Toniolo and Linder, Equation (10-)

Percentage Accurate: 35.5% → 99.3%

Time: 16.0s

Alternatives: 9

Speedup: 28.1×

• 39.5% 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: 35.5% 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: 99.3% accurate, 1.9× speedup

Math

\[\begin{array}{l} k_m = \left|k\right| \\ \begin{array}{l} \mathbf{if}\;k\_m \leq 1.15 \cdot 10^{-20}:\\ \;\;\;\;\frac{\frac{2}{t}}{\frac{k\_m}{\frac{\ell}{k\_m}}} \cdot \frac{\frac{\ell}{k\_m}}{k\_m}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\frac{k\_m}{\ell} \cdot \left(\frac{k\_m}{\ell} \cdot \left(\sin k\_m \cdot \left(t \cdot \tan k\_m\right)\right)\right)}\\ \end{array} \end{array} \]

FPCore

k_m = (fabs.f64 k)
(FPCore (t l k_m)
 :precision binary64
 (if (<= k_m 1.15e-20)
   (* (/ (/ 2.0 t) (/ k_m (/ l k_m))) (/ (/ l k_m) k_m))
   (/ 2.0 (* (/ k_m l) (* (/ k_m l) (* (sin k_m) (* t (tan k_m))))))))

C

k_m = fabs(k);
double code(double t, double l, double k_m) {
	double tmp;
	if (k_m <= 1.15e-20) {
		tmp = ((2.0 / t) / (k_m / (l / k_m))) * ((l / k_m) / k_m);
	} else {
		tmp = 2.0 / ((k_m / l) * ((k_m / l) * (sin(k_m) * (t * tan(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 <= 1.15d-20) then
        tmp = ((2.0d0 / t) / (k_m / (l / k_m))) * ((l / k_m) / k_m)
    else
        tmp = 2.0d0 / ((k_m / l) * ((k_m / l) * (sin(k_m) * (t * tan(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 <= 1.15e-20) {
		tmp = ((2.0 / t) / (k_m / (l / k_m))) * ((l / k_m) / k_m);
	} else {
		tmp = 2.0 / ((k_m / l) * ((k_m / l) * (Math.sin(k_m) * (t * Math.tan(k_m)))));
	}
	return tmp;
}

Python

k_m = math.fabs(k)
def code(t, l, k_m):
	tmp = 0
	if k_m <= 1.15e-20:
		tmp = ((2.0 / t) / (k_m / (l / k_m))) * ((l / k_m) / k_m)
	else:
		tmp = 2.0 / ((k_m / l) * ((k_m / l) * (math.sin(k_m) * (t * math.tan(k_m)))))
	return tmp

Julia

k_m = abs(k)
function code(t, l, k_m)
	tmp = 0.0
	if (k_m <= 1.15e-20)
		tmp = Float64(Float64(Float64(2.0 / t) / Float64(k_m / Float64(l / k_m))) * Float64(Float64(l / k_m) / k_m));
	else
		tmp = Float64(2.0 / Float64(Float64(k_m / l) * Float64(Float64(k_m / l) * Float64(sin(k_m) * Float64(t * tan(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 <= 1.15e-20)
		tmp = ((2.0 / t) / (k_m / (l / k_m))) * ((l / k_m) / k_m);
	else
		tmp = 2.0 / ((k_m / l) * ((k_m / l) * (sin(k_m) * (t * tan(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, 1.15e-20], N[(N[(N[(2.0 / t), $MachinePrecision] / N[(k$95$m / N[(l / k$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(N[(l / k$95$m), $MachinePrecision] / k$95$m), $MachinePrecision]), $MachinePrecision], N[(2.0 / N[(N[(k$95$m / l), $MachinePrecision] * N[(N[(k$95$m / l), $MachinePrecision] * N[(N[Sin[k$95$m], $MachinePrecision] * N[(t * N[Tan[k$95$m], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if k < 1.15e-20

   1. Initial program 36.1%

      \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) - 1\right)} \]
   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. /-lowering-/.f64 N/A

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

      2. *-commutative N/A

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

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

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

      4. metadata-eval N/A

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

      5. pow-sqr N/A

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

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

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

      7. unpow2 N/A

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

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

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

      9. unpow2 N/A

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

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

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

      11. unpow2 N/A

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

      12. *-lowering-*.f64 60.0%

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

   5. Simplified 60.0%

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

      1. remove-double-neg N/A

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

      2. times-frac N/A

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

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

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

      4. distribute-lft-neg-in N/A

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

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

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

      6. neg-sub0 N/A

         \[\leadsto \mathsf{/.f64}\left(2, \mathsf{*.f64}\left(\left(0 - \frac{t}{\ell}\right), \left(\mathsf{neg}\left(\color{blue}{\frac{\left(k \cdot k\right) \cdot \left(k \cdot k\right)}{\ell}}\right)\right)\right)\right) \]

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

         \[\leadsto \mathsf{/.f64}\left(2, \mathsf{*.f64}\left(\mathsf{\_.f64}\left(0, \left(\frac{t}{\ell}\right)\right), \left(\mathsf{neg}\left(\color{blue}{\frac{\left(k \cdot k\right) \cdot \left(k \cdot k\right)}{\ell}}\right)\right)\right)\right) \]

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

         \[\leadsto \mathsf{/.f64}\left(2, \mathsf{*.f64}\left(\mathsf{\_.f64}\left(0, \mathsf{/.f64}\left(t, \ell\right)\right), \left(\mathsf{neg}\left(\frac{\left(k \cdot k\right) \cdot \left(k \cdot k\right)}{\color{blue}{\ell}}\right)\right)\right)\right) \]

      9. neg-sub0 N/A

         \[\leadsto \mathsf{/.f64}\left(2, \mathsf{*.f64}\left(\mathsf{\_.f64}\left(0, \mathsf{/.f64}\left(t, \ell\right)\right), \left(0 - \color{blue}{\frac{\left(k \cdot k\right) \cdot \left(k \cdot k\right)}{\ell}}\right)\right)\right) \]

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

          \[\leadsto \mathsf{/.f64}\left(2, \mathsf{*.f64}\left(\mathsf{\_.f64}\left(0, \mathsf{/.f64}\left(t, \ell\right)\right), \mathsf{\_.f64}\left(0, \color{blue}{\left(\frac{\left(k \cdot k\right) \cdot \left(k \cdot k\right)}{\ell}\right)}\right)\right)\right) \]

      11. associate-*l* N/A

          \[\leadsto \mathsf{/.f64}\left(2, \mathsf{*.f64}\left(\mathsf{\_.f64}\left(0, \mathsf{/.f64}\left(t, \ell\right)\right), \mathsf{\_.f64}\left(0, \left(\frac{k \cdot \left(k \cdot \left(k \cdot k\right)\right)}{\ell}\right)\right)\right)\right) \]

      12. associate-/l* N/A

          \[\leadsto \mathsf{/.f64}\left(2, \mathsf{*.f64}\left(\mathsf{\_.f64}\left(0, \mathsf{/.f64}\left(t, \ell\right)\right), \mathsf{\_.f64}\left(0, \left(k \cdot \color{blue}{\frac{k \cdot \left(k \cdot k\right)}{\ell}}\right)\right)\right)\right) \]

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

          \[\leadsto \mathsf{/.f64}\left(2, \mathsf{*.f64}\left(\mathsf{\_.f64}\left(0, \mathsf{/.f64}\left(t, \ell\right)\right), \mathsf{\_.f64}\left(0, \mathsf{*.f64}\left(k, \color{blue}{\left(\frac{k \cdot \left(k \cdot k\right)}{\ell}\right)}\right)\right)\right)\right) \]

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

          \[\leadsto \mathsf{/.f64}\left(2, \mathsf{*.f64}\left(\mathsf{\_.f64}\left(0, \mathsf{/.f64}\left(t, \ell\right)\right), \mathsf{\_.f64}\left(0, \mathsf{*.f64}\left(k, \mathsf{/.f64}\left(\left(k \cdot \left(k \cdot k\right)\right), \color{blue}{\ell}\right)\right)\right)\right)\right) \]

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

          \[\leadsto \mathsf{/.f64}\left(2, \mathsf{*.f64}\left(\mathsf{\_.f64}\left(0, \mathsf{/.f64}\left(t, \ell\right)\right), \mathsf{\_.f64}\left(0, \mathsf{*.f64}\left(k, \mathsf{/.f64}\left(\mathsf{*.f64}\left(k, \left(k \cdot k\right)\right), \ell\right)\right)\right)\right)\right) \]

      16. *-lowering-*.f64 50.8%

          \[\leadsto \mathsf{/.f64}\left(2, \mathsf{*.f64}\left(\mathsf{\_.f64}\left(0, \mathsf{/.f64}\left(t, \ell\right)\right), \mathsf{\_.f64}\left(0, \mathsf{*.f64}\left(k, \mathsf{/.f64}\left(\mathsf{*.f64}\left(k, \mathsf{*.f64}\left(k, k\right)\right), \ell\right)\right)\right)\right)\right) \]

   7. Applied egg-rr 50.8%

      \[\leadsto \frac{2}{\color{blue}{\left(0 - \frac{t}{\ell}\right) \cdot \left(0 - k \cdot \frac{k \cdot \left(k \cdot k\right)}{\ell}\right)}} \]
   8. Step-by-step derivation

      1. associate-/r* N/A

         \[\leadsto \frac{\frac{2}{0 - \frac{t}{\ell}}}{\color{blue}{0 - k \cdot \frac{k \cdot \left(k \cdot k\right)}{\ell}}} \]

      2. sub0-neg N/A

         \[\leadsto \frac{\frac{2}{0 - \frac{t}{\ell}}}{\mathsf{neg}\left(k \cdot \frac{k \cdot \left(k \cdot k\right)}{\ell}\right)} \]

      3. distribute-frac-neg2 N/A

         \[\leadsto \mathsf{neg}\left(\frac{\frac{2}{0 - \frac{t}{\ell}}}{k \cdot \frac{k \cdot \left(k \cdot k\right)}{\ell}}\right) \]

      4. distribute-frac-neg N/A

         \[\leadsto \frac{\mathsf{neg}\left(\frac{2}{0 - \frac{t}{\ell}}\right)}{\color{blue}{k \cdot \frac{k \cdot \left(k \cdot k\right)}{\ell}}} \]

      5. distribute-frac-neg2 N/A

         \[\leadsto \frac{\frac{2}{\mathsf{neg}\left(\left(0 - \frac{t}{\ell}\right)\right)}}{\color{blue}{k} \cdot \frac{k \cdot \left(k \cdot k\right)}{\ell}} \]

      6. sub0-neg N/A

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

      7. remove-double-neg N/A

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

      8. un-div-inv N/A

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

      9. clear-num N/A

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

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

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

      11. clear-num N/A

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

      12. un-div-inv N/A

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

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

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

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

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

      15. associate-/l* N/A

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

      16. associate-*r* N/A

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

      17. clear-num N/A

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

      18. un-div-inv N/A

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

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

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

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

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

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

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

   9. Applied egg-rr 74.2%

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

       1. associate-/r/ N/A

          \[\leadsto \frac{\frac{2}{t} \cdot \ell}{\frac{\color{blue}{k \cdot k}}{\frac{\ell}{k \cdot k}}} \]

       2. associate-/r/ N/A

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

       3. times-frac N/A

          \[\leadsto \frac{\frac{2}{t}}{\frac{k \cdot k}{\ell}} \cdot \color{blue}{\frac{\ell}{k \cdot k}} \]

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

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

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

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

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

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

       7. associate-*l/ N/A

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

       8. associate-/r/ N/A

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

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

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

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

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

       11. associate-/r* N/A

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

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

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

       13. /-lowering-/.f64 82.6%

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

   11. Applied egg-rr 82.6%

       \[\leadsto \color{blue}{\frac{\frac{2}{t}}{\frac{k}{\frac{\ell}{k}}} \cdot \frac{\frac{\ell}{k}}{k}} \]

   if 1.15e-20 < k

   1. Initial program 37.8%

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

   3. Taylor expanded in t around 0

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

      1. associate-/l* N/A

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

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

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

      3. unpow2 N/A

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

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

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

      5. *-commutative N/A

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

      6. associate-/r* N/A

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

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

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

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

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

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

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

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

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

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

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

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

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

      13. unpow2 N/A

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

      14. *-lowering-*.f64 67.6%

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

   5. Simplified 67.6%

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

      1. associate-*r/ N/A

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

      2. times-frac N/A

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

      3. clear-num N/A

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

      4. un-div-inv N/A

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

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

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

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

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

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

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

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

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

      9. associate-/l* N/A

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

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

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

      11. unpow2 N/A

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

      12. associate-/l* N/A

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

      13. tan-quot N/A

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

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

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

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

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

      16. tan-lowering-tan.f64 75.4%

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

   7. Applied egg-rr 75.4%

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

      1. associate-/l* N/A

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

      2. div-inv N/A

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

      3. times-frac N/A

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

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

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

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

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

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

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

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

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

      8. associate-/r* N/A

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

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

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

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

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

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

          \[\leadsto \mathsf{/.f64}\left(2, \mathsf{*.f64}\left(\mathsf{/.f64}\left(k, \ell\right), \mathsf{/.f64}\left(\mathsf{/.f64}\left(k, \ell\right), \mathsf{/.f64}\left(\mathsf{/.f64}\left(1, t\right), \mathsf{*.f64}\left(\sin k, \color{blue}{\tan k}\right)\right)\right)\right)\right) \]

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

          \[\leadsto \mathsf{/.f64}\left(2, \mathsf{*.f64}\left(\mathsf{/.f64}\left(k, \ell\right), \mathsf{/.f64}\left(\mathsf{/.f64}\left(k, \ell\right), \mathsf{/.f64}\left(\mathsf{/.f64}\left(1, t\right), \mathsf{*.f64}\left(\mathsf{sin.f64}\left(k\right), \tan \color{blue}{k}\right)\right)\right)\right)\right) \]

