}{k \cdot \left(k \cdot \left(t \cdot {\sin k}^{2}\right)\right)}} \]
6. Step-by-step derivation

   1. associate-/l* N/A

      \[\leadsto \left(2 \cdot \left(\ell \cdot \ell\right)\right) \cdot \color{blue}{\frac{\cos k}{k \cdot \left(k \cdot \left(t \cdot {\sin k}^{2}\right)\right)}} \]

   2. associate-*r* N/A

      \[\leadsto \left(\left(2 \cdot \ell\right) \cdot \ell\right) \cdot \frac{\color{blue}{\cos k}}{k \cdot \left(k \cdot \left(t \cdot {\sin k}^{2}\right)\right)} \]

   3. associate-*l* N/A

      \[\leadsto \left(2 \cdot \ell\right) \cdot \color{blue}{\left(\ell \cdot \frac{\cos k}{k \cdot \left(k \cdot \left(t \cdot {\sin k}^{2}\right)\right)}\right)} \]

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

      \[\leadsto \mathsf{*.f64}\left(\left(2 \cdot \ell\right), \color{blue}{\left(\ell \cdot \frac{\cos k}{k \cdot \left(k \cdot \left(t \cdot {\sin k}^{2}\right)\right)}\right)}\right) \]

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

      \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(2, \ell\right), \left(\color{blue}{\ell} \cdot \frac{\cos k}{k \cdot \left(k \cdot \left(t \cdot {\sin k}^{2}\right)\right)}\right)\right) \]

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

      \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(2, \ell\right), \mathsf{*.f64}\left(\ell, \color{blue}{\left(\frac{\cos k}{k \cdot \left(k \cdot \left(t \cdot {\sin k}^{2}\right)\right)}\right)}\right)\right) \]

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

      \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(2, \ell\right), \mathsf{*.f64}\left(\ell, \mathsf{/.f64}\left(\cos k, \color{blue}{\left(k \cdot \left(k \cdot \left(t \cdot {\sin k}^{2}\right)\right)\right)}\right)\right)\right) \]

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

      \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(2, \ell\right), \mathsf{*.f64}\left(\ell, \mathsf{/.f64}\left(\mathsf{cos.f64}\left(k\right), \left(\color{blue}{k} \cdot \left(k \cdot \left(t \cdot {\sin k}^{2}\right)\right)\right)\right)\right)\right) \]

   9. associate-*r* N/A

      \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(2, \ell\right), \mathsf{*.f64}\left(\ell, \mathsf{/.f64}\left(\mathsf{cos.f64}\left(k\right), \left(\left(k \cdot k\right) \cdot \color{blue}{\left(t \cdot {\sin k}^{2}\right)}\right)\right)\right)\right) \]

   10. associate-*r* N/A

       \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(2, \ell\right), \mathsf{*.f64}\left(\ell, \mathsf{/.f64}\left(\mathsf{cos.f64}\left(k\right), \left(\left(\left(k \cdot k\right) \cdot t\right) \cdot \color{blue}{{\sin k}^{2}}\right)\right)\right)\right) \]

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

       \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(2, \ell\right), \mathsf{*.f64}\left(\ell, \mathsf{/.f64}\left(\mathsf{cos.f64}\left(k\right), \mathsf{*.f64}\left(\left(\left(k \cdot k\right) \cdot t\right), \color{blue}{\left({\sin k}^{2}\right)}\right)\right)\right)\right) \]

   12. *-commutative N/A

       \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(2, \ell\right), \mathsf{*.f64}\left(\ell, \mathsf{/.f64}\left(\mathsf{cos.f64}\left(k\right), \mathsf{*.f64}\left(\left(t \cdot \left(k \cdot k\right)\right), \left({\color{blue}{\sin k}}^{2}\right)\right)\right)\right)\right) \]

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

       \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(2, \ell\right), \mathsf{*.f64}\left(\ell, \mathsf{/.f64}\left(\mathsf{cos.f64}\left(k\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(t, \left(k \cdot k\right)\right), \left({\color{blue}{\sin k}}^{2}\right)\right)\right)\right)\right) \]

