Toniolo and Linder, Equation (10-)

Percentage Accurate: 36.6% → 94.1%

Time: 17.3s

Alternatives: 7

Speedup: 28.1×

• 40.0% of points produce a very large (infinite) output. You may want to add a precondition.

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 36.6% 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: 94.1% accurate, 1.9× speedup

Math

\[\begin{array}{l} k_m = \left|k\right| \\ \begin{array}{l} t_1 := \frac{\ell}{\frac{k\_m}{2}}\\ \mathbf{if}\;k\_m \leq 8.5 \cdot 10^{-63}:\\ \;\;\;\;\frac{\frac{\frac{t\_1}{\frac{k\_m}{\ell}} \cdot \frac{1}{t}}{k\_m}}{k\_m}\\ \mathbf{else}:\\ \;\;\;\;t\_1 \cdot \frac{\ell}{\tan k\_m \cdot \left(t \cdot \left(k\_m \cdot \sin k\_m\right)\right)}\\ \end{array} \end{array} \]

FPCore

k_m = (fabs.f64 k)
(FPCore (t l k_m)
 :precision binary64
 (let* ((t_1 (/ l (/ k_m 2.0))))
   (if (<= k_m 8.5e-63)
     (/ (/ (* (/ t_1 (/ k_m l)) (/ 1.0 t)) k_m) k_m)
     (* t_1 (/ l (* (tan k_m) (* t (* k_m (sin k_m)))))))))

C

k_m = fabs(k);
double code(double t, double l, double k_m) {
	double t_1 = l / (k_m / 2.0);
	double tmp;
	if (k_m <= 8.5e-63) {
		tmp = (((t_1 / (k_m / l)) * (1.0 / t)) / k_m) / k_m;
	} else {
		tmp = t_1 * (l / (tan(k_m) * (t * (k_m * sin(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 / 2.0d0)
    if (k_m <= 8.5d-63) then
        tmp = (((t_1 / (k_m / l)) * (1.0d0 / t)) / k_m) / k_m
    else
        tmp = t_1 * (l / (tan(k_m) * (t * (k_m * sin(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 / 2.0);
	double tmp;
	if (k_m <= 8.5e-63) {
		tmp = (((t_1 / (k_m / l)) * (1.0 / t)) / k_m) / k_m;
	} else {
		tmp = t_1 * (l / (Math.tan(k_m) * (t * (k_m * Math.sin(k_m)))));
	}
	return tmp;
}

Python

k_m = math.fabs(k)
def code(t, l, k_m):
	t_1 = l / (k_m / 2.0)
	tmp = 0
	if k_m <= 8.5e-63:
		tmp = (((t_1 / (k_m / l)) * (1.0 / t)) / k_m) / k_m
	else:
		tmp = t_1 * (l / (math.tan(k_m) * (t * (k_m * math.sin(k_m)))))
	return tmp

Julia

k_m = abs(k)
function code(t, l, k_m)
	t_1 = Float64(l / Float64(k_m / 2.0))
	tmp = 0.0
	if (k_m <= 8.5e-63)
		tmp = Float64(Float64(Float64(Float64(t_1 / Float64(k_m / l)) * Float64(1.0 / t)) / k_m) / k_m);
	else
		tmp = Float64(t_1 * Float64(l / Float64(tan(k_m) * Float64(t * Float64(k_m * sin(k_m))))));
	end
	return tmp
end

MATLAB

k_m = abs(k);
function tmp_2 = code(t, l, k_m)
	t_1 = l / (k_m / 2.0);
	tmp = 0.0;
	if (k_m <= 8.5e-63)
		tmp = (((t_1 / (k_m / l)) * (1.0 / t)) / k_m) / k_m;
	else
		tmp = t_1 * (l / (tan(k_m) * (t * (k_m * sin(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[(l / N[(k$95$m / 2.0), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[k$95$m, 8.5e-63], N[(N[(N[(N[(t$95$1 / N[(k$95$m / l), $MachinePrecision]), $MachinePrecision] * N[(1.0 / t), $MachinePrecision]), $MachinePrecision] / k$95$m), $MachinePrecision] / k$95$m), $MachinePrecision], N[(t$95$1 * N[(l / N[(N[Tan[k$95$m], $MachinePrecision] * N[(t * N[(k$95$m * N[Sin[k$95$m], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if k < 8.49999999999999969e-63

