Toniolo and Linder, Equation (10-)

Percentage Accurate: 35.5% → 97.5%

Time: 21.3s

Alternatives: 15

Speedup: 38.3×

• 40.1% of points produce a very large (infinite) output. You may want to add a precondition.

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 35.5% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 97.5% accurate, 2.0× speedup

Math

\[\begin{array}{l} \\ \frac{\frac{2}{\frac{k}{\ell}}}{t} \cdot \frac{\frac{\frac{\ell}{\sin k}}{\tan k}}{k} \end{array} \]

FPCore

(FPCore (t l k)
 :precision binary64
 (* (/ (/ 2.0 (/ k l)) t) (/ (/ (/ l (sin k)) (tan k)) k)))

C

double code(double t, double l, double k) {
	return ((2.0 / (k / l)) / t) * (((l / sin(k)) / tan(k)) / k);
}

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 / (k / l)) / t) * (((l / sin(k)) / tan(k)) / k)
end function

Java

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

Python

def code(t, l, k):
	return ((2.0 / (k / l)) / t) * (((l / math.sin(k)) / math.tan(k)) / k)

Julia

function code(t, l, k)
	return Float64(Float64(Float64(2.0 / Float64(k / l)) / t) * Float64(Float64(Float64(l / sin(k)) / tan(k)) / k))
end

MATLAB

function tmp = code(t, l, k)
	tmp = ((2.0 / (k / l)) / t) * (((l / sin(k)) / tan(k)) / k);
end

Wolfram

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

Derivation

1. Initial program 36.8%

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

   1. *-commutative N/A

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

   2. associate-*r* N/A

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

   3. associate-/r* N/A

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

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

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

3. Simplified 66.0%

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

   1. clear-num N/A

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

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

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

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

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

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

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

   5. associate-/r/ N/A

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

   6. associate-/l* N/A

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

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

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

6. Applied egg-rr 87.2%

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

7. Taylor expanded in k around 0

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

   1. associate-/r* N/A

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

   2. associate-*r/ N/A

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

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

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

   4. unpow2 N/A

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

   5. associate-/r* N/A

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

   6. associate-*r/ N/A

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

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

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

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

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

   9. /-lowering-/.f64 92.6%

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

9. Simplified 92.6%

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

    1. clear-num N/A

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

    2. associate-*r/ N/A

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

    3. associate-/r* N/A

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

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

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

    5. associate-/l/ N/A

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

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

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

    7. associate-*r/ N/A

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

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

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

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

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

    10. *-commutative N/A

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

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

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

    12. associate-/r* N/A

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

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

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

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

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

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

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

    16. tan-lowering-tan.f64 94.4%

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

11. Applied egg-rr 94.4%

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

    1. associate-*l/ N/A

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

    2. *-commutative N/A

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

    3. times-frac N/A

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

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

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

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

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

    6. associate-/l* N/A

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

    7. clear-num N/A

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

    8. un-div-inv N/A

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

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

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

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

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

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

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

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

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

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

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

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

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

    15. tan-lowering-tan.f64 98.4%

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

13. Applied egg-rr 98.4%

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

Alternative 2: 77.3% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;k \leq 6.2 \cdot 10^{-110}:\\ \;\;\;\;\frac{2}{\frac{\frac{k \cdot t}{\frac{\ell}{k}}}{\frac{\frac{1 + \left(k \cdot k\right) \cdot 0.16666666666666666}{k}}{\frac{k}{\ell}}}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{2}{k}}{\frac{k}{\frac{\ell}{t}}} \cdot \frac{\ell}{\sin k \cdot \tan k}\\ \end{array} \end{array} \]

FPCore

(FPCore (t l k)
 :precision binary64
 (if (<= k 6.2e-110)
   (/
    2.0
    (/
     (/ (* k t) (/ l k))
     (/ (/ (+ 1.0 (* (* k k) 0.16666666666666666)) k) (/ k l))))
   (* (/ (/ 2.0 k) (/ k (/ l t))) (/ l (* (sin k) (tan k))))))

C

double code(double t, double l, double k) {
	double tmp;
	if (k <= 6.2e-110) {
		tmp = 2.0 / (((k * t) / (l / k)) / (((1.0 + ((k * k) * 0.16666666666666666)) / k) / (k / l)));
	} else {
		tmp = ((2.0 / k) / (k / (l / t))) * (l / (sin(k) * tan(k)));
	}
	return tmp;
}

Fortran

real(8) function code(t, l, k)
    real(8), intent (in) :: t
    real(8), intent (in) :: l
    real(8), intent (in) :: k
    real(8) :: tmp
    if (k <= 6.2d-110) then
        tmp = 2.0d0 / (((k * t) / (l / k)) / (((1.0d0 + ((k * k) * 0.16666666666666666d0)) / k) / (k / l)))
    else
        tmp = ((2.0d0 / k) / (k / (l / t))) * (l / (sin(k) * tan(k)))
    end if
    code = tmp
end function

Java

public static double code(double t, double l, double k) {
	double tmp;
	if (k <= 6.2e-110) {
		tmp = 2.0 / (((k * t) / (l / k)) / (((1.0 + ((k * k) * 0.16666666666666666)) / k) / (k / l)));
	} else {
		tmp = ((2.0 / k) / (k / (l / t))) * (l / (Math.sin(k) * Math.tan(k)));
	}
	return tmp;
}

Python

def code(t, l, k):
	tmp = 0
	if k <= 6.2e-110:
		tmp = 2.0 / (((k * t) / (l / k)) / (((1.0 + ((k * k) * 0.16666666666666666)) / k) / (k / l)))
	else:
		tmp = ((2.0 / k) / (k / (l / t))) * (l / (math.sin(k) * math.tan(k)))
	return tmp

Julia

function code(t, l, k)
	tmp = 0.0
	if (k <= 6.2e-110)
		tmp = Float64(2.0 / Float64(Float64(Float64(k * t) / Float64(l / k)) / Float64(Float64(Float64(1.0 + Float64(Float64(k * k) * 0.16666666666666666)) / k) / Float64(k / l))));
	else
		tmp = Float64(Float64(Float64(2.0 / k) / Float64(k / Float64(l / t))) * Float64(l / Float64(sin(k) * tan(k))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(t, l, k)
	tmp = 0.0;
	if (k <= 6.2e-110)
		tmp = 2.0 / (((k * t) / (l / k)) / (((1.0 + ((k * k) * 0.16666666666666666)) / k) / (k / l)));
	else
		tmp = ((2.0 / k) / (k / (l / t))) * (l / (sin(k) * tan(k)));
	end
	tmp_2 = tmp;
end

Wolfram

code[t_, l_, k_] := If[LessEqual[k, 6.2e-110], N[(2.0 / N[(N[(N[(k * t), $MachinePrecision] / N[(l / k), $MachinePrecision]), $MachinePrecision] / N[(N[(N[(1.0 + N[(N[(k * k), $MachinePrecision] * 0.16666666666666666), $MachinePrecision]), $MachinePrecision] / k), $MachinePrecision] / N[(k / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(2.0 / k), $MachinePrecision] / N[(k / N[(l / t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(l / N[(N[Sin[k], $MachinePrecision] * N[Tan[k], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if k < 6.20000000000000014e-110

   1. Initial program 40.5%

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

      1. *-commutative N/A

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

      2. associate-*r* N/A

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

      3. associate-/r* N/A

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

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

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

   3. Simplified 66.5%

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

      1. associate-/l/ N/A

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

      2. associate-/r/ N/A

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

      3. associate-/l* N/A

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

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

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

      5. *-inverses N/A

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

      6. associate-*l* N/A

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

      7. *-commutative N/A

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

      8. times-frac N/A

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

      9. *-rgt-identity N/A

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

      10. *-inverses N/A

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

      11. *-rgt-identity N/A

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

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

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

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

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

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

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

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

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

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

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

   6. Applied egg-rr 87.0%

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

   7. Taylor expanded in k around 0

      \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(2, \mathsf{*.f64}\left(k, k\right)\right), \mathsf{/.f64}\left(t, \ell\right)\right), \mathsf{/.f64}\left(\ell, \mathsf{*.f64}\left(\mathsf{sin.f64}\left(k\right), \color{blue}{k}\right)\right)\right) \]
   8. Step-by-step derivation

   9. Simplified 74.9%

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

   10. Taylor expanded in k around 0

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

       1. unpow2 N/A

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

       2. associate-/r* N/A

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

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

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

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

          \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(2, \mathsf{*.f64}\left(k, k\right)\right), \mathsf{/.f64}\left(t, \ell\right)\right), \mathsf{/.f64}\left(\mathsf{/.f64}\left(\left(\ell + \frac{1}{6} \cdot \left({k}^{2} \cdot \ell\right)\right), k\right), k\right)\right) \]

       5. associate-*r* N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(2, \mathsf{*.f64}\left(k, k\right)\right), \mathsf{/.f64}\left(t, \ell\right)\right), \mathsf{/.f64}\left(\mathsf{/.f64}\left(\left(\ell + \left(\frac{1}{6} \cdot {k}^{2}\right) \cdot \ell\right), k\right), k\right)\right) \]

       6. distribute-rgt1-in N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(2, \mathsf{*.f64}\left(k, k\right)\right), \mathsf{/.f64}\left(t, \ell\right)\right), \mathsf{/.f64}\left(\mathsf{/.f64}\left(\left(\left(\frac{1}{6} \cdot {k}^{2} + 1\right) \cdot \ell\right), k\right), k\right)\right) \]

       7. +-commutative N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(2, \mathsf{*.f64}\left(k, k\right)\right), \mathsf{/.f64}\left(t, \ell\right)\right), \mathsf{/.f64}\left(\mathsf{/.f64}\left(\left(\left(1 + \frac{1}{6} \cdot {k}^{2}\right) \cdot \ell\right), k\right), k\right)\right) \]

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

          \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(2, \mathsf{*.f64}\left(k, k\right)\right), \mathsf{/.f64}\left(t, \ell\right)\right), \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\left(1 + \frac{1}{6} \cdot {k}^{2}\right), \ell\right), k\right), k\right)\right) \]

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

          \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(2, \mathsf{*.f64}\left(k, k\right)\right), \mathsf{/.f64}\left(t, \ell\right)\right), \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \left(\frac{1}{6} \cdot {k}^{2}\right)\right), \ell\right), k\right), k\right)\right) \]

       10. *-commutative N/A

           \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(2, \mathsf{*.f64}\left(k, k\right)\right), \mathsf{/.f64}\left(t, \ell\right)\right), \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \left({k}^{2} \cdot \frac{1}{6}\right)\right), \ell\right), k\right), k\right)\right) \]

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

           \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(2, \mathsf{*.f64}\left(k, k\right)\right), \mathsf{/.f64}\left(t, \ell\right)\right), \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\left({k}^{2}\right), \frac{1}{6}\right)\right), \ell\right), k\right), k\right)\right) \]

       12. unpow2 N/A

           \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(2, \mathsf{*.f64}\left(k, k\right)\right), \mathsf{/.f64}\left(t, \ell\right)\right), \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\left(k \cdot k\right), \frac{1}{6}\right)\right), \ell\right), k\right), k\right)\right) \]

       13. *-lowering-*.f64 67.2%

           \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(2, \mathsf{*.f64}\left(k, k\right)\right), \mathsf{/.f64}\left(t, \ell\right)\right), \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(k, k\right), \frac{1}{6}\right)\right), \ell\right), k\right), k\right)\right) \]

   12. Simplified 67.2%

       \[\leadsto \frac{\frac{2}{k \cdot k}}{\frac{t}{\ell}} \cdot \color{blue}{\frac{\frac{\left(1 + \left(k \cdot k\right) \cdot 0.16666666666666666\right) \cdot \ell}{k}}{k}} \]
   13. Step-by-step derivation

       1. associate-/l/ N/A

          \[\leadsto \frac{2}{\frac{t}{\ell} \cdot \left(k \cdot k\right)} \cdot \frac{\color{blue}{\frac{\left(1 + \left(k \cdot k\right) \cdot \frac{1}{6}\right) \cdot \ell}{k}}}{k} \]

       2. clear-num N/A

          \[\leadsto \frac{2}{\frac{t}{\ell} \cdot \left(k \cdot k\right)} \cdot \frac{1}{\color{blue}{\frac{k}{\frac{\left(1 + \left(k \cdot k\right) \cdot \frac{1}{6}\right) \cdot \ell}{k}}}} \]

       3. frac-times N/A

          \[\leadsto \frac{2 \cdot 1}{\color{blue}{\left(\frac{t}{\ell} \cdot \left(k \cdot k\right)\right) \cdot \frac{k}{\frac{\left(1 + \left(k \cdot k\right) \cdot \frac{1}{6}\right) \cdot \ell}{k}}}} \]

       4. metadata-eval N/A

          \[\leadsto \frac{2}{\color{blue}{\left(\frac{t}{\ell} \cdot \left(k \cdot k\right)\right)} \cdot \frac{k}{\frac{\left(1 + \left(k \cdot k\right) \cdot \frac{1}{6}\right) \cdot \ell}{k}}} \]

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

          \[\leadsto \mathsf{/.f64}\left(2, \color{blue}{\left(\left(\frac{t}{\ell} \cdot \left(k \cdot k\right)\right) \cdot \frac{k}{\frac{\left(1 + \left(k \cdot k\right) \cdot \frac{1}{6}\right) \cdot \ell}{k}}\right)}\right) \]

       6. clear-num N/A

          \[\leadsto \mathsf{/.f64}\left(2, \left(\left(\frac{t}{\ell} \cdot \left(k \cdot k\right)\right) \cdot \frac{1}{\color{blue}{\frac{\frac{\left(1 + \left(k \cdot k\right) \cdot \frac{1}{6}\right) \cdot \ell}{k}}{k}}}\right)\right) \]

       7. un-div-inv N/A

          \[\leadsto \mathsf{/.f64}\left(2, \left(\frac{\frac{t}{\ell} \cdot \left(k \cdot k\right)}{\color{blue}{\frac{\frac{\left(1 + \left(k \cdot k\right) \cdot \frac{1}{6}\right) \cdot \ell}{k}}{k}}}\right)\right) \]

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

          \[\leadsto \mathsf{/.f64}\left(2, \mathsf{/.f64}\left(\left(\frac{t}{\ell} \cdot \left(k \cdot k\right)\right), \color{blue}{\left(\frac{\frac{\left(1 + \left(k \cdot k\right) \cdot \frac{1}{6}\right) \cdot \ell}{k}}{k}\right)}\right)\right) \]

   14. Applied egg-rr 72.5%

       \[\leadsto \color{blue}{\frac{2}{\frac{\frac{k \cdot t}{\frac{\ell}{k}}}{\frac{\frac{1 + \left(k \cdot k\right) \cdot 0.16666666666666666}{k}}{\frac{k}{\ell}}}}} \]

   if 6.20000000000000014e-110 < k

   1. Initial program 30.6%

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

      1. *-commutative N/A

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

      2. associate-*r* N/A

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

      3. associate-/r* N/A

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

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

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

   3. Simplified 65.1%

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

      1. associate-/l/ N/A

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

      2. associate-/r/ N/A

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

      3. associate-/l* N/A

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

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

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

      5. *-inverses N/A

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

      6. associate-*l* N/A

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

      7. *-commutative N/A

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

      8. times-frac N/A

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

      9. *-rgt-identity N/A

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

      10. *-inverses N/A

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

      11. *-rgt-identity N/A

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

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

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

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

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

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

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

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

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

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

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

   6. Applied egg-rr 87.9%

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

      1. div-inv N/A

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

      2. associate-/r* N/A

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

      3. frac-times N/A

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

      4. *-rgt-identity N/A

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

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

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

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

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

      7. clear-num N/A

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

      8. un-div-inv N/A

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

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

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

      10. /-lowering-/.f64 93.8%

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

   8. Applied egg-rr 93.8%

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

Alternative 3: 74.1% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;k \leq 1.9 \cdot 10^{-109}:\\ \;\;\;\;\frac{2}{\frac{\frac{k \cdot t}{\frac{\ell}{k}}}{\frac{\frac{1 + \left(k \cdot k\right) \cdot 0.16666666666666666}{k}}{\frac{k}{\ell}}}}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{k \cdot k} \cdot \frac{\frac{\ell}{t}}{\frac{\tan k}{\frac{\ell}{\sin k}}}\\ \end{array} \end{array} \]

FPCore

(FPCore (t l k)
 :precision binary64
 (if (<= k 1.9e-109)
   (/
    2.0
    (/
     (/ (* k t) (/ l k))
     (/ (/ (+ 1.0 (* (* k k) 0.16666666666666666)) k) (/ k l))))
   (* (/ 2.0 (* k k)) (/ (/ l t) (/ (tan k) (/ l (sin k)))))))

C

double code(double t, double l, double k) {
	double tmp;
	if (k <= 1.9e-109) {
		tmp = 2.0 / (((k * t) / (l / k)) / (((1.0 + ((k * k) * 0.16666666666666666)) / k) / (k / l)));
	} else {
		tmp = (2.0 / (k * k)) * ((l / t) / (tan(k) / (l / sin(k))));
	}
	return tmp;
}

Fortran

real(8) function code(t, l, k)
    real(8), intent (in) :: t
    real(8), intent (in) :: l
    real(8), intent (in) :: k
    real(8) :: tmp
    if (k <= 1.9d-109) then
        tmp = 2.0d0 / (((k * t) / (l / k)) / (((1.0d0 + ((k * k) * 0.16666666666666666d0)) / k) / (k / l)))
    else
        tmp = (2.0d0 / (k * k)) * ((l / t) / (tan(k) / (l / sin(k))))
    end if
    code = tmp
end function

Java

public static double code(double t, double l, double k) {
	double tmp;
	if (k <= 1.9e-109) {
		tmp = 2.0 / (((k * t) / (l / k)) / (((1.0 + ((k * k) * 0.16666666666666666)) / k) / (k / l)));
	} else {
		tmp = (2.0 / (k * k)) * ((l / t) / (Math.tan(k) / (l / Math.sin(k))));
	}
	return tmp;
}

Python

def code(t, l, k):
	tmp = 0
	if k <= 1.9e-109:
		tmp = 2.0 / (((k * t) / (l / k)) / (((1.0 + ((k * k) * 0.16666666666666666)) / k) / (k / l)))
	else:
		tmp = (2.0 / (k * k)) * ((l / t) / (math.tan(k) / (l / math.sin(k))))
	return tmp

Julia

function code(t, l, k)
	tmp = 0.0
	if (k <= 1.9e-109)
		tmp = Float64(2.0 / Float64(Float64(Float64(k * t) / Float64(l / k)) / Float64(Float64(Float64(1.0 + Float64(Float64(k * k) * 0.16666666666666666)) / k) / Float64(k / l))));
	else
		tmp = Float64(Float64(2.0 / Float64(k * k)) * Float64(Float64(l / t) / Float64(tan(k) / Float64(l / sin(k)))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(t, l, k)
	tmp = 0.0;
	if (k <= 1.9e-109)
		tmp = 2.0 / (((k * t) / (l / k)) / (((1.0 + ((k * k) * 0.16666666666666666)) / k) / (k / l)));
	else
		tmp = (2.0 / (k * k)) * ((l / t) / (tan(k) / (l / sin(k))));
	end
	tmp_2 = tmp;
end

Wolfram

code[t_, l_, k_] := If[LessEqual[k, 1.9e-109], N[(2.0 / N[(N[(N[(k * t), $MachinePrecision] / N[(l / k), $MachinePrecision]), $MachinePrecision] / N[(N[(N[(1.0 + N[(N[(k * k), $MachinePrecision] * 0.16666666666666666), $MachinePrecision]), $MachinePrecision] / k), $MachinePrecision] / N[(k / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(2.0 / N[(k * k), $MachinePrecision]), $MachinePrecision] * N[(N[(l / t), $MachinePrecision] / N[(N[Tan[k], $MachinePrecision] / N[(l / N[Sin[k], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if k < 1.90000000000000001e-109

   1. Initial program 40.5%

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

      1. *-commutative N/A

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

      2. associate-*r* N/A

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

      3. associate-/r* N/A

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

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

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

   3. Simplified 66.5%

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

      1. associate-/l/ N/A

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

      2. associate-/r/ N/A

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

      3. associate-/l* N/A

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

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

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

      5. *-inverses N/A

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

      6. associate-*l* N/A

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

      7. *-commutative N/A

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

      8. times-frac N/A

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

      9. *-rgt-identity N/A

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

      10. *-inverses N/A

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

      11. *-rgt-identity N/A

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

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

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

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

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

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

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

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

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

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

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

   6. Applied egg-rr 87.0%

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

   7. Taylor expanded in k around 0

      \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(2, \mathsf{*.f64}\left(k, k\right)\right), \mathsf{/.f64}\left(t, \ell\right)\right), \mathsf{/.f64}\left(\ell, \mathsf{*.f64}\left(\mathsf{sin.f64}\left(k\right), \color{blue}{k}\right)\right)\right) \]
   8. Step-by-step derivation

   9. Simplified 74.9%

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

   10. Taylor expanded in k around 0

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

       1. unpow2 N/A

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

       2. associate-/r* N/A

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

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

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

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

          \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(2, \mathsf{*.f64}\left(k, k\right)\right), \mathsf{/.f64}\left(t, \ell\right)\right), \mathsf{/.f64}\left(\mathsf{/.f64}\left(\left(\ell + \frac{1}{6} \cdot \left({k}^{2} \cdot \ell\right)\right), k\right), k\right)\right) \]

       5. associate-*r* N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(2, \mathsf{*.f64}\left(k, k\right)\right), \mathsf{/.f64}\left(t, \ell\right)\right), \mathsf{/.f64}\left(\mathsf{/.f64}\left(\left(\ell + \left(\frac{1}{6} \cdot {k}^{2}\right) \cdot \ell\right), k\right), k\right)\right) \]

       6. distribute-rgt1-in N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(2, \mathsf{*.f64}\left(k, k\right)\right), \mathsf{/.f64}\left(t, \ell\right)\right), \mathsf{/.f64}\left(\mathsf{/.f64}\left(\left(\left(\frac{1}{6} \cdot {k}^{2} + 1\right) \cdot \ell\right), k\right), k\right)\right) \]

       7. +-commutative N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(2, \mathsf{*.f64}\left(k, k\right)\right), \mathsf{/.f64}\left(t, \ell\right)\right), \mathsf{/.f64}\left(\mathsf{/.f64}\left(\left(\left(1 + \frac{1}{6} \cdot {k}^{2}\right) \cdot \ell\right), k\right), k\right)\right) \]

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

          \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(2, \mathsf{*.f64}\left(k, k\right)\right), \mathsf{/.f64}\left(t, \ell\right)\right), \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\left(1 + \frac{1}{6} \cdot {k}^{2}\right), \ell\right), k\right), k\right)\right) \]

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

          \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(2, \mathsf{*.f64}\left(k, k\right)\right), \mathsf{/.f64}\left(t, \ell\right)\right), \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \left(\frac{1}{6} \cdot {k}^{2}\right)\right), \ell\right), k\right), k\right)\right) \]

       10. *-commutative N/A

           \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(2, \mathsf{*.f64}\left(k, k\right)\right), \mathsf{/.f64}\left(t, \ell\right)\right), \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \left({k}^{2} \cdot \frac{1}{6}\right)\right), \ell\right), k\right), k\right)\right) \]

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

           \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(2, \mathsf{*.f64}\left(k, k\right)\right), \mathsf{/.f64}\left(t, \ell\right)\right), \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\left({k}^{2}\right), \frac{1}{6}\right)\right), \ell\right), k\right), k\right)\right) \]

       12. unpow2 N/A

           \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(2, \mathsf{*.f64}\left(k, k\right)\right), \mathsf{/.f64}\left(t, \ell\right)\right), \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\left(k \cdot k\right), \frac{1}{6}\right)\right), \ell\right), k\right), k\right)\right) \]

       13. *-lowering-*.f64 67.2%

           \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(2, \mathsf{*.f64}\left(k, k\right)\right), \mathsf{/.f64}\left(t, \ell\right)\right), \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(k, k\right), \frac{1}{6}\right)\right), \ell\right), k\right), k\right)\right) \]

   12. Simplified 67.2%

       \[\leadsto \frac{\frac{2}{k \cdot k}}{\frac{t}{\ell}} \cdot \color{blue}{\frac{\frac{\left(1 + \left(k \cdot k\right) \cdot 0.16666666666666666\right) \cdot \ell}{k}}{k}} \]
   13. Step-by-step derivation

       1. associate-/l/ N/A

          \[\leadsto \frac{2}{\frac{t}{\ell} \cdot \left(k \cdot k\right)} \cdot \frac{\color{blue}{\frac{\left(1 + \left(k \cdot k\right) \cdot \frac{1}{6}\right) \cdot \ell}{k}}}{k} \]

       2. clear-num N/A

          \[\leadsto \frac{2}{\frac{t}{\ell} \cdot \left(k \cdot k\right)} \cdot \frac{1}{\color{blue}{\frac{k}{\frac{\left(1 + \left(k \cdot k\right) \cdot \frac{1}{6}\right) \cdot \ell}{k}}}} \]

       3. frac-times N/A

          \[\leadsto \frac{2 \cdot 1}{\color{blue}{\left(\frac{t}{\ell} \cdot \left(k \cdot k\right)\right) \cdot \frac{k}{\frac{\left(1 + \left(k \cdot k\right) \cdot \frac{1}{6}\right) \cdot \ell}{k}}}} \]

       4. metadata-eval N/A

          \[\leadsto \frac{2}{\color{blue}{\left(\frac{t}{\ell} \cdot \left(k \cdot k\right)\right)} \cdot \frac{k}{\frac{\left(1 + \left(k \cdot k\right) \cdot \frac{1}{6}\right) \cdot \ell}{k}}} \]

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

          \[\leadsto \mathsf{/.f64}\left(2, \color{blue}{\left(\left(\frac{t}{\ell} \cdot \left(k \cdot k\right)\right) \cdot \frac{k}{\frac{\left(1 + \left(k \cdot k\right) \cdot \frac{1}{6}\right) \cdot \ell}{k}}\right)}\right) \]

       6. clear-num N/A

          \[\leadsto \mathsf{/.f64}\left(2, \left(\left(\frac{t}{\ell} \cdot \left(k \cdot k\right)\right) \cdot \frac{1}{\color{blue}{\frac{\frac{\left(1 + \left(k \cdot k\right) \cdot \frac{1}{6}\right) \cdot \ell}{k}}{k}}}\right)\right) \]

       7. un-div-inv N/A

          \[\leadsto \mathsf{/.f64}\left(2, \left(\frac{\frac{t}{\ell} \cdot \left(k \cdot k\right)}{\color{blue}{\frac{\frac{\left(1 + \left(k \cdot k\right) \cdot \frac{1}{6}\right) \cdot \ell}{k}}{k}}}\right)\right) \]

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

          \[\leadsto \mathsf{/.f64}\left(2, \mathsf{/.f64}\left(\left(\frac{t}{\ell} \cdot \left(k \cdot k\right)\right), \color{blue}{\left(\frac{\frac{\left(1 + \left(k \cdot k\right) \cdot \frac{1}{6}\right) \cdot \ell}{k}}{k}\right)}\right)\right) \]

   14. Applied egg-rr 72.5%

       \[\leadsto \color{blue}{\frac{2}{\frac{\frac{k \cdot t}{\frac{\ell}{k}}}{\frac{\frac{1 + \left(k \cdot k\right) \cdot 0.16666666666666666}{k}}{\frac{k}{\ell}}}}} \]

   if 1.90000000000000001e-109 < k

   1. Initial program 30.6%

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

      1. *-commutative N/A

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

      2. associate-*r* N/A

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

      3. associate-/r* N/A

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

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

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

   3. Simplified 65.1%

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

      1. associate-/l/ N/A

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

      2. associate-/r/ N/A

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

      3. associate-/l* N/A

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

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

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

      5. *-inverses N/A

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

      6. associate-*l* N/A

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

      7. *-commutative N/A

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

      8. times-frac N/A

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

      9. *-rgt-identity N/A

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

      10. *-inverses N/A

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

      11. *-rgt-identity N/A

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

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

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

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

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

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

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

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

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

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

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

   6. Applied egg-rr 87.9%

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

      1. div-inv N/A

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

      2. clear-num N/A

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

      3. associate-*l* N/A

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

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

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

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

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

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

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

      7. clear-num N/A

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

      8. un-div-inv N/A

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

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

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

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

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

      11. *-commutative N/A

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

      12. associate-/l* N/A

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

      13. clear-num N/A

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

      14. un-div-inv N/A

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

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

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

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

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

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

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

      18. sin-lowering-sin.f64 85.8%

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

   8. Applied egg-rr 85.8%

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

Alternative 4: 69.7% accurate, 3.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;k \leq 3.1 \cdot 10^{+145}:\\ \;\;\;\;\frac{\frac{\frac{1 + \left(k \cdot k\right) \cdot 0.16666666666666666}{k}}{\frac{k}{\ell}}}{\frac{t}{\frac{\frac{2}{\frac{k}{\ell}}}{k}}}\\ \mathbf{else}:\\ \;\;\;\;\left(\frac{2}{k} \cdot \frac{\frac{\ell}{k}}{t}\right) \cdot \frac{\ell}{k \cdot \sin k}\\ \end{array} \end{array} \]

FPCore

(FPCore (t l k)
 :precision binary64
 (if (<= k 3.1e+145)
   (/
    (/ (/ (+ 1.0 (* (* k k) 0.16666666666666666)) k) (/ k l))
    (/ t (/ (/ 2.0 (/ k l)) k)))
   (* (* (/ 2.0 k) (/ (/ l k) t)) (/ l (* k (sin k))))))

C

double code(double t, double l, double k) {
	double tmp;
	if (k <= 3.1e+145) {
		tmp = (((1.0 + ((k * k) * 0.16666666666666666)) / k) / (k / l)) / (t / ((2.0 / (k / l)) / k));
	} else {
		tmp = ((2.0 / k) * ((l / k) / t)) * (l / (k * sin(k)));
	}
	return tmp;
}

Fortran

real(8) function code(t, l, k)
    real(8), intent (in) :: t
    real(8), intent (in) :: l
    real(8), intent (in) :: k
    real(8) :: tmp
    if (k <= 3.1d+145) then
        tmp = (((1.0d0 + ((k * k) * 0.16666666666666666d0)) / k) / (k / l)) / (t / ((2.0d0 / (k / l)) / k))
    else
        tmp = ((2.0d0 / k) * ((l / k) / t)) * (l / (k * sin(k)))
    end if
    code = tmp
end function

Java

public static double code(double t, double l, double k) {
	double tmp;
	if (k <= 3.1e+145) {
		tmp = (((1.0 + ((k * k) * 0.16666666666666666)) / k) / (k / l)) / (t / ((2.0 / (k / l)) / k));
	} else {
		tmp = ((2.0 / k) * ((l / k) / t)) * (l / (k * Math.sin(k)));
	}
	return tmp;
}

Python

def code(t, l, k):
	tmp = 0
	if k <= 3.1e+145:
		tmp = (((1.0 + ((k * k) * 0.16666666666666666)) / k) / (k / l)) / (t / ((2.0 / (k / l)) / k))
	else:
		tmp = ((2.0 / k) * ((l / k) / t)) * (l / (k * math.sin(k)))
	return tmp

Julia

function code(t, l, k)
	tmp = 0.0
	if (k <= 3.1e+145)
		tmp = Float64(Float64(Float64(Float64(1.0 + Float64(Float64(k * k) * 0.16666666666666666)) / k) / Float64(k / l)) / Float64(t / Float64(Float64(2.0 / Float64(k / l)) / k)));
	else
		tmp = Float64(Float64(Float64(2.0 / k) * Float64(Float64(l / k) / t)) * Float64(l / Float64(k * sin(k))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(t, l, k)
	tmp = 0.0;
	if (k <= 3.1e+145)
		tmp = (((1.0 + ((k * k) * 0.16666666666666666)) / k) / (k / l)) / (t / ((2.0 / (k / l)) / k));
	else
		tmp = ((2.0 / k) * ((l / k) / t)) * (l / (k * sin(k)));
	end
	tmp_2 = tmp;
end