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

       \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(2, \ell\right), \mathsf{*.f64}\left(\ell, \mathsf{/.f64}\left(\mathsf{cos.f64}\left(k\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(t, \mathsf{*.f64}\left(k, k\right)\right), \left({\sin k}^{2}\right)\right)\right)\right)\right) \]

   15. unpow2 N/A

       \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(2, \ell\right), \mathsf{*.f64}\left(\ell, \mathsf{/.f64}\left(\mathsf{cos.f64}\left(k\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(t, \mathsf{*.f64}\left(k, k\right)\right), \left(\sin k \cdot \color{blue}{\sin k}\right)\right)\right)\right)\right) \]

   16. sqr-sin-a N/A

       \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(2, \ell\right), \mathsf{*.f64}\left(\ell, \mathsf{/.f64}\left(\mathsf{cos.f64}\left(k\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(t, \mathsf{*.f64}\left(k, k\right)\right), \left(\frac{1}{2} - \color{blue}{\frac{1}{2} \cdot \cos \left(2 \cdot k\right)}\right)\right)\right)\right)\right) \]

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

       \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(2, \ell\right), \mathsf{*.f64}\left(\ell, \mathsf{/.f64}\left(\mathsf{cos.f64}\left(k\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(t, \mathsf{*.f64}\left(k, k\right)\right), \mathsf{\_.f64}\left(\frac{1}{2}, \color{blue}{\left(\frac{1}{2} \cdot \cos \left(2 \cdot k\right)\right)}\right)\right)\right)\right)\right) \]

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

       \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(2, \ell\right), \mathsf{*.f64}\left(\ell, \mathsf{/.f64}\left(\mathsf{cos.f64}\left(k\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(t, \mathsf{*.f64}\left(k, k\right)\right), \mathsf{\_.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(\frac{1}{2}, \color{blue}{\cos \left(2 \cdot k\right)}\right)\right)\right)\right)\right)\right) \]

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

       \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(2, \ell\right), \mathsf{*.f64}\left(\ell, \mathsf{/.f64}\left(\mathsf{cos.f64}\left(k\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(t, \mathsf{*.f64}\left(k, k\right)\right), \mathsf{\_.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(\frac{1}{2}, \mathsf{cos.f64}\left(\left(2 \cdot k\right)\right)\right)\right)\right)\right)\right)\right) \]

   20. *-lowering-*.f64 70.1%

       \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(2, \ell\right), \mathsf{*.f64}\left(\ell, \mathsf{/.f64}\left(\mathsf{cos.f64}\left(k\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(t, \mathsf{*.f64}\left(k, k\right)\right), \mathsf{\_.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(\frac{1}{2}, \mathsf{cos.f64}\left(\mathsf{*.f64}\left(2, k\right)\right)\right)\right)\right)\right)\right)\right) \]

7. Applied egg-rr 70.1%

   \[\leadsto \color{blue}{\left(2 \cdot \ell\right) \cdot \left(\ell \cdot \frac{\cos k}{\left(t \cdot \left(k \cdot k\right)\right) \cdot \left(0.5 - 0.5 \cdot \cos \left(2 \cdot k\right)\right)}\right)} \]

8. Taylor expanded in k around 0

   \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(2, \ell\right), \color{blue}{\left(\frac{{k}^{2} \cdot \left(\frac{-1}{2} \cdot \frac{\ell}{t} - \frac{-1}{3} \cdot \frac{\ell}{t}\right) + \frac{\ell}{t}}{{k}^{4}}\right)}\right) \]
9. Step-by-step derivation

   1. /-lowering-/.f64 N/A

      \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(2, \ell\right), \mathsf{/.f64}\left(\left({k}^{2} \cdot \left(\frac{-1}{2} \cdot \frac{\ell}{t} - \frac{-1}{3} \cdot \frac{\ell}{t}\right) + \frac{\ell}{t}\right), \color{blue}{\left({k}^{4}\right)}\right)\right) \]