   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 k around 0

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

      1. associate-/l* N/A

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

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

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

      3. metadata-eval N/A

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

      4. pow-sqr N/A

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

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

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

      6. unpow2 N/A

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

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

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

      8. unpow2 N/A

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

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

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

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

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

      11. unpow2 N/A

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

      12. *-lowering-*.f64 61.3%

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

   5. Simplified 61.3%

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

      1. clear-num N/A

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

      2. associate-/r/ N/A

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

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

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

      4. *-commutative N/A

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

      5. associate-/r* N/A

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

      6. clear-num N/A

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

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

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

      8. associate-*r/ N/A

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

      9. clear-num N/A

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

      10. un-div-inv N/A

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

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

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

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

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

      13. associate-*l* N/A

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

      14. cube-unmult N/A

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

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

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

      16. cube-unmult N/A

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

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

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

      18. *-lowering-*.f64 64.7%

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

   7. Applied egg-rr 64.7%

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

      1. *-commutative N/A

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

      2. associate-*r/ N/A

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

      3. associate-*r* N/A

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

      4. associate-/r* N/A

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

      5. clear-num N/A

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

      6. associate-/l/ N/A

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

      7. div-inv N/A

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

      8. unpow-1 N/A

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

      9. associate-/r* N/A

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

      10. unpow-1 N/A

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

      11. div-inv N/A

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

      12. unpow-1 N/A

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

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

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

   9. Applied egg-rr 72.3%

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

       1. associate-*r/ N/A

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

       2. associate-/r/ N/A

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

       3. associate-/r* N/A

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

       4. div-inv N/A

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

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

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

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

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

       7. *-commutative N/A

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

       8. clear-num N/A

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

       9. div-inv N/A

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

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

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

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

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

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

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

       13. /-lowering-/.f64 78.1%

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

   11. Applied egg-rr 78.1%

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

   if 8.49999999999999969e-63 < k

   1. Initial program 34.9%

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

   3. Taylor expanded in t around 0

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

      1. associate-*r/ N/A

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

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

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

      3. associate-*r* N/A

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

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

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

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

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

      6. unpow2 N/A

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

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

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

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

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

      9. unpow2 N/A

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

      10. associate-*l* N/A

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

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

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

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

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

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

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

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

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

      15. sin-lowering-sin.f64 79.5%

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

   5. Simplified 79.5%

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

      1. associate-*r* N/A

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

      2. unpow2 N/A

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

      3. associate-*r* N/A

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

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

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

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

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

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

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

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

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

      8. sin-lowering-sin.f64 79.6%

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

   7. Applied egg-rr 79.6%

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

      1. times-frac N/A

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

      2. *-commutative N/A

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

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

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

      4. *-commutative N/A

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

      5. associate-/r* N/A

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

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

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

      7. clear-num N/A

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

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

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

      9. quot-tan N/A

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

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

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

      11. associate-*l* N/A

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

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

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

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

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

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

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

      15. *-commutative N/A

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

      16. associate-/l* N/A

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

      17. clear-num N/A

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

      18. un-div-inv N/A

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

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

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

   9. Applied egg-rr 85.6%

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

       1. associate-/l* N/A

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

       2. associate-*r* N/A

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

       3. *-commutative N/A

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

       4. div-inv N/A

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

       5. clear-num N/A

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

       6. *-commutative N/A

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

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

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

   11. Applied egg-rr 95.5%

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

4. Final simplification 83.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;k \leq 8.5 \cdot 10^{-63}:\\ \;\;\;\;\frac{\frac{\frac{\frac{\ell}{\frac{k}{2}}}{\frac{k}{\ell}} \cdot \frac{1}{t}}{k}}{k}\\ \mathbf{else}:\\ \;\;\;\;\frac{\ell}{\frac{k}{2}} \cdot \frac{\ell}{\tan k \cdot \left(t \cdot \left(k \cdot \sin k\right)\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 93.1% accurate, 1.9× speedup

Math

\[\begin{array}{l} k_m = \left|k\right| \\ \begin{array}{l} \mathbf{if}\;k\_m \leq 3 \cdot 10^{-61}:\\ \;\;\;\;\frac{\frac{\frac{\frac{\ell}{\frac{k\_m}{2}}}{\frac{k\_m}{\ell}} \cdot \frac{1}{t}}{k\_m}}{k\_m}\\ \mathbf{else}:\\ \;\;\;\;\left(\ell \cdot 2\right) \cdot \frac{\frac{\ell}{\tan k\_m \cdot \left(t \cdot \left(k\_m \cdot \sin k\_m\right)\right)}}{k\_m}\\ \end{array} \end{array} \]

FPCore

k_m = (fabs.f64 k)
(FPCore (t l k_m)
 :precision binary64
 (if (<= k_m 3e-61)
   (/ (/ (* (/ (/ l (/ k_m 2.0)) (/ k_m l)) (/ 1.0 t)) k_m) k_m)
   (* (* l 2.0) (/ (/ l (* (tan k_m) (* t (* k_m (sin k_m))))) k_m))))

C

k_m = fabs(k);
double code(double t, double l, double k_m) {
	double tmp;
	if (k_m <= 3e-61) {
		tmp = ((((l / (k_m / 2.0)) / (k_m / l)) * (1.0 / t)) / k_m) / k_m;
	} else {
		tmp = (l * 2.0) * ((l / (tan(k_m) * (t * (k_m * sin(k_m))))) / k_m);
	}
	return tmp;
}

Fortran

k_m = abs(k)
real(8) function code(t, l, k_m)
    real(8), intent (in) :: t
    real(8), intent (in) :: l
    real(8), intent (in) :: k_m
    real(8) :: tmp
    if (k_m <= 3d-61) then
        tmp = ((((l / (k_m / 2.0d0)) / (k_m / l)) * (1.0d0 / t)) / k_m) / k_m
    else
        tmp = (l * 2.0d0) * ((l / (tan(k_m) * (t * (k_m * sin(k_m))))) / k_m)
    end if
    code = tmp
end function

Java

k_m = Math.abs(k);
public static double code(double t, double l, double k_m) {
	double tmp;
	if (k_m <= 3e-61) {
		tmp = ((((l / (k_m / 2.0)) / (k_m / l)) * (1.0 / t)) / k_m) / k_m;
	} else {
		tmp = (l * 2.0) * ((l / (Math.tan(k_m) * (t * (k_m * Math.sin(k_m))))) / k_m);
	}
	return tmp;
}

Python

k_m = math.fabs(k)
def code(t, l, k_m):
	tmp = 0
	if k_m <= 3e-61:
		tmp = ((((l / (k_m / 2.0)) / (k_m / l)) * (1.0 / t)) / k_m) / k_m
	else:
		tmp = (l * 2.0) * ((l / (math.tan(k_m) * (t * (k_m * math.sin(k_m))))) / k_m)
	return tmp

Julia

k_m = abs(k)
function code(t, l, k_m)
	tmp = 0.0
	if (k_m <= 3e-61)
		tmp = Float64(Float64(Float64(Float64(Float64(l / Float64(k_m / 2.0)) / Float64(k_m / l)) * Float64(1.0 / t)) / k_m) / k_m);
	else
		tmp = Float64(Float64(l * 2.0) * Float64(Float64(l / Float64(tan(k_m) * Float64(t * Float64(k_m * sin(k_m))))) / k_m));
	end
	return tmp
end