Wolfram

code[t_, l_, k_] := If[LessEqual[k, 3.1e+145], N[(N[(N[(N[(1.0 + N[(N[(k * k), $MachinePrecision] * 0.16666666666666666), $MachinePrecision]), $MachinePrecision] / k), $MachinePrecision] / N[(k / l), $MachinePrecision]), $MachinePrecision] / N[(t / N[(N[(2.0 / N[(k / l), $MachinePrecision]), $MachinePrecision] / k), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(2.0 / k), $MachinePrecision] * N[(N[(l / k), $MachinePrecision] / t), $MachinePrecision]), $MachinePrecision] * N[(l / N[(k * N[Sin[k], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if k < 3.09999999999999988e145

   1. Initial program 37.5%

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

      1. *-commutative N/A

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

      2. associate-*r* N/A

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

      3. associate-/r* N/A

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

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

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

   3. Simplified 66.1%

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

      1. associate-/l/ N/A

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

      2. associate-/r/ N/A

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

      3. associate-/l* N/A

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

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

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

      5. *-inverses N/A

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

      6. associate-*l* N/A

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

      7. *-commutative N/A

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

      8. times-frac N/A

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

      9. *-rgt-identity N/A

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

      10. *-inverses N/A

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

      11. *-rgt-identity N/A

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

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

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

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

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

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

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

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

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

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

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

   6. Applied egg-rr 90.1%

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

   7. Taylor expanded in k around 0

      \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(2, \mathsf{*.f64}\left(k, k\right)\right), \mathsf{/.f64}\left(t, \ell\right)\right), \mathsf{/.f64}\left(\ell, \mathsf{*.f64}\left(\mathsf{sin.f64}\left(k\right), \color{blue}{k}\right)\right)\right) \]
   8. Step-by-step derivation

   9. Simplified 71.4%

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

   10. Taylor expanded in k around 0

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

       1. unpow2 N/A

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

       2. associate-/r* N/A

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

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

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

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

          \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(2, \mathsf{*.f64}\left(k, k\right)\right), \mathsf{/.f64}\left(t, \ell\right)\right), \mathsf{/.f64}\left(\mathsf{/.f64}\left(\left(\ell + \frac{1}{6} \cdot \left({k}^{2} \cdot \ell\right)\right), k\right), k\right)\right) \]

       5. associate-*r* N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(2, \mathsf{*.f64}\left(k, k\right)\right), \mathsf{/.f64}\left(t, \ell\right)\right), \mathsf{/.f64}\left(\mathsf{/.f64}\left(\left(\ell + \left(\frac{1}{6} \cdot {k}^{2}\right) \cdot \ell\right), k\right), k\right)\right) \]

       6. distribute-rgt1-in N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(2, \mathsf{*.f64}\left(k, k\right)\right), \mathsf{/.f64}\left(t, \ell\right)\right), \mathsf{/.f64}\left(\mathsf{/.f64}\left(\left(\left(\frac{1}{6} \cdot {k}^{2} + 1\right) \cdot \ell\right), k\right), k\right)\right) \]

       7. +-commutative N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(2, \mathsf{*.f64}\left(k, k\right)\right), \mathsf{/.f64}\left(t, \ell\right)\right), \mathsf{/.f64}\left(\mathsf{/.f64}\left(\left(\left(1 + \frac{1}{6} \cdot {k}^{2}\right) \cdot \ell\right), k\right), k\right)\right) \]

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

          \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(2, \mathsf{*.f64}\left(k, k\right)\right), \mathsf{/.f64}\left(t, \ell\right)\right), \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\left(1 + \frac{1}{6} \cdot {k}^{2}\right), \ell\right), k\right), k\right)\right) \]

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

          \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(2, \mathsf{*.f64}\left(k, k\right)\right), \mathsf{/.f64}\left(t, \ell\right)\right), \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \left(\frac{1}{6} \cdot {k}^{2}\right)\right), \ell\right), k\right), k\right)\right) \]

       10. *-commutative N/A

           \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(2, \mathsf{*.f64}\left(k, k\right)\right), \mathsf{/.f64}\left(t, \ell\right)\right), \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \left({k}^{2} \cdot \frac{1}{6}\right)\right), \ell\right), k\right), k\right)\right) \]

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

           \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(2, \mathsf{*.f64}\left(k, k\right)\right), \mathsf{/.f64}\left(t, \ell\right)\right), \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\left({k}^{2}\right), \frac{1}{6}\right)\right), \ell\right), k\right), k\right)\right) \]

       12. unpow2 N/A

           \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(2, \mathsf{*.f64}\left(k, k\right)\right), \mathsf{/.f64}\left(t, \ell\right)\right), \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\left(k \cdot k\right), \frac{1}{6}\right)\right), \ell\right), k\right), k\right)\right) \]

       13. *-lowering-*.f64 65.3%

           \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(2, \mathsf{*.f64}\left(k, k\right)\right), \mathsf{/.f64}\left(t, \ell\right)\right), \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(k, k\right), \frac{1}{6}\right)\right), \ell\right), k\right), k\right)\right) \]

   12. Simplified 65.3%

       \[\leadsto \frac{\frac{2}{k \cdot k}}{\frac{t}{\ell}} \cdot \color{blue}{\frac{\frac{\left(1 + \left(k \cdot k\right) \cdot 0.16666666666666666\right) \cdot \ell}{k}}{k}} \]
   13. Step-by-step derivation

       1. *-commutative N/A

          \[\leadsto \frac{\frac{\left(1 + \left(k \cdot k\right) \cdot \frac{1}{6}\right) \cdot \ell}{k}}{k} \cdot \color{blue}{\frac{\frac{2}{k \cdot k}}{\frac{t}{\ell}}} \]

       2. div-inv N/A

          \[\leadsto \frac{\frac{\left(1 + \left(k \cdot k\right) \cdot \frac{1}{6}\right) \cdot \ell}{k}}{k} \cdot \left(\frac{2}{k \cdot k} \cdot \color{blue}{\frac{1}{\frac{t}{\ell}}}\right) \]

       3. clear-num N/A

          \[\leadsto \frac{\frac{\left(1 + \left(k \cdot k\right) \cdot \frac{1}{6}\right) \cdot \ell}{k}}{k} \cdot \left(\frac{2}{k \cdot k} \cdot \frac{\ell}{\color{blue}{t}}\right) \]

       4. frac-times N/A

          \[\leadsto \frac{\frac{\left(1 + \left(k \cdot k\right) \cdot \frac{1}{6}\right) \cdot \ell}{k}}{k} \cdot \frac{2 \cdot \ell}{\color{blue}{\left(k \cdot k\right) \cdot t}} \]

       5. associate-*r* N/A

          \[\leadsto \frac{\frac{\left(1 + \left(k \cdot k\right) \cdot \frac{1}{6}\right) \cdot \ell}{k}}{k} \cdot \frac{2 \cdot \ell}{k \cdot \color{blue}{\left(k \cdot t\right)}} \]

       6. associate-/r* N/A

          \[\leadsto \frac{\frac{\left(1 + \left(k \cdot k\right) \cdot \frac{1}{6}\right) \cdot \ell}{k}}{k} \cdot \frac{\frac{2 \cdot \ell}{k}}{\color{blue}{k \cdot t}} \]

       7. clear-num N/A

          \[\leadsto \frac{\frac{\left(1 + \left(k \cdot k\right) \cdot \frac{1}{6}\right) \cdot \ell}{k}}{k} \cdot \frac{1}{\color{blue}{\frac{k \cdot t}{\frac{2 \cdot \ell}{k}}}} \]

       8. un-div-inv N/A

          \[\leadsto \frac{\frac{\frac{\left(1 + \left(k \cdot k\right) \cdot \frac{1}{6}\right) \cdot \ell}{k}}{k}}{\color{blue}{\frac{k \cdot t}{\frac{2 \cdot \ell}{k}}}} \]

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

          \[\leadsto \mathsf{/.f64}\left(\left(\frac{\frac{\left(1 + \left(k \cdot k\right) \cdot \frac{1}{6}\right) \cdot \ell}{k}}{k}\right), \color{blue}{\left(\frac{k \cdot t}{\frac{2 \cdot \ell}{k}}\right)}\right) \]

   14. Applied egg-rr 70.4%

       \[\leadsto \color{blue}{\frac{\frac{\frac{1 + \left(k \cdot k\right) \cdot 0.16666666666666666}{k}}{\frac{k}{\ell}}}{\frac{t}{\frac{\frac{2}{\frac{k}{\ell}}}{k}}}} \]

   if 3.09999999999999988e145 < k

   1. Initial program 32.4%

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

      1. *-commutative N/A

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

      2. associate-*r* N/A

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

      3. associate-/r* N/A

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

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

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

   3. Simplified 65.3%

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

      1. associate-/l/ N/A

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

      2. associate-/r/ N/A

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

      3. associate-/l* N/A

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

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

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

      5. *-inverses N/A

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

      6. associate-*l* N/A

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

      7. *-commutative N/A

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

      8. times-frac N/A

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

      9. *-rgt-identity N/A

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

      10. *-inverses N/A

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

      11. *-rgt-identity N/A

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

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

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

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

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

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

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

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

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

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

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

   6. Applied egg-rr 69.2%

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

   7. Taylor expanded in k around 0

      \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(2, \mathsf{*.f64}\left(k, k\right)\right), \mathsf{/.f64}\left(t, \ell\right)\right), \mathsf{/.f64}\left(\ell, \mathsf{*.f64}\left(\mathsf{sin.f64}\left(k\right), \color{blue}{k}\right)\right)\right) \]
   8. Step-by-step derivation

   9. Simplified 69.2%

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

       1. associate-/r/ N/A

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

       2. associate-*l/ N/A

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

       3. associate-*l/ N/A

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

       4. associate-/l/ N/A

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

       5. associate-*r/ N/A

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

       6. associate-/l/ N/A

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

       7. *-commutative N/A

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

       8. times-frac N/A

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

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

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

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

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

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

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

       12. /-lowering-/.f64 69.4%

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

   11. Applied egg-rr 69.4%

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

4. Final simplification 70.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;k \leq 3.1 \cdot 10^{+145}:\\ \;\;\;\;\frac{\frac{\frac{1 + \left(k \cdot k\right) \cdot 0.16666666666666666}{k}}{\frac{k}{\ell}}}{\frac{t}{\frac{\frac{2}{\frac{k}{\ell}}}{k}}}\\ \mathbf{else}:\\ \;\;\;\;\left(\frac{2}{k} \cdot \frac{\frac{\ell}{k}}{t}\right) \cdot \frac{\ell}{k \cdot \sin k}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 76.0% accurate, 12.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{\frac{2}{k \cdot k}}{\frac{t}{\ell}}\\ \mathbf{if}\;k \leq 2.3 \cdot 10^{-99}:\\ \;\;\;\;\frac{\frac{\frac{2}{\frac{k}{\ell}}}{k}}{\frac{k \cdot t}{\frac{\ell}{k}}}\\ \mathbf{elif}\;k \leq 2.7 \cdot 10^{+88}:\\ \;\;\;\;t\_1 \cdot \frac{\ell \cdot \left(1 + \left(k \cdot k\right) \cdot -0.16666666666666666\right)}{k \cdot k}\\ \mathbf{else}:\\ \;\;\;\;t\_1 \cdot \left(\ell \cdot 0.16666666666666666\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (t l k)
 :precision binary64
 (let* ((t_1 (/ (/ 2.0 (* k k)) (/ t l))))
   (if (<= k 2.3e-99)
     (/ (/ (/ 2.0 (/ k l)) k) (/ (* k t) (/ l k)))
     (if (<= k 2.7e+88)
       (* t_1 (/ (* l (+ 1.0 (* (* k k) -0.16666666666666666))) (* k k)))
       (* t_1 (* l 0.16666666666666666))))))

C

double code(double t, double l, double k) {
	double t_1 = (2.0 / (k * k)) / (t / l);
	double tmp;
	if (k <= 2.3e-99) {
		tmp = ((2.0 / (k / l)) / k) / ((k * t) / (l / k));
	} else if (k <= 2.7e+88) {
		tmp = t_1 * ((l * (1.0 + ((k * k) * -0.16666666666666666))) / (k * k));
	} else {
		tmp = t_1 * (l * 0.16666666666666666);
	}
	return tmp;
}

Fortran

real(8) function code(t, l, k)
    real(8), intent (in) :: t
    real(8), intent (in) :: l
    real(8), intent (in) :: k
    real(8) :: t_1
    real(8) :: tmp
    t_1 = (2.0d0 / (k * k)) / (t / l)
    if (k <= 2.3d-99) then
        tmp = ((2.0d0 / (k / l)) / k) / ((k * t) / (l / k))
    else if (k <= 2.7d+88) then
        tmp = t_1 * ((l * (1.0d0 + ((k * k) * (-0.16666666666666666d0)))) / (k * k))
    else
        tmp = t_1 * (l * 0.16666666666666666d0)
    end if
    code = tmp
end function

Java

public static double code(double t, double l, double k) {
	double t_1 = (2.0 / (k * k)) / (t / l);
	double tmp;
	if (k <= 2.3e-99) {
		tmp = ((2.0 / (k / l)) / k) / ((k * t) / (l / k));
	} else if (k <= 2.7e+88) {
		tmp = t_1 * ((l * (1.0 + ((k * k) * -0.16666666666666666))) / (k * k));
	} else {
		tmp = t_1 * (l * 0.16666666666666666);
	}
	return tmp;
}

Python

def code(t, l, k):
	t_1 = (2.0 / (k * k)) / (t / l)
	tmp = 0
	if k <= 2.3e-99:
		tmp = ((2.0 / (k / l)) / k) / ((k * t) / (l / k))
	elif k <= 2.7e+88:
		tmp = t_1 * ((l * (1.0 + ((k * k) * -0.16666666666666666))) / (k * k))
	else:
		tmp = t_1 * (l * 0.16666666666666666)
	return tmp

Julia

function code(t, l, k)
	t_1 = Float64(Float64(2.0 / Float64(k * k)) / Float64(t / l))
	tmp = 0.0
	if (k <= 2.3e-99)
		tmp = Float64(Float64(Float64(2.0 / Float64(k / l)) / k) / Float64(Float64(k * t) / Float64(l / k)));
	elseif (k <= 2.7e+88)
		tmp = Float64(t_1 * Float64(Float64(l * Float64(1.0 + Float64(Float64(k * k) * -0.16666666666666666))) / Float64(k * k)));
	else
		tmp = Float64(t_1 * Float64(l * 0.16666666666666666));
	end
	return tmp
end

MATLAB

function tmp_2 = code(t, l, k)
	t_1 = (2.0 / (k * k)) / (t / l);
	tmp = 0.0;
	if (k <= 2.3e-99)
		tmp = ((2.0 / (k / l)) / k) / ((k * t) / (l / k));
	elseif (k <= 2.7e+88)
		tmp = t_1 * ((l * (1.0 + ((k * k) * -0.16666666666666666))) / (k * k));
	else
		tmp = t_1 * (l * 0.16666666666666666);
	end
	tmp_2 = tmp;
end

Wolfram

code[t_, l_, k_] := Block[{t$95$1 = N[(N[(2.0 / N[(k * k), $MachinePrecision]), $MachinePrecision] / N[(t / l), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[k, 2.3e-99], N[(N[(N[(2.0 / N[(k / l), $MachinePrecision]), $MachinePrecision] / k), $MachinePrecision] / N[(N[(k * t), $MachinePrecision] / N[(l / k), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[k, 2.7e+88], N[(t$95$1 * N[(N[(l * N[(1.0 + N[(N[(k * k), $MachinePrecision] * -0.16666666666666666), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(k * k), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(t$95$1 * N[(l * 0.16666666666666666), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if k < 2.2999999999999998e-99

   1. Initial program 41.0%

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

   3. Taylor expanded in k around 0

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

      1. /-lowering-/.f64 N/A

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

      2. *-commutative N/A

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

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

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

      4. metadata-eval N/A

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

      5. pow-sqr N/A

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

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

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

      7. unpow2 N/A

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

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

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

      9. unpow2 N/A

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

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

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

      11. unpow2 N/A

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

      12. *-lowering-*.f64 63.9%

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

   5. Simplified 63.9%

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

      1. associate-/r/ N/A

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

      2. associate-*r* N/A

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

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

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

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

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

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

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

      6. *-commutative N/A

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

      7. associate-*r* N/A

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

      8. *-commutative N/A

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

      9. associate-*l* N/A

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

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

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

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

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

      12. associate-*r* N/A

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

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

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

      14. *-lowering-*.f64 73.3%

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

   7. Applied egg-rr 73.3%

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

      1. associate-*l* N/A

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

      2. associate-*l* N/A

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

      3. *-commutative N/A

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

      4. *-commutative N/A

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

      5. associate-/r/ N/A

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

      6. times-frac N/A

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

      7. associate-/r* N/A

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

      8. div-inv N/A

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

      9. clear-num N/A

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

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

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

   9. Applied egg-rr 74.1%

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

       1. associate-*l/ N/A

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

       2. associate-/l* N/A

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

       3. associate-/l* N/A

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

       4. associate-/r* N/A

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

       5. *-lft-identity N/A

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

       6. associate-*l/ N/A

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

       7. clear-num N/A

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

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

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

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

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

       10. associate-/l* N/A

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

       11. clear-num N/A

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

       12. un-div-inv N/A

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

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

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

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

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

       15. associate-*l/ N/A

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

       16. *-commutative N/A

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

       17. associate-*r* N/A

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

       18. *-commutative N/A

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

       19. associate-/l* N/A

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

       20. clear-num N/A

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

       21. un-div-inv N/A

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

       22. /-lowering-/.f64 N/A

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

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

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

       24. /-lowering-/.f64 81.2%

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

   11. Applied egg-rr 81.2%

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

   if 2.2999999999999998e-99 < k < 2.70000000000000016e88

   1. Initial program 27.9%

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

      1. *-commutative N/A

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

      2. associate-*r* N/A

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

      3. associate-/r* N/A

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

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

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

   3. Simplified 67.0%

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

      1. associate-/l/ N/A

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

      2. associate-/r/ N/A

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

      3. associate-/l* N/A

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

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

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

      5. *-inverses N/A

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

      6. associate-*l* N/A

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

      7. *-commutative N/A

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

      8. times-frac N/A

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

      9. *-rgt-identity N/A

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

      10. *-inverses N/A

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

      11. *-rgt-identity N/A

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

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

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

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

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

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

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

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

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

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

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

   6. Applied egg-rr 97.5%

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

   7. Taylor expanded in k around 0

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

      1. /-lowering-/.f64 N/A

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

      2. associate-*r* N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(2, \mathsf{*.f64}\left(k, k\right)\right), \mathsf{/.f64}\left(t, \ell\right)\right), \mathsf{/.f64}\left(\left(\ell + \left(\frac{-1}{6} \cdot {k}^{2}\right) \cdot \ell\right), \left({k}^{2}\right)\right)\right) \]

      3. distribute-rgt1-in N/A

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

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

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

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

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(2, \mathsf{*.f64}\left(k, k\right)\right), \mathsf{/.f64}\left(t, \ell\right)\right), \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(\left(\frac{-1}{6} \cdot {k}^{2}\right), 1\right), \ell\right), \left({k}^{2}\right)\right)\right) \]

      6. *-commutative N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(2, \mathsf{*.f64}\left(k, k\right)\right), \mathsf{/.f64}\left(t, \ell\right)\right), \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(\left({k}^{2} \cdot \frac{-1}{6}\right), 1\right), \ell\right), \left({k}^{2}\right)\right)\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(2, \mathsf{*.f64}\left(k, k\right)\right), \mathsf{/.f64}\left(t, \ell\right)\right), \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(\left({k}^{2}\right), \frac{-1}{6}\right), 1\right), \ell\right), \left({k}^{2}\right)\right)\right) \]

      8. unpow2 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(2, \mathsf{*.f64}\left(k, k\right)\right), \mathsf{/.f64}\left(t, \ell\right)\right), \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(\left(k \cdot k\right), \frac{-1}{6}\right), 1\right), \ell\right), \left({k}^{2}\right)\right)\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(2, \mathsf{*.f64}\left(k, k\right)\right), \mathsf{/.f64}\left(t, \ell\right)\right), \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(k, k\right), \frac{-1}{6}\right), 1\right), \ell\right), \left({k}^{2}\right)\right)\right) \]

      10. unpow2 N/A

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

      11. *-lowering-*.f64 69.4%

          \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(2, \mathsf{*.f64}\left(k, k\right)\right), \mathsf{/.f64}\left(t, \ell\right)\right), \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(k, k\right), \frac{-1}{6}\right), 1\right), \ell\right), \mathsf{*.f64}\left(k, \color{blue}{k}\right)\right)\right) \]

   9. Simplified 69.4%

      \[\leadsto \frac{\frac{2}{k \cdot k}}{\frac{t}{\ell}} \cdot \color{blue}{\frac{\left(\left(k \cdot k\right) \cdot -0.16666666666666666 + 1\right) \cdot \ell}{k \cdot k}} \]

   if 2.70000000000000016e88 < k

   1. Initial program 30.9%

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

      1. *-commutative N/A

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

      2. associate-*r* N/A

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

      3. associate-/r* N/A

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

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

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

   3. Simplified 63.3%

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

      1. associate-/l/ N/A

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

      2. associate-/r/ N/A

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

      3. associate-/l* N/A

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

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

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

      5. *-inverses N/A

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

      6. associate-*l* N/A

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

      7. *-commutative N/A

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

      8. times-frac N/A

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

      9. *-rgt-identity N/A

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

      10. *-inverses N/A

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

      11. *-rgt-identity N/A

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

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

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

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

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

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

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

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

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

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

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

   6. Applied egg-rr 78.5%

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

   7. Taylor expanded in k around 0

      \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(2, \mathsf{*.f64}\left(k, k\right)\right), \mathsf{/.f64}\left(t, \ell\right)\right), \mathsf{/.f64}\left(\ell, \mathsf{*.f64}\left(\mathsf{sin.f64}\left(k\right), \color{blue}{k}\right)\right)\right) \]
   8. Step-by-step derivation

   9. Simplified 64.3%

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

   10. Taylor expanded in k around 0

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

       1. unpow2 N/A

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

       2. associate-/r* N/A

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

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

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

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

          \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(2, \mathsf{*.f64}\left(k, k\right)\right), \mathsf{/.f64}\left(t, \ell\right)\right), \mathsf{/.f64}\left(\mathsf{/.f64}\left(\left(\ell + \frac{1}{6} \cdot \left({k}^{2} \cdot \ell\right)\right), k\right), k\right)\right) \]

       5. associate-*r* N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(2, \mathsf{*.f64}\left(k, k\right)\right), \mathsf{/.f64}\left(t, \ell\right)\right), \mathsf{/.f64}\left(\mathsf{/.f64}\left(\left(\ell + \left(\frac{1}{6} \cdot {k}^{2}\right) \cdot \ell\right), k\right), k\right)\right) \]

       6. distribute-rgt1-in N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(2, \mathsf{*.f64}\left(k, k\right)\right), \mathsf{/.f64}\left(t, \ell\right)\right), \mathsf{/.f64}\left(\mathsf{/.f64}\left(\left(\left(\frac{1}{6} \cdot {k}^{2} + 1\right) \cdot \ell\right), k\right), k\right)\right) \]

       7. +-commutative N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(2, \mathsf{*.f64}\left(k, k\right)\right), \mathsf{/.f64}\left(t, \ell\right)\right), \mathsf{/.f64}\left(\mathsf{/.f64}\left(\left(\left(1 + \frac{1}{6} \cdot {k}^{2}\right) \cdot \ell\right), k\right), k\right)\right) \]

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

          \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(2, \mathsf{*.f64}\left(k, k\right)\right), \mathsf{/.f64}\left(t, \ell\right)\right), \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\left(1 + \frac{1}{6} \cdot {k}^{2}\right), \ell\right), k\right), k\right)\right) \]

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

          \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(2, \mathsf{*.f64}\left(k, k\right)\right), \mathsf{/.f64}\left(t, \ell\right)\right), \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \left(\frac{1}{6} \cdot {k}^{2}\right)\right), \ell\right), k\right), k\right)\right) \]

       10. *-commutative N/A

           \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(2, \mathsf{*.f64}\left(k, k\right)\right), \mathsf{/.f64}\left(t, \ell\right)\right), \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \left({k}^{2} \cdot \frac{1}{6}\right)\right), \ell\right), k\right), k\right)\right) \]

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

           \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(2, \mathsf{*.f64}\left(k, k\right)\right), \mathsf{/.f64}\left(t, \ell\right)\right), \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\left({k}^{2}\right), \frac{1}{6}\right)\right), \ell\right), k\right), k\right)\right) \]

       12. unpow2 N/A

           \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(2, \mathsf{*.f64}\left(k, k\right)\right), \mathsf{/.f64}\left(t, \ell\right)\right), \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\left(k \cdot k\right), \frac{1}{6}\right)\right), \ell\right), k\right), k\right)\right) \]

       13. *-lowering-*.f64 23.1%

           \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(2, \mathsf{*.f64}\left(k, k\right)\right), \mathsf{/.f64}\left(t, \ell\right)\right), \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(k, k\right), \frac{1}{6}\right)\right), \ell\right), k\right), k\right)\right) \]

   12. Simplified 23.1%

       \[\leadsto \frac{\frac{2}{k \cdot k}}{\frac{t}{\ell}} \cdot \color{blue}{\frac{\frac{\left(1 + \left(k \cdot k\right) \cdot 0.16666666666666666\right) \cdot \ell}{k}}{k}} \]

   13. Taylor expanded in k around inf

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

       1. *-commutative N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(2, \mathsf{*.f64}\left(k, k\right)\right), \mathsf{/.f64}\left(t, \ell\right)\right), \left(\ell \cdot \color{blue}{\frac{1}{6}}\right)\right) \]

       2. *-lowering-*.f64 65.7%

          \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(2, \mathsf{*.f64}\left(k, k\right)\right), \mathsf{/.f64}\left(t, \ell\right)\right), \mathsf{*.f64}\left(\ell, \color{blue}{\frac{1}{6}}\right)\right) \]

   15. Simplified 65.7%

       \[\leadsto \frac{\frac{2}{k \cdot k}}{\frac{t}{\ell}} \cdot \color{blue}{\left(\ell \cdot 0.16666666666666666\right)} \]
3. Recombined 3 regimes into one program.

4. Final simplification 76.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;k \leq 2.3 \cdot 10^{-99}:\\ \;\;\;\;\frac{\frac{\frac{2}{\frac{k}{\ell}}}{k}}{\frac{k \cdot t}{\frac{\ell}{k}}}\\ \mathbf{elif}\;k \leq 2.7 \cdot 10^{+88}:\\ \;\;\;\;\frac{\frac{2}{k \cdot k}}{\frac{t}{\ell}} \cdot \frac{\ell \cdot \left(1 + \left(k \cdot k\right) \cdot -0.16666666666666666\right)}{k \cdot k}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{2}{k \cdot k}}{\frac{t}{\ell}} \cdot \left(\ell \cdot 0.16666666666666666\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 69.7% accurate, 15.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;k \leq 1.75 \cdot 10^{+130}:\\ \;\;\;\;\frac{\frac{\frac{1 + \left(k \cdot k\right) \cdot 0.16666666666666666}{k}}{\frac{k}{\ell}}}{\frac{t}{\frac{\frac{2}{\frac{k}{\ell}}}{k}}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{2}{k \cdot k}}{\frac{t}{\ell}} \cdot \left(\ell \cdot 0.16666666666666666\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (t l k)
 :precision binary64
 (if (<= k 1.75e+130)
   (/
    (/ (/ (+ 1.0 (* (* k k) 0.16666666666666666)) k) (/ k l))
    (/ t (/ (/ 2.0 (/ k l)) k)))
   (* (/ (/ 2.0 (* k k)) (/ t l)) (* l 0.16666666666666666))))

C

double code(double t, double l, double k) {
	double tmp;
	if (k <= 1.75e+130) {
		tmp = (((1.0 + ((k * k) * 0.16666666666666666)) / k) / (k / l)) / (t / ((2.0 / (k / l)) / k));
	} else {
		tmp = ((2.0 / (k * k)) / (t / l)) * (l * 0.16666666666666666);
	}
	return tmp;
}

Fortran

real(8) function code(t, l, k)
    real(8), intent (in) :: t
    real(8), intent (in) :: l
    real(8), intent (in) :: k
    real(8) :: tmp
    if (k <= 1.75d+130) then
        tmp = (((1.0d0 + ((k * k) * 0.16666666666666666d0)) / k) / (k / l)) / (t / ((2.0d0 / (k / l)) / k))
    else
        tmp = ((2.0d0 / (k * k)) / (t / l)) * (l * 0.16666666666666666d0)
    end if
    code = tmp
end function

Java

public static double code(double t, double l, double k) {
	double tmp;
	if (k <= 1.75e+130) {
		tmp = (((1.0 + ((k * k) * 0.16666666666666666)) / k) / (k / l)) / (t / ((2.0 / (k / l)) / k));
	} else {
		tmp = ((2.0 / (k * k)) / (t / l)) * (l * 0.16666666666666666);
	}
	return tmp;
}

Python

def code(t, l, k):
	tmp = 0
	if k <= 1.75e+130:
		tmp = (((1.0 + ((k * k) * 0.16666666666666666)) / k) / (k / l)) / (t / ((2.0 / (k / l)) / k))
	else:
		tmp = ((2.0 / (k * k)) / (t / l)) * (l * 0.16666666666666666)
	return tmp

Julia

function code(t, l, k)
	tmp = 0.0
	if (k <= 1.75e+130)
		tmp = Float64(Float64(Float64(Float64(1.0 + Float64(Float64(k * k) * 0.16666666666666666)) / k) / Float64(k / l)) / Float64(t / Float64(Float64(2.0 / Float64(k / l)) / k)));
	else
		tmp = Float64(Float64(Float64(2.0 / Float64(k * k)) / Float64(t / l)) * Float64(l * 0.16666666666666666));
	end
	return tmp
end

MATLAB

function tmp_2 = code(t, l, k)
	tmp = 0.0;
	if (k <= 1.75e+130)
		tmp = (((1.0 + ((k * k) * 0.16666666666666666)) / k) / (k / l)) / (t / ((2.0 / (k / l)) / k));
	else
		tmp = ((2.0 / (k * k)) / (t / l)) * (l * 0.16666666666666666);
	end
	tmp_2 = tmp;
end

Wolfram

code[t_, l_, k_] := If[LessEqual[k, 1.75e+130], N[(N[(N[(N[(1.0 + N[(N[(k * k), $MachinePrecision] * 0.16666666666666666), $MachinePrecision]), $MachinePrecision] / k), $MachinePrecision] / N[(k / l), $MachinePrecision]), $MachinePrecision] / N[(t / N[(N[(2.0 / N[(k / l), $MachinePrecision]), $MachinePrecision] / k), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(2.0 / N[(k * k), $MachinePrecision]), $MachinePrecision] / N[(t / l), $MachinePrecision]), $MachinePrecision] * N[(l * 0.16666666666666666), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if k < 1.75e130

   1. Initial program 37.4%

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

      1. *-commutative N/A

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

      2. associate-*r* N/A

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

      3. associate-/r* N/A

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

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

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

   3. Simplified 66.2%

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

      1. associate-/l/ N/A

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

      2. associate-/r/ N/A

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

      3. associate-/l* N/A

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

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

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

      5. *-inverses N/A

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

      6. associate-*l* N/A

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

      7. *-commutative N/A

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

      8. times-frac N/A

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

      9. *-rgt-identity N/A

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

      10. *-inverses N/A

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

      11. *-rgt-identity N/A

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

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

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

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

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

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

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

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

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

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

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

   6. Applied egg-rr 90.0%

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

   7. Taylor expanded in k around 0

      \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(2, \mathsf{*.f64}\left(k, k\right)\right), \mathsf{/.f64}\left(t, \ell\right)\right), \mathsf{/.f64}\left(\ell, \mathsf{*.f64}\left(\mathsf{sin.f64}\left(k\right), \color{blue}{k}\right)\right)\right) \]
   8. Step-by-step derivation

   9. Simplified 71.6%

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

   10. Taylor expanded in k around 0

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

       1. unpow2 N/A

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

       2. associate-/r* N/A

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

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

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

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

          \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(2, \mathsf{*.f64}\left(k, k\right)\right), \mathsf{/.f64}\left(t, \ell\right)\right), \mathsf{/.f64}\left(\mathsf{/.f64}\left(\left(\ell + \frac{1}{6} \cdot \left({k}^{2} \cdot \ell\right)\right), k\right), k\right)\right) \]