      13. tan-lowering-tan.f64 99.2%

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

   9. Applied egg-rr 99.2%

      \[\leadsto \frac{2}{\color{blue}{\frac{k}{\ell} \cdot \frac{\frac{k}{\ell}}{\frac{\frac{1}{t}}{\sin k \cdot \tan k}}}} \]
   10. Step-by-step derivation

       1. div-inv N/A

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

       2. clear-num N/A

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

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

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

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

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

       5. clear-num N/A

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

       6. associate-/l/ N/A

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

       7. remove-double-div N/A

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

       8. associate-*l* N/A

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

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

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

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

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

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

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

       12. tan-lowering-tan.f64 99.3%

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

   11. Applied egg-rr 99.3%

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

4. Final simplification 87.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;k \leq 1.15 \cdot 10^{-20}:\\ \;\;\;\;\frac{\frac{2}{t}}{\frac{k}{\frac{\ell}{k}}} \cdot \frac{\frac{\ell}{k}}{k}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\frac{k}{\ell} \cdot \left(\frac{k}{\ell} \cdot \left(\sin k \cdot \left(t \cdot \tan k\right)\right)\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 95.7% accurate, 1.9× speedup

Math

\[\begin{array}{l} k_m = \left|k\right| \\ \begin{array}{l} \mathbf{if}\;k\_m \leq 1.15 \cdot 10^{-20}:\\ \;\;\;\;\frac{\frac{2}{t}}{\frac{k\_m}{\frac{\ell}{k\_m}}} \cdot \frac{\frac{\ell}{k\_m}}{k\_m}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{k\_m \cdot \frac{\frac{k\_m}{\ell}}{\frac{\frac{\ell}{t}}{\sin k\_m \cdot \tan k\_m}}}\\ \end{array} \end{array} \]

FPCore

k_m = (fabs.f64 k)
(FPCore (t l k_m)
 :precision binary64
 (if (<= k_m 1.15e-20)
   (* (/ (/ 2.0 t) (/ k_m (/ l k_m))) (/ (/ l k_m) k_m))
   (/ 2.0 (* k_m (/ (/ k_m l) (/ (/ l t) (* (sin k_m) (tan k_m))))))))

C

k_m = fabs(k);
double code(double t, double l, double k_m) {
	double tmp;
	if (k_m <= 1.15e-20) {
		tmp = ((2.0 / t) / (k_m / (l / k_m))) * ((l / k_m) / k_m);
	} else {
		tmp = 2.0 / (k_m * ((k_m / l) / ((l / t) / (sin(k_m) * tan(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 <= 1.15d-20) then
        tmp = ((2.0d0 / t) / (k_m / (l / k_m))) * ((l / k_m) / k_m)
    else
        tmp = 2.0d0 / (k_m * ((k_m / l) / ((l / t) / (sin(k_m) * tan(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 <= 1.15e-20) {
		tmp = ((2.0 / t) / (k_m / (l / k_m))) * ((l / k_m) / k_m);
	} else {
		tmp = 2.0 / (k_m * ((k_m / l) / ((l / t) / (Math.sin(k_m) * Math.tan(k_m)))));
	}
	return tmp;
}

Python

k_m = math.fabs(k)
def code(t, l, k_m):
	tmp = 0
	if k_m <= 1.15e-20:
		tmp = ((2.0 / t) / (k_m / (l / k_m))) * ((l / k_m) / k_m)
	else:
		tmp = 2.0 / (k_m * ((k_m / l) / ((l / t) / (math.sin(k_m) * math.tan(k_m)))))
	return tmp

Julia

k_m = abs(k)
function code(t, l, k_m)
	tmp = 0.0
	if (k_m <= 1.15e-20)
		tmp = Float64(Float64(Float64(2.0 / t) / Float64(k_m / Float64(l / k_m))) * Float64(Float64(l / k_m) / k_m));
	else
		tmp = Float64(2.0 / Float64(k_m * Float64(Float64(k_m / l) / Float64(Float64(l / t) / Float64(sin(k_m) * tan(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 <= 1.15e-20)
		tmp = ((2.0 / t) / (k_m / (l / k_m))) * ((l / k_m) / k_m);
	else
		tmp = 2.0 / (k_m * ((k_m / l) / ((l / t) / (sin(k_m) * tan(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, 1.15e-20], N[(N[(N[(2.0 / t), $MachinePrecision] / N[(k$95$m / N[(l / k$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(N[(l / k$95$m), $MachinePrecision] / k$95$m), $MachinePrecision]), $MachinePrecision], N[(2.0 / N[(k$95$m * N[(N[(k$95$m / l), $MachinePrecision] / N[(N[(l / t), $MachinePrecision] / N[(N[Sin[k$95$m], $MachinePrecision] * N[Tan[k$95$m], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if k < 1.15e-20

   1. Initial program 36.1%

      \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) - 1\right)} \]
   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. /-lowering-/.f64 N/A

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

      2. *-commutative N/A

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

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

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

      4. metadata-eval N/A

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

      5. pow-sqr N/A

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

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

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

      7. unpow2 N/A

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

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

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

      9. unpow2 N/A

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

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

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

      11. unpow2 N/A

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

      12. *-lowering-*.f64 60.0%

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

   5. Simplified 60.0%

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

      1. remove-double-neg N/A

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

      2. times-frac N/A

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

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

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

      4. distribute-lft-neg-in N/A

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

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

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

      6. neg-sub0 N/A

         \[\leadsto \mathsf{/.f64}\left(2, \mathsf{*.f64}\left(\left(0 - \frac{t}{\ell}\right), \left(\mathsf{neg}\left(\color{blue}{\frac{\left(k \cdot k\right) \cdot \left(k \cdot k\right)}{\ell}}\right)\right)\right)\right) \]

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

         \[\leadsto \mathsf{/.f64}\left(2, \mathsf{*.f64}\left(\mathsf{\_.f64}\left(0, \left(\frac{t}{\ell}\right)\right), \left(\mathsf{neg}\left(\color{blue}{\frac{\left(k \cdot k\right) \cdot \left(k \cdot k\right)}{\ell}}\right)\right)\right)\right) \]

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

         \[\leadsto \mathsf{/.f64}\left(2, \mathsf{*.f64}\left(\mathsf{\_.f64}\left(0, \mathsf{/.f64}\left(t, \ell\right)\right), \left(\mathsf{neg}\left(\frac{\left(k \cdot k\right) \cdot \left(k \cdot k\right)}{\color{blue}{\ell}}\right)\right)\right)\right) \]

      9. neg-sub0 N/A

         \[\leadsto \mathsf{/.f64}\left(2, \mathsf{*.f64}\left(\mathsf{\_.f64}\left(0, \mathsf{/.f64}\left(t, \ell\right)\right), \left(0 - \color{blue}{\frac{\left(k \cdot k\right) \cdot \left(k \cdot k\right)}{\ell}}\right)\right)\right) \]

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

          \[\leadsto \mathsf{/.f64}\left(2, \mathsf{*.f64}\left(\mathsf{\_.f64}\left(0, \mathsf{/.f64}\left(t, \ell\right)\right), \mathsf{\_.f64}\left(0, \color{blue}{\left(\frac{\left(k \cdot k\right) \cdot \left(k \cdot k\right)}{\ell}\right)}\right)\right)\right) \]

      11. associate-*l* N/A

          \[\leadsto \mathsf{/.f64}\left(2, \mathsf{*.f64}\left(\mathsf{\_.f64}\left(0, \mathsf{/.f64}\left(t, \ell\right)\right), \mathsf{\_.f64}\left(0, \left(\frac{k \cdot \left(k \cdot \left(k \cdot k\right)\right)}{\ell}\right)\right)\right)\right) \]

      12. associate-/l* N/A

          \[\leadsto \mathsf{/.f64}\left(2, \mathsf{*.f64}\left(\mathsf{\_.f64}\left(0, \mathsf{/.f64}\left(t, \ell\right)\right), \mathsf{\_.f64}\left(0, \left(k \cdot \color{blue}{\frac{k \cdot \left(k \cdot k\right)}{\ell}}\right)\right)\right)\right) \]

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

          \[\leadsto \mathsf{/.f64}\left(2, \mathsf{*.f64}\left(\mathsf{\_.f64}\left(0, \mathsf{/.f64}\left(t, \ell\right)\right), \mathsf{\_.f64}\left(0, \mathsf{*.f64}\left(k, \color{blue}{\left(\frac{k \cdot \left(k \cdot k\right)}{\ell}\right)}\right)\right)\right)\right) \]

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

          \[\leadsto \mathsf{/.f64}\left(2, \mathsf{*.f64}\left(\mathsf{\_.f64}\left(0, \mathsf{/.f64}\left(t, \ell\right)\right), \mathsf{\_.f64}\left(0, \mathsf{*.f64}\left(k, \mathsf{/.f64}\left(\left(k \cdot \left(k \cdot k\right)\right), \color{blue}{\ell}\right)\right)\right)\right)\right) \]

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

          \[\leadsto \mathsf{/.f64}\left(2, \mathsf{*.f64}\left(\mathsf{\_.f64}\left(0, \mathsf{/.f64}\left(t, \ell\right)\right), \mathsf{\_.f64}\left(0, \mathsf{*.f64}\left(k, \mathsf{/.f64}\left(\mathsf{*.f64}\left(k, \left(k \cdot k\right)\right), \ell\right)\right)\right)\right)\right) \]

      16. *-lowering-*.f64 50.8%

          \[\leadsto \mathsf{/.f64}\left(2, \mathsf{*.f64}\left(\mathsf{\_.f64}\left(0, \mathsf{/.f64}\left(t, \ell\right)\right), \mathsf{\_.f64}\left(0, \mathsf{*.f64}\left(k, \mathsf{/.f64}\left(\mathsf{*.f64}\left(k, \mathsf{*.f64}\left(k, k\right)\right), \ell\right)\right)\right)\right)\right) \]

   7. Applied egg-rr 50.8%

      \[\leadsto \frac{2}{\color{blue}{\left(0 - \frac{t}{\ell}\right) \cdot \left(0 - k \cdot \frac{k \cdot \left(k \cdot k\right)}{\ell}\right)}} \]
   8. Step-by-step derivation

      1. associate-/r* N/A

         \[\leadsto \frac{\frac{2}{0 - \frac{t}{\ell}}}{\color{blue}{0 - k \cdot \frac{k \cdot \left(k \cdot k\right)}{\ell}}} \]

      2. sub0-neg N/A

         \[\leadsto \frac{\frac{2}{0 - \frac{t}{\ell}}}{\mathsf{neg}\left(k \cdot \frac{k \cdot \left(k \cdot k\right)}{\ell}\right)} \]

      3. distribute-frac-neg2 N/A

         \[\leadsto \mathsf{neg}\left(\frac{\frac{2}{0 - \frac{t}{\ell}}}{k \cdot \frac{k \cdot \left(k \cdot k\right)}{\ell}}\right) \]

      4. distribute-frac-neg N/A

         \[\leadsto \frac{\mathsf{neg}\left(\frac{2}{0 - \frac{t}{\ell}}\right)}{\color{blue}{k \cdot \frac{k \cdot \left(k \cdot k\right)}{\ell}}} \]

      5. distribute-frac-neg2 N/A

         \[\leadsto \frac{\frac{2}{\mathsf{neg}\left(\left(0 - \frac{t}{\ell}\right)\right)}}{\color{blue}{k} \cdot \frac{k \cdot \left(k \cdot k\right)}{\ell}} \]

      6. sub0-neg N/A

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

      7. remove-double-neg N/A

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

      8. un-div-inv N/A

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

      9. clear-num N/A

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

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

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

      11. clear-num N/A

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

      12. un-div-inv N/A

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

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

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

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

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

      15. associate-/l* N/A

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

      16. associate-*r* N/A

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

      17. clear-num N/A

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

      18. un-div-inv N/A

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

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

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

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

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

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

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

   9. Applied egg-rr 74.2%

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

       1. associate-/r/ N/A

          \[\leadsto \frac{\frac{2}{t} \cdot \ell}{\frac{\color{blue}{k \cdot k}}{\frac{\ell}{k \cdot k}}} \]

       2. associate-/r/ N/A

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

       3. times-frac N/A

          \[\leadsto \frac{\frac{2}{t}}{\frac{k \cdot k}{\ell}} \cdot \color{blue}{\frac{\ell}{k \cdot k}} \]

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

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

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

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

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

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

       7. associate-*l/ N/A

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

       8. associate-/r/ N/A

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

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

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

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

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

       11. associate-/r* N/A

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

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

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

       13. /-lowering-/.f64 82.6%

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

   11. Applied egg-rr 82.6%

       \[\leadsto \color{blue}{\frac{\frac{2}{t}}{\frac{k}{\frac{\ell}{k}}} \cdot \frac{\frac{\ell}{k}}{k}} \]

   if 1.15e-20 < k

   1. Initial program 37.8%

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

   3. Taylor expanded in t around 0

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

      1. associate-/l* N/A

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

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

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

      3. unpow2 N/A

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

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

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

      5. *-commutative N/A

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

      6. associate-/r* N/A

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

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

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

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

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

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

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

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

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

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

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

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

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

      13. unpow2 N/A

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

      14. *-lowering-*.f64 67.6%

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

   5. Simplified 67.6%

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

      1. associate-*r/ N/A

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

      2. times-frac N/A

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

      3. clear-num N/A

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

      4. un-div-inv N/A

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

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

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

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

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

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

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

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

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

      9. associate-/l* N/A

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

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

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

      11. unpow2 N/A

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

      12. associate-/l* N/A

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

      13. tan-quot N/A

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

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

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

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

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

      16. tan-lowering-tan.f64 75.4%

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

   7. Applied egg-rr 75.4%

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

      1. div-inv N/A

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

      2. clear-num N/A

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

      3. associate-/l* N/A

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

      4. associate-*l* N/A

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

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

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

      6. clear-num N/A

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

      7. un-div-inv N/A

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

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

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

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

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

      10. associate-/r* N/A

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

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

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

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

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

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

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

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

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

      15. tan-lowering-tan.f64 92.7%

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

   9. Applied egg-rr 92.7%

      \[\leadsto \frac{2}{\color{blue}{k \cdot \frac{\frac{k}{\ell}}{\frac{\frac{\ell}{t}}{\sin k \cdot \tan k}}}} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 3: 93.4% accurate, 1.9× speedup