MATLAB

k_m = abs(k);
function tmp_2 = code(t, l, k_m)
	tmp = 0.0;
	if (k_m <= 3e-61)
		tmp = ((((l / (k_m / 2.0)) / (k_m / l)) * (1.0 / t)) / k_m) / k_m;
	else
		tmp = (l * 2.0) * ((l / (tan(k_m) * (t * (k_m * sin(k_m))))) / k_m);
	end
	tmp_2 = tmp;
end

Wolfram

k_m = N[Abs[k], $MachinePrecision]
code[t_, l_, k$95$m_] := If[LessEqual[k$95$m, 3e-61], N[(N[(N[(N[(N[(l / N[(k$95$m / 2.0), $MachinePrecision]), $MachinePrecision] / N[(k$95$m / l), $MachinePrecision]), $MachinePrecision] * N[(1.0 / t), $MachinePrecision]), $MachinePrecision] / k$95$m), $MachinePrecision] / k$95$m), $MachinePrecision], N[(N[(l * 2.0), $MachinePrecision] * N[(N[(l / N[(N[Tan[k$95$m], $MachinePrecision] * N[(t * N[(k$95$m * N[Sin[k$95$m], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / k$95$m), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if k < 3.00000000000000012e-61

   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 k around 0

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

      1. associate-/l* N/A

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

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

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

      3. metadata-eval N/A

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

      4. pow-sqr N/A

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

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

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

      6. unpow2 N/A

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

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

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

      8. unpow2 N/A

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

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

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

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

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

      11. unpow2 N/A

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

      12. *-lowering-*.f64 61.3%

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

   5. Simplified 61.3%

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

      1. clear-num N/A

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

      2. associate-/r/ N/A

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

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

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

      4. *-commutative N/A

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

      5. associate-/r* N/A

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

      6. clear-num N/A

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

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

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

      8. associate-*r/ N/A

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

      9. clear-num N/A

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

      10. un-div-inv N/A

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

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

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

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

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

      13. associate-*l* N/A

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

      14. cube-unmult N/A

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

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

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

      16. cube-unmult N/A

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

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

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

      18. *-lowering-*.f64 64.7%

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

   7. Applied egg-rr 64.7%

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

      1. *-commutative N/A

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

      2. associate-*r/ N/A

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

      3. associate-*r* N/A

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

      4. associate-/r* N/A

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

      5. clear-num N/A

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

      6. associate-/l/ N/A

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

      7. div-inv N/A

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

      8. unpow-1 N/A

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

      9. associate-/r* N/A

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

      10. unpow-1 N/A

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

      11. div-inv N/A

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

      12. unpow-1 N/A

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

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

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

   9. Applied egg-rr 72.3%

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

       1. associate-*r/ N/A

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

       2. associate-/r/ N/A

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

       3. associate-/r* N/A

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

       4. div-inv N/A

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

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

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

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

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

       7. *-commutative N/A

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

       8. clear-num N/A

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

       9. div-inv N/A

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

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

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

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

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

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

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

       13. /-lowering-/.f64 78.1%

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

   11. Applied egg-rr 78.1%

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

   if 3.00000000000000012e-61 < k

   1. Initial program 34.9%

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

   3. Taylor expanded in t around 0

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

      1. associate-*r/ N/A

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

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

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

      3. associate-*r* N/A

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

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

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

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

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

      6. unpow2 N/A

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

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

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

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

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

      9. unpow2 N/A

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

      10. associate-*l* N/A

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

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

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

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

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

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

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

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

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

      15. sin-lowering-sin.f64 79.5%

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

   5. Simplified 79.5%

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

      1. associate-*r* N/A

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

      2. unpow2 N/A

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

      3. associate-*r* N/A

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

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

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

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

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

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

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

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

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

      8. sin-lowering-sin.f64 79.6%

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

   7. Applied egg-rr 79.6%

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

      1. associate-/l* N/A

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

      2. associate-*r* N/A

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

      3. associate-*l* N/A

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

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

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

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

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

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

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

      7. *-commutative N/A

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

      8. associate-/r* N/A

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

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

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

   9. Applied egg-rr 88.3%

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

       1. associate-*r/ N/A

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

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

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

       3. associate-/l/ N/A

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

       4. un-div-inv N/A

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

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

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

       6. *-commutative N/A

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

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

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

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

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

       9. *-commutative N/A

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

       10. associate-*l* N/A

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

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

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

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

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

       13. sin-lowering-sin.f64 93.4%

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

   11. Applied egg-rr 93.4%

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

4. Final simplification 82.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;k \leq 3 \cdot 10^{-61}:\\ \;\;\;\;\frac{\frac{\frac{\frac{\ell}{\frac{k}{2}}}{\frac{k}{\ell}} \cdot \frac{1}{t}}{k}}{k}\\ \mathbf{else}:\\ \;\;\;\;\left(\ell \cdot 2\right) \cdot \frac{\frac{\ell}{\tan k \cdot \left(t \cdot \left(k \cdot \sin k\right)\right)}}{k}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 73.9% accurate, 21.0× speedup

Math

\[\begin{array}{l} k_m = \left|k\right| \\ \begin{array}{l} \mathbf{if}\;t \leq 2.9 \cdot 10^{-9}:\\ \;\;\;\;\frac{\frac{\frac{2}{k\_m} \cdot \frac{\ell}{\frac{k\_m}{\ell} \cdot t}}{k\_m}}{k\_m}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\frac{\left(k\_m \cdot k\_m\right) \cdot \frac{t \cdot \left(k\_m \cdot k\_m\right)}{\ell}}{\ell}}\\ \end{array} \end{array} \]