       5. associate-*r* N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(2, \mathsf{*.f64}\left(k, k\right)\right), \mathsf{/.f64}\left(t, \ell\right)\right), \mathsf{/.f64}\left(\mathsf{/.f64}\left(\left(\ell + \left(\frac{1}{6} \cdot {k}^{2}\right) \cdot \ell\right), k\right), k\right)\right) \]

       6. distribute-rgt1-in N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(2, \mathsf{*.f64}\left(k, k\right)\right), \mathsf{/.f64}\left(t, \ell\right)\right), \mathsf{/.f64}\left(\mathsf{/.f64}\left(\left(\left(\frac{1}{6} \cdot {k}^{2} + 1\right) \cdot \ell\right), k\right), k\right)\right) \]

       7. +-commutative N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(2, \mathsf{*.f64}\left(k, k\right)\right), \mathsf{/.f64}\left(t, \ell\right)\right), \mathsf{/.f64}\left(\mathsf{/.f64}\left(\left(\left(1 + \frac{1}{6} \cdot {k}^{2}\right) \cdot \ell\right), k\right), k\right)\right) \]

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

          \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(2, \mathsf{*.f64}\left(k, k\right)\right), \mathsf{/.f64}\left(t, \ell\right)\right), \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\left(1 + \frac{1}{6} \cdot {k}^{2}\right), \ell\right), k\right), k\right)\right) \]

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

          \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(2, \mathsf{*.f64}\left(k, k\right)\right), \mathsf{/.f64}\left(t, \ell\right)\right), \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \left(\frac{1}{6} \cdot {k}^{2}\right)\right), \ell\right), k\right), k\right)\right) \]

       10. *-commutative N/A

           \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(2, \mathsf{*.f64}\left(k, k\right)\right), \mathsf{/.f64}\left(t, \ell\right)\right), \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \left({k}^{2} \cdot \frac{1}{6}\right)\right), \ell\right), k\right), k\right)\right) \]

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

           \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(2, \mathsf{*.f64}\left(k, k\right)\right), \mathsf{/.f64}\left(t, \ell\right)\right), \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\left({k}^{2}\right), \frac{1}{6}\right)\right), \ell\right), k\right), k\right)\right) \]

       12. unpow2 N/A

           \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(2, \mathsf{*.f64}\left(k, k\right)\right), \mathsf{/.f64}\left(t, \ell\right)\right), \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\left(k \cdot k\right), \frac{1}{6}\right)\right), \ell\right), k\right), k\right)\right) \]

       13. *-lowering-*.f64 65.4%

           \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(2, \mathsf{*.f64}\left(k, k\right)\right), \mathsf{/.f64}\left(t, \ell\right)\right), \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(k, k\right), \frac{1}{6}\right)\right), \ell\right), k\right), k\right)\right) \]

   12. Simplified 65.4%

       \[\leadsto \frac{\frac{2}{k \cdot k}}{\frac{t}{\ell}} \cdot \color{blue}{\frac{\frac{\left(1 + \left(k \cdot k\right) \cdot 0.16666666666666666\right) \cdot \ell}{k}}{k}} \]
   13. Step-by-step derivation

       1. *-commutative N/A

          \[\leadsto \frac{\frac{\left(1 + \left(k \cdot k\right) \cdot \frac{1}{6}\right) \cdot \ell}{k}}{k} \cdot \color{blue}{\frac{\frac{2}{k \cdot k}}{\frac{t}{\ell}}} \]

       2. div-inv N/A

          \[\leadsto \frac{\frac{\left(1 + \left(k \cdot k\right) \cdot \frac{1}{6}\right) \cdot \ell}{k}}{k} \cdot \left(\frac{2}{k \cdot k} \cdot \color{blue}{\frac{1}{\frac{t}{\ell}}}\right) \]

       3. clear-num N/A

          \[\leadsto \frac{\frac{\left(1 + \left(k \cdot k\right) \cdot \frac{1}{6}\right) \cdot \ell}{k}}{k} \cdot \left(\frac{2}{k \cdot k} \cdot \frac{\ell}{\color{blue}{t}}\right) \]

       4. frac-times N/A

          \[\leadsto \frac{\frac{\left(1 + \left(k \cdot k\right) \cdot \frac{1}{6}\right) \cdot \ell}{k}}{k} \cdot \frac{2 \cdot \ell}{\color{blue}{\left(k \cdot k\right) \cdot t}} \]

       5. associate-*r* N/A

          \[\leadsto \frac{\frac{\left(1 + \left(k \cdot k\right) \cdot \frac{1}{6}\right) \cdot \ell}{k}}{k} \cdot \frac{2 \cdot \ell}{k \cdot \color{blue}{\left(k \cdot t\right)}} \]

       6. associate-/r* N/A

          \[\leadsto \frac{\frac{\left(1 + \left(k \cdot k\right) \cdot \frac{1}{6}\right) \cdot \ell}{k}}{k} \cdot \frac{\frac{2 \cdot \ell}{k}}{\color{blue}{k \cdot t}} \]

       7. clear-num N/A

          \[\leadsto \frac{\frac{\left(1 + \left(k \cdot k\right) \cdot \frac{1}{6}\right) \cdot \ell}{k}}{k} \cdot \frac{1}{\color{blue}{\frac{k \cdot t}{\frac{2 \cdot \ell}{k}}}} \]

       8. un-div-inv N/A

          \[\leadsto \frac{\frac{\frac{\left(1 + \left(k \cdot k\right) \cdot \frac{1}{6}\right) \cdot \ell}{k}}{k}}{\color{blue}{\frac{k \cdot t}{\frac{2 \cdot \ell}{k}}}} \]

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

          \[\leadsto \mathsf{/.f64}\left(\left(\frac{\frac{\left(1 + \left(k \cdot k\right) \cdot \frac{1}{6}\right) \cdot \ell}{k}}{k}\right), \color{blue}{\left(\frac{k \cdot t}{\frac{2 \cdot \ell}{k}}\right)}\right) \]

   14. Applied egg-rr 70.5%

       \[\leadsto \color{blue}{\frac{\frac{\frac{1 + \left(k \cdot k\right) \cdot 0.16666666666666666}{k}}{\frac{k}{\ell}}}{\frac{t}{\frac{\frac{2}{\frac{k}{\ell}}}{k}}}} \]

   if 1.75e130 < k

   1. Initial program 33.5%

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

      1. *-commutative N/A

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

      2. associate-*r* N/A

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

      3. associate-/r* N/A

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

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

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

   3. Simplified 64.6%

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

      1. associate-/l/ N/A

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

      2. associate-/r/ N/A

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

      3. associate-/l* N/A

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

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

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

      5. *-inverses N/A

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

      6. associate-*l* N/A

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

      7. *-commutative N/A

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

      8. times-frac N/A

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

      9. *-rgt-identity N/A

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

      10. *-inverses N/A

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

      11. *-rgt-identity N/A

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

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

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

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

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

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

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

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

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

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

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

   6. Applied egg-rr 71.0%

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

   7. Taylor expanded in k around 0

      \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(2, \mathsf{*.f64}\left(k, k\right)\right), \mathsf{/.f64}\left(t, \ell\right)\right), \mathsf{/.f64}\left(\ell, \mathsf{*.f64}\left(\mathsf{sin.f64}\left(k\right), \color{blue}{k}\right)\right)\right) \]
   8. Step-by-step derivation

   9. Simplified 68.2%

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

   10. Taylor expanded in k around 0

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

       1. unpow2 N/A

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

       2. associate-/r* N/A

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

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

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

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

          \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(2, \mathsf{*.f64}\left(k, k\right)\right), \mathsf{/.f64}\left(t, \ell\right)\right), \mathsf{/.f64}\left(\mathsf{/.f64}\left(\left(\ell + \frac{1}{6} \cdot \left({k}^{2} \cdot \ell\right)\right), k\right), k\right)\right) \]

       5. associate-*r* N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(2, \mathsf{*.f64}\left(k, k\right)\right), \mathsf{/.f64}\left(t, \ell\right)\right), \mathsf{/.f64}\left(\mathsf{/.f64}\left(\left(\ell + \left(\frac{1}{6} \cdot {k}^{2}\right) \cdot \ell\right), k\right), k\right)\right) \]

       6. distribute-rgt1-in N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(2, \mathsf{*.f64}\left(k, k\right)\right), \mathsf{/.f64}\left(t, \ell\right)\right), \mathsf{/.f64}\left(\mathsf{/.f64}\left(\left(\left(\frac{1}{6} \cdot {k}^{2} + 1\right) \cdot \ell\right), k\right), k\right)\right) \]

       7. +-commutative N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(2, \mathsf{*.f64}\left(k, k\right)\right), \mathsf{/.f64}\left(t, \ell\right)\right), \mathsf{/.f64}\left(\mathsf{/.f64}\left(\left(\left(1 + \frac{1}{6} \cdot {k}^{2}\right) \cdot \ell\right), k\right), k\right)\right) \]

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

          \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(2, \mathsf{*.f64}\left(k, k\right)\right), \mathsf{/.f64}\left(t, \ell\right)\right), \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\left(1 + \frac{1}{6} \cdot {k}^{2}\right), \ell\right), k\right), k\right)\right) \]

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

          \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(2, \mathsf{*.f64}\left(k, k\right)\right), \mathsf{/.f64}\left(t, \ell\right)\right), \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \left(\frac{1}{6} \cdot {k}^{2}\right)\right), \ell\right), k\right), k\right)\right) \]

       10. *-commutative N/A

           \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(2, \mathsf{*.f64}\left(k, k\right)\right), \mathsf{/.f64}\left(t, \ell\right)\right), \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \left({k}^{2} \cdot \frac{1}{6}\right)\right), \ell\right), k\right), k\right)\right) \]

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

           \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(2, \mathsf{*.f64}\left(k, k\right)\right), \mathsf{/.f64}\left(t, \ell\right)\right), \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\left({k}^{2}\right), \frac{1}{6}\right)\right), \ell\right), k\right), k\right)\right) \]

       12. unpow2 N/A

           \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(2, \mathsf{*.f64}\left(k, k\right)\right), \mathsf{/.f64}\left(t, \ell\right)\right), \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\left(k \cdot k\right), \frac{1}{6}\right)\right), \ell\right), k\right), k\right)\right) \]

       13. *-lowering-*.f64 11.3%

           \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(2, \mathsf{*.f64}\left(k, k\right)\right), \mathsf{/.f64}\left(t, \ell\right)\right), \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(k, k\right), \frac{1}{6}\right)\right), \ell\right), k\right), k\right)\right) \]

   12. Simplified 11.3%

       \[\leadsto \frac{\frac{2}{k \cdot k}}{\frac{t}{\ell}} \cdot \color{blue}{\frac{\frac{\left(1 + \left(k \cdot k\right) \cdot 0.16666666666666666\right) \cdot \ell}{k}}{k}} \]

   13. Taylor expanded in k around inf

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

       1. *-commutative N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(2, \mathsf{*.f64}\left(k, k\right)\right), \mathsf{/.f64}\left(t, \ell\right)\right), \left(\ell \cdot \color{blue}{\frac{1}{6}}\right)\right) \]

       2. *-lowering-*.f64 68.6%

          \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(2, \mathsf{*.f64}\left(k, k\right)\right), \mathsf{/.f64}\left(t, \ell\right)\right), \mathsf{*.f64}\left(\ell, \color{blue}{\frac{1}{6}}\right)\right) \]

   15. Simplified 68.6%

       \[\leadsto \frac{\frac{2}{k \cdot k}}{\frac{t}{\ell}} \cdot \color{blue}{\left(\ell \cdot 0.16666666666666666\right)} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 7: 67.8% accurate, 15.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;k \leq 2.7 \cdot 10^{+88}:\\ \;\;\;\;\frac{\frac{2 \cdot \frac{\ell}{k}}{k}}{t} \cdot \frac{\ell \cdot \left(1 + \left(k \cdot k\right) \cdot -0.16666666666666666\right)}{k \cdot k}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{2}{k \cdot k}}{\frac{t}{\ell}} \cdot \left(\ell \cdot 0.16666666666666666\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (t l k)
 :precision binary64
 (if (<= k 2.7e+88)
   (*
    (/ (/ (* 2.0 (/ l k)) k) t)
    (/ (* l (+ 1.0 (* (* k k) -0.16666666666666666))) (* k k)))
   (* (/ (/ 2.0 (* k k)) (/ t l)) (* l 0.16666666666666666))))

C

double code(double t, double l, double k) {
	double tmp;
	if (k <= 2.7e+88) {
		tmp = (((2.0 * (l / k)) / k) / t) * ((l * (1.0 + ((k * k) * -0.16666666666666666))) / (k * k));
	} else {
		tmp = ((2.0 / (k * k)) / (t / l)) * (l * 0.16666666666666666);
	}
	return tmp;
}

Fortran

real(8) function code(t, l, k)
    real(8), intent (in) :: t
    real(8), intent (in) :: l
    real(8), intent (in) :: k
    real(8) :: tmp
    if (k <= 2.7d+88) then
        tmp = (((2.0d0 * (l / k)) / k) / t) * ((l * (1.0d0 + ((k * k) * (-0.16666666666666666d0)))) / (k * k))
    else
        tmp = ((2.0d0 / (k * k)) / (t / l)) * (l * 0.16666666666666666d0)
    end if
    code = tmp
end function

Java

public static double code(double t, double l, double k) {
	double tmp;
	if (k <= 2.7e+88) {
		tmp = (((2.0 * (l / k)) / k) / t) * ((l * (1.0 + ((k * k) * -0.16666666666666666))) / (k * k));
	} else {
		tmp = ((2.0 / (k * k)) / (t / l)) * (l * 0.16666666666666666);
	}
	return tmp;
}

Python

def code(t, l, k):
	tmp = 0
	if k <= 2.7e+88:
		tmp = (((2.0 * (l / k)) / k) / t) * ((l * (1.0 + ((k * k) * -0.16666666666666666))) / (k * k))
	else:
		tmp = ((2.0 / (k * k)) / (t / l)) * (l * 0.16666666666666666)
	return tmp

Julia

function code(t, l, k)
	tmp = 0.0
	if (k <= 2.7e+88)
		tmp = Float64(Float64(Float64(Float64(2.0 * Float64(l / k)) / k) / t) * Float64(Float64(l * Float64(1.0 + Float64(Float64(k * k) * -0.16666666666666666))) / Float64(k * k)));
	else
		tmp = Float64(Float64(Float64(2.0 / Float64(k * k)) / Float64(t / l)) * Float64(l * 0.16666666666666666));
	end
	return tmp
end

MATLAB

function tmp_2 = code(t, l, k)
	tmp = 0.0;
	if (k <= 2.7e+88)
		tmp = (((2.0 * (l / k)) / k) / t) * ((l * (1.0 + ((k * k) * -0.16666666666666666))) / (k * k));
	else
		tmp = ((2.0 / (k * k)) / (t / l)) * (l * 0.16666666666666666);
	end
	tmp_2 = tmp;
end

Wolfram

code[t_, l_, k_] := If[LessEqual[k, 2.7e+88], N[(N[(N[(N[(2.0 * N[(l / k), $MachinePrecision]), $MachinePrecision] / k), $MachinePrecision] / t), $MachinePrecision] * N[(N[(l * N[(1.0 + N[(N[(k * k), $MachinePrecision] * -0.16666666666666666), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(k * k), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(2.0 / N[(k * k), $MachinePrecision]), $MachinePrecision] / N[(t / l), $MachinePrecision]), $MachinePrecision] * N[(l * 0.16666666666666666), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if k < 2.70000000000000016e88

   1. Initial program 38.2%

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

      1. *-commutative N/A

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

      2. associate-*r* N/A

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

      3. associate-/r* N/A

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

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

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

   3. Simplified 66.7%

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

      1. associate-/l/ N/A

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

      2. associate-/r/ N/A

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

      3. associate-/l* N/A

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

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

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

      5. *-inverses N/A

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

      6. associate-*l* N/A

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

      7. *-commutative N/A

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

      8. times-frac N/A

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

      9. *-rgt-identity N/A

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

      10. *-inverses N/A

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

      11. *-rgt-identity N/A

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

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

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

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

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

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

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

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

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

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

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

   6. Applied egg-rr 89.4%

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

   7. Taylor expanded in k around 0

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

      1. associate-/r* N/A

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

      2. associate-*r/ N/A

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

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

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

      4. unpow2 N/A

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

      5. associate-/r* N/A

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

      6. associate-*r/ N/A

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

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

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

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

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

      9. /-lowering-/.f64 94.3%

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

   9. Simplified 94.3%

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

   10. Taylor expanded in k around 0

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

       1. /-lowering-/.f64 N/A

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

       2. associate-*r* N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(2, \mathsf{/.f64}\left(\ell, k\right)\right), k\right), t\right), \mathsf{/.f64}\left(\left(\ell + \left(\frac{-1}{6} \cdot {k}^{2}\right) \cdot \ell\right), \left({k}^{2}\right)\right)\right) \]

       3. distribute-rgt1-in N/A

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

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

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

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

          \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(2, \mathsf{/.f64}\left(\ell, k\right)\right), k\right), t\right), \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(\left(\frac{-1}{6} \cdot {k}^{2}\right), 1\right), \ell\right), \left({k}^{2}\right)\right)\right) \]

       6. *-commutative N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(2, \mathsf{/.f64}\left(\ell, k\right)\right), k\right), t\right), \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(\left({k}^{2} \cdot \frac{-1}{6}\right), 1\right), \ell\right), \left({k}^{2}\right)\right)\right) \]

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

          \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(2, \mathsf{/.f64}\left(\ell, k\right)\right), k\right), t\right), \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(\left({k}^{2}\right), \frac{-1}{6}\right), 1\right), \ell\right), \left({k}^{2}\right)\right)\right) \]

       8. unpow2 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(2, \mathsf{/.f64}\left(\ell, k\right)\right), k\right), t\right), \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(\left(k \cdot k\right), \frac{-1}{6}\right), 1\right), \ell\right), \left({k}^{2}\right)\right)\right) \]

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

          \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(2, \mathsf{/.f64}\left(\ell, k\right)\right), k\right), t\right), \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(k, k\right), \frac{-1}{6}\right), 1\right), \ell\right), \left({k}^{2}\right)\right)\right) \]

       10. unpow2 N/A

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

       11. *-lowering-*.f64 69.7%

           \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(2, \mathsf{/.f64}\left(\ell, k\right)\right), k\right), t\right), \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(k, k\right), \frac{-1}{6}\right), 1\right), \ell\right), \mathsf{*.f64}\left(k, \color{blue}{k}\right)\right)\right) \]

   12. Simplified 69.7%

       \[\leadsto \frac{\frac{2 \cdot \frac{\ell}{k}}{k}}{t} \cdot \color{blue}{\frac{\left(\left(k \cdot k\right) \cdot -0.16666666666666666 + 1\right) \cdot \ell}{k \cdot k}} \]

   if 2.70000000000000016e88 < k

   1. Initial program 30.9%

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

      1. *-commutative N/A

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

      2. associate-*r* N/A

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

      3. associate-/r* N/A

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

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

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

   3. Simplified 63.3%

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

      1. associate-/l/ N/A

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

      2. associate-/r/ N/A

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

      3. associate-/l* N/A

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

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

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

      5. *-inverses N/A

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

      6. associate-*l* N/A

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

      7. *-commutative N/A

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

      8. times-frac N/A

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

      9. *-rgt-identity N/A

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

      10. *-inverses N/A

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

      11. *-rgt-identity N/A

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

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

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

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

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

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

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

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

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

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

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

   6. Applied egg-rr 78.5%

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

   7. Taylor expanded in k around 0

      \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(2, \mathsf{*.f64}\left(k, k\right)\right), \mathsf{/.f64}\left(t, \ell\right)\right), \mathsf{/.f64}\left(\ell, \mathsf{*.f64}\left(\mathsf{sin.f64}\left(k\right), \color{blue}{k}\right)\right)\right) \]
   8. Step-by-step derivation

   9. Simplified 64.3%

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

   10. Taylor expanded in k around 0

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

       1. unpow2 N/A

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

       2. associate-/r* N/A

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

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

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

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

          \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(2, \mathsf{*.f64}\left(k, k\right)\right), \mathsf{/.f64}\left(t, \ell\right)\right), \mathsf{/.f64}\left(\mathsf{/.f64}\left(\left(\ell + \frac{1}{6} \cdot \left({k}^{2} \cdot \ell\right)\right), k\right), k\right)\right) \]

       5. associate-*r* N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(2, \mathsf{*.f64}\left(k, k\right)\right), \mathsf{/.f64}\left(t, \ell\right)\right), \mathsf{/.f64}\left(\mathsf{/.f64}\left(\left(\ell + \left(\frac{1}{6} \cdot {k}^{2}\right) \cdot \ell\right), k\right), k\right)\right) \]

       6. distribute-rgt1-in N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(2, \mathsf{*.f64}\left(k, k\right)\right), \mathsf{/.f64}\left(t, \ell\right)\right), \mathsf{/.f64}\left(\mathsf{/.f64}\left(\left(\left(\frac{1}{6} \cdot {k}^{2} + 1\right) \cdot \ell\right), k\right), k\right)\right) \]

       7. +-commutative N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(2, \mathsf{*.f64}\left(k, k\right)\right), \mathsf{/.f64}\left(t, \ell\right)\right), \mathsf{/.f64}\left(\mathsf{/.f64}\left(\left(\left(1 + \frac{1}{6} \cdot {k}^{2}\right) \cdot \ell\right), k\right), k\right)\right) \]

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

          \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(2, \mathsf{*.f64}\left(k, k\right)\right), \mathsf{/.f64}\left(t, \ell\right)\right), \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\left(1 + \frac{1}{6} \cdot {k}^{2}\right), \ell\right), k\right), k\right)\right) \]

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

          \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(2, \mathsf{*.f64}\left(k, k\right)\right), \mathsf{/.f64}\left(t, \ell\right)\right), \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \left(\frac{1}{6} \cdot {k}^{2}\right)\right), \ell\right), k\right), k\right)\right) \]

       10. *-commutative N/A

           \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(2, \mathsf{*.f64}\left(k, k\right)\right), \mathsf{/.f64}\left(t, \ell\right)\right), \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \left({k}^{2} \cdot \frac{1}{6}\right)\right), \ell\right), k\right), k\right)\right) \]

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

           \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(2, \mathsf{*.f64}\left(k, k\right)\right), \mathsf{/.f64}\left(t, \ell\right)\right), \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\left({k}^{2}\right), \frac{1}{6}\right)\right), \ell\right), k\right), k\right)\right) \]

       12. unpow2 N/A

           \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(2, \mathsf{*.f64}\left(k, k\right)\right), \mathsf{/.f64}\left(t, \ell\right)\right), \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\left(k \cdot k\right), \frac{1}{6}\right)\right), \ell\right), k\right), k\right)\right) \]

       13. *-lowering-*.f64 23.1%

           \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(2, \mathsf{*.f64}\left(k, k\right)\right), \mathsf{/.f64}\left(t, \ell\right)\right), \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(k, k\right), \frac{1}{6}\right)\right), \ell\right), k\right), k\right)\right) \]

   12. Simplified 23.1%

       \[\leadsto \frac{\frac{2}{k \cdot k}}{\frac{t}{\ell}} \cdot \color{blue}{\frac{\frac{\left(1 + \left(k \cdot k\right) \cdot 0.16666666666666666\right) \cdot \ell}{k}}{k}} \]

   13. Taylor expanded in k around inf

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

       1. *-commutative N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(2, \mathsf{*.f64}\left(k, k\right)\right), \mathsf{/.f64}\left(t, \ell\right)\right), \left(\ell \cdot \color{blue}{\frac{1}{6}}\right)\right) \]

       2. *-lowering-*.f64 65.7%

          \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(2, \mathsf{*.f64}\left(k, k\right)\right), \mathsf{/.f64}\left(t, \ell\right)\right), \mathsf{*.f64}\left(\ell, \color{blue}{\frac{1}{6}}\right)\right) \]

   15. Simplified 65.7%

       \[\leadsto \frac{\frac{2}{k \cdot k}}{\frac{t}{\ell}} \cdot \color{blue}{\left(\ell \cdot 0.16666666666666666\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 69.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;k \leq 2.7 \cdot 10^{+88}:\\ \;\;\;\;\frac{\frac{2 \cdot \frac{\ell}{k}}{k}}{t} \cdot \frac{\ell \cdot \left(1 + \left(k \cdot k\right) \cdot -0.16666666666666666\right)}{k \cdot k}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{2}{k \cdot k}}{\frac{t}{\ell}} \cdot \left(\ell \cdot 0.16666666666666666\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 75.5% accurate, 21.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;k \leq 1.3 \cdot 10^{-100}:\\ \;\;\;\;\frac{\frac{\frac{2}{\frac{k}{\ell}}}{k}}{\frac{k \cdot t}{\frac{\ell}{k}}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{2}{k \cdot k}}{\frac{t}{\ell}} \cdot \frac{\ell}{k \cdot k}\\ \end{array} \end{array} \]

FPCore

(FPCore (t l k)
 :precision binary64
 (if (<= k 1.3e-100)
   (/ (/ (/ 2.0 (/ k l)) k) (/ (* k t) (/ l k)))
   (* (/ (/ 2.0 (* k k)) (/ t l)) (/ l (* k k)))))

C

double code(double t, double l, double k) {
	double tmp;
	if (k <= 1.3e-100) {
		tmp = ((2.0 / (k / l)) / k) / ((k * t) / (l / k));
	} else {
		tmp = ((2.0 / (k * k)) / (t / l)) * (l / (k * k));
	}
	return tmp;
}

Fortran

real(8) function code(t, l, k)
    real(8), intent (in) :: t
    real(8), intent (in) :: l
    real(8), intent (in) :: k
    real(8) :: tmp
    if (k <= 1.3d-100) then
        tmp = ((2.0d0 / (k / l)) / k) / ((k * t) / (l / k))
    else
        tmp = ((2.0d0 / (k * k)) / (t / l)) * (l / (k * k))
    end if
    code = tmp
end function

Java

public static double code(double t, double l, double k) {
	double tmp;
	if (k <= 1.3e-100) {
		tmp = ((2.0 / (k / l)) / k) / ((k * t) / (l / k));
	} else {
		tmp = ((2.0 / (k * k)) / (t / l)) * (l / (k * k));
	}
	return tmp;
}

Python

def code(t, l, k):
	tmp = 0
	if k <= 1.3e-100:
		tmp = ((2.0 / (k / l)) / k) / ((k * t) / (l / k))
	else:
		tmp = ((2.0 / (k * k)) / (t / l)) * (l / (k * k))
	return tmp

Julia

function code(t, l, k)
	tmp = 0.0
	if (k <= 1.3e-100)
		tmp = Float64(Float64(Float64(2.0 / Float64(k / l)) / k) / Float64(Float64(k * t) / Float64(l / k)));
	else
		tmp = Float64(Float64(Float64(2.0 / Float64(k * k)) / Float64(t / l)) * Float64(l / Float64(k * k)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(t, l, k)
	tmp = 0.0;
	if (k <= 1.3e-100)
		tmp = ((2.0 / (k / l)) / k) / ((k * t) / (l / k));
	else
		tmp = ((2.0 / (k * k)) / (t / l)) * (l / (k * k));
	end
	tmp_2 = tmp;
end

Wolfram

code[t_, l_, k_] := If[LessEqual[k, 1.3e-100], N[(N[(N[(2.0 / N[(k / l), $MachinePrecision]), $MachinePrecision] / k), $MachinePrecision] / N[(N[(k * t), $MachinePrecision] / N[(l / k), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(2.0 / N[(k * k), $MachinePrecision]), $MachinePrecision] / N[(t / l), $MachinePrecision]), $MachinePrecision] * N[(l / N[(k * k), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if k < 1.2999999999999999e-100

   1. Initial program 41.0%

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

   3. Taylor expanded in k around 0

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

      1. /-lowering-/.f64 N/A

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

      2. *-commutative N/A

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

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

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

      4. metadata-eval N/A

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

      5. pow-sqr N/A

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

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

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

      7. unpow2 N/A

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

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

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

      9. unpow2 N/A

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

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

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

      11. unpow2 N/A

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

      12. *-lowering-*.f64 63.9%

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

   5. Simplified 63.9%

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

      1. associate-/r/ N/A

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

      2. associate-*r* N/A

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

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

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

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

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

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

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

      6. *-commutative N/A

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

      7. associate-*r* N/A

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

      8. *-commutative N/A

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

      9. associate-*l* N/A

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

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

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

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

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

      12. associate-*r* N/A

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

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

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

      14. *-lowering-*.f64 73.3%

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

   7. Applied egg-rr 73.3%

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

      1. associate-*l* N/A

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

      2. associate-*l* N/A

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

      3. *-commutative N/A

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

      4. *-commutative N/A

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

      5. associate-/r/ N/A

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

      6. times-frac N/A

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

      7. associate-/r* N/A

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

      8. div-inv N/A

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

      9. clear-num N/A

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

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

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

   9. Applied egg-rr 74.1%

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

       1. associate-*l/ N/A

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

       2. associate-/l* N/A

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

       3. associate-/l* N/A

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

       4. associate-/r* N/A

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

       5. *-lft-identity N/A

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

       6. associate-*l/ N/A

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

       7. clear-num N/A

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

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

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

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

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

       10. associate-/l* N/A

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

       11. clear-num N/A

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

       12. un-div-inv N/A

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

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

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

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

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

       15. associate-*l/ N/A

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

       16. *-commutative N/A

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

       17. associate-*r* N/A

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

       18. *-commutative N/A

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

       19. associate-/l* N/A

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

       20. clear-num N/A

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

       21. un-div-inv N/A

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

       22. /-lowering-/.f64 N/A

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

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

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

       24. /-lowering-/.f64 81.2%

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

   11. Applied egg-rr 81.2%

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

   if 1.2999999999999999e-100 < k

   1. Initial program 29.5%

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

      1. *-commutative N/A

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

      2. associate-*r* N/A

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

      3. associate-/r* N/A

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

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

         \[\leadsto \mathsf{/.f64}\left(\left(\frac{2}{\left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) - 1\right) \cdot \left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right)}\right), \color{blue}{\tan k}\right) \]

   3. Simplified 65.1%

      \[\leadsto \color{blue}{\frac{\frac{\frac{\left(\frac{2}{k \cdot k} \cdot t\right) \cdot \frac{t}{t}}{\frac{t \cdot \frac{t}{\ell}}{\ell}}}{\sin k}}{\tan k}} \]
   4. Add Preprocessing
   5. Step-by-step derivation

      1. associate-/l/ N/A

         \[\leadsto \frac{\frac{\left(\frac{2}{k \cdot k} \cdot t\right) \cdot \frac{t}{t}}{\frac{t \cdot \frac{t}{\ell}}{\ell}}}{\color{blue}{\tan k \cdot \sin k}} \]

      2. associate-/r/ N/A

         \[\leadsto \frac{\frac{\left(\frac{2}{k \cdot k} \cdot t\right) \cdot \frac{t}{t}}{t \cdot \frac{t}{\ell}} \cdot \ell}{\color{blue}{\tan k} \cdot \sin k} \]

      3. associate-/l* N/A

         \[\leadsto \frac{\left(\frac{2}{k \cdot k} \cdot t\right) \cdot \frac{t}{t}}{t \cdot \frac{t}{\ell}} \cdot \color{blue}{\frac{\ell}{\tan k \cdot \sin k}} \]

      4. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\left(\frac{\left(\frac{2}{k \cdot k} \cdot t\right) \cdot \frac{t}{t}}{t \cdot \frac{t}{\ell}}\right), \color{blue}{\left(\frac{\ell}{\tan k \cdot \sin k}\right)}\right) \]

      5. *-inverses N/A

         \[\leadsto \mathsf{*.f64}\left(\left(\frac{\left(\frac{2}{k \cdot k} \cdot t\right) \cdot 1}{t \cdot \frac{t}{\ell}}\right), \left(\frac{\ell}{\tan k \cdot \sin k}\right)\right) \]

      6. associate-*l* N/A

         \[\leadsto \mathsf{*.f64}\left(\left(\frac{\frac{2}{k \cdot k} \cdot \left(t \cdot 1\right)}{t \cdot \frac{t}{\ell}}\right), \left(\frac{\ell}{\tan k \cdot \sin k}\right)\right) \]