Math

\[\begin{array}{l} k_m = \left|k\right| \\ \begin{array}{l} t_1 := \frac{\frac{\ell}{k\_m}}{k\_m}\\ \mathbf{if}\;k\_m \leq 5 \cdot 10^{-100}:\\ \;\;\;\;\frac{\frac{2}{t}}{\frac{k\_m}{\frac{\ell}{k\_m}}} \cdot t\_1\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \left(t\_1 \cdot \frac{\frac{\ell}{t}}{\sin k\_m \cdot \tan k\_m}\right)\\ \end{array} \end{array} \]

FPCore

k_m = (fabs.f64 k)
(FPCore (t l k_m)
 :precision binary64
 (let* ((t_1 (/ (/ l k_m) k_m)))
   (if (<= k_m 5e-100)
     (* (/ (/ 2.0 t) (/ k_m (/ l k_m))) t_1)
     (* 2.0 (* t_1 (/ (/ l t) (* (sin k_m) (tan k_m))))))))

C

k_m = fabs(k);
double code(double t, double l, double k_m) {
	double t_1 = (l / k_m) / k_m;
	double tmp;
	if (k_m <= 5e-100) {
		tmp = ((2.0 / t) / (k_m / (l / k_m))) * t_1;
	} else {
		tmp = 2.0 * (t_1 * ((l / t) / (sin(k_m) * tan(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) :: t_1
    real(8) :: tmp
    t_1 = (l / k_m) / k_m
    if (k_m <= 5d-100) then
        tmp = ((2.0d0 / t) / (k_m / (l / k_m))) * t_1
    else
        tmp = 2.0d0 * (t_1 * ((l / t) / (sin(k_m) * tan(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 t_1 = (l / k_m) / k_m;
	double tmp;
	if (k_m <= 5e-100) {
		tmp = ((2.0 / t) / (k_m / (l / k_m))) * t_1;
	} else {
		tmp = 2.0 * (t_1 * ((l / t) / (Math.sin(k_m) * Math.tan(k_m))));
	}
	return tmp;
}

Python

k_m = math.fabs(k)
def code(t, l, k_m):
	t_1 = (l / k_m) / k_m
	tmp = 0
	if k_m <= 5e-100:
		tmp = ((2.0 / t) / (k_m / (l / k_m))) * t_1
	else:
		tmp = 2.0 * (t_1 * ((l / t) / (math.sin(k_m) * math.tan(k_m))))
	return tmp

Julia

k_m = abs(k)
function code(t, l, k_m)
	t_1 = Float64(Float64(l / k_m) / k_m)
	tmp = 0.0
	if (k_m <= 5e-100)
		tmp = Float64(Float64(Float64(2.0 / t) / Float64(k_m / Float64(l / k_m))) * t_1);
	else
		tmp = Float64(2.0 * Float64(t_1 * Float64(Float64(l / t) / Float64(sin(k_m) * tan(k_m)))));
	end
	return tmp
end

MATLAB

k_m = abs(k);
function tmp_2 = code(t, l, k_m)
	t_1 = (l / k_m) / k_m;
	tmp = 0.0;
	if (k_m <= 5e-100)
		tmp = ((2.0 / t) / (k_m / (l / k_m))) * t_1;
	else
		tmp = 2.0 * (t_1 * ((l / t) / (sin(k_m) * tan(k_m))));
	end
	tmp_2 = tmp;
end

Wolfram

k_m = N[Abs[k], $MachinePrecision]
code[t_, l_, k$95$m_] := Block[{t$95$1 = N[(N[(l / k$95$m), $MachinePrecision] / k$95$m), $MachinePrecision]}, If[LessEqual[k$95$m, 5e-100], N[(N[(N[(2.0 / t), $MachinePrecision] / N[(k$95$m / N[(l / k$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * t$95$1), $MachinePrecision], N[(2.0 * N[(t$95$1 * N[(N[(l / t), $MachinePrecision] / N[(N[Sin[k$95$m], $MachinePrecision] * N[Tan[k$95$m], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if k < 5.0000000000000001e-100

   1. Initial program 35.7%

      \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) - 1\right)} \]
   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. /-lowering-/.f64 N/A

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

      2. *-commutative N/A

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

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

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

      4. metadata-eval N/A

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

      5. pow-sqr N/A

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

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

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

      7. unpow2 N/A

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

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

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

      9. unpow2 N/A

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

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

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

      11. unpow2 N/A

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

      12. *-lowering-*.f64 60.0%

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

   5. Simplified 60.0%

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

      1. remove-double-neg N/A

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

      2. times-frac N/A

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

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

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

      4. distribute-lft-neg-in N/A

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

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

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

      6. neg-sub0 N/A

         \[\leadsto \mathsf{/.f64}\left(2, \mathsf{*.f64}\left(\left(0 - \frac{t}{\ell}\right), \left(\mathsf{neg}\left(\color{blue}{\frac{\left(k \cdot k\right) \cdot \left(k \cdot k\right)}{\ell}}\right)\right)\right)\right) \]

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

         \[\leadsto \mathsf{/.f64}\left(2, \mathsf{*.f64}\left(\mathsf{\_.f64}\left(0, \left(\frac{t}{\ell}\right)\right), \left(\mathsf{neg}\left(\color{blue}{\frac{\left(k \cdot k\right) \cdot \left(k \cdot k\right)}{\ell}}\right)\right)\right)\right) \]

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

         \[\leadsto \mathsf{/.f64}\left(2, \mathsf{*.f64}\left(\mathsf{\_.f64}\left(0, \mathsf{/.f64}\left(t, \ell\right)\right), \left(\mathsf{neg}\left(\frac{\left(k \cdot k\right) \cdot \left(k \cdot k\right)}{\color{blue}{\ell}}\right)\right)\right)\right) \]

      9. neg-sub0 N/A

         \[\leadsto \mathsf{/.f64}\left(2, \mathsf{*.f64}\left(\mathsf{\_.f64}\left(0, \mathsf{/.f64}\left(t, \ell\right)\right), \left(0 - \color{blue}{\frac{\left(k \cdot k\right) \cdot \left(k \cdot k\right)}{\ell}}\right)\right)\right) \]

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

          \[\leadsto \mathsf{/.f64}\left(2, \mathsf{*.f64}\left(\mathsf{\_.f64}\left(0, \mathsf{/.f64}\left(t, \ell\right)\right), \mathsf{\_.f64}\left(0, \color{blue}{\left(\frac{\left(k \cdot k\right) \cdot \left(k \cdot k\right)}{\ell}\right)}\right)\right)\right) \]

      11. associate-*l* N/A

          \[\leadsto \mathsf{/.f64}\left(2, \mathsf{*.f64}\left(\mathsf{\_.f64}\left(0, \mathsf{/.f64}\left(t, \ell\right)\right), \mathsf{\_.f64}\left(0, \left(\frac{k \cdot \left(k \cdot \left(k \cdot k\right)\right)}{\ell}\right)\right)\right)\right) \]

      12. associate-/l* N/A

          \[\leadsto \mathsf{/.f64}\left(2, \mathsf{*.f64}\left(\mathsf{\_.f64}\left(0, \mathsf{/.f64}\left(t, \ell\right)\right), \mathsf{\_.f64}\left(0, \left(k \cdot \color{blue}{\frac{k \cdot \left(k \cdot k\right)}{\ell}}\right)\right)\right)\right) \]

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

          \[\leadsto \mathsf{/.f64}\left(2, \mathsf{*.f64}\left(\mathsf{\_.f64}\left(0, \mathsf{/.f64}\left(t, \ell\right)\right), \mathsf{\_.f64}\left(0, \mathsf{*.f64}\left(k, \color{blue}{\left(\frac{k \cdot \left(k \cdot k\right)}{\ell}\right)}\right)\right)\right)\right) \]

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

          \[\leadsto \mathsf{/.f64}\left(2, \mathsf{*.f64}\left(\mathsf{\_.f64}\left(0, \mathsf{/.f64}\left(t, \ell\right)\right), \mathsf{\_.f64}\left(0, \mathsf{*.f64}\left(k, \mathsf{/.f64}\left(\left(k \cdot \left(k \cdot k\right)\right), \color{blue}{\ell}\right)\right)\right)\right)\right) \]

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

          \[\leadsto \mathsf{/.f64}\left(2, \mathsf{*.f64}\left(\mathsf{\_.f64}\left(0, \mathsf{/.f64}\left(t, \ell\right)\right), \mathsf{\_.f64}\left(0, \mathsf{*.f64}\left(k, \mathsf{/.f64}\left(\mathsf{*.f64}\left(k, \left(k \cdot k\right)\right), \ell\right)\right)\right)\right)\right) \]

      16. *-lowering-*.f64 48.2%

          \[\leadsto \mathsf{/.f64}\left(2, \mathsf{*.f64}\left(\mathsf{\_.f64}\left(0, \mathsf{/.f64}\left(t, \ell\right)\right), \mathsf{\_.f64}\left(0, \mathsf{*.f64}\left(k, \mathsf{/.f64}\left(\mathsf{*.f64}\left(k, \mathsf{*.f64}\left(k, k\right)\right), \ell\right)\right)\right)\right)\right) \]

   7. Applied egg-rr 48.2%

      \[\leadsto \frac{2}{\color{blue}{\left(0 - \frac{t}{\ell}\right) \cdot \left(0 - k \cdot \frac{k \cdot \left(k \cdot k\right)}{\ell}\right)}} \]
   8. Step-by-step derivation

      1. associate-/r* N/A

         \[\leadsto \frac{\frac{2}{0 - \frac{t}{\ell}}}{\color{blue}{0 - k \cdot \frac{k \cdot \left(k \cdot k\right)}{\ell}}} \]

      2. sub0-neg N/A

         \[\leadsto \frac{\frac{2}{0 - \frac{t}{\ell}}}{\mathsf{neg}\left(k \cdot \frac{k \cdot \left(k \cdot k\right)}{\ell}\right)} \]

      3. distribute-frac-neg2 N/A

         \[\leadsto \mathsf{neg}\left(\frac{\frac{2}{0 - \frac{t}{\ell}}}{k \cdot \frac{k \cdot \left(k \cdot k\right)}{\ell}}\right) \]

      4. distribute-frac-neg N/A

         \[\leadsto \frac{\mathsf{neg}\left(\frac{2}{0 - \frac{t}{\ell}}\right)}{\color{blue}{k \cdot \frac{k \cdot \left(k \cdot k\right)}{\ell}}} \]

      5. distribute-frac-neg2 N/A

         \[\leadsto \frac{\frac{2}{\mathsf{neg}\left(\left(0 - \frac{t}{\ell}\right)\right)}}{\color{blue}{k} \cdot \frac{k \cdot \left(k \cdot k\right)}{\ell}} \]

      6. sub0-neg N/A

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

      7. remove-double-neg N/A

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

      8. un-div-inv N/A

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

      9. clear-num N/A

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

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

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

      11. clear-num N/A

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

      12. un-div-inv N/A

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

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

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

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

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

      15. associate-/l* N/A

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

      16. associate-*r* N/A

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

      17. clear-num N/A

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

      18. un-div-inv N/A

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

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

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

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

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

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

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

   9. Applied egg-rr 71.9%

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

       1. associate-/r/ N/A

          \[\leadsto \frac{\frac{2}{t} \cdot \ell}{\frac{\color{blue}{k \cdot k}}{\frac{\ell}{k \cdot k}}} \]

       2. associate-/r/ N/A

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

       3. times-frac N/A

          \[\leadsto \frac{\frac{2}{t}}{\frac{k \cdot k}{\ell}} \cdot \color{blue}{\frac{\ell}{k \cdot k}} \]

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

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

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

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

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

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

       7. associate-*l/ N/A

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

       8. associate-/r/ N/A

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

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

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

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

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

       11. associate-/r* N/A

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

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

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

       13. /-lowering-/.f64 81.1%

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

   11. Applied egg-rr 81.1%

       \[\leadsto \color{blue}{\frac{\frac{2}{t}}{\frac{k}{\frac{\ell}{k}}} \cdot \frac{\frac{\ell}{k}}{k}} \]

   if 5.0000000000000001e-100 < k

   1. Initial program 38.3%

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

   3. Taylor expanded in t around 0

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

      1. associate-/l* N/A

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

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

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

      3. unpow2 N/A

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

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

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

      5. *-commutative N/A

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

      6. associate-/r* N/A

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

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

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

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

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

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

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

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

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

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

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

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

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

      13. unpow2 N/A

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

      14. *-lowering-*.f64 68.6%

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

   5. Simplified 68.6%

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

      1. associate-*r/ N/A

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

      2. times-frac N/A

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

      3. clear-num N/A

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

      4. un-div-inv N/A

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

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

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

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

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

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

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

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

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

      9. associate-/l* N/A

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

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

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

      11. unpow2 N/A

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

      12. associate-/l* N/A

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

      13. tan-quot N/A

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

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

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

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

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

      16. tan-lowering-tan.f64 76.4%

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

   7. Applied egg-rr 76.4%

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

      1. associate-/l* N/A

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

      2. div-inv N/A

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

      3. times-frac N/A

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

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

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

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

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

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

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

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

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

      8. associate-/r* N/A

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

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

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

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

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

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

          \[\leadsto \mathsf{/.f64}\left(2, \mathsf{*.f64}\left(\mathsf{/.f64}\left(k, \ell\right), \mathsf{/.f64}\left(\mathsf{/.f64}\left(k, \ell\right), \mathsf{/.f64}\left(\mathsf{/.f64}\left(1, t\right), \mathsf{*.f64}\left(\sin k, \color{blue}{\tan k}\right)\right)\right)\right)\right) \]