FPCore

k_m = (fabs.f64 k)
(FPCore (t l k_m)
 :precision binary64
 (if (<= t 2.9e-9)
   (/ (/ (* (/ 2.0 k_m) (/ l (* (/ k_m l) t))) k_m) k_m)
   (/ 2.0 (/ (* (* k_m k_m) (/ (* t (* k_m k_m)) l)) l))))

C

k_m = fabs(k);
double code(double t, double l, double k_m) {
	double tmp;
	if (t <= 2.9e-9) {
		tmp = (((2.0 / k_m) * (l / ((k_m / l) * t))) / k_m) / k_m;
	} else {
		tmp = 2.0 / (((k_m * k_m) * ((t * (k_m * k_m)) / l)) / l);
	}
	return tmp;
}

Fortran

k_m = abs(k)
real(8) function code(t, l, k_m)
    real(8), intent (in) :: t
    real(8), intent (in) :: l
    real(8), intent (in) :: k_m
    real(8) :: tmp
    if (t <= 2.9d-9) then
        tmp = (((2.0d0 / k_m) * (l / ((k_m / l) * t))) / k_m) / k_m
    else
        tmp = 2.0d0 / (((k_m * k_m) * ((t * (k_m * k_m)) / l)) / l)
    end if
    code = tmp
end function

Java

k_m = Math.abs(k);
public static double code(double t, double l, double k_m) {
	double tmp;
	if (t <= 2.9e-9) {
		tmp = (((2.0 / k_m) * (l / ((k_m / l) * t))) / k_m) / k_m;
	} else {
		tmp = 2.0 / (((k_m * k_m) * ((t * (k_m * k_m)) / l)) / l);
	}
	return tmp;
}

Python

k_m = math.fabs(k)
def code(t, l, k_m):
	tmp = 0
	if t <= 2.9e-9:
		tmp = (((2.0 / k_m) * (l / ((k_m / l) * t))) / k_m) / k_m
	else:
		tmp = 2.0 / (((k_m * k_m) * ((t * (k_m * k_m)) / l)) / l)
	return tmp

Julia

k_m = abs(k)
function code(t, l, k_m)
	tmp = 0.0
	if (t <= 2.9e-9)
		tmp = Float64(Float64(Float64(Float64(2.0 / k_m) * Float64(l / Float64(Float64(k_m / l) * t))) / k_m) / k_m);
	else
		tmp = Float64(2.0 / Float64(Float64(Float64(k_m * k_m) * Float64(Float64(t * Float64(k_m * k_m)) / l)) / l));
	end
	return tmp
end

MATLAB

k_m = abs(k);
function tmp_2 = code(t, l, k_m)
	tmp = 0.0;
	if (t <= 2.9e-9)
		tmp = (((2.0 / k_m) * (l / ((k_m / l) * t))) / k_m) / k_m;
	else
		tmp = 2.0 / (((k_m * k_m) * ((t * (k_m * k_m)) / l)) / l);
	end
	tmp_2 = tmp;
end

Wolfram

k_m = N[Abs[k], $MachinePrecision]
code[t_, l_, k$95$m_] := If[LessEqual[t, 2.9e-9], N[(N[(N[(N[(2.0 / k$95$m), $MachinePrecision] * N[(l / N[(N[(k$95$m / l), $MachinePrecision] * t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / k$95$m), $MachinePrecision] / k$95$m), $MachinePrecision], N[(2.0 / N[(N[(N[(k$95$m * k$95$m), $MachinePrecision] * N[(N[(t * N[(k$95$m * k$95$m), $MachinePrecision]), $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if t < 2.89999999999999991e-9

   1. Initial program 38.6%

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

   3. Taylor expanded in k around 0

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

      1. associate-/l* N/A

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

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

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

      3. metadata-eval N/A

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

      4. pow-sqr N/A

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

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

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

      6. unpow2 N/A

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

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

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

      8. unpow2 N/A

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

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

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

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

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

      11. unpow2 N/A

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

      12. *-lowering-*.f64 60.9%

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

   5. Simplified 60.9%

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

      1. clear-num N/A

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

      2. associate-/r/ N/A

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

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

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

      4. *-commutative N/A

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

      5. associate-/r* N/A

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

      6. clear-num N/A

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

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

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

      8. associate-*r/ N/A

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

      9. clear-num N/A

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

      10. un-div-inv N/A

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

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

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

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

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

      13. associate-*l* N/A

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

      14. cube-unmult N/A

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

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

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

      16. cube-unmult N/A

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

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

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

      18. *-lowering-*.f64 65.4%

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

   7. Applied egg-rr 65.4%

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

      1. *-commutative N/A

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

      2. associate-*r/ N/A

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

      3. associate-*r* N/A

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

      4. associate-/r* N/A

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

      5. clear-num N/A

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

      6. associate-/l/ N/A

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

      7. div-inv N/A

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

      8. unpow-1 N/A

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

      9. associate-/r* N/A

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

      10. unpow-1 N/A

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

      11. div-inv N/A

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

      12. unpow-1 N/A

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

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

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

   9. Applied egg-rr 70.6%

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

       1. associate-/r/ N/A

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

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

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

       3. /-lowering-/.f64 71.7%

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

   11. Applied egg-rr 71.7%

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

   if 2.89999999999999991e-9 < t

   1. Initial program 32.6%

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

   3. Taylor expanded in k around 0

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

      1. associate-/l* N/A

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

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

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

      3. metadata-eval N/A

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

      4. pow-sqr N/A

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

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

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

      6. unpow2 N/A

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

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

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

      8. unpow2 N/A

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

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

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

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

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

      11. unpow2 N/A

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

      12. *-lowering-*.f64 62.4%

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

   5. Simplified 62.4%

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

      1. associate-*l* N/A

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

      2. associate-/l* N/A

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

      3. associate-/r* N/A

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

      4. associate-*r/ N/A

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

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

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

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

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

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

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

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

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

      9. *-commutative N/A

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

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

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

      11. *-lowering-*.f64 80.9%

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

   7. Applied egg-rr 80.9%

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

Alternative 4: 72.8% accurate, 21.0× speedup

Math

\[\begin{array}{l} k_m = \left|k\right| \\ \begin{array}{l} \mathbf{if}\;t \leq 4.5 \cdot 10^{-9}:\\ \;\;\;\;\frac{2}{\left(k\_m \cdot k\_m\right) \cdot \frac{k\_m}{\frac{\ell}{\frac{k\_m}{\frac{\ell}{t}}}}}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\frac{\left(k\_m \cdot k\_m\right) \cdot \frac{t \cdot \left(k\_m \cdot k\_m\right)}{\ell}}{\ell}}\\ \end{array} \end{array} \]