      7. *-commutative N/A

         \[\leadsto \mathsf{*.f64}\left(\left(\frac{\frac{2}{k \cdot k} \cdot \left(t \cdot 1\right)}{\frac{t}{\ell} \cdot t}\right), \left(\frac{\ell}{\tan k \cdot \sin k}\right)\right) \]

      8. times-frac N/A

         \[\leadsto \mathsf{*.f64}\left(\left(\frac{\frac{2}{k \cdot k}}{\frac{t}{\ell}} \cdot \frac{t \cdot 1}{t}\right), \left(\frac{\color{blue}{\ell}}{\tan k \cdot \sin k}\right)\right) \]

      9. *-rgt-identity N/A

         \[\leadsto \mathsf{*.f64}\left(\left(\frac{\frac{2}{k \cdot k}}{\frac{t}{\ell}} \cdot \frac{t}{t}\right), \left(\frac{\ell}{\tan k \cdot \sin k}\right)\right) \]

      10. *-inverses N/A

          \[\leadsto \mathsf{*.f64}\left(\left(\frac{\frac{2}{k \cdot k}}{\frac{t}{\ell}} \cdot 1\right), \left(\frac{\ell}{\tan k \cdot \sin k}\right)\right) \]

      11. *-rgt-identity N/A

          \[\leadsto \mathsf{*.f64}\left(\left(\frac{\frac{2}{k \cdot k}}{\frac{t}{\ell}}\right), \left(\frac{\color{blue}{\ell}}{\tan k \cdot \sin k}\right)\right) \]

      12. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\left(\frac{2}{k \cdot k}\right), \left(\frac{t}{\ell}\right)\right), \left(\frac{\color{blue}{\ell}}{\tan k \cdot \sin k}\right)\right) \]

      13. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(2, \left(k \cdot k\right)\right), \left(\frac{t}{\ell}\right)\right), \left(\frac{\ell}{\tan k \cdot \sin k}\right)\right) \]

      14. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(2, \mathsf{*.f64}\left(k, k\right)\right), \left(\frac{t}{\ell}\right)\right), \left(\frac{\ell}{\tan k \cdot \sin k}\right)\right) \]

      15. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(2, \mathsf{*.f64}\left(k, k\right)\right), \mathsf{/.f64}\left(t, \ell\right)\right), \left(\frac{\ell}{\tan k \cdot \sin k}\right)\right) \]

      16. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(2, \mathsf{*.f64}\left(k, k\right)\right), \mathsf{/.f64}\left(t, \ell\right)\right), \mathsf{/.f64}\left(\ell, \color{blue}{\left(\tan k \cdot \sin k\right)}\right)\right) \]

   6. Applied egg-rr 87.5%

      \[\leadsto \color{blue}{\frac{\frac{2}{k \cdot k}}{\frac{t}{\ell}} \cdot \frac{\ell}{\sin k \cdot \tan k}} \]

   7. Taylor expanded in k around 0

      \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(2, \mathsf{*.f64}\left(k, k\right)\right), \mathsf{/.f64}\left(t, \ell\right)\right), \color{blue}{\left(\frac{\ell}{{k}^{2}}\right)}\right) \]
   8. Step-by-step derivation

      1. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(2, \mathsf{*.f64}\left(k, k\right)\right), \mathsf{/.f64}\left(t, \ell\right)\right), \mathsf{/.f64}\left(\ell, \color{blue}{\left({k}^{2}\right)}\right)\right) \]

      2. unpow2 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(2, \mathsf{*.f64}\left(k, k\right)\right), \mathsf{/.f64}\left(t, \ell\right)\right), \mathsf{/.f64}\left(\ell, \left(k \cdot \color{blue}{k}\right)\right)\right) \]

      3. *-lowering-*.f64 61.4%

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(2, \mathsf{*.f64}\left(k, k\right)\right), \mathsf{/.f64}\left(t, \ell\right)\right), \mathsf{/.f64}\left(\ell, \mathsf{*.f64}\left(k, \color{blue}{k}\right)\right)\right) \]

   9. Simplified 61.4%

      \[\leadsto \frac{\frac{2}{k \cdot k}}{\frac{t}{\ell}} \cdot \color{blue}{\frac{\ell}{k \cdot k}} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 9: 74.7% accurate, 21.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;k \leq 1.45 \cdot 10^{+19}:\\ \;\;\;\;\frac{\frac{2 \cdot \frac{\ell}{k}}{k}}{t} \cdot \frac{\ell}{k \cdot k}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{2}{k \cdot k}}{\frac{t}{\ell}} \cdot \left(\ell \cdot 0.16666666666666666\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (t l k)
 :precision binary64
 (if (<= k 1.45e+19)
   (* (/ (/ (* 2.0 (/ l k)) k) t) (/ l (* k k)))
   (* (/ (/ 2.0 (* k k)) (/ t l)) (* l 0.16666666666666666))))

C

double code(double t, double l, double k) {
	double tmp;
	if (k <= 1.45e+19) {
		tmp = (((2.0 * (l / k)) / k) / t) * (l / (k * k));
	} else {
		tmp = ((2.0 / (k * k)) / (t / l)) * (l * 0.16666666666666666);
	}
	return tmp;
}

Fortran

real(8) function code(t, l, k)
    real(8), intent (in) :: t
    real(8), intent (in) :: l
    real(8), intent (in) :: k
    real(8) :: tmp
    if (k <= 1.45d+19) then
        tmp = (((2.0d0 * (l / k)) / k) / t) * (l / (k * k))
    else
        tmp = ((2.0d0 / (k * k)) / (t / l)) * (l * 0.16666666666666666d0)
    end if
    code = tmp
end function

Java

public static double code(double t, double l, double k) {
	double tmp;
	if (k <= 1.45e+19) {
		tmp = (((2.0 * (l / k)) / k) / t) * (l / (k * k));
	} else {
		tmp = ((2.0 / (k * k)) / (t / l)) * (l * 0.16666666666666666);
	}
	return tmp;
}

Python

def code(t, l, k):
	tmp = 0
	if k <= 1.45e+19:
		tmp = (((2.0 * (l / k)) / k) / t) * (l / (k * k))
	else:
		tmp = ((2.0 / (k * k)) / (t / l)) * (l * 0.16666666666666666)
	return tmp

Julia

function code(t, l, k)
	tmp = 0.0
	if (k <= 1.45e+19)
		tmp = Float64(Float64(Float64(Float64(2.0 * Float64(l / k)) / k) / t) * Float64(l / Float64(k * k)));
	else
		tmp = Float64(Float64(Float64(2.0 / Float64(k * k)) / Float64(t / l)) * Float64(l * 0.16666666666666666));
	end
	return tmp
end

MATLAB

function tmp_2 = code(t, l, k)
	tmp = 0.0;
	if (k <= 1.45e+19)
		tmp = (((2.0 * (l / k)) / k) / t) * (l / (k * k));
	else
		tmp = ((2.0 / (k * k)) / (t / l)) * (l * 0.16666666666666666);
	end
	tmp_2 = tmp;
end

Wolfram

code[t_, l_, k_] := If[LessEqual[k, 1.45e+19], N[(N[(N[(N[(2.0 * N[(l / k), $MachinePrecision]), $MachinePrecision] / k), $MachinePrecision] / t), $MachinePrecision] * N[(l / N[(k * k), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(2.0 / N[(k * k), $MachinePrecision]), $MachinePrecision] / N[(t / l), $MachinePrecision]), $MachinePrecision] * N[(l * 0.16666666666666666), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if k < 1.45e19

   1. Initial program 39.4%

      \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) - 1\right)} \]
   2. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \frac{2}{\left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) - 1\right) \cdot \color{blue}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right)}} \]

      2. associate-*r* N/A

         \[\leadsto \frac{2}{\left(\left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) - 1\right) \cdot \left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right)\right) \cdot \color{blue}{\tan k}} \]

      3. associate-/r* N/A

         \[\leadsto \frac{\frac{2}{\left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) - 1\right) \cdot \left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right)}}{\color{blue}{\tan k}} \]

      4. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\left(\frac{2}{\left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) - 1\right) \cdot \left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right)}\right), \color{blue}{\tan k}\right) \]

   3. Simplified 67.9%

      \[\leadsto \color{blue}{\frac{\frac{\frac{\left(\frac{2}{k \cdot k} \cdot t\right) \cdot \frac{t}{t}}{\frac{t \cdot \frac{t}{\ell}}{\ell}}}{\sin k}}{\tan k}} \]
   4. Add Preprocessing
   5. Step-by-step derivation

      1. associate-/l/ N/A

         \[\leadsto \frac{\frac{\left(\frac{2}{k \cdot k} \cdot t\right) \cdot \frac{t}{t}}{\frac{t \cdot \frac{t}{\ell}}{\ell}}}{\color{blue}{\tan k \cdot \sin k}} \]

      2. associate-/r/ N/A

         \[\leadsto \frac{\frac{\left(\frac{2}{k \cdot k} \cdot t\right) \cdot \frac{t}{t}}{t \cdot \frac{t}{\ell}} \cdot \ell}{\color{blue}{\tan k} \cdot \sin k} \]

      3. associate-/l* N/A

         \[\leadsto \frac{\left(\frac{2}{k \cdot k} \cdot t\right) \cdot \frac{t}{t}}{t \cdot \frac{t}{\ell}} \cdot \color{blue}{\frac{\ell}{\tan k \cdot \sin k}} \]

      4. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\left(\frac{\left(\frac{2}{k \cdot k} \cdot t\right) \cdot \frac{t}{t}}{t \cdot \frac{t}{\ell}}\right), \color{blue}{\left(\frac{\ell}{\tan k \cdot \sin k}\right)}\right) \]

      5. *-inverses N/A

         \[\leadsto \mathsf{*.f64}\left(\left(\frac{\left(\frac{2}{k \cdot k} \cdot t\right) \cdot 1}{t \cdot \frac{t}{\ell}}\right), \left(\frac{\ell}{\tan k \cdot \sin k}\right)\right) \]

      6. associate-*l* N/A

         \[\leadsto \mathsf{*.f64}\left(\left(\frac{\frac{2}{k \cdot k} \cdot \left(t \cdot 1\right)}{t \cdot \frac{t}{\ell}}\right), \left(\frac{\ell}{\tan k \cdot \sin k}\right)\right) \]

      7. *-commutative N/A

         \[\leadsto \mathsf{*.f64}\left(\left(\frac{\frac{2}{k \cdot k} \cdot \left(t \cdot 1\right)}{\frac{t}{\ell} \cdot t}\right), \left(\frac{\ell}{\tan k \cdot \sin k}\right)\right) \]

      8. times-frac N/A

         \[\leadsto \mathsf{*.f64}\left(\left(\frac{\frac{2}{k \cdot k}}{\frac{t}{\ell}} \cdot \frac{t \cdot 1}{t}\right), \left(\frac{\color{blue}{\ell}}{\tan k \cdot \sin k}\right)\right) \]

      9. *-rgt-identity N/A

         \[\leadsto \mathsf{*.f64}\left(\left(\frac{\frac{2}{k \cdot k}}{\frac{t}{\ell}} \cdot \frac{t}{t}\right), \left(\frac{\ell}{\tan k \cdot \sin k}\right)\right) \]

      10. *-inverses N/A

          \[\leadsto \mathsf{*.f64}\left(\left(\frac{\frac{2}{k \cdot k}}{\frac{t}{\ell}} \cdot 1\right), \left(\frac{\ell}{\tan k \cdot \sin k}\right)\right) \]

      11. *-rgt-identity N/A

          \[\leadsto \mathsf{*.f64}\left(\left(\frac{\frac{2}{k \cdot k}}{\frac{t}{\ell}}\right), \left(\frac{\color{blue}{\ell}}{\tan k \cdot \sin k}\right)\right) \]

      12. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\left(\frac{2}{k \cdot k}\right), \left(\frac{t}{\ell}\right)\right), \left(\frac{\color{blue}{\ell}}{\tan k \cdot \sin k}\right)\right) \]

      13. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(2, \left(k \cdot k\right)\right), \left(\frac{t}{\ell}\right)\right), \left(\frac{\ell}{\tan k \cdot \sin k}\right)\right) \]

      14. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(2, \mathsf{*.f64}\left(k, k\right)\right), \left(\frac{t}{\ell}\right)\right), \left(\frac{\ell}{\tan k \cdot \sin k}\right)\right) \]

      15. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(2, \mathsf{*.f64}\left(k, k\right)\right), \mathsf{/.f64}\left(t, \ell\right)\right), \left(\frac{\ell}{\tan k \cdot \sin k}\right)\right) \]

      16. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(2, \mathsf{*.f64}\left(k, k\right)\right), \mathsf{/.f64}\left(t, \ell\right)\right), \mathsf{/.f64}\left(\ell, \color{blue}{\left(\tan k \cdot \sin k\right)}\right)\right) \]

   6. Applied egg-rr 88.9%

      \[\leadsto \color{blue}{\frac{\frac{2}{k \cdot k}}{\frac{t}{\ell}} \cdot \frac{\ell}{\sin k \cdot \tan k}} \]

   7. Taylor expanded in k around 0

      \[\leadsto \mathsf{*.f64}\left(\color{blue}{\left(2 \cdot \frac{\ell}{{k}^{2} \cdot t}\right)}, \mathsf{/.f64}\left(\ell, \mathsf{*.f64}\left(\mathsf{sin.f64}\left(k\right), \mathsf{tan.f64}\left(k\right)\right)\right)\right) \]
   8. Step-by-step derivation

      1. associate-/r* N/A

         \[\leadsto \mathsf{*.f64}\left(\left(2 \cdot \frac{\frac{\ell}{{k}^{2}}}{t}\right), \mathsf{/.f64}\left(\ell, \mathsf{*.f64}\left(\mathsf{sin.f64}\left(k\right), \mathsf{tan.f64}\left(k\right)\right)\right)\right) \]

      2. associate-*r/ N/A

         \[\leadsto \mathsf{*.f64}\left(\left(\frac{2 \cdot \frac{\ell}{{k}^{2}}}{t}\right), \mathsf{/.f64}\left(\color{blue}{\ell}, \mathsf{*.f64}\left(\mathsf{sin.f64}\left(k\right), \mathsf{tan.f64}\left(k\right)\right)\right)\right) \]

      3. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\left(2 \cdot \frac{\ell}{{k}^{2}}\right), t\right), \mathsf{/.f64}\left(\color{blue}{\ell}, \mathsf{*.f64}\left(\mathsf{sin.f64}\left(k\right), \mathsf{tan.f64}\left(k\right)\right)\right)\right) \]

      4. unpow2 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\left(2 \cdot \frac{\ell}{k \cdot k}\right), t\right), \mathsf{/.f64}\left(\ell, \mathsf{*.f64}\left(\mathsf{sin.f64}\left(k\right), \mathsf{tan.f64}\left(k\right)\right)\right)\right) \]

      5. associate-/r* N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\left(2 \cdot \frac{\frac{\ell}{k}}{k}\right), t\right), \mathsf{/.f64}\left(\ell, \mathsf{*.f64}\left(\mathsf{sin.f64}\left(k\right), \mathsf{tan.f64}\left(k\right)\right)\right)\right) \]

      6. associate-*r/ N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\left(\frac{2 \cdot \frac{\ell}{k}}{k}\right), t\right), \mathsf{/.f64}\left(\ell, \mathsf{*.f64}\left(\mathsf{sin.f64}\left(k\right), \mathsf{tan.f64}\left(k\right)\right)\right)\right) \]

      7. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(\left(2 \cdot \frac{\ell}{k}\right), k\right), t\right), \mathsf{/.f64}\left(\ell, \mathsf{*.f64}\left(\mathsf{sin.f64}\left(k\right), \mathsf{tan.f64}\left(k\right)\right)\right)\right) \]

      8. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(2, \left(\frac{\ell}{k}\right)\right), k\right), t\right), \mathsf{/.f64}\left(\ell, \mathsf{*.f64}\left(\mathsf{sin.f64}\left(k\right), \mathsf{tan.f64}\left(k\right)\right)\right)\right) \]

      9. /-lowering-/.f64 94.5%

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(2, \mathsf{/.f64}\left(\ell, k\right)\right), k\right), t\right), \mathsf{/.f64}\left(\ell, \mathsf{*.f64}\left(\mathsf{sin.f64}\left(k\right), \mathsf{tan.f64}\left(k\right)\right)\right)\right) \]

   9. Simplified 94.5%

      \[\leadsto \color{blue}{\frac{\frac{2 \cdot \frac{\ell}{k}}{k}}{t}} \cdot \frac{\ell}{\sin k \cdot \tan k} \]

   10. Taylor expanded in k around 0

       \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(2, \mathsf{/.f64}\left(\ell, k\right)\right), k\right), t\right), \color{blue}{\left(\frac{\ell}{{k}^{2}}\right)}\right) \]
   11. Step-by-step derivation

       1. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(2, \mathsf{/.f64}\left(\ell, k\right)\right), k\right), t\right), \mathsf{/.f64}\left(\ell, \color{blue}{\left({k}^{2}\right)}\right)\right) \]

       2. unpow2 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(2, \mathsf{/.f64}\left(\ell, k\right)\right), k\right), t\right), \mathsf{/.f64}\left(\ell, \left(k \cdot \color{blue}{k}\right)\right)\right) \]

       3. *-lowering-*.f64 81.0%

          \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(2, \mathsf{/.f64}\left(\ell, k\right)\right), k\right), t\right), \mathsf{/.f64}\left(\ell, \mathsf{*.f64}\left(k, \color{blue}{k}\right)\right)\right) \]

   12. Simplified 81.0%

       \[\leadsto \frac{\frac{2 \cdot \frac{\ell}{k}}{k}}{t} \cdot \color{blue}{\frac{\ell}{k \cdot k}} \]

   if 1.45e19 < k

   1. Initial program 29.6%

      \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) - 1\right)} \]
   2. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \frac{2}{\left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) - 1\right) \cdot \color{blue}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right)}} \]

      2. associate-*r* N/A

         \[\leadsto \frac{2}{\left(\left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) - 1\right) \cdot \left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right)\right) \cdot \color{blue}{\tan k}} \]

      3. associate-/r* N/A

         \[\leadsto \frac{\frac{2}{\left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) - 1\right) \cdot \left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right)}}{\color{blue}{\tan k}} \]

      4. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\left(\frac{2}{\left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) - 1\right) \cdot \left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right)}\right), \color{blue}{\tan k}\right) \]

   3. Simplified 60.7%

      \[\leadsto \color{blue}{\frac{\frac{\frac{\left(\frac{2}{k \cdot k} \cdot t\right) \cdot \frac{t}{t}}{\frac{t \cdot \frac{t}{\ell}}{\ell}}}{\sin k}}{\tan k}} \]
   4. Add Preprocessing
   5. Step-by-step derivation

      1. associate-/l/ N/A

         \[\leadsto \frac{\frac{\left(\frac{2}{k \cdot k} \cdot t\right) \cdot \frac{t}{t}}{\frac{t \cdot \frac{t}{\ell}}{\ell}}}{\color{blue}{\tan k \cdot \sin k}} \]

      2. associate-/r/ N/A

         \[\leadsto \frac{\frac{\left(\frac{2}{k \cdot k} \cdot t\right) \cdot \frac{t}{t}}{t \cdot \frac{t}{\ell}} \cdot \ell}{\color{blue}{\tan k} \cdot \sin k} \]

      3. associate-/l* N/A

         \[\leadsto \frac{\left(\frac{2}{k \cdot k} \cdot t\right) \cdot \frac{t}{t}}{t \cdot \frac{t}{\ell}} \cdot \color{blue}{\frac{\ell}{\tan k \cdot \sin k}} \]

      4. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\left(\frac{\left(\frac{2}{k \cdot k} \cdot t\right) \cdot \frac{t}{t}}{t \cdot \frac{t}{\ell}}\right), \color{blue}{\left(\frac{\ell}{\tan k \cdot \sin k}\right)}\right) \]

      5. *-inverses N/A

         \[\leadsto \mathsf{*.f64}\left(\left(\frac{\left(\frac{2}{k \cdot k} \cdot t\right) \cdot 1}{t \cdot \frac{t}{\ell}}\right), \left(\frac{\ell}{\tan k \cdot \sin k}\right)\right) \]

      6. associate-*l* N/A

         \[\leadsto \mathsf{*.f64}\left(\left(\frac{\frac{2}{k \cdot k} \cdot \left(t \cdot 1\right)}{t \cdot \frac{t}{\ell}}\right), \left(\frac{\ell}{\tan k \cdot \sin k}\right)\right) \]

      7. *-commutative N/A

         \[\leadsto \mathsf{*.f64}\left(\left(\frac{\frac{2}{k \cdot k} \cdot \left(t \cdot 1\right)}{\frac{t}{\ell} \cdot t}\right), \left(\frac{\ell}{\tan k \cdot \sin k}\right)\right) \]

      8. times-frac N/A

         \[\leadsto \mathsf{*.f64}\left(\left(\frac{\frac{2}{k \cdot k}}{\frac{t}{\ell}} \cdot \frac{t \cdot 1}{t}\right), \left(\frac{\color{blue}{\ell}}{\tan k \cdot \sin k}\right)\right) \]

      9. *-rgt-identity N/A

         \[\leadsto \mathsf{*.f64}\left(\left(\frac{\frac{2}{k \cdot k}}{\frac{t}{\ell}} \cdot \frac{t}{t}\right), \left(\frac{\ell}{\tan k \cdot \sin k}\right)\right) \]

      10. *-inverses N/A

          \[\leadsto \mathsf{*.f64}\left(\left(\frac{\frac{2}{k \cdot k}}{\frac{t}{\ell}} \cdot 1\right), \left(\frac{\ell}{\tan k \cdot \sin k}\right)\right) \]

      11. *-rgt-identity N/A

          \[\leadsto \mathsf{*.f64}\left(\left(\frac{\frac{2}{k \cdot k}}{\frac{t}{\ell}}\right), \left(\frac{\color{blue}{\ell}}{\tan k \cdot \sin k}\right)\right) \]

      12. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\left(\frac{2}{k \cdot k}\right), \left(\frac{t}{\ell}\right)\right), \left(\frac{\color{blue}{\ell}}{\tan k \cdot \sin k}\right)\right) \]

      13. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(2, \left(k \cdot k\right)\right), \left(\frac{t}{\ell}\right)\right), \left(\frac{\ell}{\tan k \cdot \sin k}\right)\right) \]

      14. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(2, \mathsf{*.f64}\left(k, k\right)\right), \left(\frac{t}{\ell}\right)\right), \left(\frac{\ell}{\tan k \cdot \sin k}\right)\right) \]

      15. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(2, \mathsf{*.f64}\left(k, k\right)\right), \mathsf{/.f64}\left(t, \ell\right)\right), \left(\frac{\ell}{\tan k \cdot \sin k}\right)\right) \]

      16. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(2, \mathsf{*.f64}\left(k, k\right)\right), \mathsf{/.f64}\left(t, \ell\right)\right), \mathsf{/.f64}\left(\ell, \color{blue}{\left(\tan k \cdot \sin k\right)}\right)\right) \]

   6. Applied egg-rr 83.0%

      \[\leadsto \color{blue}{\frac{\frac{2}{k \cdot k}}{\frac{t}{\ell}} \cdot \frac{\ell}{\sin k \cdot \tan k}} \]

   7. Taylor expanded in k around 0

      \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(2, \mathsf{*.f64}\left(k, k\right)\right), \mathsf{/.f64}\left(t, \ell\right)\right), \mathsf{/.f64}\left(\ell, \mathsf{*.f64}\left(\mathsf{sin.f64}\left(k\right), \color{blue}{k}\right)\right)\right) \]
   8. Step-by-step derivation

   9. Simplified 54.8%

      \[\leadsto \frac{\frac{2}{k \cdot k}}{\frac{t}{\ell}} \cdot \frac{\ell}{\sin k \cdot \color{blue}{k}} \]

   10. Taylor expanded in k around 0

       \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(2, \mathsf{*.f64}\left(k, k\right)\right), \mathsf{/.f64}\left(t, \ell\right)\right), \color{blue}{\left(\frac{\ell + \frac{1}{6} \cdot \left({k}^{2} \cdot \ell\right)}{{k}^{2}}\right)}\right) \]
   11. Step-by-step derivation

       1. unpow2 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(2, \mathsf{*.f64}\left(k, k\right)\right), \mathsf{/.f64}\left(t, \ell\right)\right), \left(\frac{\ell + \frac{1}{6} \cdot \left({k}^{2} \cdot \ell\right)}{k \cdot \color{blue}{k}}\right)\right) \]

       2. associate-/r* N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(2, \mathsf{*.f64}\left(k, k\right)\right), \mathsf{/.f64}\left(t, \ell\right)\right), \left(\frac{\frac{\ell + \frac{1}{6} \cdot \left({k}^{2} \cdot \ell\right)}{k}}{\color{blue}{k}}\right)\right) \]

       3. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(2, \mathsf{*.f64}\left(k, k\right)\right), \mathsf{/.f64}\left(t, \ell\right)\right), \mathsf{/.f64}\left(\left(\frac{\ell + \frac{1}{6} \cdot \left({k}^{2} \cdot \ell\right)}{k}\right), \color{blue}{k}\right)\right) \]

       4. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(2, \mathsf{*.f64}\left(k, k\right)\right), \mathsf{/.f64}\left(t, \ell\right)\right), \mathsf{/.f64}\left(\mathsf{/.f64}\left(\left(\ell + \frac{1}{6} \cdot \left({k}^{2} \cdot \ell\right)\right), k\right), k\right)\right) \]

       5. associate-*r* N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(2, \mathsf{*.f64}\left(k, k\right)\right), \mathsf{/.f64}\left(t, \ell\right)\right), \mathsf{/.f64}\left(\mathsf{/.f64}\left(\left(\ell + \left(\frac{1}{6} \cdot {k}^{2}\right) \cdot \ell\right), k\right), k\right)\right) \]

       6. distribute-rgt1-in N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(2, \mathsf{*.f64}\left(k, k\right)\right), \mathsf{/.f64}\left(t, \ell\right)\right), \mathsf{/.f64}\left(\mathsf{/.f64}\left(\left(\left(\frac{1}{6} \cdot {k}^{2} + 1\right) \cdot \ell\right), k\right), k\right)\right) \]

       7. +-commutative N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(2, \mathsf{*.f64}\left(k, k\right)\right), \mathsf{/.f64}\left(t, \ell\right)\right), \mathsf{/.f64}\left(\mathsf{/.f64}\left(\left(\left(1 + \frac{1}{6} \cdot {k}^{2}\right) \cdot \ell\right), k\right), k\right)\right) \]

       8. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(2, \mathsf{*.f64}\left(k, k\right)\right), \mathsf{/.f64}\left(t, \ell\right)\right), \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\left(1 + \frac{1}{6} \cdot {k}^{2}\right), \ell\right), k\right), k\right)\right) \]

       9. +-lowering-+.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(2, \mathsf{*.f64}\left(k, k\right)\right), \mathsf{/.f64}\left(t, \ell\right)\right), \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \left(\frac{1}{6} \cdot {k}^{2}\right)\right), \ell\right), k\right), k\right)\right) \]

       10. *-commutative N/A

           \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(2, \mathsf{*.f64}\left(k, k\right)\right), \mathsf{/.f64}\left(t, \ell\right)\right), \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \left({k}^{2} \cdot \frac{1}{6}\right)\right), \ell\right), k\right), k\right)\right) \]

       11. *-lowering-*.f64 N/A

           \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(2, \mathsf{*.f64}\left(k, k\right)\right), \mathsf{/.f64}\left(t, \ell\right)\right), \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\left({k}^{2}\right), \frac{1}{6}\right)\right), \ell\right), k\right), k\right)\right) \]

       12. unpow2 N/A

           \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(2, \mathsf{*.f64}\left(k, k\right)\right), \mathsf{/.f64}\left(t, \ell\right)\right), \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\left(k \cdot k\right), \frac{1}{6}\right)\right), \ell\right), k\right), k\right)\right) \]

       13. *-lowering-*.f64 24.6%

           \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(2, \mathsf{*.f64}\left(k, k\right)\right), \mathsf{/.f64}\left(t, \ell\right)\right), \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(k, k\right), \frac{1}{6}\right)\right), \ell\right), k\right), k\right)\right) \]

   12. Simplified 24.6%

       \[\leadsto \frac{\frac{2}{k \cdot k}}{\frac{t}{\ell}} \cdot \color{blue}{\frac{\frac{\left(1 + \left(k \cdot k\right) \cdot 0.16666666666666666\right) \cdot \ell}{k}}{k}} \]

   13. Taylor expanded in k around inf

       \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(2, \mathsf{*.f64}\left(k, k\right)\right), \mathsf{/.f64}\left(t, \ell\right)\right), \color{blue}{\left(\frac{1}{6} \cdot \ell\right)}\right) \]
   14. Step-by-step derivation

       1. *-commutative N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(2, \mathsf{*.f64}\left(k, k\right)\right), \mathsf{/.f64}\left(t, \ell\right)\right), \left(\ell \cdot \color{blue}{\frac{1}{6}}\right)\right) \]

       2. *-lowering-*.f64 55.5%

          \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(2, \mathsf{*.f64}\left(k, k\right)\right), \mathsf{/.f64}\left(t, \ell\right)\right), \mathsf{*.f64}\left(\ell, \color{blue}{\frac{1}{6}}\right)\right) \]

   15. Simplified 55.5%

       \[\leadsto \frac{\frac{2}{k \cdot k}}{\frac{t}{\ell}} \cdot \color{blue}{\left(\ell \cdot 0.16666666666666666\right)} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 10: 73.5% accurate, 21.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;t \leq 3.5 \cdot 10^{-147}:\\ \;\;\;\;\frac{\frac{2}{k \cdot k}}{\frac{t}{\ell}} \cdot \frac{\ell}{k \cdot k}\\ \mathbf{else}:\\ \;\;\;\;\frac{\ell}{k} \cdot \frac{2}{\frac{k}{\frac{\ell}{k \cdot \left(k \cdot t\right)}}}\\ \end{array} \end{array} \]

FPCore

(FPCore (t l k)
 :precision binary64
 (if (<= t 3.5e-147)
   (* (/ (/ 2.0 (* k k)) (/ t l)) (/ l (* k k)))
   (* (/ l k) (/ 2.0 (/ k (/ l (* k (* k t))))))))

C

double code(double t, double l, double k) {
	double tmp;
	if (t <= 3.5e-147) {
		tmp = ((2.0 / (k * k)) / (t / l)) * (l / (k * k));
	} else {
		tmp = (l / k) * (2.0 / (k / (l / (k * (k * t)))));
	}
	return tmp;
}

Fortran

real(8) function code(t, l, k)
    real(8), intent (in) :: t
    real(8), intent (in) :: l
    real(8), intent (in) :: k
    real(8) :: tmp
    if (t <= 3.5d-147) then
        tmp = ((2.0d0 / (k * k)) / (t / l)) * (l / (k * k))
    else
        tmp = (l / k) * (2.0d0 / (k / (l / (k * (k * t)))))
    end if
    code = tmp
end function

Java

public static double code(double t, double l, double k) {
	double tmp;
	if (t <= 3.5e-147) {
		tmp = ((2.0 / (k * k)) / (t / l)) * (l / (k * k));
	} else {
		tmp = (l / k) * (2.0 / (k / (l / (k * (k * t)))));
	}
	return tmp;
}

Python

def code(t, l, k):
	tmp = 0
	if t <= 3.5e-147:
		tmp = ((2.0 / (k * k)) / (t / l)) * (l / (k * k))
	else:
		tmp = (l / k) * (2.0 / (k / (l / (k * (k * t)))))
	return tmp

Julia

function code(t, l, k)
	tmp = 0.0
	if (t <= 3.5e-147)
		tmp = Float64(Float64(Float64(2.0 / Float64(k * k)) / Float64(t / l)) * Float64(l / Float64(k * k)));
	else
		tmp = Float64(Float64(l / k) * Float64(2.0 / Float64(k / Float64(l / Float64(k * Float64(k * t))))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(t, l, k)
	tmp = 0.0;
	if (t <= 3.5e-147)
		tmp = ((2.0 / (k * k)) / (t / l)) * (l / (k * k));
	else
		tmp = (l / k) * (2.0 / (k / (l / (k * (k * t)))));
	end
	tmp_2 = tmp;
end