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

          \[\leadsto \mathsf{/.f64}\left(2, \mathsf{*.f64}\left(\mathsf{/.f64}\left(k, \ell\right), \mathsf{/.f64}\left(\mathsf{/.f64}\left(k, \ell\right), \mathsf{/.f64}\left(\mathsf{/.f64}\left(1, t\right), \mathsf{*.f64}\left(\mathsf{sin.f64}\left(k\right), \tan \color{blue}{k}\right)\right)\right)\right)\right) \]

      13. tan-lowering-tan.f64 96.0%

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

   9. Applied egg-rr 96.0%

      \[\leadsto \frac{2}{\color{blue}{\frac{k}{\ell} \cdot \frac{\frac{k}{\ell}}{\frac{\frac{1}{t}}{\sin k \cdot \tan k}}}} \]
   10. Step-by-step derivation

       1. clear-num N/A

          \[\leadsto \frac{1}{\color{blue}{\frac{\frac{k}{\ell} \cdot \frac{\frac{k}{\ell}}{\frac{\frac{1}{t}}{\sin k \cdot \tan k}}}{2}}} \]

       2. associate-/r/ N/A

          \[\leadsto \frac{1}{\frac{k}{\ell} \cdot \frac{\frac{k}{\ell}}{\frac{\frac{1}{t}}{\sin k \cdot \tan k}}} \cdot \color{blue}{2} \]

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

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

   11. Applied egg-rr 87.7%

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

4. Final simplification 83.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;k \leq 5 \cdot 10^{-100}:\\ \;\;\;\;\frac{\frac{2}{t}}{\frac{k}{\frac{\ell}{k}}} \cdot \frac{\frac{\ell}{k}}{k}\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \left(\frac{\frac{\ell}{k}}{k} \cdot \frac{\frac{\ell}{t}}{\sin k \cdot \tan k}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 74.0% accurate, 21.0× speedup

Math

\[\begin{array}{l} k_m = \left|k\right| \\ \begin{array}{l} \mathbf{if}\;t \leq 4 \cdot 10^{+14}:\\ \;\;\;\;\frac{2}{k\_m} \cdot \frac{\frac{\ell}{t}}{\frac{k\_m}{\frac{\frac{\ell}{k\_m}}{k\_m}}}\\ \mathbf{else}:\\ \;\;\;\;\ell \cdot \frac{\ell \cdot \frac{2}{k\_m}}{k\_m \cdot \left(k\_m \cdot \left(k\_m \cdot t\right)\right)}\\ \end{array} \end{array} \]

FPCore

k_m = (fabs.f64 k)
(FPCore (t l k_m)
 :precision binary64
 (if (<= t 4e+14)
   (* (/ 2.0 k_m) (/ (/ l t) (/ k_m (/ (/ l k_m) k_m))))
   (* l (/ (* l (/ 2.0 k_m)) (* k_m (* k_m (* k_m t)))))))

C

k_m = fabs(k);
double code(double t, double l, double k_m) {
	double tmp;
	if (t <= 4e+14) {
		tmp = (2.0 / k_m) * ((l / t) / (k_m / ((l / k_m) / k_m)));
	} else {
		tmp = l * ((l * (2.0 / k_m)) / (k_m * (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) :: tmp
    if (t <= 4d+14) then
        tmp = (2.0d0 / k_m) * ((l / t) / (k_m / ((l / k_m) / k_m)))
    else
        tmp = l * ((l * (2.0d0 / k_m)) / (k_m * (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 tmp;
	if (t <= 4e+14) {
		tmp = (2.0 / k_m) * ((l / t) / (k_m / ((l / k_m) / k_m)));
	} else {
		tmp = l * ((l * (2.0 / k_m)) / (k_m * (k_m * (k_m * t))));
	}
	return tmp;
}

Python

k_m = math.fabs(k)
def code(t, l, k_m):
	tmp = 0
	if t <= 4e+14:
		tmp = (2.0 / k_m) * ((l / t) / (k_m / ((l / k_m) / k_m)))
	else:
		tmp = l * ((l * (2.0 / k_m)) / (k_m * (k_m * (k_m * t))))
	return tmp

Julia

k_m = abs(k)
function code(t, l, k_m)
	tmp = 0.0
	if (t <= 4e+14)
		tmp = Float64(Float64(2.0 / k_m) * Float64(Float64(l / t) / Float64(k_m / Float64(Float64(l / k_m) / k_m))));
	else
		tmp = Float64(l * Float64(Float64(l * Float64(2.0 / k_m)) / Float64(k_m * Float64(k_m * Float64(k_m * t)))));
	end
	return tmp
end

MATLAB

k_m = abs(k);
function tmp_2 = code(t, l, k_m)
	tmp = 0.0;
	if (t <= 4e+14)
		tmp = (2.0 / k_m) * ((l / t) / (k_m / ((l / k_m) / k_m)));
	else
		tmp = l * ((l * (2.0 / k_m)) / (k_m * (k_m * (k_m * t))));
	end
	tmp_2 = tmp;
end

Wolfram

k_m = N[Abs[k], $MachinePrecision]
code[t_, l_, k$95$m_] := If[LessEqual[t, 4e+14], N[(N[(2.0 / k$95$m), $MachinePrecision] * N[(N[(l / t), $MachinePrecision] / N[(k$95$m / N[(N[(l / k$95$m), $MachinePrecision] / k$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(l * N[(N[(l * N[(2.0 / k$95$m), $MachinePrecision]), $MachinePrecision] / N[(k$95$m * N[(k$95$m * N[(k$95$m * t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if t < 4e14

   1. Initial program 41.5%

      \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) - 1\right)} \]
   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. /-lowering-/.f64 N/A

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

      2. *-commutative N/A

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

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

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

      4. metadata-eval N/A

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

      5. pow-sqr N/A

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

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

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

      7. unpow2 N/A

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

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

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

      9. unpow2 N/A

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

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

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

      11. unpow2 N/A

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

      12. *-lowering-*.f64 58.2%

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

   5. Simplified 58.2%

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

      1. remove-double-neg N/A

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

      2. times-frac N/A

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

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

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

      4. distribute-lft-neg-in N/A

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

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

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

      6. neg-sub0 N/A

         \[\leadsto \mathsf{/.f64}\left(2, \mathsf{*.f64}\left(\left(0 - \frac{t}{\ell}\right), \left(\mathsf{neg}\left(\color{blue}{\frac{\left(k \cdot k\right) \cdot \left(k \cdot k\right)}{\ell}}\right)\right)\right)\right) \]

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

         \[\leadsto \mathsf{/.f64}\left(2, \mathsf{*.f64}\left(\mathsf{\_.f64}\left(0, \left(\frac{t}{\ell}\right)\right), \left(\mathsf{neg}\left(\color{blue}{\frac{\left(k \cdot k\right) \cdot \left(k \cdot k\right)}{\ell}}\right)\right)\right)\right) \]

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

         \[\leadsto \mathsf{/.f64}\left(2, \mathsf{*.f64}\left(\mathsf{\_.f64}\left(0, \mathsf{/.f64}\left(t, \ell\right)\right), \left(\mathsf{neg}\left(\frac{\left(k \cdot k\right) \cdot \left(k \cdot k\right)}{\color{blue}{\ell}}\right)\right)\right)\right) \]

      9. neg-sub0 N/A

         \[\leadsto \mathsf{/.f64}\left(2, \mathsf{*.f64}\left(\mathsf{\_.f64}\left(0, \mathsf{/.f64}\left(t, \ell\right)\right), \left(0 - \color{blue}{\frac{\left(k \cdot k\right) \cdot \left(k \cdot k\right)}{\ell}}\right)\right)\right) \]

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

          \[\leadsto \mathsf{/.f64}\left(2, \mathsf{*.f64}\left(\mathsf{\_.f64}\left(0, \mathsf{/.f64}\left(t, \ell\right)\right), \mathsf{\_.f64}\left(0, \color{blue}{\left(\frac{\left(k \cdot k\right) \cdot \left(k \cdot k\right)}{\ell}\right)}\right)\right)\right) \]

      11. associate-*l* N/A

          \[\leadsto \mathsf{/.f64}\left(2, \mathsf{*.f64}\left(\mathsf{\_.f64}\left(0, \mathsf{/.f64}\left(t, \ell\right)\right), \mathsf{\_.f64}\left(0, \left(\frac{k \cdot \left(k \cdot \left(k \cdot k\right)\right)}{\ell}\right)\right)\right)\right) \]

      12. associate-/l* N/A

          \[\leadsto \mathsf{/.f64}\left(2, \mathsf{*.f64}\left(\mathsf{\_.f64}\left(0, \mathsf{/.f64}\left(t, \ell\right)\right), \mathsf{\_.f64}\left(0, \left(k \cdot \color{blue}{\frac{k \cdot \left(k \cdot k\right)}{\ell}}\right)\right)\right)\right) \]

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

          \[\leadsto \mathsf{/.f64}\left(2, \mathsf{*.f64}\left(\mathsf{\_.f64}\left(0, \mathsf{/.f64}\left(t, \ell\right)\right), \mathsf{\_.f64}\left(0, \mathsf{*.f64}\left(k, \color{blue}{\left(\frac{k \cdot \left(k \cdot k\right)}{\ell}\right)}\right)\right)\right)\right) \]

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

          \[\leadsto \mathsf{/.f64}\left(2, \mathsf{*.f64}\left(\mathsf{\_.f64}\left(0, \mathsf{/.f64}\left(t, \ell\right)\right), \mathsf{\_.f64}\left(0, \mathsf{*.f64}\left(k, \mathsf{/.f64}\left(\left(k \cdot \left(k \cdot k\right)\right), \color{blue}{\ell}\right)\right)\right)\right)\right) \]

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

          \[\leadsto \mathsf{/.f64}\left(2, \mathsf{*.f64}\left(\mathsf{\_.f64}\left(0, \mathsf{/.f64}\left(t, \ell\right)\right), \mathsf{\_.f64}\left(0, \mathsf{*.f64}\left(k, \mathsf{/.f64}\left(\mathsf{*.f64}\left(k, \left(k \cdot k\right)\right), \ell\right)\right)\right)\right)\right) \]

      16. *-lowering-*.f64 53.2%

          \[\leadsto \mathsf{/.f64}\left(2, \mathsf{*.f64}\left(\mathsf{\_.f64}\left(0, \mathsf{/.f64}\left(t, \ell\right)\right), \mathsf{\_.f64}\left(0, \mathsf{*.f64}\left(k, \mathsf{/.f64}\left(\mathsf{*.f64}\left(k, \mathsf{*.f64}\left(k, k\right)\right), \ell\right)\right)\right)\right)\right) \]

   7. Applied egg-rr 53.2%

      \[\leadsto \frac{2}{\color{blue}{\left(0 - \frac{t}{\ell}\right) \cdot \left(0 - k \cdot \frac{k \cdot \left(k \cdot k\right)}{\ell}\right)}} \]
   8. Step-by-step derivation

      1. associate-/r* N/A

         \[\leadsto \frac{\frac{2}{0 - \frac{t}{\ell}}}{\color{blue}{0 - k \cdot \frac{k \cdot \left(k \cdot k\right)}{\ell}}} \]

      2. sub0-neg N/A

         \[\leadsto \frac{\frac{2}{0 - \frac{t}{\ell}}}{\mathsf{neg}\left(k \cdot \frac{k \cdot \left(k \cdot k\right)}{\ell}\right)} \]

      3. distribute-frac-neg2 N/A

         \[\leadsto \mathsf{neg}\left(\frac{\frac{2}{0 - \frac{t}{\ell}}}{k \cdot \frac{k \cdot \left(k \cdot k\right)}{\ell}}\right) \]

      4. distribute-frac-neg N/A

         \[\leadsto \frac{\mathsf{neg}\left(\frac{2}{0 - \frac{t}{\ell}}\right)}{\color{blue}{k \cdot \frac{k \cdot \left(k \cdot k\right)}{\ell}}} \]

      5. distribute-frac-neg2 N/A

         \[\leadsto \frac{\frac{2}{\mathsf{neg}\left(\left(0 - \frac{t}{\ell}\right)\right)}}{\color{blue}{k} \cdot \frac{k \cdot \left(k \cdot k\right)}{\ell}} \]

      6. sub0-neg N/A

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

      7. remove-double-neg N/A

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

      8. un-div-inv N/A

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

      9. clear-num N/A

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

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

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

      11. clear-num N/A

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

      12. un-div-inv N/A

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

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

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

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

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

      15. associate-/l* N/A

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

      16. associate-*r* N/A

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

      17. clear-num N/A

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

      18. un-div-inv N/A

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

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

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

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

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

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

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

   9. Applied egg-rr 72.4%

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

       1. div-inv N/A

          \[\leadsto \frac{2 \cdot \frac{1}{\frac{t}{\ell}}}{\frac{\color{blue}{k \cdot k}}{\frac{\ell}{k \cdot k}}} \]

       2. associate-/l* N/A

          \[\leadsto \frac{2 \cdot \frac{1}{\frac{t}{\ell}}}{k \cdot \color{blue}{\frac{k}{\frac{\ell}{k \cdot k}}}} \]

       3. times-frac N/A

          \[\leadsto \frac{2}{k} \cdot \color{blue}{\frac{\frac{1}{\frac{t}{\ell}}}{\frac{k}{\frac{\ell}{k \cdot k}}}} \]