FPCore

k_m = (fabs.f64 k)
(FPCore (t l k_m)
 :precision binary64
 (if (<= t 4.5e-9)
   (/ 2.0 (* (* k_m k_m) (/ k_m (/ l (/ k_m (/ l t))))))
   (/ 2.0 (/ (* (* k_m k_m) (/ (* t (* k_m k_m)) l)) l))))

C

k_m = fabs(k);
double code(double t, double l, double k_m) {
	double tmp;
	if (t <= 4.5e-9) {
		tmp = 2.0 / ((k_m * k_m) * (k_m / (l / (k_m / (l / t)))));
	} else {
		tmp = 2.0 / (((k_m * k_m) * ((t * (k_m * k_m)) / l)) / l);
	}
	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 <= 4.5d-9) then
        tmp = 2.0d0 / ((k_m * k_m) * (k_m / (l / (k_m / (l / t)))))
    else
        tmp = 2.0d0 / (((k_m * k_m) * ((t * (k_m * k_m)) / l)) / l)
    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 <= 4.5e-9) {
		tmp = 2.0 / ((k_m * k_m) * (k_m / (l / (k_m / (l / t)))));
	} else {
		tmp = 2.0 / (((k_m * k_m) * ((t * (k_m * k_m)) / l)) / l);
	}
	return tmp;
}

Python

k_m = math.fabs(k)
def code(t, l, k_m):
	tmp = 0
	if t <= 4.5e-9:
		tmp = 2.0 / ((k_m * k_m) * (k_m / (l / (k_m / (l / t)))))
	else:
		tmp = 2.0 / (((k_m * k_m) * ((t * (k_m * k_m)) / l)) / l)
	return tmp

Julia

k_m = abs(k)
function code(t, l, k_m)
	tmp = 0.0
	if (t <= 4.5e-9)
		tmp = Float64(2.0 / Float64(Float64(k_m * k_m) * Float64(k_m / Float64(l / Float64(k_m / Float64(l / t))))));
	else
		tmp = Float64(2.0 / Float64(Float64(Float64(k_m * k_m) * Float64(Float64(t * Float64(k_m * k_m)) / l)) / l));
	end
	return tmp
end

MATLAB

k_m = abs(k);
function tmp_2 = code(t, l, k_m)
	tmp = 0.0;
	if (t <= 4.5e-9)
		tmp = 2.0 / ((k_m * k_m) * (k_m / (l / (k_m / (l / t)))));
	else
		tmp = 2.0 / (((k_m * k_m) * ((t * (k_m * k_m)) / l)) / l);
	end
	tmp_2 = tmp;
end

Wolfram

k_m = N[Abs[k], $MachinePrecision]
code[t_, l_, k$95$m_] := If[LessEqual[t, 4.5e-9], N[(2.0 / N[(N[(k$95$m * k$95$m), $MachinePrecision] * N[(k$95$m / N[(l / N[(k$95$m / N[(l / t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(2.0 / N[(N[(N[(k$95$m * k$95$m), $MachinePrecision] * N[(N[(t * N[(k$95$m * k$95$m), $MachinePrecision]), $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if t < 4.49999999999999976e-9

   1. Initial program 38.6%

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

   3. Taylor expanded in k around 0

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

      1. associate-/l* N/A

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

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

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

      3. metadata-eval N/A

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

      4. pow-sqr N/A

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

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

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

      6. unpow2 N/A

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

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

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

      8. unpow2 N/A

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

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

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

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

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

      11. unpow2 N/A

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

      12. *-lowering-*.f64 60.9%

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

   5. Simplified 60.9%

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

      1. *-commutative N/A

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

      2. div-inv N/A

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

      3. associate-*l* N/A

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

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

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

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

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

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

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

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

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

      8. associate-*l* N/A

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

      9. cube-unmult N/A

         \[\leadsto \mathsf{/.f64}\left(2, \mathsf{*.f64}\left(t, \mathsf{*.f64}\left(\mathsf{/.f64}\left(1, \mathsf{*.f64}\left(\ell, \ell\right)\right), \left(k \cdot {k}^{\color{blue}{3}}\right)\right)\right)\right) \]

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

          \[\leadsto \mathsf{/.f64}\left(2, \mathsf{*.f64}\left(t, \mathsf{*.f64}\left(\mathsf{/.f64}\left(1, \mathsf{*.f64}\left(\ell, \ell\right)\right), \mathsf{*.f64}\left(k, \color{blue}{\left({k}^{3}\right)}\right)\right)\right)\right) \]

      11. cube-unmult N/A

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

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

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

      13. *-lowering-*.f64 60.5%

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

   7. Applied egg-rr 60.5%

      \[\leadsto \frac{2}{\color{blue}{t \cdot \left(\frac{1}{\ell \cdot \ell} \cdot \left(k \cdot \left(k \cdot \left(k \cdot k\right)\right)\right)\right)}} \]
   8. Step-by-step derivation