   2. +-commutative N/A

      \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(2, \ell\right), \mathsf{/.f64}\left(\left(\frac{\ell}{t} + {k}^{2} \cdot \left(\frac{-1}{2} \cdot \frac{\ell}{t} - \frac{-1}{3} \cdot \frac{\ell}{t}\right)\right), \left({\color{blue}{k}}^{4}\right)\right)\right) \]

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

      \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(2, \ell\right), \mathsf{/.f64}\left(\mathsf{+.f64}\left(\left(\frac{\ell}{t}\right), \left({k}^{2} \cdot \left(\frac{-1}{2} \cdot \frac{\ell}{t} - \frac{-1}{3} \cdot \frac{\ell}{t}\right)\right)\right), \left({\color{blue}{k}}^{4}\right)\right)\right) \]

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

      \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(2, \ell\right), \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{/.f64}\left(\ell, t\right), \left({k}^{2} \cdot \left(\frac{-1}{2} \cdot \frac{\ell}{t} - \frac{-1}{3} \cdot \frac{\ell}{t}\right)\right)\right), \left({k}^{4}\right)\right)\right) \]

   5. unpow2 N/A

      \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(2, \ell\right), \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{/.f64}\left(\ell, t\right), \left(\left(k \cdot k\right) \cdot \left(\frac{-1}{2} \cdot \frac{\ell}{t} - \frac{-1}{3} \cdot \frac{\ell}{t}\right)\right)\right), \left({k}^{4}\right)\right)\right) \]

   6. associate-*l* N/A

      \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(2, \ell\right), \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{/.f64}\left(\ell, t\right), \left(k \cdot \left(k \cdot \left(\frac{-1}{2} \cdot \frac{\ell}{t} - \frac{-1}{3} \cdot \frac{\ell}{t}\right)\right)\right)\right), \left({k}^{4}\right)\right)\right) \]

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

      \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(2, \ell\right), \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{/.f64}\left(\ell, t\right), \mathsf{*.f64}\left(k, \left(k \cdot \left(\frac{-1}{2} \cdot \frac{\ell}{t} - \frac{-1}{3} \cdot \frac{\ell}{t}\right)\right)\right)\right), \left({k}^{4}\right)\right)\right) \]

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

      \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(2, \ell\right), \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{/.f64}\left(\ell, t\right), \mathsf{*.f64}\left(k, \mathsf{*.f64}\left(k, \left(\frac{-1}{2} \cdot \frac{\ell}{t} - \frac{-1}{3} \cdot \frac{\ell}{t}\right)\right)\right)\right), \left({k}^{4}\right)\right)\right) \]

   9. distribute-rgt-out-- N/A

      \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(2, \ell\right), \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{/.f64}\left(\ell, t\right), \mathsf{*.f64}\left(k, \mathsf{*.f64}\left(k, \left(\frac{\ell}{t} \cdot \left(\frac{-1}{2} - \frac{-1}{3}\right)\right)\right)\right)\right), \left({k}^{4}\right)\right)\right) \]

   10. metadata-eval N/A

       \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(2, \ell\right), \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{/.f64}\left(\ell, t\right), \mathsf{*.f64}\left(k, \mathsf{*.f64}\left(k, \left(\frac{\ell}{t} \cdot \frac{-1}{6}\right)\right)\right)\right), \left({k}^{4}\right)\right)\right) \]

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

       \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(2, \ell\right), \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{/.f64}\left(\ell, t\right), \mathsf{*.f64}\left(k, \mathsf{*.f64}\left(k, \mathsf{*.f64}\left(\left(\frac{\ell}{t}\right), \frac{-1}{6}\right)\right)\right)\right), \left({k}^{4}\right)\right)\right) \]

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

       \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(2, \ell\right), \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{/.f64}\left(\ell, t\right), \mathsf{*.f64}\left(k, \mathsf{*.f64}\left(k, \mathsf{*.f64}\left(\mathsf{/.f64}\left(\ell, t\right), \frac{-1}{6}\right)\right)\right)\right), \left({k}^{4}\right)\right)\right) \]