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

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

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

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

       6. associate-/r* N/A

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

       7. associate-/r/ N/A

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

       8. *-commutative N/A

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

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

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

       10. clear-num N/A

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

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

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

       12. clear-num N/A

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

       13. associate-/r* N/A

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

       14. div-inv N/A

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

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

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

       16. associate-/r* N/A

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

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

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

       18. /-lowering-/.f64 75.0%

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

   11. Applied egg-rr 75.0%

       \[\leadsto \color{blue}{\frac{2}{k} \cdot \frac{\frac{\ell}{t}}{\frac{k}{\frac{\frac{\ell}{k}}{k}}}} \]

   if 4e14 < t

   1. Initial program 23.2%

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

   3. Taylor expanded in 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. /-lowering-/.f64 N/A

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

      2. *-commutative N/A

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

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

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

      4. metadata-eval N/A

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

      5. pow-sqr N/A

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

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

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

      7. unpow2 N/A

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

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

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

      9. unpow2 N/A

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

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

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

      11. unpow2 N/A

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

      12. *-lowering-*.f64 60.6%

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

   5. Simplified 60.6%

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

      1. associate-/r/ N/A

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

      2. associate-*r* N/A

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

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

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

      4. *-commutative N/A

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

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

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

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

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

      7. associate-*l* N/A

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

      8. *-commutative N/A

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

      9. associate-*r* N/A

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

      10. *-commutative N/A

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

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

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

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

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

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

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

      14. *-lowering-*.f64 62.8%

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

   7. Applied egg-rr 62.8%

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

      1. *-commutative N/A

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

      2. associate-/r* N/A

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

      3. associate-*l/ N/A

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

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

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

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

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

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

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

      7. associate-*r* N/A

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

      8. associate-*r* N/A

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

      9. *-commutative N/A

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

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

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

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

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

      12. *-lowering-*.f64 67.0%

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

   9. Applied egg-rr 67.0%

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

4. Final simplification 72.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq 4 \cdot 10^{+14}:\\ \;\;\;\;\frac{2}{k} \cdot \frac{\frac{\ell}{t}}{\frac{k}{\frac{\frac{\ell}{k}}{k}}}\\ \mathbf{else}:\\ \;\;\;\;\ell \cdot \frac{\ell \cdot \frac{2}{k}}{k \cdot \left(k \cdot \left(k \cdot t\right)\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 73.9% accurate, 21.0× speedup

Math

\[\begin{array}{l} k_m = \left|k\right| \\ \begin{array}{l} \mathbf{if}\;t \leq 1.2 \cdot 10^{+17}:\\ \;\;\;\;\frac{2}{k\_m} \cdot \frac{\frac{\ell}{t}}{\frac{k\_m}{\frac{\frac{\ell}{k\_m}}{k\_m}}}\\ \mathbf{else}:\\ \;\;\;\;\ell \cdot \left(\frac{\ell}{k\_m} \cdot \frac{2}{k\_m \cdot \left(k\_m \cdot \left(k\_m \cdot t\right)\right)}\right)\\ \end{array} \end{array} \]

FPCore

k_m = (fabs.f64 k)
(FPCore (t l k_m)
 :precision binary64
 (if (<= t 1.2e+17)
   (* (/ 2.0 k_m) (/ (/ l t) (/ k_m (/ (/ l k_m) k_m))))
   (* l (* (/ l k_m) (/ 2.0 (* k_m (* k_m (* k_m t))))))))

C

k_m = fabs(k);
double code(double t, double l, double k_m) {
	double tmp;
	if (t <= 1.2e+17) {
		tmp = (2.0 / k_m) * ((l / t) / (k_m / ((l / k_m) / k_m)));
	} else {
		tmp = l * ((l / k_m) * (2.0 / (k_m * (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) :: tmp
    if (t <= 1.2d+17) then
        tmp = (2.0d0 / k_m) * ((l / t) / (k_m / ((l / k_m) / k_m)))
    else
        tmp = l * ((l / k_m) * (2.0d0 / (k_m * (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 tmp;
	if (t <= 1.2e+17) {
		tmp = (2.0 / k_m) * ((l / t) / (k_m / ((l / k_m) / k_m)));
	} else {
		tmp = l * ((l / k_m) * (2.0 / (k_m * (k_m * (k_m * t)))));
	}
	return tmp;
}

Python

k_m = math.fabs(k)
def code(t, l, k_m):
	tmp = 0
	if t <= 1.2e+17:
		tmp = (2.0 / k_m) * ((l / t) / (k_m / ((l / k_m) / k_m)))
	else:
		tmp = l * ((l / k_m) * (2.0 / (k_m * (k_m * (k_m * t)))))
	return tmp

Julia

k_m = abs(k)
function code(t, l, k_m)
	tmp = 0.0
	if (t <= 1.2e+17)
		tmp = Float64(Float64(2.0 / k_m) * Float64(Float64(l / t) / Float64(k_m / Float64(Float64(l / k_m) / k_m))));
	else
		tmp = Float64(l * Float64(Float64(l / k_m) * Float64(2.0 / Float64(k_m * Float64(k_m * Float64(k_m * t))))));
	end
	return tmp
end

MATLAB

k_m = abs(k);
function tmp_2 = code(t, l, k_m)
	tmp = 0.0;
	if (t <= 1.2e+17)
		tmp = (2.0 / k_m) * ((l / t) / (k_m / ((l / k_m) / k_m)));
	else
		tmp = l * ((l / k_m) * (2.0 / (k_m * (k_m * (k_m * t)))));
	end
	tmp_2 = tmp;
end

Wolfram

k_m = N[Abs[k], $MachinePrecision]
code[t_, l_, k$95$m_] := If[LessEqual[t, 1.2e+17], N[(N[(2.0 / k$95$m), $MachinePrecision] * N[(N[(l / t), $MachinePrecision] / N[(k$95$m / N[(N[(l / k$95$m), $MachinePrecision] / k$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(l * N[(N[(l / k$95$m), $MachinePrecision] * N[(2.0 / N[(k$95$m * N[(k$95$m * N[(k$95$m * t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if t < 1.2e17

   1. Initial program 41.5%

      \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) - 1\right)} \]
   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. /-lowering-/.f64 N/A

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

      2. *-commutative N/A

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

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

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

      4. metadata-eval N/A

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

      5. pow-sqr N/A

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

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

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

      7. unpow2 N/A

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

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

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

      9. unpow2 N/A

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

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

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

      11. unpow2 N/A

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

      12. *-lowering-*.f64 58.2%

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

   5. Simplified 58.2%

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

      1. remove-double-neg N/A

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

      2. times-frac N/A

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

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

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

      4. distribute-lft-neg-in N/A

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

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

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

      6. neg-sub0 N/A

         \[\leadsto \mathsf{/.f64}\left(2, \mathsf{*.f64}\left(\left(0 - \frac{t}{\ell}\right), \left(\mathsf{neg}\left(\color{blue}{\frac{\left(k \cdot k\right) \cdot \left(k \cdot k\right)}{\ell}}\right)\right)\right)\right) \]

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

         \[\leadsto \mathsf{/.f64}\left(2, \mathsf{*.f64}\left(\mathsf{\_.f64}\left(0, \left(\frac{t}{\ell}\right)\right), \left(\mathsf{neg}\left(\color{blue}{\frac{\left(k \cdot k\right) \cdot \left(k \cdot k\right)}{\ell}}\right)\right)\right)\right) \]

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

         \[\leadsto \mathsf{/.f64}\left(2, \mathsf{*.f64}\left(\mathsf{\_.f64}\left(0, \mathsf{/.f64}\left(t, \ell\right)\right), \left(\mathsf{neg}\left(\frac{\left(k \cdot k\right) \cdot \left(k \cdot k\right)}{\color{blue}{\ell}}\right)\right)\right)\right) \]

      9. neg-sub0 N/A

         \[\leadsto \mathsf{/.f64}\left(2, \mathsf{*.f64}\left(\mathsf{\_.f64}\left(0, \mathsf{/.f64}\left(t, \ell\right)\right), \left(0 - \color{blue}{\frac{\left(k \cdot k\right) \cdot \left(k \cdot k\right)}{\ell}}\right)\right)\right) \]

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

          \[\leadsto \mathsf{/.f64}\left(2, \mathsf{*.f64}\left(\mathsf{\_.f64}\left(0, \mathsf{/.f64}\left(t, \ell\right)\right), \mathsf{\_.f64}\left(0, \color{blue}{\left(\frac{\left(k \cdot k\right) \cdot \left(k \cdot k\right)}{\ell}\right)}\right)\right)\right) \]

      11. associate-*l* N/A

          \[\leadsto \mathsf{/.f64}\left(2, \mathsf{*.f64}\left(\mathsf{\_.f64}\left(0, \mathsf{/.f64}\left(t, \ell\right)\right), \mathsf{\_.f64}\left(0, \left(\frac{k \cdot \left(k \cdot \left(k \cdot k\right)\right)}{\ell}\right)\right)\right)\right) \]

      12. associate-/l* N/A

          \[\leadsto \mathsf{/.f64}\left(2, \mathsf{*.f64}\left(\mathsf{\_.f64}\left(0, \mathsf{/.f64}\left(t, \ell\right)\right), \mathsf{\_.f64}\left(0, \left(k \cdot \color{blue}{\frac{k \cdot \left(k \cdot k\right)}{\ell}}\right)\right)\right)\right) \]

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

          \[\leadsto \mathsf{/.f64}\left(2, \mathsf{*.f64}\left(\mathsf{\_.f64}\left(0, \mathsf{/.f64}\left(t, \ell\right)\right), \mathsf{\_.f64}\left(0, \mathsf{*.f64}\left(k, \color{blue}{\left(\frac{k \cdot \left(k \cdot k\right)}{\ell}\right)}\right)\right)\right)\right) \]

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

          \[\leadsto \mathsf{/.f64}\left(2, \mathsf{*.f64}\left(\mathsf{\_.f64}\left(0, \mathsf{/.f64}\left(t, \ell\right)\right), \mathsf{\_.f64}\left(0, \mathsf{*.f64}\left(k, \mathsf{/.f64}\left(\left(k \cdot \left(k \cdot k\right)\right), \color{blue}{\ell}\right)\right)\right)\right)\right) \]

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

          \[\leadsto \mathsf{/.f64}\left(2, \mathsf{*.f64}\left(\mathsf{\_.f64}\left(0, \mathsf{/.f64}\left(t, \ell\right)\right), \mathsf{\_.f64}\left(0, \mathsf{*.f64}\left(k, \mathsf{/.f64}\left(\mathsf{*.f64}\left(k, \left(k \cdot k\right)\right), \ell\right)\right)\right)\right)\right) \]

      16. *-lowering-*.f64 53.2%

          \[\leadsto \mathsf{/.f64}\left(2, \mathsf{*.f64}\left(\mathsf{\_.f64}\left(0, \mathsf{/.f64}\left(t, \ell\right)\right), \mathsf{\_.f64}\left(0, \mathsf{*.f64}\left(k, \mathsf{/.f64}\left(\mathsf{*.f64}\left(k, \mathsf{*.f64}\left(k, k\right)\right), \ell\right)\right)\right)\right)\right) \]

   7. Applied egg-rr 53.2%

      \[\leadsto \frac{2}{\color{blue}{\left(0 - \frac{t}{\ell}\right) \cdot \left(0 - k \cdot \frac{k \cdot \left(k \cdot k\right)}{\ell}\right)}} \]
   8. Step-by-step derivation

      1. associate-/r* N/A

         \[\leadsto \frac{\frac{2}{0 - \frac{t}{\ell}}}{\color{blue}{0 - k \cdot \frac{k \cdot \left(k \cdot k\right)}{\ell}}} \]

      2. sub0-neg N/A

         \[\leadsto \frac{\frac{2}{0 - \frac{t}{\ell}}}{\mathsf{neg}\left(k \cdot \frac{k \cdot \left(k \cdot k\right)}{\ell}\right)} \]

      3. distribute-frac-neg2 N/A

         \[\leadsto \mathsf{neg}\left(\frac{\frac{2}{0 - \frac{t}{\ell}}}{k \cdot \frac{k \cdot \left(k \cdot k\right)}{\ell}}\right) \]

      4. distribute-frac-neg N/A

         \[\leadsto \frac{\mathsf{neg}\left(\frac{2}{0 - \frac{t}{\ell}}\right)}{\color{blue}{k \cdot \frac{k \cdot \left(k \cdot k\right)}{\ell}}} \]

      5. distribute-frac-neg2 N/A

         \[\leadsto \frac{\frac{2}{\mathsf{neg}\left(\left(0 - \frac{t}{\ell}\right)\right)}}{\color{blue}{k} \cdot \frac{k \cdot \left(k \cdot k\right)}{\ell}} \]

      6. sub0-neg N/A

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

      7. remove-double-neg N/A

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

      8. un-div-inv N/A

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

      9. clear-num N/A

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

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

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

      11. clear-num N/A

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

      12. un-div-inv N/A

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

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

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

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

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

      15. associate-/l* N/A

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

      16. associate-*r* N/A

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

      17. clear-num N/A

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

      18. un-div-inv N/A

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

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

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

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

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

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

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

   9. Applied egg-rr 72.4%

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

       1. div-inv N/A

          \[\leadsto \frac{2 \cdot \frac{1}{\frac{t}{\ell}}}{\frac{\color{blue}{k \cdot k}}{\frac{\ell}{k \cdot k}}} \]

       2. associate-/l* N/A

          \[\leadsto \frac{2 \cdot \frac{1}{\frac{t}{\ell}}}{k \cdot \color{blue}{\frac{k}{\frac{\ell}{k \cdot k}}}} \]

       3. times-frac N/A

          \[\leadsto \frac{2}{k} \cdot \color{blue}{\frac{\frac{1}{\frac{t}{\ell}}}{\frac{k}{\frac{\ell}{k \cdot k}}}} \]