      1. associate-*r* N/A

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

      2. associate-*r* N/A

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

      3. associate-*r* N/A

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

      4. un-div-inv N/A

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

      5. associate-/l/ N/A

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

      6. clear-num N/A

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

      7. associate-/r/ N/A

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

      8. clear-num N/A

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

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

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

      10. associate-/l* N/A

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

      11. clear-num N/A

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

      12. un-div-inv N/A

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

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

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

      14. associate-/l/ N/A

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

      15. clear-num N/A

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

      16. div-inv N/A

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

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

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

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

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

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

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

      20. *-lowering-*.f64 70.4%

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

   9. Applied egg-rr 70.4%

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

   if 4.49999999999999976e-9 < t

   1. Initial program 32.6%

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

   3. Taylor expanded in k around 0

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

      1. associate-/l* N/A

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

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

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

      3. metadata-eval N/A

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

      4. pow-sqr N/A

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

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

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

      6. unpow2 N/A

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

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

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

      8. unpow2 N/A

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

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

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

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

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

      11. unpow2 N/A

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

      12. *-lowering-*.f64 62.4%

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

   5. Simplified 62.4%

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

      1. associate-*l* N/A

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

      2. associate-/l* N/A

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

      3. associate-/r* N/A

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

      4. associate-*r/ N/A

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

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

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

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

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

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

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

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

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

      9. *-commutative N/A

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

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

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

      11. *-lowering-*.f64 80.9%

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

   7. Applied egg-rr 80.9%

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

4. Final simplification 73.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq 4.5 \cdot 10^{-9}:\\ \;\;\;\;\frac{2}{\left(k \cdot k\right) \cdot \frac{k}{\frac{\ell}{\frac{k}{\frac{\ell}{t}}}}}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\frac{\left(k \cdot k\right) \cdot \frac{t \cdot \left(k \cdot k\right)}{\ell}}{\ell}}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 75.0% accurate, 28.1× speedup

Math

\[\begin{array}{l} k_m = \left|k\right| \\ \frac{\frac{\frac{\ell}{0.5} \cdot \frac{\frac{\ell}{k\_m \cdot t}}{k\_m}}{k\_m}}{k\_m} \end{array} \]

FPCore

k_m = (fabs.f64 k)
(FPCore (t l k_m)
 :precision binary64
 (/ (/ (* (/ l 0.5) (/ (/ l (* 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 / 0.5) * ((l / (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 / 0.5d0) * ((l / (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 / 0.5) * ((l / (k_m * t)) / k_m)) / k_m) / k_m;
}

Python

k_m = math.fabs(k)
def code(t, l, k_m):
	return (((l / 0.5) * ((l / (k_m * t)) / k_m)) / k_m) / k_m

Julia

k_m = abs(k)
function code(t, l, k_m)
	return Float64(Float64(Float64(Float64(l / 0.5) * Float64(Float64(l / Float64(k_m * t)) / k_m)) / k_m) / k_m)
end

MATLAB

k_m = abs(k);
function tmp = code(t, l, k_m)
	tmp = (((l / 0.5) * ((l / (k_m * t)) / k_m)) / k_m) / k_m;
end

Wolfram

k_m = N[Abs[k], $MachinePrecision]
code[t_, l_, k$95$m_] := N[(N[(N[(N[(l / 0.5), $MachinePrecision] * N[(N[(l / N[(k$95$m * t), $MachinePrecision]), $MachinePrecision] / k$95$m), $MachinePrecision]), $MachinePrecision] / k$95$m), $MachinePrecision] / k$95$m), $MachinePrecision]

Derivation

1. Initial program 36.9%

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

3. Taylor expanded in k around 0

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

   1. associate-/l* N/A

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

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

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

   3. metadata-eval N/A

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

   4. pow-sqr N/A

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

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

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

   6. unpow2 N/A

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

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

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

   8. unpow2 N/A

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

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

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

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

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

   11. unpow2 N/A

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

   12. *-lowering-*.f64 61.3%

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

5. Simplified 61.3%

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

   1. clear-num N/A

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

   2. associate-/r/ N/A

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

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

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

   4. *-commutative N/A

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

   5. associate-/r* N/A

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

   6. clear-num N/A

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

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

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

   8. associate-*r/ N/A

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

   9. clear-num N/A

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

   10. un-div-inv N/A

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

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

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

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

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

   13. associate-*l* N/A

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

   14. cube-unmult N/A

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

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

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

   16. cube-unmult N/A

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

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

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

   18. *-lowering-*.f64 65.0%

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

7. Applied egg-rr 65.0%

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

   1. *-commutative N/A

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

   2. associate-*r/ N/A

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

   3. associate-*r* N/A

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

   4. associate-/r* N/A

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

   5. clear-num N/A

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

   6. associate-/l/ N/A

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

   7. div-inv N/A

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

   8. unpow-1 N/A

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

   9. associate-/r* N/A

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

   10. unpow-1 N/A

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

   11. div-inv N/A

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

   12. unpow-1 N/A

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

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

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

9. Applied egg-rr 70.4%

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

    1. frac-times N/A

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

    2. div-inv N/A

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

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

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

    4. *-commutative N/A

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

    5. metadata-eval N/A

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

    6. metadata-eval N/A

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

    7. div-inv N/A

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

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

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

    9. metadata-eval N/A

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

    10. clear-num N/A

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

    11. un-div-inv N/A

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

    12. clear-num N/A

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

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

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

    14. associate-/l/ N/A

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

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

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

    16. *-lowering-*.f64 74.2%

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

11. Applied egg-rr 74.2%

    \[\leadsto \frac{\frac{\color{blue}{\frac{\ell}{0.5} \cdot \frac{\frac{\ell}{k \cdot t}}{k}}}{k}}{k} \]
12. Add Preprocessing

Alternative 6: 74.0% accurate, 28.1× speedup

Math

\[\begin{array}{l} k_m = \left|k\right| \\ 2 \cdot \left(\frac{\frac{\ell}{k\_m \cdot t}}{k\_m} \cdot \frac{\ell}{k\_m \cdot k\_m}\right) \end{array} \]