Wolfram

code[t_, l_, k_] := If[LessEqual[t, 3.5e-147], N[(N[(N[(2.0 / N[(k * k), $MachinePrecision]), $MachinePrecision] / N[(t / l), $MachinePrecision]), $MachinePrecision] * N[(l / N[(k * k), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(l / k), $MachinePrecision] * N[(2.0 / N[(k / N[(l / N[(k * N[(k * t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if t < 3.50000000000000004e-147

   1. Initial program 35.9%

      \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) - 1\right)} \]
   2. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \frac{2}{\left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) - 1\right) \cdot \color{blue}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right)}} \]

      2. associate-*r* N/A

         \[\leadsto \frac{2}{\left(\left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) - 1\right) \cdot \left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right)\right) \cdot \color{blue}{\tan k}} \]

      3. associate-/r* N/A

         \[\leadsto \frac{\frac{2}{\left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) - 1\right) \cdot \left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right)}}{\color{blue}{\tan k}} \]

      4. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\left(\frac{2}{\left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) - 1\right) \cdot \left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right)}\right), \color{blue}{\tan k}\right) \]

   3. Simplified 67.1%

      \[\leadsto \color{blue}{\frac{\frac{\frac{\left(\frac{2}{k \cdot k} \cdot t\right) \cdot \frac{t}{t}}{\frac{t \cdot \frac{t}{\ell}}{\ell}}}{\sin k}}{\tan k}} \]
   4. Add Preprocessing
   5. Step-by-step derivation

      1. associate-/l/ N/A

         \[\leadsto \frac{\frac{\left(\frac{2}{k \cdot k} \cdot t\right) \cdot \frac{t}{t}}{\frac{t \cdot \frac{t}{\ell}}{\ell}}}{\color{blue}{\tan k \cdot \sin k}} \]

      2. associate-/r/ N/A

         \[\leadsto \frac{\frac{\left(\frac{2}{k \cdot k} \cdot t\right) \cdot \frac{t}{t}}{t \cdot \frac{t}{\ell}} \cdot \ell}{\color{blue}{\tan k} \cdot \sin k} \]

      3. associate-/l* N/A

         \[\leadsto \frac{\left(\frac{2}{k \cdot k} \cdot t\right) \cdot \frac{t}{t}}{t \cdot \frac{t}{\ell}} \cdot \color{blue}{\frac{\ell}{\tan k \cdot \sin k}} \]

      4. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\left(\frac{\left(\frac{2}{k \cdot k} \cdot t\right) \cdot \frac{t}{t}}{t \cdot \frac{t}{\ell}}\right), \color{blue}{\left(\frac{\ell}{\tan k \cdot \sin k}\right)}\right) \]

      5. *-inverses N/A

         \[\leadsto \mathsf{*.f64}\left(\left(\frac{\left(\frac{2}{k \cdot k} \cdot t\right) \cdot 1}{t \cdot \frac{t}{\ell}}\right), \left(\frac{\ell}{\tan k \cdot \sin k}\right)\right) \]

      6. associate-*l* N/A

         \[\leadsto \mathsf{*.f64}\left(\left(\frac{\frac{2}{k \cdot k} \cdot \left(t \cdot 1\right)}{t \cdot \frac{t}{\ell}}\right), \left(\frac{\ell}{\tan k \cdot \sin k}\right)\right) \]

      7. *-commutative N/A

         \[\leadsto \mathsf{*.f64}\left(\left(\frac{\frac{2}{k \cdot k} \cdot \left(t \cdot 1\right)}{\frac{t}{\ell} \cdot t}\right), \left(\frac{\ell}{\tan k \cdot \sin k}\right)\right) \]

      8. times-frac N/A

         \[\leadsto \mathsf{*.f64}\left(\left(\frac{\frac{2}{k \cdot k}}{\frac{t}{\ell}} \cdot \frac{t \cdot 1}{t}\right), \left(\frac{\color{blue}{\ell}}{\tan k \cdot \sin k}\right)\right) \]

      9. *-rgt-identity N/A

         \[\leadsto \mathsf{*.f64}\left(\left(\frac{\frac{2}{k \cdot k}}{\frac{t}{\ell}} \cdot \frac{t}{t}\right), \left(\frac{\ell}{\tan k \cdot \sin k}\right)\right) \]

      10. *-inverses N/A

          \[\leadsto \mathsf{*.f64}\left(\left(\frac{\frac{2}{k \cdot k}}{\frac{t}{\ell}} \cdot 1\right), \left(\frac{\ell}{\tan k \cdot \sin k}\right)\right) \]

      11. *-rgt-identity N/A

          \[\leadsto \mathsf{*.f64}\left(\left(\frac{\frac{2}{k \cdot k}}{\frac{t}{\ell}}\right), \left(\frac{\color{blue}{\ell}}{\tan k \cdot \sin k}\right)\right) \]

      12. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\left(\frac{2}{k \cdot k}\right), \left(\frac{t}{\ell}\right)\right), \left(\frac{\color{blue}{\ell}}{\tan k \cdot \sin k}\right)\right) \]

      13. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(2, \left(k \cdot k\right)\right), \left(\frac{t}{\ell}\right)\right), \left(\frac{\ell}{\tan k \cdot \sin k}\right)\right) \]

      14. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(2, \mathsf{*.f64}\left(k, k\right)\right), \left(\frac{t}{\ell}\right)\right), \left(\frac{\ell}{\tan k \cdot \sin k}\right)\right) \]

      15. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(2, \mathsf{*.f64}\left(k, k\right)\right), \mathsf{/.f64}\left(t, \ell\right)\right), \left(\frac{\ell}{\tan k \cdot \sin k}\right)\right) \]

      16. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(2, \mathsf{*.f64}\left(k, k\right)\right), \mathsf{/.f64}\left(t, \ell\right)\right), \mathsf{/.f64}\left(\ell, \color{blue}{\left(\tan k \cdot \sin k\right)}\right)\right) \]

   6. Applied egg-rr 89.4%

      \[\leadsto \color{blue}{\frac{\frac{2}{k \cdot k}}{\frac{t}{\ell}} \cdot \frac{\ell}{\sin k \cdot \tan k}} \]

   7. Taylor expanded in k around 0

      \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(2, \mathsf{*.f64}\left(k, k\right)\right), \mathsf{/.f64}\left(t, \ell\right)\right), \color{blue}{\left(\frac{\ell}{{k}^{2}}\right)}\right) \]
   8. Step-by-step derivation

      1. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(2, \mathsf{*.f64}\left(k, k\right)\right), \mathsf{/.f64}\left(t, \ell\right)\right), \mathsf{/.f64}\left(\ell, \color{blue}{\left({k}^{2}\right)}\right)\right) \]

      2. unpow2 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(2, \mathsf{*.f64}\left(k, k\right)\right), \mathsf{/.f64}\left(t, \ell\right)\right), \mathsf{/.f64}\left(\ell, \left(k \cdot \color{blue}{k}\right)\right)\right) \]

      3. *-lowering-*.f64 73.2%

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(2, \mathsf{*.f64}\left(k, k\right)\right), \mathsf{/.f64}\left(t, \ell\right)\right), \mathsf{/.f64}\left(\ell, \mathsf{*.f64}\left(k, \color{blue}{k}\right)\right)\right) \]

   9. Simplified 73.2%

      \[\leadsto \frac{\frac{2}{k \cdot k}}{\frac{t}{\ell}} \cdot \color{blue}{\frac{\ell}{k \cdot k}} \]

   if 3.50000000000000004e-147 < t

   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. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(2, \mathsf{/.f64}\left(\left({k}^{4} \cdot t\right), \color{blue}{\left({\ell}^{2}\right)}\right)\right) \]

      2. *-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(2, \mathsf{/.f64}\left(\left(t \cdot {k}^{4}\right), \left({\color{blue}{\ell}}^{2}\right)\right)\right) \]

      3. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(2, \mathsf{/.f64}\left(\mathsf{*.f64}\left(t, \left({k}^{4}\right)\right), \left({\color{blue}{\ell}}^{2}\right)\right)\right) \]

      4. metadata-eval N/A

         \[\leadsto \mathsf{/.f64}\left(2, \mathsf{/.f64}\left(\mathsf{*.f64}\left(t, \left({k}^{\left(2 \cdot 2\right)}\right)\right), \left({\ell}^{2}\right)\right)\right) \]

      5. pow-sqr N/A

         \[\leadsto \mathsf{/.f64}\left(2, \mathsf{/.f64}\left(\mathsf{*.f64}\left(t, \left({k}^{2} \cdot {k}^{2}\right)\right), \left({\ell}^{2}\right)\right)\right) \]

      6. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(2, \mathsf{/.f64}\left(\mathsf{*.f64}\left(t, \mathsf{*.f64}\left(\left({k}^{2}\right), \left({k}^{2}\right)\right)\right), \left({\ell}^{2}\right)\right)\right) \]

      7. unpow2 N/A

         \[\leadsto \mathsf{/.f64}\left(2, \mathsf{/.f64}\left(\mathsf{*.f64}\left(t, \mathsf{*.f64}\left(\left(k \cdot k\right), \left({k}^{2}\right)\right)\right), \left({\ell}^{2}\right)\right)\right) \]

      8. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(2, \mathsf{/.f64}\left(\mathsf{*.f64}\left(t, \mathsf{*.f64}\left(\mathsf{*.f64}\left(k, k\right), \left({k}^{2}\right)\right)\right), \left({\ell}^{2}\right)\right)\right) \]

      9. unpow2 N/A

         \[\leadsto \mathsf{/.f64}\left(2, \mathsf{/.f64}\left(\mathsf{*.f64}\left(t, \mathsf{*.f64}\left(\mathsf{*.f64}\left(k, k\right), \left(k \cdot k\right)\right)\right), \left({\ell}^{2}\right)\right)\right) \]

      10. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(2, \mathsf{/.f64}\left(\mathsf{*.f64}\left(t, \mathsf{*.f64}\left(\mathsf{*.f64}\left(k, k\right), \mathsf{*.f64}\left(k, k\right)\right)\right), \left({\ell}^{2}\right)\right)\right) \]

      11. unpow2 N/A

          \[\leadsto \mathsf{/.f64}\left(2, \mathsf{/.f64}\left(\mathsf{*.f64}\left(t, \mathsf{*.f64}\left(\mathsf{*.f64}\left(k, k\right), \mathsf{*.f64}\left(k, k\right)\right)\right), \left(\ell \cdot \color{blue}{\ell}\right)\right)\right) \]

      12. *-lowering-*.f64 58.9%

          \[\leadsto \mathsf{/.f64}\left(2, \mathsf{/.f64}\left(\mathsf{*.f64}\left(t, \mathsf{*.f64}\left(\mathsf{*.f64}\left(k, k\right), \mathsf{*.f64}\left(k, k\right)\right)\right), \mathsf{*.f64}\left(\ell, \color{blue}{\ell}\right)\right)\right) \]

   5. Simplified 58.9%

      \[\leadsto \frac{2}{\color{blue}{\frac{t \cdot \left(\left(k \cdot k\right) \cdot \left(k \cdot k\right)\right)}{\ell \cdot \ell}}} \]
   6. Step-by-step derivation

      1. associate-/r/ N/A

         \[\leadsto \frac{2}{t \cdot \left(\left(k \cdot k\right) \cdot \left(k \cdot k\right)\right)} \cdot \color{blue}{\left(\ell \cdot \ell\right)} \]

      2. associate-*r* N/A

         \[\leadsto \left(\frac{2}{t \cdot \left(\left(k \cdot k\right) \cdot \left(k \cdot k\right)\right)} \cdot \ell\right) \cdot \color{blue}{\ell} \]

      3. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\left(\frac{2}{t \cdot \left(\left(k \cdot k\right) \cdot \left(k \cdot k\right)\right)} \cdot \ell\right), \color{blue}{\ell}\right) \]

      4. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\left(\frac{2}{t \cdot \left(\left(k \cdot k\right) \cdot \left(k \cdot k\right)\right)}\right), \ell\right), \ell\right) \]

      5. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(2, \left(t \cdot \left(\left(k \cdot k\right) \cdot \left(k \cdot k\right)\right)\right)\right), \ell\right), \ell\right) \]

      6. *-commutative N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(2, \left(\left(\left(k \cdot k\right) \cdot \left(k \cdot k\right)\right) \cdot t\right)\right), \ell\right), \ell\right) \]

      7. associate-*r* N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(2, \left(\left(\left(\left(k \cdot k\right) \cdot k\right) \cdot k\right) \cdot t\right)\right), \ell\right), \ell\right) \]

      8. *-commutative N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(2, \left(\left(k \cdot \left(\left(k \cdot k\right) \cdot k\right)\right) \cdot t\right)\right), \ell\right), \ell\right) \]

      9. associate-*l* N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(2, \left(k \cdot \left(\left(\left(k \cdot k\right) \cdot k\right) \cdot t\right)\right)\right), \ell\right), \ell\right) \]

      10. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(2, \mathsf{*.f64}\left(k, \left(\left(\left(k \cdot k\right) \cdot k\right) \cdot t\right)\right)\right), \ell\right), \ell\right) \]

      11. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(2, \mathsf{*.f64}\left(k, \mathsf{*.f64}\left(\left(\left(k \cdot k\right) \cdot k\right), t\right)\right)\right), \ell\right), \ell\right) \]

      12. associate-*r* N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(2, \mathsf{*.f64}\left(k, \mathsf{*.f64}\left(\left(k \cdot \left(k \cdot k\right)\right), t\right)\right)\right), \ell\right), \ell\right) \]

      13. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(2, \mathsf{*.f64}\left(k, \mathsf{*.f64}\left(\mathsf{*.f64}\left(k, \left(k \cdot k\right)\right), t\right)\right)\right), \ell\right), \ell\right) \]

      14. *-lowering-*.f64 68.4%

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(2, \mathsf{*.f64}\left(k, \mathsf{*.f64}\left(\mathsf{*.f64}\left(k, \mathsf{*.f64}\left(k, k\right)\right), t\right)\right)\right), \ell\right), \ell\right) \]

   7. Applied egg-rr 68.4%

      \[\leadsto \color{blue}{\left(\frac{2}{k \cdot \left(\left(k \cdot \left(k \cdot k\right)\right) \cdot t\right)} \cdot \ell\right) \cdot \ell} \]
   8. Step-by-step derivation

      1. associate-*l* N/A

         \[\leadsto \frac{2}{k \cdot \left(\left(k \cdot \left(k \cdot k\right)\right) \cdot t\right)} \cdot \color{blue}{\left(\ell \cdot \ell\right)} \]

      2. associate-*l* N/A

         \[\leadsto \frac{2}{k \cdot \left(k \cdot \left(\left(k \cdot k\right) \cdot t\right)\right)} \cdot \left(\ell \cdot \ell\right) \]

      3. *-commutative N/A

         \[\leadsto \frac{2}{k \cdot \left(\left(\left(k \cdot k\right) \cdot t\right) \cdot k\right)} \cdot \left(\ell \cdot \ell\right) \]

      4. *-commutative N/A

         \[\leadsto \frac{2}{\left(\left(\left(k \cdot k\right) \cdot t\right) \cdot k\right) \cdot k} \cdot \left(\ell \cdot \ell\right) \]

      5. associate-/r/ N/A

         \[\leadsto \frac{2}{\color{blue}{\frac{\left(\left(\left(k \cdot k\right) \cdot t\right) \cdot k\right) \cdot k}{\ell \cdot \ell}}} \]

      6. times-frac N/A

         \[\leadsto \frac{2}{\frac{\left(\left(k \cdot k\right) \cdot t\right) \cdot k}{\ell} \cdot \color{blue}{\frac{k}{\ell}}} \]

      7. associate-/r* N/A

         \[\leadsto \frac{\frac{2}{\frac{\left(\left(k \cdot k\right) \cdot t\right) \cdot k}{\ell}}}{\color{blue}{\frac{k}{\ell}}} \]

      8. div-inv N/A

         \[\leadsto \frac{2}{\frac{\left(\left(k \cdot k\right) \cdot t\right) \cdot k}{\ell}} \cdot \color{blue}{\frac{1}{\frac{k}{\ell}}} \]

      9. clear-num N/A

         \[\leadsto \frac{2}{\frac{\left(\left(k \cdot k\right) \cdot t\right) \cdot k}{\ell}} \cdot \frac{\ell}{\color{blue}{k}} \]

      10. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\left(\frac{2}{\frac{\left(\left(k \cdot k\right) \cdot t\right) \cdot k}{\ell}}\right), \color{blue}{\left(\frac{\ell}{k}\right)}\right) \]

   9. Applied egg-rr 68.1%

      \[\leadsto \color{blue}{\frac{2}{\frac{k \cdot \left(k \cdot k\right)}{\frac{\ell}{t}}} \cdot \frac{\ell}{k}} \]
   10. Step-by-step derivation

       1. clear-num N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(2, \left(\frac{1}{\frac{\frac{\ell}{t}}{k \cdot \left(k \cdot k\right)}}\right)\right), \mathsf{/.f64}\left(\ell, k\right)\right) \]

       2. associate-/l/ N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(2, \left(\frac{1}{\frac{\ell}{\left(k \cdot \left(k \cdot k\right)\right) \cdot t}}\right)\right), \mathsf{/.f64}\left(\ell, k\right)\right) \]

       3. associate-*r* N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(2, \left(\frac{1}{\frac{\ell}{k \cdot \left(\left(k \cdot k\right) \cdot t\right)}}\right)\right), \mathsf{/.f64}\left(\ell, k\right)\right) \]

       4. associate-*r* N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(2, \left(\frac{1}{\frac{\ell}{k \cdot \left(k \cdot \left(k \cdot t\right)\right)}}\right)\right), \mathsf{/.f64}\left(\ell, k\right)\right) \]

       5. associate-/l/ N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(2, \left(\frac{1}{\frac{\frac{\ell}{k \cdot \left(k \cdot t\right)}}{k}}\right)\right), \mathsf{/.f64}\left(\ell, k\right)\right) \]

       6. clear-num N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(2, \left(\frac{k}{\frac{\ell}{k \cdot \left(k \cdot t\right)}}\right)\right), \mathsf{/.f64}\left(\ell, k\right)\right) \]

       7. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(2, \mathsf{/.f64}\left(k, \left(\frac{\ell}{k \cdot \left(k \cdot t\right)}\right)\right)\right), \mathsf{/.f64}\left(\ell, k\right)\right) \]

       8. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(2, \mathsf{/.f64}\left(k, \mathsf{/.f64}\left(\ell, \left(k \cdot \left(k \cdot t\right)\right)\right)\right)\right), \mathsf{/.f64}\left(\ell, k\right)\right) \]

       9. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(2, \mathsf{/.f64}\left(k, \mathsf{/.f64}\left(\ell, \mathsf{*.f64}\left(k, \left(k \cdot t\right)\right)\right)\right)\right), \mathsf{/.f64}\left(\ell, k\right)\right) \]

       10. *-lowering-*.f64 73.7%

           \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(2, \mathsf{/.f64}\left(k, \mathsf{/.f64}\left(\ell, \mathsf{*.f64}\left(k, \mathsf{*.f64}\left(k, t\right)\right)\right)\right)\right), \mathsf{/.f64}\left(\ell, k\right)\right) \]

   11. Applied egg-rr 73.7%

       \[\leadsto \frac{2}{\color{blue}{\frac{k}{\frac{\ell}{k \cdot \left(k \cdot t\right)}}}} \cdot \frac{\ell}{k} \]
3. Recombined 2 regimes into one program.

4. Final simplification 73.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq 3.5 \cdot 10^{-147}:\\ \;\;\;\;\frac{\frac{2}{k \cdot k}}{\frac{t}{\ell}} \cdot \frac{\ell}{k \cdot k}\\ \mathbf{else}:\\ \;\;\;\;\frac{\ell}{k} \cdot \frac{2}{\frac{k}{\frac{\ell}{k \cdot \left(k \cdot t\right)}}}\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 75.6% accurate, 21.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;k \leq 1.45 \cdot 10^{+19}:\\ \;\;\;\;\frac{\ell}{k} \cdot \frac{\frac{2}{k}}{\frac{k \cdot t}{\frac{\ell}{k}}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{2}{k \cdot k}}{\frac{t}{\ell}} \cdot \left(\ell \cdot 0.16666666666666666\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (t l k)
 :precision binary64
 (if (<= k 1.45e+19)
   (* (/ l k) (/ (/ 2.0 k) (/ (* k t) (/ l k))))
   (* (/ (/ 2.0 (* k k)) (/ t l)) (* l 0.16666666666666666))))

C

double code(double t, double l, double k) {
	double tmp;
	if (k <= 1.45e+19) {
		tmp = (l / k) * ((2.0 / k) / ((k * t) / (l / k)));
	} else {
		tmp = ((2.0 / (k * k)) / (t / l)) * (l * 0.16666666666666666);
	}
	return tmp;
}

Fortran

real(8) function code(t, l, k)
    real(8), intent (in) :: t
    real(8), intent (in) :: l
    real(8), intent (in) :: k
    real(8) :: tmp
    if (k <= 1.45d+19) then
        tmp = (l / k) * ((2.0d0 / k) / ((k * t) / (l / k)))
    else
        tmp = ((2.0d0 / (k * k)) / (t / l)) * (l * 0.16666666666666666d0)
    end if
    code = tmp
end function

Java

public static double code(double t, double l, double k) {
	double tmp;
	if (k <= 1.45e+19) {
		tmp = (l / k) * ((2.0 / k) / ((k * t) / (l / k)));
	} else {
		tmp = ((2.0 / (k * k)) / (t / l)) * (l * 0.16666666666666666);
	}
	return tmp;
}

Python

def code(t, l, k):
	tmp = 0
	if k <= 1.45e+19:
		tmp = (l / k) * ((2.0 / k) / ((k * t) / (l / k)))
	else:
		tmp = ((2.0 / (k * k)) / (t / l)) * (l * 0.16666666666666666)
	return tmp

Julia

function code(t, l, k)
	tmp = 0.0
	if (k <= 1.45e+19)
		tmp = Float64(Float64(l / k) * Float64(Float64(2.0 / k) / Float64(Float64(k * t) / Float64(l / k))));
	else
		tmp = Float64(Float64(Float64(2.0 / Float64(k * k)) / Float64(t / l)) * Float64(l * 0.16666666666666666));
	end
	return tmp
end

MATLAB

function tmp_2 = code(t, l, k)
	tmp = 0.0;
	if (k <= 1.45e+19)
		tmp = (l / k) * ((2.0 / k) / ((k * t) / (l / k)));
	else
		tmp = ((2.0 / (k * k)) / (t / l)) * (l * 0.16666666666666666);
	end
	tmp_2 = tmp;
end

Wolfram

code[t_, l_, k_] := If[LessEqual[k, 1.45e+19], N[(N[(l / k), $MachinePrecision] * N[(N[(2.0 / k), $MachinePrecision] / N[(N[(k * t), $MachinePrecision] / N[(l / k), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(2.0 / N[(k * k), $MachinePrecision]), $MachinePrecision] / N[(t / l), $MachinePrecision]), $MachinePrecision] * N[(l * 0.16666666666666666), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if k < 1.45e19

   1. Initial program 39.4%

      \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) - 1\right)} \]
   2. Add Preprocessing

   3. Taylor expanded in k around 0

      \[\leadsto \mathsf{/.f64}\left(2, \color{blue}{\left(\frac{{k}^{4} \cdot t}{{\ell}^{2}}\right)}\right) \]
   4. Step-by-step derivation

      1. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(2, \mathsf{/.f64}\left(\left({k}^{4} \cdot t\right), \color{blue}{\left({\ell}^{2}\right)}\right)\right) \]

      2. *-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(2, \mathsf{/.f64}\left(\left(t \cdot {k}^{4}\right), \left({\color{blue}{\ell}}^{2}\right)\right)\right) \]

      3. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(2, \mathsf{/.f64}\left(\mathsf{*.f64}\left(t, \left({k}^{4}\right)\right), \left({\color{blue}{\ell}}^{2}\right)\right)\right) \]

      4. metadata-eval N/A

         \[\leadsto \mathsf{/.f64}\left(2, \mathsf{/.f64}\left(\mathsf{*.f64}\left(t, \left({k}^{\left(2 \cdot 2\right)}\right)\right), \left({\ell}^{2}\right)\right)\right) \]

      5. pow-sqr N/A

         \[\leadsto \mathsf{/.f64}\left(2, \mathsf{/.f64}\left(\mathsf{*.f64}\left(t, \left({k}^{2} \cdot {k}^{2}\right)\right), \left({\ell}^{2}\right)\right)\right) \]

      6. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(2, \mathsf{/.f64}\left(\mathsf{*.f64}\left(t, \mathsf{*.f64}\left(\left({k}^{2}\right), \left({k}^{2}\right)\right)\right), \left({\ell}^{2}\right)\right)\right) \]

      7. unpow2 N/A

         \[\leadsto \mathsf{/.f64}\left(2, \mathsf{/.f64}\left(\mathsf{*.f64}\left(t, \mathsf{*.f64}\left(\left(k \cdot k\right), \left({k}^{2}\right)\right)\right), \left({\ell}^{2}\right)\right)\right) \]

      8. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(2, \mathsf{/.f64}\left(\mathsf{*.f64}\left(t, \mathsf{*.f64}\left(\mathsf{*.f64}\left(k, k\right), \left({k}^{2}\right)\right)\right), \left({\ell}^{2}\right)\right)\right) \]

      9. unpow2 N/A

         \[\leadsto \mathsf{/.f64}\left(2, \mathsf{/.f64}\left(\mathsf{*.f64}\left(t, \mathsf{*.f64}\left(\mathsf{*.f64}\left(k, k\right), \left(k \cdot k\right)\right)\right), \left({\ell}^{2}\right)\right)\right) \]

      10. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(2, \mathsf{/.f64}\left(\mathsf{*.f64}\left(t, \mathsf{*.f64}\left(\mathsf{*.f64}\left(k, k\right), \mathsf{*.f64}\left(k, k\right)\right)\right), \left({\ell}^{2}\right)\right)\right) \]

      11. unpow2 N/A

          \[\leadsto \mathsf{/.f64}\left(2, \mathsf{/.f64}\left(\mathsf{*.f64}\left(t, \mathsf{*.f64}\left(\mathsf{*.f64}\left(k, k\right), \mathsf{*.f64}\left(k, k\right)\right)\right), \left(\ell \cdot \color{blue}{\ell}\right)\right)\right) \]

      12. *-lowering-*.f64 63.4%

          \[\leadsto \mathsf{/.f64}\left(2, \mathsf{/.f64}\left(\mathsf{*.f64}\left(t, \mathsf{*.f64}\left(\mathsf{*.f64}\left(k, k\right), \mathsf{*.f64}\left(k, k\right)\right)\right), \mathsf{*.f64}\left(\ell, \color{blue}{\ell}\right)\right)\right) \]

   5. Simplified 63.4%

      \[\leadsto \frac{2}{\color{blue}{\frac{t \cdot \left(\left(k \cdot k\right) \cdot \left(k \cdot k\right)\right)}{\ell \cdot \ell}}} \]
   6. Step-by-step derivation

      1. associate-/r/ N/A

         \[\leadsto \frac{2}{t \cdot \left(\left(k \cdot k\right) \cdot \left(k \cdot k\right)\right)} \cdot \color{blue}{\left(\ell \cdot \ell\right)} \]

      2. associate-*r* N/A

         \[\leadsto \left(\frac{2}{t \cdot \left(\left(k \cdot k\right) \cdot \left(k \cdot k\right)\right)} \cdot \ell\right) \cdot \color{blue}{\ell} \]

      3. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\left(\frac{2}{t \cdot \left(\left(k \cdot k\right) \cdot \left(k \cdot k\right)\right)} \cdot \ell\right), \color{blue}{\ell}\right) \]

      4. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\left(\frac{2}{t \cdot \left(\left(k \cdot k\right) \cdot \left(k \cdot k\right)\right)}\right), \ell\right), \ell\right) \]

      5. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(2, \left(t \cdot \left(\left(k \cdot k\right) \cdot \left(k \cdot k\right)\right)\right)\right), \ell\right), \ell\right) \]

      6. *-commutative N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(2, \left(\left(\left(k \cdot k\right) \cdot \left(k \cdot k\right)\right) \cdot t\right)\right), \ell\right), \ell\right) \]

      7. associate-*r* N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(2, \left(\left(\left(\left(k \cdot k\right) \cdot k\right) \cdot k\right) \cdot t\right)\right), \ell\right), \ell\right) \]

      8. *-commutative N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(2, \left(\left(k \cdot \left(\left(k \cdot k\right) \cdot k\right)\right) \cdot t\right)\right), \ell\right), \ell\right) \]

      9. associate-*l* N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(2, \left(k \cdot \left(\left(\left(k \cdot k\right) \cdot k\right) \cdot t\right)\right)\right), \ell\right), \ell\right) \]

      10. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(2, \mathsf{*.f64}\left(k, \left(\left(\left(k \cdot k\right) \cdot k\right) \cdot t\right)\right)\right), \ell\right), \ell\right) \]

      11. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(2, \mathsf{*.f64}\left(k, \mathsf{*.f64}\left(\left(\left(k \cdot k\right) \cdot k\right), t\right)\right)\right), \ell\right), \ell\right) \]

      12. associate-*r* N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(2, \mathsf{*.f64}\left(k, \mathsf{*.f64}\left(\left(k \cdot \left(k \cdot k\right)\right), t\right)\right)\right), \ell\right), \ell\right) \]

      13. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(2, \mathsf{*.f64}\left(k, \mathsf{*.f64}\left(\mathsf{*.f64}\left(k, \left(k \cdot k\right)\right), t\right)\right)\right), \ell\right), \ell\right) \]

      14. *-lowering-*.f64 72.1%

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(2, \mathsf{*.f64}\left(k, \mathsf{*.f64}\left(\mathsf{*.f64}\left(k, \mathsf{*.f64}\left(k, k\right)\right), t\right)\right)\right), \ell\right), \ell\right) \]

   7. Applied egg-rr 72.1%

      \[\leadsto \color{blue}{\left(\frac{2}{k \cdot \left(\left(k \cdot \left(k \cdot k\right)\right) \cdot t\right)} \cdot \ell\right) \cdot \ell} \]
   8. Step-by-step derivation

      1. associate-*l* N/A

         \[\leadsto \frac{2}{k \cdot \left(\left(k \cdot \left(k \cdot k\right)\right) \cdot t\right)} \cdot \color{blue}{\left(\ell \cdot \ell\right)} \]

      2. associate-*l* N/A

         \[\leadsto \frac{2}{k \cdot \left(k \cdot \left(\left(k \cdot k\right) \cdot t\right)\right)} \cdot \left(\ell \cdot \ell\right) \]

      3. *-commutative N/A

         \[\leadsto \frac{2}{k \cdot \left(\left(\left(k \cdot k\right) \cdot t\right) \cdot k\right)} \cdot \left(\ell \cdot \ell\right) \]

      4. *-commutative N/A

         \[\leadsto \frac{2}{\left(\left(\left(k \cdot k\right) \cdot t\right) \cdot k\right) \cdot k} \cdot \left(\ell \cdot \ell\right) \]

      5. associate-/r/ N/A

         \[\leadsto \frac{2}{\color{blue}{\frac{\left(\left(\left(k \cdot k\right) \cdot t\right) \cdot k\right) \cdot k}{\ell \cdot \ell}}} \]

      6. times-frac N/A

         \[\leadsto \frac{2}{\frac{\left(\left(k \cdot k\right) \cdot t\right) \cdot k}{\ell} \cdot \color{blue}{\frac{k}{\ell}}} \]

      7. associate-/r* N/A

         \[\leadsto \frac{\frac{2}{\frac{\left(\left(k \cdot k\right) \cdot t\right) \cdot k}{\ell}}}{\color{blue}{\frac{k}{\ell}}} \]

      8. div-inv N/A

         \[\leadsto \frac{2}{\frac{\left(\left(k \cdot k\right) \cdot t\right) \cdot k}{\ell}} \cdot \color{blue}{\frac{1}{\frac{k}{\ell}}} \]

      9. clear-num N/A

         \[\leadsto \frac{2}{\frac{\left(\left(k \cdot k\right) \cdot t\right) \cdot k}{\ell}} \cdot \frac{\ell}{\color{blue}{k}} \]