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

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

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

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

       6. associate-/r* N/A

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

       7. associate-/r/ N/A

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

       8. *-commutative N/A

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

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

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

       10. clear-num N/A

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

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

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

       12. clear-num N/A

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

       13. associate-/r* N/A

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

       14. div-inv N/A

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

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

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

       16. associate-/r* N/A

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

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

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

       18. /-lowering-/.f64 75.0%

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

   11. Applied egg-rr 75.0%

       \[\leadsto \color{blue}{\frac{2}{k} \cdot \frac{\frac{\ell}{t}}{\frac{k}{\frac{\frac{\ell}{k}}{k}}}} \]

   if 1.2e17 < t

   1. Initial program 23.2%

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

   3. Taylor expanded in 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. /-lowering-/.f64 N/A

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

      2. *-commutative N/A

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

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

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

      4. metadata-eval N/A

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

      5. pow-sqr N/A

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

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

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

      7. unpow2 N/A

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

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

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

      9. unpow2 N/A

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

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

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

      11. unpow2 N/A

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

      12. *-lowering-*.f64 60.6%

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

   5. Simplified 60.6%

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

      1. associate-/r/ N/A

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

      2. associate-*r* N/A

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

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

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

      4. *-commutative N/A

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

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

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

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

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

      7. associate-*l* N/A

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

      8. *-commutative N/A

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

      9. associate-*r* N/A

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

      10. *-commutative N/A

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

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

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

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

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

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

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

      14. *-lowering-*.f64 62.8%

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

   7. Applied egg-rr 62.8%

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

      1. associate-*r/ N/A

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

      2. times-frac N/A

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

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

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

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

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

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

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

      6. associate-*r* N/A

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

      7. associate-*r* N/A

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

      8. *-commutative N/A

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

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

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

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

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

      11. *-lowering-*.f64 67.0%

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

   9. Applied egg-rr 67.0%

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

4. Final simplification 72.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq 1.2 \cdot 10^{+17}:\\ \;\;\;\;\frac{2}{k} \cdot \frac{\frac{\ell}{t}}{\frac{k}{\frac{\frac{\ell}{k}}{k}}}\\ \mathbf{else}:\\ \;\;\;\;\ell \cdot \left(\frac{\ell}{k} \cdot \frac{2}{k \cdot \left(k \cdot \left(k \cdot t\right)\right)}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 76.3% accurate, 28.1× speedup

Math

\[\begin{array}{l} k_m = \left|k\right| \\ \frac{\frac{2}{t}}{\frac{k\_m}{\frac{\ell}{k\_m}}} \cdot \frac{\frac{\ell}{k\_m}}{k\_m} \end{array} \]

FPCore

k_m = (fabs.f64 k)
(FPCore (t l k_m)
 :precision binary64
 (* (/ (/ 2.0 t) (/ k_m (/ l k_m))) (/ (/ l k_m) k_m)))

C

k_m = fabs(k);
double code(double t, double l, double k_m) {
	return ((2.0 / t) / (k_m / (l / k_m))) * ((l / 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 / t) / (k_m / (l / k_m))) * ((l / 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 / t) / (k_m / (l / k_m))) * ((l / k_m) / k_m);
}

Python

k_m = math.fabs(k)
def code(t, l, k_m):
	return ((2.0 / t) / (k_m / (l / k_m))) * ((l / k_m) / k_m)

Julia

k_m = abs(k)
function code(t, l, k_m)
	return Float64(Float64(Float64(2.0 / t) / Float64(k_m / Float64(l / k_m))) * Float64(Float64(l / k_m) / k_m))
end

MATLAB

k_m = abs(k);
function tmp = code(t, l, k_m)
	tmp = ((2.0 / t) / (k_m / (l / k_m))) * ((l / k_m) / k_m);
end

Wolfram

k_m = N[Abs[k], $MachinePrecision]
code[t_, l_, k$95$m_] := N[(N[(N[(2.0 / t), $MachinePrecision] / N[(k$95$m / N[(l / k$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(N[(l / k$95$m), $MachinePrecision] / k$95$m), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 36.6%

   \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) - 1\right)} \]
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. /-lowering-/.f64 N/A

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

   2. *-commutative N/A

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

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

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

   4. metadata-eval N/A

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

   5. pow-sqr N/A

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

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

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

   7. unpow2 N/A

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

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

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

   9. unpow2 N/A

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

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

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

   11. unpow2 N/A

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

   12. *-lowering-*.f64 58.8%

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

5. Simplified 58.8%

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

   1. remove-double-neg N/A

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

   2. times-frac N/A

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

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

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

   4. distribute-lft-neg-in N/A

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

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

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

   6. neg-sub0 N/A

      \[\leadsto \mathsf{/.f64}\left(2, \mathsf{*.f64}\left(\left(0 - \frac{t}{\ell}\right), \left(\mathsf{neg}\left(\color{blue}{\frac{\left(k \cdot k\right) \cdot \left(k \cdot k\right)}{\ell}}\right)\right)\right)\right) \]

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

      \[\leadsto \mathsf{/.f64}\left(2, \mathsf{*.f64}\left(\mathsf{\_.f64}\left(0, \left(\frac{t}{\ell}\right)\right), \left(\mathsf{neg}\left(\color{blue}{\frac{\left(k \cdot k\right) \cdot \left(k \cdot k\right)}{\ell}}\right)\right)\right)\right) \]

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

      \[\leadsto \mathsf{/.f64}\left(2, \mathsf{*.f64}\left(\mathsf{\_.f64}\left(0, \mathsf{/.f64}\left(t, \ell\right)\right), \left(\mathsf{neg}\left(\frac{\left(k \cdot k\right) \cdot \left(k \cdot k\right)}{\color{blue}{\ell}}\right)\right)\right)\right) \]

   9. neg-sub0 N/A

      \[\leadsto \mathsf{/.f64}\left(2, \mathsf{*.f64}\left(\mathsf{\_.f64}\left(0, \mathsf{/.f64}\left(t, \ell\right)\right), \left(0 - \color{blue}{\frac{\left(k \cdot k\right) \cdot \left(k \cdot k\right)}{\ell}}\right)\right)\right) \]

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

       \[\leadsto \mathsf{/.f64}\left(2, \mathsf{*.f64}\left(\mathsf{\_.f64}\left(0, \mathsf{/.f64}\left(t, \ell\right)\right), \mathsf{\_.f64}\left(0, \color{blue}{\left(\frac{\left(k \cdot k\right) \cdot \left(k \cdot k\right)}{\ell}\right)}\right)\right)\right) \]

   11. associate-*l* N/A

       \[\leadsto \mathsf{/.f64}\left(2, \mathsf{*.f64}\left(\mathsf{\_.f64}\left(0, \mathsf{/.f64}\left(t, \ell\right)\right), \mathsf{\_.f64}\left(0, \left(\frac{k \cdot \left(k \cdot \left(k \cdot k\right)\right)}{\ell}\right)\right)\right)\right) \]

   12. associate-/l* N/A

       \[\leadsto \mathsf{/.f64}\left(2, \mathsf{*.f64}\left(\mathsf{\_.f64}\left(0, \mathsf{/.f64}\left(t, \ell\right)\right), \mathsf{\_.f64}\left(0, \left(k \cdot \color{blue}{\frac{k \cdot \left(k \cdot k\right)}{\ell}}\right)\right)\right)\right) \]

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

       \[\leadsto \mathsf{/.f64}\left(2, \mathsf{*.f64}\left(\mathsf{\_.f64}\left(0, \mathsf{/.f64}\left(t, \ell\right)\right), \mathsf{\_.f64}\left(0, \mathsf{*.f64}\left(k, \color{blue}{\left(\frac{k \cdot \left(k \cdot k\right)}{\ell}\right)}\right)\right)\right)\right) \]

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

       \[\leadsto \mathsf{/.f64}\left(2, \mathsf{*.f64}\left(\mathsf{\_.f64}\left(0, \mathsf{/.f64}\left(t, \ell\right)\right), \mathsf{\_.f64}\left(0, \mathsf{*.f64}\left(k, \mathsf{/.f64}\left(\left(k \cdot \left(k \cdot k\right)\right), \color{blue}{\ell}\right)\right)\right)\right)\right) \]

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

       \[\leadsto \mathsf{/.f64}\left(2, \mathsf{*.f64}\left(\mathsf{\_.f64}\left(0, \mathsf{/.f64}\left(t, \ell\right)\right), \mathsf{\_.f64}\left(0, \mathsf{*.f64}\left(k, \mathsf{/.f64}\left(\mathsf{*.f64}\left(k, \left(k \cdot k\right)\right), \ell\right)\right)\right)\right)\right) \]

   16. *-lowering-*.f64 51.8%

       \[\leadsto \mathsf{/.f64}\left(2, \mathsf{*.f64}\left(\mathsf{\_.f64}\left(0, \mathsf{/.f64}\left(t, \ell\right)\right), \mathsf{\_.f64}\left(0, \mathsf{*.f64}\left(k, \mathsf{/.f64}\left(\mathsf{*.f64}\left(k, \mathsf{*.f64}\left(k, k\right)\right), \ell\right)\right)\right)\right)\right) \]

7. Applied egg-rr 51.8%

   \[\leadsto \frac{2}{\color{blue}{\left(0 - \frac{t}{\ell}\right) \cdot \left(0 - k \cdot \frac{k \cdot \left(k \cdot k\right)}{\ell}\right)}} \]
8. Step-by-step derivation

   1. associate-/r* N/A

      \[\leadsto \frac{\frac{2}{0 - \frac{t}{\ell}}}{\color{blue}{0 - k \cdot \frac{k \cdot \left(k \cdot k\right)}{\ell}}} \]

   2. sub0-neg N/A

      \[\leadsto \frac{\frac{2}{0 - \frac{t}{\ell}}}{\mathsf{neg}\left(k \cdot \frac{k \cdot \left(k \cdot k\right)}{\ell}\right)} \]

   3. distribute-frac-neg2 N/A

      \[\leadsto \mathsf{neg}\left(\frac{\frac{2}{0 - \frac{t}{\ell}}}{k \cdot \frac{k \cdot \left(k \cdot k\right)}{\ell}}\right) \]

   4. distribute-frac-neg N/A

      \[\leadsto \frac{\mathsf{neg}\left(\frac{2}{0 - \frac{t}{\ell}}\right)}{\color{blue}{k \cdot \frac{k \cdot \left(k \cdot k\right)}{\ell}}} \]

   5. distribute-frac-neg2 N/A

      \[\leadsto \frac{\frac{2}{\mathsf{neg}\left(\left(0 - \frac{t}{\ell}\right)\right)}}{\color{blue}{k} \cdot \frac{k \cdot \left(k \cdot k\right)}{\ell}} \]

   6. sub0-neg N/A

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

   7. remove-double-neg N/A

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

   8. un-div-inv N/A

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

   9. clear-num N/A

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

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

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

   11. clear-num N/A

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

   12. un-div-inv N/A

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

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

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

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

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

   15. associate-/l* N/A

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

   16. associate-*r* N/A

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

   17. clear-num N/A

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

   18. un-div-inv N/A

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

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

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

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

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

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

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

9. Applied egg-rr 69.4%

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

    1. associate-/r/ N/A

       \[\leadsto \frac{\frac{2}{t} \cdot \ell}{\frac{\color{blue}{k \cdot k}}{\frac{\ell}{k \cdot k}}} \]

    2. associate-/r/ N/A

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

    3. times-frac N/A

       \[\leadsto \frac{\frac{2}{t}}{\frac{k \cdot k}{\ell}} \cdot \color{blue}{\frac{\ell}{k \cdot k}} \]

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

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

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

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

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

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

    7. associate-*l/ N/A

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

    8. associate-/r/ N/A

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

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

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

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

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

    11. associate-/r* N/A

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

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

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

    13. /-lowering-/.f64 75.5%

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

11. Applied egg-rr 75.5%

    \[\leadsto \color{blue}{\frac{\frac{2}{t}}{\frac{k}{\frac{\ell}{k}}} \cdot \frac{\frac{\ell}{k}}{k}} \]
12. Add Preprocessing

Alternative 7: 71.9% accurate, 28.1× speedup

Math

\[\begin{array}{l} k_m = \left|k\right| \\ \ell \cdot \left(\frac{\ell}{k\_m} \cdot \frac{2}{k\_m \cdot \left(k\_m \cdot \left(k\_m \cdot t\right)\right)}\right) \end{array} \]

FPCore

k_m = (fabs.f64 k)
(FPCore (t l k_m)
 :precision binary64
 (* l (* (/ l k_m) (/ 2.0 (* k_m (* k_m (* k_m t)))))))

C

k_m = fabs(k);
double code(double t, double l, double k_m) {
	return l * ((l / k_m) * (2.0 / (k_m * (k_m * (k_m * t)))));
}

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 / k_m) * (2.0d0 / (k_m * (k_m * (k_m * t)))))
end function

Java

k_m = Math.abs(k);
public static double code(double t, double l, double k_m) {
	return l * ((l / k_m) * (2.0 / (k_m * (k_m * (k_m * t)))));
}

Python

k_m = math.fabs(k)
def code(t, l, k_m):
	return l * ((l / k_m) * (2.0 / (k_m * (k_m * (k_m * t)))))

Julia

k_m = abs(k)
function code(t, l, k_m)
	return Float64(l * Float64(Float64(l / k_m) * Float64(2.0 / Float64(k_m * Float64(k_m * Float64(k_m * t))))))
end

MATLAB

k_m = abs(k);
function tmp = code(t, l, k_m)
	tmp = l * ((l / k_m) * (2.0 / (k_m * (k_m * (k_m * t)))));
end

Wolfram

k_m = N[Abs[k], $MachinePrecision]
code[t_, l_, k$95$m_] := N[(l * N[(N[(l / k$95$m), $MachinePrecision] * N[(2.0 / N[(k$95$m * N[(k$95$m * N[(k$95$m * t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 36.6%

   \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) - 1\right)} \]
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. /-lowering-/.f64 N/A