   13. pow-lowering-pow.f64 49.2%

       \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(2, \ell\right), \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{/.f64}\left(\ell, t\right), \mathsf{*.f64}\left(k, \mathsf{*.f64}\left(k, \mathsf{*.f64}\left(\mathsf{/.f64}\left(\ell, t\right), \frac{-1}{6}\right)\right)\right)\right), \mathsf{pow.f64}\left(k, \color{blue}{4}\right)\right)\right) \]

10. Simplified 49.2%

    \[\leadsto \left(2 \cdot \ell\right) \cdot \color{blue}{\frac{\frac{\ell}{t} + k \cdot \left(k \cdot \left(\frac{\ell}{t} \cdot -0.16666666666666666\right)\right)}{{k}^{4}}} \]

11. Taylor expanded in k around inf

    \[\leadsto \color{blue}{\frac{-1}{3} \cdot \frac{{\ell}^{2}}{{k}^{2} \cdot t}} \]
12. Step-by-step derivation

    1. associate-*r/ N/A

       \[\leadsto \frac{\frac{-1}{3} \cdot {\ell}^{2}}{\color{blue}{{k}^{2} \cdot t}} \]

    2. associate-/l/ N/A

       \[\leadsto \frac{\frac{\frac{-1}{3} \cdot {\ell}^{2}}{t}}{\color{blue}{{k}^{2}}} \]

    3. associate-*r/ N/A

       \[\leadsto \frac{\frac{-1}{3} \cdot \frac{{\ell}^{2}}{t}}{{\color{blue}{k}}^{2}} \]

    4. *-commutative N/A

       \[\leadsto \frac{\frac{{\ell}^{2}}{t} \cdot \frac{-1}{3}}{{\color{blue}{k}}^{2}} \]

    5. metadata-eval N/A

       \[\leadsto \frac{\frac{{\ell}^{2}}{t} \cdot \left(\frac{-1}{6} \cdot 2\right)}{{k}^{2}} \]

    6. associate-*l* N/A

       \[\leadsto \frac{\left(\frac{{\ell}^{2}}{t} \cdot \frac{-1}{6}\right) \cdot 2}{{\color{blue}{k}}^{2}} \]

    7. metadata-eval N/A

       \[\leadsto \frac{\left(\frac{{\ell}^{2}}{t} \cdot \left(\frac{-1}{2} - \frac{-1}{3}\right)\right) \cdot 2}{{k}^{2}} \]

    8. distribute-rgt-out-- N/A

       \[\leadsto \frac{\left(\frac{-1}{2} \cdot \frac{{\ell}^{2}}{t} - \frac{-1}{3} \cdot \frac{{\ell}^{2}}{t}\right) \cdot 2}{{k}^{2}} \]

    9. *-commutative N/A

       \[\leadsto \frac{2 \cdot \left(\frac{-1}{2} \cdot \frac{{\ell}^{2}}{t} - \frac{-1}{3} \cdot \frac{{\ell}^{2}}{t}\right)}{{\color{blue}{k}}^{2}} \]

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

        \[\leadsto \mathsf{/.f64}\left(\left(2 \cdot \left(\frac{-1}{2} \cdot \frac{{\ell}^{2}}{t} - \frac{-1}{3} \cdot \frac{{\ell}^{2}}{t}\right)\right), \color{blue}{\left({k}^{2}\right)}\right) \]

13. Simplified 24.0%

    \[\leadsto \color{blue}{\frac{\left(\ell \cdot \frac{\ell}{t}\right) \cdot -0.3333333333333333}{k \cdot k}} \]
14. Add Preprocessing

Reproduce

herbie shell --seed 2024154 
(FPCore (t l k)
  :name "Toniolo and Linder, Equation (10-)"
  :precision binary64
  (/ 2.0 (* (* (* (/ (pow t 3.0) (* l l)) (sin k)) (tan k)) (- (+ 1.0 (pow (/ k t) 2.0)) 1.0))))