      10. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\left(\frac{2}{\frac{\left(\left(k \cdot k\right) \cdot t\right) \cdot k}{\ell}}\right), \color{blue}{\left(\frac{\ell}{k}\right)}\right) \]

   9. Applied egg-rr 75.3%

      \[\leadsto \color{blue}{\frac{2}{\frac{k \cdot \left(k \cdot k\right)}{\frac{\ell}{t}}} \cdot \frac{\ell}{k}} \]
   10. Step-by-step derivation

       1. associate-/l* N/A

          \[\leadsto \mathsf{*.f64}\left(\left(\frac{2}{k \cdot \frac{k \cdot k}{\frac{\ell}{t}}}\right), \mathsf{/.f64}\left(\ell, k\right)\right) \]

       2. associate-/r* N/A

          \[\leadsto \mathsf{*.f64}\left(\left(\frac{\frac{2}{k}}{\frac{k \cdot k}{\frac{\ell}{t}}}\right), \mathsf{/.f64}\left(\color{blue}{\ell}, k\right)\right) \]

       3. *-lft-identity N/A

          \[\leadsto \mathsf{*.f64}\left(\left(\frac{\frac{2}{k}}{\frac{1 \cdot \left(k \cdot k\right)}{\frac{\ell}{t}}}\right), \mathsf{/.f64}\left(\ell, k\right)\right) \]

       4. associate-*l/ N/A

          \[\leadsto \mathsf{*.f64}\left(\left(\frac{\frac{2}{k}}{\frac{1}{\frac{\ell}{t}} \cdot \left(k \cdot k\right)}\right), \mathsf{/.f64}\left(\ell, k\right)\right) \]

       5. clear-num N/A

          \[\leadsto \mathsf{*.f64}\left(\left(\frac{\frac{2}{k}}{\frac{t}{\ell} \cdot \left(k \cdot k\right)}\right), \mathsf{/.f64}\left(\ell, k\right)\right) \]

       6. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\left(\frac{2}{k}\right), \left(\frac{t}{\ell} \cdot \left(k \cdot k\right)\right)\right), \mathsf{/.f64}\left(\color{blue}{\ell}, k\right)\right) \]

       7. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(2, k\right), \left(\frac{t}{\ell} \cdot \left(k \cdot k\right)\right)\right), \mathsf{/.f64}\left(\ell, k\right)\right) \]

       8. associate-*l/ N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(2, k\right), \left(\frac{t \cdot \left(k \cdot k\right)}{\ell}\right)\right), \mathsf{/.f64}\left(\ell, k\right)\right) \]

       9. *-commutative N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(2, k\right), \left(\frac{\left(k \cdot k\right) \cdot t}{\ell}\right)\right), \mathsf{/.f64}\left(\ell, k\right)\right) \]

       10. associate-*r* N/A

           \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(2, k\right), \left(\frac{k \cdot \left(k \cdot t\right)}{\ell}\right)\right), \mathsf{/.f64}\left(\ell, k\right)\right) \]

       11. *-commutative N/A

           \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(2, k\right), \left(\frac{\left(k \cdot t\right) \cdot k}{\ell}\right)\right), \mathsf{/.f64}\left(\ell, k\right)\right) \]

       12. associate-/l* N/A

           \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(2, k\right), \left(\left(k \cdot t\right) \cdot \frac{k}{\ell}\right)\right), \mathsf{/.f64}\left(\ell, k\right)\right) \]

       13. clear-num N/A

           \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(2, k\right), \left(\left(k \cdot t\right) \cdot \frac{1}{\frac{\ell}{k}}\right)\right), \mathsf{/.f64}\left(\ell, k\right)\right) \]

       14. un-div-inv N/A

           \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(2, k\right), \left(\frac{k \cdot t}{\frac{\ell}{k}}\right)\right), \mathsf{/.f64}\left(\ell, k\right)\right) \]

       15. /-lowering-/.f64 N/A

           \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(2, k\right), \mathsf{/.f64}\left(\left(k \cdot t\right), \left(\frac{\ell}{k}\right)\right)\right), \mathsf{/.f64}\left(\ell, k\right)\right) \]

       16. *-lowering-*.f64 N/A

           \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(2, k\right), \mathsf{/.f64}\left(\mathsf{*.f64}\left(k, t\right), \left(\frac{\ell}{k}\right)\right)\right), \mathsf{/.f64}\left(\ell, k\right)\right) \]

       17. /-lowering-/.f64 80.1%

           \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(2, k\right), \mathsf{/.f64}\left(\mathsf{*.f64}\left(k, t\right), \mathsf{/.f64}\left(\ell, k\right)\right)\right), \mathsf{/.f64}\left(\ell, k\right)\right) \]

   11. Applied egg-rr 80.1%

       \[\leadsto \color{blue}{\frac{\frac{2}{k}}{\frac{k \cdot t}{\frac{\ell}{k}}}} \cdot \frac{\ell}{k} \]

   if 1.45e19 < k

   1. Initial program 29.6%

      \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) - 1\right)} \]
   2. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \frac{2}{\left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) - 1\right) \cdot \color{blue}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right)}} \]

      2. associate-*r* N/A

         \[\leadsto \frac{2}{\left(\left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) - 1\right) \cdot \left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right)\right) \cdot \color{blue}{\tan k}} \]

      3. associate-/r* N/A

         \[\leadsto \frac{\frac{2}{\left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) - 1\right) \cdot \left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right)}}{\color{blue}{\tan k}} \]

      4. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\left(\frac{2}{\left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) - 1\right) \cdot \left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right)}\right), \color{blue}{\tan k}\right) \]

   3. Simplified 60.7%

      \[\leadsto \color{blue}{\frac{\frac{\frac{\left(\frac{2}{k \cdot k} \cdot t\right) \cdot \frac{t}{t}}{\frac{t \cdot \frac{t}{\ell}}{\ell}}}{\sin k}}{\tan k}} \]
   4. Add Preprocessing
   5. Step-by-step derivation

      1. associate-/l/ N/A

         \[\leadsto \frac{\frac{\left(\frac{2}{k \cdot k} \cdot t\right) \cdot \frac{t}{t}}{\frac{t \cdot \frac{t}{\ell}}{\ell}}}{\color{blue}{\tan k \cdot \sin k}} \]

      2. associate-/r/ N/A

         \[\leadsto \frac{\frac{\left(\frac{2}{k \cdot k} \cdot t\right) \cdot \frac{t}{t}}{t \cdot \frac{t}{\ell}} \cdot \ell}{\color{blue}{\tan k} \cdot \sin k} \]

      3. associate-/l* N/A

         \[\leadsto \frac{\left(\frac{2}{k \cdot k} \cdot t\right) \cdot \frac{t}{t}}{t \cdot \frac{t}{\ell}} \cdot \color{blue}{\frac{\ell}{\tan k \cdot \sin k}} \]

      4. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\left(\frac{\left(\frac{2}{k \cdot k} \cdot t\right) \cdot \frac{t}{t}}{t \cdot \frac{t}{\ell}}\right), \color{blue}{\left(\frac{\ell}{\tan k \cdot \sin k}\right)}\right) \]

      5. *-inverses N/A

         \[\leadsto \mathsf{*.f64}\left(\left(\frac{\left(\frac{2}{k \cdot k} \cdot t\right) \cdot 1}{t \cdot \frac{t}{\ell}}\right), \left(\frac{\ell}{\tan k \cdot \sin k}\right)\right) \]

      6. associate-*l* N/A

         \[\leadsto \mathsf{*.f64}\left(\left(\frac{\frac{2}{k \cdot k} \cdot \left(t \cdot 1\right)}{t \cdot \frac{t}{\ell}}\right), \left(\frac{\ell}{\tan k \cdot \sin k}\right)\right) \]

      7. *-commutative N/A

         \[\leadsto \mathsf{*.f64}\left(\left(\frac{\frac{2}{k \cdot k} \cdot \left(t \cdot 1\right)}{\frac{t}{\ell} \cdot t}\right), \left(\frac{\ell}{\tan k \cdot \sin k}\right)\right) \]

      8. times-frac N/A

         \[\leadsto \mathsf{*.f64}\left(\left(\frac{\frac{2}{k \cdot k}}{\frac{t}{\ell}} \cdot \frac{t \cdot 1}{t}\right), \left(\frac{\color{blue}{\ell}}{\tan k \cdot \sin k}\right)\right) \]

      9. *-rgt-identity N/A

         \[\leadsto \mathsf{*.f64}\left(\left(\frac{\frac{2}{k \cdot k}}{\frac{t}{\ell}} \cdot \frac{t}{t}\right), \left(\frac{\ell}{\tan k \cdot \sin k}\right)\right) \]

      10. *-inverses N/A

          \[\leadsto \mathsf{*.f64}\left(\left(\frac{\frac{2}{k \cdot k}}{\frac{t}{\ell}} \cdot 1\right), \left(\frac{\ell}{\tan k \cdot \sin k}\right)\right) \]

      11. *-rgt-identity N/A

          \[\leadsto \mathsf{*.f64}\left(\left(\frac{\frac{2}{k \cdot k}}{\frac{t}{\ell}}\right), \left(\frac{\color{blue}{\ell}}{\tan k \cdot \sin k}\right)\right) \]

      12. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\left(\frac{2}{k \cdot k}\right), \left(\frac{t}{\ell}\right)\right), \left(\frac{\color{blue}{\ell}}{\tan k \cdot \sin k}\right)\right) \]

      13. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(2, \left(k \cdot k\right)\right), \left(\frac{t}{\ell}\right)\right), \left(\frac{\ell}{\tan k \cdot \sin k}\right)\right) \]

      14. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(2, \mathsf{*.f64}\left(k, k\right)\right), \left(\frac{t}{\ell}\right)\right), \left(\frac{\ell}{\tan k \cdot \sin k}\right)\right) \]

      15. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(2, \mathsf{*.f64}\left(k, k\right)\right), \mathsf{/.f64}\left(t, \ell\right)\right), \left(\frac{\ell}{\tan k \cdot \sin k}\right)\right) \]

      16. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(2, \mathsf{*.f64}\left(k, k\right)\right), \mathsf{/.f64}\left(t, \ell\right)\right), \mathsf{/.f64}\left(\ell, \color{blue}{\left(\tan k \cdot \sin k\right)}\right)\right) \]

   6. Applied egg-rr 83.0%

      \[\leadsto \color{blue}{\frac{\frac{2}{k \cdot k}}{\frac{t}{\ell}} \cdot \frac{\ell}{\sin k \cdot \tan k}} \]

   7. Taylor expanded in k around 0

      \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(2, \mathsf{*.f64}\left(k, k\right)\right), \mathsf{/.f64}\left(t, \ell\right)\right), \mathsf{/.f64}\left(\ell, \mathsf{*.f64}\left(\mathsf{sin.f64}\left(k\right), \color{blue}{k}\right)\right)\right) \]
   8. Step-by-step derivation

   9. Simplified 54.8%

      \[\leadsto \frac{\frac{2}{k \cdot k}}{\frac{t}{\ell}} \cdot \frac{\ell}{\sin k \cdot \color{blue}{k}} \]

   10. Taylor expanded in k around 0

       \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(2, \mathsf{*.f64}\left(k, k\right)\right), \mathsf{/.f64}\left(t, \ell\right)\right), \color{blue}{\left(\frac{\ell + \frac{1}{6} \cdot \left({k}^{2} \cdot \ell\right)}{{k}^{2}}\right)}\right) \]
   11. Step-by-step derivation

       1. unpow2 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(2, \mathsf{*.f64}\left(k, k\right)\right), \mathsf{/.f64}\left(t, \ell\right)\right), \left(\frac{\ell + \frac{1}{6} \cdot \left({k}^{2} \cdot \ell\right)}{k \cdot \color{blue}{k}}\right)\right) \]

       2. associate-/r* N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(2, \mathsf{*.f64}\left(k, k\right)\right), \mathsf{/.f64}\left(t, \ell\right)\right), \left(\frac{\frac{\ell + \frac{1}{6} \cdot \left({k}^{2} \cdot \ell\right)}{k}}{\color{blue}{k}}\right)\right) \]

       3. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(2, \mathsf{*.f64}\left(k, k\right)\right), \mathsf{/.f64}\left(t, \ell\right)\right), \mathsf{/.f64}\left(\left(\frac{\ell + \frac{1}{6} \cdot \left({k}^{2} \cdot \ell\right)}{k}\right), \color{blue}{k}\right)\right) \]

       4. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(2, \mathsf{*.f64}\left(k, k\right)\right), \mathsf{/.f64}\left(t, \ell\right)\right), \mathsf{/.f64}\left(\mathsf{/.f64}\left(\left(\ell + \frac{1}{6} \cdot \left({k}^{2} \cdot \ell\right)\right), k\right), k\right)\right) \]

       5. associate-*r* N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(2, \mathsf{*.f64}\left(k, k\right)\right), \mathsf{/.f64}\left(t, \ell\right)\right), \mathsf{/.f64}\left(\mathsf{/.f64}\left(\left(\ell + \left(\frac{1}{6} \cdot {k}^{2}\right) \cdot \ell\right), k\right), k\right)\right) \]

       6. distribute-rgt1-in N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(2, \mathsf{*.f64}\left(k, k\right)\right), \mathsf{/.f64}\left(t, \ell\right)\right), \mathsf{/.f64}\left(\mathsf{/.f64}\left(\left(\left(\frac{1}{6} \cdot {k}^{2} + 1\right) \cdot \ell\right), k\right), k\right)\right) \]

       7. +-commutative N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(2, \mathsf{*.f64}\left(k, k\right)\right), \mathsf{/.f64}\left(t, \ell\right)\right), \mathsf{/.f64}\left(\mathsf{/.f64}\left(\left(\left(1 + \frac{1}{6} \cdot {k}^{2}\right) \cdot \ell\right), k\right), k\right)\right) \]

       8. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(2, \mathsf{*.f64}\left(k, k\right)\right), \mathsf{/.f64}\left(t, \ell\right)\right), \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\left(1 + \frac{1}{6} \cdot {k}^{2}\right), \ell\right), k\right), k\right)\right) \]

       9. +-lowering-+.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(2, \mathsf{*.f64}\left(k, k\right)\right), \mathsf{/.f64}\left(t, \ell\right)\right), \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \left(\frac{1}{6} \cdot {k}^{2}\right)\right), \ell\right), k\right), k\right)\right) \]

       10. *-commutative N/A

           \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(2, \mathsf{*.f64}\left(k, k\right)\right), \mathsf{/.f64}\left(t, \ell\right)\right), \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \left({k}^{2} \cdot \frac{1}{6}\right)\right), \ell\right), k\right), k\right)\right) \]

       11. *-lowering-*.f64 N/A

           \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(2, \mathsf{*.f64}\left(k, k\right)\right), \mathsf{/.f64}\left(t, \ell\right)\right), \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\left({k}^{2}\right), \frac{1}{6}\right)\right), \ell\right), k\right), k\right)\right) \]

       12. unpow2 N/A

           \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(2, \mathsf{*.f64}\left(k, k\right)\right), \mathsf{/.f64}\left(t, \ell\right)\right), \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\left(k \cdot k\right), \frac{1}{6}\right)\right), \ell\right), k\right), k\right)\right) \]

       13. *-lowering-*.f64 24.6%

           \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(2, \mathsf{*.f64}\left(k, k\right)\right), \mathsf{/.f64}\left(t, \ell\right)\right), \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(k, k\right), \frac{1}{6}\right)\right), \ell\right), k\right), k\right)\right) \]

   12. Simplified 24.6%

       \[\leadsto \frac{\frac{2}{k \cdot k}}{\frac{t}{\ell}} \cdot \color{blue}{\frac{\frac{\left(1 + \left(k \cdot k\right) \cdot 0.16666666666666666\right) \cdot \ell}{k}}{k}} \]

   13. Taylor expanded in k around inf

       \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(2, \mathsf{*.f64}\left(k, k\right)\right), \mathsf{/.f64}\left(t, \ell\right)\right), \color{blue}{\left(\frac{1}{6} \cdot \ell\right)}\right) \]
   14. Step-by-step derivation

       1. *-commutative N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(2, \mathsf{*.f64}\left(k, k\right)\right), \mathsf{/.f64}\left(t, \ell\right)\right), \left(\ell \cdot \color{blue}{\frac{1}{6}}\right)\right) \]

       2. *-lowering-*.f64 55.5%

          \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(2, \mathsf{*.f64}\left(k, k\right)\right), \mathsf{/.f64}\left(t, \ell\right)\right), \mathsf{*.f64}\left(\ell, \color{blue}{\frac{1}{6}}\right)\right) \]

   15. Simplified 55.5%

       \[\leadsto \frac{\frac{2}{k \cdot k}}{\frac{t}{\ell}} \cdot \color{blue}{\left(\ell \cdot 0.16666666666666666\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 73.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;k \leq 1.45 \cdot 10^{+19}:\\ \;\;\;\;\frac{\ell}{k} \cdot \frac{\frac{2}{k}}{\frac{k \cdot t}{\frac{\ell}{k}}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{2}{k \cdot k}}{\frac{t}{\ell}} \cdot \left(\ell \cdot 0.16666666666666666\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 12: 74.9% accurate, 21.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;k \leq 1.45 \cdot 10^{+19}:\\ \;\;\;\;\frac{\ell}{k} \cdot \frac{2}{\frac{k}{\frac{\ell}{k \cdot \left(k \cdot t\right)}}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{2}{k \cdot k}}{\frac{t}{\ell}} \cdot \left(\ell \cdot 0.16666666666666666\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (t l k)
 :precision binary64
 (if (<= k 1.45e+19)
   (* (/ l k) (/ 2.0 (/ k (/ l (* k (* k t))))))
   (* (/ (/ 2.0 (* k k)) (/ t l)) (* l 0.16666666666666666))))

C

double code(double t, double l, double k) {
	double tmp;
	if (k <= 1.45e+19) {
		tmp = (l / k) * (2.0 / (k / (l / (k * (k * t)))));
	} else {
		tmp = ((2.0 / (k * k)) / (t / l)) * (l * 0.16666666666666666);
	}
	return tmp;
}

Fortran

real(8) function code(t, l, k)
    real(8), intent (in) :: t
    real(8), intent (in) :: l
    real(8), intent (in) :: k
    real(8) :: tmp
    if (k <= 1.45d+19) then
        tmp = (l / k) * (2.0d0 / (k / (l / (k * (k * t)))))
    else
        tmp = ((2.0d0 / (k * k)) / (t / l)) * (l * 0.16666666666666666d0)
    end if
    code = tmp
end function

Java

public static double code(double t, double l, double k) {
	double tmp;
	if (k <= 1.45e+19) {
		tmp = (l / k) * (2.0 / (k / (l / (k * (k * t)))));
	} else {
		tmp = ((2.0 / (k * k)) / (t / l)) * (l * 0.16666666666666666);
	}
	return tmp;
}

Python

def code(t, l, k):
	tmp = 0
	if k <= 1.45e+19:
		tmp = (l / k) * (2.0 / (k / (l / (k * (k * t)))))
	else:
		tmp = ((2.0 / (k * k)) / (t / l)) * (l * 0.16666666666666666)
	return tmp

Julia

function code(t, l, k)
	tmp = 0.0
	if (k <= 1.45e+19)
		tmp = Float64(Float64(l / k) * Float64(2.0 / Float64(k / Float64(l / Float64(k * Float64(k * t))))));
	else
		tmp = Float64(Float64(Float64(2.0 / Float64(k * k)) / Float64(t / l)) * Float64(l * 0.16666666666666666));
	end
	return tmp
end

MATLAB

function tmp_2 = code(t, l, k)
	tmp = 0.0;
	if (k <= 1.45e+19)
		tmp = (l / k) * (2.0 / (k / (l / (k * (k * t)))));
	else
		tmp = ((2.0 / (k * k)) / (t / l)) * (l * 0.16666666666666666);
	end
	tmp_2 = tmp;
end

Wolfram

code[t_, l_, k_] := If[LessEqual[k, 1.45e+19], N[(N[(l / k), $MachinePrecision] * N[(2.0 / N[(k / N[(l / N[(k * N[(k * t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(2.0 / N[(k * k), $MachinePrecision]), $MachinePrecision] / N[(t / l), $MachinePrecision]), $MachinePrecision] * N[(l * 0.16666666666666666), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if k < 1.45e19

   1. Initial program 39.4%

      \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) - 1\right)} \]
   2. Add Preprocessing

   3. Taylor expanded in k around 0

      \[\leadsto \mathsf{/.f64}\left(2, \color{blue}{\left(\frac{{k}^{4} \cdot t}{{\ell}^{2}}\right)}\right) \]
   4. Step-by-step derivation

      1. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(2, \mathsf{/.f64}\left(\left({k}^{4} \cdot t\right), \color{blue}{\left({\ell}^{2}\right)}\right)\right) \]

      2. *-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(2, \mathsf{/.f64}\left(\left(t \cdot {k}^{4}\right), \left({\color{blue}{\ell}}^{2}\right)\right)\right) \]

      3. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(2, \mathsf{/.f64}\left(\mathsf{*.f64}\left(t, \left({k}^{4}\right)\right), \left({\color{blue}{\ell}}^{2}\right)\right)\right) \]

      4. metadata-eval N/A

         \[\leadsto \mathsf{/.f64}\left(2, \mathsf{/.f64}\left(\mathsf{*.f64}\left(t, \left({k}^{\left(2 \cdot 2\right)}\right)\right), \left({\ell}^{2}\right)\right)\right) \]

      5. pow-sqr N/A

         \[\leadsto \mathsf{/.f64}\left(2, \mathsf{/.f64}\left(\mathsf{*.f64}\left(t, \left({k}^{2} \cdot {k}^{2}\right)\right), \left({\ell}^{2}\right)\right)\right) \]

      6. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(2, \mathsf{/.f64}\left(\mathsf{*.f64}\left(t, \mathsf{*.f64}\left(\left({k}^{2}\right), \left({k}^{2}\right)\right)\right), \left({\ell}^{2}\right)\right)\right) \]

      7. unpow2 N/A

         \[\leadsto \mathsf{/.f64}\left(2, \mathsf{/.f64}\left(\mathsf{*.f64}\left(t, \mathsf{*.f64}\left(\left(k \cdot k\right), \left({k}^{2}\right)\right)\right), \left({\ell}^{2}\right)\right)\right) \]

      8. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(2, \mathsf{/.f64}\left(\mathsf{*.f64}\left(t, \mathsf{*.f64}\left(\mathsf{*.f64}\left(k, k\right), \left({k}^{2}\right)\right)\right), \left({\ell}^{2}\right)\right)\right) \]

      9. unpow2 N/A

         \[\leadsto \mathsf{/.f64}\left(2, \mathsf{/.f64}\left(\mathsf{*.f64}\left(t, \mathsf{*.f64}\left(\mathsf{*.f64}\left(k, k\right), \left(k \cdot k\right)\right)\right), \left({\ell}^{2}\right)\right)\right) \]

      10. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(2, \mathsf{/.f64}\left(\mathsf{*.f64}\left(t, \mathsf{*.f64}\left(\mathsf{*.f64}\left(k, k\right), \mathsf{*.f64}\left(k, k\right)\right)\right), \left({\ell}^{2}\right)\right)\right) \]

      11. unpow2 N/A

          \[\leadsto \mathsf{/.f64}\left(2, \mathsf{/.f64}\left(\mathsf{*.f64}\left(t, \mathsf{*.f64}\left(\mathsf{*.f64}\left(k, k\right), \mathsf{*.f64}\left(k, k\right)\right)\right), \left(\ell \cdot \color{blue}{\ell}\right)\right)\right) \]

      12. *-lowering-*.f64 63.4%

          \[\leadsto \mathsf{/.f64}\left(2, \mathsf{/.f64}\left(\mathsf{*.f64}\left(t, \mathsf{*.f64}\left(\mathsf{*.f64}\left(k, k\right), \mathsf{*.f64}\left(k, k\right)\right)\right), \mathsf{*.f64}\left(\ell, \color{blue}{\ell}\right)\right)\right) \]

   5. Simplified 63.4%

      \[\leadsto \frac{2}{\color{blue}{\frac{t \cdot \left(\left(k \cdot k\right) \cdot \left(k \cdot k\right)\right)}{\ell \cdot \ell}}} \]
   6. Step-by-step derivation

      1. associate-/r/ N/A

         \[\leadsto \frac{2}{t \cdot \left(\left(k \cdot k\right) \cdot \left(k \cdot k\right)\right)} \cdot \color{blue}{\left(\ell \cdot \ell\right)} \]

      2. associate-*r* N/A

         \[\leadsto \left(\frac{2}{t \cdot \left(\left(k \cdot k\right) \cdot \left(k \cdot k\right)\right)} \cdot \ell\right) \cdot \color{blue}{\ell} \]

      3. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\left(\frac{2}{t \cdot \left(\left(k \cdot k\right) \cdot \left(k \cdot k\right)\right)} \cdot \ell\right), \color{blue}{\ell}\right) \]

      4. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\left(\frac{2}{t \cdot \left(\left(k \cdot k\right) \cdot \left(k \cdot k\right)\right)}\right), \ell\right), \ell\right) \]

      5. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(2, \left(t \cdot \left(\left(k \cdot k\right) \cdot \left(k \cdot k\right)\right)\right)\right), \ell\right), \ell\right) \]

      6. *-commutative N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(2, \left(\left(\left(k \cdot k\right) \cdot \left(k \cdot k\right)\right) \cdot t\right)\right), \ell\right), \ell\right) \]

      7. associate-*r* N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(2, \left(\left(\left(\left(k \cdot k\right) \cdot k\right) \cdot k\right) \cdot t\right)\right), \ell\right), \ell\right) \]

      8. *-commutative N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(2, \left(\left(k \cdot \left(\left(k \cdot k\right) \cdot k\right)\right) \cdot t\right)\right), \ell\right), \ell\right) \]

      9. associate-*l* N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(2, \left(k \cdot \left(\left(\left(k \cdot k\right) \cdot k\right) \cdot t\right)\right)\right), \ell\right), \ell\right) \]

      10. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(2, \mathsf{*.f64}\left(k, \left(\left(\left(k \cdot k\right) \cdot k\right) \cdot t\right)\right)\right), \ell\right), \ell\right) \]

      11. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(2, \mathsf{*.f64}\left(k, \mathsf{*.f64}\left(\left(\left(k \cdot k\right) \cdot k\right), t\right)\right)\right), \ell\right), \ell\right) \]

      12. associate-*r* N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(2, \mathsf{*.f64}\left(k, \mathsf{*.f64}\left(\left(k \cdot \left(k \cdot k\right)\right), t\right)\right)\right), \ell\right), \ell\right) \]

      13. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(2, \mathsf{*.f64}\left(k, \mathsf{*.f64}\left(\mathsf{*.f64}\left(k, \left(k \cdot k\right)\right), t\right)\right)\right), \ell\right), \ell\right) \]

      14. *-lowering-*.f64 72.1%

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(2, \mathsf{*.f64}\left(k, \mathsf{*.f64}\left(\mathsf{*.f64}\left(k, \mathsf{*.f64}\left(k, k\right)\right), t\right)\right)\right), \ell\right), \ell\right) \]

   7. Applied egg-rr 72.1%

      \[\leadsto \color{blue}{\left(\frac{2}{k \cdot \left(\left(k \cdot \left(k \cdot k\right)\right) \cdot t\right)} \cdot \ell\right) \cdot \ell} \]
   8. Step-by-step derivation

      1. associate-*l* N/A

         \[\leadsto \frac{2}{k \cdot \left(\left(k \cdot \left(k \cdot k\right)\right) \cdot t\right)} \cdot \color{blue}{\left(\ell \cdot \ell\right)} \]

      2. associate-*l* N/A

         \[\leadsto \frac{2}{k \cdot \left(k \cdot \left(\left(k \cdot k\right) \cdot t\right)\right)} \cdot \left(\ell \cdot \ell\right) \]

      3. *-commutative N/A

         \[\leadsto \frac{2}{k \cdot \left(\left(\left(k \cdot k\right) \cdot t\right) \cdot k\right)} \cdot \left(\ell \cdot \ell\right) \]

      4. *-commutative N/A

         \[\leadsto \frac{2}{\left(\left(\left(k \cdot k\right) \cdot t\right) \cdot k\right) \cdot k} \cdot \left(\ell \cdot \ell\right) \]

      5. associate-/r/ N/A

         \[\leadsto \frac{2}{\color{blue}{\frac{\left(\left(\left(k \cdot k\right) \cdot t\right) \cdot k\right) \cdot k}{\ell \cdot \ell}}} \]

      6. times-frac N/A

         \[\leadsto \frac{2}{\frac{\left(\left(k \cdot k\right) \cdot t\right) \cdot k}{\ell} \cdot \color{blue}{\frac{k}{\ell}}} \]

      7. associate-/r* N/A

         \[\leadsto \frac{\frac{2}{\frac{\left(\left(k \cdot k\right) \cdot t\right) \cdot k}{\ell}}}{\color{blue}{\frac{k}{\ell}}} \]

      8. div-inv N/A

         \[\leadsto \frac{2}{\frac{\left(\left(k \cdot k\right) \cdot t\right) \cdot k}{\ell}} \cdot \color{blue}{\frac{1}{\frac{k}{\ell}}} \]

      9. clear-num N/A

         \[\leadsto \frac{2}{\frac{\left(\left(k \cdot k\right) \cdot t\right) \cdot k}{\ell}} \cdot \frac{\ell}{\color{blue}{k}} \]

      10. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\left(\frac{2}{\frac{\left(\left(k \cdot k\right) \cdot t\right) \cdot k}{\ell}}\right), \color{blue}{\left(\frac{\ell}{k}\right)}\right) \]

   9. Applied egg-rr 75.3%

      \[\leadsto \color{blue}{\frac{2}{\frac{k \cdot \left(k \cdot k\right)}{\frac{\ell}{t}}} \cdot \frac{\ell}{k}} \]
   10. Step-by-step derivation

       1. clear-num N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(2, \left(\frac{1}{\frac{\frac{\ell}{t}}{k \cdot \left(k \cdot k\right)}}\right)\right), \mathsf{/.f64}\left(\ell, k\right)\right) \]

       2. associate-/l/ N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(2, \left(\frac{1}{\frac{\ell}{\left(k \cdot \left(k \cdot k\right)\right) \cdot t}}\right)\right), \mathsf{/.f64}\left(\ell, k\right)\right) \]

       3. associate-*r* N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(2, \left(\frac{1}{\frac{\ell}{k \cdot \left(\left(k \cdot k\right) \cdot t\right)}}\right)\right), \mathsf{/.f64}\left(\ell, k\right)\right) \]

       4. associate-*r* N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(2, \left(\frac{1}{\frac{\ell}{k \cdot \left(k \cdot \left(k \cdot t\right)\right)}}\right)\right), \mathsf{/.f64}\left(\ell, k\right)\right) \]

       5. associate-/l/ N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(2, \left(\frac{1}{\frac{\frac{\ell}{k \cdot \left(k \cdot t\right)}}{k}}\right)\right), \mathsf{/.f64}\left(\ell, k\right)\right) \]

       6. clear-num N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(2, \left(\frac{k}{\frac{\ell}{k \cdot \left(k \cdot t\right)}}\right)\right), \mathsf{/.f64}\left(\ell, k\right)\right) \]

       7. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(2, \mathsf{/.f64}\left(k, \left(\frac{\ell}{k \cdot \left(k \cdot t\right)}\right)\right)\right), \mathsf{/.f64}\left(\ell, k\right)\right) \]

       8. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(2, \mathsf{/.f64}\left(k, \mathsf{/.f64}\left(\ell, \left(k \cdot \left(k \cdot t\right)\right)\right)\right)\right), \mathsf{/.f64}\left(\ell, k\right)\right) \]

       9. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(2, \mathsf{/.f64}\left(k, \mathsf{/.f64}\left(\ell, \mathsf{*.f64}\left(k, \left(k \cdot t\right)\right)\right)\right)\right), \mathsf{/.f64}\left(\ell, k\right)\right) \]

       10. *-lowering-*.f64 80.1%

           \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(2, \mathsf{/.f64}\left(k, \mathsf{/.f64}\left(\ell, \mathsf{*.f64}\left(k, \mathsf{*.f64}\left(k, t\right)\right)\right)\right)\right), \mathsf{/.f64}\left(\ell, k\right)\right) \]