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

   2. *-commutative N/A

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

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

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

   4. metadata-eval N/A

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

   5. pow-sqr N/A

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

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

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

   7. unpow2 N/A

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

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

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

   9. unpow2 N/A

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

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

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

   11. unpow2 N/A

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

   12. *-lowering-*.f64 58.8%

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

5. Simplified 58.8%

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

   1. associate-/r/ N/A

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

   2. associate-*r* N/A

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

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

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

   4. *-commutative N/A

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

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

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

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

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

   7. associate-*l* N/A

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

   8. *-commutative N/A

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

   9. associate-*r* N/A

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

   10. *-commutative N/A

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

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

       \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\ell, \mathsf{/.f64}\left(2, \mathsf{*.f64}\left(k, \left(t \cdot \left(k \cdot \left(k \cdot k\right)\right)\right)\right)\right)\right), \ell\right) \]

   12. *-lowering-*.f64 N/A

       \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\ell, \mathsf{/.f64}\left(2, \mathsf{*.f64}\left(k, \mathsf{*.f64}\left(t, \left(k \cdot \left(k \cdot k\right)\right)\right)\right)\right)\right), \ell\right) \]

   13. *-lowering-*.f64 N/A

       \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\ell, \mathsf{/.f64}\left(2, \mathsf{*.f64}\left(k, \mathsf{*.f64}\left(t, \mathsf{*.f64}\left(k, \left(k \cdot k\right)\right)\right)\right)\right)\right), \ell\right) \]

   14. *-lowering-*.f64 67.5%

       \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\ell, \mathsf{/.f64}\left(2, \mathsf{*.f64}\left(k, \mathsf{*.f64}\left(t, \mathsf{*.f64}\left(k, \mathsf{*.f64}\left(k, k\right)\right)\right)\right)\right)\right), \ell\right) \]

7. Applied egg-rr 67.5%

   \[\leadsto \color{blue}{\left(\ell \cdot \frac{2}{k \cdot \left(t \cdot \left(k \cdot \left(k \cdot k\right)\right)\right)}\right) \cdot \ell} \]
8. Step-by-step derivation

   1. associate-*r/ N/A

      \[\leadsto \mathsf{*.f64}\left(\left(\frac{\ell \cdot 2}{k \cdot \left(t \cdot \left(k \cdot \left(k \cdot k\right)\right)\right)}\right), \ell\right) \]

   2. times-frac N/A

      \[\leadsto \mathsf{*.f64}\left(\left(\frac{\ell}{k} \cdot \frac{2}{t \cdot \left(k \cdot \left(k \cdot k\right)\right)}\right), \ell\right) \]

   3. *-lowering-*.f64 N/A

      \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\left(\frac{\ell}{k}\right), \left(\frac{2}{t \cdot \left(k \cdot \left(k \cdot k\right)\right)}\right)\right), \ell\right) \]

   4. /-lowering-/.f64 N/A

      \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(\ell, k\right), \left(\frac{2}{t \cdot \left(k \cdot \left(k \cdot k\right)\right)}\right)\right), \ell\right) \]

   5. /-lowering-/.f64 N/A

      \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(\ell, k\right), \mathsf{/.f64}\left(2, \left(t \cdot \left(k \cdot \left(k \cdot k\right)\right)\right)\right)\right), \ell\right) \]

   6. associate-*r* N/A

      \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(\ell, k\right), \mathsf{/.f64}\left(2, \left(\left(t \cdot k\right) \cdot \left(k \cdot k\right)\right)\right)\right), \ell\right) \]

   7. associate-*r* N/A

      \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(\ell, k\right), \mathsf{/.f64}\left(2, \left(\left(\left(t \cdot k\right) \cdot k\right) \cdot k\right)\right)\right), \ell\right) \]

   8. *-commutative N/A

      \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(\ell, k\right), \mathsf{/.f64}\left(2, \left(\left(\left(k \cdot t\right) \cdot k\right) \cdot k\right)\right)\right), \ell\right) \]

   9. *-lowering-*.f64 N/A

      \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(\ell, k\right), \mathsf{/.f64}\left(2, \mathsf{*.f64}\left(\left(\left(k \cdot t\right) \cdot k\right), k\right)\right)\right), \ell\right) \]

   10. *-lowering-*.f64 N/A

       \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(\ell, k\right), \mathsf{/.f64}\left(2, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\left(k \cdot t\right), k\right), k\right)\right)\right), \ell\right) \]

   11. *-lowering-*.f64 69.6%

       \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(\ell, k\right), \mathsf{/.f64}\left(2, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(k, t\right), k\right), k\right)\right)\right), \ell\right) \]

9. Applied egg-rr 69.6%

   \[\leadsto \color{blue}{\left(\frac{\ell}{k} \cdot \frac{2}{\left(\left(k \cdot t\right) \cdot k\right) \cdot k}\right)} \cdot \ell \]

10. Final simplification 69.6%

    \[\leadsto \ell \cdot \left(\frac{\ell}{k} \cdot \frac{2}{k \cdot \left(k \cdot \left(k \cdot t\right)\right)}\right) \]
11. Add Preprocessing

Alternative 8: 71.1% accurate, 28.1× speedup

Math

\[\begin{array}{l} k_m = \left|k\right| \\ \ell \cdot \left(\frac{2}{k\_m \cdot t} \cdot \frac{\ell}{k\_m \cdot \left(k\_m \cdot k\_m\right)}\right) \end{array} \]

FPCore

k_m = (fabs.f64 k)
(FPCore (t l k_m)
 :precision binary64
 (* l (* (/ 2.0 (* k_m t)) (/ l (* k_m (* k_m k_m))))))

C

k_m = fabs(k);
double code(double t, double l, double k_m) {
	return l * ((2.0 / (k_m * t)) * (l / (k_m * (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 * ((2.0d0 / (k_m * t)) * (l / (k_m * (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 * ((2.0 / (k_m * t)) * (l / (k_m * (k_m * k_m))));
}

Python

k_m = math.fabs(k)
def code(t, l, k_m):
	return l * ((2.0 / (k_m * t)) * (l / (k_m * (k_m * k_m))))

Julia

k_m = abs(k)
function code(t, l, k_m)
	return Float64(l * Float64(Float64(2.0 / Float64(k_m * t)) * Float64(l / Float64(k_m * Float64(k_m * k_m)))))
end

MATLAB

k_m = abs(k);
function tmp = code(t, l, k_m)
	tmp = l * ((2.0 / (k_m * t)) * (l / (k_m * (k_m * k_m))));
end

Wolfram

k_m = N[Abs[k], $MachinePrecision]
code[t_, l_, k$95$m_] := N[(l * N[(N[(2.0 / N[(k$95$m * t), $MachinePrecision]), $MachinePrecision] * N[(l / N[(k$95$m * N[(k$95$m * k$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 36.6%

   \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) - 1\right)} \]
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. /-lowering-/.f64 N/A

      \[\leadsto \mathsf{/.f64}\left(2, \mathsf{/.f64}\left(\left({k}^{4} \cdot t\right), \color{blue}{\left({\ell}^{2}\right)}\right)\right) \]

   2. *-commutative N/A

      \[\leadsto \mathsf{/.f64}\left(2, \mathsf{/.f64}\left(\left(t \cdot {k}^{4}\right), \left({\color{blue}{\ell}}^{2}\right)\right)\right) \]

   3. *-lowering-*.f64 N/A

      \[\leadsto \mathsf{/.f64}\left(2, \mathsf{/.f64}\left(\mathsf{*.f64}\left(t, \left({k}^{4}\right)\right), \left({\color{blue}{\ell}}^{2}\right)\right)\right) \]

   4. metadata-eval N/A

      \[\leadsto \mathsf{/.f64}\left(2, \mathsf{/.f64}\left(\mathsf{*.f64}\left(t, \left({k}^{\left(2 \cdot 2\right)}\right)\right), \left({\ell}^{2}\right)\right)\right) \]

   5. pow-sqr N/A

      \[\leadsto \mathsf{/.f64}\left(2, \mathsf{/.f64}\left(\mathsf{*.f64}\left(t, \left({k}^{2} \cdot {k}^{2}\right)\right), \left({\ell}^{2}\right)\right)\right) \]

   6. *-lowering-*.f64 N/A

      \[\leadsto \mathsf{/.f64}\left(2, \mathsf{/.f64}\left(\mathsf{*.f64}\left(t, \mathsf{*.f64}\left(\left({k}^{2}\right), \left({k}^{2}\right)\right)\right), \left({\ell}^{2}\right)\right)\right) \]

   7. unpow2 N/A

      \[\leadsto \mathsf{/.f64}\left(2, \mathsf{/.f64}\left(\mathsf{*.f64}\left(t, \mathsf{*.f64}\left(\left(k \cdot k\right), \left({k}^{2}\right)\right)\right), \left({\ell}^{2}\right)\right)\right) \]

   8. *-lowering-*.f64 N/A

      \[\leadsto \mathsf{/.f64}\left(2, \mathsf{/.f64}\left(\mathsf{*.f64}\left(t, \mathsf{*.f64}\left(\mathsf{*.f64}\left(k, k\right), \left({k}^{2}\right)\right)\right), \left({\ell}^{2}\right)\right)\right) \]

   9. unpow2 N/A

      \[\leadsto \mathsf{/.f64}\left(2, \mathsf{/.f64}\left(\mathsf{*.f64}\left(t, \mathsf{*.f64}\left(\mathsf{*.f64}\left(k, k\right), \left(k \cdot k\right)\right)\right), \left({\ell}^{2}\right)\right)\right) \]

   10. *-lowering-*.f64 N/A

       \[\leadsto \mathsf{/.f64}\left(2, \mathsf{/.f64}\left(\mathsf{*.f64}\left(t, \mathsf{*.f64}\left(\mathsf{*.f64}\left(k, k\right), \mathsf{*.f64}\left(k, k\right)\right)\right), \left({\ell}^{2}\right)\right)\right) \]

   11. unpow2 N/A

       \[\leadsto \mathsf{/.f64}\left(2, \mathsf{/.f64}\left(\mathsf{*.f64}\left(t, \mathsf{*.f64}\left(\mathsf{*.f64}\left(k, k\right), \mathsf{*.f64}\left(k, k\right)\right)\right), \left(\ell \cdot \color{blue}{\ell}\right)\right)\right) \]

   12. *-lowering-*.f64 58.8%

       \[\leadsto \mathsf{/.f64}\left(2, \mathsf{/.f64}\left(\mathsf{*.f64}\left(t, \mathsf{*.f64}\left(\mathsf{*.f64}\left(k, k\right), \mathsf{*.f64}\left(k, k\right)\right)\right), \mathsf{*.f64}\left(\ell, \color{blue}{\ell}\right)\right)\right) \]

5. Simplified 58.8%

   \[\leadsto \frac{2}{\color{blue}{\frac{t \cdot \left(\left(k \cdot k\right) \cdot \left(k \cdot k\right)\right)}{\ell \cdot \ell}}} \]
6. Step-by-step derivation

   1. associate-/r/ N/A

      \[\leadsto \frac{2}{t \cdot \left(\left(k \cdot k\right) \cdot \left(k \cdot k\right)\right)} \cdot \color{blue}{\left(\ell \cdot \ell\right)} \]

   2. associate-*r* N/A

      \[\leadsto \left(\frac{2}{t \cdot \left(\left(k \cdot k\right) \cdot \left(k \cdot k\right)\right)} \cdot \ell\right) \cdot \color{blue}{\ell} \]

   3. *-lowering-*.f64 N/A

      \[\leadsto \mathsf{*.f64}\left(\left(\frac{2}{t \cdot \left(\left(k \cdot k\right) \cdot \left(k \cdot k\right)\right)} \cdot \ell\right), \color{blue}{\ell}\right) \]

   4. *-commutative N/A

      \[\leadsto \mathsf{*.f64}\left(\left(\ell \cdot \frac{2}{t \cdot \left(\left(k \cdot k\right) \cdot \left(k \cdot k\right)\right)}\right), \ell\right) \]

   5. *-lowering-*.f64 N/A

      \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\ell, \left(\frac{2}{t \cdot \left(\left(k \cdot k\right) \cdot \left(k \cdot k\right)\right)}\right)\right), \ell\right) \]

   6. /-lowering-/.f64 N/A

      \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\ell, \mathsf{/.f64}\left(2, \left(t \cdot \left(\left(k \cdot k\right) \cdot \left(k \cdot k\right)\right)\right)\right)\right), \ell\right) \]

   7. associate-*l* N/A

      \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\ell, \mathsf{/.f64}\left(2, \left(t \cdot \left(k \cdot \left(k \cdot \left(k \cdot k\right)\right)\right)\right)\right)\right), \ell\right) \]

   8. *-commutative N/A

      \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\ell, \mathsf{/.f64}\left(2, \left(t \cdot \left(\left(k \cdot \left(k \cdot k\right)\right) \cdot k\right)\right)\right)\right), \ell\right) \]

   9. associate-*r* N/A

      \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\ell, \mathsf{/.f64}\left(2, \left(\left(t \cdot \left(k \cdot \left(k \cdot k\right)\right)\right) \cdot k\right)\right)\right), \ell\right) \]

   10. *-commutative N/A

       \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\ell, \mathsf{/.f64}\left(2, \left(k \cdot \left(t \cdot \left(k \cdot \left(k \cdot k\right)\right)\right)\right)\right)\right), \ell\right) \]

   11. *-lowering-*.f64 N/A

       \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\ell, \mathsf{/.f64}\left(2, \mathsf{*.f64}\left(k, \left(t \cdot \left(k \cdot \left(k \cdot k\right)\right)\right)\right)\right)\right), \ell\right) \]

   12. *-lowering-*.f64 N/A

       \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\ell, \mathsf{/.f64}\left(2, \mathsf{*.f64}\left(k, \mathsf{*.f64}\left(t, \left(k \cdot \left(k \cdot k\right)\right)\right)\right)\right)\right), \ell\right) \]