FPCore

k_m = (fabs.f64 k)
(FPCore (t l k_m)
 :precision binary64
 (* 2.0 (* (/ (/ l (* k_m t)) k_m) (/ l (* k_m k_m)))))

C

k_m = fabs(k);
double code(double t, double l, double k_m) {
	return 2.0 * (((l / (k_m * t)) / 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 * (((l / (k_m * t)) / 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 * (((l / (k_m * t)) / k_m) * (l / (k_m * k_m)));
}

Python

k_m = math.fabs(k)
def code(t, l, k_m):
	return 2.0 * (((l / (k_m * t)) / k_m) * (l / (k_m * k_m)))

Julia

k_m = abs(k)
function code(t, l, k_m)
	return Float64(2.0 * Float64(Float64(Float64(l / Float64(k_m * t)) / k_m) * Float64(l / Float64(k_m * k_m))))
end

MATLAB

k_m = abs(k);
function tmp = code(t, l, k_m)
	tmp = 2.0 * (((l / (k_m * t)) / k_m) * (l / (k_m * k_m)));
end

Wolfram

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

Derivation

1. Initial program 36.9%

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

3. Taylor expanded in k around 0

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

   1. associate-/l* N/A

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

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

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

   3. metadata-eval N/A

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

   4. pow-sqr N/A

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

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

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

   6. unpow2 N/A

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

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

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

   8. unpow2 N/A

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

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

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

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

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

   11. unpow2 N/A

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

   12. *-lowering-*.f64 61.3%

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

5. Simplified 61.3%

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

   1. clear-num N/A

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

   2. associate-/r/ N/A

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

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

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

   4. *-commutative N/A

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

   5. associate-/r* N/A

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

   6. clear-num N/A

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

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

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

   8. associate-*r/ N/A

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

   9. clear-num N/A

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

   10. un-div-inv N/A

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

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

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

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

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

   13. associate-*l* N/A

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

   14. cube-unmult N/A

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

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

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

   16. cube-unmult N/A

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

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

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

   18. *-lowering-*.f64 65.0%

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

7. Applied egg-rr 65.0%

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

   1. associate-/l/ N/A

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

   2. associate-/r* N/A

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

   3. div-inv N/A

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

   4. clear-num N/A

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

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

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

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

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

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

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

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

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

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

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

   10. /-lowering-/.f64 66.1%

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

9. Applied egg-rr 66.1%

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

    1. associate-*l/ N/A

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

    2. associate-*r* N/A

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

    3. times-frac N/A

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

    4. clear-num N/A

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

    5. associate-*r/ N/A

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

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

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

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

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

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

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

    9. clear-num N/A

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

    10. un-div-inv N/A

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

    11. clear-num N/A

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

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

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

    13. associate-/l/ N/A

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

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

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

    15. *-lowering-*.f64 72.9%

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

11. Applied egg-rr 72.9%

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

12. Final simplification 72.9%

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

Alternative 7: 29.4% accurate, 38.3× speedup

Math

\[\begin{array}{l} k_m = \left|k\right| \\ \frac{-0.3333333333333333 \cdot \left(\ell \cdot \frac{\ell}{t}\right)}{k\_m \cdot k\_m} \end{array} \]

FPCore

k_m = (fabs.f64 k)
(FPCore (t l k_m)
 :precision binary64
 (/ (* -0.3333333333333333 (* l (/ l t))) (* k_m k_m)))

C

k_m = fabs(k);
double code(double t, double l, double k_m) {
	return (-0.3333333333333333 * (l * (l / t))) / (k_m * k_m);
}

Fortran

k_m = abs(k)
real(8) function code(t, l, k_m)
    real(8), intent (in) :: t
    real(8), intent (in) :: l
    real(8), intent (in) :: k_m
    code = ((-0.3333333333333333d0) * (l * (l / t))) / (k_m * k_m)
end function

Java

k_m = Math.abs(k);
public static double code(double t, double l, double k_m) {
	return (-0.3333333333333333 * (l * (l / t))) / (k_m * k_m);
}

Python

k_m = math.fabs(k)
def code(t, l, k_m):
	return (-0.3333333333333333 * (l * (l / t))) / (k_m * k_m)

Julia

k_m = abs(k)
function code(t, l, k_m)
	return Float64(Float64(-0.3333333333333333 * Float64(l * Float64(l / t))) / Float64(k_m * k_m))
end

MATLAB

k_m = abs(k);
function tmp = code(t, l, k_m)
	tmp = (-0.3333333333333333 * (l * (l / t))) / (k_m * k_m);
end

Wolfram

k_m = N[Abs[k], $MachinePrecision]
code[t_, l_, k$95$m_] := N[(N[(-0.3333333333333333 * N[(l * N[(l / t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(k$95$m * k$95$m), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 36.9%

   \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) - 1\right)} \]
2. Step-by-step derivation

   1. associate-/r* N/A

      \[\leadsto \frac{\frac{2}{\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k}}{\color{blue}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) - 1}} \]

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

      \[\leadsto \mathsf{/.f64}\left(\left(\frac{2}{\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k}\right), \color{blue}{\left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) - 1\right)}\right) \]

3. Simplified 49.8%

   \[\leadsto \color{blue}{\frac{\frac{\frac{\frac{\frac{2}{t}}{\frac{\frac{t \cdot t}{\ell}}{\ell}}}{\sin k}}{\tan k}}{k \cdot \frac{\frac{k}{t}}{t}}} \]
4. Add Preprocessing

5. Taylor expanded in k around 0

   \[\leadsto \mathsf{/.f64}\left(\color{blue}{\left(\frac{\frac{-1}{3} \cdot \frac{{k}^{2} \cdot {\ell}^{2}}{{t}^{3}} + 2 \cdot \frac{{\ell}^{2}}{{t}^{3}}}{{k}^{2}}\right)}, \mathsf{*.f64}\left(k, \mathsf{/.f64}\left(\mathsf{/.f64}\left(k, t\right), t\right)\right)\right) \]
6. Step-by-step derivation