   11. Applied egg-rr 80.1%

       \[\leadsto \frac{2}{\color{blue}{\frac{k}{\frac{\ell}{k \cdot \left(k \cdot t\right)}}}} \cdot \frac{\ell}{k} \]

   if 1.45e19 < k

   1. Initial program 29.6%

      \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) - 1\right)} \]
   2. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \frac{2}{\left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) - 1\right) \cdot \color{blue}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right)}} \]

      2. associate-*r* N/A

         \[\leadsto \frac{2}{\left(\left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) - 1\right) \cdot \left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right)\right) \cdot \color{blue}{\tan k}} \]

      3. associate-/r* N/A

         \[\leadsto \frac{\frac{2}{\left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) - 1\right) \cdot \left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right)}}{\color{blue}{\tan k}} \]

      4. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\left(\frac{2}{\left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) - 1\right) \cdot \left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right)}\right), \color{blue}{\tan k}\right) \]

   3. Simplified 60.7%

      \[\leadsto \color{blue}{\frac{\frac{\frac{\left(\frac{2}{k \cdot k} \cdot t\right) \cdot \frac{t}{t}}{\frac{t \cdot \frac{t}{\ell}}{\ell}}}{\sin k}}{\tan k}} \]
   4. Add Preprocessing
   5. Step-by-step derivation

      1. associate-/l/ N/A

         \[\leadsto \frac{\frac{\left(\frac{2}{k \cdot k} \cdot t\right) \cdot \frac{t}{t}}{\frac{t \cdot \frac{t}{\ell}}{\ell}}}{\color{blue}{\tan k \cdot \sin k}} \]

      2. associate-/r/ N/A

         \[\leadsto \frac{\frac{\left(\frac{2}{k \cdot k} \cdot t\right) \cdot \frac{t}{t}}{t \cdot \frac{t}{\ell}} \cdot \ell}{\color{blue}{\tan k} \cdot \sin k} \]

      3. associate-/l* N/A

         \[\leadsto \frac{\left(\frac{2}{k \cdot k} \cdot t\right) \cdot \frac{t}{t}}{t \cdot \frac{t}{\ell}} \cdot \color{blue}{\frac{\ell}{\tan k \cdot \sin k}} \]

      4. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\left(\frac{\left(\frac{2}{k \cdot k} \cdot t\right) \cdot \frac{t}{t}}{t \cdot \frac{t}{\ell}}\right), \color{blue}{\left(\frac{\ell}{\tan k \cdot \sin k}\right)}\right) \]

      5. *-inverses N/A

         \[\leadsto \mathsf{*.f64}\left(\left(\frac{\left(\frac{2}{k \cdot k} \cdot t\right) \cdot 1}{t \cdot \frac{t}{\ell}}\right), \left(\frac{\ell}{\tan k \cdot \sin k}\right)\right) \]

      6. associate-*l* N/A

         \[\leadsto \mathsf{*.f64}\left(\left(\frac{\frac{2}{k \cdot k} \cdot \left(t \cdot 1\right)}{t \cdot \frac{t}{\ell}}\right), \left(\frac{\ell}{\tan k \cdot \sin k}\right)\right) \]

      7. *-commutative N/A

         \[\leadsto \mathsf{*.f64}\left(\left(\frac{\frac{2}{k \cdot k} \cdot \left(t \cdot 1\right)}{\frac{t}{\ell} \cdot t}\right), \left(\frac{\ell}{\tan k \cdot \sin k}\right)\right) \]

      8. times-frac N/A

         \[\leadsto \mathsf{*.f64}\left(\left(\frac{\frac{2}{k \cdot k}}{\frac{t}{\ell}} \cdot \frac{t \cdot 1}{t}\right), \left(\frac{\color{blue}{\ell}}{\tan k \cdot \sin k}\right)\right) \]

      9. *-rgt-identity N/A

         \[\leadsto \mathsf{*.f64}\left(\left(\frac{\frac{2}{k \cdot k}}{\frac{t}{\ell}} \cdot \frac{t}{t}\right), \left(\frac{\ell}{\tan k \cdot \sin k}\right)\right) \]

      10. *-inverses N/A

          \[\leadsto \mathsf{*.f64}\left(\left(\frac{\frac{2}{k \cdot k}}{\frac{t}{\ell}} \cdot 1\right), \left(\frac{\ell}{\tan k \cdot \sin k}\right)\right) \]

      11. *-rgt-identity N/A

          \[\leadsto \mathsf{*.f64}\left(\left(\frac{\frac{2}{k \cdot k}}{\frac{t}{\ell}}\right), \left(\frac{\color{blue}{\ell}}{\tan k \cdot \sin k}\right)\right) \]

      12. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\left(\frac{2}{k \cdot k}\right), \left(\frac{t}{\ell}\right)\right), \left(\frac{\color{blue}{\ell}}{\tan k \cdot \sin k}\right)\right) \]

      13. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(2, \left(k \cdot k\right)\right), \left(\frac{t}{\ell}\right)\right), \left(\frac{\ell}{\tan k \cdot \sin k}\right)\right) \]

      14. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(2, \mathsf{*.f64}\left(k, k\right)\right), \left(\frac{t}{\ell}\right)\right), \left(\frac{\ell}{\tan k \cdot \sin k}\right)\right) \]

      15. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(2, \mathsf{*.f64}\left(k, k\right)\right), \mathsf{/.f64}\left(t, \ell\right)\right), \left(\frac{\ell}{\tan k \cdot \sin k}\right)\right) \]

      16. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(2, \mathsf{*.f64}\left(k, k\right)\right), \mathsf{/.f64}\left(t, \ell\right)\right), \mathsf{/.f64}\left(\ell, \color{blue}{\left(\tan k \cdot \sin k\right)}\right)\right) \]

   6. Applied egg-rr 83.0%

      \[\leadsto \color{blue}{\frac{\frac{2}{k \cdot k}}{\frac{t}{\ell}} \cdot \frac{\ell}{\sin k \cdot \tan k}} \]

   7. Taylor expanded in k around 0

      \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(2, \mathsf{*.f64}\left(k, k\right)\right), \mathsf{/.f64}\left(t, \ell\right)\right), \mathsf{/.f64}\left(\ell, \mathsf{*.f64}\left(\mathsf{sin.f64}\left(k\right), \color{blue}{k}\right)\right)\right) \]
   8. Step-by-step derivation

   9. Simplified 54.8%

      \[\leadsto \frac{\frac{2}{k \cdot k}}{\frac{t}{\ell}} \cdot \frac{\ell}{\sin k \cdot \color{blue}{k}} \]

   10. Taylor expanded in k around 0

       \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(2, \mathsf{*.f64}\left(k, k\right)\right), \mathsf{/.f64}\left(t, \ell\right)\right), \color{blue}{\left(\frac{\ell + \frac{1}{6} \cdot \left({k}^{2} \cdot \ell\right)}{{k}^{2}}\right)}\right) \]
   11. Step-by-step derivation

       1. unpow2 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(2, \mathsf{*.f64}\left(k, k\right)\right), \mathsf{/.f64}\left(t, \ell\right)\right), \left(\frac{\ell + \frac{1}{6} \cdot \left({k}^{2} \cdot \ell\right)}{k \cdot \color{blue}{k}}\right)\right) \]

       2. associate-/r* N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(2, \mathsf{*.f64}\left(k, k\right)\right), \mathsf{/.f64}\left(t, \ell\right)\right), \left(\frac{\frac{\ell + \frac{1}{6} \cdot \left({k}^{2} \cdot \ell\right)}{k}}{\color{blue}{k}}\right)\right) \]

       3. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(2, \mathsf{*.f64}\left(k, k\right)\right), \mathsf{/.f64}\left(t, \ell\right)\right), \mathsf{/.f64}\left(\left(\frac{\ell + \frac{1}{6} \cdot \left({k}^{2} \cdot \ell\right)}{k}\right), \color{blue}{k}\right)\right) \]

       4. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(2, \mathsf{*.f64}\left(k, k\right)\right), \mathsf{/.f64}\left(t, \ell\right)\right), \mathsf{/.f64}\left(\mathsf{/.f64}\left(\left(\ell + \frac{1}{6} \cdot \left({k}^{2} \cdot \ell\right)\right), k\right), k\right)\right) \]

       5. associate-*r* N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(2, \mathsf{*.f64}\left(k, k\right)\right), \mathsf{/.f64}\left(t, \ell\right)\right), \mathsf{/.f64}\left(\mathsf{/.f64}\left(\left(\ell + \left(\frac{1}{6} \cdot {k}^{2}\right) \cdot \ell\right), k\right), k\right)\right) \]

       6. distribute-rgt1-in N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(2, \mathsf{*.f64}\left(k, k\right)\right), \mathsf{/.f64}\left(t, \ell\right)\right), \mathsf{/.f64}\left(\mathsf{/.f64}\left(\left(\left(\frac{1}{6} \cdot {k}^{2} + 1\right) \cdot \ell\right), k\right), k\right)\right) \]

       7. +-commutative N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(2, \mathsf{*.f64}\left(k, k\right)\right), \mathsf{/.f64}\left(t, \ell\right)\right), \mathsf{/.f64}\left(\mathsf{/.f64}\left(\left(\left(1 + \frac{1}{6} \cdot {k}^{2}\right) \cdot \ell\right), k\right), k\right)\right) \]

       8. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(2, \mathsf{*.f64}\left(k, k\right)\right), \mathsf{/.f64}\left(t, \ell\right)\right), \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\left(1 + \frac{1}{6} \cdot {k}^{2}\right), \ell\right), k\right), k\right)\right) \]

       9. +-lowering-+.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(2, \mathsf{*.f64}\left(k, k\right)\right), \mathsf{/.f64}\left(t, \ell\right)\right), \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \left(\frac{1}{6} \cdot {k}^{2}\right)\right), \ell\right), k\right), k\right)\right) \]

       10. *-commutative N/A

           \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(2, \mathsf{*.f64}\left(k, k\right)\right), \mathsf{/.f64}\left(t, \ell\right)\right), \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \left({k}^{2} \cdot \frac{1}{6}\right)\right), \ell\right), k\right), k\right)\right) \]

       11. *-lowering-*.f64 N/A

           \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(2, \mathsf{*.f64}\left(k, k\right)\right), \mathsf{/.f64}\left(t, \ell\right)\right), \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\left({k}^{2}\right), \frac{1}{6}\right)\right), \ell\right), k\right), k\right)\right) \]

       12. unpow2 N/A

           \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(2, \mathsf{*.f64}\left(k, k\right)\right), \mathsf{/.f64}\left(t, \ell\right)\right), \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\left(k \cdot k\right), \frac{1}{6}\right)\right), \ell\right), k\right), k\right)\right) \]

       13. *-lowering-*.f64 24.6%

           \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(2, \mathsf{*.f64}\left(k, k\right)\right), \mathsf{/.f64}\left(t, \ell\right)\right), \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(k, k\right), \frac{1}{6}\right)\right), \ell\right), k\right), k\right)\right) \]

   12. Simplified 24.6%

       \[\leadsto \frac{\frac{2}{k \cdot k}}{\frac{t}{\ell}} \cdot \color{blue}{\frac{\frac{\left(1 + \left(k \cdot k\right) \cdot 0.16666666666666666\right) \cdot \ell}{k}}{k}} \]

   13. Taylor expanded in k around inf

       \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(2, \mathsf{*.f64}\left(k, k\right)\right), \mathsf{/.f64}\left(t, \ell\right)\right), \color{blue}{\left(\frac{1}{6} \cdot \ell\right)}\right) \]
   14. Step-by-step derivation

       1. *-commutative N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(2, \mathsf{*.f64}\left(k, k\right)\right), \mathsf{/.f64}\left(t, \ell\right)\right), \left(\ell \cdot \color{blue}{\frac{1}{6}}\right)\right) \]

       2. *-lowering-*.f64 55.5%

          \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(2, \mathsf{*.f64}\left(k, k\right)\right), \mathsf{/.f64}\left(t, \ell\right)\right), \mathsf{*.f64}\left(\ell, \color{blue}{\frac{1}{6}}\right)\right) \]

   15. Simplified 55.5%

       \[\leadsto \frac{\frac{2}{k \cdot k}}{\frac{t}{\ell}} \cdot \color{blue}{\left(\ell \cdot 0.16666666666666666\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 73.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;k \leq 1.45 \cdot 10^{+19}:\\ \;\;\;\;\frac{\ell}{k} \cdot \frac{2}{\frac{k}{\frac{\ell}{k \cdot \left(k \cdot t\right)}}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{2}{k \cdot k}}{\frac{t}{\ell}} \cdot \left(\ell \cdot 0.16666666666666666\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 13: 70.2% accurate, 21.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;k \leq 1.45 \cdot 10^{+19}:\\ \;\;\;\;\ell \cdot \left(\ell \cdot \frac{2}{k \cdot \left(t \cdot \left(k \cdot \left(k \cdot k\right)\right)\right)}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{2}{k \cdot k}}{\frac{t}{\ell}} \cdot \left(\ell \cdot 0.16666666666666666\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (t l k)
 :precision binary64
 (if (<= k 1.45e+19)
   (* l (* l (/ 2.0 (* k (* t (* k (* k k)))))))
   (* (/ (/ 2.0 (* k k)) (/ t l)) (* l 0.16666666666666666))))

C

double code(double t, double l, double k) {
	double tmp;
	if (k <= 1.45e+19) {
		tmp = l * (l * (2.0 / (k * (t * (k * (k * k))))));
	} else {
		tmp = ((2.0 / (k * k)) / (t / l)) * (l * 0.16666666666666666);
	}
	return tmp;
}

Fortran

real(8) function code(t, l, k)
    real(8), intent (in) :: t
    real(8), intent (in) :: l
    real(8), intent (in) :: k
    real(8) :: tmp
    if (k <= 1.45d+19) then
        tmp = l * (l * (2.0d0 / (k * (t * (k * (k * k))))))
    else
        tmp = ((2.0d0 / (k * k)) / (t / l)) * (l * 0.16666666666666666d0)
    end if
    code = tmp
end function

Java

public static double code(double t, double l, double k) {
	double tmp;
	if (k <= 1.45e+19) {
		tmp = l * (l * (2.0 / (k * (t * (k * (k * k))))));
	} else {
		tmp = ((2.0 / (k * k)) / (t / l)) * (l * 0.16666666666666666);
	}
	return tmp;
}

Python

def code(t, l, k):
	tmp = 0
	if k <= 1.45e+19:
		tmp = l * (l * (2.0 / (k * (t * (k * (k * k))))))
	else:
		tmp = ((2.0 / (k * k)) / (t / l)) * (l * 0.16666666666666666)
	return tmp

Julia

function code(t, l, k)
	tmp = 0.0
	if (k <= 1.45e+19)
		tmp = Float64(l * Float64(l * Float64(2.0 / Float64(k * Float64(t * Float64(k * Float64(k * k)))))));
	else
		tmp = Float64(Float64(Float64(2.0 / Float64(k * k)) / Float64(t / l)) * Float64(l * 0.16666666666666666));
	end
	return tmp
end

MATLAB

function tmp_2 = code(t, l, k)
	tmp = 0.0;
	if (k <= 1.45e+19)
		tmp = l * (l * (2.0 / (k * (t * (k * (k * k))))));
	else
		tmp = ((2.0 / (k * k)) / (t / l)) * (l * 0.16666666666666666);
	end
	tmp_2 = tmp;
end

Wolfram

code[t_, l_, k_] := If[LessEqual[k, 1.45e+19], N[(l * N[(l * N[(2.0 / N[(k * N[(t * N[(k * N[(k * k), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(2.0 / N[(k * k), $MachinePrecision]), $MachinePrecision] / N[(t / l), $MachinePrecision]), $MachinePrecision] * N[(l * 0.16666666666666666), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if k < 1.45e19

   1. Initial program 39.4%

      \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) - 1\right)} \]
   2. Add Preprocessing

   3. Taylor expanded in k around 0

      \[\leadsto \mathsf{/.f64}\left(2, \color{blue}{\left(\frac{{k}^{4} \cdot t}{{\ell}^{2}}\right)}\right) \]
   4. Step-by-step derivation

      1. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(2, \mathsf{/.f64}\left(\left({k}^{4} \cdot t\right), \color{blue}{\left({\ell}^{2}\right)}\right)\right) \]

      2. *-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(2, \mathsf{/.f64}\left(\left(t \cdot {k}^{4}\right), \left({\color{blue}{\ell}}^{2}\right)\right)\right) \]

      3. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(2, \mathsf{/.f64}\left(\mathsf{*.f64}\left(t, \left({k}^{4}\right)\right), \left({\color{blue}{\ell}}^{2}\right)\right)\right) \]

      4. metadata-eval N/A

         \[\leadsto \mathsf{/.f64}\left(2, \mathsf{/.f64}\left(\mathsf{*.f64}\left(t, \left({k}^{\left(2 \cdot 2\right)}\right)\right), \left({\ell}^{2}\right)\right)\right) \]

      5. pow-sqr N/A

         \[\leadsto \mathsf{/.f64}\left(2, \mathsf{/.f64}\left(\mathsf{*.f64}\left(t, \left({k}^{2} \cdot {k}^{2}\right)\right), \left({\ell}^{2}\right)\right)\right) \]

      6. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(2, \mathsf{/.f64}\left(\mathsf{*.f64}\left(t, \mathsf{*.f64}\left(\left({k}^{2}\right), \left({k}^{2}\right)\right)\right), \left({\ell}^{2}\right)\right)\right) \]

      7. unpow2 N/A

         \[\leadsto \mathsf{/.f64}\left(2, \mathsf{/.f64}\left(\mathsf{*.f64}\left(t, \mathsf{*.f64}\left(\left(k \cdot k\right), \left({k}^{2}\right)\right)\right), \left({\ell}^{2}\right)\right)\right) \]

      8. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(2, \mathsf{/.f64}\left(\mathsf{*.f64}\left(t, \mathsf{*.f64}\left(\mathsf{*.f64}\left(k, k\right), \left({k}^{2}\right)\right)\right), \left({\ell}^{2}\right)\right)\right) \]

      9. unpow2 N/A

         \[\leadsto \mathsf{/.f64}\left(2, \mathsf{/.f64}\left(\mathsf{*.f64}\left(t, \mathsf{*.f64}\left(\mathsf{*.f64}\left(k, k\right), \left(k \cdot k\right)\right)\right), \left({\ell}^{2}\right)\right)\right) \]

      10. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(2, \mathsf{/.f64}\left(\mathsf{*.f64}\left(t, \mathsf{*.f64}\left(\mathsf{*.f64}\left(k, k\right), \mathsf{*.f64}\left(k, k\right)\right)\right), \left({\ell}^{2}\right)\right)\right) \]

      11. unpow2 N/A

          \[\leadsto \mathsf{/.f64}\left(2, \mathsf{/.f64}\left(\mathsf{*.f64}\left(t, \mathsf{*.f64}\left(\mathsf{*.f64}\left(k, k\right), \mathsf{*.f64}\left(k, k\right)\right)\right), \left(\ell \cdot \color{blue}{\ell}\right)\right)\right) \]

      12. *-lowering-*.f64 63.4%

          \[\leadsto \mathsf{/.f64}\left(2, \mathsf{/.f64}\left(\mathsf{*.f64}\left(t, \mathsf{*.f64}\left(\mathsf{*.f64}\left(k, k\right), \mathsf{*.f64}\left(k, k\right)\right)\right), \mathsf{*.f64}\left(\ell, \color{blue}{\ell}\right)\right)\right) \]

   5. Simplified 63.4%

      \[\leadsto \frac{2}{\color{blue}{\frac{t \cdot \left(\left(k \cdot k\right) \cdot \left(k \cdot k\right)\right)}{\ell \cdot \ell}}} \]
   6. Step-by-step derivation

      1. associate-/r/ N/A

         \[\leadsto \frac{2}{t \cdot \left(\left(k \cdot k\right) \cdot \left(k \cdot k\right)\right)} \cdot \color{blue}{\left(\ell \cdot \ell\right)} \]

      2. associate-*r* N/A

         \[\leadsto \left(\frac{2}{t \cdot \left(\left(k \cdot k\right) \cdot \left(k \cdot k\right)\right)} \cdot \ell\right) \cdot \color{blue}{\ell} \]

      3. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\left(\frac{2}{t \cdot \left(\left(k \cdot k\right) \cdot \left(k \cdot k\right)\right)} \cdot \ell\right), \color{blue}{\ell}\right) \]

      4. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\left(\frac{2}{t \cdot \left(\left(k \cdot k\right) \cdot \left(k \cdot k\right)\right)}\right), \ell\right), \ell\right) \]

      5. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(2, \left(t \cdot \left(\left(k \cdot k\right) \cdot \left(k \cdot k\right)\right)\right)\right), \ell\right), \ell\right) \]

      6. *-commutative N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(2, \left(\left(\left(k \cdot k\right) \cdot \left(k \cdot k\right)\right) \cdot t\right)\right), \ell\right), \ell\right) \]

      7. associate-*r* N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(2, \left(\left(\left(\left(k \cdot k\right) \cdot k\right) \cdot k\right) \cdot t\right)\right), \ell\right), \ell\right) \]

      8. *-commutative N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(2, \left(\left(k \cdot \left(\left(k \cdot k\right) \cdot k\right)\right) \cdot t\right)\right), \ell\right), \ell\right) \]

      9. associate-*l* N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(2, \left(k \cdot \left(\left(\left(k \cdot k\right) \cdot k\right) \cdot t\right)\right)\right), \ell\right), \ell\right) \]

      10. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(2, \mathsf{*.f64}\left(k, \left(\left(\left(k \cdot k\right) \cdot k\right) \cdot t\right)\right)\right), \ell\right), \ell\right) \]

      11. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(2, \mathsf{*.f64}\left(k, \mathsf{*.f64}\left(\left(\left(k \cdot k\right) \cdot k\right), t\right)\right)\right), \ell\right), \ell\right) \]

      12. associate-*r* N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(2, \mathsf{*.f64}\left(k, \mathsf{*.f64}\left(\left(k \cdot \left(k \cdot k\right)\right), t\right)\right)\right), \ell\right), \ell\right) \]

      13. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(2, \mathsf{*.f64}\left(k, \mathsf{*.f64}\left(\mathsf{*.f64}\left(k, \left(k \cdot k\right)\right), t\right)\right)\right), \ell\right), \ell\right) \]

      14. *-lowering-*.f64 72.1%

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(2, \mathsf{*.f64}\left(k, \mathsf{*.f64}\left(\mathsf{*.f64}\left(k, \mathsf{*.f64}\left(k, k\right)\right), t\right)\right)\right), \ell\right), \ell\right) \]

   7. Applied egg-rr 72.1%

      \[\leadsto \color{blue}{\left(\frac{2}{k \cdot \left(\left(k \cdot \left(k \cdot k\right)\right) \cdot t\right)} \cdot \ell\right) \cdot \ell} \]

   if 1.45e19 < k

   1. Initial program 29.6%

      \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) - 1\right)} \]
   2. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \frac{2}{\left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) - 1\right) \cdot \color{blue}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right)}} \]

      2. associate-*r* N/A

         \[\leadsto \frac{2}{\left(\left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) - 1\right) \cdot \left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right)\right) \cdot \color{blue}{\tan k}} \]

      3. associate-/r* N/A

         \[\leadsto \frac{\frac{2}{\left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) - 1\right) \cdot \left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right)}}{\color{blue}{\tan k}} \]

      4. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\left(\frac{2}{\left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) - 1\right) \cdot \left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right)}\right), \color{blue}{\tan k}\right) \]

   3. Simplified 60.7%

      \[\leadsto \color{blue}{\frac{\frac{\frac{\left(\frac{2}{k \cdot k} \cdot t\right) \cdot \frac{t}{t}}{\frac{t \cdot \frac{t}{\ell}}{\ell}}}{\sin k}}{\tan k}} \]
   4. Add Preprocessing
   5. Step-by-step derivation

      1. associate-/l/ N/A

         \[\leadsto \frac{\frac{\left(\frac{2}{k \cdot k} \cdot t\right) \cdot \frac{t}{t}}{\frac{t \cdot \frac{t}{\ell}}{\ell}}}{\color{blue}{\tan k \cdot \sin k}} \]

      2. associate-/r/ N/A

         \[\leadsto \frac{\frac{\left(\frac{2}{k \cdot k} \cdot t\right) \cdot \frac{t}{t}}{t \cdot \frac{t}{\ell}} \cdot \ell}{\color{blue}{\tan k} \cdot \sin k} \]

      3. associate-/l* N/A

         \[\leadsto \frac{\left(\frac{2}{k \cdot k} \cdot t\right) \cdot \frac{t}{t}}{t \cdot \frac{t}{\ell}} \cdot \color{blue}{\frac{\ell}{\tan k \cdot \sin k}} \]

      4. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\left(\frac{\left(\frac{2}{k \cdot k} \cdot t\right) \cdot \frac{t}{t}}{t \cdot \frac{t}{\ell}}\right), \color{blue}{\left(\frac{\ell}{\tan k \cdot \sin k}\right)}\right) \]

      5. *-inverses N/A

         \[\leadsto \mathsf{*.f64}\left(\left(\frac{\left(\frac{2}{k \cdot k} \cdot t\right) \cdot 1}{t \cdot \frac{t}{\ell}}\right), \left(\frac{\ell}{\tan k \cdot \sin k}\right)\right) \]

      6. associate-*l* N/A

         \[\leadsto \mathsf{*.f64}\left(\left(\frac{\frac{2}{k \cdot k} \cdot \left(t \cdot 1\right)}{t \cdot \frac{t}{\ell}}\right), \left(\frac{\ell}{\tan k \cdot \sin k}\right)\right) \]

      7. *-commutative N/A

         \[\leadsto \mathsf{*.f64}\left(\left(\frac{\frac{2}{k \cdot k} \cdot \left(t \cdot 1\right)}{\frac{t}{\ell} \cdot t}\right), \left(\frac{\ell}{\tan k \cdot \sin k}\right)\right) \]

      8. times-frac N/A

         \[\leadsto \mathsf{*.f64}\left(\left(\frac{\frac{2}{k \cdot k}}{\frac{t}{\ell}} \cdot \frac{t \cdot 1}{t}\right), \left(\frac{\color{blue}{\ell}}{\tan k \cdot \sin k}\right)\right) \]

      9. *-rgt-identity N/A

         \[\leadsto \mathsf{*.f64}\left(\left(\frac{\frac{2}{k \cdot k}}{\frac{t}{\ell}} \cdot \frac{t}{t}\right), \left(\frac{\ell}{\tan k \cdot \sin k}\right)\right) \]

      10. *-inverses N/A

          \[\leadsto \mathsf{*.f64}\left(\left(\frac{\frac{2}{k \cdot k}}{\frac{t}{\ell}} \cdot 1\right), \left(\frac{\ell}{\tan k \cdot \sin k}\right)\right) \]

      11. *-rgt-identity N/A

          \[\leadsto \mathsf{*.f64}\left(\left(\frac{\frac{2}{k \cdot k}}{\frac{t}{\ell}}\right), \left(\frac{\color{blue}{\ell}}{\tan k \cdot \sin k}\right)\right) \]

      12. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\left(\frac{2}{k \cdot k}\right), \left(\frac{t}{\ell}\right)\right), \left(\frac{\color{blue}{\ell}}{\tan k \cdot \sin k}\right)\right) \]

      13. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(2, \left(k \cdot k\right)\right), \left(\frac{t}{\ell}\right)\right), \left(\frac{\ell}{\tan k \cdot \sin k}\right)\right) \]

      14. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(2, \mathsf{*.f64}\left(k, k\right)\right), \left(\frac{t}{\ell}\right)\right), \left(\frac{\ell}{\tan k \cdot \sin k}\right)\right) \]

      15. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(2, \mathsf{*.f64}\left(k, k\right)\right), \mathsf{/.f64}\left(t, \ell\right)\right), \left(\frac{\ell}{\tan k \cdot \sin k}\right)\right) \]

      16. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(2, \mathsf{*.f64}\left(k, k\right)\right), \mathsf{/.f64}\left(t, \ell\right)\right), \mathsf{/.f64}\left(\ell, \color{blue}{\left(\tan k \cdot \sin k\right)}\right)\right) \]

   6. Applied egg-rr 83.0%

      \[\leadsto \color{blue}{\frac{\frac{2}{k \cdot k}}{\frac{t}{\ell}} \cdot \frac{\ell}{\sin k \cdot \tan k}} \]

   7. Taylor expanded in k around 0

      \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(2, \mathsf{*.f64}\left(k, k\right)\right), \mathsf{/.f64}\left(t, \ell\right)\right), \mathsf{/.f64}\left(\ell, \mathsf{*.f64}\left(\mathsf{sin.f64}\left(k\right), \color{blue}{k}\right)\right)\right) \]
   8. Step-by-step derivation

   9. Simplified 54.8%

      \[\leadsto \frac{\frac{2}{k \cdot k}}{\frac{t}{\ell}} \cdot \frac{\ell}{\sin k \cdot \color{blue}{k}} \]

   10. Taylor expanded in k around 0

       \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(2, \mathsf{*.f64}\left(k, k\right)\right), \mathsf{/.f64}\left(t, \ell\right)\right), \color{blue}{\left(\frac{\ell + \frac{1}{6} \cdot \left({k}^{2} \cdot \ell\right)}{{k}^{2}}\right)}\right) \]
   11. Step-by-step derivation

       1. unpow2 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(2, \mathsf{*.f64}\left(k, k\right)\right), \mathsf{/.f64}\left(t, \ell\right)\right), \left(\frac{\ell + \frac{1}{6} \cdot \left({k}^{2} \cdot \ell\right)}{k \cdot \color{blue}{k}}\right)\right) \]

       2. associate-/r* N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(2, \mathsf{*.f64}\left(k, k\right)\right), \mathsf{/.f64}\left(t, \ell\right)\right), \left(\frac{\frac{\ell + \frac{1}{6} \cdot \left({k}^{2} \cdot \ell\right)}{k}}{\color{blue}{k}}\right)\right) \]

       3. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(2, \mathsf{*.f64}\left(k, k\right)\right), \mathsf{/.f64}\left(t, \ell\right)\right), \mathsf{/.f64}\left(\left(\frac{\ell + \frac{1}{6} \cdot \left({k}^{2} \cdot \ell\right)}{k}\right), \color{blue}{k}\right)\right) \]

       4. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(2, \mathsf{*.f64}\left(k, k\right)\right), \mathsf{/.f64}\left(t, \ell\right)\right), \mathsf{/.f64}\left(\mathsf{/.f64}\left(\left(\ell + \frac{1}{6} \cdot \left({k}^{2} \cdot \ell\right)\right), k\right), k\right)\right) \]

       5. associate-*r* N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(2, \mathsf{*.f64}\left(k, k\right)\right), \mathsf{/.f64}\left(t, \ell\right)\right), \mathsf{/.f64}\left(\mathsf{/.f64}\left(\left(\ell + \left(\frac{1}{6} \cdot {k}^{2}\right) \cdot \ell\right), k\right), k\right)\right) \]

       6. distribute-rgt1-in N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(2, \mathsf{*.f64}\left(k, k\right)\right), \mathsf{/.f64}\left(t, \ell\right)\right), \mathsf{/.f64}\left(\mathsf{/.f64}\left(\left(\left(\frac{1}{6} \cdot {k}^{2} + 1\right) \cdot \ell\right), k\right), k\right)\right) \]

       7. +-commutative N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(2, \mathsf{*.f64}\left(k, k\right)\right), \mathsf{/.f64}\left(t, \ell\right)\right), \mathsf{/.f64}\left(\mathsf{/.f64}\left(\left(\left(1 + \frac{1}{6} \cdot {k}^{2}\right) \cdot \ell\right), k\right), k\right)\right) \]

       8. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(2, \mathsf{*.f64}\left(k, k\right)\right), \mathsf{/.f64}\left(t, \ell\right)\right), \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\left(1 + \frac{1}{6} \cdot {k}^{2}\right), \ell\right), k\right), k\right)\right) \]

       9. +-lowering-+.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(2, \mathsf{*.f64}\left(k, k\right)\right), \mathsf{/.f64}\left(t, \ell\right)\right), \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \left(\frac{1}{6} \cdot {k}^{2}\right)\right), \ell\right), k\right), k\right)\right) \]