   13. *-lowering-*.f64 N/A

       \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\ell, \mathsf{/.f64}\left(2, \mathsf{*.f64}\left(k, \mathsf{*.f64}\left(t, \mathsf{*.f64}\left(k, \left(k \cdot k\right)\right)\right)\right)\right)\right), \ell\right) \]

   14. *-lowering-*.f64 67.5%

       \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\ell, \mathsf{/.f64}\left(2, \mathsf{*.f64}\left(k, \mathsf{*.f64}\left(t, \mathsf{*.f64}\left(k, \mathsf{*.f64}\left(k, k\right)\right)\right)\right)\right)\right), \ell\right) \]

7. Applied egg-rr 67.5%

   \[\leadsto \color{blue}{\left(\ell \cdot \frac{2}{k \cdot \left(t \cdot \left(k \cdot \left(k \cdot k\right)\right)\right)}\right) \cdot \ell} \]
8. Step-by-step derivation

   1. associate-*r/ N/A

      \[\leadsto \mathsf{*.f64}\left(\left(\frac{\ell \cdot 2}{k \cdot \left(t \cdot \left(k \cdot \left(k \cdot k\right)\right)\right)}\right), \ell\right) \]

   2. *-commutative N/A

      \[\leadsto \mathsf{*.f64}\left(\left(\frac{2 \cdot \ell}{k \cdot \left(t \cdot \left(k \cdot \left(k \cdot k\right)\right)\right)}\right), \ell\right) \]

   3. associate-*r* N/A

      \[\leadsto \mathsf{*.f64}\left(\left(\frac{2 \cdot \ell}{\left(k \cdot t\right) \cdot \left(k \cdot \left(k \cdot k\right)\right)}\right), \ell\right) \]

   4. times-frac N/A

      \[\leadsto \mathsf{*.f64}\left(\left(\frac{2}{k \cdot t} \cdot \frac{\ell}{k \cdot \left(k \cdot k\right)}\right), \ell\right) \]

   5. *-lowering-*.f64 N/A

      \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\left(\frac{2}{k \cdot t}\right), \left(\frac{\ell}{k \cdot \left(k \cdot k\right)}\right)\right), \ell\right) \]

   6. /-lowering-/.f64 N/A

      \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(2, \left(k \cdot t\right)\right), \left(\frac{\ell}{k \cdot \left(k \cdot k\right)}\right)\right), \ell\right) \]

   7. *-lowering-*.f64 N/A

      \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(2, \mathsf{*.f64}\left(k, t\right)\right), \left(\frac{\ell}{k \cdot \left(k \cdot k\right)}\right)\right), \ell\right) \]

   8. /-lowering-/.f64 N/A

      \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(2, \mathsf{*.f64}\left(k, t\right)\right), \mathsf{/.f64}\left(\ell, \left(k \cdot \left(k \cdot k\right)\right)\right)\right), \ell\right) \]

   9. *-lowering-*.f64 N/A

      \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(2, \mathsf{*.f64}\left(k, t\right)\right), \mathsf{/.f64}\left(\ell, \mathsf{*.f64}\left(k, \left(k \cdot k\right)\right)\right)\right), \ell\right) \]

   10. *-lowering-*.f64 69.3%

       \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(2, \mathsf{*.f64}\left(k, t\right)\right), \mathsf{/.f64}\left(\ell, \mathsf{*.f64}\left(k, \mathsf{*.f64}\left(k, k\right)\right)\right)\right), \ell\right) \]

9. Applied egg-rr 69.3%

   \[\leadsto \color{blue}{\left(\frac{2}{k \cdot t} \cdot \frac{\ell}{k \cdot \left(k \cdot k\right)}\right)} \cdot \ell \]

10. Final simplification 69.3%

    \[\leadsto \ell \cdot \left(\frac{2}{k \cdot t} \cdot \frac{\ell}{k \cdot \left(k \cdot k\right)}\right) \]
11. Add Preprocessing

Alternative 9: 69.8% accurate, 28.1× speedup

Math

\[\begin{array}{l} k_m = \left|k\right| \\ \ell \cdot \left(\ell \cdot \frac{2}{k\_m \cdot \left(t \cdot \left(k\_m \cdot \left(k\_m \cdot k\_m\right)\right)\right)}\right) \end{array} \]

FPCore

k_m = (fabs.f64 k)
(FPCore (t l k_m)
 :precision binary64
 (* l (* l (/ 2.0 (* k_m (* t (* k_m (* k_m k_m))))))))

C

k_m = fabs(k);
double code(double t, double l, double k_m) {
	return l * (l * (2.0 / (k_m * (t * (k_m * (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 * (2.0d0 / (k_m * (t * (k_m * (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 * (2.0 / (k_m * (t * (k_m * (k_m * k_m))))));
}

Python

k_m = math.fabs(k)
def code(t, l, k_m):
	return l * (l * (2.0 / (k_m * (t * (k_m * (k_m * k_m))))))

Julia

k_m = abs(k)
function code(t, l, k_m)
	return Float64(l * Float64(l * Float64(2.0 / Float64(k_m * Float64(t * Float64(k_m * Float64(k_m * k_m)))))))
end

MATLAB

k_m = abs(k);
function tmp = code(t, l, k_m)
	tmp = l * (l * (2.0 / (k_m * (t * (k_m * (k_m * k_m))))));
end

Wolfram

k_m = N[Abs[k], $MachinePrecision]
code[t_, l_, k$95$m_] := N[(l * N[(l * N[(2.0 / N[(k$95$m * N[(t * N[(k$95$m * N[(k$95$m * k$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 36.6%

   \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) - 1\right)} \]
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. /-lowering-/.f64 N/A

      \[\leadsto \mathsf{/.f64}\left(2, \mathsf{/.f64}\left(\left({k}^{4} \cdot t\right), \color{blue}{\left({\ell}^{2}\right)}\right)\right) \]

   2. *-commutative N/A

      \[\leadsto \mathsf{/.f64}\left(2, \mathsf{/.f64}\left(\left(t \cdot {k}^{4}\right), \left({\color{blue}{\ell}}^{2}\right)\right)\right) \]

   3. *-lowering-*.f64 N/A

      \[\leadsto \mathsf{/.f64}\left(2, \mathsf{/.f64}\left(\mathsf{*.f64}\left(t, \left({k}^{4}\right)\right), \left({\color{blue}{\ell}}^{2}\right)\right)\right) \]

   4. metadata-eval N/A

      \[\leadsto \mathsf{/.f64}\left(2, \mathsf{/.f64}\left(\mathsf{*.f64}\left(t, \left({k}^{\left(2 \cdot 2\right)}\right)\right), \left({\ell}^{2}\right)\right)\right) \]

   5. pow-sqr N/A

      \[\leadsto \mathsf{/.f64}\left(2, \mathsf{/.f64}\left(\mathsf{*.f64}\left(t, \left({k}^{2} \cdot {k}^{2}\right)\right), \left({\ell}^{2}\right)\right)\right) \]

   6. *-lowering-*.f64 N/A

      \[\leadsto \mathsf{/.f64}\left(2, \mathsf{/.f64}\left(\mathsf{*.f64}\left(t, \mathsf{*.f64}\left(\left({k}^{2}\right), \left({k}^{2}\right)\right)\right), \left({\ell}^{2}\right)\right)\right) \]

   7. unpow2 N/A

      \[\leadsto \mathsf{/.f64}\left(2, \mathsf{/.f64}\left(\mathsf{*.f64}\left(t, \mathsf{*.f64}\left(\left(k \cdot k\right), \left({k}^{2}\right)\right)\right), \left({\ell}^{2}\right)\right)\right) \]

   8. *-lowering-*.f64 N/A

      \[\leadsto \mathsf{/.f64}\left(2, \mathsf{/.f64}\left(\mathsf{*.f64}\left(t, \mathsf{*.f64}\left(\mathsf{*.f64}\left(k, k\right), \left({k}^{2}\right)\right)\right), \left({\ell}^{2}\right)\right)\right) \]

   9. unpow2 N/A

      \[\leadsto \mathsf{/.f64}\left(2, \mathsf{/.f64}\left(\mathsf{*.f64}\left(t, \mathsf{*.f64}\left(\mathsf{*.f64}\left(k, k\right), \left(k \cdot k\right)\right)\right), \left({\ell}^{2}\right)\right)\right) \]

   10. *-lowering-*.f64 N/A

       \[\leadsto \mathsf{/.f64}\left(2, \mathsf{/.f64}\left(\mathsf{*.f64}\left(t, \mathsf{*.f64}\left(\mathsf{*.f64}\left(k, k\right), \mathsf{*.f64}\left(k, k\right)\right)\right), \left({\ell}^{2}\right)\right)\right) \]

   11. unpow2 N/A

       \[\leadsto \mathsf{/.f64}\left(2, \mathsf{/.f64}\left(\mathsf{*.f64}\left(t, \mathsf{*.f64}\left(\mathsf{*.f64}\left(k, k\right), \mathsf{*.f64}\left(k, k\right)\right)\right), \left(\ell \cdot \color{blue}{\ell}\right)\right)\right) \]

   12. *-lowering-*.f64 58.8%

       \[\leadsto \mathsf{/.f64}\left(2, \mathsf{/.f64}\left(\mathsf{*.f64}\left(t, \mathsf{*.f64}\left(\mathsf{*.f64}\left(k, k\right), \mathsf{*.f64}\left(k, k\right)\right)\right), \mathsf{*.f64}\left(\ell, \color{blue}{\ell}\right)\right)\right) \]

5. Simplified 58.8%

   \[\leadsto \frac{2}{\color{blue}{\frac{t \cdot \left(\left(k \cdot k\right) \cdot \left(k \cdot k\right)\right)}{\ell \cdot \ell}}} \]
6. Step-by-step derivation

   1. associate-/r/ N/A

      \[\leadsto \frac{2}{t \cdot \left(\left(k \cdot k\right) \cdot \left(k \cdot k\right)\right)} \cdot \color{blue}{\left(\ell \cdot \ell\right)} \]

   2. associate-*r* N/A

      \[\leadsto \left(\frac{2}{t \cdot \left(\left(k \cdot k\right) \cdot \left(k \cdot k\right)\right)} \cdot \ell\right) \cdot \color{blue}{\ell} \]

   3. *-lowering-*.f64 N/A

      \[\leadsto \mathsf{*.f64}\left(\left(\frac{2}{t \cdot \left(\left(k \cdot k\right) \cdot \left(k \cdot k\right)\right)} \cdot \ell\right), \color{blue}{\ell}\right) \]

   4. *-commutative N/A

      \[\leadsto \mathsf{*.f64}\left(\left(\ell \cdot \frac{2}{t \cdot \left(\left(k \cdot k\right) \cdot \left(k \cdot k\right)\right)}\right), \ell\right) \]

   5. *-lowering-*.f64 N/A

      \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\ell, \left(\frac{2}{t \cdot \left(\left(k \cdot k\right) \cdot \left(k \cdot k\right)\right)}\right)\right), \ell\right) \]

   6. /-lowering-/.f64 N/A

      \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\ell, \mathsf{/.f64}\left(2, \left(t \cdot \left(\left(k \cdot k\right) \cdot \left(k \cdot k\right)\right)\right)\right)\right), \ell\right) \]

   7. associate-*l* N/A

      \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\ell, \mathsf{/.f64}\left(2, \left(t \cdot \left(k \cdot \left(k \cdot \left(k \cdot k\right)\right)\right)\right)\right)\right), \ell\right) \]

   8. *-commutative N/A

      \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\ell, \mathsf{/.f64}\left(2, \left(t \cdot \left(\left(k \cdot \left(k \cdot k\right)\right) \cdot k\right)\right)\right)\right), \ell\right) \]

   9. associate-*r* N/A

      \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\ell, \mathsf{/.f64}\left(2, \left(\left(t \cdot \left(k \cdot \left(k \cdot k\right)\right)\right) \cdot k\right)\right)\right), \ell\right) \]

   10. *-commutative N/A

       \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\ell, \mathsf{/.f64}\left(2, \left(k \cdot \left(t \cdot \left(k \cdot \left(k \cdot k\right)\right)\right)\right)\right)\right), \ell\right) \]

   11. *-lowering-*.f64 N/A

       \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\ell, \mathsf{/.f64}\left(2, \mathsf{*.f64}\left(k, \left(t \cdot \left(k \cdot \left(k \cdot k\right)\right)\right)\right)\right)\right), \ell\right) \]

   12. *-lowering-*.f64 N/A

       \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\ell, \mathsf{/.f64}\left(2, \mathsf{*.f64}\left(k, \mathsf{*.f64}\left(t, \left(k \cdot \left(k \cdot k\right)\right)\right)\right)\right)\right), \ell\right) \]

   13. *-lowering-*.f64 N/A

       \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\ell, \mathsf{/.f64}\left(2, \mathsf{*.f64}\left(k, \mathsf{*.f64}\left(t, \mathsf{*.f64}\left(k, \left(k \cdot k\right)\right)\right)\right)\right)\right), \ell\right) \]

   14. *-lowering-*.f64 67.5%

       \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\ell, \mathsf{/.f64}\left(2, \mathsf{*.f64}\left(k, \mathsf{*.f64}\left(t, \mathsf{*.f64}\left(k, \mathsf{*.f64}\left(k, k\right)\right)\right)\right)\right)\right), \ell\right) \]

7. Applied egg-rr 67.5%

   \[\leadsto \color{blue}{\left(\ell \cdot \frac{2}{k \cdot \left(t \cdot \left(k \cdot \left(k \cdot k\right)\right)\right)}\right) \cdot \ell} \]

8. Final simplification 67.5%

   \[\leadsto \ell \cdot \left(\ell \cdot \frac{2}{k \cdot \left(t \cdot \left(k \cdot \left(k \cdot k\right)\right)\right)}\right) \]
9. Add Preprocessing

Reproduce

herbie shell --seed 2024138 
(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))))