   1. *-commutative N/A

      \[\leadsto \mathsf{/.f64}\left(\left(\frac{\frac{{k}^{2} \cdot {\ell}^{2}}{{t}^{3}} \cdot \frac{-1}{3} + 2 \cdot \frac{{\ell}^{2}}{{t}^{3}}}{{k}^{2}}\right), \mathsf{*.f64}\left(k, \mathsf{/.f64}\left(\mathsf{/.f64}\left(k, t\right), t\right)\right)\right) \]

   2. associate-/l* N/A

      \[\leadsto \mathsf{/.f64}\left(\left(\frac{\left({k}^{2} \cdot \frac{{\ell}^{2}}{{t}^{3}}\right) \cdot \frac{-1}{3} + 2 \cdot \frac{{\ell}^{2}}{{t}^{3}}}{{k}^{2}}\right), \mathsf{*.f64}\left(k, \mathsf{/.f64}\left(\mathsf{/.f64}\left(k, t\right), t\right)\right)\right) \]

   3. associate-*r* N/A

      \[\leadsto \mathsf{/.f64}\left(\left(\frac{{k}^{2} \cdot \left(\frac{{\ell}^{2}}{{t}^{3}} \cdot \frac{-1}{3}\right) + 2 \cdot \frac{{\ell}^{2}}{{t}^{3}}}{{k}^{2}}\right), \mathsf{*.f64}\left(k, \mathsf{/.f64}\left(\mathsf{/.f64}\left(k, t\right), t\right)\right)\right) \]

   4. *-commutative N/A

      \[\leadsto \mathsf{/.f64}\left(\left(\frac{{k}^{2} \cdot \left(\frac{-1}{3} \cdot \frac{{\ell}^{2}}{{t}^{3}}\right) + 2 \cdot \frac{{\ell}^{2}}{{t}^{3}}}{{k}^{2}}\right), \mathsf{*.f64}\left(k, \mathsf{/.f64}\left(\mathsf{/.f64}\left(k, t\right), t\right)\right)\right) \]

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

      \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\left({k}^{2} \cdot \left(\frac{-1}{3} \cdot \frac{{\ell}^{2}}{{t}^{3}}\right) + 2 \cdot \frac{{\ell}^{2}}{{t}^{3}}\right), \left({k}^{2}\right)\right), \mathsf{*.f64}\left(\color{blue}{k}, \mathsf{/.f64}\left(\mathsf{/.f64}\left(k, t\right), t\right)\right)\right) \]

7. Simplified 28.3%

   \[\leadsto \frac{\color{blue}{\frac{\frac{\ell \cdot \ell}{t \cdot \left(t \cdot t\right)} \cdot \left(2 + -0.3333333333333333 \cdot \left(k \cdot k\right)\right)}{k \cdot k}}}{k \cdot \frac{\frac{k}{t}}{t}} \]

8. Taylor expanded in k around inf

   \[\leadsto \color{blue}{\frac{-1}{3} \cdot \frac{{\ell}^{2}}{{k}^{2} \cdot t}} \]
9. Step-by-step derivation

   1. associate-*r/ N/A

      \[\leadsto \frac{\frac{-1}{3} \cdot {\ell}^{2}}{\color{blue}{{k}^{2} \cdot t}} \]

   2. times-frac N/A

      \[\leadsto \frac{\frac{-1}{3}}{{k}^{2}} \cdot \color{blue}{\frac{{\ell}^{2}}{t}} \]

   3. associate-*l/ N/A

      \[\leadsto \frac{\frac{-1}{3} \cdot \frac{{\ell}^{2}}{t}}{\color{blue}{{k}^{2}}} \]

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

      \[\leadsto \mathsf{/.f64}\left(\left(\frac{-1}{3} \cdot \frac{{\ell}^{2}}{t}\right), \color{blue}{\left({k}^{2}\right)}\right) \]

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

      \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\frac{-1}{3}, \left(\frac{{\ell}^{2}}{t}\right)\right), \left({\color{blue}{k}}^{2}\right)\right) \]

   6. unpow2 N/A

      \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\frac{-1}{3}, \left(\frac{\ell \cdot \ell}{t}\right)\right), \left({k}^{2}\right)\right) \]

   7. associate-/l* N/A

      \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\frac{-1}{3}, \left(\ell \cdot \frac{\ell}{t}\right)\right), \left({k}^{2}\right)\right) \]

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

      \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\frac{-1}{3}, \mathsf{*.f64}\left(\ell, \left(\frac{\ell}{t}\right)\right)\right), \left({k}^{2}\right)\right) \]

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

      \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\frac{-1}{3}, \mathsf{*.f64}\left(\ell, \mathsf{/.f64}\left(\ell, t\right)\right)\right), \left({k}^{2}\right)\right) \]

   10. unpow2 N/A

       \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\frac{-1}{3}, \mathsf{*.f64}\left(\ell, \mathsf{/.f64}\left(\ell, t\right)\right)\right), \left(k \cdot \color{blue}{k}\right)\right) \]

   11. *-lowering-*.f64 29.8%

       \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\frac{-1}{3}, \mathsf{*.f64}\left(\ell, \mathsf{/.f64}\left(\ell, t\right)\right)\right), \mathsf{*.f64}\left(k, \color{blue}{k}\right)\right) \]

10. Simplified 29.8%

    \[\leadsto \color{blue}{\frac{-0.3333333333333333 \cdot \left(\ell \cdot \frac{\ell}{t}\right)}{k \cdot k}} \]
11. Add Preprocessing

Reproduce

herbie shell --seed 2024155 
(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))))