       10. *-commutative N/A

           \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(2, \mathsf{*.f64}\left(k, k\right)\right), \mathsf{/.f64}\left(t, \ell\right)\right), \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \left({k}^{2} \cdot \frac{1}{6}\right)\right), \ell\right), k\right), k\right)\right) \]

       11. *-lowering-*.f64 N/A

           \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(2, \mathsf{*.f64}\left(k, k\right)\right), \mathsf{/.f64}\left(t, \ell\right)\right), \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\left({k}^{2}\right), \frac{1}{6}\right)\right), \ell\right), k\right), k\right)\right) \]

       12. unpow2 N/A

           \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(2, \mathsf{*.f64}\left(k, k\right)\right), \mathsf{/.f64}\left(t, \ell\right)\right), \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\left(k \cdot k\right), \frac{1}{6}\right)\right), \ell\right), k\right), k\right)\right) \]

       13. *-lowering-*.f64 24.6%

           \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(2, \mathsf{*.f64}\left(k, k\right)\right), \mathsf{/.f64}\left(t, \ell\right)\right), \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(k, k\right), \frac{1}{6}\right)\right), \ell\right), k\right), k\right)\right) \]

   12. Simplified 24.6%

       \[\leadsto \frac{\frac{2}{k \cdot k}}{\frac{t}{\ell}} \cdot \color{blue}{\frac{\frac{\left(1 + \left(k \cdot k\right) \cdot 0.16666666666666666\right) \cdot \ell}{k}}{k}} \]

   13. Taylor expanded in k around inf

       \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(2, \mathsf{*.f64}\left(k, k\right)\right), \mathsf{/.f64}\left(t, \ell\right)\right), \color{blue}{\left(\frac{1}{6} \cdot \ell\right)}\right) \]
   14. Step-by-step derivation

       1. *-commutative N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(2, \mathsf{*.f64}\left(k, k\right)\right), \mathsf{/.f64}\left(t, \ell\right)\right), \left(\ell \cdot \color{blue}{\frac{1}{6}}\right)\right) \]

       2. *-lowering-*.f64 55.5%

          \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(2, \mathsf{*.f64}\left(k, k\right)\right), \mathsf{/.f64}\left(t, \ell\right)\right), \mathsf{*.f64}\left(\ell, \color{blue}{\frac{1}{6}}\right)\right) \]

   15. Simplified 55.5%

       \[\leadsto \frac{\frac{2}{k \cdot k}}{\frac{t}{\ell}} \cdot \color{blue}{\left(\ell \cdot 0.16666666666666666\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 67.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;k \leq 1.45 \cdot 10^{+19}:\\ \;\;\;\;\ell \cdot \left(\ell \cdot \frac{2}{k \cdot \left(t \cdot \left(k \cdot \left(k \cdot k\right)\right)\right)}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{2}{k \cdot k}}{\frac{t}{\ell}} \cdot \left(\ell \cdot 0.16666666666666666\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 14: 61.5% accurate, 32.4× speedup

Math

\[\begin{array}{l} \\ \frac{\frac{2}{k \cdot k}}{\frac{t}{\ell}} \cdot \left(\ell \cdot 0.16666666666666666\right) \end{array} \]

FPCore

(FPCore (t l k)
 :precision binary64
 (* (/ (/ 2.0 (* k k)) (/ t l)) (* l 0.16666666666666666)))

C

double code(double t, double l, double k) {
	return ((2.0 / (k * k)) / (t / l)) * (l * 0.16666666666666666);
}

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 / (k * k)) / (t / l)) * (l * 0.16666666666666666d0)
end function

Java

public static double code(double t, double l, double k) {
	return ((2.0 / (k * k)) / (t / l)) * (l * 0.16666666666666666);
}

Python

def code(t, l, k):
	return ((2.0 / (k * k)) / (t / l)) * (l * 0.16666666666666666)

Julia

function code(t, l, k)
	return Float64(Float64(Float64(2.0 / Float64(k * k)) / Float64(t / l)) * Float64(l * 0.16666666666666666))
end

MATLAB

function tmp = code(t, l, k)
	tmp = ((2.0 / (k * k)) / (t / l)) * (l * 0.16666666666666666);
end

Wolfram

code[t_, l_, k_] := N[(N[(N[(2.0 / N[(k * k), $MachinePrecision]), $MachinePrecision] / N[(t / l), $MachinePrecision]), $MachinePrecision] * N[(l * 0.16666666666666666), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 36.8%

   \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) - 1\right)} \]
2. Step-by-step derivation

   1. *-commutative N/A

      \[\leadsto \frac{2}{\left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) - 1\right) \cdot \color{blue}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right)}} \]

   2. associate-*r* N/A

      \[\leadsto \frac{2}{\left(\left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) - 1\right) \cdot \left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right)\right) \cdot \color{blue}{\tan k}} \]

   3. associate-/r* N/A

      \[\leadsto \frac{\frac{2}{\left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) - 1\right) \cdot \left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right)}}{\color{blue}{\tan k}} \]

   4. /-lowering-/.f64 N/A

      \[\leadsto \mathsf{/.f64}\left(\left(\frac{2}{\left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) - 1\right) \cdot \left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right)}\right), \color{blue}{\tan k}\right) \]

3. Simplified 66.0%

   \[\leadsto \color{blue}{\frac{\frac{\frac{\left(\frac{2}{k \cdot k} \cdot t\right) \cdot \frac{t}{t}}{\frac{t \cdot \frac{t}{\ell}}{\ell}}}{\sin k}}{\tan k}} \]
4. Add Preprocessing
5. Step-by-step derivation

   1. associate-/l/ N/A

      \[\leadsto \frac{\frac{\left(\frac{2}{k \cdot k} \cdot t\right) \cdot \frac{t}{t}}{\frac{t \cdot \frac{t}{\ell}}{\ell}}}{\color{blue}{\tan k \cdot \sin k}} \]

   2. associate-/r/ N/A

      \[\leadsto \frac{\frac{\left(\frac{2}{k \cdot k} \cdot t\right) \cdot \frac{t}{t}}{t \cdot \frac{t}{\ell}} \cdot \ell}{\color{blue}{\tan k} \cdot \sin k} \]

   3. associate-/l* N/A

      \[\leadsto \frac{\left(\frac{2}{k \cdot k} \cdot t\right) \cdot \frac{t}{t}}{t \cdot \frac{t}{\ell}} \cdot \color{blue}{\frac{\ell}{\tan k \cdot \sin k}} \]

   4. *-lowering-*.f64 N/A

      \[\leadsto \mathsf{*.f64}\left(\left(\frac{\left(\frac{2}{k \cdot k} \cdot t\right) \cdot \frac{t}{t}}{t \cdot \frac{t}{\ell}}\right), \color{blue}{\left(\frac{\ell}{\tan k \cdot \sin k}\right)}\right) \]

   5. *-inverses N/A

      \[\leadsto \mathsf{*.f64}\left(\left(\frac{\left(\frac{2}{k \cdot k} \cdot t\right) \cdot 1}{t \cdot \frac{t}{\ell}}\right), \left(\frac{\ell}{\tan k \cdot \sin k}\right)\right) \]

   6. associate-*l* N/A

      \[\leadsto \mathsf{*.f64}\left(\left(\frac{\frac{2}{k \cdot k} \cdot \left(t \cdot 1\right)}{t \cdot \frac{t}{\ell}}\right), \left(\frac{\ell}{\tan k \cdot \sin k}\right)\right) \]

   7. *-commutative N/A

      \[\leadsto \mathsf{*.f64}\left(\left(\frac{\frac{2}{k \cdot k} \cdot \left(t \cdot 1\right)}{\frac{t}{\ell} \cdot t}\right), \left(\frac{\ell}{\tan k \cdot \sin k}\right)\right) \]

   8. times-frac N/A

      \[\leadsto \mathsf{*.f64}\left(\left(\frac{\frac{2}{k \cdot k}}{\frac{t}{\ell}} \cdot \frac{t \cdot 1}{t}\right), \left(\frac{\color{blue}{\ell}}{\tan k \cdot \sin k}\right)\right) \]

   9. *-rgt-identity N/A

      \[\leadsto \mathsf{*.f64}\left(\left(\frac{\frac{2}{k \cdot k}}{\frac{t}{\ell}} \cdot \frac{t}{t}\right), \left(\frac{\ell}{\tan k \cdot \sin k}\right)\right) \]

   10. *-inverses N/A

       \[\leadsto \mathsf{*.f64}\left(\left(\frac{\frac{2}{k \cdot k}}{\frac{t}{\ell}} \cdot 1\right), \left(\frac{\ell}{\tan k \cdot \sin k}\right)\right) \]

   11. *-rgt-identity N/A

       \[\leadsto \mathsf{*.f64}\left(\left(\frac{\frac{2}{k \cdot k}}{\frac{t}{\ell}}\right), \left(\frac{\color{blue}{\ell}}{\tan k \cdot \sin k}\right)\right) \]

   12. /-lowering-/.f64 N/A

       \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\left(\frac{2}{k \cdot k}\right), \left(\frac{t}{\ell}\right)\right), \left(\frac{\color{blue}{\ell}}{\tan k \cdot \sin k}\right)\right) \]

   13. /-lowering-/.f64 N/A

       \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(2, \left(k \cdot k\right)\right), \left(\frac{t}{\ell}\right)\right), \left(\frac{\ell}{\tan k \cdot \sin k}\right)\right) \]

   14. *-lowering-*.f64 N/A

       \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(2, \mathsf{*.f64}\left(k, k\right)\right), \left(\frac{t}{\ell}\right)\right), \left(\frac{\ell}{\tan k \cdot \sin k}\right)\right) \]

   15. /-lowering-/.f64 N/A

       \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(2, \mathsf{*.f64}\left(k, k\right)\right), \mathsf{/.f64}\left(t, \ell\right)\right), \left(\frac{\ell}{\tan k \cdot \sin k}\right)\right) \]

   16. /-lowering-/.f64 N/A

       \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(2, \mathsf{*.f64}\left(k, k\right)\right), \mathsf{/.f64}\left(t, \ell\right)\right), \mathsf{/.f64}\left(\ell, \color{blue}{\left(\tan k \cdot \sin k\right)}\right)\right) \]

6. Applied egg-rr 87.3%

   \[\leadsto \color{blue}{\frac{\frac{2}{k \cdot k}}{\frac{t}{\ell}} \cdot \frac{\ell}{\sin k \cdot \tan k}} \]

7. Taylor expanded in k around 0

   \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(2, \mathsf{*.f64}\left(k, k\right)\right), \mathsf{/.f64}\left(t, \ell\right)\right), \mathsf{/.f64}\left(\ell, \mathsf{*.f64}\left(\mathsf{sin.f64}\left(k\right), \color{blue}{k}\right)\right)\right) \]
8. Step-by-step derivation

9. Simplified 71.2%

   \[\leadsto \frac{\frac{2}{k \cdot k}}{\frac{t}{\ell}} \cdot \frac{\ell}{\sin k \cdot \color{blue}{k}} \]

10. Taylor expanded in k around 0

    \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(2, \mathsf{*.f64}\left(k, k\right)\right), \mathsf{/.f64}\left(t, \ell\right)\right), \color{blue}{\left(\frac{\ell + \frac{1}{6} \cdot \left({k}^{2} \cdot \ell\right)}{{k}^{2}}\right)}\right) \]
11. Step-by-step derivation

    1. unpow2 N/A

       \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(2, \mathsf{*.f64}\left(k, k\right)\right), \mathsf{/.f64}\left(t, \ell\right)\right), \left(\frac{\ell + \frac{1}{6} \cdot \left({k}^{2} \cdot \ell\right)}{k \cdot \color{blue}{k}}\right)\right) \]

    2. associate-/r* N/A

       \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(2, \mathsf{*.f64}\left(k, k\right)\right), \mathsf{/.f64}\left(t, \ell\right)\right), \left(\frac{\frac{\ell + \frac{1}{6} \cdot \left({k}^{2} \cdot \ell\right)}{k}}{\color{blue}{k}}\right)\right) \]

    3. /-lowering-/.f64 N/A

       \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(2, \mathsf{*.f64}\left(k, k\right)\right), \mathsf{/.f64}\left(t, \ell\right)\right), \mathsf{/.f64}\left(\left(\frac{\ell + \frac{1}{6} \cdot \left({k}^{2} \cdot \ell\right)}{k}\right), \color{blue}{k}\right)\right) \]

    4. /-lowering-/.f64 N/A

       \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(2, \mathsf{*.f64}\left(k, k\right)\right), \mathsf{/.f64}\left(t, \ell\right)\right), \mathsf{/.f64}\left(\mathsf{/.f64}\left(\left(\ell + \frac{1}{6} \cdot \left({k}^{2} \cdot \ell\right)\right), k\right), k\right)\right) \]

    5. associate-*r* N/A

       \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(2, \mathsf{*.f64}\left(k, k\right)\right), \mathsf{/.f64}\left(t, \ell\right)\right), \mathsf{/.f64}\left(\mathsf{/.f64}\left(\left(\ell + \left(\frac{1}{6} \cdot {k}^{2}\right) \cdot \ell\right), k\right), k\right)\right) \]

    6. distribute-rgt1-in N/A

       \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(2, \mathsf{*.f64}\left(k, k\right)\right), \mathsf{/.f64}\left(t, \ell\right)\right), \mathsf{/.f64}\left(\mathsf{/.f64}\left(\left(\left(\frac{1}{6} \cdot {k}^{2} + 1\right) \cdot \ell\right), k\right), k\right)\right) \]

    7. +-commutative N/A

       \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(2, \mathsf{*.f64}\left(k, k\right)\right), \mathsf{/.f64}\left(t, \ell\right)\right), \mathsf{/.f64}\left(\mathsf{/.f64}\left(\left(\left(1 + \frac{1}{6} \cdot {k}^{2}\right) \cdot \ell\right), k\right), k\right)\right) \]

    8. *-lowering-*.f64 N/A

       \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(2, \mathsf{*.f64}\left(k, k\right)\right), \mathsf{/.f64}\left(t, \ell\right)\right), \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\left(1 + \frac{1}{6} \cdot {k}^{2}\right), \ell\right), k\right), k\right)\right) \]

    9. +-lowering-+.f64 N/A

       \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(2, \mathsf{*.f64}\left(k, k\right)\right), \mathsf{/.f64}\left(t, \ell\right)\right), \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \left(\frac{1}{6} \cdot {k}^{2}\right)\right), \ell\right), k\right), k\right)\right) \]

    10. *-commutative N/A

        \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(2, \mathsf{*.f64}\left(k, k\right)\right), \mathsf{/.f64}\left(t, \ell\right)\right), \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \left({k}^{2} \cdot \frac{1}{6}\right)\right), \ell\right), k\right), k\right)\right) \]

    11. *-lowering-*.f64 N/A

        \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(2, \mathsf{*.f64}\left(k, k\right)\right), \mathsf{/.f64}\left(t, \ell\right)\right), \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\left({k}^{2}\right), \frac{1}{6}\right)\right), \ell\right), k\right), k\right)\right) \]

    12. unpow2 N/A

        \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(2, \mathsf{*.f64}\left(k, k\right)\right), \mathsf{/.f64}\left(t, \ell\right)\right), \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\left(k \cdot k\right), \frac{1}{6}\right)\right), \ell\right), k\right), k\right)\right) \]

    13. *-lowering-*.f64 57.8%

        \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(2, \mathsf{*.f64}\left(k, k\right)\right), \mathsf{/.f64}\left(t, \ell\right)\right), \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(k, k\right), \frac{1}{6}\right)\right), \ell\right), k\right), k\right)\right) \]

12. Simplified 57.8%

    \[\leadsto \frac{\frac{2}{k \cdot k}}{\frac{t}{\ell}} \cdot \color{blue}{\frac{\frac{\left(1 + \left(k \cdot k\right) \cdot 0.16666666666666666\right) \cdot \ell}{k}}{k}} \]

13. Taylor expanded in k around inf

    \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(2, \mathsf{*.f64}\left(k, k\right)\right), \mathsf{/.f64}\left(t, \ell\right)\right), \color{blue}{\left(\frac{1}{6} \cdot \ell\right)}\right) \]
14. Step-by-step derivation

    1. *-commutative N/A

       \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(2, \mathsf{*.f64}\left(k, k\right)\right), \mathsf{/.f64}\left(t, \ell\right)\right), \left(\ell \cdot \color{blue}{\frac{1}{6}}\right)\right) \]

    2. *-lowering-*.f64 61.7%

       \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(2, \mathsf{*.f64}\left(k, k\right)\right), \mathsf{/.f64}\left(t, \ell\right)\right), \mathsf{*.f64}\left(\ell, \color{blue}{\frac{1}{6}}\right)\right) \]

15. Simplified 61.7%

    \[\leadsto \frac{\frac{2}{k \cdot k}}{\frac{t}{\ell}} \cdot \color{blue}{\left(\ell \cdot 0.16666666666666666\right)} \]
16. Add Preprocessing

Alternative 15: 56.8% accurate, 38.3× speedup

Math

\[\begin{array}{l} \\ 0.3333333333333333 \cdot \frac{\frac{\frac{\ell \cdot \ell}{k}}{k}}{t} \end{array} \]

FPCore

(FPCore (t l k)
 :precision binary64
 (* 0.3333333333333333 (/ (/ (/ (* l l) k) k) t)))

C

double code(double t, double l, double k) {
	return 0.3333333333333333 * ((((l * l) / k) / k) / t);
}

Fortran

real(8) function code(t, l, k)
    real(8), intent (in) :: t
    real(8), intent (in) :: l
    real(8), intent (in) :: k
    code = 0.3333333333333333d0 * ((((l * l) / k) / k) / t)
end function

Java

public static double code(double t, double l, double k) {
	return 0.3333333333333333 * ((((l * l) / k) / k) / t);
}

Python

def code(t, l, k):
	return 0.3333333333333333 * ((((l * l) / k) / k) / t)

Julia

function code(t, l, k)
	return Float64(0.3333333333333333 * Float64(Float64(Float64(Float64(l * l) / k) / k) / t))
end

MATLAB

function tmp = code(t, l, k)
	tmp = 0.3333333333333333 * ((((l * l) / k) / k) / t);
end

Wolfram

code[t_, l_, k_] := N[(0.3333333333333333 * N[(N[(N[(N[(l * l), $MachinePrecision] / k), $MachinePrecision] / k), $MachinePrecision] / t), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 36.8%

   \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) - 1\right)} \]
2. Step-by-step derivation

   1. *-commutative N/A

      \[\leadsto \frac{2}{\left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) - 1\right) \cdot \color{blue}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right)}} \]

   2. associate-*r* N/A

      \[\leadsto \frac{2}{\left(\left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) - 1\right) \cdot \left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right)\right) \cdot \color{blue}{\tan k}} \]

   3. associate-/r* N/A

      \[\leadsto \frac{\frac{2}{\left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) - 1\right) \cdot \left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right)}}{\color{blue}{\tan k}} \]

   4. /-lowering-/.f64 N/A

      \[\leadsto \mathsf{/.f64}\left(\left(\frac{2}{\left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) - 1\right) \cdot \left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right)}\right), \color{blue}{\tan k}\right) \]

3. Simplified 66.0%

   \[\leadsto \color{blue}{\frac{\frac{\frac{\left(\frac{2}{k \cdot k} \cdot t\right) \cdot \frac{t}{t}}{\frac{t \cdot \frac{t}{\ell}}{\ell}}}{\sin k}}{\tan k}} \]
4. Add Preprocessing
5. Step-by-step derivation

   1. associate-/l/ N/A

      \[\leadsto \frac{\frac{\left(\frac{2}{k \cdot k} \cdot t\right) \cdot \frac{t}{t}}{\frac{t \cdot \frac{t}{\ell}}{\ell}}}{\color{blue}{\tan k \cdot \sin k}} \]

   2. associate-/r/ N/A

      \[\leadsto \frac{\frac{\left(\frac{2}{k \cdot k} \cdot t\right) \cdot \frac{t}{t}}{t \cdot \frac{t}{\ell}} \cdot \ell}{\color{blue}{\tan k} \cdot \sin k} \]

   3. associate-/l* N/A

      \[\leadsto \frac{\left(\frac{2}{k \cdot k} \cdot t\right) \cdot \frac{t}{t}}{t \cdot \frac{t}{\ell}} \cdot \color{blue}{\frac{\ell}{\tan k \cdot \sin k}} \]

   4. *-lowering-*.f64 N/A

      \[\leadsto \mathsf{*.f64}\left(\left(\frac{\left(\frac{2}{k \cdot k} \cdot t\right) \cdot \frac{t}{t}}{t \cdot \frac{t}{\ell}}\right), \color{blue}{\left(\frac{\ell}{\tan k \cdot \sin k}\right)}\right) \]

   5. *-inverses N/A

      \[\leadsto \mathsf{*.f64}\left(\left(\frac{\left(\frac{2}{k \cdot k} \cdot t\right) \cdot 1}{t \cdot \frac{t}{\ell}}\right), \left(\frac{\ell}{\tan k \cdot \sin k}\right)\right) \]

   6. associate-*l* N/A

      \[\leadsto \mathsf{*.f64}\left(\left(\frac{\frac{2}{k \cdot k} \cdot \left(t \cdot 1\right)}{t \cdot \frac{t}{\ell}}\right), \left(\frac{\ell}{\tan k \cdot \sin k}\right)\right) \]

   7. *-commutative N/A

      \[\leadsto \mathsf{*.f64}\left(\left(\frac{\frac{2}{k \cdot k} \cdot \left(t \cdot 1\right)}{\frac{t}{\ell} \cdot t}\right), \left(\frac{\ell}{\tan k \cdot \sin k}\right)\right) \]

   8. times-frac N/A

      \[\leadsto \mathsf{*.f64}\left(\left(\frac{\frac{2}{k \cdot k}}{\frac{t}{\ell}} \cdot \frac{t \cdot 1}{t}\right), \left(\frac{\color{blue}{\ell}}{\tan k \cdot \sin k}\right)\right) \]

   9. *-rgt-identity N/A

      \[\leadsto \mathsf{*.f64}\left(\left(\frac{\frac{2}{k \cdot k}}{\frac{t}{\ell}} \cdot \frac{t}{t}\right), \left(\frac{\ell}{\tan k \cdot \sin k}\right)\right) \]

   10. *-inverses N/A

       \[\leadsto \mathsf{*.f64}\left(\left(\frac{\frac{2}{k \cdot k}}{\frac{t}{\ell}} \cdot 1\right), \left(\frac{\ell}{\tan k \cdot \sin k}\right)\right) \]

   11. *-rgt-identity N/A

       \[\leadsto \mathsf{*.f64}\left(\left(\frac{\frac{2}{k \cdot k}}{\frac{t}{\ell}}\right), \left(\frac{\color{blue}{\ell}}{\tan k \cdot \sin k}\right)\right) \]

   12. /-lowering-/.f64 N/A

       \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\left(\frac{2}{k \cdot k}\right), \left(\frac{t}{\ell}\right)\right), \left(\frac{\color{blue}{\ell}}{\tan k \cdot \sin k}\right)\right) \]

   13. /-lowering-/.f64 N/A

       \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(2, \left(k \cdot k\right)\right), \left(\frac{t}{\ell}\right)\right), \left(\frac{\ell}{\tan k \cdot \sin k}\right)\right) \]

   14. *-lowering-*.f64 N/A

       \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(2, \mathsf{*.f64}\left(k, k\right)\right), \left(\frac{t}{\ell}\right)\right), \left(\frac{\ell}{\tan k \cdot \sin k}\right)\right) \]

   15. /-lowering-/.f64 N/A

       \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(2, \mathsf{*.f64}\left(k, k\right)\right), \mathsf{/.f64}\left(t, \ell\right)\right), \left(\frac{\ell}{\tan k \cdot \sin k}\right)\right) \]

   16. /-lowering-/.f64 N/A

       \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(2, \mathsf{*.f64}\left(k, k\right)\right), \mathsf{/.f64}\left(t, \ell\right)\right), \mathsf{/.f64}\left(\ell, \color{blue}{\left(\tan k \cdot \sin k\right)}\right)\right) \]

6. Applied egg-rr 87.3%

   \[\leadsto \color{blue}{\frac{\frac{2}{k \cdot k}}{\frac{t}{\ell}} \cdot \frac{\ell}{\sin k \cdot \tan k}} \]

7. Taylor expanded in k around 0

   \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(2, \mathsf{*.f64}\left(k, k\right)\right), \mathsf{/.f64}\left(t, \ell\right)\right), \mathsf{/.f64}\left(\ell, \mathsf{*.f64}\left(\mathsf{sin.f64}\left(k\right), \color{blue}{k}\right)\right)\right) \]
8. Step-by-step derivation

9. Simplified 71.2%

   \[\leadsto \frac{\frac{2}{k \cdot k}}{\frac{t}{\ell}} \cdot \frac{\ell}{\sin k \cdot \color{blue}{k}} \]

10. Taylor expanded in k around 0

    \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(2, \mathsf{*.f64}\left(k, k\right)\right), \mathsf{/.f64}\left(t, \ell\right)\right), \color{blue}{\left(\frac{\ell + \frac{1}{6} \cdot \left({k}^{2} \cdot \ell\right)}{{k}^{2}}\right)}\right) \]
11. Step-by-step derivation

    1. unpow2 N/A

       \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(2, \mathsf{*.f64}\left(k, k\right)\right), \mathsf{/.f64}\left(t, \ell\right)\right), \left(\frac{\ell + \frac{1}{6} \cdot \left({k}^{2} \cdot \ell\right)}{k \cdot \color{blue}{k}}\right)\right) \]

    2. associate-/r* N/A

       \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(2, \mathsf{*.f64}\left(k, k\right)\right), \mathsf{/.f64}\left(t, \ell\right)\right), \left(\frac{\frac{\ell + \frac{1}{6} \cdot \left({k}^{2} \cdot \ell\right)}{k}}{\color{blue}{k}}\right)\right) \]

    3. /-lowering-/.f64 N/A

       \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(2, \mathsf{*.f64}\left(k, k\right)\right), \mathsf{/.f64}\left(t, \ell\right)\right), \mathsf{/.f64}\left(\left(\frac{\ell + \frac{1}{6} \cdot \left({k}^{2} \cdot \ell\right)}{k}\right), \color{blue}{k}\right)\right) \]

    4. /-lowering-/.f64 N/A

       \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(2, \mathsf{*.f64}\left(k, k\right)\right), \mathsf{/.f64}\left(t, \ell\right)\right), \mathsf{/.f64}\left(\mathsf{/.f64}\left(\left(\ell + \frac{1}{6} \cdot \left({k}^{2} \cdot \ell\right)\right), k\right), k\right)\right) \]

    5. associate-*r* N/A

       \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(2, \mathsf{*.f64}\left(k, k\right)\right), \mathsf{/.f64}\left(t, \ell\right)\right), \mathsf{/.f64}\left(\mathsf{/.f64}\left(\left(\ell + \left(\frac{1}{6} \cdot {k}^{2}\right) \cdot \ell\right), k\right), k\right)\right) \]

    6. distribute-rgt1-in N/A

       \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(2, \mathsf{*.f64}\left(k, k\right)\right), \mathsf{/.f64}\left(t, \ell\right)\right), \mathsf{/.f64}\left(\mathsf{/.f64}\left(\left(\left(\frac{1}{6} \cdot {k}^{2} + 1\right) \cdot \ell\right), k\right), k\right)\right) \]

    7. +-commutative N/A

       \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(2, \mathsf{*.f64}\left(k, k\right)\right), \mathsf{/.f64}\left(t, \ell\right)\right), \mathsf{/.f64}\left(\mathsf{/.f64}\left(\left(\left(1 + \frac{1}{6} \cdot {k}^{2}\right) \cdot \ell\right), k\right), k\right)\right) \]

    8. *-lowering-*.f64 N/A

       \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(2, \mathsf{*.f64}\left(k, k\right)\right), \mathsf{/.f64}\left(t, \ell\right)\right), \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\left(1 + \frac{1}{6} \cdot {k}^{2}\right), \ell\right), k\right), k\right)\right) \]

    9. +-lowering-+.f64 N/A

       \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(2, \mathsf{*.f64}\left(k, k\right)\right), \mathsf{/.f64}\left(t, \ell\right)\right), \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \left(\frac{1}{6} \cdot {k}^{2}\right)\right), \ell\right), k\right), k\right)\right) \]

    10. *-commutative N/A

        \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(2, \mathsf{*.f64}\left(k, k\right)\right), \mathsf{/.f64}\left(t, \ell\right)\right), \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \left({k}^{2} \cdot \frac{1}{6}\right)\right), \ell\right), k\right), k\right)\right) \]

    11. *-lowering-*.f64 N/A

        \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(2, \mathsf{*.f64}\left(k, k\right)\right), \mathsf{/.f64}\left(t, \ell\right)\right), \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\left({k}^{2}\right), \frac{1}{6}\right)\right), \ell\right), k\right), k\right)\right) \]

    12. unpow2 N/A

        \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(2, \mathsf{*.f64}\left(k, k\right)\right), \mathsf{/.f64}\left(t, \ell\right)\right), \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\left(k \cdot k\right), \frac{1}{6}\right)\right), \ell\right), k\right), k\right)\right) \]

    13. *-lowering-*.f64 57.8%

        \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(2, \mathsf{*.f64}\left(k, k\right)\right), \mathsf{/.f64}\left(t, \ell\right)\right), \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(k, k\right), \frac{1}{6}\right)\right), \ell\right), k\right), k\right)\right) \]

12. Simplified 57.8%

    \[\leadsto \frac{\frac{2}{k \cdot k}}{\frac{t}{\ell}} \cdot \color{blue}{\frac{\frac{\left(1 + \left(k \cdot k\right) \cdot 0.16666666666666666\right) \cdot \ell}{k}}{k}} \]

13. Taylor expanded in k around inf

    \[\leadsto \color{blue}{\frac{1}{3} \cdot \frac{{\ell}^{2}}{{k}^{2} \cdot t}} \]
14. Step-by-step derivation

    1. *-lowering-*.f64 N/A

       \[\leadsto \mathsf{*.f64}\left(\frac{1}{3}, \color{blue}{\left(\frac{{\ell}^{2}}{{k}^{2} \cdot t}\right)}\right) \]

    2. associate-/r* N/A

       \[\leadsto \mathsf{*.f64}\left(\frac{1}{3}, \left(\frac{\frac{{\ell}^{2}}{{k}^{2}}}{\color{blue}{t}}\right)\right) \]

    3. /-lowering-/.f64 N/A

       \[\leadsto \mathsf{*.f64}\left(\frac{1}{3}, \mathsf{/.f64}\left(\left(\frac{{\ell}^{2}}{{k}^{2}}\right), \color{blue}{t}\right)\right) \]

    4. unpow2 N/A

       \[\leadsto \mathsf{*.f64}\left(\frac{1}{3}, \mathsf{/.f64}\left(\left(\frac{{\ell}^{2}}{k \cdot k}\right), t\right)\right) \]

    5. associate-/r* N/A

       \[\leadsto \mathsf{*.f64}\left(\frac{1}{3}, \mathsf{/.f64}\left(\left(\frac{\frac{{\ell}^{2}}{k}}{k}\right), t\right)\right) \]

    6. /-lowering-/.f64 N/A

       \[\leadsto \mathsf{*.f64}\left(\frac{1}{3}, \mathsf{/.f64}\left(\mathsf{/.f64}\left(\left(\frac{{\ell}^{2}}{k}\right), k\right), t\right)\right) \]

    7. /-lowering-/.f64 N/A

       \[\leadsto \mathsf{*.f64}\left(\frac{1}{3}, \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(\left({\ell}^{2}\right), k\right), k\right), t\right)\right) \]

    8. unpow2 N/A

       \[\leadsto \mathsf{*.f64}\left(\frac{1}{3}, \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(\left(\ell \cdot \ell\right), k\right), k\right), t\right)\right) \]

    9. *-lowering-*.f64 57.3%

       \[\leadsto \mathsf{*.f64}\left(\frac{1}{3}, \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\ell, \ell\right), k\right), k\right), t\right)\right) \]

15. Simplified 57.3%

    \[\leadsto \color{blue}{0.3333333333333333 \cdot \frac{\frac{\frac{\ell \cdot \ell}{k}}{k}}{t}} \]
16. Add Preprocessing

Reproduce

herbie shell --seed 2024191 
(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))))